**Remark 8** We notice that these seven categories can be divided into two groups of strategies. The first takes into account the ESG scores when building an investment portfolio (exclusion, selection, thematic and integration). The primary objective of these strategies remains the financial performance of the portfolio. The second group places a high priority on ethical behavior (norm-based screening, engagement and impact investing) and can be related to signaling theory. In this approach, investors send a negative signal to the market and to companies when they apply norm-based screening or engage with a company<sup>85</sup>. Conversely, investors send a positive signal when they implement impact investing.

### 1.5.2 The market share of ESG investing

In this section, we present a global view of ESG investment market growth<sup>86</sup> based on the Global Sustainable Investment Reviews (GSIA, 2017, 2019, 2021). For each report, Figures 1.16, 1.17 and 1.18 show the AUM of responsible investments, the corresponding market share, the global growth and the breakdown by country. According to GSIA (2021), sustainable investments represented $35.3 bn of assets under management at the beginning of 2020, i.e a market share of 36%. They continue to grow in most regions<sup>87</sup>, with Canada experiencing the largest increase (48% growth), followed by the United States (42% growth). On a regional analysis, the market share of sustainable investments is 62% in Canada, 42% in Europe, 38% in Australasia, 33% in the United States, and 24% in Japan.

<table><caption>Table 1.10: ESG asset growth by region</caption><thead><tr><th rowspan="2">Region</th><th colspan="4">Asset under management (in $ bn)</th><th colspan="4">Asset growth (in %)</th></tr><tr><th>2016</th><th>2018</th><th>2020</th><th>2022</th><th>2016</th><th>2018</th><th>2020</th><th>2022</th></tr></thead><tbody><tr><td>Europe</td><td>12 040</td><td>14 075</td><td>12 017</td><td>14 054</td><td>12%</td><td>11%</td><td>-13%</td><td>31%</td></tr><tr><td>Canada</td><td>1 086</td><td>1 699</td><td>2 423</td><td>2 358</td><td>49%</td><td>42%</td><td>48%</td><td>-5%</td></tr><tr><td>Australasia</td><td>516</td><td>734</td><td>906</td><td>1 220</td><td>248%</td><td>46%</td><td>25%</td><td>30%</td></tr><tr><td>Japan</td><td>474</td><td>2 180</td><td>2 874</td><td>4 289</td><td>6 692%</td><td>307%</td><td>34%</td><td>59%</td></tr><tr><td>Sub-total</td><td>14 116</td><td>18 688</td><td>18 220</td><td>21 921</td><td>21%</td><td>32%</td><td>-3%</td><td>20%</td></tr><tr><td>United States</td><td>8 723</td><td>11 995</td><td>17 081</td><td>8 400</td><td>33%</td><td>38%</td><td>42%</td><td>-51%</td></tr><tr><td>Total</td><td>22 839</td><td>30 683</td><td>35 301</td><td>30 321</td><td>25%</td><td>34%</td><td>15%</td><td>-14%</td></tr></tbody></table>

Source: GSIA (2015, 2017, 2019, 2021, 2023) & Author's calculations.

The 2022 GSIR paints a different picture of the ESG investment market (Table 1.10). We see strong growth in Europe (+31% between 2020 and 2022), Australasia (+30%) and Japan (+59%). Over the same period, there is a decline in North America, particularly in the United States. As there is a change in the assets collected by the US SIF in the United States, it is difficult to know whether the decline is due to the change in methodology or whether it is a setback in the US due to the ESG backlash<sup>88</sup>.

<footnote n="85"><sup>85</sup>Because the investor is not satisfied with the company's environmental, social or governance policies.</footnote>

<footnote n="86"><sup>86</sup>The figures concerning specific segments (mutual funds, ETFs, green financing, etc.) are given in Chapters 2 and 10.</footnote>

<footnote n="87"><sup>87</sup>The case of Europe (13% decline in the growth of sustainable investment assets between 2018 and 2020) is due to a change in the measurement methodology from which European data are collected and to European regulations, in particular the SFDR. We must therefore be cautious before drawing conclusions. In fact, these data do not take into account how ESG factors are truly implemented.</footnote>

<footnote n="88"><sup>88</sup>GSIA (2023, page 8) gives the following reasons: “The US SIF modified their methodology for the latest edition of the US SIF Trends report, which is used as the basis for the Global Sustainable Investment Review. The revised</footnote>
<figure><img src="image_19.png" /><figcaption>Figure 1.16: Sustainable investment assets at the start of 2016</figcaption></figure>

Source: GSIA (2017).

<figure><img src="image_20.png" /><figcaption>Figure 1.17: Sustainable investment assets at the start of 2018</figcaption></figure>

Source: GSIA (2019).
<figure><img src="image_21.png" /><figcaption>Figure 1.18: Sustainable investment assets at the start of 2020</figcaption></figure>

Source: GSIA (2021).

<figure><img src="image_22.png" /><figcaption>Figure 1.19: Asset values of ESG strategies between 2014 and 2022</figcaption></figure>

Source: GSIA (2015, 2017, 2019, 2021, 2023) & Author's calculations.
Figure 1.19 shows changes in asset values by ESG category. For many years, negative screening and exclusion dominated the other strategies. In 2020, ESG integration has become the most implemented approach. We also notice that some categories are less represented: thematic investing, best-in-class/selection and impact investing. However, we observe that thematic investing is the category with the highest growth in assets. Again, the 2022 figures are not comparable to the 2020 figures due to the change in methodology<fnref n="89" />. However, we note that engagement takes the lead, followed by integration and exclusion (Table 1.11).

<table><caption>Table 1.11: ESG asset growth by strategy</caption><thead><tr><th rowspan="2">#</th><th rowspan="2">ESG strategy</th><th colspan="3">Asset growth</th><th colspan="2">AUM (in $ bn)</th></tr><tr><th>2014-2016</th><th>2016-2018</th><th>2018-2020</th><th>2020</th><th>2022</th></tr></thead><tbody><tr><td>1</td><td>Exclusion</td><td>11.7%</td><td>14.6%</td><td>-24.0%</td><td>15 030</td><td>3 840</td></tr><tr><td>2</td><td>Values/Norms-based</td><td>19.0%</td><td>-13.1%</td><td>-11.5%</td><td>4 140</td><td>1 807</td></tr><tr><td>3</td><td>Selection</td><td>7.6%</td><td>50.1%</td><td>-24.9%</td><td>1 384</td><td>574</td></tr><tr><td>4</td><td>Thematic Investing</td><td>55.1%</td><td>92.0%</td><td>91.4%</td><td>1 948</td><td>598</td></tr><tr><td>5</td><td>Integration</td><td>17.4%</td><td>30.2%</td><td>43.6%</td><td>25 195</td><td>5 588</td></tr><tr><td>6</td><td>Engagement</td><td>18.9%</td><td>8.3%</td><td>6.8%</td><td>10 504</td><td>8 053</td></tr><tr><td>7</td><td>Impact Investing</td><td>56.8%</td><td>33.7%</td><td>-20.8%</td><td>352</td><td>55</td></tr></tbody><tfoot><tr><td colspan="2">Total</td><td></td><td></td><td></td><td>58 553</td><td>20 515</td></tr></tfoot></table>

Source: GSIA (2015, 2017, 2019, 2021, 2023) & Author's calculations.

If we focus on asset ownership, the ESG market is driven primarily by institutional investors. They currently account for 75% of the market, with the remaining 25% being retail assets. However, we are seeing a fundamental shift as the split in 2012 was 89% vs. 11% (GSIA, 2021, page 13). This is particularly true in Europe, where retail investors are increasingly interested in understanding the impact of their investments on the world, as demonstrated by both market developments and academic research (GSIA, 2023, page 22). It seems that this trend is also present in Japan, but not in the UK and the US. We also note that institutional and retail investors have different beliefs about ESG. For example, Giglio et al. (2023a) found that retail investors generally expect ESG investments to underperform the market, while institutional investors in Europe believe that ESG can improve the risk management of their portfolios.

## 1.6 Conclusion

This little-boring introduction provides a global overview of the sustainable finance landscape. As a student, you need to understand who does what, the outlines of the various regulations, and some figures about sustainable finance. However, a course in sustainable finance cannot be summarized by

<footnote>methodology does not include the AUM of investors who stated that they practice firmwide ESG integration but did not provide information on any specific ESG criteria they used (such as biodiversity, human rights or tobacco) in their investment decision-making. Additionally, several asset managers in the 2022 Trends Information Request reported far lower sustainable investing AUM than they had in 2020 (in some cases, in the magnitude of billions and trillions of dollars). The modified methodology and change in money manager reporting led to a significant drop in total reported ESG AUM, as well as ESG AUM for specific criteria.”</footnote>

<footnote n="89">⁸⁹"In previous years, respondents have been able to select multiple sustainable investing strategies. This has meant that the value across the categories has often exceeded the total ESG AUM (150-200%). In this year's report respondents from the US and Canada selected the predominant category and were required to validate selection with provision of AUM incorporating this strategy. As a result of this change the value across the categories does not meet the total ESG AUM, representing around 66%." (GSIA, 2023, page 8).</footnote>
a set of acronyms (Figure 1.20). We also need to understand the data we are manipulating. Because the data are very noisy and not exhaustive, probability and quantitative modeling is important for two main reasons. The first is to estimate the limits of what professionals do. The second reason is to measure the relationships between the various factors from an ex-ante point of view and to make some convictions, because data and numbers can lie to us.

<figure><img src="image_23.png" /><figcaption>Figure 1.20: The ESG world of acronyms</figcaption></figure>
<figure><img src="image_24.png" /></figure>
# Part I

## ESG Risk
<figure><img src="image_25.png" /></figure>
# Chapter 2

## ESG Scoring

To develop ESG analysis, we need extra-financial data provided by companies, reporting frameworks, academic institutions, research centers, international bodies, etc. In general, this heterogeneous data is collected by ESG data providers. These data are available for two levels of use. First, they are widely used by ESG analysts to assess the sustainability risks of companies and countries. Second, they provide the raw material for ESG scoring systems. Like credit scoring models, such systems are of paramount importance for risk assessment and decision making. In the case of credit scoring, the decision is whether or not to grant credit. Therefore, credit scoring models are at the center of the lending process. In the case of ESG scoring, the issue is a bit different. Of course, we can use ESG scores to decide whether or not to invest in a company, but exclusion is not the only strategy, as we saw in the previous chapter. For example, ESG scores are fully embedded in the strategy of ESG integration. In this approach, they play the role of screening rules for portfolio selection. ESG scoring is therefore more than a traditional scoring model. Nevertheless, the analogy between ESG scoring and credit scoring remains essential and poses several challenges in terms of performance evaluation, score consistency and backtesting. This is particularly true as ESG risk ratings are generated from these scoring systems. From this perspective, the concept of ESG model validation takes on a new dimension. We recall that any internal risk model must comply with an independent model validation process that is highly binding and formal from a regulatory standpoint (FRB, 2011; EBA, 2014). Moreover, the validation process is not limited to credit, market, operational and liquidity risks. For example, it also applies to compliance risks: statistical models developed for anti-money laundering detection, transaction monitoring, anomaly detection scenarios, list filtering, etc. The systematic validation approach of any model (risk-based, behavioral, rules-based and AI-based) has recently been reinforced in the US with the interagency guidance on model risk management (FRB, 2021). As ESG scoring models become more widely used by financial institutions, we can easily predict that they will be regulated in the near future, just like the other risk models. Therefore, in this chapter, we adopt the basic idea that a scoring system needs to be validated from an ex-post viewpoint. To do so, we make extensive use of the mathematical and statistical tools available to us in the fields of scoring theory and Markov-based rating methods (Roncalli, 2020a, Section 3.3.3 and Chapter 15). This chapter is then organized as follows. Section 1 presents the ESG data and the sources of extra-financial information. The construction of ESG scores and the performance evaluation of ESG scoring models are discussed in Section 2. Finally, we examine ESG rating systems and assess the consistency of ESG migration matrices.
## 2.1 Data and variables

In this section, we list the most important variables or indicators that are used in an ESG scoring system. For that, we distinguish between sovereign and corporate data since the sources are not the same and their access is more or less easy. As the adage says that “we can only measure what we can define”, we must first specify the meaning of the three ESG factors, because the objective of an ESG score is to measure the risk and opportunities of an entity with respect to environmental, social and governance dimensions. If we consider the definition on page 1, we notice that each factor is defined by encompassing several issues:

**E** climate change mitigation, climate change adaptation, preservation of biodiversity, pollution prevention, circular economy;

**S** inequality, inclusiveness, labor relations, investment in human capital and communities, human rights;

**G** management structure, employee relations, executive remuneration.

Of course, this list is non-exhaustive and must be adapted to show the difference between sovereign and corporate entities. Let us consider an example. According to the Universal Declaration of Human Rights<fnref n="1" />, States have obligations and duties under international law to respect, protect and fulfill human rights:

> "The obligation to respect means that States must refrain from interfering with or curtail-ing the enjoyment of human rights. The obligation to protect requires States to protect individuals and groups against human rights abuses. The obligation to fulfill means that States must take positive action to facilitate the enjoyment of basic human rights. "

For a sovereign, the issue of human rights concerns both social (e.g., access to health and education, labor rights) and governance (e.g., safeguarding of civil and political rights) pillars, while it is more related to the social pillar (e.g., ethical supply chain, employment conditions) of a company. Therefore, it is certainly easier to define the three dimensions with examples.

### 2.1.1 Sovereign ESG data

#### The World Bank framework

The World Bank database dedicated to sovereign ESG indicators is certainly the easiest way to understand the most common topics relevant to ESG analysis. The database is available at the following webpage: https://datatopics.worldbank.org/esg. It contains 67 ESG indicators grouped into 17 themes. In Table 2.1, we have reported these categories and the corresponding number of indicators. For example, the **E** pillar is made up of 5 categories, such as emissions and pollution that contains 5 indicators. In total, we have 5 **E**, 6 **S** and 6 **G** themes. We observe that global warming and its consequences are the main drivers of the environmental pillar. If we analyse the 27 indicators (Table 2.2), two categories are related to the measurement of climate change (emissions & pollution, environment/climate risk & resilience), one category is related to the mitigation risks of climate change (natural capital endowment & management, energy use & security), and the last category concerns the impact of climate change on food security. If we focus on the **S** pillar, the sources of social risks are related to inclusiveness and inequality (education & skills, poverty & inequality, health & nutrition, access to services), in particular the literacy rate, the school enrollment,

<footnote n="1">¹See https://www.ohchr.org/en/ohchr_homepage.</footnote>
<table><caption>Table 2.1: The World Bank database of sovereign ESG indicators</caption><thead><tr><th>Environmental (27)</th><th>Social (22)</th><th>Governance (18)</th></tr></thead><tbody><tr><td><ul><li>Emissions & pollution (5)</li><li>Natural capital endowment & management (6)</li><li>Energy use & security (7)</li><li>Environment/climate risk & resilience (6)</li><li>Food security (3)</li></ul></td><td><ul><li>Education & skills (3)</li><li>Employment (3)</li><li>Demography (3)</li><li>Poverty & inequality (4)</li><li>Health & nutrition (5)</li><li>Access to services (4)</li></ul></td><td><ul><li>Human rights (2)</li><li>Government effectiveness (2)</li><li>Stability & rule of law (4)</li><li>Economic environment (3)</li><li>Gender (4)</li><li>Innovation (3)</li></ul></td></tr></tbody></table>

<caption>Table 2.2: Indicators of the environmental pillar (World Bank database)</caption>

* **Emissions & pollution** (1) CO2 emissions (metric tons per capita); (2) GHG net emissions/removals by LULUCF (Mt of CO2 equivalent); (3) Methane emissions (metric tons of CO2 equivalent per capita); (4) Nitrous oxide emissions (metric tons of CO2 equivalent per capita); (5) PM2.5 air pollution, mean annual exposure (micrograms per cubic meter);

* **Natural capital endowment & management:** (1) Adjusted savings: natural resources depletion (% of GNI); (2) Adjusted savings: net forest depletion (% of GNI); (3) Annual freshwater withdrawals, total (% of internal resources); (4) Forest area (% of land area); (5) Mammal species, threatened; (6) Terrestrial and marine protected areas (% of total territorial area);

* **Energy use & security:** (1) Electricity production from coal sources (% of total); (2) Energy imports, net (% of energy use); (3) Energy intensity level of primary energy (MJ/$2011 PPP GDP); (4) Energy use (kg of oil equivalent per capita); (5) Fossil fuel energy consumption (% of total); (6) Renewable electricity output (% of total electricity output); (7) Renewable energy consumption (% of total final energy consumption);

* **Environment/climate risk & resilience:** (1) Cooling degree days (projected change in number of degree Celsius); (2) Droughts, floods, extreme temperatures (% of population, average 1990-2009); (3) Heat Index 35 (projected change in days); (4) Maximum 5-day rainfall, 25-year return level (projected change in mm); (5) Mean drought index (projected change, unitless); (6) Population density (people per sq. km of land area)

* **Food security:** (1) Agricultural land (% of land area); (2) Agriculture, forestry, and fishing, value added (% of GDP); (3) Food production index (2004-2006 = 100);

Source: https://datatopics.worldbank.org/esg/framework.html.
<caption>Table 2.3: Indicators of the social pillar (World Bank database)</caption>

* **Education & skills:** (1) Government expenditure on education, total (% of government expenditure); (2) Literacy rate, adult total (% of people ages 15 and above); (3) School enrollment, primary (% gross);

* **Employment:** (1) Children in employment, total (% of children ages 7-14); (2) Labor force participation rate, total (% of total population ages 15-64) (modeled ILO estimate); (3) Unemployment, total (% of total labor force) (modeled ILO estimate);

* **Demography:** (1) Fertility rate, total (births per woman); (2) Life expectancy at birth, total (years); (3) Population ages 65 and above (% of total population);

* **Poverty & inequality:** (1) Annualized average growth rate in per capita real survey mean consumption or income, total population (%); (2) Gini index (World Bank estimate); (3) Income share held by lowest 20%; (4) Poverty headcount ratio at national poverty lines (% of population);

* **Health & nutrition:** (1) Cause of death, by communicable diseases and maternal, prenatal and nutrition conditions (% of total); (2) Hospital beds (per 1,000 people); (3) Mortality rate, under-5 (per 1,000 live births); (4) Prevalence of overweight (% of adults); (5) Prevalence of undernourishment (% of population);

* **Access to services:** (1) Access to clean fuels and technologies for cooking (% of population); (2) Access to electricity (% of population); (3) People using safely managed drinking water services (% of population); (4) People using safely managed sanitation services (% of population);

<caption>Table 2.4: Indicators of the governance pillar (World Bank database)</caption>

* **Human rights:** (1) Strength of legal rights index (0 = weak to 12 = strong); (2) Voice and accountability (estimate);

* **Government effectiveness:** (1) Government effectiveness (estimate); (2) Regulatory quality (estimate);

* **Stability & rule of law:** (1) Control of corruption (estimate); (2) Net migration; (3) Political stability and absence of violence/terrorism (estimate); (4) Rule of law (estimate)

* **Economic environment:** (1) Ease of doing business index (1 = most business-friendly regulations); (2) GDP growth (annual %); (3) Individuals using the internet (% of population);

* **Gender:** (1) Proportion of seats held by women in national parliaments (%); (2) Ratio of female to male labor force participation rate (%) (modeled ILO estimate); (3) School enrollment, primary and secondary (gross), gender parity index (GPI); (4) Unmet need for contraception (% of married women ages 15-49);

* **Innovation:** (1) Patent applications, residents; (2) Research and development expenditure (% of GDP); (3) Scientific and technical journal articles;
the Gini index<fnref n="2" />, the income share held by the lowest 20%, etc. We also notice that the integration of some indicators from the categories employment and demography is disturbing. Indeed, we may wonder how the fertility rate is related to the social pillar. For instance, does a high fertility rate increase or decrease social risk? The G pillar includes two classical governance categories (government effectiveness, stability & rule of law), two economic categories (economic development, innovation) and two social-based categories (human rights, gender). In this last case, the frontier between social and governance is blurred. For instance, we can classify the four indicators of the gender category in the social pillar as a non-discrimination category.

**Remark 9** The definition of each indicator can be found on the website https://datatopics.worldbank.org/esg/framework.html. Most of these variables are intuitive and easy to understand. Some of them are more technical and less comprehensible, especially some technical variables of the governance pillar. Therefore, we report in footnotes the definition provided by the World Bank for the following indicators: strength of legal rights index<fnref n="3" />; voice and accountability<fnref n="4" />, government effectiveness<fnref n="5" />; regulatory quality<fnref n="6" />; rule of law<fnref n="7" />.

Certainly, one of the difficulties when building an ESG score is the data gathering, which requires the use of many internal and external sources. In the case of the World Bank framework, the data comes from<fnref n="8" />:

* National accounts statistics collected by Eurostat, OECD (https://www.oecd-ilibrary.org/statistics), United Nations Statistics Division (UNSD, https://unstats.un.org) and the World Bank;

* Internal departments and specialized databases of the World Bank: World Bank Open Data (https://data.worldbank.org), Business Enabling Environment (BEE), Climate Change Knowledge Portal (CCKP, https://climateknowledgeportal.worldbank.org), Global Database of Shared Prosperity (GDSP), Global Electrification Database (GEP), and Poverty and Inequality Platform (PIP, https://pip.worldbank.org);

* International organizations: Emission Database for Global Atmospheric Research (EDGAR, https://edgar.jrc.ec.europa.eu), Food and Agriculture Organization (FAO, https://www.fao.org/faostat), International Energy Agency (IEA, https://www.iea.org/data-and-statistics), International Labour Organization (ILO, https://ilostat.ilo.org), International Renewable Energy Agency (IRENA, https://www.irena.org/Data), UNESCO Institute for Statistics (UIS, http://uis.unesco.org), United Nations Population Division

<footnote n="2">²The Gini index is a measure of income inequality among individuals. It is based on the comparison of cumulative proportions of the population against cumulative proportions of income they receive, and it ranges between 0 in the case of perfect equality and 1 in the case of perfect inequality. Its computation is derived from the Lorenz curve.</footnote>

<footnote n="3">³"Strength of legal rights index measures the degree to which collateral and bankruptcy laws protect the rights of borrowers and lenders and thus facilitate lending."</footnote>

<footnote n="4">⁴"Voice and accountability captures perceptions of the extent to which a country's citizens are able to participate in selecting their government, as well as freedom of expression, freedom of association, and a free media."</footnote>

<footnote n="5">⁵"Government effectiveness captures perceptions of the quality of public services, the quality of the civil service and the degree of its independence from political pressures, the quality of policy formulation and implementation, and the credibility of the government's commitment to such policies."</footnote>

<footnote n="6">⁶"Regulatory quality captures perceptions of the ability of the government to formulate and implement sound policies and regulations that permit and promote private sector development."</footnote>

<footnote n="7">⁷"Rule of law captures perceptions of the extent to which agents have confidence in and abide by the rules of society, and in particular the quality of contract enforcement, property rights, the police, and the courts, as well as the likelihood of crime and violence."</footnote>

<footnote n="8">⁸The list is not exhaustive.</footnote>
(DESA, https://population.un.org), World Health Organization (WHO, https://www.who.int/data), World Intellectual Property Organization (WIPO, https://www.wipo.int/ipstats);

* National agencies and non-governmental organizations: Climate Watch (https://www.climatewatchdata.org), Netherlands Environmental Assessment Agency (PBL, https://www.pbl.nl), World Database on Protected Areas (WDPA, https://www.protectedplanet.net/en/thematic-areas/wdpa);

* Academic resources: Kaufmann et al. (2010), Cohen et al. (2017), and the international disasters database (EM-DAT) of the Centre for Research on the Epidemiology of Disasters (CRED, Université Catholique de Louvain).

Some of these databases are more relevant than others. If we would like to focus on a small number, our preferences are CCKP, EDGAR and Climate Watch for the **E** pillar, FAO, ILO and WHO for the **S** pillar, and the Worldwide Governance Indicators (WGI, https://info.worldbank.org/governance/wgi produced by Daniel Kaufmann and Aart Kraay for the **G** pillar.

<table><caption>Table 2.5: Sovereign ESG taxonomy</caption><thead><tr><th>Environmental</th><th>Social</th><th>Governance</th></tr></thead><tbody><tr><td><ul><li>Biodiversity & land use</li><li>CO2e emissions</li><li>Compliance with environmental standards</li><li>Energy security & renewables</li><li>Emissions reduction targeting</li><li>Food security</li><li>Fossil fuel dependency</li><li>Green economy</li><li>Physical risk exposure</li><li>Pollution & waste management</li><li>Temperature</li><li>Transition risk</li><li>Water management</li></ul></td><td><ul><li>Civil unrest</li><li>Demography</li><li>Education</li><li>Gender</li><li>Health</li><li>Income inequality & poverty</li><li>Labour rights & working conditions</li><li>Living standards</li><li>Migration</li><li>Human rights & local communities</li><li>Non-discrimination</li><li>Social cohesion</li><li>Water and electricity access</li></ul></td><td><ul><li>Business & economic environment</li><li>Corruption & money laundering</li><li>Governance effectiveness</li><li>Infrastructure and mobility</li><li>International relations</li><li>Justice</li><li>National security</li><li>Political stability & institutional strength</li><li>Personal freedom & civil liberties</li><li>Rights of shareholders</li><li>Rule of law</li></ul></td></tr></tbody></table>

Source: Author's research based on the works of Bouyé and Menville (2021), Gratcheva et al. (2020) and Semet et al. (2021).
## Other frameworks

Most ESG rating agencies provide sovereign ESG data. The most known are FTSE (Beyond Ratings), Moody's (Vigeo-Eiris), MSCI, Sustainalytics and RepRisk. One of the most comprehensive databases is certainly Verisk Mapplecroft (https://www.maplecroft.com), which is a global company covering country risk. If we make the union of the different categories, we obtain a taxonomy that looks like the one in Table 2.5. There are many categories, much greater than for the World Bank framework or the PRI taxonomy<fnref n="9" />. If we consider the indicators, the number of variables is large, much greater than 400. Nevertheless, as explained by Bouyé and Menville (2021) and Semet et al. (2021), they are highly correlated. If we perform a principal component analysis, there are few independent dimensions (less than 10). In fact, many of these indicators are correlated to the GDP. For instance, Gratcheva et al. (2020) found that the average correlation between sovereign ESG scores and national income is equal to 81% for aggregate ESG, 51% for E pillar, 85% for S pillar, and 70% for G pillar. If we consider correlations for ESG providers, the lowest correlations are obtained for the E pillar of ISS (7%), MSCI (10%) and V.E (23%), and the G pillar of RepRisk (37%) and V.E (39%), but they are generally high. Therefore, although all these providers use very different indicators, we notice a relative convergence between them<fnref n="10" />.

<table><caption>Table 2.6: Correlation of ESG scores with country's national income (GNI per capita)</caption><thead><tr><th>Factor</th><th>ESG</th><th>E</th><th>S</th><th>G</th></tr></thead><tbody><tr><td>ISS</td><td>68%</td><td>7%</td><td>86%</td><td>77%</td></tr><tr><td>FTSE (Beyond Ratings)</td><td>91%</td><td>74%</td><td>88%</td><td>84%</td></tr><tr><td>MSCI</td><td>84%</td><td>10%</td><td>90%</td><td>77%</td></tr><tr><td>RepRisk</td><td>78%</td><td>79%</td><td>75%</td><td>37%</td></tr><tr><td>RobecoSAM</td><td>89%</td><td>82%</td><td>85%</td><td>85%</td></tr><tr><td>Sustainalytics</td><td>95%</td><td>83%</td><td>94%</td><td>93%</td></tr><tr><td>V.E</td><td>60%</td><td>23%</td><td>79%</td><td>39%</td></tr><tr><td>Total</td><td>81%</td><td>51%</td><td>85%</td><td>70%</td></tr></tbody></table>

Source: Gratcheva et al. (2020, Table 3.1, page 32).

## The mushrooming growth of data

We observe among ESG data providers a mushrooming of indicators and data sources. This concerns the first well-established variables. For a very long time, income inequality was mainly measured by the Gini coefficient or the Lorenz curve, even if there were many other academic measures. It seems that data providers have recently rediscovered and embraced the academic literature. Thus, income inequality may also be measured by the Palma ratio, the S80/S20 (or 20:20) ratio, the Atkinson index, the percentile ratios (P90/P10, P90/P50, P50/P10), the Pietra index, the coefficient of variation or the Theil index. Nevertheless, the mushrooming growth of data mainly concerns non-economic variables. We provide some examples in Figures 2.1–2.4 with palm oil production and

<footnote n="9">⁹PRI (2019a) identifies 4 environmental factors (natural resources, physical risks, energy transition risk, energy security), 4 social factors (demographic change, education and human capital, living standards and income inequality, social cohesion) and 4 governance factors (institutional strength, political stability, government effectiveness, regulatory effectiveness).</footnote>

<footnote n="10">¹⁰Gratcheva et al. (2020, Table 2.3, page 27) found that the average cross-correlation between these providers is equal to 85% for the ESG score, 42% for the environmental score, 85% for the social score and 71% for the governance score. These results are confirmed by the study of Bouyé and Menville (2021, Table 4, page 14).</footnote>
<figure><img src="image_26.png" /><figcaption>Figure 2.1: Palm oil production (2019)</figcaption></figure>

Source: Our World in Data, https://ourworldindata.org/palm-oil.

<figure><img src="image_27.png" /><figcaption>Figure 2.2: Palm oil imports (2019)</figcaption></figure>

Source: Our World in Data, https://ourworldindata.org/palm-oil.
<figure><img src="image_28.png" /><figcaption>Figure 2.3: Share of global annual deforestation (2015)</figcaption></figure>

Source: Our World in Data, https://ourworldindata.org/deforestation.

<figure><img src="image_29.png" /><figcaption>Figure 2.4: Threatened mammal species (2018)</figcaption></figure>

Source: Our World in Data, https://ourworldindata.org/biodiversity.
consumption, deforestation and threatened mammal species. In particular, we notice the increasing use of geo-location data, real-time data, or satellite data, for example, the data provided by the World Resources Institute (WRI) and its different data platforms (https://www.wri.org/data/data-platforms). The most interesting are Ocean Watch (data on ocean economies and management), Aqueduct (cutting-edge data to identify and evaluate water risks), Global Forest Watch (data on forest economies and management) and LandMark (global data of indigenous and community lands). For instance, we can collect data on coastal eutrophication risk, mangrove extent change, coral reef locations, seagrass, salt marshes, soil erosion, chlorophyll-a concentration, etc.

<figure><img src="image_30.png" /><figcaption>Figure 2.5: Global living planet index</figcaption></figure>

Source: https://livingplanetindex.org/latest_results & Author's calculations.

One of the hot topics is currently the biodiversity. A quick search on the web produces dozen of internet pages<fnref n="11" />. Financial institutions have also launched another initiative: Finance For Biodiversity Pledge (https://www.financeforbiodiversity.org). The UN Biodiversity Conference (COP 15), which is organized by the CBD in Montreal, Canada from 7 to 19 December 2022, has certainly given a special impulse, and may explain this new interest. However, biodiversity loss<fnref n="12" /> is a very old topic and has been scientifically documented since the 1990s (Cardinale et al., 2012). According to Almond et al. (2022), biodiversity, as measured by the Living Planet Index<fnref n="13" />, has decreased by 69%

<footnote n="11">¹¹For example, “Why biodiversity is moving to top of mind for investors” (Lazard Asset Management, February 2022), “Investors grapple with complexities of biodiversity” (Financial Times, September 2022), “Asset Management, a lever for preserving biodiversity” (BNP Paribas, September 2022), “Biodiversity quickly rises up the ESG investing agenda” (Financial Times, September 2022), “Biodiversity: why investors should care” (Pictet AM, October 2022), “More asset owners and managers sign biodiversity pledge” (Pensions&Investments, October 2022), etc.</footnote>

<footnote n="12">¹²Biodiversity loss describes the decline in the number, genetic variability, and variety of species, and the biological communities in a given area.</footnote>

<footnote n="13">¹³The LPI is computed using a subset of 31 821 populations of 5 230 species and a statistical model (Westveer et al., 2022, page 28-31).</footnote>
on average since 1970, but with a lot of heterogeneity across regions<fnref n="14" />. Even if the biodiversity loss has decreased these last years (Figure 2.5), this will inevitably result in negative consequences on global wealth in the long run. The seminal work of Costanza et al. (1997) estimated that the annual economic value of natural capital is on average two times the annual economic value of global GNP, explaining that “ecosystem services provide a significant portion of the total contribution to human welfare on this planet.”

Box 2.1: Ecological diversity indexes

Let $p = (p_1, \dots, p_{n_S})$ be the proportion vector of species where $p_i$ is the relative abundance$^a$ of the $i$th specie. The Hill diversity coefficient of order $\eta \ge 0$ is defined as:

$$D^{\eta}(p) = \left( \sum_{i=1}^{n_S} p_i^{\eta} \right)^{1/(1-\eta)}$$

We can show that $1 \le D^{\eta}(p) \le n_S$ and the bounds are reached for the 1- and $n$-diversity distributions$^b \pi_n^-$ and $\pi_n^+$. The Hill number measures the “effective number of species”, meaning that the system holds a diversity equivalent to $D^{\eta}(p)$ equally distributed species. The parameter $\eta$ defines the sensitivity of the true diversity to rare versus. abundant species by modifying the weight given to specie abundances. When $\eta = 0$, $D^{\eta}(p)$ is equal the current number $n_S$ of species or the richness of species. When $\eta \to 1$, we obtain the Shannon diversity index $\mathcal{I}^*(p)$, which is equal to the exponential of the Shannon entropy $\mathcal{I}(p)$:

$$D^1(p) = \mathcal{I}^*(p) = \exp(\mathcal{I}(p)) = \exp\left(-\sum_{i=1}^{n_S} p_i \ln p_i\right)$$

When $\eta = 2$, we obtain:

$$D^2(p) = \left( \sum_{i=1}^{n_S} p_i^2 \right)^{-1}$$

We recognize the inverse of the Herfindahl index $H(p) = \sum_{i=1}^{n_S} p_i^2$ (also called the Simpson index $\lambda(p)$ in ecology). Finally, when $\eta \to \infty$, the Hill index converges to the proportional abundance of the most abundant specie:

$$D^{\infty}(p) = \max_{i} p_i$$

$D^{\infty}(p)$ is then equal to the infinite norm of $p$.

The sudden interest of financial institutions in biodiversity may be explained by climate change, but also by a greater awareness of its critical functions (food security, health, etc.). Moreover, the seventh mandatory PAI indicator requires reporting the share of investments that negatively affect biodiversity sensitive areas<fnref n="15" />, and the sixth objective of the EUTR is dedicated to the “protec-

<footnote>$^a$It is equal to number of individuals in the $i$th specie relative to the total number of individuals in the population.</footnote>

<footnote>$^b$See their definition on page 1150.</footnote>

<footnote n="14">¹⁴This figure is respectively equal to -18% in Europe and Central Asia, -20% in North America, -55% in Asia and the Pacific, -66% in Africa, and -94% in Latin America.</footnote>

<footnote n="15">¹⁵See Table 1.8 on page 38.</footnote>
<figure><img src="image_31.png" /><figcaption>Figure 2.6: Aggregate national RLI</figcaption></figure>

Source: https://www.iucnredlist.org/search & IUCN (2022).

tion and restoration of biodiversity and ecosystems.” All this obviously creates a high demand for biodiversity data and new opportunities for data providers, but as mentioned by Bowker (2000), biodiversity implies data diversity. Again, we are dealing with a huge amount of data<fnref n="16" />. For example, Icerberg Data Lab<fnref n="17" /> (corporate biodiversity footprint or CBF), Carbon 4 (global biodiversity score<fnref n="18" /> or GBS), CDC Biodiversité (global biodiversity score for financial institutions<fnref n="19" /> or GBSFI), ISS ESG<fnref n="20" /> (mean species abundance or MSA, potentially disappeared fraction of species or PDF) and Verisk Maplecroft (biodiversity and protected areas index score<fnref n="21" /> or BPAI) have already developed biodiversity scores. For countries, most biodiversity data are open source<fnref n="22" /> (Stephenson and Stengel, 2020):

* the Red List Index (RLI, https://www.iucnredlist.org)

The RLI is an index of extinction risk for species of plants and animals. It is computed<fnref n="23" /> by the IUCN and is available for five taxonomic groups: birds, mammals, amphibians, cycads and warm-water reef-forming corals. It can be disaggregated in various ways: subset of species (pollinator species, forest-specialist species, invasive alien species, etc.), country, region, etc.

<footnote n="16">¹⁶And also with a lot of diversity measures (Bandeira et al., 2013; Ohlmann et al., 2019).</footnote>

<footnote n="17">¹⁷https://icebergdatabab.com.</footnote>

<footnote n="18">¹⁸https://www.carbon4finance.com/product/biodiversity-impacts.</footnote>

<footnote n="19">¹⁹https://www.cdc-biodiversite.fr/le-global-biodiversity-score.</footnote>

<footnote n="20">²⁰https://www.issgovernance.com/esg/biodiversity-impact-assessment-tool.</footnote>

<footnote n="21">²¹https://www.maplecroft.com/insights/analysis/mining-operations-face-growing-biodiversity-risks.</footnote>

<footnote n="22">²²See the Guide on Biodiversity Measurement Approaches produced by the Finance for Biodiversity Pledge for a comparison of commercial and open source databases (https://www.financeforbiodiversity.org/publications/guide-on-biodiversity-measurement-approaches).</footnote>

<footnote n="23">²³The methodology is described in Butchart et al. (2007).</footnote>
For instance, we report in Figure 2.6 the aggregate RLI for Brazil, China, France, Poland, La Réunion and US.

* World Database on Protected Areas (WDPA, https://www.protectedplanet.net)

* Integrated Biodiversity Assessment Tool (IBAT, https://www.ibat-alliance.org), including the Species Threat Abatement and Restoration metric (STAR)

* Exploring Natural Capital Opportunities, Risks and Exposure (ENCORE, https://encore.naturalcapital.finance)

* Etc.

**Remark 10** Biodiversity risk is a key element for impact investing. We refer to Chapter 5 for an extensive study of this risk (Section 5.4 on page 365).

## 2.1.2 Corporate ESG data

Compared to sovereign ESG data, the collection, understanding and use of corporate ESG data is much more complicated and requires a lot of time and resources. In the first case, we have about 200 countries in the world, many international organizations that produce country data for decades, and vast academic research on this topic. For instance, the economic literature on income inequality starts in the early twentieth century with the seminal publications of Lorenz (1905) and Gini (1921). Since that time, the number of research studies on income inequality has grown exponentially<fnref n="24" />. In the case of corporate ESG data, data production has just become very recently, and the data dimension is not comparable. According to the World Federation of Exchanges (WFE), there are nearly 58 200 listed companies in the world at the end of Q1 2022. Moreover, data collection is not easier because it has concerned private data for a very long time. It is only recently that extra-financial reporting frameworks for corporate issuers exist, and most of them are voluntary. Finally, the last issue when using corporate ESG data is that most of indicators does not have a universal definition. In a nutshell, there are 3 main challenges and barriers to corporate ESG data:

1. Data coverage (how to collect data for all the listed companies?);

2. Data sourcing (where to find the data?);

3. Data quality (what is the accuracy of the collected data?)

### Main indicators

As we have previously seen, we must distinguish several levels of data. Indeed, raw data are generally transformed into ESG metrics, and these metrics are used to define an ESG score. The main difficulty is collecting the raw data. In the case of corporates, this process is time-consuming and manual. The main sources of raw data are:

1. Corporate publications (self-reporting)

(a) Annual reports

(b) Corporate sustainability reports

<footnote n="24">²⁴According to Google scholar, there are more than 5 700 published papers on this topic from 1950 to 1980, and already about 270 before 1950.</footnote>
<figure><img src="image_32.png" /><figcaption>Figure 2.7: From raw data to ESG pillars</figcaption></figure>

## 2. Financial and regulatory filings (standardized reporting)

(a) Mandatory reports (SFDR, CSRD, EUTR, etc.)

(b) Non-mandatory frameworks (PRI, TCFD, CDP, etc.)

For instance, the CDP database is the basic raw material used by all ESG rating agencies for measuring the carbon footprint of issuers. It can be viewed as the entry point and gives a first picture. Then, rating agencies will complete these data by gathering information from annual reports and other sources such as:

3. News and other media

4. NGO reports and websites

5. Company assessment and due diligence questionnaire (DDQ)

For example, S&P Global uses a 230-pages company questionnaire<fnref n="25" /> called *Corporate Sustainability Assessment*. At the end of the collection data process, we may have missing data, noisy data or heterogeneous data. Therefore, the data are completed or adjusted by considering internal statistical models, e.g., industry-based clustering methods. This can be considered as a sixth source of data:

6. Internal models

Once these raw data are collected and cleaned, they can be used to calculate ESG metrics (second level of ESG variables). They are then grouped to define ESG indicators (third level of ESG variables), which are combined to form the basic ESG themes (fourth level of ESG variables). In the sequel, an ESG criterion is a generic term to name ESG variables: it may be an ESG metric, an ESG indicator or an ESG theme. Finally, ESG Pillars are generally based on a few number of ESG themes. This slicing method of ESG variables is illustrated in Figure 2.7. ESG rating agencies do not publicly disclose the raw data they use. Generally, they stop on the theme stage, sometimes on the indicator stage. To better understand the slotting method, we report an example of ESG criteria in Table 2.7. In this example, the **E** and **S** pillars are made up of 9 environmental themes, 7 social themes and 8 governance themes. Here, we do not have access to the ESG indicators. We

<footnote n="25">²⁵It can be downloaded at https://portal.csa.spglobal.com/survey/documents/CSA_Companion.pdf.</footnote>
notice that some criteria are global and concerns all the issuers (e.g., carbon emissions), but other are specific to a given industry (e.g., green cars for the automobile sector, green financing for the banking sector). The choice of the themes/indicators will be done by the ESG rating agency, and the distinction between the two levels is not always obvious. For instance, if the board diversity is measured by the male-female ratio<fnref n="26" />, the ESG theme is measured by a single ESG indicator. In this case, it is difficult to make a distinction between the two levels. Another issue concerns the classification of ESG themes. Let us consider the supply chain for example. It is a social issue if we would like to measure whether or not the suppliers of the company respect human rights and labor standards, but it can be an environmental issue if we would like to measure the impact of the suppliers on climate change and pollution. In Table 2.7, we also observe that the choice of ESG themes may be subjective. For example, we can merge the two categories Pollution and Waste disposal into one category Pollution & waste disposal, we can name Corporate ethics instead of Corporate behaviour, we can split Biodiversity into two categories (fauna & wildlife conservation; flora & land management), etc.

<table><caption>Table 2.7: An example of ESG criteria</caption><thead><tr><th>Environmental</th><th>Social</th><th>Governance</th></tr></thead><tbody><tr><td><ul><li>Biodiversity</li><li>Carbon emissions</li><li>Green cars*</li><li>Green financing*</li><li>Energy use</li><li>Pollution</li><li>Renewable energy</li><li>Waste disposal</li><li>Water use</li></ul></td><td><ul><li>Access to medicine</li><li>Community involvement & human rights</li><li>Customer concern & responsibility</li><li>Diversity</li><li>Employment conditions & labor standards</li><li>Gender equality</li><li>Supply chain</li></ul></td><td><ul><li>Audit and control</li><li>Board diversity</li><li>Board independence</li><li>Corporate behaviour</li><li>CSR strategy</li><li>Executive compensation</li><li>Management compensation</li><li>Shareholder's rights</li></ul></td></tr></tbody></table>

In what follows, we give some insight into the themes and indicators used by rating agencies. This public information about the ESG criteria has been collected from their website, and varies considerably between providers.

* Bloomberg rates 11 800 public companies. They use more than 120 ESG indicators and 2 000+ data points.

* ISS ESG rates about 10 000 issuers. They use more than 800 indicators and apply approximately 100 indicators per company<fnref n="27" />.

<footnote n="26">²⁶It is the ratio of men to women or the proportion of women in the company board.</footnote>

<footnote n="27">²⁷E.g., climate change strategy, eco-efficiency, energy management, environmental impact of product portfolio, environmental management, water risk and impact for the (E) pillar; equal opportunities, freedom of association, health and safety, human rights, product responsibility, social impact of product portfolio, supply chain management, taxes for the (S) pillar; business ethics, compliance, independence of the board, voting rights, shareholder participation, remuneration for the (G) pillar.</footnote>
<table><caption>Table 2.8: MSCI ESG key issue hierarchy</caption><thead><tr><th>Pillar</th><th>#</th><th>Theme</th><th>#</th><th>Indicator</th></tr></thead><tbody><tr><td rowspan="12">Environment</td><td rowspan="4">1</td><td rowspan="4">Climate Change</td><td>1</td><td>Carbon Emissions</td></tr><tr><td>2</td><td>Product Carbon Footprint</td></tr><tr><td>3</td><td>Financing Environmental Impact</td></tr><tr><td>4</td><td>Climate Change Vulnerability</td></tr><tr><td rowspan="3">2</td><td rowspan="3">Natural Capital</td><td>5</td><td>Water Stress</td></tr><tr><td>6</td><td>Biodiversity & Land Use</td></tr><tr><td>7</td><td>Raw Material Sourcing</td></tr><tr><td rowspan="3">3</td><td rowspan="3">Pollution & Waste</td><td>8</td><td>Toxic Emissions & Waste</td></tr><tr><td>9</td><td>Packaging Material & Waste</td></tr><tr><td>10</td><td>Electronic Waste</td></tr><tr><td rowspan="3">4</td><td rowspan="3">Environmental Opportunities</td><td>11</td><td>Opportunities in Clean Tech</td></tr><tr><td>12</td><td>Opportunities in Green Building</td></tr><tr><td>13</td><td>Opportunities in Renewable Energy</td></tr><tr><td rowspan="12">Social</td><td rowspan="4">5</td><td rowspan="4">Human Capital</td><td>14</td><td>Labor Management</td></tr><tr><td>15</td><td>Health & Safety</td></tr><tr><td>16</td><td>Human Capital Development</td></tr><tr><td>17</td><td>Supply Chain Labor Standards</td></tr><tr><td rowspan="6">6</td><td rowspan="6">Product Liability</td><td>18</td><td>Product Safety & Quality</td></tr><tr><td>19</td><td>Chemical Safety</td></tr><tr><td>20</td><td>Consumer Financial Protection</td></tr><tr><td>21</td><td>Privacy & Data Security</td></tr><tr><td>22</td><td>Responsible Investment</td></tr><tr><td>23</td><td>Health & Demographic Risk</td></tr><tr><td rowspan="2">7</td><td rowspan="2">Stakeholder Opposition</td><td>24</td><td>Controversial Sourcing</td></tr><tr><td>25</td><td>Community Relations</td></tr><tr><td rowspan="4">8</td><td rowspan="4">Social Opportunities</td><td>26</td><td>Access to Communications</td></tr><tr><td>27</td><td>Access to Finance</td></tr><tr><td>28</td><td>Access to Health Care</td></tr><tr><td>29</td><td>Opportunities in Nutrition & Health</td></tr><tr><td rowspan="10">Governance</td><td rowspan="4">9</td><td rowspan="4">Corporate Governance</td><td>30</td><td>Ownership & Control</td></tr><tr><td>31</td><td>Board</td></tr><tr><td>32</td><td>Pay</td></tr><tr><td>33</td><td>Accounting</td></tr><tr><td rowspan="2">10</td><td rowspan="2">Corporate Behavior</td><td>34</td><td>Business Ethics</td></tr><tr><td>35</td><td>Tax Transparency</td></tr></tbody></table>

Source: MSCI (2022, Exhibit 2, page 4).
<table><caption>Table 2.9: Refinitiv materiality matrix</caption><thead><tr><th>Pillar</th><th>#</th><th>Theme</th><th>Metrics</th><th>#</th><th>Indicator</th></tr></thead><tbody><tr><td rowspan="14">Environment</td><td rowspan="4">1</td><td rowspan="4">Emissions</td><td rowspan="4">28</td><td>1</td><td>Emissions</td></tr><tr><td>2</td><td>Waste</td></tr><tr><td>3</td><td>Biodiversity</td></tr><tr><td>4</td><td>Environmental Management Systems</td></tr><tr><td rowspan="2">2</td><td rowspan="2">Innovation</td><td rowspan="2">20</td><td>5</td><td>Product Innovation</td></tr><tr><td>6</td><td>Green Revenues, Green R&D and Green CapEx</td></tr><tr><td rowspan="4">3</td><td rowspan="4">Resource Use</td><td rowspan="4">20</td><td>7</td><td>Water</td></tr><tr><td>8</td><td>Energy</td></tr><tr><td>9</td><td>Sustainable Packaging</td></tr><tr><td>10</td><td>Environmental Supply Chain</td></tr><tr><td rowspan="12">Social</td><td>4</td><td>Community</td><td>14</td><td>11</td><td>Community</td></tr><tr><td>5</td><td>Human Rights</td><td>8</td><td>12</td><td>Human Rights</td></tr><tr><td rowspan="3">6</td><td rowspan="3">Product Responsibility</td><td rowspan="3">10</td><td>13</td><td>Responsible Marketing</td></tr><tr><td>14</td><td>Product Quality</td></tr><tr><td>15</td><td>Data privacy</td></tr><tr><td rowspan="4">7</td><td rowspan="4">Workforce</td><td rowspan="4">30</td><td>16</td><td>Diversity &amp; Inclusion</td></tr><tr><td>17</td><td>Career Development &amp; Training</td></tr><tr><td>18</td><td>Working Conditions</td></tr><tr><td>19</td><td>Health &amp; Safety</td></tr><tr><td rowspan="6">Governance</td><td rowspan="2">8</td><td rowspan="2">CSR Strategy</td><td rowspan="2">9</td><td>20</td><td>CSR Strategy</td></tr><tr><td>21</td><td>ESG Reporting &amp; Transparency</td></tr><tr><td rowspan="2">9</td><td rowspan="2">Management</td><td rowspan="2">35</td><td>22</td><td>Structure (independence, diversity, com- mittees)</td></tr><tr><td>23</td><td>Compensation</td></tr><tr><td rowspan="2">10</td><td rowspan="2">Shareholders</td><td rowspan="2">12</td><td>24</td><td>Shareholder Rights</td></tr><tr><td>25</td><td>Takeover Defenses</td></tr></tbody></table>

Source: Refinitiv (2022, page 10).

* FTSE Russell rates about 7200 securities. They use more than 300 indicators and 14 themes: biodiversity, climate change, pollution and resources, supply chain and water security for the **E** pillar; customer responsibility, health and safety, human rights and community, labor standards and supply chain for the **S** pillar; anti-corruption, corporate governance, risk management and tax transparency for the **G** pillar. Each theme contains 10 to 35 indicators, and an average of 125 indicators are applied per company.

* Moody's V.E rates more than 5000 companies. They consider six pillars (corporate governance, business behavior, environment, human rights, human resources, community involvement) and 38 ESG indicators<fnref n="28" />.

<footnote n="28">²⁸Moody's has also developed a methodology for assessing ESG risks in credit analysis based on 15 themes: carbon transition, physical climate risks, water management, waste & pollution, natural capital for the **E** pillar; customer relations, human capital, demographic & societal trends, health & safety, responsible production for the **S** pillar; financial strategy & risk management, management credibility & track record, organizational structure, compliance & reporting, board structure & procedures for the **G** pillars.</footnote>
* MSCI (2022) rates 10 000 companies (14 000 issuers including subsidiaries) and 680 000 securities globally. Using 1000+ data points, they consider two families of metrics: 80 exposure metrics (how exposed is the company to each material issue?) and 250+ management metrics (how is the company managing each material issue?). These metrics are then combined into 35 key issues selected annually for each industry. These key metrics are reported in Table 2.8 and are combined to build 10 main themes.

* Refinitiv (2022) rates 12 000 public and private companies. They consider 10 themes: resource use, emissions and innovation for the **E** pillar; workforce, human rights, community and product responsibility for the **S** pillar; management, shareholders and responsibility (CSR) strategy for the **G** pillar. These themes are built using 186 metrics and 630+ data points. Table 2.9 shows the materiality matrix of themes, indicators and the number of metrics per theme.

* S&P Dow Jones Indices uses between 16 to 27 criteria scores, a questionnaire-based analysis process with 80-120 industry-specific questions and 1 000 data points.

* Sustainalytics rates more than 16 300 companies. They consider 20 material ESG issues, based on 350+ indicators.

**Remark 11** Contrary to sovereign issuers, raw data for corporate issuers are more difficult to find, because they are not in open source data or they can only be manually collected (e.g., annual reporting). The ESG Data Cartography<fnref n="29" />, which has been developed by the Louis Bachelier Institute, proposes a comprehensive list of ESG data with 140+ data sources. The user can filter the databases by accessibility (free, open source, partially free and proprietary).

**Exercise 1** Berg et al. (2022) consider a common taxonomy based on 64 indicators to compare the different ESG rating providers: access to basic services; access to healthcare; animal welfare; anti-competitive practices; audit; biodiversity; board; board diversity; business ethics; chairperson-CEO separation; child labor; climate risk management; clinical trials; collective bargaining; community & society; corporate governance; corruption; customer relationship; diversity; ESG incentives; electromagnetic fields; employee development; employee turnover; energy; environmental fines; environmental management system; environmental policy; environmental reporting; financial inclusion; forests; GHG emissions; GHG policies; GMOs; Global Compact membership; green buildings; green products; HIV programs; hazardous waste; health & safety; human rights; indigenous rights; labor practices; lobbying; non-GHG air emissions; ozone-depleting gases; packaging; philanthropy; privacy & IT; product safety; public health; remuneration; reporting quality; resource efficiency; responsible marketing; shareholders; site closure; supply chain; sustainable finance; systemic risk; taxes; toxic spills; unions; waste; water. For each pillar, give the list of indicators that fall in the category. We consider a very basic ESG classification matrix with 12 themes:

<table><thead><tr><th></th><th>E</th><th>S</th><th>G</th></tr></thead><tbody><tr><td>Global warming</td><td>Health</td><td>Board</td></tr><tr><td>Green opportunities</td><td>Human rights</td><td>Corporate ethics</td></tr><tr><td>Natural resource</td><td>Workforce</td><td>CSR strategy</td></tr><tr><td>Transition risk</td><td>Social responsibility</td><td>Shareholder</td></tr></tbody></table>

For each indicator, associate the right ESG theme<fnref n="30" />.

<footnote n="29">²⁹The website is https://www.institutlouisbachelier.org/en/esg-data-cartography.</footnote>

<footnote n="30">³⁰There may be no or several valid answers.</footnote>
## The race for alternative data

Alternative data corresponds to data that is not available through traditional channels (corporate publications, sustainable reporting, etc.). It includes non-structured data such as images or textual contents. The case of ESG ratings mainly concerns three types of data:

* Internet traffic, browsing activity, web scraping, product reviews, social media and sentiment data;

* Satellite imagery, geotracking data, sensor data<fnref n="31" />;

* Supply-chain data.

Brière et al. (2022) discuss several uses of alternative data sets. The most famous application is the tracking and measurement of ESG controversies. A controversy risk occurs when allegations concerning a company could lead to reputational risk<fnref n="32" /> and financial losses. Everybody knows the famous quotes of Warren Buffett about building and destroying a reputation:

> "It takes 20 years to build a reputation and five minutes to ruin it. If you think about that, you'll do things differently. [...] We can afford to lose money — even a lot of money. But we can't afford to lose reputation — even a shred of reputation. [...] Should you find yourself in a chronically leaking boat, energy devoted to changing vessels is likely to be more productive than energy devoted to patching leaks. [...] Lose money for the firm, and I will be understanding. Lose a shred of reputation for the firm, and I will be ruthless."

The allegations can be reported by media, NGOs, social networks and stakeholders. Data providers generally use text mining and natural language processing (NLP) to analyze an enormous amount of information, detect controversial events and measure the severity of the reputational risk. For example, Refinitiv (2022) completes the traditional ESG score with a controversy score for the 10 ESG themes presented in Table 2.9 on page 69. This score is updated on a weekly basis. The ESG controversies score is calculated based on 23 ESG controversy topics:

* Community: (1) anti-competition controversy, (2) business ethics controversies, (3) intellectual property controversies, (4) critical countries controversies, (5) public health controversies, (6) tax fraud controversies;

* Human rights: (7) child labour controversies, (8) human rights controversies;

* Management: (9) management compensation controversies count;

* Product responsibility: (10) consumer controversies, (11) customer health and safety controversies; (12) privacy controversies; (13) product access controversies; (14) responsible marketing controversies; (15) responsible R&D controversies;

* Resource use: (16) environmental controversies;

* Shareholders: (17) accounting controversies count, (18) insider dealings controversies, (19) shareholder rights controversies;

<footnote n="31">³¹E.g., temperature, humidity, pressure, chemical levels.</footnote>

<footnote n="32">³²Some famous examples are the *Mexico oil spill* (BP, 2010), *dieselgate* affair (Volkswagen, 2015), the gender pay gap (BBC, 2017), the Cambridge Analytica scandal (Facebook, 2018), the opioid epidemic (Purdue Pharma, 2019), the Ehpad scandal (Orpea, 2021), the Pegasus software (NSO, 2021), the greenwashing (DWS, 2022), etc.</footnote>
* Workforce: (20) diversity and opportunity controversies; (21) employee health and safety controversies; (22) wages or working conditions controversies; (23) strikes.

One of the most famous controversy data providers is the Swiss company RepRisk (https://www.reprisk.com), which was created in Zurich in 1998. They are specialized in ESG data science and machine learning. In November 2021, they published their comprehensive methodology (RepRisk, 2022) and Jupyter Notebooks. To identify and classify ESG risks consistent with how key international standards and norms define ESG, they consider a 3-step process:

1. Daily, they collect 500 000+ documents from 100 000+ sources<fnref n="33" /> in 23 languages;

2. These documents are scraped from online sources and fed to machine learning (ML) applications, which predict relevant and unique ESG risk incidents. Results are sent to the ML reducer, in particular, irrelevant results are discarded and predictions fed to the multilingual queue;

3. Then, documents are sorted in priority order. A team of 150+ human analysts confirm and correct ML predictions, assess severity, reach, and novelty, and write risk incident summaries; Final results are incorporated into RepRisk databases<fnref n="34" />.

**Exercise 2** RepRisk (2022) uses 73 controversial topics: abusive/illegal fishing; access to products and services; agricultural commodity speculation; airborne pollutants; alcohol; animal transportation; arctic drilling; asbestos; automatic and semi-automatic weapons; biological weapons; chemical weapons; cluster munitions; coal-fired power plants; conflict minerals; coral reefs; cyberattack; deep sea drilling; depleted uranium munitions; diamonds; drones; economic impact; endangered species; energy management; epidemics/pandemics; forest burning; frocking; fur and exotic animal skins; gambling; gender inequality; genetically modified organisms (GMOs); genocide/ethnic cleansing; greenhouse gas (GHG) emissions; health impact; high conservation value forests; human trafficking; hydropower (dams); illegal logging; indigenous people; involuntary resettlement; land ecosystems; land grabbing; land mines; lobbying; marijuana/cannabis; marine/coastal ecosystems; migrant labor; monocultures; mountaintop removal mining; negligence; nuclear power; nuclear weapons; offshore drilling; oil sands; opioids; palm oil; plastics; pornography; predatory lending; privacy violations; protected areas; racism/racial inequality; rare earths; salaries and benefits; sand mining and dredging; seabed mining; security services; ship breaking and scrapping; soy; tax havens; tobacco; wastewater management; water management; water scarcity. For each topic, associate the right ESG pillar.

Besides controversy risk, text mining and NLP techniques became recently an essential ML tool for different ESG applications. For example, they are more and more used for assessing company disclosures and verifying their credibility. Friederich et al. (2021) use the language model BERT<fnref n="35" /> to automatically identify disclosures of climate-related risks from corporates' annual reports. In a similar way, Bingler et al. (2022) analyse climate risk disclosures along the TCFD categories and conclude that “the firms' TCFD support is mostly cheap talk and that firms cherry-pick to report primarily non-material climate risk information.” Always using the same machine learning model

<footnote n="33">³³These include government agencies, news sites, newsletters, NGOs, print media, regulators, research firms, social media blogs, think tanks and twitter messages.</footnote>

<footnote n="34">³⁴As of July 2022, the RepRisk dataset includes more than 205 000 companies that are associated with risk incidents. Of these 205 000 companies, approximately 7% are listed companies and 93% are non-listed companies (RepRisk, 2022, page 5).</footnote>

<footnote n="35">³⁵Bidirectional Encoder Representations from Transformers (BERT) is a transformer-based machine learning technique for NLP pre-training developed by Google.</footnote>
BERT, Kölbel et al. (2022) consider the impact of climate risk disclosures on the CDS market and find that disclosing transition risks increases CDS spreads, which is not the case for physical risks.

<figure><img src="image_33.png" /><figcaption>Figure 2.8: Geolocation of world power plants by energy source</figcaption></figure>

Source: Global Power Database version 1.3 (June 2021).

Another application of alternative data is the estimation of physical risk exposures. They correspond to the potential financial losses that companies can suffer, and includes droughts, floods, storms, etc. This risk is more difficult to quantify, and its evaluation requires multidisciplinary methodologies: climate modeling, physical asset geolocation, financial loss estimation, etc. In this case, asset tracking is really the crux of physical risk modeling. An example of spatial data is provided in Figure 2.8. This type of geolocalized data is extensively used by Le Guenedal et al. (2021, 2022) when developing a fully integrated methodology to measure cyclone-related physical risk. Until now, most of the models have been developed for countries and regions (Burke et al., 2021) (Burke et al., 2021). When dealing with corporate ESG, data providers generally use input-output matrices in order to compute the risk exposure or contribution of each firm. It is for example the case for biodiversity risk. Nevertheless, we have recently observed some initiatives to provide geospatial data and asset tracking directly at the company level. Even if these solutions are not yet mature, they are very promising<fnref n="36" />.

**Remark 12** Apart from controversies and physical risk, we also notice a third application of alternative data, which consists in building more reactive or real-time ESG scores. Ben Dor et al. (2022) propose to monitor planned sustainability-related corporate activities based on firms’ actions, rather than relying solely on their announcements. For that, they use job postings and NLP to identify ESG-related openings and ESG-related activities of firms. This technique can be used to understand the dynamics of sustainability within a firm.

<footnote n="36">³⁶You can visit the website of the French technology company Kayrros (https://www.kayrros.com), which received the Financial TimesâĂŹ Tech Champions award for its innovation in the IT & Software sector. Kayrros uses satellite observation and AI to analyse trends in emissions and deforestation.</footnote>
<figure><img src="image_34.png" /><figcaption>Figure 2.9: ESG rating disagreement</figcaption></figure>

Source: Berg et al. (2022).

“This graph illustrates the ESG rating divergence. The horizontal axis indicates the value of the Sustainalytics rating as a benchmark for each firm (n = 924). Rating values by the other five raters are plotted on the vertical axis in different colors*. For each rater, the distribution of values has been normalized to zero mean and unit variance. The Sustainalytics rating has discrete values that show up visually as vertical lines where several companies have the same rating value.”

## The divergence of corporate ESG ratings

Corporate ESG data are very different than sovereign ESG data in terms of standardization. Therefore, we must expect more discrepancies in ESG ratings. In Figure 2.9, we have reported one of the most famous illustrations<fnref n="37" /> about this rating disagreement extracted from the pioneering research of Berg et al. (2022). These authors investigate the divergence of ESG ratings from six prominent rating agencies: KLD, Moody's ESG, MSCI, Refinitiv, S&P Global and Sustainalytics. Using the ESG metrics of these data providers, they reconstruct synthetic ratings based on a common taxonomy of 64 indicators<fnref n="38" />. They identify three sources of divergence:

1. “Measurement divergence refers to situation where rating agencies measure the same indicator using different ESG metrics;

2. Scope divergence refers to situation where ratings are based on different set of ESG indicators;

<footnote n="37">³⁷We have the following mapping: *S&P Global, Moody's ESG, KLD, Refinitiv, MSCI.</footnote>

<footnote n="38">³⁸See Exercise 1 on page 70 for the list of ESG indicators</footnote>
3. **Weight** divergence emerges when rating agencies take different views on the relative importance of ESG indicators.”

They find that measurement contributes to 56% of the divergence, scope 38% and weight 6%.

Since this publication, the standardization issue of data and methodologies has been an ongoing discussion among practitioners and academics. For instance, Billio et al. (2021) analyze ESG ratings and indexes agreement and find that “it is extremely difficult to measure the ability of a fund manager if financial performances are strongly conditioned by the chosen ESG benchmark” and “disagreement in the scores provided by the rating agencies disperses the effect of preferences of ESG investors on asset prices.” In Table 2.10, we report their rank correlation matrix. On average, they obtain a mean correlation of 58% for corporate ESG ratings vs. 85% for sovereign ESG scores (Gratcheva et al., 2020).

<table><caption>Table 2.10: Rank correlation among ESG ratings</caption><thead><tr><th></th><th>MSCI</th><th>Refinitiv</th><th>S&P Global</th></tr></thead><tbody><tr><td>MSCI</td><td>100%</td><td></td><td></td></tr><tr><td>Refinitiv</td><td>43%</td><td>100%</td><td></td></tr><tr><td>S&P Global</td><td>45%</td><td>69%</td><td>100%</td></tr><tr><td>Sustainalytics</td><td>53%</td><td>64%</td><td>69%</td><td>100%</td></tr></tbody></table>

Source: Billio et al. (2021, Table 3, page 1432).

## 2.2 Scoring system

A scoring model is a mathematical model that forms the basis for risk stratification. For example, credit scoring refers to statistical models used to measure the creditworthiness of a company or individual (Roncalli, 2020a). In particular, the Altman Z score is probably the most famous score for predicting the bankruptcy of commercial companies (Altman, 1968). However, scoring models can be found in many areas. For example, anti-money laundering (AML) scoring is a rating model used to assess the risk profile of customers (Chen et al., 2018). The goal of trauma and field triage scoring systems is to predict the severity of injury or estimate the prognosis of trauma patients (Senkowski and McKenney, 1999). The Apgar score assesses the physical condition of newborns shortly after birth (Finster et al., 2005). In the case of medicine, we find many scoring systems: ACR score (rheumatoid arthritis symptoms), Alvarado score (appendicitis), Framingham and QRISK scores (cardiovascular risk), Geneva score (pulmonary embolism), etc.

At first glance, we might think that ESG scoring is an extension of credit scoring using extra-financial data instead of financial data. And there are many similarities between the two concepts: ESG ratings vs. credit ratings, ESG materiality vs. credit materiality, ESG risk vs. credit risk, etc. However, from a mathematical point of view, they are two different concepts. In fact, ESG scoring is an unsupervised approach to risk materiality, while credit scoring is a supervised approach to risk materiality<fnref n="39" />. In the case of credit, we want to measure the one-year probability of default.

<footnote n="39">³⁹Unsupervised learning is a branch of statistical learning in which the test data does not contain a response variable. It is in contrast to supervised learning, where the goal is to predict the value of the response variable *Y* given a set of explanatory variables *X*. In unsupervised learning, we only know the *X* values, because the *Y* values do not exist or are not observed. Supervised and unsupervised learning methods are also called “learning with/without a teacher” (Hastie et al., 2009). This metaphor means that in supervised learning we have access to the correct answer provided by the supervisor (or teacher). In unsupervised learning, we have no feedback on the correct answer. For example,</footnote>
Therefore, credit scoring models are calibrated with a historical database of borrower default events. The response variable is then a binary variable that is 1 if the borrower has defaulted and 0 otherwise. In the case of ESG, we want to measure the sustainability of issuers, but we face an endogenous puzzle because the ESG score is already the sustainability measure. The big problem with ESG scoring systems is how to define the response variable. In most cases, the scoring model is not calibrated and is a simple rule-based method. For this reason, we generally consider ESG scoring to be an unsupervised statistical approach. This has several drawbacks in terms of performance evaluation, score consistency and backtesting. Nevertheless, we will see that we can define some proxies for the response variable and use traditional statistical tools to assess the quality of ESG scores.

### 2.2.1 ESG and scoring theory

The goal of scoring models is to produce a numeric score $S$. This score can take values between a lower bound $S^-$ and an upper bound $S^+$. We generally assume that a high value of $S$ is a good risk, while a low value of $S$ is a bad risk. Scoring systems are then used for two main types of decisions: selection and exclusion. A selection process consists of selecting the good risks so that $S \ge s_1$, while an exclusion process consists of excluding the bad risks so that $S \le s_2$. The choice of the thresholds $s_1$ and $s_2$ is important and depends on the process. For example, suppose the score is between 0 and 1000. We can formulate the following rule:

$$ \begin{cases} S \ge 500 \Rightarrow \text{good risk} \Rightarrow \text{selection} \\ S \le 500 \Rightarrow \text{bad risk} \Rightarrow \text{exclusion} \end{cases} $$

Obviously, separating the population into two groups (bad and good risks) is a difficult task. Another approach is to use stricter decision rules:

$$ \begin{cases} S \ge 600 \Rightarrow \text{good risk} \Rightarrow \text{selection} \\ S \le 300 \Rightarrow \text{bad risk} \Rightarrow \text{exclusion} \end{cases} $$

Here we see that there is an asymmetry in the definition of good and bad risks. The choice of the threshold is as important as the construction of the score. However, we have an endogenous problem because the threshold depends on the model itself. In the case of ESG, scores can be designed to define an exclusion process or a selection process, but in most cases they are used to perform both or they are used to define an integration process. This implies that the previous binary choice is replaced by the preference ordering:

$$ S_1 \succ S_2 \Leftrightarrow \text{issuer #1 is a better ESG risk than issuer #2} $$

This approach to scoring is different from the traditional approach. We face a problem here because ESG scores are generally the result of an unsupervised statistical method. Therefore, the development of ESG scoring has been done outside the framework of scoring theory. In fact, unsupervised statistical methods are generally used for clustering and classification. They are then adapted to a multimodal statistical problem, such as a binary statistical model (bad risk versus good risk). ESG scoring is more of an expert system than a true scoring model. In what follows, we try to provide a theoretical framework for ESG scoring, but we must be careful. This theoretical framework is limited for all the reasons mentioned above.

**Remark 13** The fact that ESG scoring is designed as an unsupervised statistical process makes it difficult to assess the quality of ESG rating systems, particularly with respect to materiality.

<footnote>linear regression is a typical supervised learning model, while principal component analysis is an unsupervised learning approach.</footnote>
## 2.2.2 Tree-based scoring methods

### Tree structure

To understand a tree-based scoring model, we first consider the one-level tree structure. Let $X_1, \dots, X_m$ be $m$ features. These metrics are linearly combined to obtain a score:

$$ \mathbf{S} = \sum_{j=1}^{m} \omega_{j} X_{j} $$

where $\omega_j$ is the weight of the $j^{th}$ metric. Generally, the weights are normalized such that $\sum_{j=1}^{m} \omega_j = 1$. This is the most simple scoring model. For instance, the original bankruptcy score of Altman (1968) was equal to:

$$ Z = 1.2 \cdot X_1 + 1.4 \cdot X_2 + 3.3 \cdot X_3 + 0.6 \cdot X_4 + 1.0 \cdot X_5 $$

where the variables $X_j$ represent the following financial ratios:

<table><thead><tr><th>X<sub>j</sub></th><th>Ratio</th></tr></thead><tbody><tr><td>X<sub>1</sub></td><td>Working capital / Total assets</td></tr><tr><td>X<sub>2</sub></td><td>Retained earnings / Total assets</td></tr><tr><td>X<sub>3</sub></td><td>Earnings before interest and tax / Total assets</td></tr><tr><td>X<sub>4</sub></td><td>Market value of equity / Total liabilities</td></tr><tr><td>X<sub>5</sub></td><td>Sales / Total assets</td></tr></tbody></table>

If we note $Z_i$ the score of the firm $i$, we can calculate the normalized score $Z_i^* = (Z_i - m_z) / \sigma_z$ where $m_z$ and $\sigma_z$ are the mean and standard deviation of the observed scores. $Z_i^*$ can then be compared to the quantile of the Gaussian distribution or the empirical distribution. A low value of $Z_i^*$ (for instance $Z_i^* < -2.5$) indicates that the firm has a high probability of default, while companies with high scores above (for instance $Z_i^* > 3$) are not likely to go bankrupt.

We can extend the previous approach to a two-level tree structure. We first begin to compute intermediary scores:

$$ \mathbf{S}_k^{(1)} = \sum_{j=1}^{m} \omega_{j,k}^{(1)} X_j $$

Then we obtain a set of $m_{(1)}$ intermediary scores ($k = 1, \dots, m_{(1)}$), which are combined to obtain the final score:

$$ \mathbf{S} := \mathbf{S}_1^{(0)} = \sum_{k=1}^{m_{(1)}} \omega_k^{(0)} \mathbf{S}_k^{(1)} $$

The exponents (0) and (1) indicate the level of the tree. An example of two-level tree structure is given in Figure 2.10. For the first level, we have:

$$ \left\{ \begin{array}{l} \mathbf{S}_1^{(1)} = 0.5 \cdot X_1 + 0.25 \cdot X_2 + 0.25 \cdot X_3 \\ \mathbf{S}_2^{(1)} = 0.5 \cdot X_4 + 0.5 \cdot X_5 \\ \mathbf{S}_3^{(1)} = X_6 \end{array} \right. $$

The final score is the average of the three intermediary scores:

$$ \mathbf{S} = \frac{\mathbf{S}_1^{(1)} + \mathbf{S}_2^{(1)} + \mathbf{S}_3^{(1)}}{3} $$
<figure><img src="image_35.png" /><figcaption>Figure 2.10: A two-level tree</figcaption></figure>

* Level 1: $\omega_{1,1}^{(1)} = 50\%$; $\omega_{2,1}^{(1)} = 25\%$; $\omega_{3,1}^{(1)} = 25\%$; $\omega_{4,2}^{(1)} = 50\%$; $\omega_{5,2}^{(1)} = 50\%$; $\omega_{6,3}^{(1)} = 100\%$;

* Level 0: $\omega_{1}^{(0)} = \omega_{2}^{(0)} = \omega_{3}^{(0)} = 33.33\%$;

<figure><img src="image_36.png" /><figcaption>Figure 2.11: A two-level overlapping graph</figcaption></figure>

* Level 1: $\omega_{1,1}^{(1)} = 50\%$; $\omega_{2,1}^{(1)} = 25\%$; $\omega_{3,1}^{(1)} = 25\%$; $\omega_{3,2}^{(1)} = 25\%$; $\omega_{4,2}^{(1)} = 25\%$; $\omega_{5,2}^{(1)} = 50\%$; $\omega_{6,3}^{(1)} = 100\%$;

* Level 0: $\omega_{1}^{(0)} = \omega_{2}^{(0)} = \omega_{3}^{(0)} = 33.33\%$;

This tree is a non-overlapping graph because each child node is related to a single parent node, otherwise it is an overlapping graph (but it is not a tree). For example, if we assume that the score $S_2^{(1)}$ also depends on the metric $X_3$, we obtain the overlapping graph structure given in Figure 2.11. In this case, the first level becomes:

$$ \left\{ \begin{array}{l} S_1^{(1)} = 0.5 \cdot X_1 + 0.25 \cdot X_2 + 0.25 \cdot X_3 \\ S_2^{(1)} = 0.25 \cdot X_3 + 0.25 \cdot X_4 + 0.5 \cdot X_5 \\ S_3^{(1)} = X_6 \end{array} \right. $$
Box 2.2: Tree and graph theory

A scoring tree is a special case of a tree data structure, which is defined as a collection of nodes that are organized in a hierarchical structure (see Figure 2.A). A tree is also a connected graph without any circuits⁸. Therefore, the terminology of trees derives from the graph theory. A node (or a vertex) is the basic unit that may contain data and links to other nodes. A connection between two nodes is called an edge. In a tree, edges are directed and are also called arcs or arrows. For instance, our example has 13 nodes $V = (A, \dots, M)$ and 12 edges $\mathcal{E} = (\{K, H\}, \{L, J\}, \dots, \{C, A\}, \{D, A\})$. Mathematically, the tree $\mathcal{T}$ is defined by the set $V$ of nodes (or vertices) and the set $\mathcal{E}$ of edges: $\mathcal{T} = (V, \mathcal{E})$. In a tree, the first node is called the root node ($A$). Any node within a tree can be viewed as a root of its own subtree. By definition, The subtree $\mathcal{T}_{(v)}$ with $v$ as its root is also a tree consisting of $v$ and its descendants. $\mathcal{T}_{(v)}$ is defined by $(\mathcal{V}_{(v)}, \mathcal{E}_{(v)})$, where $\mathcal{V}_{(v)}$ and $\mathcal{E}_{(v)}$ are the sets of vertices and edges of the subgraph. Each edge (or directed path) has a child and a parent (or an internal node). For example, $C$ is the parent node of $(H, I)$ and $H$ is a child node of $C$. Our example tree has then 6 parent nodes $\mathcal{P} = (A, B, C, D, H, J)$ and 12 children$^b$. $B$ has three children $(E, F, G)$ and $B$ is a child of $A$. Child nodes with the same parent are sibling nodes, while a leaf node (or external node) is a node without child nodes. In the example tree, $(B, C, D)$, $(E, F, G)$, $(H, I)$ are siblings. The leaf nodes are $(E, F, G, I, K, L, M)$. The degree of a node is its number of children. It is equal to 3 for $(A, B)$, 2 for $(C, J)$ and 1 for $(H, D)$ and 0 for the leaves. The degree of a tree is equal to the maximum degree of nodes. Our example tree has a degree 3. The level of a node refers to the distance between the node and the root. We deduce that the root node is at level 0. Children of the root are at level 1 $(B, C, D)$. Level 2 corresponds to the nodes $(E, F, G, H, I, J)$ and we have 3 nodes at level 3 $(K, L, M)$. The depth of a tree is the level of the deepest node. It is equal to 3 in our example tree.

<figure><img src="image_37.png" /><figcaption>Figure 2.A: Tree data structure</figcaption></figure>

<footnote>ᵃWe have the following properties:</footnote>

<footnote>* There is one and only one path between every pair of nodes in a tree;</footnote>

<footnote>* A tree with $n$ nodes has exactly $n-1$ edges;</footnote>

<footnote>* A graph is a tree if and only if it is minimally connected;</footnote>

<footnote>* Any connected graph with $n$ nodes and $n-1$ edges is a tree.</footnote>

<footnote>ᵇIn a tree, the number of children is exactly equal to the number of edges.</footnote>
The two-level tree structure can be extended to multi-level tree structures. Let L be the number of levels. At level $ℓ$, the value of the $k^{th}$ node is given by:

$$ \mathbf{S}_k^{(\ell)} = \sum_{j=1}^{m_{(\ell+1)}} \omega_{j,k}^{(\ell)} \mathbf{S}_j^{(\ell+1)} \quad (2.1) $$

where $m_{(\ell+1)}$ is the number of scores at level $ℓ+1$, $\mathbf{S}_j^{(\ell+1)}$ is the $j^{th}$ score at level $ℓ+1$ and $\omega_{j,k}^{(\ell)}$ is the weight of the $j^{th}$ score at level $ℓ+1$ for the $k^{th}$ score at level $ℓ$. By construction, the scores at level L are exactly equal to the features: $\mathbf{S}_j^{(L)} = X_j$. We verify that the final score $\mathbf{S}$ corresponds to the root score $\mathbf{S}_1^{(0)}$. It can be computed by Algorithm 1. If we would like to target a specific level $ℓ^*$, we replace the for statement $t = 1 : L$ by $t = 1 : L + 1 - ℓ^*$.

<caption>**Algorithm 1** Recursive tree-based algorithm for computing the final score</caption>

Compute the final score $\mathbf{S}_1^{(1)}$
**Input:** L the number of levels, $(X_1, \dots, X_m)$ the vector of the metrics and $\{\omega_{j,k}^{(\ell)}\}$ the weight tensor
**Initialize** $m_{(L)} = m$
**for** $j = 1 : m_{(L)}$ **do**
  $\mathbf{S}_j^{(L)} \leftarrow X_j$
  **end for**
  **for** $t = 1 : L$ **do**
    {Change the value of L by $L - ℓ^*$ if you target the level $ℓ^*$}
    $ℓ \leftarrow L - t$
    **for** $k = 1 : m_{(ℓ)}$ **do**
      $\mathbf{S}_k^{(ℓ)} \leftarrow 0$
      **for** $j = 1 : m_{(ℓ+1)}$ **do**
        $\mathbf{S}_k^{(ℓ)} \leftarrow \mathbf{S}_k^{(ℓ)} + \omega_{j,k}^{(ℓ)} \mathbf{S}_j^{(ℓ+1)}$
      **end for**
      **end for**
    **end for**
    $\mathbf{S} \leftarrow \mathbf{S}_1^{(0)}$
    **return** $\mathbf{S}$

The multi-level tree structure is very popular for computing ESG scores. For instance, the final ESG score corresponds to level 0. It is the weighted average of the **E**, **S** and **G** scores, which form the first level. Each pillar depends on a number of ESG themes, which constitutes the second level. As we have seen previously, an ESG theme is made up of several indicators. These last ones are located at the third level of the scoring tree. The computation of indicators requires some ESG metrics. Therefore, an ESG scoring model has at least four levels. For example, we have reported in Figure 2.12 an example of a tree from the MSCI scoring model. *Carbon emissions management and exposure* are two metrics (level 4). They are combined to form the indicator *carbon emissions* (level 3). MSCI uses this indicator and four others to define the *climate change* theme (level 2). It is one of the four themes of the environmental pillar (level 1).
<figure><img src="image_38.png" /><figcaption>Figure 2.12: An example of ESG scoring tree (MSCI methodology)</figcaption></figure>

Source: MSCI (2020).

## Score normalization

**Why we need to normalize** Let $\omega_{(\ell)}$ be the $m_{(\ell+1)} \times m_{(\ell)}$ matrix, whose elements are $\omega_{j,k}^{(\ell)}$ for $j = 1, \dots, m_{(\ell+1)}$ and $k = 1, \dots, m_{(\ell)}$. We note $\mathbf{S}^{(\ell)} = (\mathbf{S}_1^{(\ell)}, \dots, \mathbf{S}_{m_{(\ell)}}^{(\ell)})$ the vector of scores at the tree level $\ell$. We have:

$$ \mathbf{S}^{(\ell)} = \omega_{(\ell)}^{\top} \mathbf{S}^{(\ell+1)} $$

At level 1, we obtain $\mathbf{S}_1^{(1)} = \omega_{(1)}^{\top} \mathbf{S}^{(2)}$. Since we have $\mathbf{S}^{(2)} = \omega_{(2)}^{\top} \mathbf{S}^{(3)}$, we deduce that $\mathbf{S}_1^{(1)} = \omega_{(1)}^{\top} \omega_{(2)}^{\top} \mathbf{S}^{(3)}$. By iterating the previous equation and noting that $\mathbf{S}^{(L)} = X$, the final score is equal to:

$$ \mathbf{S} = \omega^{\top} X \qquad (2.2) $$

where:

$$ \omega = \omega_{(L-1)} \cdots \omega_{(1)} \omega_{(0)} $$

If we are interested in an intermediary score, we proceed in a similar way and we have:

$$ \mathbf{S}_k^{(\ell)} = \mathbf{e}_k^{\top} \mathbf{S}^{(\ell)} \\ = \omega^{\top} X $$

where:

$$ \omega = \omega_{(L-1)} \cdots \omega_{(\ell-1)} \omega_{(\ell)} $$

We conclude that all the scores are a weighted average of initial metrics.
Let us consider the scoring tree given in Figure 2.11. We have:

$$ \omega_{(1)} = \begin{bmatrix} 0.5 & 0 & 0 \\ 0.25 & 0 & 0 \\ 0.25 & 0.25 & 0 \\ 0 & 0.25 & 0 \\ 0 & 0.5 & 0 \\ 0 & 0 & 1 \end{bmatrix} $$

and:

$$ \omega_{(0)} = \frac{1}{3} \begin{bmatrix} 1 \\ 1 \\ 1 \end{bmatrix} $$

We deduce that:

$$ \omega = \omega_{(1)}\omega_{(0)} = \frac{1}{12} \begin{bmatrix} 2 \\ 1 \\ 2 \\ 1 \\ 2 \\ 4 \end{bmatrix} $$

The expression of the final score is:

$$ \mathbf{S} = \frac{2X_1 + X_2 + 2X_3 + X_4 + 2X_5 + 4X_3}{12} $$

Let us assume that $X$ follows a multivariate probability distribution $\mathbf{F}$. We deduce that $\mathbf{S}$ follows the univariate probability distribution $\mathbf{G}$ defined by:

$$ \begin{aligned} \mathbf{G}(s) &= \operatorname{Pr}\{\mathbf{S} \le s\} \\ &= \operatorname{Pr}\{\omega^{\top} X \le s\} \\ &= \int \dots \int \mathbf{1}\{\omega^{\top} x \le s\} \mathrm{d}\mathbf{F}(x) \\ &= \int \dots \int \mathbf{1}\left\{\sum_{j=1}^{m} \omega_j x_j \le s\right\} \mathrm{d}\mathbf{F}(x_1, \dots, x_m) \\ &= \int \dots \int \mathbf{1}\left\{\sum_{j=1}^{m} \omega_j x_j \le s\right\} \mathrm{d}\mathbf{C}(\mathbf{F}_1(x_1), \dots, \mathbf{F}_m(x_m)) \end{aligned} $$

Therefore, the distribution $\mathbf{G}$ depends on the copula function $\mathbf{C}$ and the marginals $(\mathbf{F}_1, \dots, \mathbf{F}_m)$ of $\mathbf{F}$.

We first investigate the independent case. It follows that:

$$ \mathbf{G}(s) = \int \dots \int \mathbf{1}\left\{\sum_{j=1}^{m} \omega_j x_j \le s\right\} \prod_{j=1}^{m} \mathrm{d}\mathbf{F}_j(x_j) $$
We deduce that $\mathbf{G}$ is a convolution probability distribution. In some cases, it corresponds to a well-known probability distribution. For example, if $X_j \sim \mathcal{N}(\mu_j, \sigma_j^2)$, we have $\omega_j X_j \sim \mathcal{N}(\omega_j \mu_j, \omega_j^2 \sigma_j^2)$. We deduce that:

$$ \mathbf{S} \sim \mathcal{N} \left( \sum_{j=1}^{m} \omega_j \mu_j, \sum_{j=1}^{m} \omega_j^2 \sigma_j^2 \right) \equiv \mathcal{N} \left( \omega^\top \mu, \omega^\top \Sigma \omega \right) $$

where $\mu = (\mu_1, \dots, \mu_m)$ and $\Sigma = \text{diag}(\sigma_1^2, \dots, \sigma_m^2)$. In Figure 2.13, we have reported the density function of the intermediary and final scores for the tree 2.10 when $X_j \sim \mathcal{N}(0, 1)$. The four scores $\mathbf{S}_1^{(1)}, \mathbf{S}_2^{(1)}, \mathbf{S}_3^{(1)}$ and $\mathbf{S}$ are Gaussian, centered at 0 with different standard deviations$^{40}$. We face here an issue because we cannot compare the different scores and it is impossible to have a homogeneous rule to assess whether a score is good or not.

<figure><img src="image_39.png" /><figcaption>Figure 2.13: Probability distribution of the scores (Tree 2.10)</figcaption></figure>

**Example 3** We assume that $X_1 \sim \mathcal{U}_{[0,1]}$ and $X_2 \sim \mathcal{U}_{[0,1]}$ are two independent random variables. We consider the score $\mathbf{S}$ defined as:

$$ \mathbf{S} = \frac{X_1 + X_2}{2} $$

We have $\mathbf{S} \in [0, 1]$. In Figure 2.14, we consider a geometric interpretation of the probability mass function $\text{Pr}\{\mathbf{S} \le s\} = \text{Pr}\{X_1 + X_2 \le 2s\}$. We distinguish two cases. The first case (a) corresponds to $s \le 0.5$. The probability mass corresponds then to the right triangle with vertices $(0, 0)$, $(0, 2s)$ and $(2s, 0)$. It is equal to one-half the square, whose length is $2s$. The second case (b) corresponds to $0.5 \le s \le 1$. Here the probability mass is equal to the area of the polygon shape, whose vertices

<footnote>$^{40}$They are equal to 0.6124 for $\mathbf{S}_1^{(1)}$, 0.7071 for $\mathbf{S}_2^{(1)}$, 1 for $\mathbf{S}_3^{(1)}$ and 0.4564 for $\mathbf{S}$.</footnote>
<figure><img src="image_40.png" /><figcaption>Figure 2.14: Geometric interpretation of the probability mass function (Example 3)</figcaption></figure>

are (0,0), (1,0), (1,2s-1), (2s-1,1) and (0,1). This is equivalent to computing the area of the unit square minus the right triangle with vertices (1,2s-1), (1,1) and (2s-1,1). We notice that the area of the right triangle is equal to one-half the square, whose length is 2-2s. We deduce that:

$$ \mathrm{Pr}\{\mathbf{S} \leq s\} = \begin{cases} \frac{1}{2}(2s)^2 & \text{if } 0 \leq s \leq \frac{1}{2} \\ 1 - \frac{1}{2}(2-2s)^2 & \text{if } \frac{1}{2} \leq s \leq 1 \end{cases} $$

Finally, we obtain:

$$ \mathbf{G}(s) = \begin{cases} 2s^2 & \text{if } 0 \le s \le \frac{1}{2} \\ -1 + 4s - 2s^2 & \text{if } \frac{1}{2} \le s \le 1 \end{cases} $$

The density function is then:

$$ g(s) = \begin{cases} 4s & \text{if } 0 \le s \le \frac{1}{2} \\ 4 - 4s & \text{if } \frac{1}{2} \le s \le 1 \end{cases} $$

The previous results can be extended to the case $m > 2$. We can show that the score follows the Bates distribution$^{41}$:

$$ \mathbf{S} = \frac{X_1 + X_2 + \cdots + X_m}{m} \sim \mathfrak{B}ates(m) $$

In Figures 2.15 and 2.16, we report the density and distribution functions of the score for several values of $m$. We verify that the shape of the score depends on the number $m$ of features. In particular, we observe that the score tends to the Dirac distribution $\delta(x - 1/2)$.

Let us now investigate the dependent case. If we assume that $X \sim \mathcal{N}(\mu, \Sigma)$, the score $\mathbf{S} = \omega^\top X$ is normally distributed: $\mathbf{S} \sim \mathcal{N}(\omega^\top \mu, \omega^\top \Sigma \omega)$. We consider that $\mu_j = 0, \sigma_j = 1$ and $\rho_{j,k} = \rho$ for $j \neq k$. Since the covariance matrix is the constant correlation matrix $C_m(\rho)$, we deduce that

<footnote>$^{41}$See Exercise 2.4.1 on page 132.</footnote>
<figure><img src="image_41.png" /><figcaption>Figure 2.16: Cumulative distribution function of $\mathcal{S}$ (uniform distribution)</figcaption></figure>

<figure><img src="image_42.png" /></figure>
$\mathbb{E}[\mathbf{S}] = 0$ and$^{42}$:

$$
\begin{align*}
\operatorname{var}(\mathbf{S}) &= \omega^\top \mathbb{C}_m(\rho) \omega \\
&= \sum_{j=1}^{m} \sum_{k=1}^{m} \omega_j \omega_k \rho_{j,k} + \left( \sum_{j=1}^{m} \omega_j^2 \rho - \sum_{j=1}^{m} \omega_j^2 \rho \right) \\
&= \rho \sum_{j=1}^{m} \sum_{k=1}^{m} \omega_j \omega_k + (1 - \rho) \sum_{j=1}^{m} \omega_j^2 \\
&= \rho \mathcal{S}^2(w) + (1 - \rho) \mathcal{H}(w)
\end{align*}
$$

where $\mathcal{S}(w) = \sum_{j=1}^{m} \omega_j$ is the sum index and $\mathcal{H}(\omega) = \sum_{j=1}^{m} \omega_j^2$ is the Herfindahl index. Generally, the weights are normalized and we obtain $\mathbf{S} \sim \mathcal{N}(0, \sigma_{\mathbf{S}}^2)$ where:

$$
\sigma_{\mathbf{S}} = \sqrt{\rho + (1 - \rho) \mathcal{H}(\omega)}
$$

Since $0 \le \mathcal{H}(\omega) \le 1$, $\sigma_{\mathcal{S}} \in [\sqrt{\rho}, 1]$. When the weights are equal ($\omega_j = 1/m$), the previous formula reduces to$^{43}$:

$$
\sigma_{\mathbf{S}} = \sqrt{\rho + \frac{(1 - \rho)}{m}}
$$

Let us build the confidence interval of the score at the confidence level $\alpha$. We have $\text{Pr}\{\mathbf{S} \in [\mathbf{S}^{-}(m, \alpha), \mathbf{S}^{+}(m, \alpha)]\} = \alpha$. Since the expectation of the score is equal to zero, we consider an interval centered at 0. We deduce that:

$$
\mathbf{S}^{+}(m, \alpha) = \Phi^{-1}\left(\frac{1+\alpha}{2}\right) \sqrt{\rho + \frac{(1-\rho)}{m}}
$$

and $\mathbf{S}^{-}(m, \alpha) = -\mathbf{S}^{+}(m, \alpha)$.

In Figure 2.17, we illustrate the shrinkage issue of the score when $\alpha$ is set to 99.75%. At this
confidence level, a standard normal random variable lies between -3 and +3. This is the range that
we have in mind when we build a z-score. We observe that the shrinkage begins when the score is
made up of three features and a correlation lower than 80%. The shrinkage issue increases with the
number of features. Let us consider for example an ESG score that depends on 20 ESG metrics,
that have an average correlation of 20%. While the range of the features is between -3 and +3, the
aggregate ESG score lies between -1.5 and +1.5, meaning that the support of the score has been
divided by a factor of two!

**How to normalize?** The previous analysis implies that we must normalize the raw data and the scores at the different levels such that they follow the same probability distribution $\mathbf{F}_{\mathbf{S}}$. Equation (2.1) is no longer valid and we deduce that the node values of the multi-level tree structure are equal to:

$$
\mathbf{S}_k^{(\ell)} = \varphi \left( \sum_{j=1}^{m_{(\ell+1)}} \omega_{j,k}^{(\ell)} \mathbf{S}_j^{(\ell+1)} \right) \qquad (2.3)
$$

where $\varphi(s)$ is the normalization function. We also have $\mathbf{S}_j^{(L)} = \varphi(X_j)$.

Let $X$ be a variable to normalize and $\{x_1, \dots, x_n\}$ a sample of $n$ observations. In practice, there are three main approaches:

<footnote>$^{42}$We note $\mathcal{S}(w) = \sum_{j=1}^{m} \omega_j$. We deduce that $(\sum_{j=1}^{m} \omega_j)^2 = \sum_{j=1}^{m} \sum_{k=1}^{m} \omega_j \omega_k = \mathcal{S}^2(w)$.</footnote>

<footnote>$^{43}$We have $\mathcal{H}(\omega) = \sum_{j=1}^{m} (1/m)^2 = 1/m$.</footnote>
<figure><img src="image_43.png" /><figcaption>Figure 2.17: Upper and lower bounds of the aggregate score when $\alpha = 99.75\%$</figcaption></figure>

1. The first one is the m-score (or min-max) normalization:

$$m_i = \frac{x_i - x^{-}}{x^{+} - x^{-}}$$

where $x^{-} = \min x_i$ and $x^{+} = \max x_i$. This is the most naive approach to obtain a 0/1 normalization.

2. The second approach is the q-score (or quantile) normalization:

$$q_i = \mathbf{H}(x_i)$$

where $\mathbf{H}$ is the distribution function of $X$. When the distribution of $X$ is unknown, we replace $\mathbf{H}$ by the empirical distribution $\hat{\mathbf{H}}$: $q_i = \hat{\mathbf{H}}(x_i)$. In both cases, we obtain a 0/1 normalization.

3. The third method is the famous z-score normalization:

$$z_i = \frac{x_i - \mu}{\sigma}$$

where $\mu$ and $\sigma$ are the mathematical expectation and standard deviation of $X$. Again, when the distribution of $X$ is unknown, we use the empirical mean and standard deviation of the sample:

$$z_i = \frac{x_i - \hat{\mu}}{\hat{\sigma}}$$

By construction, we have $z_i \in (-\infty, \infty)$. Nevertheless, we have seen that $\Pr\{-3 \le N(0,1) \le 3\} \approx 99.75\%$. Generally, we consider that the z-score method produces a $-3/+3$ normalization.
Among the three approaches, only the second approach satisfies the consistency property. Indeed, if $X \sim \mathbf{H}$ and is continuous, we know that $Y = \mathbf{H}(X)$ is a uniform random variable$^{44}$. $Y = \mathbf{H}(X)$ defines the probability integral transform, which plays an important role in statistics and probability. The min-max approach is consistent only if $X \sim \mathcal{U}_{[x^-, x^+]}$, whereas the z-score normalization is consistent if the original data are normally distributed.

Box 2.3: Computing the empirical distribution $\hat{\mathbf{H}}$

Let $\{x_1, x_2, \dots, x_n\}$ be the sample. We have:

$$q_i = \hat{\mathbf{H}}(x_i) = \Pr\{X \le x_i\} = \frac{\#\{x_j \le x_i\}}{n_q}$$

We can use two normalization factors: $n_q = n$ or $n_q = n + 1$. For example, if $n = 4$, we have $q_i \in \{0.25, 0.5, 0.75, 1\}$ if $n_q = n$, and $q_i \in \{0.2, 0.4, 0.6, 0.8\}$ if $n_q = n + 1$. The second solution is better because $q_i \in ]0, 1[$. Therefore, we can transform $q_i$ into a random variable $Y$ with probability distribution $\mathbf{G}$ by considering the inverse probability integral transform:

$$Y = \mathbf{G}^{-1}(q_i) \sim \mathbf{G}$$

For example, we can transform a $q$-score into a $z$-score by considering the Gaussian quantile function:

$$z = \Phi^{-1}(q)$$

With the second solution, we are sure that $z \notin (-\infty, \infty)$.

To obtain an $a/b$ normalization, we consider the following property:

$$\mathcal{U}_{[a,b]} = a + (b - a) \mathcal{U}_{[0,1]}$$

Therefore, we apply the following transform to obtain the new score:

$$q' = a + (b - a) q$$

The $q$-score is distributed according to the uniform distribution. It has the advantage to be normalized between 0 and 1. However, it has the disadvantage to be a flat score, meaning that the extreme scores have the same probability to occur as the mean score. The $z$-score is distributed according to the Gaussian distribution. It has a bell-curve shape, meaning that the extreme scores have a lower probability to occur than the mean score. Therefore, it is interesting to combine the two properties to obtain a discriminant score between 0 and 1. This is done by considering the $b$-score using a Beta distribution $\mathcal{B}(\alpha, \beta)$. In this case, we have:

$$b_i = \mathfrak{B}^{-1}(\mathbf{H}(x_i); \alpha, \beta)$$

$$\begin{aligned}
\Pr\{Y \le y\} &= \Pr\{\mathbf{H}(X) \le y\} \\
&= \Pr\{X \le \mathbf{H}^{-1}(y)\} \\
&= \mathbf{H}(\mathbf{H}^{-1}(y)) \\
&= y
\end{aligned}$$

We deduce that $Y \sim \mathcal{U}_{[0,1]}$.

<footnote>$^{44}$We have $Y \in [0, 1]$ and:</footnote>
When $\alpha = \beta$, This creates a symmetric score around 0.5. To obtain a left-skewed score (when having more best-in-class issuers than worst-in-class issuers), we use $\alpha > \beta$. The case $\alpha < \beta$ produces a right-skewed score (when having more worst-in-class issuers than best-in-class issuers). The standard values for parameters are $\alpha = \beta = 2$.

**Example 4** The data are normally distributed with mean $\mu = 5$ and standard deviation $\sigma = 2$. To map these data into a 0/1 score, we consider the following transform:

$$s := \varphi(x) = \mathfrak{B}^{-1}\left(\Phi\left(\frac{x-5}{2}\right); \alpha, \beta\right)$$

In Figure 2.18, we report the transform function $\varphi(x)$ and the final distribution for three sets of parameters $(\alpha, \beta)$.

<figure><img src="image_44.png" /><figcaption>Figure 2.18: Transforming data into b-score using the Beta distribution</figcaption></figure>

**Remark 14** ESG data providers do not publish their statistical methodology for computing ESG scores<fnref n="45" />. It is then difficult to know which normalization approach is used. Nevertheless, they give the scale of the scores. Bloomberg, S&P Global and Sustainalytics use a range from 0 to 100, Refinitiv from 1 to 100, MSCI from 0 to 10, ISS ESG from 1 to 10, etc. Some asset managers use a scale between -3 and +3.

$$S = \frac{2}{1 + e^{-z}} - 1 \in [-1, 1]$$

<footnote n="45">⁴⁵Only S&P Dow Jones Indices (2022, page 8) indicates that “the normalization is performed by a sigmoid-function on a standard z-score”:</footnote>
**Example 5** We consider the raw data of 9 companies in Table 2.11. These companies belong to the same industry. The first variable measures the carbon intensity of scope 1 + 2 in 2020, while the second variable is the variation of carbon emissions between 2015 and 2020. We would like to create the score $S = 70\% \cdot X_1 + 30\% \cdot X_2$.

<table><caption>Table 2.11: Raw data of 9 companies (carbon emissions and carbon momentum)</caption><thead><tr><th>Firm</th><th>Carbon intensity<br>(in tCO<sub>2</sub>e/$ mn)</th><th>Carbon momentum<br>(in %)</th></tr></thead><tbody><tr><td>1</td><td>94.0</td><td>-3.0</td></tr><tr><td>2</td><td>38.6</td><td>-5.5</td></tr><tr><td>3</td><td>30.6</td><td>5.6</td></tr><tr><td>4</td><td>74.4</td><td>-1.3</td></tr><tr><td>5</td><td>97.1</td><td>-16.8</td></tr><tr><td>6</td><td>57.1</td><td>-3.5</td></tr><tr><td>7</td><td>132.4</td><td>8.5</td></tr><tr><td>8</td><td>92.5</td><td>-9.1</td></tr><tr><td>9</td><td>64.9</td><td>-4.6</td></tr></tbody></table>

To create the synthetic score, we must analyze the data. An ESG investor prefers to be exposed to low-carbon companies than to high-carbon companies. Similarly, he favors firms that have reduced their carbon emissions in the past. If we consider the raw variables as the two features, a high value of the score will indicate a worst-in-class company while a low value of the score will indicate a best-in-class company. If we prefer that high scores correspond to best-in-class scores, we need to take the opposite of these data. We consider the first choice. In Table 2.12, we report the computation of the score by using a q-score 0/100 normalization. For instance, company #7 has the highest carbon emission and then the highest score $q_1 = 90$. It is followed by company #5 which has a score of 80. We verify that the mean of $q_1$ is equal to $(0 + 100)/2$ and its standard deviation is approximately$^{46}$ equal to $\sqrt{(100 - 0)^2/12}$. For the carbon momentum, the best issuer is company #5 with a trend of -16.8%, and its q-score is equal to 10. We compute then the score $s = 0.7 \cdot q_1 + 0.3 \cdot q_2$. This indicates that the ESG investor is more sensitive to carbon intensity than carbon intensity. Said differently, he would like to build a score primarily based on the carbon intensity, but he would like to penalize companies that increase their carbon emissions. For the first firm, we obtain $s = 0.7 \times 70 + 0.3 \times 60 = 67$. We notice that the standard deviation of the variable $s$ is equal to 20.60, which is lower than 27.39. Again, we have to normalize the variable $s$ and we obtain the final score. We also report the rank $R$ and we obtain the following ordering:

$$ \#2 \succ \#3 \succ \#6 \succ \#9 \succ \#8 \succ \#4 \succ \#5 \succ \#1 \succ \#7 $$

We deduce that the best-in-class and worst-in-class issuers are respectively companies #2 and #7. If we consider the z-score normalization, the results are given in Table 2.13. For example, $z_1$ and $z_2$ are equal to $(92.5 - 75.73)/31.95 = 0.525$ and $(-9.10 + 3.30)/7.46 = -0.778$ for company #8. We deduce that $s = 0.7 \times 0.525 + 0.3 \times (-0.778) = 0.134$. Again we observe that the variable $s$ is not normalized since its standard deviation is not equal to 1. We deduce that $S = (0.134 - 0.000)/0.759 = 0.177$. Finally, we obtain the following ordering:

$$ \#2 \succ \#3 \succ \#6 \succ \#9 \succ \#5 \succ \#4 \succ \#8 \succ \#1 \succ \#7 $$

We conclude that the two approaches produce two different rankings.
<table><caption>Table 2.12: Computation of the score $\mathcal{S} \equiv 70\% \cdot X_1 + 30\% \cdot X_2$ ($q$-score 0/100 normalization)</caption><thead><tr><th>#</th><th>X<sub>1</sub></th><th>q<sub>1</sub></th><th>X<sub>2</sub></th><th>q<sub>2</sub></th><th>s</th><th>S</th><th>Re</th></tr></thead><tbody><tr><td>1</td><td>94.00</td><td>70.00</td><td>-3.00</td><td>60.00</td><td>67.00</td><td>80.00</td><td>8</td></tr><tr><td>2</td><td>38.60</td><td>20.00</td><td>-5.50</td><td>30.00</td><td>23.00</td><td>10.00</td><td>1</td></tr><tr><td>3</td><td>30.60</td><td>10.00</td><td>5.60</td><td>80.00</td><td>31.00</td><td>20.00</td><td>2</td></tr><tr><td>4</td><td>74.40</td><td>50.00</td><td>-1.30</td><td>70.00</td><td>56.00</td><td>60.00</td><td>6</td></tr><tr><td>5</td><td>97.10</td><td>80.00</td><td>-16.80</td><td>10.00</td><td>59.00</td><td>70.00</td><td>7</td></tr><tr><td>6</td><td>57.10</td><td>30.00</td><td>-3.50</td><td>50.00</td><td>36.00</td><td>30.00</td><td>3</td></tr><tr><td>7</td><td>132.40</td><td>90.00</td><td>8.50</td><td>90.00</td><td>90.00</td><td>90.00</td><td>9</td></tr><tr><td>8</td><td>92.50</td><td>60.00</td><td>-9.10</td><td>20.00</td><td>48.00</td><td>50.00</td><td>5</td></tr><tr><td>9</td><td>64.90</td><td>40.00</td><td>-4.60</td><td>40.00</td><td>40.00</td><td>40.00</td><td>4</td></tr><tr><td>Mean</td><td>75.73</td><td>50.00</td><td>-3.30</td><td>50.00</td><td>50.00</td><td>50.00</td><td></td></tr><tr><td>Std-dev.</td><td>31.95</td><td>27.39</td><td>7.46</td><td>27.39</td><td>20.60</td><td>27.39</td><td></td></tr></tbody></table>

<table><caption>Table 2.13: Computation of the score $\mathcal{S} \equiv 70\% \cdot X_1 + 30\% \cdot X_2$ ($z$-score normalization)</caption><thead><tr><th>#</th><th>X<sub>1</sub></th><th>z<sub>1</sub></th><th>X<sub>2</sub></th><th>z<sub>2</sub></th><th>s</th><th>S</th><th>Re</th></tr></thead><tbody><tr><td>1</td><td>94.00</td><td>0.572</td><td>-3.00</td><td>0.040</td><td>0.412</td><td>0.543</td><td>8</td></tr><tr><td>2</td><td>38.60</td><td>-1.162</td><td>-5.50</td><td>-0.295</td><td>-0.902</td><td>-1.188</td><td>1</td></tr><tr><td>3</td><td>30.60</td><td>-1.413</td><td>5.60</td><td>1.193</td><td>-0.631</td><td>-0.831</td><td>2</td></tr><tr><td>4</td><td>74.40</td><td>-0.042</td><td>-1.30</td><td>0.268</td><td>0.051</td><td>0.067</td><td>6</td></tr><tr><td>5</td><td>97.10</td><td>0.669</td><td>-16.80</td><td>-1.810</td><td>-0.075</td><td>-0.099</td><td>5</td></tr><tr><td>6</td><td>57.10</td><td>-0.583</td><td>-3.50</td><td>-0.027</td><td>-0.416</td><td>-0.548</td><td>3</td></tr><tr><td>7</td><td>132.40</td><td>1.774</td><td>8.50</td><td>1.582</td><td>1.716</td><td>2.261</td><td>9</td></tr><tr><td>8</td><td>92.50</td><td>0.525</td><td>-9.10</td><td>-0.778</td><td>0.134</td><td>0.177</td><td>7</td></tr><tr><td>9</td><td>64.90</td><td>-0.339</td><td>-4.60</td><td>-0.174</td><td>-0.290</td><td>-0.382</td><td>4</td></tr><tr><td>Mean</td><td>75.73</td><td>0.000</td><td>-3.30</td><td>0.000</td><td>0.000</td><td>0.000</td><td></td></tr><tr><td>Std-dev.</td><td>31.95</td><td>1.000</td><td>7.46</td><td>1.000</td><td>0.759</td><td>1.000</td><td></td></tr></tbody></table>

<table><caption>Table 2.14: Comparison of the different scoring methods</caption><thead><tr><th>#</th><th>q<br>S<br>Re</th><th>z<br>S<br>Re</th><th>qz<br>S<br>Re</th><th>zq<br>S<br>Re</th><th>bz<br>S<br>Re</th><th>bz<sup>*</sup><br>S<br>Re</th></tr></thead><tbody><tr><td>1</td><td>80.00</td><td>0.54</td><td>76.27</td><td>0.84</td><td>0.66</td><td>0.81</td></tr><tr><td>2</td><td>10.00</td><td>-1.19</td><td>9.19</td><td>-1.28</td><td>0.20</td><td>0.30</td></tr><tr><td>3</td><td>20.00</td><td>-0.83</td><td>21.37</td><td>-0.84</td><td>0.29</td><td>0.38</td></tr><tr><td>4</td><td>60.00</td><td>0.07</td><td>54.13</td><td>0.25</td><td>0.52</td><td>0.70</td></tr><tr><td>5</td><td>70.00</td><td>-0.10</td><td>56.65</td><td>0.52</td><td>0.51</td><td>0.64</td></tr><tr><td>6</td><td>30.00</td><td>-0.55</td><td>24.42</td><td>-0.52</td><td>0.34</td><td>0.50</td></tr><tr><td>7</td><td>90.00</td><td>2.26</td><td>98.04</td><td>1.28</td><td>0.93</td><td>0.96</td></tr><tr><td>8</td><td>50.00</td><td>0.18</td><td>60.39</td><td>0.00</td><td>0.56</td><td>0.72</td></tr><tr><td>9</td><td>40.00</td><td>-0.38</td><td>30.96</td><td>-0.25</td><td>0.39</td><td>0.56</td></tr><tr><td>Mean</td><td>50.00</td><td>0.00</td><td>47.94</td><td>0.00</td><td>0.49</td><td>0.62</td></tr><tr><td>Std-dev.</td><td>27.39</td><td>1.00</td><td>28.79</td><td>0.82</td><td>0.22</td><td>0.21</td></tr></tbody></table>
In Table 2.14, we compare the scores and the ranks obtained for different scoring schemes. Besides the q- and z-scores, we consider the following transforms:

* The $qz$-score is defined as:

$$qz = c \cdot \Phi(z) \in [0, c]$$

where $c = 100$ is the scaling factor.

* The $zq$-score is defined as:

$$zq = \Phi^{-1}\left(\frac{q}{c}\right) \in [-3, 3]$$

* The $bz$-score is defined as:

$$bz = \mathfrak{B}^{-1}(\Phi(z); \alpha, \beta) \in [0, 1]$$

where $\alpha = \beta = 2$.

* The $bz^*$-score is a modification of the $bz$-score by using $\alpha = 2.5$ and $\beta = 1.5$.

We verify that the $bz^*$-score is left-skewed and the mean is above $1/2$.

**Remark 15** Most ESG scoring systems are sector neutral, meaning that the normalization is done at the sector (or industry) level, not at the issuer universe level. ESG scores are then **relative** scores (with respect to the sector/industry), not **absolute** scores. This is the concept of best-in-class/worst-in-class issuers. A best-in-class company is then not a best-in-universe issuer. Let us consider the example where the score of corporate A is $S_A = +2$ and the score of corporate B is $S_B = +1$. We have:

* If A and B belong to the same sector, we have $A \succ B$;

* If A and B belong to two different sectors, we may have $A \succ B$ or $B \succ A$.

The preference ordering $\succ$ is then partial and not total.

**An example with the CEO pay ratio** The CEO pay ratio is calculated by dividing the CEO's compensation by the pay of the median employee. It is one of the key metrics for the G pillar. It has been imposed by the Dodd-Frank Act, which requires that publicly traded companies disclose:

1. the median total annual compensation of all employees other than the CEO;

2. the ratio of the CEO's annual total compensation to that of the median employee;

3. the wage ratio of the CEO to the median employee.

According to the American Federation of Labor and Congress of Industrial Organizations (AFL-CIO, https://aflcio.org), the average S&P 500 company's CEO-to-worker pay ratio was 324-to-1 in 2021. In Table 2.15, we have reported some data collected by this organization (P is the median worker pay (in $) and R is the CEO pay ratio).

Computing the scores for the pay ratio is a real challenge, because the probability distribution of the pay ratio has both a high skew and a big kurtosis. To illustrate these issues, Figure 2.19 shows the histogram of the CEO pay ratios for all US public companies$^{47}$ for the fiscal year 2021. If we

<footnote>$^{46}$Since we have only 9 observations, we observe a small difference between the true and the sample values.</footnote>

<footnote>$^{47}$We use the database Fiscal 2021 CEO Pay Ratios by Mark Siciliano, which can be downloaded at the University of Alabama: https://ir.ua.edu/handle/123456789/8639.</footnote>
<table><caption>Table 2.15: Examples of CEO pay ratio (June 2021)</caption><thead><tr><th>Company name</th><th>P</th><th>R</th><th>Company name</th><th>P</th><th>R</th></tr></thead><tbody><tr><td>Abercrombie & Fitch</td><td>1 954</td><td>4,293</td><td>Netflix</td><td>202 931</td><td>190</td></tr><tr><td>McDonald's</td><td>9 291</td><td>1,939</td><td>BlackRock</td><td>133 644</td><td>182</td></tr><tr><td>Coca-Cola</td><td>11 285</td><td>1,657</td><td>Pfizer</td><td>98 972</td><td>181</td></tr><tr><td>Gap</td><td>6 177</td><td>1,558</td><td>Goldman Sachs</td><td>138 854</td><td>178</td></tr><tr><td>Alphabet</td><td>258 708</td><td>1,085</td><td>MSCI</td><td>55 857</td><td>165</td></tr><tr><td>Walmart</td><td>22 484</td><td>983</td><td>Verisk Analytics</td><td>77 055</td><td>117</td></tr><tr><td>Estee Lauder</td><td>30 733</td><td>697</td><td>Facebook</td><td>247 883</td><td>94</td></tr><tr><td>Ralph Lauren</td><td>21 358</td><td>570</td><td>Invesco</td><td>125 282</td><td>92</td></tr><tr><td>NIKE</td><td>25 386</td><td>550</td><td>Boeing</td><td>158 869</td><td>90</td></tr><tr><td>Citigroup</td><td>52 988</td><td>482</td><td>Citrix Systems</td><td>181 769</td><td>80</td></tr><tr><td>PepsiCo</td><td>45 896</td><td>368</td><td>Harley-Davidson</td><td>187 157</td><td>59</td></tr><tr><td>Microsoft</td><td>172 512</td><td>249</td><td>Amazon.com</td><td>28 848</td><td>58</td></tr><tr><td>Apple</td><td>57 596</td><td>201</td><td>Berkshire Hathaway</td><td>65 740</td><td>6</td></tr></tbody></table>

Source: https://aflcio.org (June 2021)

compute the z-score directly from the pay ratio, we obtain the blue histogram in Figure 2.20. This z-score has a mean around 0 and a standard deviation of 1, but we have $z \in (-0.338, 38.669)$. If we do the same exercise with the logarithm of the CEO pay ratio, we obtain the red histogram. In this case, we have $z \in (-3.561, 4.545)$. This is better, but it is not perfect. This example demonstrates that we must conduct a deep analysis of each data before applying a blind scoring approach. Most ESG data are skewed with fat tails, some of them are binary, others take discrete values, etc. In this context, data analysis is essential to choose the right normalization and scoring transform.

**Exercise 6** The database of the CEO pay ratios for all US public companies contains several sector/industry variables. Compute the z-score of the CEO pay ratio and its logarithm at the sector level. Identify the most problematic sectors. Same question if we consider the industry level.

### 2.2.3 Other statistical methods

Since ESG scoring is an unsupervised statistical method, the use of alternative statistical methods is limited. Indeed, we can use K-means clustering, hierarchical clustering, and principal component analysis to perform dimension reduction of the variables or to create synthetic features. However, it is difficult to go further with these methods. The use of classical parametric supervised methods, such as discriminant analysis and binary choice models, is not well suited except when the goal is to identify bad ESG risks. They are not used in practice. In fact, the only alternative statistical method that adds some value is the lasso method, which is described in Appendix A.2.3 on page 1171. The underlying idea is to perform the penalized linear regression $\mathbf{S} = \beta^\top X + U$ subject to the $L_1$-norm constraint $\|\beta\|_1 \le \tau$ and identify the most important variables of the ESG scoring system. The lasso regression is then used as an ex-post statistical method to estimate the contribution of each variable. In practice, we find that a complex ESG scoring system based on more than 100 variables can be replicated with fewer variables.

### 2.2.4 Performance evaluation criteria

This section is dedicated to the performance assessment of a score. Backtesting an ESG score is relatively close to backtesting a credit score. Even if the tree-based scoring model is an unsuper-
<figure><img src="image_45.png" /><figcaption>Figure 2.20: Histogram of z-score applied to the CEO pay ratio</figcaption></figure>

<figure><img src="image_46.png" /></figure>
vised learning approach, we have seen that it is possible to build supervised models by using a control variable. The response function can be exogenous such as the controversy index $C(t)$ or the controversy indicator $I(t) = 1\{C(t) > 0\}$. It can also be endogenous by considering control variables based on the score $S(t + \delta)$ in the future. Examples are the best-in-class risk indicator $G(t, \delta) = 1\{S(t + \delta) \ge s^*\}$ or the worst-in-class risk indicator $B(t, \delta) = 1\{S(t + \delta) \le s^*\}$. They are noted $G$ and $B$ by analogy with the risk theory that distinguishes good and bad risk. For instance, credit scoring models are mainly based on bad risk detection. This is also the same thing in the case of responsible investing. Nevertheless, we might also want to have a statistical model, whose main objective is to select the good issuers. In the case of the ESG momentum strategy, the response variable depends on the improvement of the ESG score. It can be defined as$^{48}$ $M(t, \delta) = 1\{S(t + \delta) - S(t) \ge \Delta_s^*\}$.

**Remark 16** In this section, the score $S$ is not necessarily the final ESG score. It corresponds to any score or screening rule derived from the ESG scoring model. For instance, it may correspond to a selection score, an exclusion score, etc.

We notice that most control variables are binary. Therefore, we can use the classical tool of credit scoring and follow Roncalli (2020a, Chapter 15) to assess the performance of scoring models. In the first paragraph, we use information theory to know if the scoring system is informative or not. The second paragraph presents the graphical tools to measure the classification accuracy of the score. We then define the different statistical measures to estimate the score performance. Finally, we consider the assessment tools when there is no response function.

## Shannon entropy

**Definition and properties** The entropy is a measure of unpredictability or uncertainty of a random variable. Let $(X, Y)$ be a random vector where $p_{i,j} = \text{Pr}\{X = x_i, Y = y_j\}$, $p_i = \text{Pr}\{X = x_i\}$ and $p_j = \text{Pr}\{Y = y_j\}$. The Shannon entropy of the discrete random variable $X$ is given by$^{49}$:

$$ \mathcal{I}(X) = - \sum_{i=1}^{n} p_i \ln p_i $$

We have the property $0 \le \mathcal{I}(X) \le \ln n$. The entropy is equal to zero if there is a state $i$ such that $p_i = 1$ and is equal to $\ln n$ in the case of the uniform distribution ($p_i = 1/n$). The Shannon entropy is a measure of the average information of the system. The lower the Shannon entropy, the more informative the system. For a random vector $(X, Y)$, we have:

$$ \mathcal{I}(X, Y) = - \sum_{i=1}^{n} \sum_{j=1}^{n} p_{i,j} \ln p_{i,j} $$

We deduce that the conditional information of $Y$ given $X$ is equal to:

$$ \begin{aligned} \mathcal{I}(Y | X) &= \mathbb{E}[\mathcal{I}(Y | X = x)] \\ &= - \sum_{i=1}^{n} \sum_{j=1}^{n} p_{i,j} \ln \frac{p_{i,j}}{p_i} \\ &= \mathcal{I}(X, Y) - \mathcal{I}(X) \end{aligned} $$

We have the following properties:

<footnote>$^{48}$Alternative measures are $M(t, \delta) = 1\{S(t + \delta) - S(t) > \Delta_s^*|S(t) \ge s^*\}$ when we prefer to select companies among the best-in-class issuers and $M(t, \delta) = 1\{S(t + \delta) - S(t) > \Delta_s^*|S(t) \le s^*\}$ when we want to detect worst-in-class issuers that will improve their ESG score.</footnote>

<footnote>$^{49}$We use the convention $p_i \ln p_i = 0$ when $p_i$ is equal to zero.</footnote>
* if X and Y are independent, we have $\mathcal{I}(Y | X) = \mathcal{I}(Y)$ and $\mathcal{I}(X, Y) = \mathcal{I}(Y) + \mathcal{I}(X)$;

* if X and Y are perfectly dependent, we have $\mathcal{I}(Y | X) = 0$ and $\mathcal{I}(X, Y) = \mathcal{I}(X)$.

The amount of information obtained about one random variable, through the other random variable is measured by the mutual information:

$$ \begin{aligned} \mathcal{I}(X \cap Y) &= \mathcal{I}(Y) + \mathcal{I}(X) - \mathcal{I}(X, Y) \\ &= \sum_{i=1}^{n} \sum_{j=1}^{n} p_{i,j} \ln \frac{p_{i,j}}{p_i p_j} \end{aligned} $$

<figure><img src="image_47.png" /><figcaption>Figure 2.21: Examples of Shannon entropy calculation</figcaption></figure>

Figure 2.21 shows some examples of Shannon entropy calculation. For each example, we indicate the probabilities $p_{i,j}$ and the values taken by $\mathcal{I}(X)$, $\mathcal{I}(Y)$, $\mathcal{I}(X, Y)$ and $\mathcal{I}(X \cap Y)$. The top/left panel corresponds to a diffuse system. The value of $\mathcal{I}(X, Y)$ is maximum, meaning that the system is extremely disordered. The top/right panel represents a highly ordered system in the bivariate case and a diffuse system in the univariate case. We have $\mathcal{I}(X | Y) = \mathcal{I}(Y | X) = 0$, implying that the knowledge of X is sufficient to find the state of Y. Generally, the system is not perfectly ordered or perfectly disordered. For instance, in the case of the system described in the bottom/left panel, the knowledge of X informs us about the state of Y. Indeed, if X is in the third state, then we
know that Y cannot be in the first or sixth state. Another example is provided in the bottom/right panel.

**Remark 17** If we apply the Shannon entropy to the transition matrix of a Markov chain, we set $X = \mathcal{R}(s)$ and $Y = \mathcal{R}(t)$ where $\mathcal{R}(t)$ is the state variable at the date t. We obtain:

$$ \mathcal{I}(\mathcal{R}(t) | \mathcal{R}(s)) = - \sum_{i=1}^{K} \pi_{i}^{*} \sum_{j=1}^{K} p_{i,j}^{(t-s)} \ln p_{i,j}^{(t-s)} $$

where $p_{i,j} = \Pr\{\mathcal{R}(t+1) = j | \mathcal{R}(t) = i\}$, $\mathbb{K} = \{1, 2, \dots, K\}$ is the state space of the Markov chain and $\pi^*$ is the associated stationary distribution.

**Application to scoring** Let $\mathcal{S}$ and $Y$ be the score and the control variable. For instance, $Y$ is a binary random variable that may indicate a bad ESG risk ($Y = 0$) or a good ESG risk ($Y = 1$). $Y$ may also correspond to classes defined by some quantiles. With Shannon entropy, we can measure the information of the system $(\mathcal{S}, Y)$. We can also compare two scores $\mathcal{S}_1$ and $\mathcal{S}_2$ by using the statistical measures $\mathcal{I}(\mathcal{S}_1 \cap Y)$ and $\mathcal{I}(\mathcal{S}_2 \cap Y)$. Let $\mathcal{S}_3$ be the aggregated score obtained from the two individual scores $\mathcal{S}_1$ and $\mathcal{S}_2$. We can calculate the information contribution of each score with respect to the global score. Therefore, we can verify that a score really adds an information.

We consider the following decision rule:

$$ \begin{cases} \mathcal{S} \le 0 \Rightarrow \mathcal{S}^* = 0 \\ \mathcal{S} > 0 \Rightarrow \mathcal{S}^* = 1 \end{cases} $$

We note $n_{i,j}$ the number of observations such that $\mathcal{S}^* = i$ and $Y = j$. We obtain the following system $(\mathcal{S}^*, Y)$:

<table><thead><tr><th></th><th>Y = 0</th><th>Y = 1</th></tr></thead><tbody><tr><th><b>S</b><sup>*</sup> = 0</th><td>n<sub>0,0</sub></td><td>n<sub>0,1</sub></td></tr><tr><th><b>S</b><sup>*</sup> = 1</th><td>n<sub>1,0</sub></td><td>n<sub>1,1</sub></td></tr></tbody></table>

where $n = n_{0,0} + n_{0,1} + n_{1,0} + n_{1,1}$ is the total number of observations. The hit rate is the ratio of good bets:

$$ H = \frac{n_{0,0} + n_{1,1}}{n} $$

This statistic can be viewed as an information measure of the system $(\mathcal{S}, Y)$. When there are more states, we can consider the Shannon entropy. In Figure 2.22, we report the contingency table of two scores $\mathcal{S}_1$ and $\mathcal{S}_2$ for 100 observations$^{50}$. We have $\mathcal{I}(\mathcal{S}_1 \cap Y) = 0.763$ and $\mathcal{I}(\mathcal{S}_2 \cap Y) = 0.636$. We deduce that $\mathcal{S}_1$ is more informative than $\mathcal{S}_2$.

## Graphical methods

We assume that the control variable $Y$ can takes two values: $Y = 0$ corresponds to a bad risk (or bad signal) while $Y = 1$ corresponds to a good risk (or good signal). Gouriéroux (1992) introduced three graphical tools for assessing the quality of a score: the performance curve, the selection curve and the discrimination curve$^{51}$. In the following, we assume that the probability $\Pr\{Y = 1 | \mathcal{S} \ge s\}$ is increasing with respect to the level$^{52}$ $s \in [0, 1]$, which corresponds to the rate of acceptance. We deduce that the decision rule is the following:

<footnote>$^{50}$Each score is divided into 6 intervals $(s_1, \dots, s_6)$ while the dependent variable is divided into 5 intervals $(y_1, \dots, y_5)$.</footnote>

<footnote>$^{51}$See also Gouriéroux and Jasiak (2007).</footnote>

<footnote>$^{52}$We assume that the score is based on the 0/1 normalization, but this assumption is not important since we can always map a general score into a 0/1 score.</footnote>
<table><caption>Figure 2.22: Scorecards $\mathcal{S}_1$ and $\mathcal{S}_2$ (100 observations)</caption><thead><tr><th></th><th>y<sub>1</sub></th><th>y<sub>2</sub></th><th>y<sub>3</sub></th><th>y<sub>4</sub></th><th>y<sub>5</sub></th></tr></thead><tbody><tr><th>s<sub>1</sub></th><td>10</td><td>9</td><td></td><td></td><td></td></tr><tr><th>s<sub>2</sub></th><td>7</td><td>9</td><td></td><td></td><td></td></tr><tr><th>s<sub>3</sub></th><td>3</td><td></td><td>7</td><td>2</td><td></td></tr><tr><th>s<sub>4</sub></th><td></td><td>2</td><td>10</td><td>4</td><td>5</td></tr><tr><th>s<sub>5</sub></th><td></td><td></td><td></td><td>10</td><td>2</td></tr><tr><th>s<sub>6</sub></th><td></td><td></td><td>3</td><td>4</td><td>13</td></tr></tbody></table><p>I(\mathcal{S}<sub>1</sub>) = 1.767<br>I(Y) = 1.609<br>I(\mathcal{S}<sub>1</sub>, Y) = 2.614<br>I(\mathcal{S}<sub>1</sub> ∩ Y) = 0.763</p><table><thead><tr><th></th><th>y<sub>1</sub></th><th>y<sub>2</sub></th><th>y<sub>3</sub></th><th>y<sub>4</sub></th><th>y<sub>5</sub></th></tr></thead><tbody><tr><th>s<sub>1</sub></th><td>7</td><td>10</td><td></td><td></td><td></td></tr><tr><th>s<sub>2</sub></th><td>10</td><td>8</td><td></td><td></td><td></td></tr><tr><th>s<sub>3</sub></th><td></td><td></td><td>5</td><td>4</td><td>3</td></tr><tr><th>s<sub>4</sub></th><td>3</td><td></td><td>10</td><td>6</td><td>4</td></tr><tr><th>s<sub>5</sub></th><td>2</td><td></td><td></td><td>5</td><td>8</td></tr><tr><th>s<sub>6</sub></th><td></td><td></td><td>5</td><td>5</td><td>5</td></tr></tbody></table><p>I(\mathcal{S}<sub>2</sub>) = 1.771<br>I(Y) = 1.609<br>I(\mathcal{S}<sub>2</sub>, Y) = 2.745<br>I(\mathcal{S}<sub>2</sub> ∩ Y) = 0.636</p>

* if the score of the observation is above the threshold $s$, the observation is selected;

* if the score of the observation is below the threshold $s$, the observation is not selected.

If $s$ is equal to one, we select no observation. If $s$ is equal to zero, we select all the observations. In a scoring system, the threshold $s$ is given. Below, we assume that $s$ is varying and we analyze the relevance of the score with respect to this parameter.

**Performance curve, selection curve and discriminant curve** The performance curve is the parametric function $y = \mathcal{P}(x)$ defined by:

$$ \left\{ \begin{aligned} x(s) &= \Pr\{\mathcal{S} \geq s\} \\ y(s) &= \frac{\Pr\{Y = 0 | \mathcal{S} \geq s\}}{\Pr\{Y = 0\}} \end{aligned} \right. $$

where $x(s)$ corresponds to the proportion of selected observations and $y(s)$ corresponds to the ratio between the proportion of selected bad risks and the proportion of bad risks in the population. The score is efficient if the ratio is below one. If $y(s) > 1$, the score selects more bad risks than those we can find in the population$^{53}$. If $y(s) = 1$, the score is random and the performance is equal to zero. In this case, the selected population is representative of the total population.

The selection curve is the parametric curve $y = \mathcal{S}(x)$ defined by:

$$ \left\{ \begin{aligned} x(s) &= \Pr\{\mathcal{S} \geq s\} \\ y(s) &= \Pr\{\mathcal{S} \geq s | Y = 0\} \end{aligned} \right. $$

where $y(s)$ corresponds to the ratio of observations that are wrongly selected. By construction, we would like that the curve $y = \mathcal{S}(x)$ is located below the bisecting line $y = x$ in order to verify that $\Pr\{\mathcal{S} \geq s | Y = 0\} < \Pr\{\mathcal{S} \geq s\}$.

<footnote>$^{53}$In this case, we have $\Pr\{Y = 0 | \mathcal{S} \geq s\} > \Pr\{Y = 0\}$.</footnote>
**Remark 18** The performance and selection curves are related as follows<fnref n="54" />:

$$S(x) = xP(x)$$

The discriminant curve is the parametric curve $y = D(x)$ defined by:

$$D(x) = g_1(g_0^{-1}(x))$$

where:

$$g_y(s) = \Pr\{\mathbf{S} \geq s | Y = y\}$$

It represents the proportion of good risks in the selected population with respect to the proportion of bad risks in the selected population. The score is said to be discriminant if the curve $y = D(x)$ is located above the bisecting line $y = x$.

**Some properties** We first notice that the previous parametric curves do not depend on the probability distribution of the score $\mathbf{S}$, but only on the ranking of the observations. They are then invariant if we apply an increasing function to the score. Gouriéroux (1992) also established the following properties:

1. the performance curve (respectively, the selection curve) is located below the line $y = 1$ (respectively, the bisecting line $y = x$) if and only if $\text{cov}(f(Y), g(\mathbf{S})) \geq 0$ for any increasing functions $f$ and $g$;

2. the performance curve is increasing if and only if:

$$\text{cov}(f(Y), g(\mathbf{S}) | \mathbf{S} \geq s) \geq 0$$

for any increasing functions $f$ and $g$, and any threshold level $s$;

3. the selection curve is convex if and only if $\mathbb{E}[f(Y) | \mathbf{S} = s]$ is increasing with respect to the threshold level $s$ for any increasing function $f$.

**Remark 19** The first property is the least restrictive. It allows us to verify that the score $\mathbf{S}$ is better than a random score. We can show that $(3) \Rightarrow (2) \Rightarrow (1)$. The last property is then the most restrictive.

A score is perfect or optimal if there is a threshold level $s^*$ such that $\text{Pr}\{Y = 1 | \mathbf{S} \geq s^*\} = 1$ and $\text{Pr}\{Y = 0 | \mathbf{S} < s^*\} = 1$. It separates the population between good and bad risks. Graphically, the selection curve of a perfect score is equal to:

$$y = 1 \{x > \text{Pr}\{Y = 1\}\} \cdot \left(1 + \frac{x - 1}{\text{Pr}\{Y = 0\}}\right)$$

$$\begin{align*}
\text{Pr}\{\mathbf{S} \geq s | Y = 0\} &= \frac{\text{Pr}\{\mathbf{S} \geq s, Y = 0\}}{\text{Pr}\{Y = 0\}} \\
&= \text{Pr}\{\mathbf{S} \geq s\} \cdot \frac{\text{Pr}\{\mathbf{S} \geq s, Y = 0\}}{\text{Pr}\{\mathbf{S} \geq s\} \text{Pr}\{Y = 0\}} \\
&= \text{Pr}\{\mathbf{S} \geq s\} \cdot \frac{\text{Pr}\{Y = 0 | \mathbf{S} \geq s\}}{\text{Pr}\{Y = 0\}}
\end{align*}$$

<footnote n="54">⁵⁴We have:</footnote>
Using the relationship $S(x) = xP(x)$, we deduce that the performance curve of a perfect score is given by:

$$y = 1 \{x > \Pr\{Y = 1\}\} \cdot \left( \frac{x - \Pr\{Y = 1\}}{x \cdot \Pr\{Y = 0\}} \right)$$

For the discriminant curve, a perfect score satisfies $D(x) = 1$. When the score is random, we have $S(x) = D(x) = x$ and $P(x) = 1$. In Figure 2.23, we have reported the performance, selection and discriminant curves of a given score $S$. We also show the curves obtained with an optimal (or perfect) score and a random score. A score must be located in the area between the curve computed with a random score and the curve computed with a perfect score, except if the score ranks the observations in a worst way than a random score.

Gouriéroux (1992) also established two properties for comparing two scores $S_1$ and $S_2$:

* the score $S_1$ is more performing on the population $P_1$ than the score $S_2$ on the population $P_2$ if and only if the performance (or selection) curve of $(S_1, P_1)$ is below the performance (or selection) curve of $(S_2, P_2)$;

* the score $S_1$ is more discriminatory on the population $P_1$ than the score $S_2$ on the population $P_2$ if and only if the discriminant curve of $(S_1, P_1)$ is above the discriminant curve of $(S_2, P_2)$.

Figure 2.24 illustrates the case where the score $S_1$ is better than the score $S_2$. However, the order is only partial. Most of the time, the two scores cannot be globally compared. An example is provided in Figure 2.25. The second score is not very good to distinguish good and bad risks when it takes small values, but it is close to a perfect score when it takes high values.

<figure><img src="image_48.png" /><figcaption>Figure 2.23: Performance, selection and discriminant curves</figcaption></figure>
<figure><img src="image_49.png" /><figcaption>Figure 2.25: Illustration of the partial ordering between two scores</figcaption></figure>

<figure><img src="image_50.png" /></figure>
## Statistical methods

Since the quantitative tools for comparing two scores are numerous, we focus on two non-parametric measures: the Kolmogorov-Smirnov test and the Gini coefficient.

**Kolmogorov-Smirnov test** We consider the cumulative distribution functions:

$$F_0(s) = \Pr\{S \le s | Y = 0\}$$

and:

$$F_1(s) = \Pr\{S \le s | Y = 1\}$$

The score $S$ is relevant if we have the stochastic dominance order $F_0 \succ F_1$. In this case, the score quality is measured by the Kolmogorov-Smirnov statistic:

$$KS = \max_s |F_0(s) - F_1(s)|$$

It takes the value 1 if the score is perfect. The KS statistic may be used to verify that the score is not random. We then test the assumption $H_0: KS = 0$ by using the tabulated critical values$^{55}$. In Figure 2.26, we give an example with 5000 observations. The KS statistic is equal to 36%, which implies that $H_0$ is rejected at the confidence level 1%.

## Gini coefficient

**The Lorenz curve** The Gini coefficient is the statistic, which is the most used for measuring the performance of a score. It is related to the concept of Lorenz curve, which is a graphical representation of the concentration. Let $X$ and $Y$ be two random variables. The Lorenz curve $y = \mathcal{L}(x)$ is the parametric curve defined by:

$$\begin{cases} x = \Pr\{X \le x\} \\ y = \Pr\{Y \le y | X \le x\} \end{cases}$$

In economics, $x$ represents the proportion of individuals that are ranked by income while $y$ represents the proportion of income. In this case, the Lorenz curve is a graphical representation of the distribution of income and is used for illustrating inequality of the wealth distribution between individuals. For example, we observe that 70% of individuals have only 34% of total income in Figure 2.27.

**Definition of the Gini coefficient** The Lorenz curve has two limit cases. If the wealth is perfectly concentrated, one individual holds 100% of the total wealth. If the wealth is perfectly allocated between all the individuals, the corresponding Lorenz curve is the bisecting line. We define the Gini coefficient by:

$$Gini(\mathcal{L}) = \frac{A}{A + B}$$

<table><thead><tr><th>n</th><th>10</th><th>50</th><th>100</th><th>500</th><th>5000</th></tr></thead><tbody><tr><th>CV</th><td>40.9%</td><td>18.8%</td><td>13.4%</td><td>6.0%</td><td>1.9%</td></tr></tbody></table>

More generally, the null hypothesis is rejected at the confidence level $\alpha$ if we have:

$$KS > \sqrt{\frac{1}{2n} \ln\left(\frac{2}{\alpha}\right)}$$

<footnote>$^{55}$The critical values at the 5% confidence level are equal to:</footnote>
<figure><img src="image_51.png" /><figcaption>Figure 2.27: An example of Lorenz curve</figcaption></figure>

<figure><img src="image_52.png" /></figure>
where A is the area between the Lorenz curve and the curve of perfect equality, and B is the area between the curve of perfect concentration and the Lorenz curve. By construction, we have $0 \le Gini(\mathcal{L}) \le 1$. The Gini coefficient is equal to zero in the case of perfect equality and one in the case of perfect concentration. We have:

$$Gini(\mathcal{L}) = 1 - 2 \int_{0}^{1} \mathcal{L}(x) \, dx$$

**Application to scoring** We can interpret the selection curve as a Lorenz curve. We recall that $\mathbf{F}(s) = \text{Pr}\{\mathbf{S} \le s\}$, $\mathbf{F}_0(s) = \text{Pr}\{\mathbf{S} \le s | Y = 0\}$ and $\mathbf{F}_1(s) = \text{Pr}\{\mathbf{S} \le s | Y = 1\}$. The selection curve is defined by the following parametric coordinates:

$$\begin{cases} x(s) = 1 - \mathbf{F}(s) \\ y(s) = 1 - \mathbf{F}_0(s) \end{cases}$$

The selection curve measures the capacity of the score for not selecting bad risks. We could also build the Lorenz curve that measures the capacity of the score for selecting good risks:

$$\begin{cases} x(s) = \text{Pr}\{\mathbf{S} \ge s\} = 1 - \mathbf{F}(s) \\ y(s) = \text{Pr}\{\mathbf{S} \ge s | Y = 1\} = 1 - \mathbf{F}_1(s) \end{cases}$$

It is called the precision curve. Another popular graphical tool is the receiver operating characteristic (or ROC) curve (Powers, 2011), which is defined by:

$$\begin{cases} x(s) = \text{Pr}\{\mathbf{S} \ge s | Y = 0\} = 1 - \mathbf{F}_0(s) \\ y(s) = \text{Pr}\{\mathbf{S} \ge s | Y = 1\} = 1 - \mathbf{F}_1(s) \end{cases}$$

An example for a given score $\mathbf{S}$ is provided in Figure 2.28. For all the three curves, we can calculate the Gini coefficient. Since the precision and ROC curves are located above the bisecting line, the Gini coefficient associated to the Lorenz curve $\mathcal{L}$ becomes$^{56}$:

$$Gini(\mathcal{L}) = 2 \int_{0}^{1} \mathcal{L}(x) \, dx - 1$$

The Gini coefficient of the score $\mathbf{S}$ is then computed as follows:

$$Gini^{*}(\mathbf{S}) = \frac{Gini(\mathcal{L})}{Gini(\mathcal{L}^{*})}$$

where $\mathcal{L}^*$ is the Lorenz curve associated to the perfect score.

**Remark 20** The Gini coefficient is not necessarily the same for the three curves. However, if the population is homogeneous, we generally obtain very similar figures$^{57}$.

$$Gini(\text{ROC}) = 2 \times \text{AUC}(\text{ROC}) - 1$$

<table><thead><tr><th>Curve</th><th><i>Gini</i>(<i>L</i>)</th><th><i>Gini</i>(<i>L</i><sup>*</sup>)</th><th><i>Gini</i><sup>*</sup>(<i>S</i>)</th></tr></thead><tbody><tr><td>Selection</td><td>20.41%</td><td>40.02%</td><td>51.01%</td></tr><tr><td>Precision</td><td>30.62%</td><td>59.98%</td><td>51.05%</td></tr><tr><td>ROC</td><td>51.03%</td><td>100.00%</td><td>51.03%</td></tr></tbody></table>

<footnote>$^{56}$An alternative to the Gini coefficient is the AUC measure, which corresponds to the area under the ROC curve. However, they give the same information since they are related by the equation:</footnote>

<footnote>$^{57}$For instance, we obtain the following results with the score $\mathbf{S}$ that has been used in Figure 2.28:</footnote>
<figure><img src="image_53.png" /><figcaption>Figure 2.28: Selection, precision and ROC curves</figcaption></figure>

**Choice of the optimal cut-off** The choice of the optimal cut-off $s^*$ depends on the objective function. For instance, we can calibrate $s^*$ in order to achieve a minimum universe size of ESG assets. We can also fix a given selection rate. From a statistical point of view, we must distinguish the construction of the scoring model and the decision rule. In statistical learning, we generally consider three datasets: the training set, the validation set and the test set. The training set is used for calibrating the model and its parameters whereas the validation set helps to avoid overfitting. But the decision rule is based on the test set.

**Confusion matrix** A confusion matrix is a special case of contingency matrix. Each row of the matrix represents the frequency in a predicted class while each column represents the frequency in an actual class. Using the test set, it takes the following form:

<table><thead><tr><th></th><th>Y = 0</th><th>Y = 1</th></tr></thead><tbody><tr><th>S &lt; s</th><td>n<sub>0,0</sub></td><td>n<sub>0,1</sub></td></tr><tr><th>S ≥ s</th><td>n<sub>1,0</sub></td><td>n<sub>1,1</sub></td></tr><tr><td></td><td colspan="2">n<sub>0</sub> = n<sub>0,0</sub> + n<sub>1,0</sub> &nbsp;&nbsp;&nbsp; n<sub>1</sub> = n<sub>0,1</sub> + n<sub>1,1</sub></td></tr></tbody></table>

where $n_{i,j}$ represents the number of observations of the cell $(i, j)$. The interpretation of the confusion matrix is given in Table 2.16. The cells ($S < s, Y = 0$) and ($S \ge s, Y = 1$) correspond to observations that are well-classified: true negative (TN) and true positive (TP). The cells ($S \ge s, Y = 0$) and ($S < s, Y = 1$) correspond to two types of errors:

1. A false positive (FP) can induce a future loss, because the risk can materialize: this is a type I error;

2. A false negative (FN) potentially corresponds to an opportunity cost: this is a type II error.
<table><caption>Table 2.16: Interpretation of the confusion matrix</caption><thead><tr><th></th><th>Y = 0</th><th>Y = 1</th></tr></thead><tbody><tr><th><i>S</i> &lt; <i>s</i></th><td>It is rejected<br>and it is a bad risk<br>(true negative)</td><td>It is rejected,<br>but it is a good risk<br>(false negative)</td></tr><tr><th><i>S</i> ≥ <i>s</i></th><td>It is accepted,<br>but it is a bad risk<br>(false positive)</td><td>It is accepted<br>and it is a good risk<br>(true positive)</td></tr><tr><th></th><td>(negative)</td><td>(positive)</td></tr></tbody></table>

**Classification ratios** Binary classification defines many metrics for measuring the performance of the classifier<fnref n="58" /> (Fawcett, 2006):

$$
\begin{align*}
\text{True Positive Rate} \quad \text{TPR} &= \frac{\text{TP}}{\text{TP} + \text{FN}} \\
\text{False Negative Rate} \quad \text{FNR} &= \frac{\text{FN}}{\text{FN} + \text{TP}} = 1 - \text{TPR} \\
\text{True Negative Rate} \quad \text{TNR} &= \frac{\text{TN}}{\text{TN} + \text{FP}} \\
\text{False Positive Rate} \quad \text{FPR} &= \frac{\text{FP}}{\text{FP} + \text{TN}} = 1 - \text{TNR}
\end{align*}
$$

The true positive rate (TPR) is also known as the sensitivity or the recall. It measures the proportion of real good risks that are correctly predicted good risk. Fawcett (2006) also defines the precision or the positive predictive value (PPV):

$$ \text{PPV} = \frac{\text{TP}}{\text{TP} + \text{FP}} $$

It measures the proportion of predicted good risks that are correctly real good risk. Besides these metrics, statisticians also use three generic metrics:

1. The accuracy considers the classification of both negatives and positives:

$$ \text{ACC} = \frac{\text{TP} + \text{TN}}{\text{P} + \text{N}} = \frac{\text{TP} + \text{TN}}{\text{TP} + \text{FN} + \text{TN} + \text{FP}} $$

2. The $F_1$ score is the harmonic mean of precision and sensitivity:

$$
\begin{align*}
F_1 &= \frac{2}{\frac{1}{\text{precision}} + \frac{1}{\text{sensitivity}}} \\
    &= \frac{2 \cdot \text{PPV} \cdot \text{TPR}}{\text{PPV} + \text{TPR}}
\end{align*}
$$

<table><thead><tr><th></th><th>Y = 0</th><th>Y = 1</th></tr></thead><tbody><tr><th><i>S</i> &lt; <i>s</i></th><td>TN</td><td>FN</td></tr><tr><th><i>S</i> ≥ <i>s</i></th><td>FP</td><td>TP</td></tr><tr><th></th><td>N = TN + FP</td><td>P = FN + TP</td></tr></tbody></table>

<footnote n="58">⁵⁸We rewrite the confusion matrix as follows:</footnote>
3. The $\phi$ coefficient or the Matthews correlation coefficient (MCC) is a measure of association between $\mathbf{S}$ and $Y$:

$$ \phi = \text{MCC} = \frac{\text{TP} \cdot \text{TN} - \text{FP} \cdot \text{FN}}{\sqrt{(\text{TP} + \text{FP}) \cdot (\text{TP} + \text{FN}) \cdot (\text{TN} + \text{FP}) \cdot (\text{TN} + \text{FN})}} $$

$\mathbf{S}$ and $Y$ are positively associated if most of the observations fall along the diagonal cells.

<table><caption>Table 2.17: Confusion matrix of three scoring systems and three cut-off values $s$</caption><thead><tr><th>Score</th><th>s = 100</th><th>s = 200</th><th>s = 500</th></tr></thead><tbody><tr><td>S<sub>1</sub></td><td>386 616<br>1 614 7 384</td><td>698 1 304<br>1 302 6 696</td><td>1 330 3 672<br>670 4 328</td></tr><tr><td>S<sub>2</sub></td><td>372 632<br>1 628 7 368</td><td>700 1 304<br>1 300 6 696</td><td>1 386 3 616<br>614 4 384</td></tr><tr><td>S<sub>3</sub></td><td>382 616<br>1 618 7 384</td><td>656 1 344<br>1 344 6 656</td><td>1 378 3 624<br>622 4 376</td></tr><tr><td>Perfect</td><td>1 000 0<br>1 000 8 000</td><td>2 000 0<br>0 8 000</td><td>2 000 3 000<br>0 5 000</td></tr></tbody></table>

**Example 7** We consider three scoring systems that have been calibrated on a training set. These systems produce a score between 0 and 1000. A low value predicts a bad risk while a high value predicts a good risk. In order to calibrate the cut-off, we consider a test set, which is composed of 10000 new observations. In Table 2.17, we report the confusion matrix of each scoring system for different cut-off values (100, 200 and 500).

<table><caption>Table 2.18: Binary classification ratios (in %) of the three scoring systems</caption><thead><tr><th>Score</th><th>s</th><th>TPR</th><th>FNR</th><th>TNR</th><th>FPR</th><th>PPV</th><th>ACC</th><th>F<sub>1</sub></th></tr></thead><tbody><tr><td rowspan="3">S<sub>1</sub></td><td>100</td><td>92.3</td><td>7.7</td><td>19.3</td><td>80.7</td><td>82.1</td><td>77.7</td><td>86.9</td></tr><tr><td>200</td><td>83.7</td><td>16.3</td><td>34.9</td><td>65.1</td><td>83.7</td><td>73.9</td><td>83.7</td></tr><tr><td>500</td><td>54.1</td><td>45.9</td><td>66.5</td><td>33.5</td><td>86.6</td><td>56.6</td><td>66.6</td></tr><tr><td rowspan="3">S<sub>2</sub></td><td>100</td><td>92.1</td><td>7.9</td><td>18.6</td><td>81.4</td><td>81.9</td><td>77.4</td><td>86.7</td></tr><tr><td>200</td><td>83.7</td><td>16.3</td><td>35.0</td><td>65.0</td><td>83.7</td><td>74.0</td><td>83.7</td></tr><tr><td>500</td><td>54.8</td><td>45.2</td><td>69.3</td><td>30.7</td><td>87.7</td><td>57.7</td><td>67.5</td></tr><tr><td rowspan="3">S<sub>3</sub></td><td>100</td><td>92.3</td><td>7.7</td><td>19.1</td><td>80.9</td><td>82.0</td><td>77.7</td><td>86.9</td></tr><tr><td>200</td><td>83.2</td><td>16.8</td><td>32.8</td><td>67.2</td><td>83.2</td><td>73.1</td><td>83.2</td></tr><tr><td>500</td><td>54.7</td><td>45.3</td><td>68.9</td><td>31.1</td><td>87.6</td><td>57.5</td><td>67.3</td></tr><tr><td rowspan="3">Perfect</td><td>100</td><td>100.0</td><td>0.0</td><td>50.0</td><td>50.0</td><td>88.9</td><td>90.0</td><td>94.1</td></tr><tr><td>200</td><td>100.0</td><td>0.0</td><td>100.0</td><td>0.0</td><td>100.0</td><td>100.0</td><td>100.0</td></tr><tr><td>500</td><td>62.5</td><td>37.5</td><td>100.0</td><td>0.0</td><td>100.0</td><td>70.0</td><td>76.9</td></tr><tr><td rowspan="3">Best<br>scoring<br>system</td><td>100</td><td>S<sub>1</sub>/S<sub>3</sub></td><td>S<sub>1</sub>/S<sub>3</sub></td><td>S<sub>1</sub></td><td>S<sub>1</sub></td><td>S<sub>1</sub></td><td>S<sub>1</sub></td><td>S<sub>1</sub></td></tr><tr><td>200</td><td>S<sub>1</sub>/S<sub>2</sub></td><td>S<sub>1</sub>/S<sub>2</sub></td><td>S<sub>2</sub></td><td>S<sub>2</sub></td><td>S<sub>2</sub></td><td>S<sub>2</sub></td><td>S<sub>2</sub></td></tr><tr><td>500</td><td>S<sub>2</sub></td><td>S<sub>2</sub></td><td>S<sub>2</sub></td><td>S<sub>2</sub></td><td>S<sub>2</sub></td><td>S<sub>2</sub></td><td>S<sub>2</sub></td></tr></tbody></table>

Using confusion matrices given in Table 2.17, we calculate the different classification ratios and report them in Table 2.18. In addition to the three scoring systems, we have also considered a
perfect score in order to show what the best value is for each classification ratio, and we indicate the best scoring system. We notice that it depends on the ratio and on the value of the cut-off. For instance, if we want to maximize the true positive ratio or minimize the false negative ratio, $S_1$ is the best scoring system for low value of $s$ while $S_2$ is better when $s$ is equal to 500. For the other ratios, $S_1$ seems to be the best scoring system when $s = 100$, otherwise $S_2$ dominates $S_1$ and $S_3$ when $s = 200$ or $s = 500$.

**Remark 21** We recall that $F_0(s) = \Pr\{S \le s | Y = 0\}$ and $F_1(s) = \Pr\{S \le s | Y = 1\}$. We deduce that $TNR = F_0(s)$, $FNR = F_1(s)$, $FPR = 1 - F_0(s)$ and $TPR = 1 - F_1(s)$. Therefore, the ROC curve is the parametric curve, where the x-coordinates are the false positive rates and the y-coordinates are the true positive rates. Generally, we note $\alpha$ and $\beta$ the type I and II errors. We may also interpret the ROC curve as the relationship of $1 - \beta(s)$ with respect to $\alpha(s)$.

## Backtesting of unsupervised scoring systems

To understand how to implement a backtesting procedure, we take the example of credit scoring models. Let $S_i(t)$ be the credit score of individual/company $i$ at time $t$. The response variable is the default indicator variable $Y_i(t)$:

$$Y_i(t) = 1\{\tau_i \ge t + \delta\} = 1\{D_i(t + \delta) = 0\}$$

where $\tau_i$ and $D_i$ are the default time and the default indicator function, and $\delta$ is the time horizon (e.g., one year). In this problem, $S_i(t)$ is used to predict the default time $\tau_i$ and the decision process is:

$$\begin{cases} S_i(t) \le s^* \Rightarrow i \text{ is a bad risk} \Rightarrow Y_i(t) = 0 \\ S_i(t) > s^* \Rightarrow i \text{ is not a bad risk} \Rightarrow Y_i(t) = 1 \end{cases}$$

The calibration problem for the credit scoring model is therefore:

$$\Pr\{Y_i(t) = 0\} = f(S_i(t))$$

where $f$ is an increasing function. We obtain a binary choice model that can be calibrated using discriminant analysis, clustering methods or logit/probit regression. The backtesting procedure consists of verifying that $\hat{Y}_i(t) = 1\{S_i(t) > s^*\}$ is a good estimator of $Y_i(t)$. Building a backtesting procedure for ESG scores then requires defining an ad hoc response variable.

**Static analysis** Let $S_i(t)$ be the ESG score of company $i$ at time $t$. The endogenous response variable can be defined as follows:

<table><thead><tr><th>Scoring system</th><th>Risk class</th><th>Y<sub>i</sub>(t)</th></tr></thead><tbody><tr><td>Best-in-class oriented</td><td>Good risk</td><td>1{S<sub>i</sub>(t + δ) ≥ s<sup>*</sup>}</td></tr><tr><td>Worst-in-class oriented</td><td>Bad risk</td><td>1{S<sub>i</sub>(t + δ) ≤ s<sup>*</sup>}</td></tr></tbody></table>

where $s^*$ is the best-in-class/worst-in-class threshold to determine. $Y_i(t)$ is endogenous because it depends on the future value of the score. Here, the backtesting procedure can be seen as a stability test of the ESG scoring system. An alternative is to use an exogenous response variable based on controversies. For example, to predict bad risks, we can use the binary response $Y_i(t) = 1\{C_i(t + \delta) \ge 0\}$ where $C_i(t)$ is the controversy index.

**Dynamic analysis** The static analysis can be extended to the dynamic case. Instead of using the ESG score, we consider the past momentum $M_i(t, h) = S_i(t) - S_i(t - h)$ where $h$ is typically the year, while the response variable is based on the future momentum $S_i(t + \delta) - S_i(t)$.
**Illustration using an ESG scoring system** We consider the scoring system of an ESG rating agency. We normalize the published scores in order to obtain z-scores and apply the backtesting procedure to these latter numbers. We consider four risk classes:

<table><thead><tr><th>Risk class</th><th>Definition</th><th>Y<sub>i</sub>(t)</th></tr></thead><tbody><tr><td>Worst-in-class</td><td>S<sub>i</sub>(t) ≤ F̂<sup>-1</sup>(20%)</td><td>1{S<sub>i</sub>(t + δ) ≤ s<sup>*</sup>}</td></tr><tr><td>Bad risk</td><td>S<sub>i</sub>(t) ≤ S̄</td><td>1{S<sub>i</sub>(t + δ) ≤ s<sup>*</sup>}</td></tr><tr><td>Good risk</td><td>S<sub>i</sub>(t) ≥ S̄</td><td>1{S<sub>i</sub>(t + δ) ≥ s<sup>*</sup>}</td></tr><tr><td>Best-in-class</td><td>S<sub>i</sub>(t) ≥ F̂<sup>-1</sup>(80%)</td><td>1{S<sub>i</sub>(t + δ) ≥ s<sup>*</sup>}</td></tr></tbody></table>

where $\hat{F}$ is the empirical distribution of the score and $\bar{S}$ is the average of scores. For each risk class, we compute the classification ratios ACC, $F_1$, and $\phi$ and the Shannon entropy $\mathcal{I}(S \cap Y)$ with respect to the cut-off value $s$. The results, expressed in %, are shown in Figures 2.29 to 2.32 when we consider the MSCI World and MSCI EM universes.

<table><caption>Table 2.19: Optimal cut-off $s^*$ (MSCI World)</caption><thead><tr><th rowspan="2">Risk class</th><th colspan="4">δ = 3 months</th><th colspan="4">δ = 12 months</th></tr><tr><th>ACC</th><th>F<sub>1</sub></th><th>φ</th><th>I(S ∩ Y)</th><th>ACC</th><th>F<sub>1</sub></th><th>φ</th><th>I(S ∩ Y)</th></tr></thead><tbody><tr><td>Worst-in-class</td><td>-0.91</td><td>-0.61</td><td>-0.68</td><td>-0.58</td><td>-0.96</td><td>-0.58</td><td>-0.67</td><td>-0.54</td></tr><tr><td>Bad risk</td><td>-0.01</td><td>0.18</td><td>0.02</td><td>0.02</td><td>0.01</td><td>0.24</td><td>0.04</td><td>0.05</td></tr><tr><td>Good risk</td><td>-0.02</td><td>-0.18</td><td>0.01</td><td>0.02</td><td>-0.01</td><td>-0.20</td><td>0.03</td><td>0.04</td></tr><tr><td>Best-in-class</td><td>1.05</td><td>0.79</td><td>0.85</td><td>0.77</td><td>1.08</td><td>0.76</td><td>0.83</td><td>0.72</td></tr></tbody></table>

<table><caption>Table 2.20: Optimal cut-off $s^*$ (MSCI EM)</caption><thead><tr><th rowspan="2">Risk class</th><th colspan="4">δ = 3 months</th><th colspan="4">δ = 12 months</th></tr><tr><th>ACC</th><th>F<sub>1</sub></th><th>φ</th><th>I(S ∩ Y)</th><th>ACC</th><th>F<sub>1</sub></th><th>φ</th><th>I(S ∩ Y)</th></tr></thead><tbody><tr><td>Worst-in-class</td><td>-1.87</td><td>-1.19</td><td>-1.29</td><td>-1.15</td><td>-2.00</td><td>-1.17</td><td>-1.30</td><td>-1.12</td></tr><tr><td>Bad risk</td><td>0.13</td><td>0.23</td><td>-0.03</td><td>-0.10</td><td>0.16</td><td>0.28</td><td>-0.05</td><td>-0.14</td></tr><tr><td>Good risk</td><td>0.13</td><td>-0.15</td><td>-0.03</td><td>-0.14</td><td>0.16</td><td>-0.22</td><td>-0.05</td><td>-0.24</td></tr><tr><td>Best-in-class</td><td>0.48</td><td>0.14</td><td>0.22</td><td>0.11</td><td>0.56</td><td>0.13</td><td>0.24</td><td>0.11</td></tr></tbody></table>

We can then estimate the optimal cut-off $s^*$, which is the value $s$ that maximizes the back-testing metric. Results are given in Tables 2.19 and 2.20. Theoretically, the optimal cut-off is $s^* = \Phi^{-1}(20\%) = -0.8416$ for the worst-in-class category, $s^* = \mathbb{E}[\mathcal{N}(0,1)] = 0$ for the bad-risk and good-risk categories and $s^* = \Phi^{-1}(80\%) = 0.8416$ for the best-in-class category, because the backtesting procedure concerns $z$-scores. For the MSCI World universe, the estimated cut-offs are not that far from the theoretical cut-offs. Moreover, we do not observe a large difference between the three-month and the twelve-month horizons. For the MSCI World universe, we face two problems. First, the backtesting of the worst-in-class category implies an optimal cut-off that is well below the theoretical cut-off, meaning that the worst-in-class universe is not adequately defined. Second, the scoring system is not really able to discriminate between bad risk and worst-in-class categories because the optimal cut-off for the latter category is close to zero.

**Remark 22** The previous example illustrates that the robustness of an ESG scoring system can vary depending on the investment universe.
<figure><img src="image_54.png" /><figcaption>Figure 2.30: Backtesting of ESG scores (best-in-class & good risk, MSCI World)</figcaption></figure>

<figure><img src="image_55.png" /></figure>
<figure><img src="image_56.png" /><figcaption>Figure 2.32: Backtesting of ESG scores (best-in-class & good risk, MSCI EM)</figcaption></figure>

<figure><img src="image_57.png" /></figure>
## 2.3 Rating system

As we have seen, a scoring model provides an automatic and statistical score. It is a pure quantitative approach. There may be the intervention of an analyst, but it is limited to data quality checks or forcing of input data<fnref n="59" />. A rating (or a notation) is different from a score, because it implies a quality scale. Since it implies a value judgement, a rating is generally produced by an analyst. For example, this is the case of credit ratings, which are made by an analyst who takes into account several quantitative scores, qualitative data, private and meeting information. Nevertheless, the balance between quantitative and qualitative judgements depends on the type of issuers. For retail borrowers, the rating is mainly explained by the scoring model. For blue chip and mega-cap companies, the rating highly depends on the qualitative assessment of the credit risk. If we consider ESG risk, the rating process shares similar patterns. The ESG score is generally the key component of the ESG rating. It is validated by an extra-financial analyst and it may be *forced* based on his qualitative information and experience. This explains why extra-financial analysis is organized with respect to sectors. An ESG analyst, who is specialized in a given sector, can then have a better view of all the ratings produced for this sector. This is particularly true for the strategic sectors: automobiles, coal, ciment, oil & gas, fertilizers & agricultural chemicals, utilities, etc. Nevertheless, there is a strong difference between credit and ESG rating processes from an investor viewpoint. Indeed, the number of rated companies for ESG analysis bears no comparison with the number of rated companies for credit analysis, because only few corporates have access to bond markets. On the contrary, a comprehensive ESG rating system must encompass all the securities and assets, notably all listed corporates and some private companies. In this context, the impact and the implication of the extra-financial analyst decrease with the firm size. This is why there is a strong small size bias in ESG rating systems.

### 2.3.1 Definition

ESG rating systems are based on the terminology of credit ratings (Box 2.4). For example, MSCI (2022) uses a 7-grade rating scale based on the grades AAA, AA, A, BBB, BB, B and CCC (Table 2.21). The number of grades, the rating symbols (letter, numeric) and the ordering of the system (low/worst rating to high/best rating vs. high/best to low/worst rating) differs from one provider to another provider<fnref n="60" />:

* Amundi: A (high), B,... to G (low) — 7-grade scale

* FTSE Russell: 0 (low), 1,... to 5 (high) — 6-grade scale

* ISS ESG: 1 (high), 2,... to 10 (low) — 10-grade scale

* MSCI: AAA (high), AA,... to CCC (low) — 7-grade scale

* Refinitiv: A+ (high), A, A-, B+,... to D- (low) — 12-grade scale

* RepRisk: AAA (high), AA,... to D (low) — 8-grade scale

* Sustainalytics: 1 (low), 2,... to 5 (high) — 5-grade scale

We notice the high heterogeneity of rating scales. Nevertheless, we observe that they are less granular than those used by credit rating agencies. On average, an ESG rating system is made up of 7 grades vs. 20 grades for a credit rating system.

<footnote n="59">⁵⁹In some cases, the analyst may also validate the score.</footnote>

<footnote n="60">⁶⁰In this list, we have included the asset manager Amundi, because ESG ratings are not only built by ESG rating agencies. Some investors (asset owners and managers) have defined their own internal ESG ratings.</footnote>
Box 2.4: Terminology of credit ratings

A rating system is a symbolic or numeric classification according to grade, which indicates a degree or step in a scale. For example, the credit rating systems of S&P, Moody's and Fitch is reported in Table 2.B. They are all based on a rating scale of 20 grades<sup>a</sup>. The symbolic rank AA+ (or BBB) is then a grade or a rating in the S&P classification. A notch means the difference between a particular rating and the next lower. For example, in the case of Moody's, the difference between Baa1 and Baa2 constitutes one Notch, whereas the difference between Aaa and Aa2 corresponds to two notches. When a credit rating agency revises the credit risk of a company, it may upgrade its rating by one notch (+1 notch), two notches (+2 notches), etc. or it may downgrade its rating by one notch (-1 notch), two notches (-2 notches), etc.

<table><caption>Table 2.B: Credit rating system of S&P, Moody's and Fitch</caption><thead><tr><th></th><th>Prime<br>Maximum Safety</th><th colspan="3">High Grade<br>High Quality</th><th colspan="3">Upper<br>Medium Grade</th><th colspan="3">Lower<br>Medium Grade</th></tr></thead><tbody><tr><th>S&P/Fitch<br>Moody's</th><td>AAA<br>Aaa</td><td>AA+<br>Aa1</td><td>AA<br>Aa2</td><td>AA-<br>Aa3</td><td>A+<br>A1</td><td>A<br>A2</td><td>A-<br>A3</td><td>BBB+<br>Baa1</td><td>BBB<br>Baa2</td><td>BBB-<br>Baa3</td></tr><tr><th>S&P/Fitch<br>Moody's</th><td>BB+<br>Ba1</td><td>BB<br>Ba2</td><td>BB-<br>Ba3</td><td></td><td>B+<br>B1</td><td>B<br>B2</td><td>B-<br>B3</td><td>CCC+<br>Caa1</td><td>CCC<br>Caa2</td><td>Extremely<br>Speculative<br>CC<br>Ca</td></tr></tbody></table>

<table><caption>Table 2.21: ESG rating system of Moody's</caption><thead><tr><th>Leader</th><th>Average</th><th>Laggard</th></tr></thead><tbody><tr><td>AAA AA</td><td>A BBB BB</td><td>B CCC</td></tr></tbody></table>

Source: MSCI (2022, Exhibit 8, page 12).

**Remark 23** In the sequel, we use the 7-grade scale based on the ratings AAA, AA, A, BBB, BB, B and CCC. We think that it is easier to manipulate and understand from a pedagogical point of view. A company rated AAA is a good company (or a good ESG risk) and a company rated CCC is a bad company (or a bad ESG risk).

### 2.3.2 ESG rating process

The construction of ESG ratings follows the same process than credit ratings (Figure 2.33). We need to define the map function that converts an ESG score into an ESG rating. In the case of credit risk, the estimate of the one-year probability of default is converted into credit ratings<sup>61</sup>. In the case of ESG risk, the ESG score is converted into an ESG rating such that the best scores correspond to the best ratings and the worst scores correspond to the worst ratings.

<footnote><sup>a</sup>Or 21 grades if we include the issuer default. Nevertheless, D is not considered as a rating.</footnote>

<footnote n="61"><sup>61</sup>For instance, a CCC-rated company has a one-year probability of default of 25%; a B-rated company has a 5% probability to default in the next year; for a BB-rated company, the one-year probability of default is equal to 1%; etc.</footnote>
<figure><img src="image_58.png" /><figcaption>Figure 2.33: From ESG score to ESG rating</figcaption></figure>

The first step consists in specifying the map function:

$$ \begin{array}{rcl} \mathcal{M} \mathit{ap}: & \Omega_{\mathcal{S}} & \to \Omega_{\mathcal{R}} \\ & \mathcal{S} & \mapsto \mathcal{R} = \mathcal{M} \mathit{ap}(\mathcal{S}) \end{array} $$

where $\Omega_{\mathcal{S}}$ is the support of ESG scores, $\Omega_{\mathcal{R}}$ is the ordered state space of ESG ratings and $\mathcal{R}$ is the ESG rating. By construction, $\mathcal{M} \mathit{ap}$ is a monotone function in order to preserve the preference ordering. In the case where $\mathcal{M} \mathit{ap}$ is increasing, we verify that:

$$ \mathcal{S}_2 > \mathcal{S}_1 \Leftrightarrow \mathcal{M} \mathit{ap}(\mathcal{S}_2) > \mathcal{M} \mathit{ap}(\mathcal{S}_1) $$

The second step is the validation (and the possible forcing) of the rating by the analyst.

Let us see some examples. MSCI (2022, page 12) explains that they use a uniform map function where $\Omega_{\mathcal{S}} = [0, 10]$ and $\Omega_{\mathcal{R}} = \{\text{CCC}, \text{B}, \text{BB}, \text{BBB}, \text{A}, \text{AA}, \text{AAA}\}$. The score is then divided into 7 equally-sized intervals and we have:

$$ \mathcal{M} \mathit{ap}(s) = \begin{cases} \text{CCC} & \text{if } \mathcal{S} \in [0, 10/7] \quad (0 - 1.429) \\ \text{B} & \text{if } \mathcal{S} \in [10/7, 20/7] \quad (1.429 - 2.857) \\ \text{BB} & \text{if } \mathcal{S} \in [20/7, 30/7] \quad (2.857 - 4.286) \\ \text{BBB} & \text{if } \mathcal{S} \in [30/7, 40/7] \quad (4.286 - 5.714) \\ \text{A} & \text{if } \mathcal{S} \in [40/7, 50/7] \quad (5.714 - 7.143) \\ \text{AA} & \text{if } \mathcal{S} \in [50/7, 60/7] \quad (7.143 - 8.571) \\ \text{AAA} & \text{if } \mathcal{S} \in [60/7, 10] \quad (8.571 - 10) \end{cases} $$

For instance, if the ESG score of the company is equal to 5, we assign the grade BBB, a score of 8 corresponds to the grade AA, etc. Refinitiv (2022, page 7) also considers a uniform map function, implying that $\Omega_{\mathcal{S}}$ is divided by 12 equally-sized intervals:

"[...] 'D' score (D-, D and D+) indicates poor relative ESG performance and insufficient degree of transparency in reporting material ESG data publicly. 'C' score (C-, C and C+) indicates satisfactory relative ESG performance and moderate degree of transparency in reporting material ESG data publicly. 'B' score (B-, B and B+) indicates good relative ESG performance and above average degree of transparency in reporting material ESG data publicly. 'A' score (A-, A and A+) indicates excellent relative ESG performance and high degree of transparency in reporting material ESG data publicly."

We assume that the map function is an increasing piecewise function, $\mathcal{S} \sim \mathcal{F}$ and $\mathcal{S} \in (s^-, s^+)$. We note $s_1^*, \dots, s_{K-1}^*$ the knots of the piecewise function, $K$ the number of ratings and $\Omega_{\mathcal{R}} = \{R_1, \dots, R_K\}$ the set of grades. We set $s_0^* = s^-$ and $s_K^* = s^+$. We deduce that:

$$ \begin{align*} p_k &= \Pr\{\mathcal{R} = R_k\} \\ &= \Pr\{s_{k-1}^* \le \mathcal{S} < s_k^*\} \\ &= \mathbf{F}(s_k^*) - \mathbf{F}(s_{k-1}^*) \end{align*} $$
Using this equation, we can then compute the frequency distribution of the ratings. The set of frequencies {$p_1, \dots, p_K$} is denoted by $\mathbb{P}$. If we don't know the distribution $\mathbf{F}$, we consider the empirical distribution $\hat{\mathbf{F}}$ and the estimated frequency is equal to $\hat{p}_k = \hat{\mathbf{F}}(s_k^*) - \hat{\mathbf{F}}(s_{k-1}^*)$. If we would like to build a rating system with pre-defined frequencies ($p_1, \dots, p_K$), we have to solve the following equation:

$$ \mathbf{F}(s_k^*) - \mathbf{F}(s_{k-1}^*) = p_k $$

We deduce that:

$$ \begin{aligned} \mathbf{F}(s_k^*) &= p_k + \mathbf{F}(s_{k-1}^*) \\ &= p_k + p_{k-1} + \mathbf{F}(s_{k-2}^*) \\ &= \left( \sum_{j=1}^{k} p_j \right) + \mathbf{F}(s_0^*) \end{aligned} $$

Since $\mathbf{F}(s_0^*) = 0$, we conclude that:

$$ s_k^* = \mathbf{F}^{-1} \left( \sum_{j=1}^{k} p_j \right) $$

**Remark 24** The discrete probability space of the rating system is denoted by $(\Omega_R, \Omega_R, \mathbb{P})$ and we have:

$$ \mathcal{E} := \Omega_R \times \mathbb{P} = \{(R_1, p_1), \dots, (R_K, p_K)\} $$

Let us consider a uniform score $\mathbf{S} \sim \mathcal{U}_{[a,b]}$. We have $\mathbf{F}(s) = (s-a)/(b-a)$. The rating system consists in $K$ equally-sized intervals. The knots of the map function are then equal to:

$$ s_k^* = a + \frac{(b-a)}{K}k $$

It follows that:

$$ \begin{aligned} p_k &= \mathbf{F}\left(a + \frac{(b-a)}{K}k\right) - \mathbf{F}\left(a + \frac{(b-a)}{K}(k-1)\right) \\ &= \frac{a + \frac{(b-a)}{K}k - a}{b-a} - \frac{a + \frac{(b-a)}{K}(k-1) - a}{b-a} \\ &= \frac{1}{K} \end{aligned} $$

We obtain a trivial result: the rating frequencies are all equal. In the case where we impose pre-defined frequencies ($p_1, \dots, p_K$), the knots of the map function are equal to$^{62}$:

$$ s_k^* = a + (b-a) \left( \sum_{j=1}^{k} p_j \right) $$

If we consider a 0/100 uniform score, we deduce that $s_k^* = 100 \cdot \sum_{j=1}^k p_j$. For example, if $\Omega_R \times \mathbb{P} = \{(CCC, 5\%), (B, 10\%), (BB, 20\%), (BBB, 30\%), (A, 20\%), (AA, 10\%), (AAA, 5\%)\}$, we obtain the trivial piecewise function where$^{63}$ $s_{CCC}^* = 5$, $s_B^* = 15$, $s_{BB}^* = 35$, $s_{BBB}^* = 65$, $s_A^* = 85$ and $s_{AA}^* = 95$.
<figure><img src="image_59.png" /><figcaption>Figure 2.34: Map function of a z-score (equal-space ratings)</figcaption></figure>

For a z-score system, we assume that $\mathcal{S} \sim \mathcal{N}(0, 1)$ and we obtain:

$$p_k = \Phi(s_k^*) - \Phi(s_{k-1}^*)$$

If we consider the 7-grade rating system with the classical knots $(-2.5, -1.5, -0.5, 0.5, 1.5, 2.5)$, we obtain the map function given in Figure 2.34 where$^{64}$:

$$p_k = \Phi(-3.5 + k) - \Phi(-4.5 + k)$$

The rating system with equal frequencies is obtained by using the following knots:

$$s_k^* = \Phi^{-1}\left(\frac{k}{K}\right) \quad \text{for } k = 1, \dots, K$$

In the case $K = 7$, the map function is given in Figure 2.35.

<figure><img src="image_60.png" /><figcaption>Figure 2.35: Map function of a z-score (equal-frequency ratings)</figcaption></figure>

**Remark 25** We recall that the role of the ESG analyst is to verify the consistency of the rated companies. This is why we generally observe forced ratings (or scores).

<footnote>$^{62}$We have $\mathbf{F}(s) = u \Leftrightarrow (s-a)/(b-a) = u$. We deduce that $s = \mathbf{F}^{-1}(u) = a + (b-a)u$.</footnote>

<footnote>$^{63}$By construction, the knot $s_{AAA}^* = 100$ is not necessarily to be defined because we always have $s_{AAA}^* = s^+$.</footnote>

<footnote>$^{64}$We obtain the following results: $p_{CCC} = \Phi(-2.5) = 0.62\%$, $p_B = \Phi(-1.5) - \Phi(-2.5) = 6.06\%$, $p_{BB} = \Phi(-0.5) - \Phi(-1.5) = 24.17\%$, $p_{BBB} = \Phi(0.5) - \Phi(-0.5) = 38.29\%$, $p_A = \Phi(1.5) - \Phi(0.5) = 24.17\%$, $p_{AA} = \Phi(2.5) - \Phi(1.5) = 6.06\%$ and $p_{AAA} = 1 - \Phi(2.5) = 0.62\%$.</footnote>
Box 2.5: Asymmetric rating system

In a symmetric rating system, the probability of the $k^{th}$ rating class is equal to the probability of the $(K-k+1)^{th}$ rating class$^a$, e.g., $p_{CCC} = p_{AAA}$, $p_B = p_A$ and $p_{BB} = p_A$. To obtain an asymmetric rating system, the first approach is to define the frequencies $p_k$ such that $\exists k : p_k \neq p_{K-k+1}$. We note $P_{worst} = \sum_{k=1}^{[K/2]} p_k$ and $P_{best} = \sum_{[K/2+1]}^{K} p_k$ the probability to be below and average the median rating. The rating process is said to be a *losing-oriented* system if $P_{worst} \ge P_{best}$, otherwise it is a *winning-oriented* system. This means that companies with bad ESG risk are more prevalent than companies with good ESG risk. For instance, $\Omega_R \times \mathbb{P} = \{(CCC, 5\%), (B, 10\%), (BB, 25\%), (BBB, 40\%), (A, 15\%), (AA, 4\%), (AAA, 1\%)\}$ is a *losing-oriented* system. The choice of an asymmetric rating system may be motivated by the underlying ESG strategy. For instance, implementing an exclusion ESG policy is not equivalent to considering a selection ESG policy. The investor may then want to adapt his rating system to take into account the objective of the strategy. The second approach to obtain an asymmetric rating system is to consider a $b$-score normalization with $\alpha \neq \beta$. Some examples are provided in Table 2.C.

<table><caption>Table 2.C: Frequency distribution of ESG ratings (in %)</caption><thead><tr><th colspan="2">Parameters</th><th colspan="7">Rating</th></tr><tr><th>α</th><th>β</th><th>CCC</th><th>B</th><th>BB</th><th>BBB</th><th>A</th><th>AA</th><th>AAA</th></tr></thead><tbody><tr><td>1</td><td>1</td><td>14.3</td><td>14.3</td><td>14.3</td><td>14.3</td><td>14.3</td><td>14.3</td><td>14.3</td></tr><tr><td>2</td><td>2</td><td>5.5</td><td>14.3</td><td>19.5</td><td>21.3</td><td>19.5</td><td>14.3</td><td>5.5</td></tr><tr><td>3</td><td>3</td><td>2.3</td><td>12.1</td><td>22.3</td><td>26.4</td><td>22.3</td><td>12.1</td><td>2.3</td></tr><tr><td>0.25</td><td>0.25</td><td>33.9</td><td>7.5</td><td>5.9</td><td>5.5</td><td>5.9</td><td>7.5</td><td>33.9</td></tr><tr><td>2.5</td><td>1.5</td><td>1.5</td><td>6.4</td><td>12.4</td><td>18.1</td><td>22.3</td><td>23.2</td><td>16.0</td></tr><tr><td>1.5</td><td>2.5</td><td>16.0</td><td>23.2</td><td>22.3</td><td>18.1</td><td>12.4</td><td>6.4</td><td>1.5</td></tr><tr><td>0.75</td><td>1</td><td>23.2</td><td>15.8</td><td>13.9</td><td>12.8</td><td>12.0</td><td>11.4</td><td>10.9</td></tr></tbody></table>

### 2.3.3 Rating migration matrix

One important issue concerns the consistency of the rating system. In particular, we may wonder whether it is relevant to use an equal-frequency, an equal-space or an asymmetric rating scheme. In the case of credit rating systems, we generally observe that medium risk classes have a higher frequency than extreme (low/high) risk classes. For instance, there are more BBB-rated companies than CCC-rated companies, the less frequent class is by far the AAA rating (less than 1%), etc. In the case of ESG rating systems, there is no consensus. Therefore, to assess the consistency and robustness of the rating system, we need to use probabilistic tools based on transition probability matrices (Norris, 1997).

#### Discrete-time modeling

**Markov chain model** We consider a time-homogeneous Markov chain $\mathcal{R}$, whose transition matrix is $P = (p_{i,j})$. We note $\Omega_R = \{R_1, \dots, R_K\}$ the state space of the chain and $\mathbb{K} = \{1, \dots, K\}$ the corresponding index set. $p_{i,j}$ is the probability that the entity migrates from rating $R_i$ to rating $R_j$.

<footnote>$^a$We reiterate that we only consider rating systems that satisfy a comprehensive preference ordering: $\forall k : \mathcal{R}_k \succ \mathcal{R}_{k-1}$.</footnote>
The matrix $P$ satisfies the following properties:

* $\forall i, j \in \mathbb{K}, p_{i,j} \ge 0$;

* $\forall i \in \mathbb{K}, \sum_{j=1}^{K} p_{i,j} = 1$.

Let $\mathcal{R}(t)$ be the value of the state at time $t$. We define $p(s, i; t, j)$ as the probability that the entity reaches the state $R_j$ at time $t$ given that it has reached the state $R_i$ at time $s$. We have:

$$p(s, i; t, j) = \Pr\{\mathcal{R}(t) = R_j \mid \mathcal{R}(s) = R_i\} = p_{i,j}^{(t-s)}$$

This probability only depends on the duration between $s$ and $t$ because of the Markov property. Therefore, we can restrict the analysis by calculating the $n$-step transition probability:

$$p_{i,j}^{(n)} = \Pr\{\mathcal{R}(t+n) = R_j \mid \mathcal{R}(t) = R_i\}$$

and the associated $n$-step transition matrix $P^{(n)} = (p_{i,j}^{(n)})$. For $n = 2$, we obtain:

$$\begin{aligned}
p_{i,j}^{(2)} &= \Pr\{\mathcal{R}(t+2) = R_j \mid \mathcal{R}(t) = R_i\} \\
&= \sum_{k=1}^{K} \Pr\{\mathcal{R}(t+2) = R_j, \mathcal{R}(t+1) = R_k \mid \mathcal{R}(t) = R_i\} \\
&= \sum_{k=1}^{K} \Pr\{\mathcal{R}(t+2) = R_j \mid \mathcal{R}(t+1) = R_k\} \cdot \Pr\{\mathcal{R}(t+1) = R_k \mid \mathcal{R}(t) = R_i\} \\
&= \sum_{k=1}^{K} p_{i,k} \cdot p_{k,j}
\end{aligned}$$

In a similar way, we obtain:

$$p_{i,j}^{(n+m)} = \sum_{k=1}^{K} p_{i,k}^{(n)} \cdot p_{k,j}^{(m)} \quad \forall n, m > 0 \qquad (2.4)$$

This equation is called the forward Chapman-Kolmogorov equation. In matrix form, we have:

$$P^{(n+m)} = P^{(n)} \cdot P^{(m)}$$

with the convention $P^{(0)} = I$. In particular, we have:

$$\begin{aligned}
P^{(n)} &= P^{(n-1)} \cdot P^{(1)} \\
&= P^{(n-2)} \cdot P^{(1)} \cdot P^{(1)} \\
&= \prod_{t=1}^{n} P^{(1)} \\
&= P^{n}
\end{aligned}$$

We deduce that:

$$p(t, i; t+n, j) = p_{i,j}^{(n)} = e_i^\top P^n e_j \qquad (2.5)$$

When we apply this framework to ESG risk, $\mathcal{R}(t)$ denotes the rating (or the risk class) of the company at time $t$ and $p_{i,j}$ is the one-period transition probability from rating $R_i$ to rating $R_j$.
<table><caption>Table 2.22: ESG migration matrix #1 (one-year transition probability in %)</caption><thead><tr><th></th><th>AAA</th><th>AA</th><th>A</th><th>BBB</th><th>BB</th><th>B</th><th>CCC</th></tr></thead><tbody><tr><th>AAA</th><td>92.76</td><td>5.66</td><td>0.90</td><td>0.45</td><td>0.23</td><td>0.00</td><td>0.00</td></tr><tr><th>AA</th><td>4.15</td><td>82.73</td><td>11.86</td><td>0.89</td><td>0.30</td><td>0.07</td><td>0.00</td></tr><tr><th>A</th><td>0.18</td><td>15.47</td><td>72.98</td><td>10.46</td><td>0.82</td><td>0.09</td><td>0.00</td></tr><tr><th>BBB</th><td>0.07</td><td>1.32</td><td>19.60</td><td>69.49</td><td>9.03</td><td>0.42</td><td>0.07</td></tr><tr><th>BB</th><td>0.04</td><td>0.19</td><td>1.55</td><td>19.36</td><td>70.88</td><td>7.75</td><td>0.23</td></tr><tr><th>B</th><td>0.00</td><td>0.05</td><td>0.24</td><td>1.43</td><td>21.54</td><td>74.36</td><td>2.38</td></tr><tr><th>CCC</th><td>0.00</td><td>0.00</td><td>0.22</td><td>0.44</td><td>2.21</td><td>13.24</td><td>83.89</td></tr></tbody></table>

In Table 2.22, we report an example of transition probability matrix. We read the figures as follows<sup>65</sup>: a company rated AAA has a one-year probability of 92.76% to remain AAA; its probability to become AA is 5.66%; a company rated CCC has a probability of 16.11% to improve its rating, etc. In Tables 2.23 and 2.24, we have reported the two-year and five-year transition matrices. We detail below the calculation of $p_{AAA,AAA}^{(2)}$:

$$ 
\begin{aligned}
p_{AAA,AAA}^{(2)} &= p_{AAA,AAA} \times p_{AAA,AAA} + p_{AAA,AA} \times p_{AA,AAA} + p_{AAA,A} \times p_{A,AAA} + p_{AAA,BBB} \times p_{BBB,AAA} + \\
&\quad p_{AAA,BB} \times p_{BB,AAA} + p_{AAA,B} \times p_{B,AAA} + p_{AAA,CCC} \times p_{CCC,AAA} \\
&= 0.9276^2 + 0.0566 \times 0.0415 + 0.0090 \times 0.0018 + 0.0045 \times 0.0007 + 0.0023 \times 0.0004 \\
&= 86.28\%
\end{aligned}
$$

<table><caption>Table 2.23: Two-year transition probability in % (migration matrix #1)</caption><thead><tr><th></th><th>AAA</th><th>AA</th><th>A</th><th>BBB</th><th>BB</th><th>B</th><th>CCC</th></tr></thead><tbody><tr><th>AAA</th><td>86.28</td><td>10.08</td><td>2.25</td><td>0.92</td><td>0.44</td><td>0.02</td><td>0.00</td></tr><tr><th>AA</th><td>7.30</td><td>70.52</td><td>18.68</td><td>2.67</td><td>0.66</td><td>0.15</td><td>0.00</td></tr><tr><th>A</th><td>0.95</td><td>24.24</td><td>57.16</td><td>15.20</td><td>2.19</td><td>0.25</td><td>0.01</td></tr><tr><th>BBB</th><td>0.21</td><td>5.06</td><td>28.22</td><td>52.11</td><td>12.93</td><td>1.33</td><td>0.14</td></tr><tr><th>BB</th><td>0.09</td><td>0.79</td><td>6.07</td><td>27.45</td><td>53.68</td><td>11.37</td><td>0.55</td></tr><tr><th>B</th><td>0.01</td><td>0.18</td><td>0.98</td><td>6.26</td><td>31.47</td><td>57.28</td><td>3.82</td></tr><tr><th>CCC</th><td>0.00</td><td>0.05</td><td>0.50</td><td>1.32</td><td>6.31</td><td>21.13</td><td>70.70</td></tr></tbody></table>

<table><caption>Table 2.24: Five-year transition probability in % (migration matrix #1)</caption><thead><tr><th></th><th>AAA</th><th>AA</th><th>A</th><th>BBB</th><th>BB</th><th>B</th><th>CCC</th></tr></thead><tbody><tr><th>AAA</th><td>70.45</td><td>18.69</td><td>6.97</td><td>2.61</td><td>1.08</td><td>0.18</td><td>0.01</td></tr><tr><th>AA</th><td>13.13</td><td>50.21</td><td>26.03</td><td>7.90</td><td>2.22</td><td>0.48</td><td>0.03</td></tr><tr><th>A</th><td>4.35</td><td>33.20</td><td>37.78</td><td>17.99</td><td>5.52</td><td>1.08</td><td>0.09</td></tr><tr><th>BBB</th><td>1.50</td><td>16.49</td><td>32.49</td><td>30.90</td><td>14.61</td><td>3.63</td><td>0.38</td></tr><tr><th>BB</th><td>0.50</td><td>5.98</td><td>17.83</td><td>30.10</td><td>31.35</td><td>12.85</td><td>1.39</td></tr><tr><th>B</th><td>0.15</td><td>1.90</td><td>7.40</td><td>18.95</td><td>35.11</td><td>31.26</td><td>5.23</td></tr><tr><th>CCC</th><td>0.05</td><td>0.64</td><td>2.55</td><td>6.93</td><td>17.96</td><td>38.54</td><td>43.33</td></tr></tbody></table>

<footnote n="65"><sup>65</sup>The rows represent the initial rating whereas the columns indicate the final rating.</footnote>
Box 2.6: Stationary distribution of a Markov chain

We note $\pi_k^{(n)}$ the probability of the state $R_k$ at time $n$:

$$\pi_k^{(n)} = \Pr\{\mathcal{R}(n) = R_k\}$$

and $\pi^{(n)} = (\pi_1^{(n)}, \dots, \pi_K^{(n)})$ the probability distribution. By construction, we have:

$$\pi^{(n+1)} = P^{\top}\pi^{(n)}$$

The Markov chain $\mathcal{R}$ admits a stationary distribution $\pi^*$ if$^a$:

$$\pi^* = P^{\top}\pi^*$$

In this case, $\pi_k^*$ is the limiting probability of state $R_k$:

$$\lim_{n \to \infty} p_{i,k}^{(n)} = \pi_k^* \quad \forall i$$

We can interpret $\pi_k^*$ as the average duration spent by the chain $\mathcal{R}$ in the state $R_k$. Let $\mathcal{T}_k$ be the return period$^b$ of state $R_k$:

$$\mathcal{T}_k = \inf \{n : \mathcal{R}(n) = R_k \mid \mathcal{R}(0) = R_k\}$$

The average return period is then equal to:

$$\tau_k := \mathbb{E}[\mathcal{T}_k] = \frac{1}{\pi_k^*}$$

We compute the stationary distribution$^{66}$ and we obtain:

$$\pi^* = (17.78\%, 29.59\%, 25.12\%, 15.20\%, 8.35\%, 3.29\%, 0.67\%)$$

The average return periods are then equal to 5.6, 3.4, 4.0, 6.6, 12.0, 30.4 and 149.0 years. The interpretation of these results is the following. In the long term, the probability to observe a AAA-rated company is equal to 17.78%, while the probability to observe a CCC-rated company is equal to 0.67%. The probability $\pi_k^*$ is then the long-term equivalent of the current (or sample) frequency $p_k$. Similarly, the expected time to reach the worst-in-class state is equal to 149 years. These results show that the rating system #1 is clearly a winning-oriented system, where more than 70% of corporates are expected to have a rating above BBB.

<footnote>$^a$Not all Markov chains behave in this way, meaning that $\pi^*$ does not necessarily exist.</footnote>

<footnote>$^b$ $\mathcal{T}_k$ is a stopping time. It is also called the *first-passage time*.</footnote>

<footnote>$^{66}$There are three numerical approaches to compute $\pi^*$. The first one is to approximate $P^{(\infty)}$ by $P^{(n)}$ with $n$ sufficient large ($n > 100$) and take any rows of the matrix $P^{(\infty)}$. The second method is to compute the eigendecomposition $V\Lambda V^{-1}$ of $P^{\top}$ and return the left eigenvector whose eigenvalue is exactly equal to 1. This second approach uses the fact that $\pi^* = P^{\top}\pi^*$ defines an eigenvalue problem $(P^{\top} - \lambda I_K) \pi^* = 0$ with $\lambda = 1$. Finally, the third method directly solves the equation $(P^{\top} - I_K) \pi^* = 0$ by computing an orthonormal basis for the null space of $P^{\top} - I_K$. For the last two methods, we normalize the solution such that $1^{\top}\pi^* = 1$.</footnote>
<table><caption>Table 2.25: ESG migration matrix #2 (one-month transition probability in %)</caption><thead><tr><th></th><th>AAA</th><th>AA</th><th>A</th><th>BBB</th><th>BB</th><th>B</th><th>CCC</th></tr></thead><tbody><tr><th>AAA</th><td>93.50</td><td>5.00</td><td>0.50</td><td>0.50</td><td>0.50</td><td>0.00</td><td>0.00</td></tr><tr><th>AA</th><td>2.00</td><td>93.00</td><td>4.00</td><td>0.50</td><td>0.50</td><td>0.00</td><td>0.00</td></tr><tr><th>A</th><td>0.00</td><td>3.00</td><td>93.00</td><td>3.90</td><td>0.10</td><td>0.00</td><td>0.00</td></tr><tr><th>BBB</th><td>0.00</td><td>0.10</td><td>2.80</td><td>94.00</td><td>3.00</td><td>0.10</td><td>0.00</td></tr><tr><th>BB</th><td>0.00</td><td>0.00</td><td>0.10</td><td>3.50</td><td>94.50</td><td>1.80</td><td>0.10</td></tr><tr><th>B</th><td>0.00</td><td>0.00</td><td>0.00</td><td>0.10</td><td>3.70</td><td>96.00</td><td>0.20</td></tr><tr><th>CCC</th><td>0.00</td><td>0.00</td><td>0.00</td><td>0.40</td><td>0.50</td><td>0.60</td><td>98.50</td></tr></tbody></table>

In Table 2.25, we now consider the ESG migration matrix #2, which has been computed on a monthly basis. If we would like to compare the two rating systems, we can compute the one-year probability transition matrix (Table 2.26). We observe that the two transition matrices are very different. Indeed, the second rating system is more reactive than the first rating system. If we compute the stationary distribution of the second rating system, we obtain:

$$ \pi^* = (3.11\%, 10.10\%, 17.46\%, 27.76\%, 25.50\%, 12.68\%, 3.39\%) $$

implying that the average return periods are equal to 32.2, 9.9, 5.7, 3.6, 3.9, 7.9 and 29.5 years. These results show that the rating system #2 is a balanced system.

<table><caption>Table 2.26: One-year probability transition in % (migration matrix #2)</caption><thead><tr><th></th><th>AAA</th><th>AA</th><th>A</th><th>BBB</th><th>BB</th><th>B</th><th>CCC</th></tr></thead><tbody><tr><th>AAA</th><td>48.06</td><td>29.71</td><td>10.34</td><td>6.42</td><td>4.95</td><td>0.49</td><td>0.03</td></tr><tr><th>AA</th><td>11.65</td><td>49.25</td><td>24.10</td><td>9.60</td><td>4.87</td><td>0.49</td><td>0.03</td></tr><tr><th>A</th><td>2.02</td><td>17.51</td><td>49.67</td><td>24.72</td><td>5.52</td><td>0.54</td><td>0.03</td></tr><tr><th>BBB</th><td>0.27</td><td>3.53</td><td>17.46</td><td>55.50</td><td>20.21</td><td>2.88</td><td>0.16</td></tr><tr><th>BB</th><td>0.03</td><td>0.60</td><td>4.21</td><td>23.43</td><td>57.45</td><td>13.27</td><td>1.01</td></tr><tr><th>B</th><td>0.00</td><td>0.08</td><td>0.74</td><td>5.94</td><td>27.10</td><td>64.18</td><td>1.96</td></tr><tr><th>CCC</th><td>0.00</td><td>0.07</td><td>0.57</td><td>4.22</td><td>5.77</td><td>5.85</td><td>83.51</td></tr></tbody></table>

<table><caption>Table 2.27: One-month probability transition in % (migration matrix #1)</caption><thead><tr><th></th><th>AAA</th><th>AA</th><th>A</th><th>BBB</th><th>BB</th><th>B</th><th>CCC</th></tr></thead><tbody><tr><th>AAA</th><td>99.36</td><td>0.53</td><td>0.05</td><td>0.04</td><td>0.02</td><td>0.00</td><td>0.00</td></tr><tr><th>AA</th><td>0.39</td><td>98.31</td><td>1.26</td><td>0.01</td><td>0.03</td><td>0.01</td><td>0.00</td></tr><tr><th>A</th><td>-0.02</td><td>1.65</td><td>97.14</td><td>1.21</td><td>0.02</td><td>0.01</td><td>0.00</td></tr><tr><th>BBB</th><td>0.01</td><td>-0.07</td><td>2.28</td><td>96.72</td><td>1.06</td><td>-0.01</td><td>0.01</td></tr><tr><th>BB</th><td>0.00</td><td>0.02</td><td>-0.12</td><td>2.29</td><td>96.92</td><td>0.88</td><td>0.01</td></tr><tr><th>B</th><td>0.00</td><td>0.00</td><td>0.04</td><td>-0.15</td><td>2.45</td><td>97.42</td><td>0.25</td></tr><tr><th>CCC</th><td>0.00</td><td>0.00</td><td>0.02</td><td>0.04</td><td>0.05</td><td>1.37</td><td>98.53</td></tr></tbody></table>

**Remark 26** Another approach to analyze the two rating systems is to compute the monthly transition matrix associated to the migration matrix #1. In this case, we have to find the matrix M such
that $M^{(12)} = P$. The solution<sup>67</sup> is given by $M = P^{1/12}$ and reported in Table 2.27. We can compare it with the matrix in Table 2.25. Because $M$ has some negative probabilities, it is not a transition matrix, which indicates that the rating system #1 does not satisfy the Markov property<sup>68</sup>.

Box 2.7: Mean hitting time

Let $\mathcal{A} \subset \mathbb{K}$ be a given subset. The first hitting time of $\mathcal{A}$ is given by:

$$ \mathcal{T}(\mathcal{A}) = \inf \{n : \mathcal{R}(n) \in \mathcal{A}\} $$

$\mathcal{T}(\mathcal{A})$ measures how long it takes to reach the target states $j \in \mathcal{A}$. We can show that it is a stopping time. The mean first hitting (or passage) time to target $\mathcal{A}$ from state $k$ is defined as:

$$ \tau_k(\mathcal{A}) = \mathbb{E}[\mathcal{T}(\mathcal{A}) | \mathcal{R}(0) = R_k] $$

Let $\tau(\mathcal{A}) = (\tau_1(\mathcal{A}), \dots, \tau_K(\mathcal{A}))$ be the vector of mean first hitting times. Norris (1997) showed that:

$$ \tau_k(\mathcal{A}) = 1 + \sum_{j=1}^{K} p_{k,j} \tau_j(\mathcal{A}) $$

By construction, we have $\tau_k(\mathcal{A}) = 0$ if $k \in \mathcal{A}$. In fact, $\tau_k(\mathcal{A})$ is the minimal non negative solution to the previous system. It follows that $\|\tau(\mathcal{A})\| = \sum_{k=1}^{K} |\tau_k(\mathcal{A})| = \sum_{k=1}^{K} \tau_k(\mathcal{A})$ because $\tau_k(\mathcal{A}) \ge 0$. We deduce that:

$$ \tau(\mathcal{A}) = \begin{array}{ll} \arg\min & \sum_{k=1}^{K} x_k \\ \text{s.t.} & \begin{cases} x_k = 0 & \text{if } k \in \mathcal{A} \\ x_k = 1 + \sum_{j=1}^{K} p_{k,j}x_j & \text{if } k \notin \mathcal{A} \\ x_k \ge 0 & \end{cases} \end{array} $$

We obtain a linear programming problem with $K + 1$ equally constraints and $K$ lower bounds:

$$ \tau(\mathcal{A}) = \begin{array}{ll} \arg\min & \sum_{k=1}^{K} x_k \\ \text{s.t.} & \begin{cases} x_k = 0 & \text{if } k \in \mathcal{A} \\ \sum_{j \notin \mathcal{A}} p_{k,j} x_j = -1 & \\ \sum_{j \notin \mathcal{A} \cup \{k\}} p_{k,j} x_j + (p_{k,k} - 1) x_k = -1 & \text{if } k \notin \mathcal{A} \\ x_k \ge 0 & \end{cases} \end{array} $$

<footnote n="67"><sup>67</sup>Since $f(x) = x^\alpha$ with $\alpha > 1$ is a transcendental function, we use the Schur decomposition $P = QTQ^*$ to compute numerically the matrix $M$. Using Appendix A.1.1, we deduce that $M = QT^{1/12}Q^*$.</footnote>

<footnote n="68"><sup>68</sup>The Markov property of ESG ratings is discussed later on page 127.</footnote>
Let $\mathcal{B} = \{AAA, AA, A\}$ and $\mathcal{W} = \{BB, B, CCC\}$ be the best-in-class and worst-in-class sets. We obtain the following mean hitting times (in years) for the two rating systems:

<table><thead><tr><th rowspan="2">Rating system</th><th colspan="4">W-target</th><th colspan="4">B-target</th></tr><tr><th>AAA</th><th>AA</th><th>A</th><th>BBB</th><th>BBB</th><th>BB</th><th>B</th><th>CCC</th></tr></thead><tbody><tr><td>#1</td><td>79.21</td><td>70.04</td><td>62.34</td><td>46.54</td><td>7.50</td><td>13.28</td><td>17.58</td><td>22.68</td></tr><tr><td>#2</td><td>10.24</td><td>9.92</td><td>9.13</td><td>6.68</td><td>8.68</td><td>11.99</td><td>14.26</td><td>17.54</td></tr></tbody></table>

**Estimation of the transition matrix** Using Bayes theorem, we have:

$$ 
\begin{aligned}
p_{i,j} &= \Pr\{\mathcal{R}(t+1) = R_j | \mathcal{R}(t) = R_i\} \\
&= \frac{\Pr\{\mathcal{R}(t+1) = R_j, \mathcal{R}(t) = R_i\}}{\Pr\{\mathcal{R}(t) = R_i\}}
\end{aligned}
$$

We reiterate that<fnref n="69" /> $\mathcal{R}(t) = R_k \Leftrightarrow \mathcal{S}(t) \in [s_{k-1}^*, s_k^*]$. We have seen that:

$$ \Pr\{\mathcal{R}(t) = R_k\} = \mathbf{F}(s_k^*) - \mathbf{F}(s_{k-1}^*) = p_k $$

where $\mathbf{F}(s)$ is the probability distribution of the score $\mathcal{S}(t)$. We assume that<fnref n="70" />:

$$ \Pr\{\mathcal{S}(t) \le s, \mathcal{S}(t+1) \le s'\} = \mathbf{C}(\mathbf{F}(s), \mathbf{F}(s')) $$

where $\mathbf{C}$ is the copula function of the random vector $(\mathcal{S}(t), \mathcal{S}(t+1))$. We deduce that:

$$ 
\begin{aligned}
\Pr\{\mathcal{R}(t+1) = R_j, \mathcal{R}(t) = R_i\} &= \Pr\{s_{i-1}^* \le \mathcal{S}(t) \le s_i^*, s_{j-1}^* \le \mathcal{S}(t+1) \le s_j^*\} \\
&= \mathbf{C}(\mathbf{F}(s_i^*), \mathbf{F}(s_j^*)) - \mathbf{C}(\mathbf{F}(s_{i-1}^*), \mathbf{F}(s_j^*)) - \\
&\qquad \mathbf{C}(\mathbf{F}(s_i^*), \mathbf{F}(s_{j-1}^*)) + \mathbf{C}(\mathbf{F}(s_{i-1}^*), \mathbf{F}(s_{j-1}^*))
\end{aligned}
$$

Finally, we obtain:

$$ p_{i,j} = \frac{\mathbf{C}(\mathbf{F}(s_i^*), \mathbf{F}(s_j^*)) - \mathbf{C}(\mathbf{F}(s_{i-1}^*), \mathbf{F}(s_j^*)) - \mathbf{C}(\mathbf{F}(s_i^*), \mathbf{F}(s_{j-1}^*)) + \mathbf{C}(\mathbf{F}(s_{i-1}^*), \mathbf{F}(s_{j-1}^*))}{\mathbf{F}(s_i^*) - \mathbf{F}(s_{i-1}^*)} $$

This is the theoretical expression of the probability transition $p_{i,j}$. In practice, we do not know the probability functions $\mathbf{F}$ and $\mathbf{C}$. Therefore, we can estimate them and the estimated value of $p_{i,j}$ is equal to:

$$ \hat{p}_{i,j} = \frac{\hat{\mathbf{C}}(\hat{\mathbf{F}}(s_i^*), \hat{\mathbf{F}}(s_j^*)) - \hat{\mathbf{C}}(\hat{\mathbf{F}}(s_{i-1}^*), \hat{\mathbf{F}}(s_j^*)) - \hat{\mathbf{C}}(\hat{\mathbf{F}}(s_i^*), \hat{\mathbf{F}}(s_{j-1}^*)) + \hat{\mathbf{C}}(\hat{\mathbf{F}}(s_{i-1}^*), \hat{\mathbf{F}}(s_{j-1}^*))}{\hat{\mathbf{F}}(s_i^*) - \hat{\mathbf{F}}(s_{i-1}^*)} $$

This parametric estimation approach is interesting when we specify the parametric functions $\mathbf{F}(s; \theta_1)$ and $\mathbf{C}(s, s'; \theta_2)$, and we estimate the parameters $\theta_1$ and $\theta_2$.

Generally, we have no idea about the probability functions $\mathbf{F}$ and $\mathbf{C}$. We can then adopt a non-parametric estimation approach. The first idea is to replace $\mathbf{F}$ and $\mathbf{C}$ by the empirical distribution of $\mathcal{S}(t)$ and the empirical copula of $(\mathcal{S}(t), \mathcal{S}(t+1))$. In practice, we can simplify this approach by estimating directly the empirical probability. Thanks to the Bayes theorem, we have:

$$ \hat{p}_{i,j}(t) = \frac{\#\{\mathcal{R}(t+1) = R_j, \mathcal{R}(t) = R_i\}}{\#\{\mathcal{R}(t) = R_i\}} $$

<footnote n="69">⁶⁹In this analysis, we have the following correspondence: $R_1 = CCC, R_2 = B, \dots, R_K = AAA$.</footnote>

<footnote n="70">⁷⁰There is no reason that the probability distribution of $\mathcal{S}(t+1)$ is different than this of $\mathcal{S}(t)$.</footnote>
We consider a cohort of issuers for a given period $[t, t+1]$. Let $n_{i,·}(t)$ be the number of issuers rated $R_i$ at the beginning of the period $t$. Let $n_{i,j}(t)$ be the number of issuers rated $R_i$ at the beginning of the period $t$ and $R_j$ at the end of the period $t$. We deduce that $\hat{p}_{i,j}$ is the ratio between the two quantities:

$$ \hat{p}_{i,j}(t) = \frac{n_{i,j}(t)}{n_{i,·}(t)} $$

When the period is the year YYYY, the cohort starts on 1 January YYYY and ends on 31 December YYYY. If we have several annual cohorts, we can average the empirical probabilities:

$$ \hat{p}_{i,j} = \frac{1}{T} \sum_{t=1}^{T} \hat{p}_{i,j}(t) = \frac{1}{T} \sum_{t=1}^{T} \frac{n_{i,j}(t)}{n_{i,·}(t)} $$

Another approach is to use the pooling method:

$$ \hat{p}_{i,j} = \frac{\sum_{t=1}^{T} n_{i,j}(t)}{\sum_{t=1}^{T} n_{i,·}(t)} $$

The two approaches give different results. In the first case, each annual cohort has the same weight. In the second case, the approach puts more weight on the year which is more representative<fnref n="71" />.

<table><caption>Table 2.28: Number of observations $n_{i,j}$ (migration matrix #1)</caption><thead><tr><th>n<sub>i,j</sub></th><th>AAA</th><th>AA</th><th>A</th><th>BBB</th><th>BB</th><th>B</th><th>CCC</th><th>n<sub>i,·</sub>(t)</th><th>hat<sub>i,·</sub>(t)</th></tr></thead><tbody><tr><th>AAA</th><td>2 050</td><td>125</td><td>20</td><td>10</td><td>5</td><td>0</td><td>0</td><td>2 210</td><td>3.683%</td></tr><tr><th>AA</th><td>280</td><td>5 580</td><td>800</td><td>60</td><td>20</td><td>5</td><td>0</td><td>6 745</td><td>11.242%</td></tr><tr><th>A</th><td>20</td><td>1 700</td><td>8 020</td><td>1 150</td><td>90</td><td>10</td><td>0</td><td>10 990</td><td>18.317%</td></tr><tr><th>BBB</th><td>10</td><td>190</td><td>2 820</td><td>10 000</td><td>1 300</td><td>60</td><td>10</td><td>14 390</td><td>23.983%</td></tr><tr><th>BB</th><td>5</td><td>25</td><td>200</td><td>2 500</td><td>9 150</td><td>1 000</td><td>30</td><td>12 910</td><td>21.517%</td></tr><tr><th>B</th><td>0</td><td>5</td><td>25</td><td>150</td><td>2 260</td><td>7 800</td><td>250</td><td>10 490</td><td>17.483%</td></tr><tr><th>CCC</th><td>0</td><td>0</td><td>5</td><td>10</td><td>50</td><td>300</td><td>1 900</td><td>2 265</td><td>3.775%</td></tr><tr><th>n<sub>.,j</sub>(t)</th><td>2 365</td><td>7 625</td><td>11 890</td><td>13 850</td><td>12 875</td><td>9 175</td><td>2 190</td><td>60 000</td><td></td></tr><tr><th>hat<sub>.,j</sub>(t)</th><td>3.942%</td><td>12.708%</td><td>19.817%</td><td>23.133%</td><td>21.458%</td><td>15.292%</td><td>3.650%</td><td>100.00%</td><td></td></tr></tbody></table>

In Table 2.28, we report all the information<fnref n="72" /> for estimating the migration matrix #1. We have used the pooling method with 60 000 observations. For 2 050 observations, the initial rating on 1 January YYYY is AAA and the final rating on 31 December YYYY is AAA. We observe 125 cases where a AAA-rated issuer has been downgraded by one notch. If we compute the sum, we obtain 2 210 AAA-rated observations at the beginning of the year and 2 365 AAA-rated observations at the end of the year. We can then compute the transition probabilities: $\hat{p}_{AAA,AAA} = \frac{2 050}{2 210} = 92.76\%$, $\hat{p}_{AAA,AA} = \frac{125}{2 210} = 5.66\%$, ..., $\hat{p}_{CCC,CCC} = \frac{1 900}{2 265} = 83.89\%$.

Previously, we have seen that the stationary distribution of the migration matrix #1 is equal to:

$$ \pi^* = (17.78\%, 29.59\%, 25.12\%, 15.20\%, 8.35\%, 3.29\%, 0.67\%) $$

<footnote n="71">⁷¹From a theoretical viewpoint, this second method is biased. However, it is extensively used in particular when the number of observations is low for each period.</footnote>

<footnote n="72">⁷²We have $n(t) = \sum_{i=1}^{K} n_{i,·}(t)$, $\hat{p}_{i,·}(t) = n_{i,·}(t) / n(t)$, $n'(t) = \sum_{j=1}^{K} n_{j,·}(t)$, $\hat{p}_{j,·}(t) = n_{j,·}(t) / n'(t)$.</footnote>