Since each sector contributes differently to emissions, their respective transitions will proceed at different speeds based on technological maturity.

## Power and electricity

According to Energy Transitions Commission (2023a), the three main challenges are increasing the total electricity supply from around 30 000 TWh today to over 100 000 TWh by mid-century, extending the transmission and distribution networks from approximately 70 million km to up to 200 million km, and producing 500-800 Mt of green hydrogen per year.

## Hydrogen

Hydrogen will play a key role in many aspects of the transition process. The $80 billion investment will be allocated to hydrogen production and distribution. Of this amount, $40 billion will be used to produce green and blue hydrogen, as well as to convert gray hydrogen. The remaining $40 billion will fund the construction of pipelines, refueling stations, exchange terminals, and storage facilities.

## Sustainable agriculture and land use requirements

Although most land is used for agriculture, more land is needed for wind and solar farms. The demand for land under the renewable energy system far exceeds the demand under the fossil fuel system but is still far less than the demand for agriculture. Agriculture is the primary cause of deforestation and must adopt greenhouse gas-efficient farming practices and encourage changes in consumer behavior.

## Buildings

For the buildings sector to become green, the power sector must first initiate a transition. This means retrofitting older buildings and constructing new, carbon-efficient ones. Of the $500 billion invested in the buildings sector each year, $230 billion will be allocated for upgrading buildings with new green technologies, $130 billion for purchasing renewable heat, and $150 billion for installing new heat pumps.

## Mobility and transport

The investment required for the transport sector's transition falls into three categories. First, shifting from internal combustion engines (ICES) to electric vehicles (EVs) will require $130 billion annually for developing charging and refueling infrastructure. Another $70 billion will be spent on manufacturing facilities for sustainable aircraft and batteries. Finally, $40 billion will be spent on greening the shipping system through new infrastructure, vessels, and investments.

## Industry

The industrial sector relies on carbon-intensive materials such as steel and cement, which require decarbonization through carbon capture and storage (CCS) facilities and conversion processes. Of the $70 billion investment, $10 billion will fund decarbonizing steel production, and an additional $10 billion will fund decarbonizing cement plants. Another $40 billion will fund the development and integration of CCS and other decarbonization technologies, while the final $10 billion will fund the deployment of low-carbon technologies in smelters and refineries.
## Water management

According to Energy Transitions Commission (2023b), global water consumption is expected to reach 4 trillion cubic meters per year by 2050. Of that total, 70% will be used for agriculture, 58 billion cubic meters will be used for clean energy production, and 37 billion cubic meters will be used for fossil fuels. Water will also be used for nuclear power generation (14 billion cubic meters per year), hydrogen production by electrolysis (11 billion cubic meters per year), carbon capture (19–29 billion cubic meters per year), and cleaning solar panels. This is in addition to the approximately 4 billion cubic meters used for mining each year. The high demand for water in energy production could lead to increased water stress and scarcity. To alleviate this issue, solutions such as reducing water use, especially for mining, improving efficiency, and exploring ways to obtain more water through desalination or recycling must be implemented. Energy and water are closely linked. The energy sector needs water for clean energy transformation, and the water sector needs energy for development, water recycling, desalination, transport, and distribution. According to IEA (2017), global energy use in the water sector is expected to double by 2040.

## Waste management and circular economy

When discussing the transition to a low-carbon economy, waste management is an important consideration. Waste is generated at every stage of the transition, including food waste from agriculture and waste from solar panels, wind farms, and mining operations. According to Energy Transitions Commission (2023b), the energy transition could generate up to 13 billion tonnes of waste by 2050. Various solutions can be implemented to manage this waste and create a more circular economy. The most important solution is introducing recycling, which reduces waste and increases the supply of materials. McKinsey (2022) suggests other solutions, such as allocating landfill space for organic waste, investing in digesters and composters, and using new technologies to capture methane.

## Material and resource requirements

The transition to clean energy will create a significant demand for various materials. Some of these materials, such as steel, copper, aluminium, nickel and chromium, are essential components of many clean technologies. Others, such as steel, play a critical role in sectors unrelated to energy, such as construction. Certain materials are specific to particular technologies. For example, cobalt is essential for electric vehicles and batteries, while polysilicon is crucial for solar photovoltaics (Table 10.22). Between 2022 and 2050, total material demand could reach 6.5 billion tonnes, with steel, copper and aluminium accounting for around 95% of this total. While current global mineral resources appear sufficient to meet demand until 2050, especially as new reserves are expected to emerge in response to market signals, some materials may face depletion risks and struggle to meet cumulative demand. These challenges can be mitigated through technological innovation, expanding existing mines, developing new ones and increasing investment. However, one of the most effective strategies remains recycling. For cobalt, graphite, and lithium, recycled materials could account for over half of the total demand required for the energy transition. For copper and aluminium, recycled materials could provide 30–40% of total demand (Energy Transitions Commission, 2023b).

Figure 10.21 shows the calculated demand and primary supply<fnref n="44" /> in 2030 calculated by Energy Transitions Commission (2023b). A material-specific analysis is provided in Box 10.5. The sup-

<footnote n="44">⁴⁴The legend is as follows: S-22 represents the supply in 2022; S-30 is the projected supply in 2030; U-30 represents the primary demand in 2030, including the demand for the energy transition; and L-30 is the primary demand in 2030, assuming maximum efficiency and improvements in recycling (Energy Transitions Commission, 2023b).</footnote>
<figure><img src="image_494.png" /><figcaption>Figure 10.21: Demand and primary supply in 2030</figcaption></figure>

Source: Energy Transitions Commission (2023b), Material Factsheets.

ply/demand balance for copper and nickel appears achievable by 2030. However, this can only be achieved for cobalt, lithium and neodymium by increasing efficiency and improving recycling.

**Remark 131** The supply of materials can be threatened by many factors, such as a lack of diversification in the mining, refining and processing stages. The mining stage is dominated by a few countries, for example the Democratic Republic of Congo for cobalt, and China for rare earths. The refining and processing stages are even more concentrated, with China dominating five materials: cobalt, copper, lithium, nickel, and rare earths. This increases the risk of supply disruptions due to political instability, geopolitical tensions or physical risks, such as drought. The extraction and production of these materials also has a significant environmental and social impact. This includes threats to local ecosystems, deforestation, pollution, toxicity concerns and significant water consumption due to mining in arid areas, as well as an impact on local communities where good practices and workers' rights are not always respected.

### 10.4.3 Tracking net-zero progress

In July 2023, the International Energy Agency published a report tracking the progress of clean energy across 50 components (Tables 10.23 and 10.24). Only three components are on track according to the Net Zero by 2050 scenario: solar PV in the electricity sector, electric vehicles in transport, and lighting in buildings. Solar PV experienced annual growth of 26% in 2022, while electric vehicle sales surged by 55% the same year. Although few components are fully 'on track', significant progress is evident in parts of the energy system, driven largely by the availability of clean technologies and falling costs. However, progress remains uneven across regions and sectors. For instance, most
<table><caption>Table 10.22: Mineral requirements for clean energy technologies</caption><thead><tr><th></th><th>Aluminium</th><th>Chromium</th><th>Copper</th><th>Cobalt</th><th>Graphite</th><th>Lithium</th><th>Neodymium</th></tr></thead><tbody><tr><td>Bioenergy CSP</td><td>●</td><td>●</td><td>●</td><td>●</td><td>●</td><td>●</td><td>●</td></tr><tr><td>Electricity Networks</td><td>●</td><td>●</td><td>●</td><td>●</td><td>●</td><td>●</td><td>●</td></tr><tr><td>EVs and Battery storage</td><td>●</td><td>●</td><td>●</td><td>●</td><td>●</td><td>●</td><td>●</td></tr><tr><td>Geothermal</td><td>●</td><td>●</td><td>●</td><td>●</td><td>●</td><td>●</td><td>●</td></tr><tr><td>Hydrogen</td><td>●</td><td>●</td><td>●</td><td>●</td><td>●</td><td>●</td><td>●</td></tr><tr><td>Hydropower</td><td>●</td><td>●</td><td>●</td><td>●</td><td>●</td><td>●</td><td>●</td></tr><tr><td>Nuclear</td><td>●</td><td>●</td><td>●</td><td>●</td><td>●</td><td>●</td><td>●</td></tr><tr><td>Solar PV</td><td>●</td><td>●</td><td>●</td><td>●</td><td>●</td><td>●</td><td>●</td></tr><tr><td>Wind</td><td>●</td><td>●</td><td>●</td><td>●</td><td>●</td><td>●</td><td>●</td></tr></tbody></table><table><thead><tr><th></th><th>Nickel</th><th>Platinum</th><th>Polysilicon</th><th>REEs</th><th>Silver</th><th>Steel</th><th>Uranium</th><th>Zinc</th></tr></thead><tbody><tr><td>Bioenergy CSP</td><td>●</td><td>●</td><td>●</td><td>●</td><td>●</td><td>●</td><td>●</td><td>●</td></tr><tr><td>Electricity Networks</td><td>●</td><td>●</td><td>●</td><td>●</td><td>●</td><td>●</td><td>●</td><td>●</td></tr><tr><td>EVs and Battery storage</td><td>●</td><td>●</td><td>●</td><td>●</td><td>●</td><td>●</td><td>●</td><td>●</td></tr><tr><td>Geothermal</td><td>●</td><td>●</td><td>●</td><td>●</td><td>●</td><td>●</td><td>●</td><td>●</td></tr><tr><td>Hydrogen</td><td>●</td><td>●</td><td>●</td><td>●</td><td>●</td><td>●</td><td>●</td><td>●</td></tr><tr><td>Hydropower</td><td>●</td><td>●</td><td>●</td><td>●</td><td>●</td><td>●</td><td>●</td><td>●</td></tr><tr><td>Nuclear</td><td>●</td><td>●</td><td>●</td><td>●</td><td>●</td><td>●</td><td>●</td><td>●</td></tr><tr><td>Solar PV</td><td>●</td><td>●</td><td>●</td><td>●</td><td>●</td><td>●</td><td>●</td><td>●</td></tr><tr><td>Wind</td><td>●</td><td>●</td><td>●</td><td>●</td><td>●</td><td>●</td><td>●</td><td>●</td></tr></tbody></table>

Source: IEA (2022a, page 45) & Energy Transitions Commission (2023b, Exhibit 1.4, page 20).
Box 10.5: Supply and demand issues by material

* Demand for steel and aluminium is largely driven by non-energy transition sources. Between 2022 and 2050, the expected demand for steel is 170 Mt, less than 10% of current annual production. On the other hand, aluminium demand averages 30 Mt, which is 30% of current annual production. The supply of steel and aluminium will increase steadily until 2030, with much of the supply coming from recycling. However, the production of steel and aluminium is highly carbon intensive, so measures will need to be taken to decarbonize these materials.

* Demand for cobalt comes mainly from electric vehicles and is expected to increase 2-3 times by 2030. The current supply of 170 kt, mainly from the Democratic Republic of Congo (70%), cannot meet the future demand of 420 kt. Increased efficiency and new technologies could reduce demand by up to 40%, resulting in cobalt demand of 260 kt in 2030. Supply could increase to 245 kt due to expansion in Indonesia.

* Copper is probably the material with the highest demand, as it is required in most clean energy technologies. Current supply is less than the overwhelming future demand due to various challenges such as depletion of existing mines and lack of investment to expand production. However, a potential shift could reduce copper demand by 4 Mt in 2030 (from 37 Mt to 33 Mt) and measures on recycling and efficiency can increase supply in 2030 and close the gap (34 Mt).

* Graphite is a key component in battery production. The increasing demand for batteries in the transition period makes the current supply inadequate (1 Mt against 7 Mt). The possible use of silicon can help to slightly reduce the demand of 1 Mt. The supply gap can be closed somewhat by taking into account synthetic graphite, which will increase the graphite supply to 4 Mt in 2030, 40% lower than the lower bound demand of 6 Mt.

* Lithium demand is driven by electric vehicle batteries and will increase to a demand of 760 kt, completely overtaking the current supply of 125 kt. The use of Na-ion chemistries and a shift in battery sizes could help reduce demand by 25% to 560 kt. Global supply is expected to rise sharply to 510 kt by 2030, which is better than current supply but still short of demand.

* Neodymium is used in permanent magnets for EV motors and wind turbines. The current supply of neodymium does not match the primary demand in 2030 (50 kt in 2022 vs. 125 kt in 2030), even taking into account the potential reduction of 30% due to new technologies. However, there is potential to close the gap between supply and demand with the new mining projects underway in several countries, leading to a supply of 88 kt in 2030.

* Demand for nickel is high and diversified, ranging from electric vehicles and battery storage to geothermal and hydrogen electrolysis. The current supply of nickel does not match the growing demand, but the strong possibility of nickel-free batteries could help to reduce this demand from 5.5 Mt to 3.9 Mt in 2030 and allow supply (4.6 Mt) to exceed demand, thanks to the rapid and significant expansion of mining in Indonesia.
electric vehicle sales and carbon capture capacity are concentrated in China, the US, and Europe. More broadly, despite some progress, far more action is needed to achieve a low-carbon economy by 2050. This is the main conclusion of recent reports (IEA, 2024b; McKinsey, 2024b).

<table><caption>Table 10.23: What's on track (energy system overview)</caption><tr><td>Energy Efficiency</td><td>●</td></tr><tr><td>Behavioural Changes</td><td>●</td></tr><tr><td>Electrification</td><td>●</td></tr><tr><td>Renewables</td><td>●</td></tr><tr><td>Bioenergy</td><td>●</td></tr><tr><td>Hydrogen</td><td>●</td></tr><tr><td>Carbon Capture, Utilisation and Storage</td><td>●</td></tr><tr><td>Innovation</td><td>●</td></tr><tr><td>International Collaboration</td><td>●</td></tr><tr><td>Digitalisation</td><td>●</td></tr></table>

<table><caption>Table 10.24: What's on track (sector analysis)</caption><tbody><tr><td>Transversal Technologies & Infrastructure</td><td>●</td><td>Electricity</td><td>●</td></tr><tr><td>CO<sub>2</sub> Transport and Storage</td><td>●</td><td>Coal</td><td>●</td></tr><tr><td>CO<sub>2</sub> Capture and Utilisation</td><td>●</td><td>Natural Gas</td><td>●</td></tr><tr><td>Bioenergy with Carbon Capture and Storage</td><td>●</td><td>Solar PV</td><td>●</td></tr><tr><td>Direct Air Capture</td><td>●</td><td>Wind</td><td>●</td></tr><tr><td>Electrolysers</td><td>●</td><td>Hydroelectricity</td><td>●</td></tr><tr><td>District Heating</td><td>●</td><td>Demand Response</td><td>●</td></tr><tr><td>Data Centres and Transmission Networks</td><td>●</td><td>Nuclear Power</td><td>●</td></tr><tr><td>Transport</td><td>●</td><td>Grid-scale Storage</td><td>●</td></tr><tr><td>Cars and Vans</td><td>●</td><td>Smart Grids</td><td>●</td></tr><tr><td>Trucks and Buses</td><td>●</td><td>Energy</td><td>●</td></tr><tr><td>Rail</td><td>●</td><td>Oil & Natural Gas Supply</td><td>●</td></tr><tr><td>Aviation</td><td>●</td><td>Methane Abatement</td><td>●</td></tr><tr><td>International Shipping</td><td>●</td><td>Gas Flaring</td><td>●</td></tr><tr><td>Electric Vehicles</td><td>●</td><td>Biofuels</td><td>●</td></tr><tr><td>Industry</td><td>●</td><td>Buildings</td><td>●</td></tr><tr><td>Steel</td><td>●</td><td>Heating</td><td>●</td></tr><tr><td>Chemicals</td><td>●</td><td>Space Cooling</td><td>●</td></tr><tr><td>Cement</td><td>●</td><td>Lighting</td><td>●</td></tr><tr><td>Aluminium</td><td>●</td><td>Appliances & Equipment</td><td>●</td></tr><tr><td>Paper</td><td>●</td><td>Building Envelopes</td><td>●</td></tr><tr><td>Light Industry</td><td>●</td><td>Heat Pumps</td><td>●</td></tr></tbody></table>

● on track, ● more efforts needed, ● not on track

Source: IEA (2023a).
## 10.5 Exercises

### 10.5.1 Compensating and equivalent variation

We consider the following utility function:

$$u(x_1, x_2) = x_1^{\alpha} x_2^{\beta}$$

where $x_1$ and $x_2$ are the quantities of the goods 1 and 2, $\alpha > 0$ and $\beta > 0$. The budget constraint is:

$$p_1x_1 + p_2x_2 = y$$

where $p_1$ and $p_2$ are the prices of the goods and $y$ is the income.

1. Write the utility maximization problem of the consumer. Find the optimal solution $(x_1^*, x_2^*)$ and the maximum utility $u^*$.

2. Given a utility level $u$, we want to find the optimal solution that achieves this utility at a minimum cost. Write the cost minimization problem of the consumer. Find the optimal solution $(x_1^*, x_2^*)$ and the minimum cost $y^*$.

3. Compare the Marshallian and Hicksian demand functions.

4. Let us assume the equilibrium $(x_1^*, x_2^*, u^*)$. We assume that the price of good 1 decreases from $p_1$ to $\tilde{p}_1$. Find the solution $(\tilde{x}_1^*, \tilde{x}_2^*)$ and the minimum cost $\tilde{y}^*$ that correspond to utility $u^*$. Compute the compensation variation:

$$CV = y - \tilde{y}^*$$

5. Let us assume the equilibrium $(\bar{x}_1^*, \bar{x}_2^*, \bar{u}^*)$ when the price of good 1 decreases from $p_1$ to $\bar{p}_1$. Find the solution $(\bar{x}_1^*, \bar{x}_2^*)$ and the minimum cost $\bar{y}^*$ that correspond to utility $\bar{u}^*$ and price system $(p_1, p_2)$. Compute the equivalent variation:

$$EV = \bar{y}^* - y$$

6. Numerical application: $\alpha = 0.6, \beta = 0.4, p_1 = p_2 = 1, y = 3$ and $\tilde{p}_1 = \bar{p}_1 = 0.75$.

7. Numerical application: $\alpha = 0.5, \beta = 0.5, p_1 = p_2 = 1, y = 3$ and $\tilde{p}_1 = \bar{p}_1 = 1.75$.
Chapter 11

Climate Portfolio Construction

With the 2015 Paris Agreement, the development of ESG investing, and the emergence of net-
zero investment policies, climate risk is undoubtedly the most important issue and challenge for
asset owners and managers today and in the coming years. Building portfolios to manage climate
risk began in 2014, when asset owners AP4 and FRR worked with asset manager Amundi and
index provider MSCI to define an investment strategy to help hedge climate risk (Andersson et
al., 2016). Together, they defined the concept of a low-carbon portfolio. The 2015–2020 period
corresponds to the growth of ESG investing and the adoption of sustainable finance by many asset
owners and managers. During this period, climate investing is part of ESG investing, and climate
portfolio construction remains relatively marginal, essentially involving passive management. On
the contrary, ESG investing has seen a major development with the adoption of ESG scores in
active management. Since 2020, we have seen a new trend. The line between ESG investing and
climate investing is becoming increasingly blurred, and the two issues are now separate. One of the
reasons for this is the emergence of net-zero investment policies, which have profoundly changed the
investment decisions of asset owners. This separation has accelerated with COP26. The proliferation
of net zero alliances (GFANZ, NZAOA, NZAM, NZBA, etc.), the commitments made by financial
institutions (asset managers, banks, pension funds, insurance companies, etc.), and the push for
regulation<fnref n="1" /> are all contributing to the shift from ESG investing to climate investing.

This chapter is dedicated to portfolio construction when we integrate climate risk measures. It
is therefore closely related to Chapter 9, as we use carbon footprint and green footprint metrics.
It is also related to Chapter 10 on transition risk, as the goal of climate investing is to reduce the
transition risk of investment portfolios. Integrating physical risk is more complicated today, because
we do not have the right metrics to assess physical risk at the corporate or security level. Finally,
it is related to Chapter 2, which is dedicated to the impact of ESG investing on asset prices and
portfolio returns, because we use the same tools and methodologies. We will therefore make extensive
use of portfolio optimization. A comprehensive review of portfolio optimization is presented in the
first section. We distinguish between allocations to equity and fixed income portfolios because they
require two different approaches. The reason is that we generally measure equity risk in terms of
volatility risk, while bond risk is multi-dimensional and must at least integrate duration and credit
risk. The second section is a guide to building a low-carbon portfolio. While there are several
approaches, the choice of Scope emissions is certainly the most important decision and has a major
impact on asset allocation. Finally, the last section focuses on net-zero investing and lists many
challenges to defining an investment portfolio that is aligned with a net-zero emissions scenario.

<footnote n="1">¹Examples include the Net-Zero Industry Act of the European Commission, the work of the NGFS on climate scenarios, or the climate stress tests organized by the ECB.</footnote>
## 11.1 Portfolio optimization in practice

Before studying portfolio allocation in the context of climate risk, we first begin to remind some basics about portfolio optimization. As mentioned by Perrin and Roncalli (2020), the success of mean-variance optimization is due to the appealing properties of the quadratic utility function, and it is easy to solve numerically quadratic programming problems<fnref n="2" />. This is why most of the portfolio allocation problems that we will encounter in this chapter will be cast into a QP problem, whose standard formulation is<fnref n="3" />:

$$ 
\begin{aligned}
x^* &= \arg \min \frac{1}{2} x^\top Q x - x^\top R \\
\text{s.t.} \quad & \begin{cases} Ax = B \\ Cx \le D \\ x^- \le x \le x^+ \end{cases} 
\end{aligned} 
\qquad (11.1) 
$$

where $x$ is a $n \times 1$ vector, $Q$ is a $n \times n$ matrix, $R$ is a $n \times 1$ vector, $A$ is a $n_A \times n$ matrix, $B$ is a $n_A \times 1$ vector, $C$ is a $n_C \times n$ matrix, $D$ is a $n_C \times 1$ vector, and $x^-$ and $x^+$ are two $n \times 1$ vectors. If $n_A = 0$, there is no equality constraints. Similarly, there is no inequality constraints if $n_C = 0$. If there is no lower bounds or/and upper bounds, this implies that $x^- = -\infty \cdot \mathbf{1}_n$ and $x^+ = \infty \cdot \mathbf{1}_n$. From a numerical viewpoint, QP solvers generally replace these bounds by $x^- = -c \cdot \mathbf{1}_n$ and $x^+ = c \cdot \mathbf{1}_n$ where $c$ is a large floating-point number<fnref n="4" /> (e.g., $c = 10^{200}$).

### 11.1.1 Equity portfolios

#### Basic optimization problems

We consider a universe of $n$ assets. We note $w$ the vector of portfolio weights. Let $\mu$ and $\Sigma$ be the vector of expected returns and the covariance matrix of asset returns. The long-only mean-variance optimization problem is given by:

$$ 
\begin{aligned}
w^* &= \arg \min \frac{1}{2} w^\top \Sigma w - \gamma w^\top \mu \\
\text{s.t.} \quad & \begin{cases} \mathbf{1}_n^\top w = 1 \\ \mathbf{0}_n \le w \le \mathbf{1}_n \end{cases}
\end{aligned} 
$$

where $\gamma$ is the risk-tolerance coefficient, the equality constraint is the budget constraint ($\sum_{i=1}^n w_i = 1$) and the bounds correspond to the no short-selling restriction ($w_i \ge 0$). We recognize a QP problem where $Q = \Sigma$, $R = \gamma\mu$, $A = \mathbf{1}_n^\top$, $B = 1$, $w^- = \mathbf{0}_n$ and $w^+ = \mathbf{1}$. In this problem, we have one equality constraint ($n_A = 1$) and zero inequality constraint<fnref n="5" /> ($n_C = 1$).

In many problems, we will have to manage the portfolio with respect to a benchmark. In this case, the objective function depends on the tracking error risk variance $\sigma^2(w|b) = (w-b)^\top \Sigma (w-b)$ where $b$ is the vector of benchmark weights. In Section 3.1.1 on page 148, we have seen that the tracking error optimization problem is defined as:

$$ 
\begin{aligned}
w^* &= \arg \min \frac{1}{2} w^\top \Sigma w - w^\top (\gamma \mu + \Sigma b) \\
\text{s.t.} \quad & \begin{cases} \mathbf{1}_n^\top w = 1 \\ \mathbf{0}_n \le w \le \mathbf{1}_n \end{cases}
\end{aligned} 
$$

<footnote n="2">²See Appendix A.1.2 on page 1144.</footnote>

<footnote n="3">³The objective function is a quadratic form and is noted $QF(x; Q, R, \mathbf{0}_n)$.</footnote>

<footnote n="4">⁴The largest finite floating-point number in IEEE double precision is equal to $(2 - 2^{-52}) \times 2^{1023}$.</footnote>

<footnote n="5">⁵We do not take into account the bounds.</footnote>
This is exactly the same QP problem as previously except that $R = \gamma\mu + \Sigma b$. If the objective of the portfolio manager is to minimize the tracking error risk, we obtain $R = \Sigma b$.

## Specification of the constraints

We can extend the previous framework by considering more constraints. For instance, we consider a sector weight constraint:

$$s_j^- \le \sum_{i \in \mathcal{S}ector_j} w_i \le s_j^+$$

We notice that:

$$\sum_{i \in \mathcal{S}ector_j} w_i = \mathbf{s}_j^\top w$$

where $\mathbf{s}_j$ is the $n \times 1$ sector-mapping vector whose elements are $\mathbf{s}_{i,j} = 1 \{i \in \mathcal{S}ector_j\}$. We deduce that the sector constraint can be written as:

$$s_j^- \le \sum_{i \in \mathcal{S}ector_j} w_i \le s_j^+ \Leftrightarrow \left\{ \begin{array}{l} s_j^- \le \mathbf{s}_j^\top w \\ \mathbf{s}_j^\top w \le s_j^+ \end{array} \right. \Leftrightarrow \left\{ \begin{array}{l} -\mathbf{s}_j^\top w \le -s_j^- \\ \mathbf{s}_j^\top w \le s_j^+ \end{array} \right.$$

It follows that the inequality constraint $Cw \le D$ is defined by the following system:

$$\underbrace{\left( \begin{array}{c} -\mathbf{s}_j^\top \\ \mathbf{s}_j \end{array} \right)}_C w \le \underbrace{\left( \begin{array}{c} -s_j^- \\ s_j^+ \end{array} \right)}_D$$

In this case, $C$ is a $2 \times n$ matrix and $D$ is a $2 \times 1$ vector. The previous analysis can be extended when there are many sectors.

We denote by $\mathcal{S}$ a vector of scores (e.g., ESG scores) and we would like to impose that the (linear) score of the portfolio is greater than a threshold $\mathcal{S}^*$:

$$\sum_{i=1}^{n} w_i \mathcal{S}_i \ge \mathcal{S}^*$$

The QP form of this constraint is:

$$-\mathcal{S}^\top w \le -\mathcal{S}^*$$

Let us now assume that we would like to apply this constraint to a sector. In this case, we have:

$$\sum_{i \in \mathcal{S}ector_j} w_i \mathcal{S}_i \ge \mathcal{S}_j^* \Leftrightarrow \sum_{i=1}^{n} 1 \{i \in \mathcal{S}ector_j\} w_i \mathcal{S}_i \ge \mathcal{S}_j^* \\ \Leftrightarrow \sum_{i=1}^{n} \mathbf{s}_{i,j} w_i \mathcal{S}_i \ge \mathcal{S}_j^* \\ \Leftrightarrow \sum_{i=1}^{n} w_i (\mathbf{s}_{i,j} \mathcal{S}_i) \ge \mathcal{S}_j^* \\ \Leftrightarrow (\mathbf{s}_j \circ \mathcal{S})^\top w \ge \mathcal{S}_j^*$$

where $a \circ b$ is the Hadamard product: $(a \circ b)_i = a_i b_i$. The QP form of the sector-specific score constraint is defined by $C = -(\mathbf{s}_j \circ \mathcal{S})^\top$ and $D = -\mathcal{S}_j^*$.
**Example 52** We consider a capitalization-weighted equity index, which is composed of 8 stocks. The weights are equal to 23%, 19%, 17%, 13%, 9%, 8%, 6% and 5%. We assume that the stock volatilities are equal to 22%, 20%, 25%, 18%, 35%, 23%, 13% and 29%. The correlation matrix is given by:

$$ C = \begin{pmatrix} 100\% & & & & & & & \\ & 100\% & & & & & & \\ & & 70\% & 75\% & 100\% & & & \\ & & & 60\% & 65\% & 80\% & 100\% & & \\ & & & & 70\% & 50\% & 70\% & 85\% & 100\% \\ & & & & & 50\% & 60\% & 70\% & 80\% & 60\% & 100\% \\ & & & & & & 70\% & 50\% & 70\% & 75\% & 80\% & 50\% & 100\% \\ & & & & & & & 60\% & 65\% & 70\% & 75\% & 65\% & 70\% & 80\% & 100\% \end{pmatrix} $$

The ESG score, carbon intensity and sector of the eight stocks are the following:

<table><thead><tr><th>Stock</th><th>#1</th><th>#2</th><th>#3</th><th>#4</th><th>#5</th><th>#6</th><th>#7</th><th>#8</th></tr></thead><tbody><tr><td><b>S</b></td><td>-1.20</td><td>0.80</td><td>2.75</td><td>1.60</td><td>-2.75</td><td>-1.30</td><td>0.90</td><td>-1.70</td></tr><tr><td><b>CI</b></td><td>125</td><td>75</td><td>254</td><td>822</td><td>109</td><td>17</td><td>341</td><td>741</td></tr><tr><td><b>Sector</b></td><td>1</td><td>1</td><td>2</td><td>2</td><td>1</td><td>2</td><td>1</td><td>2</td></tr></tbody></table>

The objective function is minimizing the tracking error risk. We deduce that the equivalent QP problem is:

$$ \begin{align*} w^* &= \arg \min \frac{1}{2} w^\top Q w - w^\top R \\ \text{s.t.} \quad & \begin{cases} Aw = B \\ Cw \leq D \\ w^- \leq w \leq w^+ \end{cases} \end{align*} $$

where<fnref n="6" />:

$$ Q = \Sigma = \begin{pmatrix} 484.00 & 352.00 & 385.00 & 237.60 & 539.00 & 253.00 & 200.20 & 382.80 \\ 352.00 & 400.00 & 375.00 & 234.00 & 350.00 & 276.00 & 130.00 & 377.00 \\ 385.00 & 375.00 & 625.00 & 360.00 & 612.50 & 402.50 & 227.50 & 507.50 \\ 237.60 & 234.00 & 360.00 & 324.00 & 535.50 & 331.20 & 175.50 & 391.50 \\ 539.00 & 350.00 & 612.50 & 535.50 & 1225.00 & 483.00 & 364.00 & 659.75 \\ 253.00 & 276.00 & 402.50 & 331.20 & 483.00 & 529.00 & 149.50 & 466.90 \\ 200.20 & 130.00 & 227.50 & 175.50 & 364.00 & 149.50 & 169.00 & 301.60 \\ 382.80 & 377.00 & 507.50 & 391.50 & 659.75 & 466.90 & 301.60 & 841.00 \end{pmatrix} \times 10^{-4} $$

and:

$$ R = \Sigma b = \begin{pmatrix} 3.74 \\ 3.31 \\ 4.39 \\ 3.07 \\ 5.68 \\ 3.40 \\ 2.02 \\ 4.54 \end{pmatrix} \times 10^{-2} $$

<footnote n="6">⁶We have $\Sigma_{i,j} = C_{i,j}\sigma_i\sigma_j$.</footnote>
We assume that the portfolio is long-only. It follows that $w^{-} = 0_{8}$ and $w^{+} = 1_{8}$. To satisfy the budget constraint $\sum_{i=1}^{8} w_{i} = 1$, we have a first linear equation $A_{0}w = B_{0}$ where $A_{0} = 1_{8}^{\top}$ and $B_{0} = 1$. We consider three type of constraints:

* We impose a relative reduction of the benchmark carbon intensity:

$$ \mathcal{CI}(w) \leq (1 - \mathcal{R}) \mathcal{CI}(b) $$

where $\mathcal{R}$ is the reduction rate. Since $\mathcal{CI}(w) = \mathcal{CI}^{\top}w$, we deduce the following inequality constraint $C_{1}w \leq D_{1}$ where $C_{1} = \mathcal{CI}^{\top}$ and $D_{1} = (1 - \mathcal{R}) \mathcal{CI}(b)$.

* We impose an absolute increase of the benchmark ESG score:

$$ \mathcal{S}(w) \geq \mathcal{S}(b) + \Delta\mathcal{S}^{*} $$

Since $\mathcal{S}(w) = \mathcal{S}^{\top}w$, we deduce the following inequality constraint $C_{2}w \leq D_{2}$ where $C_{2} = -\mathcal{S}^{\top}$ and $D_{2} = -(\mathcal{S}(b) + \Delta\mathcal{S}^{*})$.

* We impose the sector neutrality of the portfolio meaning that:

$$ \sum_{i \in \mathcal{S}ector_{j}} w_{i} = \sum_{i \in \mathcal{S}ector_{j}} b_{i} $$

We note:

$$ A_{1} = \mathbf{s}_{1}^{\top} = (1 \ 1 \ 0 \ 0 \ 1 \ 0 \ 1 \ 0) $$

and:

$$ A_{2} = \mathbf{s}_{2}^{\top} = (0 \ 0 \ 1 \ 1 \ 0 \ 1 \ 0 \ 1) $$

We compute $B_{1} = \mathbf{s}_{1}^{\top}b = \sum_{i \in \mathcal{S}ector_{1}} b_{i}$ and $B_{2} = \mathbf{s}_{2}^{\top}b = \sum_{i \in \mathcal{S}ector_{2}} b_{i}$. The sector neutrality constraint can be written as:

$$ \begin{pmatrix} A_{1} \\ A_{2} \end{pmatrix} w = \begin{pmatrix} B_{1} \\ B_{2} \end{pmatrix} $$

Let us now combine the different constraints. For that, we use the block matrix notation, which is particularly convenient when manipulating nested QP problems. The set #1 of constraint corresponds to the reduction of the carbon intensity, the set #2 corresponds to the ESG score improvement, the set #3 combines the two constraints and we add the sector neutrality in the set #4 of constraints.

<table><thead><tr><th>Set of constraints</th><th>Carbon intensity</th><th>ESG score</th><th>Sector neutrality</th><th>A</th><th>B</th><th>C</th><th>D</th></tr></thead><tbody><tr><td>#1</td><td>✓</td><td></td><td></td><td>A<sub>0</sub></td><td>B<sub>0</sub></td><td>C<sub>1</sub></td><td>D<sub>1</sub></td></tr><tr><td>#2</td><td></td><td>✓</td><td></td><td>A<sub>0</sub></td><td>B<sub>0</sub></td><td>C<sub>2</sub></td><td>D<sub>2</sub></td></tr><tr><td>#3</td><td>✓</td><td>✓</td><td></td><td>A<sub>0</sub></td><td>B<sub>0</sub></td><td>C<sub>1</sub><br>C<sub>2</sub></td><td>D<sub>1</sub><br>D<sub>2</sub></td></tr><tr><td>#4</td><td>✓</td><td>✓</td><td>✓</td><td>A<sub>0</sub><br>A<sub>1</sub><br>A<sub>2</sub></td><td>B<sub>0</sub><br>B<sub>1</sub><br>B<sub>2</sub></td><td>C<sub>1</sub><br>C<sub>2</sub></td><td>D<sub>1</sub><br>D<sub>2</sub></td></tr></tbody></table>
We can now solve the QP problem with the right specification of matrices A, B, C and D for each set of constraints. Results are reported below for the following parameters<fnref n="7" />: **R** = 30% and **ΔS**\* = 0.50. We indicate the optimal weights **w**\*, the tracking error volatility σ (w\* | b), the carbon intensity **CI** (w\*), the effective reduction rate<fnref n="8" /> **R** (w\* | b), the ESG score **S** (w\*), the ESG score variation **S** (w\*) − **S** (b), the weights<fnref n="9" /> w\* (**Sector**₁) and w\* (**Sector**₂) of sectors 1 and 2.

<table><thead><tr><th></th><th>Benchmark</th><th>Set #1</th><th>Set #2</th><th>Set #3</th><th>Set #4</th></tr></thead><tbody><tr><td rowspan="7">Weights (in %)</td><td>w<sub>1</sub><sup>*</sup></td><td>23.00</td><td>18.17</td><td>25.03</td><td>8.64</td><td>12.04</td></tr><tr><td>w<sub>2</sub><sup>*</sup></td><td>19.00</td><td>24.25</td><td>14.25</td><td>29.27</td><td>23.76</td></tr><tr><td>w<sub>3</sub><sup>*</sup></td><td>17.00</td><td>16.92</td><td>21.95</td><td>26.80</td><td>30.55</td></tr><tr><td>w<sub>4</sub><sup>*</sup></td><td>13.00</td><td>2.70</td><td>27.30</td><td>1.48</td><td>2.25</td></tr><tr><td>w<sub>5</sub><sup>*</sup></td><td>9.00</td><td>12.31</td><td>3.72</td><td>10.63</td><td>8.51</td></tr><tr><td>w<sub>6</sub><sup>*</sup></td><td>8.00</td><td>11.23</td><td>1.34</td><td>6.30</td><td>10.20</td></tr><tr><td>w<sub>7</sub><sup>*</sup></td><td>6.00</td><td>11.28</td><td>1.68</td><td>16.87</td><td>12.69</td></tr><tr><td>w<sub>8</sub><sup>*</sup></td><td>5.00</td><td>3.15</td><td>4.74</td><td>0.00</td><td>0.00</td></tr><tr><td rowspan="7">Statistics</td><td>σ (w<sup>*</sup> | b) (in %)</td><td>0.00</td><td>0.50</td><td>1.18</td><td>1.90</td><td>2.12</td></tr><tr><td><b>CI</b> (w<sup>*</sup>)</td><td>261.72</td><td>183.20</td><td>367.25</td><td>183.20</td><td>183.20</td></tr><tr><td><b>R</b> (w<sup>*</sup> | b) (in %)</td><td></td><td>30.00</td><td>-40.32</td><td>30.00</td><td>30.00</td></tr><tr><td><b>S</b> (w<sup>*</sup>)</td><td>0.17</td><td>0.05</td><td>0.67</td><td>0.67</td><td>0.67</td></tr><tr><td><b>S</b> (w<sup>*</sup>) − <b>S</b> (b)</td><td></td><td>-0.12</td><td>0.50</td><td>0.50</td><td>0.50</td></tr><tr><td>w<sup>*</sup> (<b>S</b>ector<sub>1</sub>) (in %)</td><td>57.00</td><td>66.00</td><td>44.67</td><td>65.41</td><td>57.00</td></tr><tr><td>w<sup>*</sup> (<b>S</b>ector<sub>2</sub>) (in %)</td><td>43.00</td><td>34.00</td><td>55.33</td><td>34.59</td><td>43.00</td></tr></tbody></table>

If we only reduce the carbon intensity, the tracking error cost is 50 bps. Nevertheless, the ESG score of the optimized portfolio is below the ESG score of the benchmark (0.05 vs. 0.17). If we improve the ESG score by 0.50, the tracking error cost is 118 bps. This optimal portfolio has a carbon footprint which is higher than this of the benchmark (367.25 vs. 261.72). If we combine the two constraints, the tracking error cost is close to 2%. We observe a reallocation between the sectors. For instance, sector 1 represents 65.41% of the portfolio while its weight is equal to 57% in the benchmark. Finally, the tracking error volatility of the optimal portfolio is equal to 2.12% when we add the sector neutrality constraint.

## Dealing with constraints on relative weights

Let us assume that we would like to reduce the carbon footprint at the sector level. In this case, we denote by **CI** (w; **Sector**<sub>j</sub>) the carbon intensity of the jth sector within the portfolio w:

$$ \mathbf{CI}(w; \mathbf{Sector}_j) = \sum_{i \in \mathbf{Sector}_j} \tilde{w}_i \mathbf{CI}_i $$

where $\tilde{w}_i$ is the normalized weight in the sector bucket:

$$ \tilde{w}_i = \frac{w_i}{\sum_{k \in \mathbf{Sector}_j} w_k} $$

$$ \mathbf{R}(w | b) = \frac{\mathbf{CI}(b) - \mathbf{CI}(w)}{\mathbf{CI}(b)} $$

$$ w(\mathbf{Sector}_j) = \sum_{i \in \mathbf{Sector}_j} w_i = \mathbf{s}_j^\top w $$

<footnote n="7">⁷We have $B_1 = 0.57$, $B_2 = 0.43$, $D_1 = 183.2040$ and $D_2 = -0.6690$.</footnote>

<footnote n="8">⁸We have:</footnote>

<footnote n="9">⁹We have:</footnote>
Another expression of $\mathcal{CI}(w; \mathbf{Sector}_j)$ is:

$$ \mathcal{CI}(w; \mathbf{Sector}_j) = \frac{\sum_{i \in \mathbf{Sector}_j} w_i \mathcal{CI}_i}{\sum_{i \in \mathbf{Sector}_j} w_i} = \frac{(\mathbf{s}_j \circ \mathcal{CI})^\top w}{\mathbf{s}_j^\top w} $$

If we consider the constraint $\mathcal{CI}(w; \mathbf{Sector}_j) \le \mathcal{CI}_j^*$, we obtain:

$$ \begin{aligned} (*) & \Leftrightarrow \mathcal{CI}(w; \mathbf{Sector}_j) \le \mathcal{CI}_j^* \\ & \Leftrightarrow (\mathbf{s}_j \circ \mathcal{CI})^\top w \le \mathcal{CI}_j^* (\mathbf{s}_j^\top w) \\ & \Leftrightarrow ((\mathbf{s}_j \circ \mathcal{CI}) - \mathcal{CI}_j^* \mathbf{s}_j)^\top w \le 0 \\ & \Leftrightarrow (\mathbf{s}_j \circ (\mathcal{CI} - \mathcal{CI}_j^*))^\top w \le 0 \end{aligned} \quad (11.2) $$

The QP form is then $C = (\mathbf{s}_j \circ (\mathcal{CI} - \mathcal{CI}_j^*))^\top$ and $D = 0$. We remark that we obtain a classical reduction constraint where the vector of carbon intensities is replaced by the vector of deviations $\mathcal{CI} - \mathcal{CI}_j^*$.

**Remark 132** We have defined an absolute threshold $\mathcal{CI}_j^*$, but we can implement a relative threshold by setting $\mathcal{CI}_j^* = (1 - \mathbf{R}_j) \mathcal{CI}(b; \mathbf{Sector}_j)$.

Let us consider again Example 52 on page 1042. We would like to reduce the carbon footprint of the benchmark by 30% and impose the sector neutrality. In this case, the linear equality constraint $Aw = B$ is defined by:

$$ A = \begin{pmatrix} 1 & 1 & 1 & 1 & 1 & 1 & 1 \\ 1 & 1 & 0 & 0 & 1 & 0 & 1 \\ 0 & 0 & 1 & 1 & 0 & 1 & 0 \end{pmatrix} $$

and:

$$ B = \begin{pmatrix} 100\% \\ 57\% \\ 43\% \end{pmatrix} $$

For the linear inequality constraint $Cw \le D$, we obtain:

$$ C = ( 125 \ 75 \ 254 \ 822 \ 109 \ 17 \ 341 \ 741 ) $$

and:

$$ D = 183.2040 $$

The optimal solution is $w^* = (21.54\%, 18.50\%, 21.15\%, 3.31\%, 10.02\%, 15.26\%, 6.94\%, 3.27\%)$. The tracking error volatility $\sigma(w^* | b)$ is equal to 112 bps. We verify that the carbon intensity of $w^*$ is equal to 183.20, which corresponds to a reduction rate of 30% compared to the carbon intensity of the benchmark — $\mathcal{CI}(b) = 261.72$. Nevertheless, we observe that the sector breakdown of the carbon footprint is the following:

$$ \left\{ \begin{array}{l} \mathcal{CI}(w^*; \mathbf{Sector}_1) = 132.25 \\ \mathcal{CI}(w^*; \mathbf{Sector}_2) = 250.74 \end{array} \right. $$

versus

$$ \left\{ \begin{array}{l} \mathcal{CI}(b; \mathbf{Sector}_1) = 128.54 \\ \mathcal{CI}(b; \mathbf{Sector}_2) = 438.26 \end{array} \right. $$

It follows that the 30% global reduction is explained by an increase of 2.89% of the carbon footprint for the first sector and a decrease of 42.79% of the carbon footprint for the second sector. We deduce that the decarbonization of this portfolio is only supported by the second sector. This is
why we impose that the first sector must reduce its carbon footprint by $\mathbf{R}_1 = 20\%$. In this case, the inequality system becomes:

$$ C = \begin{pmatrix} \mathcal{C}\mathcal{I}^{\top} \\ (\mathbf{s}_1 \circ (\mathcal{C}\mathcal{I} - (1 - \mathbf{R}_1)\mathcal{C}\mathcal{I}(b; \mathbf{S} \mathbf{e}ctor_1)))^{\top} \end{pmatrix} = \begin{pmatrix} 125 & 75 & 254 & 822 & 109 & 17 & 341 & 741 \\ 22.1649 & -27.8351 & 0 & 0 & 6.1649 & 0 & 238.1649 & 0 \end{pmatrix} $$

and:

$$ D = \begin{pmatrix} 183.2040 \\ 0 \end{pmatrix} $$

Solving this new QP problem gives the following optimal portfolio:

$$ w^* = (22.70\%, 22.67\%, 19.23\%, 5.67\%, 11.39\%, 14.50\%, 0.24\%, 3.61\%) $$

The tracking error volatility $\sigma(w^*|b)$ becomes 144 bps. We verify that the carbon intensity of $w^*$ is equal to 183.20. Moreover, we obtain $\mathcal{C}\mathcal{I}(w^*; \mathbf{S} \mathbf{e}ctor_1) = 102.84$ and $\mathcal{C}\mathcal{I}(w^*; \mathbf{S} \mathbf{e}ctor_2) = 289.74$, i.e., a reduction of 20% and 33.89% respectively.

### 11.1.2 Bond portfolios

#### Risk measure of a bond portfolio

There is a large consensus to use the historical volatility as a risk measure for an equity portfolio, even if it is not perfect and has many drawbacks. For instance, skewness and kurtosis risks are not taken into account, correlations are time-varying, we generally observe jumps in the time series of stock returns, volatility is heteroscedastic, etc. In the case of fixed-income portfolios, the historical volatility is not a relevant risk measure, because the bond volatility depends on duration, credit spread, liquidity, etc. Let us consider a zero-coupon bond, whose price and maturity date are $B(t, T)$ and $T$. Following Roncalli (2013, page 225), we have:

$$ B_t(t, T) = e^{-(r(t)+s(t))(T-t)+L(t)} $$

where $r(t)$, $s(t)$ and $L(t)$ are the interest rate, the credit spread and the liquidity premium. We deduce that:

$$ \begin{aligned} \mathrm{d} \ln B(t, T) &= -(T-t) \mathrm{d}r(t) - (T-t) \mathrm{d}s(t) + \mathrm{d}L(t) \\ &= -\mathrm{D} \mathrm{d}r(t) - (\mathrm{D} \mathrm{s}(t)) \frac{\mathrm{d}s(t)}{\mathrm{s}(t)} + \mathrm{d}L(t) \\ &= -\mathrm{D} \mathrm{d}r(t) - \mathrm{DTS}(t) \frac{\mathrm{d}s(t)}{\mathrm{s}(t)} + \mathrm{d}L(t) \end{aligned} $$

where $D = T - t$ is the remaining maturity (or duration) and $\mathrm{DTS}(t)$ is the duration-times-spread factor. If we assume that $r(t)$, $s(t)$ and $L(t)$ are independent, the risk of the defaultable bond is equal to:

$$ \sigma^2(\mathrm{d} \ln B(t, T)) = D^2 \sigma^2(\mathrm{d}r(t)) + \mathrm{DTS}(t)^2 \sigma^2\left(\frac{\mathrm{d}s(t)}{\mathrm{s}(t)}\right) + \sigma^2(\mathrm{d}L(t)) $$

It can then be decomposed into three components: an interest-rate risk component, a credit risk component and a liquidity risk component. In the case where the three volatility risks (interest rate, credit spread and liquidity) are constant, we obtain:

$$ \sigma^2(\mathrm{d} \ln B(t, T)) = D^2 \sigma_r^2 + \mathrm{DTS}(t)^2 \sigma_s^2 + \sigma_L^2 $$
We see that the bond risk depends not only on interest rate, credit and liquidity volatilities but it varies with respect to the duration and the level of the credit spread. Therefore, we confirm that the historical volatility of a bond price is not a relevant risk measure.

**Remark 133** In the sequel, we do not consider the liquidity risk, since it is generally measured by proprietary liquidity scores (Ben Slimane and De Jong, 2017).

Basic optimization problems

In practice, the duration risk is measured using the modified duration (MD) metric. We have:

$$MD(w) = \sum_{i=1}^{n} w_i MD_i$$

The portfolio DTS is equal to:

$$DTS(w) = \sum_{i=1}^{n} w_i DTS_i$$

Generally, the construction of a bond portfolio uses the clustering approach. The bond universe is divided into clusters based on the currency, the sector, the credit quality and the maturity band. For instance, the cluster (EUR, Financials, AAA to A-, 1Y-3Y) corresponds to euro-denominated bonds from the financials sector, whose rating is greater than A- and the duration is between one and three years. A cluster is then a generalization of the sector concept. Nevertheless, we will continue to use the term sector instead of the term cluster in the sequel when we will build bond portfolios. Let $MD_j(w)$ and $DTS_j(w)$ be the MD and DTS contributions<fnref n="10" /> of the jth sector. We have $MD_j(w) = \sum_{i \in S_{sector_j}} w_i MD_i$ and $DTS_j(w) = \sum_{i \in S_{sector_j}} w_i DTS_i$. Bond portfolio optimization consists in maximizing the expected carry of the portfolio under a set of numerous constraints:

$$\Omega = \{j = 1, \dots, n_{S_{sector}} : MD_j(w) \approx MD_j^*, DTS_j(w) \approx DTS_j^*\}$$

Therefore, the objective function of the optimization problem has the following form:

$$w^* = \arg \min \frac{\varphi_{MD}}{2} \sum_{j=1}^{n_{S_{sector}}} (MD_j(w) - MD_j^*)^2 + \frac{\varphi_{DTS}}{2} \sum_{j=1}^{n_{S_{sector}}} (DTS_j(w) - DTS_j^*)^2 - \gamma \sum_{i=1}^{n} w_i C_i \quad (11.3)$$

where $\varphi_{MD} \ge 0$ and $\varphi_{DTS} \ge 0$ indicate the relative weight of each risk component, $C_i$ is the expected carry<fnref n="11" /> of bond i and $\gamma$ is the risk-tolerance coefficient. In order to solve this optimization problem, the goal is to transform Equation (11.3) into a QP problem:

$$\begin{array}{l} w^* = \arg \min \mathcal{QF}(w; Q, R, c) \qquad (11.4) \\ \text{s.t.} \quad \begin{cases} \mathbf{1}_n^\top w = 1 \\ \mathbf{0}_n \le w \le \mathbf{1}_n \end{cases} \end{array}$$

where $\mathcal{QF}(w; Q, R, c)$ is the quadratic form of the objective function.

<footnote n="10">¹⁰Be careful, they do not measure the MD and DTS of the sector, which are measured with relative weights within the sector.</footnote>

<footnote n="11">¹¹The expected carry of a bond corresponds to the carry of the bond plus its expected mark-to-market return minus the expected credit loss.</footnote>
We use the properties of quadratic forms presented in Section A.1.3 on page 1146. We have<fnref n="12" />:

$$
\begin{align*}
\frac{1}{2} (\mathrm{MD}_j(w) - \mathrm{MD}_j^\star)^2 &= \frac{1}{2} \left( \sum_{i \in \mathcal{S}_{\mathrm{sector}}_j} w_i \mathrm{MD}_i - \mathrm{MD}_j^\star \right)^2 \\
&= \frac{1}{2} \left( \sum_{i=1}^n \mathbf{s}_{i,j} w_i \mathrm{MD}_i - \mathrm{MD}_j^\star \right)^2 \\
&= \frac{1}{2} \left( \sum_{i=1}^n \mathbf{s}_{i,j} \mathrm{MD}_i w_i \right)^2 - w^\top (\mathbf{s}_j \circ \mathrm{MD}) \mathrm{MD}_j^\star + \frac{1}{2} \mathrm{MD}_j^{\star 2} \\
&= \mathcal{Q}\mathcal{F} \left( w; \mathcal{T}(\mathbf{s}_j \circ \mathrm{MD}), (\mathbf{s}_j \circ \mathrm{MD}) \mathrm{MD}_j^\star, \frac{1}{2} \mathrm{MD}_j^{\star 2} \right)
\end{align*}
$$

where MD = (MD₁, ..., MDₙ) is the vector of modified durations. It follows that:

$$
\frac{1}{2} \sum_{j=1}^{n_{\mathcal{S}_{\text{sector}}}} (\mathrm{MD}_j(w) - \mathrm{MD}_j^\star)^2 = \mathcal{Q}\mathcal{F}(w; Q_{\mathrm{MD}}, R_{\mathrm{MD}}, c_{\mathrm{MD}})
$$

where:

$$
\left\{
\begin{aligned}
Q_{\text{MD}} &= \sum_{j=1}^{n_{\mathcal{S}_{\text{sector}}}} \mathcal{T}(\mathbf{s}_j \circ \text{MD}) \\
R_{\text{MD}} &= \sum_{j=1}^{n_{\mathcal{S}_{\text{sector}}}} (\mathbf{s}_j \circ \text{MD}) \text{MD}_j^* \\
c_{\text{MD}} &= \frac{1}{2} \sum_{j=1}^{n_{\mathcal{S}_{\text{sector}}}} \text{MD}_j^{*2}
\end{aligned}
\right.
$$

In a similar way, we have:

$$
\frac{1}{2} \sum_{j=1}^{n_{\mathcal{S}_{\text{sector}}}} (\text{DTS}_j(w) - \text{DTS}_j^\star)^2 = \mathcal{Q}\mathcal{F}(w; Q_{\text{DTS}}, R_{\text{DTS}}, c_{\text{DTS}})
$$

where:

$$
\left\{
\begin{aligned}
Q_{\text{DTS}} &= \sum_{j=1}^{n_{\mathcal{S}_{\text{sector}}}} \mathcal{T}(\mathbf{s}_j \circ \text{DTS}) \\
R_{\text{MD}} &= \sum_{j=1}^{n_{\mathcal{S}_{\text{sector}}}} (\mathbf{s}_j \circ \text{DTS}) \text{DTS}_j^* \\
c_{\text{DTS}} &= \frac{1}{2} \sum_{j=1}^{n_{\mathcal{S}_{\text{sector}}}} \text{DTS}_j^{*2}
\end{aligned}
\right.
$$

where DTS = (DTS₁, ..., DTSₙ) is the vector of DTS values. Since we have −γ ∑n i=1 wᵢCᵢ = QF(w; 0ₙ,ₙ, γC, 0) where C = (C₁, ..., Cₙ) is the vector of expected carry values, we conclude that
the function to optimize is:

$$
\begin{align*}
\mathcal{Q}\mathcal{F}(w; Q, R, c) ={}& \varphi_{\text{MD}}\mathcal{Q}\mathcal{F}(w; Q_{\text{MD}}, R_{\text{MD}}, c_{\text{MD}}) + \varphi_{\text{DTS}}\mathcal{Q}\mathcal{F}(w; Q_{\text{DTS}}, R_{\text{DTS}}, c_{\text{DTS}}) + \\
& \mathcal{Q}\mathcal{F}(w; \mathbf{0}_{n,n}, \gamma C, 0)
\end{align*}
$$

<footnote n="12">¹²We remind that $\mathcal{T}(u) = uu^\top$.</footnote>
where:

$$
\left\{
\begin{array}{l}
Q = \varphi_{\text{MD}} Q_{\text{MD}} + \varphi_{\text{DTS}} Q_{\text{DTS}} \\
R = \gamma C + \varphi_{\text{MD}} R_{\text{MD}} + \varphi_{\text{DTS}} R_{\text{DTS}} \\
c = \varphi_{\text{MD}} c_{\text{MD}} + \varphi_{\text{DTS}} c_{\text{DTS}}
\end{array}
\right.
$$

We can extend the previous analysis when there is a benchmark. In this case, the thresh-
old values MD\*<sub>j</sub> and DTS\*<sub>j</sub> are equal to the MD and DTS contributions of the benchmark:
MD\*<sub>j</sub> = &Sigma;<sub>i&isin;S<sub>sector<sub>j</sub></sub></sub> b<sub>i</sub> MD<sub>i</sub> and DTS\*<sub>j</sub> = &Sigma;<sub>i&isin;S<sub>sector<sub>j</sub></sub></sub> b<sub>i</sub> DTS<sub>i</sub>. The MD- and DTS-based tracking
error variances are then equal to:

$$
\mathcal{R}_{\text{MD}}(w | b) = \sigma_{\text{MD}}^2(w | b) = \sum_{j=1}^{n_{\mathcal{S}_{\text{sector}}}} \left( \sum_{i \in \mathcal{S}_{\text{sector}_j}} (w_i - b_i) \text{MD}_i \right)^2
$$

and:

$$
\mathcal{R}_{\text{DTS}}(w | b) = \sigma_{\text{DTS}}^2(w | b) = \sum_{j=1}^{n_{\mathcal{S}_{\text{sector}}}} \left( \sum_{i \in \mathcal{S}_{\text{sector}_j}} (w_i - b_i) \text{DTS}_i \right)^2
$$

Generally, we also consider a third active risk component called the active share risk<fnref n="13" />:

$$
\mathcal{R}_{\text{AS}}(w | b) = \sigma_{\text{AS}}^2(w | b) = \sum_{i=1}^{n} (w_i - b_i)^2
$$

The basic optimization problem becomes:

$$
\begin{equation}
\begin{split}
w^* &= \arg \min \frac{1}{2} \mathcal{R}(w|b) - \gamma \sum_{i=1}^{n} (w_i - b_i) C_i \\
    &\qquad \text{s.t.} \quad \begin{cases} \mathbf{1}_n^\top w = 1 \\ \mathbf{0}_n \le w \le \mathbf{1}_n \end{cases}
\end{split}
\tag{11.5}
\end{equation}
$$

where the synthetic risk measure is equal to:

$$
\mathcal{R}(w | b) = \varphi_{\text{AS}} \mathcal{R}_{\text{AS}}(w | b) + \varphi_{\text{MD}} \mathcal{R}_{\text{MD}}(w | b) + \varphi_{\text{DTS}} \mathcal{R}_{\text{DTS}}(w | b)
$$

By using the properties $Q\mathcal{F}(x-y; Q, R, c) = Q\mathcal{F}\left(x; Q, R + Qy, \frac{1}{2}y^\top Qy + y^\top R + c\right)$, $T(s_j \circ \text{MD})b = (s_j \circ \text{MD})(s_j \circ \text{MD})^\top b = (s_j \circ \text{MD})\text{MD}_j^*$, $T(s_j \circ \text{DTS})b = (s_j \circ \text{DTS})\text{DTS}_j^*$, $b^\top T(s_j \circ \text{MD})b = \text{MD}_j^{*2}$ and $b^\top T(s_j \circ \text{DTS})b = \text{DTS}_j^{*2}$, we have:

$$
\left\{
\begin{array}{l}
\frac{1}{2} \mathcal{R}_{\text{AS}}(w | b) = \mathcal{Q}\mathcal{F}(w; Q_{\text{AS}}(b), R_{\text{AS}}(b), c_{\text{AS}}(b)) \\
\frac{1}{2} \mathcal{R}_{\text{MD}}(w | b) = \mathcal{Q}\mathcal{F}(w; Q_{\text{MD}}(b), R_{\text{MD}}(b), c_{\text{MD}}(b)) \\
\frac{1}{2} \mathcal{R}_{\text{DTS}}(w | b) = \mathcal{Q}\mathcal{F}(w; Q_{\text{DTS}}(b), R_{\text{DTS}}(b), c_{\text{DTS}}(b))
\end{array}
\right.
$$

$$
\mathrm{AS}(w | b) = \frac{1}{2} \sum_{i=1}^{n} |w_i - b_i|
$$

We prefer to use the L₂-norm which is more tractable from a numerical viewpoint.

<footnote n="13">¹³The traditional definition of the active share is:</footnote>
where $Q_{AS}(b) = I_n$, $R_{AS}(b) = b$, $c_{AS}(b) = \frac{1}{2}b^\top b$, $Q_{MD}(b) = Q_{MD}$, $R_{MD}(b) = Q_{MD}b = R_{MD}$, $c_{MD}(b) = \frac{1}{2}b^\top Q_{MD}b = c_{MD}$, $Q_{DTS}(b) = Q_{DTS}$, $R_{DTS}(b) = Q_{DTS}b = R_{DTS}$, and $c_{DTS}(b) = \frac{1}{2}b^\top Q_{DTS}b = c_{DTS}$. We conclude that Equation (11.5) becomes:

$$ w^* = \arg \min_{\text{s.t.}} \begin{array}{l} \mathcal{Q}\mathcal{F}(w; Q(b), R(b), c(b)) \\ \left\{ \begin{array}{l} \mathbf{1}_n^\top w = 1 \\ \mathbf{0}_n \le w \le \mathbf{1}_n \end{array} \right. \end{array} \qquad (11.6) $$

where¹⁴:

$$ \left\{ \begin{array}{l} Q(b) = \varphi_{AS}Q_{AS}(b) + \varphi_{MD}Q_{MD}(b) + \varphi_{DTS}Q_{DTS}(b) \\ R(b) = \gamma C + \varphi_{AS}R_{AS}(b) + \varphi_{MD}R_{MD}(b) + \varphi_{DTS}R_{DTS}(b) \\ c(b) = \gamma b^\top C + \varphi_{AS}c_{AS}(b) + \varphi_{MD}c_{MD}(b) + \varphi_{DTS}c_{DTS}(b) \end{array} \right. $$

**Remark 134** We notice that the optimization problem (11.4) is a special case of the optimization problem (11.6) when $\varphi_{AS}$ and $b$ are set to 0 and $\mathbf{0}_n$.

**Example 53** We consider an investment universe of five bonds with the following characteristics: the benchmark is $b = (30\%, 25\%, 20\%, 15\%, 10\%)$; the modified durations (in years) are $MD = (3, 7, 6, 12, 2)$; the DTS values (in bps) are $DTS = (100, 250, 70, 400, 150)$; the carry vector (in bps) is $C = (200, 300, 150, 250, 600)$; the first, third and fourth bonds belong to the first sector while the second and fourth bonds belong to the second sector. We assume that $\varphi_{AS} = 1$, $\varphi_{MD} = 1$, $\varphi_{DTS} = 0.001$ and $\gamma = 20\%$.

We proceed by steps in order to obtain the final quadratic form. First, we first compute the modified duration and the duration-times-spread factor of the benchmark. We obtain $MD(b) = 5.85$ years and $DTS(b) = 181.50$ bps. The sector contributions are $MD_1(b) = 3.90$ and $DTS_1(b) = 104.00$ for sector $\mathcal{S}$ector$_1$, and $MD_2(b) = 1.95$ and $DTS_2(b) = 77.50$ for sector $\mathcal{S}$ector$_2$. The target values are then $MD_1^* = 3.90$, $MD_2^* = 1.95$, $DTS_1^* = 104.00$ and $DTS_2^* = 77.50$. Second, we compute the matrices $Q_{MD}$, $R_{MD}$, $c_{MD}$, $Q_{DTS}$, $R_{DTS}$, and $c_{DTS}$. Third, using the benchmark composition $b$, we deduce the matrices $Q_{AS}(b)$, $R_{AS}(b)$, $c_{AS}(b)$, $Q_{MD}(b)$, $R_{MD}(b)$, $c_{MD}(b)$, $Q_{DTS}(b)$, $R_{DTS}(b)$ and $c_{DTS}(b)$. Finally, we calculate the matrices $Q(b)$, $R(b)$ and $c(b)$. All these matrices are reported in Table 11.1 on page 1051. If we compute $\mathcal{Q}\mathcal{F}(w_{ew}; Q(b), R(b), c(b))$ where $w_{ew}$ is the equally-weighted portfolio, we obtain -7.37812. Another way to check that we obtain the good results is to verify that $\mathcal{Q}\mathcal{F}(b; Q(b), R(b), c(b)) = 0$.

## Illustration

**Example 54** We consider an investment universe of nine corporate bonds with the following characteristics:

<table><thead><tr><th>Issuer</th><th>#1</th><th>#2</th><th>#3</th><th>#4</th><th>#5</th><th>#6</th><th>#7</th><th>#8</th><th>#9</th></tr></thead><tbody><tr><td>b<sub>i</sub> (in %)</td><td>21</td><td>19</td><td>16</td><td>12</td><td>11</td><td>8</td><td>6</td><td>4</td><td>3</td></tr><tr><td>CI<sub>i</sub> (in tCO<sub>2</sub>e/$ mn)</td><td>111</td><td>52</td><td>369</td><td>157</td><td>18</td><td>415</td><td>17</td><td>253</td><td>900</td></tr><tr><td>MD<sub>i</sub> (in years)</td><td>3.16</td><td>6.48</td><td>3.54</td><td>9.23</td><td>6.40</td><td>2.30</td><td>8.12</td><td>7.96</td><td>5.48</td></tr><tr><td>DTS<sub>i</sub> (in bps)</td><td>107</td><td>255</td><td>75</td><td>996</td><td>289</td><td>45</td><td>620</td><td>285</td><td>125</td></tr><tr><td>Sector</td><td>1</td><td>1</td><td>1</td><td>2</td><td>2</td><td>2</td><td>3</td><td>3</td><td>3</td></tr></tbody></table>

We assume that the portfolio weights can not deviate too far from the benchmark and impose the following constraint: $0.25 \times b_i \le w_i \le 4 \times b_i$. We have $\varphi_{AS} = 100$, $\varphi_{MD} = 25$ and $\varphi_{DTS} = 0.001$.
<table><caption>Table 11.1: Quadratic form (Example 53)</caption><thead><tr><th colspan="5">Q<sub>AS</sub> (b) = Q<sub>AS</sub></th><th>R<sub>AS</sub></th><th>R<sub>AS</sub> (b)</th><th>c<sub>AS</sub></th><th>c<sub>AS</sub> (b)</th></tr></thead><tbody><tr><td>1</td><td>0</td><td>0</td><td>0</td><td>0</td><td>0.00</td><td>0.30</td><td></td><td></td></tr><tr><td>0</td><td>1</td><td>0</td><td>0</td><td>0</td><td>0.00</td><td>0.25</td><td></td><td></td></tr><tr><td>0</td><td>0</td><td>1</td><td>0</td><td>0</td><td>0.00</td><td>0.20</td><td>0.0000</td><td>0.1125</td></tr><tr><td>0</td><td>0</td><td>0</td><td>1</td><td>0</td><td>0.00</td><td>0.15</td><td></td><td></td></tr><tr><td>0</td><td>0</td><td>0</td><td>0</td><td>1</td><td>0.00</td><td>0.10</td><td></td><td></td></tr></tbody><thead><tr><th colspan="5">Q<sub>MD</sub> (b) = Q<sub>MD</sub></th><th>R<sub>MD</sub></th><th>R<sub>MD</sub> (b)</th><th>c<sub>MD</sub></th><th>c<sub>MD</sub> (b)</th></tr></thead><tbody><tr><td>9</td><td>0</td><td>18</td><td>36</td><td>0</td><td>11.70</td><td>11.70</td><td></td><td></td></tr><tr><td>0</td><td>49</td><td>0</td><td>0</td><td>14</td><td>13.65</td><td>13.65</td><td></td><td></td></tr><tr><td>18</td><td>0</td><td>36</td><td>72</td><td>0</td><td>23.40</td><td>23.40</td><td>9.5062</td><td>9.5062</td></tr><tr><td>36</td><td>0</td><td>72</td><td>144</td><td>0</td><td>46.80</td><td>46.80</td><td></td><td></td></tr><tr><td>0</td><td>14</td><td>0</td><td>0</td><td>4</td><td>3.90</td><td>3.90</td><td></td><td></td></tr></tbody><thead><tr><th colspan="5">Q<sub>DTS</sub> (b) = Q<sub>DTS</sub></th><th>R<sub>DTS</sub></th><th>R<sub>DTS</sub> (b)</th><th>c<sub>DTS</sub></th><th>c<sub>DTS</sub> (b)</th></tr></thead><tbody><tr><td>10000</td><td>0</td><td>7000</td><td>40000</td><td>0</td><td>10400</td><td>10400</td><td></td><td></td></tr><tr><td>0</td><td>62500</td><td>0</td><td>0</td><td>37500</td><td>19375</td><td>19375</td><td></td><td></td></tr><tr><td>7000</td><td>0</td><td>4900</td><td>28000</td><td>0</td><td>7280</td><td>7280</td><td>8411.1250</td><td>8411.1250</td></tr><tr><td>40000</td><td>0</td><td>28000</td><td>160000</td><td>0</td><td>41600</td><td>41600</td><td></td><td></td></tr><tr><td>0</td><td>37500</td><td>0</td><td>0</td><td>22500</td><td>11625</td><td>11625</td><td></td><td></td></tr></tbody><thead><tr><th colspan="5">Q (b)</th><th>R (b)</th><th></th><th>c (b)</th><th></th></tr></thead><tbody><tr><td>20.0</td><td>0.0</td><td>25.0</td><td>76.0</td><td>0.0</td><td>62.400</td><td></td><td></td><td></td></tr><tr><td>0.0</td><td>112.5</td><td>0.0</td><td>0.0</td><td>51.5</td><td>93.275</td><td></td><td></td><td></td></tr><tr><td>25.0</td><td>0.0</td><td>41.9</td><td>100.0</td><td>0.0</td><td>60.880</td><td></td><td>70.530</td><td></td></tr><tr><td>76.0</td><td>0.0</td><td>100.0</td><td>305.0</td><td>0.0</td><td>138.550</td><td></td><td></td><td></td></tr><tr><td>0.0</td><td>51.5</td><td>0.0</td><td>0.0</td><td>27.5</td><td>135.625</td><td></td><td></td><td></td></tr></tbody></table>

The objective is to reduce the carbon footprint of the benchmark and minimizing the tracking error risk. In this case, the carry vector $\mathcal{C}$ is set to the zero vector and the optimization problem is defined as:

$$ w^*(\mathcal{R}) = \arg \min \frac{1}{2} w^\top Q(b) w - w^\top R(b) \\ \text{s.t.} \quad \begin{cases} \mathbf{1}_9^\top w = 1 \\ \mathcal{CI}^\top w \le (1 - \mathcal{R}) \mathcal{CI}(b) \\ \frac{b}{4} \le w \le 4b \end{cases} $$

where $\mathcal{R}$ is the reduction rate. Since the bonds are ordering by sectors, $Q(b)$ is a block diagonal matrix:

$$ Q(b) = \begin{pmatrix} Q_1 & \mathbf{0}_{3 \times 3} & \mathbf{0}_{3 \times 3} \\ \mathbf{0}_{3 \times 3} & Q_2 & \mathbf{0}_{3 \times 3} \\ \mathbf{0}_{3 \times 3} & \mathbf{0}_{3 \times 3} & Q_3 \end{pmatrix} \times 10^3 $$

where:

$$ Q_1 = \begin{pmatrix} 0.3611 & 0.5392 & 0.2877 \\ 0.5392 & 1.2148 & 0.5926 \\ 0.2877 & 0.5926 & 0.4189 \end{pmatrix}, \quad Q_2 = \begin{pmatrix} 3.2218 & 1.7646 & 0.5755 \\ 1.7646 & 1.2075 & 0.3810 \\ 0.5755 & 0.3810 & 0.2343 \end{pmatrix} $$

<footnote n="14">¹⁴We have $-\gamma \sum_{i=1}^{n} (w_i - b_i) C_i = QF(w; \mathbf{0}_{n,n}, \gamma C, \gamma b^\top C)$.</footnote>
<figure><img src="image_495.png" /><figcaption>Figure 11.1: Relationship between the reduction rate and the tracking risk (Example 54)</figcaption></figure>

and:

$$Q_3 = \begin{pmatrix} 2.1328 & 1.7926 & 1.1899 \\ 1.7926 & 1.7653 & 1.1261 \\ 1.1899 & 1.1261 & 0.8664 \end{pmatrix}$$

The vector $R(b)$ is equal to $(2.2431, 4.3886, 2.4004, 6.2678, 3.7506, 1.2972, 2.3537, 2.1195, 1.4243) \times 10^2$. In Tables 11.2 and 11.3, we report the optimal weights and the risk statistics. Increasing the reduction rate implies to take more active risk in terms of active share, modified duration and duration-times-spread factor. For instance, if we target $\mathcal{R} = 50\%$, the MD-based tracking risk is 2.4 months while the DTS-based tracking risk is 30.11 bps. The asset- and sector-based<fnref n="15" /> active shares are respectively equal to 21.21% and 28.31%. In Figure 11.1, we show the relationship between the reduction rate and these different risk statistics. We notice that there is no solution to the optimization problem when $\mathcal{R} \ge 65.73\%$.

<table><caption>Table 11.2: Weights in % of optimized bond portfolios (Example 54)</caption><thead><tr><th>Portfolio</th><th>#1</th><th>#2</th><th>#3</th><th>#4</th><th>#5</th><th>#6</th><th>#7</th><th>#8</th><th>#9</th></tr></thead><tbody><tr><td>b</td><td>21.00</td><td>19.00</td><td>16.00</td><td>12.00</td><td>11.00</td><td>8.00</td><td>6.00</td><td>4.00</td><td>3.00</td></tr><tr><td>w* (10%)</td><td>21.92</td><td>19.01</td><td>15.53</td><td>11.72</td><td>11.68</td><td>7.82</td><td>6.68</td><td>4.71</td><td>0.94</td></tr><tr><td>w* (30%)</td><td>26.29</td><td>20.24</td><td>10.90</td><td>10.24</td><td>16.13</td><td>3.74</td><td>9.21</td><td>2.50</td><td>0.75</td></tr><tr><td>w* (50%)</td><td>27.48</td><td>23.97</td><td>4.00</td><td>6.94</td><td>22.70</td><td>2.00</td><td>11.15</td><td>1.00</td><td>0.75</td></tr></tbody></table>

<footnote n="15">¹⁵The sector-based active share is defined as $AS_{Sector}(w | b) = \frac{1}{2} \sum_{j=1}^{n} s_{sector} | \sum_{i \in Sector_j} (w_i - b_i)|$.</footnote>
<table><caption>Table 11.3: Risk statistics of optimized bond portfolios (Example 54)</caption><thead><tr><th>Portfolio</th><th>AS<sub>Sector</sub><br>(in %)</th><th>MD (w)<br>(in years)</th><th>DTS (w)<br>(in bps)</th><th>σ<sub>AS</sub> (w | b)<br>(in %)</th><th>σ<sub>MD</sub> (w | b)<br>(in years)</th><th>σ<sub>DTS</sub> (w | b)<br>(in bps)</th><th>CI (w)<br>gCO<sub>2</sub>e/$</th></tr></thead><tbody><tr><td>b</td><td>0.00</td><td>5.43</td><td>290.18</td><td>0.00</td><td>0.00</td><td>0.00</td><td>184.39</td></tr><tr><td>w<sup>*</sup> (10%)</td><td>3.00</td><td>5.45</td><td>293.53</td><td>2.62</td><td>0.02</td><td>3.80</td><td>165.95</td></tr><tr><td>w<sup>*</sup> (30%)</td><td>14.87</td><td>5.58</td><td>303.36</td><td>10.98</td><td>0.10</td><td>14.49</td><td>129.07</td></tr><tr><td>w<sup>*</sup> (50%)</td><td>28.31</td><td>5.73</td><td>302.14</td><td>21.21</td><td>0.19</td><td>30.11</td><td>92.19</td></tr></tbody></table>

### 11.1.3 Advanced optimization problems

In this section, we go beyond the previous approaches in order to solve more complex portfolio optimization problems. First, we consider the case of large bond universes when there are more than 10 000 bonds in the index. In this case, we generally use linear programming (LP) algorithms. Second, we show how to incorporate some standard nonlinear constraints in the QP/LP framework. Finally, we consider mathematical problems when the variable to optimize is not a vector of weights, but the numbers of shares. In particular, this last problem occurs when we would like to build an investible fixed-income portfolio.

#### Large bond universe

Quadratic problem algorithms are efficient when the dimension of the problem is relative small, say, when $n \le 5000$. The issue is not the convergence of the algorithm, but more the manipulation of the Hessian matrix $Q$ of the quadratic form. Indeed, since $Q$ is a $n \times n$ matrix, we have to store $n^2$ floating-point numbers. For instance, $Q$ has 25 millions of elements when $n$ is equal to 5 000. In the case of bond portfolio optimization, we have seen that $Q$ is a block matrix with the following structure<fnref n="16" />:

$$Q = \begin{pmatrix} Q_1 & \mathbf{0}_{n_{(1)} \times n_{(2)}} & \mathbf{0} \\ \mathbf{0}_{n_{(2)} \times n_{(1)}} & Q_2 & \cdots \\ \mathbf{0} & \mathbf{0} & Q_{n_{\mathcal{S}ector}} \end{pmatrix}$$

where $Q_j$ is the $n_{(j)} \times n_{(j)}$ matrix associated to the $j^{th}$ sector and $n_{(j)}$ is the number of assets that belong to the $j^{th}$ sector. By construction, we have $\sum_{j=1}^{n_{\mathcal{S}ector}} n_{(j)} = n$. Therefore, the number of non-zero entries is equal to:

$$\eta(Q) = \sum_{j=1}^{n_{\mathcal{S}ector}} n_{(j)}^2 \ll n^2$$

Let us assume that the number of assets are equally distributed among the sectors; $n_{(j)} = n/n_{\mathcal{S}ector}$. We obtain:

$$\eta(Q) = \sum_{j=1}^{n_{\mathcal{S}ector}} \left( \frac{n}{n_{\mathcal{S}ector}} \right)^2 = \frac{n^2}{n_{\mathcal{S}ector}}$$

We deduce that $Q$ is a highly sparse matrix since the number of non-zero elements is a fraction of the full number of elements. If $n = 5000$ and $n_{\mathcal{S}ector} = 10$, the number of non-zero elements is equal to 2.5 millions of elements. We reiterate that the bucket concept in bond portfolio optimization generalizes the sector concept. For instance, if we consider ten sectors, three currencies and five

<footnote n="16">¹⁶We assume that the bonds are ordered according to the sectors.</footnote>
maturity bands, the number of buckets is equal to 10 × 3 × 5 = 150. If $n = 5 000$, the number of non-zero elements is approximatively equal to 167 000 of elements. This is equivalent to manipulate a dense matrix of dimension 409 × 409. It follows that portfolio optimization with large bond universe remains feasible by using sparse quadratic programming solvers.

As of 31/01/2023, the Bloomberg Global Aggregate Total Return Index has 28 799 securities, while the ICE BOFA Global Broad Market Index has 33 575 securities. For these two bond universes, we reach the limits of QP optimization from a numerical viewpoint. For instance, the number of non-zero entries is equal to more than 8 million floating-point numbers for the Hessian Q matrix when $n$ is equal to 35 000 and we have 150 equally-distributed buckets. In the case where we only consider sectors as buckets, we have more than 120 million floating-point numbers for 10 equally-distributed sectors. Moreover, the number of bonds are generally concentrated in some sectors, e.g., Financials. This explains that bond portfolio managers prefer to use another approach than the QP framework when performing portfolio optimization with such large investment universes.

<figure><img src="image_496.png" /><figcaption>Figure 11.2: Relationship between $\mathcal{L}_1$-norm and $\mathcal{L}_2$-norm</figcaption></figure>

The underlying idea is to replace $\mathcal{L}_2$-norm risk measures by $\mathcal{L}_1$-norm risk measures. We recall that $f(x) = \sqrt{x^2} = |x|$ when $x$ is a scalar. In the general case where $x$ is a $n \times 1$ vector, we have $\|x\|_1 = \sum_{i=1}^n |x_i|$ and $\|x\|_2 = \sqrt{\sum_{i=1}^n x_i^2} = \sqrt{x^\top x}$. We notice that $\|x\|_2^2 \le \sum_{i=1}^n |x_i| \sum_{i=1}^n |x_i| = \|x\|_1^2$ and $\|x\|_1^2 = \mathbf{1}_n^\top |x| \le \|\mathbf{1}_n\|_2 \cdot \|x\|_2 \le \sqrt{n} \|x\|_2$, implying that $\|x\|_2 \le \|x\|_1 \le \sqrt{n} \|x\|_2$. This means that there is not a simple relationship between the two norms. Nevertheless, if the vector $x$ is “homogenous”, we can use the following approximation $\|x\|_2 \approx \beta_0 + \beta_1 \|x\|_1$ where $0 < \beta_1 \le 1$. In Figure 11.2, we report the scatter plot of $(\|x\|_1, \|x\|_2)$ and the least square fitting model $\|x\|_2 = \hat{\beta}_0 + \hat{\beta}_1 \|x\|_1$ when $x_i$ is the random variate, whose probability distribution is $F(x_i) = \text{Pr}\{U_{[0,1]} \le \Delta^{-1}x_i\}$. The estimated values $(\hat{\beta}_0, \hat{\beta}_1)$ depend then on the dimension $n$ of the vector $x$ and the parameter $\Delta$.
We exploit the previous relationship and replace the synthetic risk measure by:

$$ \mathcal{D}(w | b) = \varphi'_{\text{AS}} \mathcal{D}_{\text{AS}}(w | b) + \varphi'_{\text{MD}} \mathcal{D}_{\text{MD}}(w | b) + \varphi'_{\text{DTS}} \mathcal{D}_{\text{DTS}}(w | b) $$

where:

$$ \mathcal{D}_{\text{AS}}(w | b) = \frac{1}{2} \sum_{i=1}^{n} |w_i - b_i| = \frac{1}{2} \|w - b\|_1 $$

$$ \mathcal{D}_{\text{MD}}(w | b) = \sum_{j=1}^{n \mathcal{S}_{\text{sector}}} \left| \sum_{i \in \mathcal{S}_{\text{sector}}_{j}} (w_i - b_i) \text{MD}_i \right| = \sum_{j=1}^{n \mathcal{S}_{\text{sector}}} \| \mathbf{s}_j \circ (w-b) \circ \text{MD} \|_1 $$

$$ \mathcal{D}_{\text{DTS}}(w | b) = \sum_{j=1}^{n \mathcal{S}_{\text{sector}}} \left| \sum_{i \in \mathcal{S}_{\text{sector}}_{j}} (w_i - b_i) \text{DTS}_i \right| = \sum_{j=1}^{n \mathcal{S}_{\text{sector}}} \| \mathbf{s}_j \circ (w-b) \circ \text{DTS} \|_1 $$

The optimization problem becomes then:

$$ w^* = \arg \min_{\text{s.t.}} \mathcal{D}(w | b) - \gamma \sum_{i=1}^{n} (w_i - b_i) C_i \quad (11.7) $$

$$ \begin{cases} \mathbf{1}_n^\top w = 1 \\ \mathbf{0}_n \le w \le \mathbf{1}_n \end{cases} $$

Using the absolute value trick, we transform the nonlinear objective function into a linear function:

$$ w^* = \arg \min \frac{1}{2} \varphi'_{\text{AS}} \sum_{i=1}^{n} \tau_{i,w} + \varphi'_{\text{MD}} \sum_{j=1}^{n \mathcal{S}_{\text{sector}}} \tau_{j,\text{MD}} + \varphi'_{\text{DTS}} \sum_{j=1}^{n \mathcal{S}_{\text{sector}}} \tau_{j,\text{DTS}} - \gamma \sum_{i=1}^{n} (w_i - b_i) C_i $$

$$ \text{s.t.} \quad \left\{ \begin{array}{l} \mathbf{1}_n^\top w = 1 \\ \mathbf{0}_n \le w \le \mathbf{1}_n \\ |w_i - b_i| \le \tau_{i,w} \\ \left| \sum_{i \in \mathcal{S}_{\text{sector}}_{j}} (w_i - b_i) \text{MD}_i \right| \le \tau_{j,\text{MD}} \\ \left| \sum_{i \in \mathcal{S}_{\text{sector}}_{j}} (w_i - b_i) \text{DTS}_i \right| \le \tau_{j,\text{DTS}} \\ \tau_{i,w} \ge 0, \tau_{j,\text{MD}} \ge 0, \tau_{j,\text{DTS}} \ge 0 \end{array} \right. $$

We notice that:

$$ |w_i - b_i| \le \tau_{i,w} \Leftrightarrow \begin{cases} w_i - \tau_{i,w} \le b_i \\ -w_i - \tau_{i,w} \le -b_i \end{cases} $$

and:

$$ (*) \quad \Leftrightarrow \left| \sum_{i \in \mathcal{S}_{\text{sector}}_{j}} (w_i - b_i) \text{MD}_i \right| \le \tau_{j,\text{MD}} $$

$$ \Leftrightarrow -\tau_{j,\text{MD}} \le \sum_{i \in \mathcal{S}_{\text{sector}}_{j}} (w_i - b_i) \text{MD}_i \le \tau_{j,\text{MD}} $$

$$ \Leftrightarrow -\tau_{j,\text{MD}} + \sum_{i \in \mathcal{S}_{\text{sector}}_{j}} b_i \text{MD}_i \le \sum_{i \in \mathcal{S}_{\text{sector}}_{j}} w_i \text{MD}_i \le \tau_{j,\text{MD}} + \sum_{i \in \mathcal{S}_{\text{sector}}_{j}} b_i \text{MD}_i $$

$$ \Leftrightarrow -\tau_{j,\text{MD}} + \text{MD}_j^* \le (\mathbf{s}_j \circ \text{MD})^\top w \le \tau_{j,\text{MD}} + \text{MD}_j^* $$

$$ \Leftrightarrow \left\{ \begin{array}{l} (\mathbf{s}_j \circ \text{MD})^\top w - \tau_{j,\text{MD}} \le \text{MD}_j^* \\ -(\mathbf{s}_j \circ \text{MD})^\top w - \tau_{j,\text{MD}} \le -\text{MD}_j^* \end{array} \right. $$
We can now formulate problem (11.7) as a standard linear programming (LP) problem:

$$
x^* = \arg \min c^\top x \tag{11.8} \\
\text{s.t.} \quad \begin{cases} Ax = B \\ Cx \le D \\ x^- \le x \le x^+ \end{cases}
$$

where:

$$
x = \begin{pmatrix} w \\ \tau_w \\ \tau_{\text{MD}} \\ \tau_{\text{DTS}} \end{pmatrix}
$$

is a vector of dimension $n_x = 2 \times (n + n_{\mathbf{Sector}})$. The vector $c$ is equal to:

$$
c = \begin{pmatrix} -\gamma C \\ \frac{1}{2} \varphi'_{\text{AS}} \mathbf{1}_n \\ \varphi'_{\text{MD}} \mathbf{1}_{n_{\mathbf{Sector}}} \\ \varphi'_{\text{DTS}} \mathbf{1}_{n_{\mathbf{Sector}}} \end{pmatrix}
$$

The equality constraint is defined by $A = (\mathbf{1}_n^\top \quad \mathbf{0}_n^\top \quad \mathbf{0}_{n_{\mathbf{Sector}}^\top} \quad \mathbf{0}_{n_{\mathbf{Sector}}^\top})$ and $B = 1$. We also have:

$$
C x \le D \Leftrightarrow \left( \begin{array}{cccc} I_n & -I_n & \mathbf{0}_{n,n_{\mathbf{Sector}}} & \mathbf{0}_{n,n_{\mathbf{Sector}}} \\ -I_n & -I_n & \mathbf{0}_{n,n_{\mathbf{Sector}}} & \mathbf{0}_{n,n_{\mathbf{Sector}}} \\ C_{\text{MD}} & \mathbf{0}_{n_{\mathbf{Sector}},n} & -I_{n_{\mathbf{Sector}}} & \mathbf{0}_{n_{\mathbf{Sector}},n_{\mathbf{Sector}}} \\ -C_{\text{MD}} & \mathbf{0}_{n_{\mathbf{Sector}},n} & -I_{n_{\mathbf{Sector}}} & \mathbf{0}_{n_{\mathbf{Sector}},n_{\mathbf{Sector}}} \\ C_{\text{DTS}} & \mathbf{0}_{n_{\mathbf{Sector}},n} & \mathbf{0}_{n_{\mathbf{Sector}},n_{\mathbf{Sector}}} & -I_{n_{\mathbf{Sector}}} \\ -C_{\text{DTS}} & \mathbf{0}_{n_{\mathbf{Sector}},n} & \mathbf{0}_{n_{\mathbf{Sector}},n_{\mathbf{Sector}}} & -I_{n_{\mathbf{Sector}}} \end{array} \right) x \le \left( \begin{array}{c} b \\ -b \\ \text{MD}^* \\ -\text{MD}^* \\ \text{DTS}^* \\ -\text{DTS}^* \end{array} \right)
$$

where $C_{\text{MD}}$ and $C_{\text{DTS}}$ are two $n_{\text{Sector}} \times n$ matrices, whose $(j, i)$-elements are $s_{i,j}$ $MD_i$ and $s_{i,j}$ $DTS_i$, $MD^* = (MD_1^*, \dots, MD_{n_{\text{Sector}}^*})$ and $DTS^* = (DTS_1^*, \dots, DTS_{n_{\text{Sector}}^*})$. Finally, the bounds are $x^- = \mathbf{0}_{n_x}$ and $x^+ = \infty \cdot \mathbf{1}_{n_x}$.

**Remark 135** In the case where there are additional constraints, they are cast into LP constraints using the following equivalence:

$$
\left\{
\begin{array}{l}
Aw = B \\
Cw \leq D
\end{array}
\right.
\Leftrightarrow
\left\{
\begin{array}{l}
(A \quad \mathbf{0}_{n_A, n_x-n}) x = B \\
(C \quad \mathbf{0}_{n_A, n_x-n}) x \leq D
\end{array}
\right.
$$

**Remark 136** While the dimension of the *QP* problem was *n*, the dimension of the *LP* problem is $2 \times (n + n_{\mathbf{Sector}})$. This is not an issue since *LP* algorithms are efficient to solve optimization problems with million of decision variables.

**Example 55** We consider a toy example with four corporate bonds:

<table>
  <thead>
    <tr>
      <th>Issuer</th>
      <th>#1</th>
      <th>#2</th>
      <th>#3</th>
      <th>#4</th>
    </tr>
  </thead>
  <tbody>
    <tr>
      <td>b<sub>i</sub> (in %)</td>
      <td>35</td>
      <td>15</td>
      <td>20</td>
      <td>30</td>
    </tr>
    <tr>
      <td><b>CT</b><sub>i</sub> (in tCO<sub>2</sub>e/$ mn)</td>
      <td>117</td>
      <td>284</td>
      <td>162.5</td>
      <td>359</td>
    </tr>
    <tr>
      <td>MD<sub>i</sub> (in years)</td>
      <td>3.0</td>
      <td>5.0</td>
      <td>2.0</td>
      <td>6.0</td>
    </tr>
    <tr>
      <td>DTS<sub>i</sub> (in bps)</td>
      <td>100</td>
      <td>150</td>
      <td>200</td>
      <td>250</td>
    </tr>
    <tr>
      <td><b>S</b>ector</td>
      <td>1</td>
      <td>1</td>
      <td>2</td>
      <td>2</td>
    </tr>
  </tbody>
</table>

We would like to reduce the carbon footprint by 20%, and we set $\varphi'_{\text{AS}} = 100$, $\varphi'_{\text{MD}} = 25$ and $\varphi'_{\text{DTS}} = 1$.

<footnote n="17">¹⁷We remark that the constraint $w \le \mathbf{1}_n$ is already embedded into the linear equality constraint.</footnote>
We have $n = 4$, $n_{Sector} = 2$ and:

$$x = \underbrace{(w_1, w_2, w_3, w_4)}_{w}, \underbrace{\tau_{w_1}, \tau_{w_2}, \tau_{w_3}, \tau_{w_4}}_{\tau_w}, \underbrace{\tau_{MD_1}, \tau_{MD_2}}_{\tau_{MD}}, \underbrace{\tau_{DTS_1}, \tau_{DTS_2}}_{\tau_{DTS}}$$

Since the vector $C$ is equal to $\mathbf{0}_4$, we obtain:

$$c = (0, 0, 0, 0, 50, 50, 50, 50, 25, 25, 1, 1)$$

The equality system $Ax = B$ is defined by:

$$A = (1 \ 1 \ 1 \ 1 \ 0 \ 0 \ 0 \ 0 \ 0 \ 0 \ 0 \ 0 \ 0)$$

and $B = 1$, while the inequality system $Cx \le D$ is given by:

$$C = \begin{pmatrix}
\begin{array}{cccc}
I_4 & & & \\
-I_4 & -I_4 & & \\
& & -I_4 & \\
& & & \mathbf{0}_{4,4}
\end{array}
&
\begin{array}{cccc}
-I_4 & & & \\
-I_4 & -I_4 & & \\
& & -I_4 & \\
& & & \mathbf{0}_{4,4}
\end{array}
&
\begin{array}{cccc}
0 & & & \\
0 & 0 & & \\
& & 0 & \\
& & & \mathbf{0}_{4,4}
\end{array}
&
\begin{array}{cccc}
0 & & & \\
0 & 0 & & \\
& & 0 & \\
& & & \mathbf{0}_{4,4}
\end{array}
\end{pmatrix}$$

and:

$$D = (0.35, 0.15, 0.2, 0.3, -0.35, -0.15, -0.2, -0.3, 1.8, 2.2, -1.8, -2.2, 57.5, 115, -57.5, -115, 179)$$

The last row of $Cx \le D$ corresponds to the carbon footprint constraint. Indeed, we have $CI(b) = 223.75$ tCO$_2$e/$mn and $(1 - R)CI(b) = 0.80 \times 223.75 = 179.00$ tCO$_2$e/$mn$. We can now solve the LP program, and we obtain the following solution: $w^* = (47.34\%, 0\%, 33.3\%, 19.36\%)$, $\tau_w^* = (12.34\%, 15\%, 13.3\%, 10.64\%)$, $\tau_{MD}^* = (0.3798, 0.3725)$ and $\tau_{DTS}^* = (10.1604, 0)$.

We consider Example 54 on page 1050. By assuming that $\phi'_{AS} = \phi_{AS} = 100$, $\phi'_{MD} = \phi_{MD} = 25$ and $\phi'_{DTS} = \phi'_{DTS} = 0.001$, we obtain the results given in Table 11.4 and 11.5. By construction, these results differ from those obtained in 11.2 and 11.3 on page 1052. However, we observe a consistency between the two solutions. In particular, the overweighting and underweighting directions are respected by the QP and LP algorithms.

<table><caption>Table 11.4: Weights in % of optimized bond portfolios (Example 54)</caption><thead><tr><th>Portfolio</th><th>#1</th><th>#2</th><th>#3</th><th>#4</th><th>#5</th><th>#6</th><th>#7</th><th>#8</th><th>#9</th></tr></thead><tbody><tr><td>b</td><td>21.00</td><td>19.00</td><td>16.00</td><td>12.00</td><td>11.00</td><td>8.00</td><td>6.00</td><td>4.00</td><td>3.00</td></tr><tr><td>w<sup>*</sup> (10%)</td><td>21.70</td><td>19.00</td><td>16.00</td><td>12.00</td><td>11.00</td><td>8.00</td><td>7.46</td><td>4.00</td><td>0.84</td></tr><tr><td>w<sup>*</sup> (30%)</td><td>34.44</td><td>19.00</td><td>4.00</td><td>11.65</td><td>11.98</td><td>6.65</td><td>7.52</td><td>4.00</td><td>0.75</td></tr><tr><td>w<sup>*</sup> (50%)</td><td>33.69</td><td>19.37</td><td>4.00</td><td>3.91</td><td>24.82</td><td>2.00</td><td>10.46</td><td>1.00</td><td>0.75</td></tr></tbody></table>

**Remark 137** The linear programming approach is used by Barahhou et al. (2022) and Ben Slimane et al. (2023a) to implement portfolio alignment and net-zero investment policies<fnref n="18" />. The reason is that the investment universe of corporate bonds is too large to be manipulated by the QP algorithm.

<footnote n="18">¹⁸See Section 11.2 on page 1059 for using linear programming for decarbonized bond portfolios and Section 11.3 on page 1083 for net-zero bond portfolios.</footnote>
<table><caption>Table 11.5: Risk statistics of optimized bond portfolios (Example 54)</caption><thead><tr><th>Portfolio</th><th>AS<sub>Sector</sub><br>(in %)</th><th>MD (w)<br>(in years)</th><th>DTS (w)<br>(in bps)</th><th>σ<sub>AS</sub> (w | b)<br>(in %)</th><th>σ<sub>MD</sub> (w | b)<br>(in years)</th><th>σ<sub>DTS</sub> (w | b)<br>(in bps)</th><th>CI (w)<br>gCO<sub>2</sub>e/$</th></tr></thead><tbody><tr><td>b</td><td>0.00</td><td>5.43</td><td>290.18</td><td>0.00</td><td>0.00</td><td>0.00</td><td>184.39</td></tr><tr><td>w<sup>*</sup> (10%)</td><td>2.16</td><td>5.45</td><td>297.28</td><td>2.16</td><td>0.02</td><td>7.10</td><td>165.95</td></tr><tr><td>w<sup>*</sup> (30%)</td><td>15.95</td><td>5.43</td><td>300.96</td><td>15.95</td><td>0.00</td><td>13.20</td><td>129.07</td></tr><tr><td>w<sup>*</sup> (50%)</td><td>31.34</td><td>5.43</td><td>268.66</td><td>31.34</td><td>0.00</td><td>65.12</td><td>92.19</td></tr></tbody></table>

## Nonlinear constraints

A first idea to deal with nonlinear constraints is to use a general optimization solver, such as the sequential quadratic programming algorithm. However, this algorithm is not efficient when the dimension of the problem is large. In most cases, nonlinear constraints are standard in portfolio optimization. They concern the turnover, the transaction costs, the leverage limit, etc. Therefore, we can use the quadratic property of the objective function to solve the nonlinear constrained optimization problem. The underlying idea is to preserve the QP or LP structure of the problem by considering sequential optimization. Among the different algorithms, the most famous is the alternating direction method of multipliers<fnref n="19" /> (ADMM), which can be used to solve many nonlinear portfolio optimization problems (Perrin and Roncalli, 2020). In particular, Lezmi et al. (2022) used it to solve multi-period portfolio allocation and some classes of net-zero optimization problems.

## Investible portfolios

The previous optimization problems are defined with respect to the unknown variable $w = (w_1, \dots, w_n)$, where $w_i$ is the portfolio weight of Security $i$. In this approach, $w \in \mathbb{R}^n$ and the budget constraint is $\sum_{i=1}^n w_i = 1$. This type of problem falls under the umbrella of continuous portfolio optimization. In some cases, it does not make sense to define the portfolio allocation by considering continuous weights because the solution is far from investible. Therefore, the unknown variable becomes $q = (q_1, \dots, q_n)$, where $q_i$ is the number of shares of Security $i$. Let $P_i$ be the price and $Q_i = q_i P_i$ be the tradable amount. In this approach, $q \in \mathbb{N}^n$ and the budget constraint is $\sum_{i=1}^n q_i P_i = \sum_{i=1}^n Q_i = A$ where $A$ is the investment amount. This type of problem falls under the umbrella of discrete portfolio optimization and can be solved using genetic algorithms (Ben Slimane, 2021) or mixed integer programming algorithms (Ben Slimane and Menchaoui, 2023).

Given an optimal solution $w^*$, the amount invested in Security $i$ is equal to $Q_i = w_i^* A$. For some markets, $Q_i$ may be far from the optimal investible solution, which is equal to $q_i^* P_i$. This is for example the case of corporate bonds<fnref n="20" />. Let us introduce some definitions. The minimum tradable lot size is the minimum order size that can be bought or sold for a bond, while the incremental lot size is the minimum multiple that must be added to the minimum tradable lot to buy or sell a bond. For example, if the minimum tradable lot size is 10 000 and the incremental lot size is 1 000, the order sizes are $Q = 10\,000 + 1\,000 \cdot k$ where $k \in \mathbb{N}$. For EUR and USD corporate bonds, the minimum tradable size is typically €100 000 and $2 000, respectively. Suppose we want to invest 5% of a million euros in a security. If the lot size of the security is €540, we have the choice of buying 92 or 93 shares, that is €49 680 or €50 220. Both solutions are close to the desired amount of €50 000. If the minimum tradeable amount of the security is €100 000, we have the choice of investing 0 or €100 000. Both solutions are far from the desired amount of €50 000.

<footnote n="19">¹⁹The ADMM algorithm is described on page 1145.</footnote>

<footnote n="20">²⁰For equity and sovereign bond portfolios, it is common practice to use continuous portfolio optimization.</footnote>
## 11.2 Portfolio decarbonization

The objective of portfolio decarbonization is to construct a portfolio $w$ with a low carbon metric, which is generally the carbon intensity. Most of the time, the fund manager needs a reference portfolio $b$ in order to measure the reduction $\mathcal{R}(w | b)$. This reference portfolio can be the capitalization-weighted benchmark of the investment universe, a model portfolio, the managed portfolio one year ago, etc. When an explicit objective is defined, portfolio decarbonization can be viewed as a portfolio management constraint:

$$ \mathcal{CI}(w) \leq (1 - \mathcal{R}) \mathcal{CI}(b) $$

In this case, the decarbonization issue is solved by using constrained portfolio optimization. This framework can be extended when the global decarbonization constraint is replaced by a set of sector-specific decarbonization constraints. At first sight, portfolio decarbonization is then an application of basic portfolio optimization. Nevertheless, there are many implementation differences that make the exercise more difficult than we can imagine. In particular, portfolio decarbonization is ultimately an exclusion process and can be put in the category of exclusion ESG strategies. In some sense, portfolio decarbonization is a disguised form of divestment, which impacts the worst-in-class issuers that have the higher carbon intensities.

### 11.2.1 Global reduction of the carbon footprint

#### Equity portfolios

Let $b$ be the reference portfolio (e.g., a benchmark). The optimization problem is defined as:

$$ w^* = \arg\min \frac{1}{2} (w - b)^{\top} \Sigma (w - b) \quad (11.9) $$

$$ \text{s.t.} \quad \begin{cases} \mathbf{1}_n^{\top} w = 1 \\ w \in \Omega \\ \mathbf{0}_n \le w \le \mathbf{1}_n \\ \mathcal{CI}(w) \le (1 - \mathcal{R}) \mathcal{CI}(b) \end{cases} $$

where $\mathcal{R}$ is the reduction rate and $w \in \Omega$ is a set of additional weight constraints (e.g., weight or sector deviation). Therefore, we minimize the tracking error variance of the investment portfolio $w$ with respect to the reference portfolio $b$ by imposing a long-only constraint and reducing the carbon intensity of the benchmark. Since we both impose a constraint and minimize the tracking error risk, the portfolio $w$ has fewer stocks than the benchmark $b$. In fact, the number of stocks depends on several parameters: the reduction rate $\mathcal{R}$, the number $n$ of stocks in the benchmark and the covariance matrix. This implies that portfolio $w$ is less diversified than benchmark $b$. In order to explicitly control the number of removed stocks, Andersson et al. (2016) proposed a second portfolio decarbonization approach by eliminating the $m$ worst performing issuers in terms of carbon intensity.

Let $\mathcal{CI}_{i:n}$ be the order statistics of $(\mathcal{CI}_1, \dots, \mathcal{CI}_n)$ such that:

$$ \min \mathcal{CI}_i = \mathcal{CI}_{1:n} \le \mathcal{CI}_{2:n} \le \dots \le \mathcal{CI}_{i:n} \le \dots \le \mathcal{CI}_{n:n} = \max \mathcal{CI}_i $$

The carbon intensity bound $\mathcal{CI}^{(m,n)}$ is defined as $\mathcal{CI}^{(m,n)} = \mathcal{CI}_{n-m+1:n}$ where $\mathcal{CI}_{n-m+1:n}$ is the $(n-m+1)$-th order statistic of $(\mathcal{CI}_1, \dots, \mathcal{CI}_n)$. Eliminating the $m$ worst performing assets is equivalent to imposing the following constraint: $\mathcal{CI}_i \ge \mathcal{CI}^{(m,n)} \Rightarrow w_i = 0$. We then obtain the
following optimization problem:

$$w^* = \arg\min \frac{1}{2} (w-b)^\top \Sigma (w-b) \quad (11.10)$$

$$\text{s.t.} \quad \begin{cases} \mathbf{1}_n^\top w = 1 \\ w \in \Omega \\ \mathbf{0}_n \le w \le 1 \quad \{\mathbf{CI} < \mathbf{CI}^{(m,n)}\} \end{cases}$$

Finally, a third method consists in re-weighting the remaining assets:

$$w_i^* = \frac{1 \{\mathbf{CI}_i < \mathbf{CI}^{(m,n)}\} \cdot b_i}{\sum_{k=1}^n 1 \{\mathbf{CI}_k < \mathbf{CI}^{(m,n)}\} \cdot b_k} \quad (11.11)$$

**Remark 138** Problem (11.9) is called the “threshold” approach, whereas problems (11.10) and (11.11) are known as the “order-statistic” and “naive” approach.

The QP form of problem (11.9) is: $Q = \Sigma$, $R = \Sigma b$, $A = \mathbf{1}_n^\top$, $B = 1$, $C = \mathbf{CI}^\top$, $D = (1 - \mathbf{R}) \mathbf{CI}(b)$, $w^- = \mathbf{0}_n$ and $w^+ = \mathbf{1}_n$. For problem (11.10), the matrices $C$ and $D$ vanishes and the upper bound becomes $w^+ = 1 \{\mathbf{CI} < \mathbf{CI}^{(m,n)}\}$, which is a vector of zeros and ones.

**Example 56** We consider a capitalization-weighted equity index, which is composed of eight stocks. Their weights are equal to 20%, 19%, 17%, 13%, 12%, 8%, 6% and 5%. The carbon intensities (expressed in tCO₂e/$ mn) are respectively equal to 100.5, 97.2, 250.4, 352.3, 27.1, 54.2, 78.6 and 426.7. To evaluate the risk of the portfolio, we use the market one-factor model: the beta $\beta_i$ of each stock is equal to 0.30, 1.80, 0.85, 0.83, 1.47, 0.94, 1.67 and 1.08, the idiosyncratic volatilities $\tilde{\sigma}_i$ are respectively equal to 10%, 5%, 6%, 12%, 15%, 4%, 8% and 7%, and the estimated market volatility $\sigma_m$ is 18%.

In order to solve the different optimization problems, we need to compute the covariance matrix:

$$\Sigma = \beta \beta^\top \sigma_m^2 + D$$

where $\beta$ is the vector of beta coefficients, $\sigma_m^2$ is the variance of the market portfolio and $D = \text{diag}(\tilde{\sigma}_1^2, \dots, \tilde{\sigma}_n^2)$ is the diagonal matrix, whose elements are the idiosyncratic variances. The specification of the $D$ matrix in the optimization problem (11.9) also requires to compute the carbon intensity of the benchmark and we obtain $\mathbf{CI}(b) = 160.57 \text{ tCO}_2\text{/S mn}$. We report the different results in Tables 11.6, 11.7 and 11.8. With the threshold approach, we can target an explicit value of the reduction rate. For instance, if $\mathbf{R}$ is set to 10%, the tracking error volatility $\sigma(w^*|b)$ is equal to 30 bps. In the case of the order-statistic approach, it is not possible to have an optimized portfolio, whose tracking error volatility is less than 37 bps. This figure is obtained when we exclude the eight stock, which has the largest carbon intensity. In this case, the reduction rate is equal to 9.62%. If we exclude the two assets that have the largest carbon footprint (asset #4 and #8), the tracking error volatility and the reduction rate are equal to 1.68% and 29.33%. If we use the order-statistic approach, it is then not possible to target an intermediary reduction rate, for instance 20% in our example. For a given value of $m$, the naive implies a lower reduction rate and a higher tracking risk compared to the order-statistic approach. For example, if $m$ is equal to 3, their values are respectively equal to 2.25% and 54.05% versus 3.04% and 51.26%. In Figure 11.3, we show the efficient decarbonization frontier which indicates the relationship between the reduction rate and
<table><caption>Table 11.6: Optimal decarbonization portfolios (Example 56, threshold approach)</caption><thead><tr><th><i>R</i></th><th>0</th><th>10</th><th>20</th><th>30</th><th>40</th><th>50</th><th><i>CI</i><sub>i</sub></th></tr></thead><tbody><tr><th><i>w</i><sup>*</sup><sub>1</sub></th><td>20.00</td><td>20.54</td><td>21.14</td><td>21.86</td><td>22.58</td><td>22.96</td><td>100.5</td></tr><tr><th><i>w</i><sup>*</sup><sub>2</sub></th><td>19.00</td><td>19.33</td><td>19.29</td><td>18.70</td><td>18.11</td><td>17.23</td><td>97.2</td></tr><tr><th><i>w</i><sup>*</sup><sub>3</sub></th><td>17.00</td><td>15.67</td><td>12.91</td><td>8.06</td><td>3.22</td><td>0.00</td><td>250.4</td></tr><tr><th><i>w</i><sup>*</sup><sub>4</sub></th><td>13.00</td><td>12.28</td><td>10.95</td><td>8.74</td><td>6.53</td><td>3.36</td><td>352.3</td></tr><tr><th><i>w</i><sup>*</sup><sub>5</sub></th><td>12.00</td><td>12.26</td><td>12.60</td><td>13.07</td><td>13.53</td><td>14.08</td><td>27.1</td></tr><tr><th><i>w</i><sup>*</sup><sub>6</sub></th><td>8.00</td><td>11.71</td><td>16.42</td><td>22.57</td><td>28.73</td><td>34.77</td><td>54.2</td></tr><tr><th><i>w</i><sup>*</sup><sub>7</sub></th><td>6.00</td><td>6.36</td><td>6.69</td><td>7.00</td><td>7.30</td><td>7.59</td><td>78.6</td></tr><tr><th><i>w</i><sup>*</sup><sub>8</sub></th><td>5.00</td><td>1.86</td><td>0.00</td><td>0.00</td><td>0.00</td><td>0.00</td><td>426.7</td></tr><tr><th>σ(<i>w</i><sup>*</sup> | <i>b</i>)</th><td>0.00</td><td>30.01</td><td>61.90</td><td>104.10</td><td>149.65</td><td>196.87</td><td></td></tr><tr><th><i>CI</i>(<i>w</i>)</th><td>160.57</td><td>144.52</td><td>128.46</td><td>112.40</td><td>96.34</td><td>80.29</td><td></td></tr><tr><th><i>R</i>(<i>w</i> | <i>b</i>)</th><td>0.00</td><td>10.00</td><td>20.00</td><td>30.00</td><td>40.00</td><td>50.00</td><td></td></tr></tbody></table>

The reduction rate and weights are expressed in %, while the tracking error volatility is measured in bps.

<table><caption>Table 11.7: Optimal decarbonization portfolios (Example 56, order-statistic approach)</caption><thead><tr><th><i>m</i></th><th>0</th><th>1</th><th>2</th><th>3</th><th>4</th><th>5</th><th>6</th><th>7</th><th><i>CI</i><sub>i</sub></th></tr></thead><tbody><tr><th><i>w</i><sup>*</sup><sub>1</sub></th><td>20.00</td><td>20.40</td><td>22.35</td><td>26.46</td><td>0.00</td><td>0.00</td><td>0.00</td><td>0.00</td><td>100.5</td></tr><tr><th><i>w</i><sup>*</sup><sub>2</sub></th><td>19.00</td><td>19.90</td><td>20.07</td><td>20.83</td><td>7.57</td><td>0.00</td><td>0.00</td><td>0.00</td><td>97.2</td></tr><tr><th><i>w</i><sup>*</sup><sub>3</sub></th><td>17.00</td><td>17.94</td><td>21.41</td><td>0.00</td><td>0.00</td><td>0.00</td><td>0.00</td><td>0.00</td><td>250.4</td></tr><tr><th><i>w</i><sup>*</sup><sub>4</sub></th><td>13.00</td><td>13.24</td><td>0.00</td><td>0.00</td><td>0.00</td><td>0.00</td><td>0.00</td><td>0.00</td><td>352.3</td></tr><tr><th><i>w</i><sup>*</sup><sub>5</sub></th><td>12.00</td><td>12.12</td><td>12.32</td><td>12.79</td><td>13.04</td><td>14.26</td><td>18.78</td><td>100.00</td><td>27.1</td></tr><tr><th><i>w</i><sup>*</sup><sub>6</sub></th><td>8.00</td><td>10.04</td><td>17.14</td><td>32.38</td><td>74.66</td><td>75.12</td><td>81.22</td><td>0.00</td><td>54.2</td></tr><tr><th><i>w</i><sup>*</sup><sub>7</sub></th><td>6.00</td><td>6.37</td><td>6.70</td><td>7.53</td><td>4.73</td><td>10.62</td><td>0.00</td><td>0.00</td><td>78.6</td></tr><tr><th><i>w</i><sup>*</sup><sub>8</sub></th><td>5.00</td><td>0.00</td><td>0.00</td><td>0.00</td><td>0.00</td><td>0.00</td><td>0.00</td><td>0.00</td><td>426.7</td></tr><tr><th>σ(<i>w</i><sup>*</sup> | <i>b</i>)</th><td>0.00</td><td>0.37</td><td>1.68</td><td>2.25</td><td>3.98</td><td>4.04</td><td>4.30</td><td>15.41</td><td></td></tr><tr><th><i>CI</i>(<i>w</i>)</th><td>160.57</td><td>145.12</td><td>113.48</td><td>73.78</td><td>55.08</td><td>52.93</td><td>49.11</td><td>27.10</td><td></td></tr><tr><th><i>R</i>(<i>w</i> | <i>b</i>)</th><td>0.00</td><td>9.62</td><td>29.33</td><td>54.05</td><td>65.70</td><td>67.04</td><td>69.42</td><td>83.12</td><td></td></tr></tbody></table>

The reduction rate, weights, and tracking error volatility are expressed in %.

<table><caption>Table 11.8: Optimal decarbonization portfolios (Example 56, naive approach)</caption><thead><tr><th><i>m</i></th><th>0</th><th>1</th><th>2</th><th>3</th><th>4</th><th>5</th><th>6</th><th>7</th><th><i>CI</i><sub>i</sub></th></tr></thead><tbody><tr><th><i>w</i><sup>*</sup><sub>1</sub></th><td>20.00</td><td>21.05</td><td>24.39</td><td>30.77</td><td>0.00</td><td>0.00</td><td>0.00</td><td>0.00</td><td>100.5</td></tr><tr><th><i>w</i><sup>*</sup><sub>2</sub></th><td>19.00</td><td>20.00</td><td>23.17</td><td>29.23</td><td>42.22</td><td>0.00</td><td>0.00</td><td>0.00</td><td>97.2</td></tr><tr><th><i>w</i><sup>*</sup><sub>3</sub></th><td>17.00</td><td>17.89</td><td>20.73</td><td>0.00</td><td>0.00</td><td>0.00</td><td>0.00</td><td>0.00</td><td>250.4</td></tr><tr><th><i>w</i><sup>*</sup><sub>4</sub></th><td>13.00</td><td>13.68</td><td>0.00</td><td>0.00</td><td>0.00</td><td>0.00</td><td>0.00</td><td>0.00</td><td>352.3</td></tr><tr><th><i>w</i><sup>*</sup><sub>5</sub></th><td>12.00</td><td>12.63</td><td>14.63</td><td>18.46</td><td>26.67</td><td>46.15</td><td>60.00</td><td>100.00</td><td>27.1</td></tr><tr><th><i>w</i><sup>*</sup><sub>6</sub></th><td>8.00</td><td>8.42</td><td>9.76</td><td>12.31</td><td>17.78</td><td>30.77</td><td>40.00</td><td>0.00</td><td>54.2</td></tr><tr><th><i>w</i><sup>*</sup><sub>7</sub></th><td>6.00</td><td>6.32</td><td>7.32</td><td>9.23</td><td>13.33</td><td>23.08</td><td>0.00</td><td>0.00</td><td>78.6</td></tr><tr><th><i>w</i><sup>*</sup><sub>8</sub></th><td>5.00</td><td>0.00</td><td>0.00</td><td>0.00</td><td>0.00</td><td>0.00</td><td>0.00</td><td>0.00</td><td>426.7</td></tr><tr><th>σ(<i>w</i><sup>*</sup> | <i>b</i>)</th><td>0.00</td><td>0.39</td><td>1.85</td><td>3.04</td><td>9.46</td><td>8.08</td><td>8.65</td><td>15.41</td><td></td></tr><tr><th><i>CI</i>(<i>w</i>)</th><td>160.57</td><td>146.57</td><td>113.95</td><td>78.26</td><td>68.38</td><td>47.32</td><td>37.94</td><td>27.10</td><td></td></tr><tr><th><i>R</i>(<i>w</i> | <i>b</i>)</th><td>0.00</td><td>8.72</td><td>29.04</td><td>51.26</td><td>57.41</td><td>70.53</td><td>76.37</td><td>83.12</td><td></td></tr></tbody></table>

The reduction rate, weights, and tracking error volatility are expressed in %.
<figure><img src="image_497.png" /><figcaption>Figure 11.4: Efficient decarbonization frontier of the interpolated naive approach (Example 56)</figcaption></figure>

<figure><img src="image_498.png" /></figure>
the tracking error volatility. In the case of the threshold approach, it is a continuous curve, which is not the case for the two other approaches.

In the case of the naive approach, we can extend the method to a given reduction rate $\mathcal{R}$. Let us denote by $w^*_{(m)}$ the optimal portfolio when we exclude $m$ assets. We consider the index $m$ such that$^{21}$:

$$ \mathcal{R}(w^*_{(m-1)} | b) \le \mathcal{R} \le \mathcal{R}(w^*_{(m)} | b) $$

We define the portfolio $w^*$ as the linear interpolation between $w^*_{(m-1)}$ and $w^*_{(m)}$:

$$ w^* = \alpha w^*_{(m-1)} + (1 - \alpha) w^*_{(m)} $$

where:

$$ \alpha = \frac{\mathcal{R}(w^*_{(m)} | b) - \mathcal{R}}{\mathcal{R}(w^*_{(m)} | b) - \mathcal{R}(w^*_{(m-1)} | b)} $$

We deduce that:

$$ \begin{aligned} \mathcal{R}(w^* | b) &= 1 - \frac{w^{*\top} \mathcal{C} \mathcal{I}}{b^{\top} \mathcal{C} \mathcal{I}} \\ &= 1 - \frac{\left( \alpha w^*_{(m-1)} + (1-\alpha) w^*_{(m)} \right)^{\top} \mathcal{C} \mathcal{I}}{b^{\top} \mathcal{C} \mathcal{I}} \\ &= (\alpha + (1-\alpha)) - \left( \alpha \frac{w^{*\top}_{(m-1)} \mathcal{C} \mathcal{I}}{b^{\top} \mathcal{C} \mathcal{I}} + (1-\alpha) \frac{w^{*\top}_{(m)} \mathcal{C} \mathcal{I}}{b^{\top} \mathcal{C} \mathcal{I}} \right) \\ &= \alpha \mathcal{R}(w^*_{(m-1)} | b) + (1-\alpha) \mathcal{R}(w^*_{(m)} | b) \\ &= \mathcal{R} \end{aligned} $$

In Figure 11.4, we report the efficient decarbonization frontier when we apply the interpolated naive method to Exercise. We notice that the tracking error volatility is not necessarily an increasing function of the reduction rate. This is certainly the main drawback of the naive approach. The monotonicity property is also not always satisfied when we consider the order-statistic approach$^{22}$.

## Bond portfolios

In the case of bond portfolios, we can use the same framework as equity portfolios except that we use the synthetic risk measure $\mathcal{R}(w | b)$ instead of the tracking error variance. We can implement the threshold or order-statistic approach, but the naive approach is not really appropriate since it may induce high deformation of the MD and DTS profiles.

**Example 57** We consider a debt-weighted bond index, which is composed of eight bonds. Their weights are equal to 20%, 19%, 17%, 13%, 12%, 8%, 6% and 5%. The carbon intensities (expressed in tCO$_2$e/$mn) are respectively equal to 100.5, 97.2, 250.4, 352.3, 27.1, 54.2, 78.6 and 426.7. To evaluate the risk of the portfolio, we use the modified duration which is respectively equal to 3.1, 6.6, 7.2, 5, 4.7, 2.1, 8.1 and 2.6 years, and the duration-times-spread factor, which is respectively equal to 100, 155, 575, 436, 159, 145, 804 and 365 bps. There are two sectors. Bonds #1, #3, #4 and #8 belong to Sector$_1$ while Bonds #2, #5, #6 and #7 belong to Sector$_2$.

Results are given in Tables 11.9 and 11.10. We see the impact of the decarbonization constraint on the modified duration or the DTS factor on the portfolio.

<footnote>$^{21}$We have $\mathcal{R}(w^*_{(0)}) = 0$ and $w^*_{(0)} = b$.</footnote>

<footnote>$^{22}$See Exercise 11.4.2 on page 1124.</footnote>
<table><caption>Table 11.9: Optimal decarbonization portfolios (Example 57, threshold approach)</caption><thead><tr><th>R</th><th>0</th><th>10</th><th>20</th><th>30</th><th>40</th><th>50</th><th>CI<sub>i</sub></th></tr></thead><tbody><tr><td>w<sub>1</sub><sup>*</sup></td><td>20.00</td><td>21.62</td><td>23.93</td><td>26.72</td><td>30.08</td><td>33.44</td><td>100.5</td></tr><tr><td>w<sub>2</sub><sup>*</sup></td><td>19.00</td><td>18.18</td><td>16.98</td><td>14.18</td><td>7.88</td><td>1.58</td><td>97.2</td></tr><tr><td>w<sub>3</sub><sup>*</sup></td><td>17.00</td><td>18.92</td><td>21.94</td><td>22.65</td><td>16.82</td><td>11.00</td><td>250.4</td></tr><tr><td>w<sub>4</sub><sup>*</sup></td><td>13.00</td><td>11.34</td><td>5.35</td><td>0.00</td><td>0.00</td><td>0.00</td><td>352.3</td></tr><tr><td>w<sub>5</sub><sup>*</sup></td><td>12.00</td><td>13.72</td><td>16.14</td><td>21.63</td><td>33.89</td><td>46.14</td><td>27.1</td></tr><tr><td>w<sub>6</sub><sup>*</sup></td><td>8.00</td><td>9.60</td><td>10.47</td><td>10.06</td><td>7.21</td><td>4.36</td><td>54.2</td></tr><tr><td>w<sub>7</sub><sup>*</sup></td><td>6.00</td><td>5.56</td><td>5.19</td><td>4.75</td><td>4.11</td><td>3.48</td><td>78.6</td></tr><tr><td>w<sub>8</sub><sup>*</sup></td><td>5.00</td><td>1.05</td><td>0.00</td><td>0.00</td><td>0.00</td><td>0.00</td><td>426.7</td></tr><tr><td>AS<sub>Sector</sub></td><td>0.00</td><td>6.87</td><td>15.49</td><td>24.07</td><td>31.97</td><td>47.58</td><td></td></tr><tr><td>MD (w)</td><td>5.48</td><td>5.49</td><td>5.45</td><td>5.29</td><td>4.90</td><td>4.51</td><td></td></tr><tr><td>DTS (w)</td><td>301.05</td><td>292.34</td><td>282.28</td><td>266.12</td><td>236.45</td><td>206.78</td><td></td></tr><tr><td>σ<sub>AS</sub> (w | b)</td><td>0.00</td><td>5.57</td><td>12.31</td><td>19.82</td><td>30.04</td><td>43.58</td><td></td></tr><tr><td>σ<sub>MD</sub> (w | b)</td><td>0.00</td><td>0.01</td><td>0.04</td><td>0.17</td><td>0.49</td><td>0.81</td><td></td></tr><tr><td>σ<sub>DTS</sub> (w | b)</td><td>0.00</td><td>8.99</td><td>19.29</td><td>35.74</td><td>65.88</td><td>96.01</td><td></td></tr><tr><td>CI (w)</td><td>160.57</td><td>144.52</td><td>128.46</td><td>112.40</td><td>96.34</td><td>80.29</td><td></td></tr><tr><td>R (w | b)</td><td>0.00</td><td>10.00</td><td>20.00</td><td>30.00</td><td>40.00</td><td>50.00</td><td></td></tr></tbody></table>

The reduction rate, weights, and active share metrics are expressed in %, the MD metrics are measured in years, and the DTS metrics are computed in bps.

<table><caption>Table 11.10: Optimal decarbonization portfolios (Example 57, order-statistic approach)</caption><thead><tr><th>m</th><th>0</th><th>1</th><th>2</th><th>3</th><th>4</th><th>5</th><th>6</th><th>7</th><th>CI<sub>i</sub></th></tr></thead><tbody><tr><td>w<sub>1</sub><sup>*</sup></td><td>20.00</td><td>20.83</td><td>24.62</td><td>64.64</td><td>0.00</td><td>0.00</td><td>0.00</td><td>0.00</td><td>100.5</td></tr><tr><td>w<sub>2</sub><sup>*</sup></td><td>19.00</td><td>18.60</td><td>18.13</td><td>21.32</td><td>3.32</td><td>0.00</td><td>0.00</td><td>0.00</td><td>97.2</td></tr><tr><td>w<sub>3</sub><sup>*</sup></td><td>17.00</td><td>17.79</td><td>26.30</td><td>0.00</td><td>0.00</td><td>0.00</td><td>0.00</td><td>0.00</td><td>250.4</td></tr><tr><td>w<sub>4</sub><sup>*</sup></td><td>13.00</td><td>14.53</td><td>0.00</td><td>0.00</td><td>0.00</td><td>0.00</td><td>0.00</td><td>0.00</td><td>352.3</td></tr><tr><td>w<sub>5</sub><sup>*</sup></td><td>12.00</td><td>12.89</td><td>13.96</td><td>6.00</td><td>36.57</td><td>41.27</td><td>41.27</td><td>100.00</td><td>27.1</td></tr><tr><td>w<sub>6</sub><sup>*</sup></td><td>8.00</td><td>9.74</td><td>11.85</td><td>0.00</td><td>60.11</td><td>58.73</td><td>58.73</td><td>0.00</td><td>54.2</td></tr><tr><td>w<sub>7</sub><sup>*</sup></td><td>6.00</td><td>5.62</td><td>5.15</td><td>8.03</td><td>0.00</td><td>0.00</td><td>0.00</td><td>0.00</td><td>78.6</td></tr><tr><td>w<sub>8</sub><sup>*</sup></td><td>5.00</td><td>0.00</td><td>0.00</td><td>0.00</td><td>0.00</td><td>0.00</td><td>0.00</td><td>0.00</td><td>426.7</td></tr><tr><td>AS<sub>Sector</sub></td><td>0.00</td><td>5.78</td><td>19.72</td><td>49.00</td><td>76.68</td><td>80.00</td><td>80.00</td><td>88.00</td><td></td></tr><tr><td>MD (w)</td><td>5.48</td><td>5.52</td><td>5.54</td><td>4.77</td><td>3.27</td><td>3.17</td><td>3.17</td><td>4.70</td><td></td></tr><tr><td>DTS (w)</td><td>301.05</td><td>295.08</td><td>284.71</td><td>171.82</td><td>150.45</td><td>150.78</td><td>150.78</td><td>159.00</td><td></td></tr><tr><td>σ<sub>AS</sub> (w | b)</td><td>0.00</td><td>5.73</td><td>17.94</td><td>50.85</td><td>66.96</td><td>68.63</td><td>68.63</td><td>95.33</td><td></td></tr><tr><td>σ<sub>MD</sub> (w | b)</td><td>0.00</td><td>0.03</td><td>0.04</td><td>0.63</td><td>2.66</td><td>2.64</td><td>2.64</td><td>3.21</td><td></td></tr><tr><td>σ<sub>DTS</sub> (w | b)</td><td>0.00</td><td>6.21</td><td>16.87</td><td>128.04</td><td>197.22</td><td>197.29</td><td>197.29</td><td>199.22</td><td></td></tr><tr><td>CI (w)</td><td>160.57</td><td>147.94</td><td>122.46</td><td>93.63</td><td>45.72</td><td>43.02</td><td>43.02</td><td>27.10</td><td></td></tr><tr><td>R (w | b)</td><td>0.00</td><td>7.87</td><td>23.74</td><td>41.69</td><td>71.53</td><td>73.21</td><td>73.21</td><td>83.12</td><td></td></tr></tbody></table>

The reduction rate, weights and active share metrics are expressed in %, the MD metrics are measured in years, and the DTS metrics are computed in bps.
### 11.2.2 Sector-specific constraints

Sectors are very important in portfolio construction and allocation. For instance, performance attribution is generally done by distinguishing three components: sector allocation, asset selection and interaction effect. Since the goal of portfolio decarbonization is not to take active bets, this explains that most of allocation constraints concern sectors. Therefore, it is necessary to understand sector classification systems in asset management. In Box 11.1, we describe the GICS system, which is the most known and used sector taxonomy.

#### Sector scenario

The previous approach can be extended by considering one decarbonization scenario per sector instead of a global reduction rate:

$$ \mathcal{CI}(w; \mathbf{Sector}_j) \leq (1 - \mathbf{R}_j) \mathcal{CI}(b; \mathbf{Sector}_j) $$

for $j = 1, \dots, n_{\mathbf{Sector}}$. On page 1045, we have seen that these constraints can be written as:

$$ (\mathbf{s}_j \circ (\mathcal{CI} - \mathcal{CI}^*_j))^{\top} w \leq 0 $$

where $\mathcal{CI}^*_j = (1 - \mathbf{R}_j) \mathcal{CI}(b; \mathbf{Sector}_j)$. The QP form $Cw \leq D$ is then:

$$ C = \begin{pmatrix} (\mathbf{s}_1 \circ (\mathcal{CI} - \mathcal{CI}^*_1))^{\top} \\ \vdots \\ (\mathbf{s}_j \circ (\mathcal{CI} - \mathcal{CI}^*_j))^{\top} \\ \vdots \\ (\mathbf{s}_{n_{\mathbf{Sector}}} \circ (\mathcal{CI} - \mathcal{CI}^*_{n_{\mathbf{Sector}}}))^{\top} \end{pmatrix} $$

and:

$$ D = \begin{pmatrix} (1 - \mathbf{R}_1) \mathcal{CI}(b; \mathbf{Sector}_1) \\ \vdots \\ (1 - \mathbf{R}_j) \mathcal{CI}(b; \mathbf{Sector}_j) \\ \vdots \\ (1 - \mathbf{R}_{n_{\mathbf{Sector}}}) \mathcal{CI}(b; \mathbf{Sector}_{n_{\mathbf{Sector}}}) \end{pmatrix} $$

We can also apply these constraints on a subset of sectors and combine them with a global reduction constraint.

In Table 9.25 on page 930, we have seen that the reduction rates of the IEA NZE scenario depend on the sector. In particular, the Electricity sector must decarbonize itself faster than the other sectors. For 2030, using 2020 as the base year, the reduction rates are 56.9% for the Electricity sector, 18.8% for the Industry sector, 20.0% for the Transport sector, 36.7% for the Buildings sector and 52.4% for the other sectors. The global reduction rate is set to 36.6%. In Table 9.16 on page 913, we have reported the carbon intensity of the MSCI World index. It is equal to 163 tCO₂e/$ mn if we consider the scope 1 + 2 emissions. By 2030, we can then use the following global threshold:

$$ \mathcal{CI}^* = (1 - \mathbf{R}) \mathcal{CI}(b) = (1 - 0.366) \times 163 = 103.3 \text{ tCO}_2\text{e/}$ mn

If we use the sector scenarios, we first have to map the IEA sectors on to the GICS sectors. For instance, the Utilities sector corresponds to the Electricity sector. Since the carbon intensity of this sector is equal to 1794 tCO₂e/$ mn, we can use the following specific threshold:

$$ \mathcal{CI}^*_j = (1 - \mathbf{R}_j) \mathcal{CI}(b; \mathbf{Sector}_j) = (1 - 0.569) \times 1794 = 773.2 \text{ tCO}_2\text{e/}$ mn
Box 11.1: Global Industry Classification Standard (GICS)

The Global Industry Classification Standard (GICS) is an industry taxonomy developed in 1999 by MSCI and Standard & Poor's (S&P) and based on a 4-level structure:

<table><thead><tr><th>Level</th><th>1</th><th>2</th><th>3</th><th>4</th></tr></thead><tbody><tr><th>Name</th><td>Sector</td><td>Industry group</td><td>Industry</td><td>Sub-industry</td></tr><tr><th>Code</th><td>AA</td><td>AABB</td><td>AABBCC</td><td>AABBCCDD</td></tr></tbody></table>

GICS is used as a basis for S&P and MSCI market indexes. The level codes are generated using 2-, 4-, 6- and 8-number classifications, whose root (AA, BB, CC and DD) is a multiple of ten (10, 20, ..., 90). For instance, the Energy sector has the following structure:

* Energy (1010)

- Energy (1010)

* Energy equipment & services (101010)

- Oil & gas drilling (10101010): drilling contractors or owners of drilling rigs that contract their services for drilling wells;

- Oil & gas equipment & services (10101020): manufacturers of equipment, including drilling rigs and equipment, and providers of supplies and services to companies involved in the drilling, evaluation and completion of oil and gas wells;

* Oil, gas & consumable fuels (101020)

- Integrated oil & gas (10102010): integrated oil companies engaged in the exploration & production of oil and gas;

- Oil & gas exploration & production (10102020): companies engaged in the exploration and production of oil and gas not classified elsewhere;

- Oil & gas refining & marketing (10102030): companies engaged in the refining and marketing of oil, gas and/or refined products;

- Oil & gas storage & transportation (10102040): companies engaged in the storage and/or transportation of oil, gas and/or refined products;

- Coal & consumable fuels (10102050): companies primarily involved in the production and mining of coal, related products and other consumable fuels related to the generation of energy.

The Energy sector has then one industry group, two industries and seven sub-industries. Here is the number of items per sector:

<table><thead><tr><th>Name</th><th>Sector</th><th>Industry Group</th><th>Industry</th><th>Sub-industry</th></tr></thead><tbody><tr><td>Energy</td><td>1</td><td>1</td><td>2</td><td>7</td></tr><tr><td>Materials</td><td>1</td><td>1</td><td>5</td><td>17</td></tr><tr><td>Industrials</td><td>1</td><td>3</td><td>14</td><td>27</td></tr><tr><td>Consumer Discretionary</td><td>1</td><td>4</td><td>10</td><td>27</td></tr><tr><td>Consumer Staples</td><td>1</td><td>3</td><td>6</td><td>12</td></tr><tr><td>Health Care</td><td>1</td><td>2</td><td>6</td><td>10</td></tr><tr><td>Financials</td><td>1</td><td>3</td><td>6</td><td>18</td></tr><tr><td>Information Technology</td><td>1</td><td>3</td><td>6</td><td>12</td></tr><tr><td>Communication Services</td><td>1</td><td>2</td><td>5</td><td>10</td></tr><tr><td>Utilities</td><td>1</td><td>1</td><td>5</td><td>6</td></tr><tr><td>Real Estate</td><td>1</td><td>2</td><td>10</td><td>17</td></tr></tbody><tfoot><tr><td>Total</td><td>11</td><td>25</td><td>75</td><td>163</td></tr></tfoot></table>

Finally, we report the level 3 classification in Tables 11.11 and 11.12.
<table><caption>Table 11.11: GICS level 3 classification (March 2023)</caption><thead><tr><th>Sector</th><th>Industry Group</th><th>Industry</th></tr></thead><tbody><tr><td>10 Energy</td><td>1010 Energy</td><td>101010 Energy Equipment & Services</td></tr><tr><td>15 Materials</td><td>1510 Materials</td><td>101020 Oil, Gas & Consumable Fuels<br>151010 Chemicals<br>151020 Construction Materials<br>151030 Containers & Packaging<br>151040 Metals & Mining<br>151050 Paper & Forest Products</td></tr><tr><td>20 Industrials</td><td>2010 Capital Goods</td><td>201010 Aerospace & Defense<br>201020 Building Products<br>201030 Construction & Engineering<br>201040 Electrical Equipment<br>201050 Industrial Conglomerates<br>201060 Machinery<br>201070 Trading Companies & Distributors</td></tr><tr><td>2020 Commercial & Professional Services</td><td>202010 Commercial Services & Supplies</td><td>202020 Professional Services</td></tr><tr><td>2030 Transportation</td><td>203010 Air Freight & Logistics<br>203020 Passenger Airlines<br>203030 Marine Transportation<br>203040 Ground Transportation<br>203050 Transportation Infrastructure</td><td>203010 Air Freight & Logistics<br>203020 Passenger Airlines<br>203030 Marine Transportation<br>203040 Ground Transportation<br>203050 Transportation Infrastructure</td></tr><tr><td>25 Consumer Discretionary</td><td>2510 Automobiles & Components</td><td>251010 Automobile Components<br>251020 Automobiles</td></tr><tr><td>2520 Consumer Durables & Apparel</td><td>252010 Household Durables<br>252020 Leisure Products</td><td>252010 Household Durables<br>252020 Leisure Products</td></tr><tr><td>2530 Consumer Services</td><td>253010 Hotels, Restaurants & Leisure</td><td>252030 Textiles, Apparel & Luxury Goods</td></tr><tr><td>2550 Consumer Discretionary Distribution & Retail</td><td>255010 Diversified Consumer Services<br>255030 Broadline Retail<br>255040 Specialty Retail</td><td>253020 Diversified Consumer Services<br>255030 Broadline Retail<br>255040 Specialty Retail</td></tr><tr><td>30 Consumer Staples</td><td>3010 Consumer Staples Dist. & Retail</td><td>301010 Consumer Staples Distribution & Retail</td></tr><tr><td>3020 Food, Beverage & Tobacco</td><td>302010 Beverages</td><td>302020 Food Products<br>302030 Tobacco</td></tr><tr><td>3030 Household & Personal Products</td><td>303010 Household Products<br>303020 Personal Care Products</td><td>303010 Household Products<br>303020 Personal Care Products</td></tr></tbody></table>
<table><caption>Table 11.12: GICS level 3 classification (March 2023)</caption><thead><tr><th>Sector</th><th>Industry Group</th><th>Industry</th></tr></thead><tbody><tr><td rowspan="4">35 Health Care</td><td rowspan="4">3510 Health Care Equipment & Services</td><td>351010 Health Care Equipment & Supplies</td></tr><tr><td>351020 Health Care Providers & Services</td></tr><tr><td>351030 Health Care Technology</td></tr><tr><td>3520 Pharmaceutical, Biotechnology & Life Sciences</td></tr><tr><td rowspan="5">40 Financials</td><td rowspan="5">4010 Banks</td><td>352010 Biotechnology</td></tr><tr><td>352020 Pharmaceuticals</td></tr><tr><td>352030 Life Sciences Tools & Services</td></tr><tr><td>401010 Banks</td></tr><tr><td>402010 Financial Services</td></tr><tr><td rowspan="4">45 Information Technology</td><td rowspan="4">4510 Software & Services</td><td>402010 Consumer Finance</td></tr><tr><td>402030 Capital Markets</td></tr><tr><td>402040 Mortgage Real Estate Investment Trusts (REITs)</td></tr><tr><td>4030 Insurance</td></tr><tr><td rowspan="5">50 Communication Services</td><td rowspan="5">5010 Telecommunication Services</td><td>403010 Insurance</td></tr><tr><td>451020 IT Services</td></tr><tr><td>451030 Software</td></tr><tr><td>452010 Communications Equipment</td></tr><tr><td>452020 Technology Hardware, Storage & Peripherals</td></tr><tr><td rowspan="5">55 Utilities</td><td rowspan="5">5510 Utilities</td><td>452030 Electronic Equipment, Instruments & Components</td></tr><tr><td>453010 Semiconductors & Semiconductor Equipment</td></tr><tr><td>501010 Diversified Telecommunication Services</td></tr><tr><td>501020 Wireless Telecommunication Services</td></tr><tr><td>5020 Media</td></tr><tr><td rowspan="5">60 Real Estate</td><td rowspan="5">6010 Equity Real Estate Investment Trusts (REITs)</td><td>502010 Entertainment</td></tr><tr><td>502030 Interactive Media & Services</td></tr><tr><td>551010 Electric Utilities</td></tr><tr><td>551020 Gas Utilities</td></tr><tr><td>551030 Multi-Utilities</td></tr><tr><td rowspan="6">6020 Real Estate Management & Dev.</td><td rowspan="6">602010 Real Estate Management & Development</td><td>551040 Water Utilities</td></tr><tr><td>551050 Independent Power and Renewable Electricity Producers</td></tr><tr><td>601010 Diversified REITs</td></tr><tr><td>601025 Industrial REITs</td></tr><tr><td>601030 Hotel & Resort REITs</td></tr><tr><td>601040 Office REITs</td></tr><tr><td>601050 Health Care REITs</td></tr><tr><td>601060 Residential REITs</td></tr><tr><td>601070 Retail REITs</td></tr><tr><td>601080 Specialized REITs</td></tr></tbody></table>
Table 11.13 shows the different sector thresholds for the different scope emissions. For that, we have assumed the following mapping between the GICS sectors and the IEA sectors: Communication Services → Other, Consumer Discretionary → Other, Consumer Staples → Other, Energy → Electricity, Financials → Other, Health Care → Other, Industrials → Industry, Information Technology → Other, Materials → Buildings, Real Estate → Buildings and Utilities → Electricity. This mapping is simplistic since the Industrials GICS sector is composed of three industry groups: capital goods, commercial & professional services and transportation. Therefore, we can obtain a more accurate mapping if we use the GICS level II classification.

<table><caption>Table 11.13: Carbon intensity and threshold in tCO₂e/$ mn by GICS sector (MSCI World, 2030)</caption><thead><tr><th rowspan="2">Sector</th><th colspan="4">CI (b; Sector<sub>j</sub>)</th><th rowspan="2">R<sub>j</sub><br>(in %)</th><th colspan="4">CI<sub>j</sub><sup>*</sup></th></tr><tr><th>SC<sub>1</sub></th><th>SC<sub>1-2</sub></th><th>SC<sub>1-3</sub><sup>up</sup></th><th>SC<sub>1-3</sub></th><th>SC<sub>1</sub></th><th>SC<sub>1-2</sub></th><th>SC<sub>1-3</sub><sup>up</sup></th><th>SC<sub>1-3</sub></th></tr></thead><tbody><tr><td>Communication Services</td><td>2</td><td>28</td><td>134</td><td>172</td><td>52.4</td><td>1</td><td>13</td><td>64</td><td>82</td></tr><tr><td>Consumer Discretionary</td><td>23</td><td>65</td><td>206</td><td>590</td><td>52.4</td><td>11</td><td>31</td><td>98</td><td>281</td></tr><tr><td>Consumer Staples</td><td>28</td><td>55</td><td>401</td><td>929</td><td>52.4</td><td>13</td><td>26</td><td>191</td><td>442</td></tr><tr><td>Energy</td><td>632</td><td>698</td><td>1006</td><td>6823</td><td>56.9</td><td>272</td><td>301</td><td>434</td><td>2941</td></tr><tr><td>Financials</td><td>13</td><td>19</td><td>52</td><td>244</td><td>52.4</td><td>6</td><td>9</td><td>25</td><td>116</td></tr><tr><td>Health Care</td><td>10</td><td>22</td><td>120</td><td>146</td><td>52.4</td><td>5</td><td>10</td><td>57</td><td>70</td></tr><tr><td>Industrials</td><td>111</td><td>130</td><td>298</td><td>1662</td><td>18.8</td><td>90</td><td>106</td><td>242</td><td>1350</td></tr><tr><td>Information Technology</td><td>7</td><td>23</td><td>112</td><td>239</td><td>52.4</td><td>3</td><td>11</td><td>53</td><td>114</td></tr><tr><td>Materials</td><td>478</td><td>702</td><td>1113</td><td>2957</td><td>36.7</td><td>303</td><td>445</td><td>704</td><td>1872</td></tr><tr><td>Real Estate</td><td>22</td><td>101</td><td>167</td><td>571</td><td>36.7</td><td>14</td><td>64</td><td>106</td><td>361</td></tr><tr><td>Utilities</td><td>1744</td><td>1794</td><td>2053</td><td>2840</td><td>56.9</td><td>752</td><td>773</td><td>885</td><td>1224</td></tr><tr><td>MSCI World</td><td>130</td><td>163</td><td>310</td><td>992</td><td>36.6</td><td>82</td><td>103</td><td>196</td><td>629</td></tr></tbody></table>

Source: MSCI (2022), Trucost (2022) & Author's calculations.

**Remark 139** The mapping between the different classifications is not always obvious. Indeed, asset owners and managers generally use the Global Industry Classification System (GICS), while the European Union and the United Nations have their own classification systems: the Statistical Classification of Economic Activities in the European Community (NACE) and the International Standard Industrial Classification (ISIC). Le Guenedal et al. (2022) perform a mapping between the four classification systems: GICS, NACE, ISIC and IEA. At the more granular level, they obtain a perfect match only in 70% of cases.

## Sector and weight deviation constraints

**Equity portfolio** If we consider problem (11.9), we notice that the set of weight constraints are $\Omega_0 \cap \Omega$. The first set of constraint $\Omega_0 = \{w : \mathbf{1}_n^\top w = 1, \mathbf{0}_n \le w \le \mathbf{1}_n\}$ is necessary to obtain a long-only portfolio, but is not sufficient to define a realistic portfolio because it can be concentrated in few number of assets. Therefore, portfolio managers and index providers generally impose some additional constraints to obtain a diversified portfolio:

1. To control the weight deviation between portfolio $w$ and benchmark $b$, we can use:

$$ \Omega := C_1(m_w^-, m_w^+) = \{w : m_w^- b \le w \le m_w^+ b\} $$

where $m_w^- \in [0, 1]$ and $m_w^+ \in [1, \infty]$. In this case, the portfolio's weight $w_i$ can only deviate from the benchmark's weight $b_i$ by lower and upper ratios $m_w^-$ and $m_w^+$. Typical figures are $m_w^- = 1/2$ and $m_w^+ = 2$.
2. Another approach consists in controlling the sector deviations. In this case, we can use a relative deviation allowance<fnref n="23" />:

$$ \Omega := C_2(m_s^-, m_s^+) = \left\{ \forall j : m_s^- \sum_{i \in \mathcal{S}ector_j} b_i \le \sum_{i \in \mathcal{S}ector_j} w_i \le m_s^+ \sum_{i \in \mathcal{S}ector_j} b_i \right\} $$

where $m_s^- \in [0, 1[$ and $m_s^+ \in [1, \infty[$. Typical figures are $m_s^- = 1/2$, $m_s^+ = 2$ and $\delta_s^+ = 5\%$.

In the sequel, we define 4 sets of constraints: $C_0$ imposes long-only constraints, $C_1$ ($m_w^-, m_w^+$) adds stock weight constraints, $C_2$ ($m_s$) adds sector relative allocation constraints with $m_s^- = 1/m_s$ and $m_s^+ = m_s$, and $C_3$ ($m_w^-, m_w^+, m_s$) = $C_1$ ($m_w^-, m_w^+$) $\cap C_2$ ($m_s$) combines $C_1$ and $C_2$.

**Bond portfolio** In the case of bond portfolio optimization, the current problem is already highly constrained at the sector level, because the objective function controls MD and DTS tracking risks at the sector level. Therefore, we need to use another constraints $\Omega_0 \cap \Omega$, where the first set $\Omega_0 := C_0 = \{w : \mathbf{1}_n^T w = 1, \mathbf{0}_n \le w \le \mathbf{1}_n\}$ defines a long-only portfolio and the second set $\Omega$ controls the deviation of risk metrics between portfolio $w$ and benchmark $b$:

1. We can neutralize the modified duration at the portfolio level:

$$ \Omega := C'_1 = \{w : \text{MD}(w) = \text{MD}(b)\} = \left\{ w : \sum_{i=1}^{n} (x_i - b_i) \text{MD}_i = 0 \right\} $$

2. We can neutralize the duration-times-spread factor at the portfolio level:

$$ \Omega := C'_2 = \{w : \text{DTS}(w) = \text{DTS}(b)\} = \left\{ w : \sum_{i=1}^{n} (x_i - b_i) \text{DTS}_i = 0 \right\} $$

3. We can constraint the portfolio to have the same weight per maturity bucket as the benchmark:

$$ \Omega := C'_3 = \left\{ w : \sum_{i \in \mathcal{B}ucket_j} (x_i - b_i) = 0 \right\} $$

where $\mathcal{B}ucket_j$ is the $j^{th}$ maturity bucket. Typical maturity buckets (expressed in years) are: 0-1, 1-3, 3-5, 5-7, 7-10 and 10+.

4. Instead of maturity buckets, we can consider rating categories:

$$ \Omega := C'_4 = \left\{ w : \sum_{i \in \mathcal{R}_j} (x_i - b_i) = 0 \right\} $$

where $\mathcal{R}_j$ is the $j^{th}$ rating category. For investment grade bonds, typical rating categories are: AAA-AA (AAA, AA+, AA and AA-), A (A+, A and A-) and BBB (BBB+, BBB, BBB-).

$$ \Omega = \left\{ \forall j : \left| \sum_{i \in \mathcal{S}ector_j} (w_i - b_i) \right| \le \delta_s^+ \right\} $$

where $\delta_s^+ \in [0, 1]$.

<footnote n="23">²³An alternative approach is to use an absolute deviation allowance:</footnote>
## HCIS constraint

The CTB and PAB labels require that the exposure to sectors highly exposed to climate change is at least equal to the exposure in the investment universe. TEG (2019a) distinguishes two types of sectors:

1. High climate impact sectors (HCIS);

2. Low climate impact sectors (LCIS).

According to TEG (2019a), the first category is made up of “sectors that are key to the low-carbon transition.” They correspond to the following NACE classes: A. Agriculture, Forestry, and Fishing; B. Mining and Quarrying; C. Manufacturing; D. Electricity, Gas, Steam, and Air Conditioning Supply; E. Water Supply; Sewerage, Waste Management, and Remediation Activities; F. Construction; G. Wholesale and Retail Trade; Repair of Motor Vehicles and Motorcycles; H. Transportation and Storage; L. Real Estate Activities. Let $HCIS(w) = \sum_{i \in HCIS} w_i$ be the HCIS weight of portfolio w. In order to comply with CTB and PAB labels, we must verify that:

$$HCIS(w) \geq HCIS(b) \qquad (11.12)$$

TEG (2019b, Appendix B, pages 26-170) has published a mapping between the NACE classes and several sector classification structures: BICS (Bloomberg), GICS (MSCI and S&P), ICB (FTSE) and TRBC (Refinitiv). In the case of the GICS taxonomy, about 70% of sub-industries are classified as high climate impact sectors. The HCIS constraint has been criticized because this figure is very high. This would mean that almost all activities are critical for building a low-carbon economy. Therefore, only two sectors are classified in low climate impact sectors (Communication Services and Financials), but more than half of the Health Care and Information Technology sub-industries are viewed as high climate impact sectors. The original idea of the HCIS constraint was to continue financing the sectors that are essential for reaching a low-carbon economy (e.g., Energy and Utilities) and at the same time promoting investments in green issuers instead of brown issuers in these sectors. Nevertheless, the constraint (11.12) is not very restrictive with this broad HCIS measure. Moreover, this constraint encourages substitutions between sectors or industries and not substitutions between issuers within a same sector. Therefore, the trade-off is not necessarily between green electricity and brown electricity, but for example between electricity generation and health care equipment.

In Table 11.14, we report the weight and the carbon intensity of each sector when applying the HCIS constraint to the MSCI World index. The carbon intensity of the MSCI World index is equal to 992 tCO₂e/US mn when we consider the scope $SC_{1-3}$ and 1498 after the HCIS filter. We verify that the HCIS investment universe has a higher carbon intensity than the reference universe. Nevertheless, the HCIS universe represents 59.79% of the MSCI World index, which is a very high figure. This means that more than half of stocks have a high climate impact. This explains that the HCIS constraint is not really used in portfolio allocation. Indeed, it is difficult to justify that 70% of the Health Care sector, 50% of the Information Technology sector but 0% of the Financials sector are classified as high climate impact sectors.

### 11.2.3 Empirical results

We apply the previous framework to the MSCI World index and the ICE BofA Global Corporate index. The numerical results are those obtained by Barahhou et al. (2022).
<table><caption>Table 11.14: Weight and carbon intensity when applying the HCIS filter (MSCI World, June 2022)</caption><thead><tr><th rowspan="2">Sector</th><th>Index</th><th>HCIS</th><th colspan="2">SC<sub>1</sub></th><th colspan="2">SC<sub>1-2</sub></th><th colspan="2">SC<sub>1-3</sub><sup>up</sup></th><th colspan="2">SC<sub>1-3</sub></th></tr><tr><th>b<sub>j</sub></th><th>b'<sub>j</sub></th><th>CI</th><th>CI'</th><th>CI</th><th>CI'</th><th>CI</th><th>CI'</th><th>CI</th><th>CI'</th></tr></thead><tbody><tr><td>Communication Services</td><td>7.58</td><td>0.00</td><td>2</td><td></td><td>28</td><td></td><td>134</td><td></td><td>172</td><td></td></tr><tr><td>Consumer Discretionary</td><td>10.56</td><td>8.01</td><td>23</td><td>14</td><td>65</td><td>31</td><td>206</td><td>189</td><td>590</td><td>462</td></tr><tr><td>Consumer Staples</td><td>7.80</td><td>7.80</td><td>28</td><td>28</td><td>55</td><td>55</td><td>401</td><td>401</td><td>929</td><td>929</td></tr><tr><td>Energy</td><td>4.99</td><td>4.99</td><td>632</td><td>632</td><td>698</td><td>698</td><td>1006</td><td>1006</td><td>6823</td><td>6823</td></tr><tr><td>Financials</td><td>13.56</td><td>0.00</td><td>13</td><td></td><td>19</td><td></td><td>52</td><td></td><td>244</td><td></td></tr><tr><td>Health Care</td><td>14.15</td><td>9.98</td><td>10</td><td>13</td><td>22</td><td>26</td><td>120</td><td>141</td><td>146</td><td>177</td></tr><tr><td>Industrials</td><td>9.90</td><td>7.96</td><td>111</td><td>132</td><td>130</td><td>151</td><td>298</td><td>332</td><td>1662</td><td>1921</td></tr><tr><td>Information Technology</td><td>21.08</td><td>10.67</td><td>7</td><td>12</td><td>23</td><td>30</td><td>112</td><td>165</td><td>239</td><td>390</td></tr><tr><td>Materials</td><td>4.28</td><td>4.28</td><td>478</td><td>478</td><td>702</td><td>702</td><td>1113</td><td>1113</td><td>2957</td><td>2957</td></tr><tr><td>Real Estate</td><td>2.90</td><td>2.90</td><td>22</td><td>22</td><td>101</td><td>101</td><td>167</td><td>167</td><td>571</td><td>571</td></tr><tr><td>Utilities</td><td>3.21</td><td>3.21</td><td>1744</td><td>1744</td><td>1794</td><td>1794</td><td>2053</td><td>2053</td><td>2840</td><td>2840</td></tr></tbody><tfoot><tr><td>MSCI World</td><td>100.00</td><td>59.79</td><td>130</td><td>210</td><td>163</td><td>252</td><td>310</td><td>458</td><td>992</td><td>1498</td></tr></tfoot></table>

The weights are expressed in %. Column $b_j$ represents the sector weight in the benchmark, while column $b'_j$ indicates the sector weight when applying the HCIS filter. The carbon intensities are measured in tCO₂e/$ mn. Column $CI$ is the WACI measure of the sector, while column $CI'$ takes into account the HCIS filter.

Source: MSCI (2022), Trucost (2022) & Author's calculations.

## Equity portfolios

In Table 11.13 on page 1069, we have reported the carbon intensity of the MSCI World index and its sectors. We obtain 130 tCO₂e/$ mn for scope **SC**₁, 163 tCO₂e/$ mn if we include scope 2, 310 tCO₂e/$ mn if we add the upstream scope 3, and finally 992tCO₂e/$ mn if we consider the full scope **SC**₁⁻³. We observe a high discrepancy between sectors. Low-carbon sectors are Communication Services, Financials, Health Care and Information Technology, whereas high-carbon sectors are Energy, Materials and Utilities. We foresee that decarbonizing a portfolio implies reducing the exposure to high-carbon sectors and increasing the exposure to low-carbon sectors. When imposing sector constraints, for instance sector neutrality, the trade-off will be between issuers of the same sector. In Figures 11.5 and 11.6, we show the empirical distribution of carbon intensities for each sector (logarithmic scale). Portfolio decarbonization consists in underweighting the issuers with high carbon intensities and overweighing the issuers with high carbon intensities. The number of outliers or extreme cases depends on the scope. Therefore, portfolio decarbonization will be more or less difficult depending on the definition of scope emissions.

Barahhou et al. (2022) consider the basic optimization problem:

$$ 
\begin{aligned}
w^* &= \arg \min \frac{1}{2} (w-b)^{\top} \Sigma (w-b) \\
\text{s.t.} \quad & \begin{cases} \mathcal{CI}(w) \le (1-\mathcal{R}) \mathcal{CI}(b) \\ w \in \Omega_0 \cap \Omega \end{cases}
\end{aligned}
$$

and measure the impact of constraints $\Omega_0 \cap \Omega$ on the tracking error volatility. Figure 11.7 corresponds to the $C_0$ constraint. The tracking risk increases when we include scope 2 or upstream scope 3, whereas downstream scope 3 reduces it because of its large dispersion. Sector allocation is given in Table 11.15. We observe that portfolio decarbonization is a strategy that is long on Financials sector and short on Energy, Materials and Utilities sectors.
<figure><img src="image_499.png" /><figcaption>Figure 11.5: Boxplot of carbon intensity per sector (MSCI World, June 2022, scope $SC_{1-2}$)</figcaption></figure>

Source: MSCI (2022), Trucost (2022) & Barahhou et al. (2022).

<figure><img src="image_500.png" /><figcaption>Figure 11.6: Boxplot of carbon intensity per sector (MSCI World, June 2022, scope $SC_{1-3}$)</figcaption></figure>

Source: MSCI (2022), Trucost (2022) & Barahhou et al. (2022).
<figure><img src="image_501.png" /><figcaption>Figure 11.7: Impact of the carbon scope on the tracking error volatility (MSCI World, June 2022, $C_0$ constraint)</figcaption></figure>

Source: MSCI (2022), Trucost (2022) & Barahhou et al. (2022).

<table><caption>Table 11.15: Sector allocation in % (MSCI World, June 2022, $C_0$ constraint, scope $SC_{1-3}$)</caption><thead><tr><th rowspan="2">Sector</th><th rowspan="2">Index</th><th colspan="7">Reduction rate <i>R</i></th></tr><tr><th>30%</th><th>40%</th><th>50%</th><th>60%</th><th>70%</th><th>80%</th><th>90%</th></tr></thead><tbody><tr><td>Communication Services</td><td>7.58</td><td>7.95</td><td>8.15</td><td>8.42</td><td>8.78</td><td>9.34</td><td>10.13</td><td>12.27</td></tr><tr><td>Consumer Discretionary</td><td>10.56</td><td>10.69</td><td>10.69</td><td>10.65</td><td>10.52</td><td>10.23</td><td>9.62</td><td>6.74</td></tr><tr><td>Consumer Staples</td><td>7.80</td><td>7.80</td><td>7.69</td><td>7.48</td><td>7.11</td><td>6.35</td><td>5.03</td><td>1.77</td></tr><tr><td>Energy</td><td>4.99</td><td>4.14</td><td>3.65</td><td>3.10</td><td>2.45</td><td>1.50</td><td>0.49</td><td>0.00</td></tr><tr><td>Financials</td><td>13.56</td><td>14.53</td><td>15.17</td><td>15.94</td><td>16.90</td><td>18.39</td><td>20.55</td><td>28.62</td></tr><tr><td>Health Care</td><td>14.15</td><td>14.74</td><td>15.09</td><td>15.50</td><td>16.00</td><td>16.78</td><td>17.77</td><td>17.69</td></tr><tr><td>Industrials</td><td>9.90</td><td>9.28</td><td>9.01</td><td>8.71</td><td>8.36</td><td>7.79</td><td>7.21</td><td>6.03</td></tr><tr><td>Information Technology</td><td>21.08</td><td>21.68</td><td>22.03</td><td>22.39</td><td>22.88</td><td>23.51</td><td>24.12</td><td>24.02</td></tr><tr><td>Materials</td><td>4.28</td><td>3.78</td><td>3.46</td><td>3.06</td><td>2.56</td><td>1.85</td><td>1.14</td><td>0.24</td></tr><tr><td>Real Estate</td><td>2.90</td><td>3.12</td><td>3.27</td><td>3.41</td><td>3.57</td><td>3.72</td><td>3.71</td><td>2.51</td></tr><tr><td>Utilities</td><td>3.21</td><td>2.28</td><td>1.79</td><td>1.36</td><td>0.90</td><td>0.54</td><td>0.24</td><td>0.12</td></tr></tbody></table>

Source: MSCI (2022), Trucost (2022) & Barahhou et al. (2022).
If we impose the classical weight constraint $C_1(1/3, 3)$, which is very popular in index portfolio management, we observe a high increase in the tracking error volatility (Panel 1, Figure 11.8). Moreover, we generally have no solution when $\mathcal{R} > 60\%$. The issue comes from the lower bound, which is way to narrow. Indeed, portfolio decarbonization is an exclusion process. By imposing a lower bound, we then limit portfolio decarbonization. For instance, we obtain similar results between constraint $C_1(0, 3)$ and constraint $C_0$ (Panel 2, Figure 11.8). Nevertheless, we must be careful when choosing $m_w^+$, because a low value can lead to infeasible solutions. For instance, this is the case of constraint $C_1(0, 1.25)$, as shown in Panel 4, Figure 11.8.

The impact of sector constraints is less important than the impact of weight constraints. For instance, constraint $C_2(2)$ does not increase the tracking error volatility with respect to constraint $C_0$ (Panel 1, Figure 11.9). If we impose sector neutrality (constraint $C_2(1)$), we observe a small increase of the tracking risk. Indeed, for low reduction rates (less than 50%), the tracking error volatility remains below 40 bps (Panel 2, Figure 11.9). At first sight, it may be surprising that weight constraints are more binding than sector constraints. Indeed, we generally consider that the sector contribution is greater than the idiosyncratic contribution. Therefore, we expect that the inter-class dispersion largely dominates the intra-class variance. Nevertheless, this viewpoint is biased because it considers homogeneous sectors. In our case, we use level 1 of the GICS classification. The concept of sector is then very heterogeneous. Within a particular sector, we can have low-carbon and high-carbon issuers as we have seen in Figures 11.5 and 11.6.

The combination of weight and sectoral constraints is a more difficult exercise as shown in the bottom panels in Figure 11.9. This is why portfolio managers generally use less restrictive rules. Thus, $C_3(0, 10, 2)$ constraint has became the standard approach in the ETF market when building climate benchmarks. As shown in Figure 11.10, the impact of this constraint is relatively low.

<figure><img src="image_502.png" /><figcaption>Figure 11.8: Impact of $C_1$ constraint on the tracking error volatility (MSCI World, June 2022)</figcaption></figure>

Source: MSCI (2022), Trucost (2022) & Barahhou et al. (2022).
<figure><img src="image_503.png" /><figcaption>Figure 11.9: Impact of $C_2$ and $C_3$ constraints (MSCI World, June 2022)</figcaption></figure>

Source: MSCI (2022), Trucost (2022) & Barahhou et al. (2022).

<figure><img src="image_504.png" /><figcaption>Figure 11.10: Tracking error volatility with $C_3$ (0, 10, 2) constraint (MSCI World, June 2022)</figcaption></figure>

Source: MSCI (2022), Trucost (2022) & Barahhou et al. (2022).
The efficiency of the order-statistic and naive approaches depends on the distribution of carbon intensities. For instance, if the carbon intensity of the index is concentrated in one stock, then removing this stock reduces dramatically the carbon footprint of the portfolio. In Figure 11.11, we have reported the relative carbon footprint contribution of the *m* worst performing assets:

$$CFC^{(m,n)} = \frac{\sum_{i=1}^{n} 1 \{CI_i \geq CI^{(m,n)}\} \cdot b_i CI_i}{CI(b)}$$

where $CI^{(m,n)} = CI_{n-m+1:n}$ is the $(n-m+1)$-th order statistic. By construction, $CFC^{(m,n)}$ is an increasing function with respect to *m* and we have the following properties: $CFC^{(0,n)} = 0$ and $CFC^{(n,n)} = 1$. The fifty worst performing stocks<fnref n="24" /> represent respectively about 50% and 40% of $SC_1$ and $SC_{1-3}$ carbon intensities of the MSCI World index. The previous analysis can be extended by considering another metric to perform the rank ordering, for instance the absolute contribution of stock *i*:

$$CFC^{(m,n)} = \frac{\sum_{i=1}^{n} 1 \{CIC_i \geq CIC^{(m,n)}\} \cdot b_i CI_i}{CI(b)}$$

where $CIC_i = b_i CI_i$ and $CIC^{(m,n)} = CIC_{n-m+1:n}$. In this approach, we do not consider $CI_i$ but the product $b_i CI_i$ to define the exclusion process. Results are given in Figure 11.12. The 50 worst performing stocks represent then 70% and 55% of $SC_1$ and $SC_{1-3}$ carbon intensities.

<figure><img src="image_505.png" /><figcaption>Figure 11.11: Carbon footprint contribution $CFC^{(m,n)}$ in % (MSCI World, June 2022, first approach)</figcaption></figure>

Source: MSCI (2022), Trucost (2022) & Author's calculations.

<footnote n="24">²⁴The MSCI World index was composed of 1513 stocks in June 2022.</footnote>
<figure><img src="image_506.png" /><figcaption>Figure 11.12: Carbon footprint contribution $CFC^{(m,n)}$ in % (MSCI World, June 2022, second approach)</figcaption></figure>

Source: MSCI (2022), Trucost (2022) & Author's calculations.

<table><caption>Table 11.16: Carbon footprint contribution $CFC^{(m,n)}$ in % (MSCI World, June 2022, second approach, $SC_{1-3}$)</caption><thead><tr><th rowspan="2">Sector</th><th colspan="8">m</th></tr><tr><th>1</th><th>5</th><th>10</th><th>25</th><th>50</th><th>75</th><th>100</th><th>200</th></tr></thead><tbody><tr><td>Communication Services</td><td></td><td></td><td></td><td></td><td></td><td>0.44</td><td>0.44</td><td>0.73</td></tr><tr><td>Consumer Discretionary</td><td></td><td></td><td></td><td>0.78</td><td>1.37</td><td>2.44</td><td>2.93</td><td>4.28</td></tr><tr><td>Consumer Staples</td><td></td><td>2.46</td><td>2.46</td><td>2.46</td><td>3.75</td><td>4.44</td><td>4.92</td><td>5.62</td></tr><tr><td>Energy</td><td></td><td>9.61</td><td>17.35</td><td>23.78</td><td>29.56</td><td>31.78</td><td>33.02</td><td>33.89</td></tr><tr><td>Financials</td><td></td><td></td><td></td><td></td><td></td><td>0.72</td><td>1.53</td><td>1.88</td></tr><tr><td>Health Care</td><td></td><td></td><td></td><td></td><td></td><td></td><td>0.21</td><td>0.37</td></tr><tr><td>Industrials</td><td></td><td></td><td>2.16</td><td>5.59</td><td>7.13</td><td>8.70</td><td>9.48</td><td>13.05</td></tr><tr><td>Information Technology</td><td></td><td></td><td></td><td>0.98</td><td>1.58</td><td>1.94</td><td>2.15</td><td>3.30</td></tr><tr><td>Materials</td><td>4.08</td><td>4.08</td><td>4.08</td><td>5.81</td><td>7.31</td><td>8.81</td><td>9.59</td><td>10.75</td></tr><tr><td>Real Estate</td><td></td><td></td><td></td><td></td><td>0.77</td><td>0.77</td><td>0.77</td><td>0.85</td></tr><tr><td>Utilities</td><td></td><td></td><td></td><td>0.81</td><td>3.20</td><td>3.89</td><td>5.24</td><td>7.98</td></tr><tr><td>Total</td><td>4.08</td><td>16.15</td><td>26.06</td><td>40.21</td><td>54.66</td><td>63.94</td><td>70.29</td><td>82.70</td></tr></tbody></table>

Source: MSCI (2022), Trucost (2022) & Author's calculations.
<table><caption>Table 11.17: Weight contribution $WC^{(m,n)}$ in % (MSCI World, June 2022, second approach, $SC_{1-3}$)</caption><thead><tr><th rowspan="2">Sector</th><th>b<sub>j</sub></th><th colspan="8">m</th></tr><tr><th>(in %)</th><th>1</th><th>5</th><th>10</th><th>25</th><th>50</th><th>75</th><th>100</th><th>200</th></tr></thead><tbody><tr><td>Communication Services</td><td>7.58</td><td></td><td></td><td></td><td></td><td></td><td>0.08</td><td>0.08</td><td>3.03</td></tr><tr><td>Consumer Discretionary</td><td>10.56</td><td></td><td></td><td></td><td>0.58</td><td>1.79</td><td>2.44</td><td>4.51</td><td>5.89</td></tr><tr><td>Consumer Staples</td><td>7.80</td><td></td><td>0.70</td><td>0.70</td><td>0.70</td><td>1.90</td><td>2.50</td><td>2.84</td><td>3.84</td></tr><tr><td>Energy</td><td>4.99</td><td></td><td>1.71</td><td>2.25</td><td>2.96</td><td>3.62</td><td>3.99</td><td>4.33</td><td>4.65</td></tr><tr><td>Financials</td><td>13.56</td><td></td><td></td><td></td><td></td><td></td><td>0.74</td><td>1.17</td><td>2.33</td></tr><tr><td>Health Care</td><td>14.15</td><td></td><td></td><td></td><td></td><td></td><td></td><td>0.95</td><td>1.34</td></tr><tr><td>Industrials</td><td>9.90</td><td></td><td></td><td>0.06</td><td>0.32</td><td>0.70</td><td>0.96</td><td>1.20</td><td>4.12</td></tr><tr><td>Information Technology</td><td>21.08</td><td></td><td></td><td></td><td>0.16</td><td>4.70</td><td>8.42</td><td>8.78</td><td>11.62</td></tr><tr><td>Materials</td><td>4.28</td><td>0.29</td><td>0.29</td><td>0.29</td><td>0.47</td><td>0.88</td><td>1.10</td><td>1.40</td><td>1.87</td></tr><tr><td>Real Estate</td><td>2.90</td><td></td><td></td><td></td><td></td><td>0.05</td><td>0.05</td><td>0.05</td><td>0.23</td></tr><tr><td>Utilities</td><td>3.21</td><td></td><td></td><td></td><td>0.31</td><td>0.86</td><td>1.04</td><td>1.31</td><td>2.33</td></tr></tbody><tfoot><tr><td>Total</td><td></td><td>0.29</td><td>2.71</td><td>3.30</td><td>5.49</td><td>14.50</td><td>21.32</td><td>26.63</td><td>41.24</td></tr></tfoot></table>

Source: MSCI (2022), Trucost (2022) & Author's calculations.

In Table 11.16, we have reported the relative carbon intensity contribution $CFC^{(m,n)}$ for several values of $m$. For example, one stock contributes to 4.08% of the $SC_{1-3}$ carbon intensity$^{25}$, while the fifty worst performing stocks have a contribution of 54.66%. Table 11.16 also shows the sector allocation of $CFC^{(m,n)}$. We verify that the Energy sector is the most important contributor and represents about 50% on average when $m \le 100$. In Table 11.17, we calculate the weight contribution:

$$WC^{(m,n)} = \sum_{i=1}^{n} 1 \{CIC_i \ge CIC^{(m,n)}\} \cdot b_i$$

We notice that the fifty worst performing stocks represent then 14.50% of the MSCI World index in terms of allocation, but 54.66% of its carbon footprint$^{26}$.

**Remark 140** The previous figures highly depend on the scope definition and the ordering approach. For instance, if we consider scope $SC_{1-2}$, the fifty worst performing stocks represent then 12.65% of the MSCI World index in terms of allocation, but 60.58% of its carbon footprint. If we consider the first ordering approach based on the carbon intensity $CI_i$ and not the absolute contribution $CIC_i$, the figures become respectively 2.57% and 45.34%.

We implement the order-statistic optimization problem:

$$ \begin{aligned} w^* &= \arg \min \frac{1}{2} (w-b)^{\top} \Sigma (w-b) \\ \text{s.t.} \quad & \begin{cases} \mathbf{1}_n^{\top} w = 1 \\ \mathbf{0}_n \le w \le w^{(m,n)} \end{cases} \end{aligned} $$

where the upper bound $w^{(m,n)}$ is equal to $1 \{CI < CIC^{(m,n)}\}$ for the first ordering approach and $1 \{CIC < CIC^{(m,n)}\}$ for the second ordering approach$^{27}$. We compare the optimization method

<footnote>$^{25}$We remind that the $SC_{1-3}$ carbon intensity of the MSCI World index is equal to 992 tCO$_2$e/$\$ mn.</footnote>

<footnote>$^{26}$If we consider the first approach, 3.21% of the allocation explains 39.15% of the carbon footprint.</footnote>

<footnote>$^{27}$We have $CIC = (CIC_1, \dots, CIC_n)$ where $CIC_i = b_i CI_i$.</footnote>
<figure><img src="image_507.png" /><figcaption>Figure 11.13: Tracking error volatility (MSCI World, June 2022, SC1-3, first ordering method)</figcaption></figure>

Source: MSCI (2022), Trucost (2022) & Author's calculations.

<figure><img src="image_508.png" /><figcaption>Figure 11.14: Tracking error volatility (MSCI World, June 2022, SC1-3, second ordering method)</figcaption></figure>

Source: MSCI (2022), Trucost (2022) & Author's calculations.
with the naive method:

$$w_i^* = \frac{e_i b_i}{\sum_{k=1}^{n} e_k b_k}$$

where $e_i$ is defined as $1\{CI_i < CI^{(m,n)}\}$ for the first ordering approach and $1\{CIC_i < CIC^{(m,n)}\}$ for the second ordering approach. Results are given in Figures 11.13 and 11.14. Compared to the threshold method, the order-statistic and naive solutions are less efficient with a higher tracking error volatility. We also observe that the second ordering approach is not robust, because it may remove stocks, whose carbon intensity contribution is mainly explained by their weights.

## Bond portfolios

In the case of corporate bonds, Barahhou et al. (2022) solve the following optimization problem:

$$w^* = \arg \min \frac{1}{2} \sum_{i=1}^{n} |w_i - b_i| + 50 \sum_{j=1}^{n_{Sector}} \left| \sum_{i \in Sector_j} (w_i - b_i) \text{DTS}_i \right| \\ \text{s.t.} \quad \begin{cases} CI(w) \le (1 - R) CI(b) \\ w \in C_0 \cap C'_1 \cap C'_3 \cap C'_4 \end{cases}$$

We remind that $C'_1$ constraint neutralizes the modified duration at the portfolio level, whereas $C'_3$ and $C'_4$ constraints requires the portfolio to have the same weights as the benchmark per maturity bucket and rating category. The tracking risk measures $D_{AS}(w|b)$ and $D_{DTS}(w|b)$ are reported in Figures 11.15 and 11.16. We observe that the tracking risk is low when we consider the DTS component, whereas it is significant when we focus on the active share component.

<table><caption>Table 11.18: Sector allocation in % (ICE Global Corp., June 2022, scope $SC_{1-3}$)</caption><thead><tr><th rowspan="2">Sector</th><th rowspan="2">Index</th><th colspan="7">Reduction rate <i>R</i></th></tr><tr><th>30%</th><th>40%</th><th>50%</th><th>60%</th><th>70%</th><th>80%</th><th>90%</th></tr></thead><tbody><tr><td>Communication Services</td><td>7.34</td><td>7.35</td><td>7.34</td><td>7.37</td><td>7.43</td><td>7.43</td><td>7.31</td><td>7.30</td></tr><tr><td>Consumer Discretionary</td><td>5.97</td><td>5.97</td><td>5.96</td><td>5.94</td><td>5.93</td><td>5.46</td><td>4.48</td><td>3.55</td></tr><tr><td>Consumer Staples</td><td>6.04</td><td>6.04</td><td>6.04</td><td>6.04</td><td>6.04</td><td>6.02</td><td>5.39</td><td>4.06</td></tr><tr><td>Energy</td><td>6.49</td><td>5.49</td><td>4.42</td><td>3.84</td><td>3.69</td><td>3.23</td><td>2.58</td><td>2.52</td></tr><tr><td>Financials</td><td>33.91</td><td>34.64</td><td>35.66</td><td>35.96</td><td>36.09</td><td>37.36</td><td>38.86</td><td>39.00</td></tr><tr><td>Health Care</td><td>7.50</td><td>7.50</td><td>7.50</td><td>7.50</td><td>7.50</td><td>7.50</td><td>7.52</td><td>7.48</td></tr><tr><td>Industrials</td><td>8.92</td><td>9.38</td><td>9.62</td><td>10.19</td><td>11.34</td><td>12.07</td><td>13.55</td><td>18.13</td></tr><tr><td>Information Technology</td><td>5.57</td><td>5.57</td><td>5.59</td><td>5.59</td><td>5.60</td><td>5.60</td><td>5.52</td><td>5.27</td></tr><tr><td>Materials</td><td>3.44</td><td>3.43</td><td>3.31</td><td>3.18</td><td>3.12</td><td>2.64</td><td>2.25</td><td>1.86</td></tr><tr><td>Real Estate</td><td>4.76</td><td>4.74</td><td>4.74</td><td>4.74</td><td>4.74</td><td>4.66</td><td>4.61</td><td>3.93</td></tr><tr><td>Utilities</td><td>10.06</td><td>9.89</td><td>9.82</td><td>9.64</td><td>8.52</td><td>8.04</td><td>7.92</td><td>6.88</td></tr></tbody></table>

Source: ICE (2022), Trucost (2022) & Barahhou et al. (2022).

Table 11.18 shows the sector allocation when considering scope $SC_{1-3}$. Like for equities, the decarbonization process is a strategy that is long on the Financials sector and short on Materials and Utilities sectors. Health care, Communication Services, Consumer Discretionary, and Information Technology weights are very close to their benchmark's. The case of the Industrials sector may be disturbing, but the deviation highly depends on the scope.
<figure><img src="image_509.png" /><figcaption>Figure 11.15: Impact of the carbon scope on the active share in % (ICE Global Corp., June 2022)</figcaption></figure>

Source: ICE (2022), Trucost (2022) & Barahhou et al. (2022).

<figure><img src="image_510.png" /><figcaption>Figure 11.16: Impact of the carbon scope on the DTS risk in bps (ICE Global Corp., June 2022)</figcaption></figure>

Source: ICE (2022), Trucost (2022) & Barahhou et al. (2022).
## 11.3 Net-zero investing

The emergence of net-zero emissions policies is one of the hottest topics in finance today. In particular, net-zero emissions policies have gained significant traction in recent years with the proliferation of net-zero alliances (GFANZ, NZAOA, NZAM, NZBA, etc.) and their commitments. This issue is significantly changing portfolio allocation and investment frameworks for both passive and active investors. Indeed, it implies that investors need to implement dynamic decarbonization pathways with continuous reference to business-as-usual benchmarks. In addition, a net-zero portfolio must be considered to finance the transition. The greenness or green intensity of the portfolio is therefore important. A net-zero investment policy therefore has two main dimensions: decarbonizing the portfolio and financing the transition. This means that net-zero investing is not only a carbon footprint issue, but also a green footprint issue.

According to Ben Slimane et al. (2023b), there are two approaches to implementing a net-zero investment policy. The integrated approach combines the decarbonization and financing dimensions in an allocation process that considers both carbon intensity for the decarbonization dimension and green intensity for the financing dimension. In the core-satellite strategy, the decarbonization dimension is managed within the core portfolio, while the objective of the satellite strategy is to finance the transition to a low-carbon economy.

### 11.3.1 Integrated approach

#### Choice of the decarbonization scenario

**The carbon emissions/intensity approach** A decarbonization scenario is defined as a function that relates a decarbonization rate to a time index $t$:

$$f : \mathbb{R}^{+} \to [0, 1] \\ t \mapsto \mathcal{R}(t_0, t)$$

where $t_0$ is the base year and $\mathcal{R}(t_0, t_0^-) = 0$. In general, we assume that $\mathcal{R}(t_0, t)$ is a nondecreasing function of time $t$. When considering a decarbonization pathway, we need to distinguish between two different concepts: economic decarbonization and financial decarbonization. In the first case, the variable of interest is the level of carbon emissions, while in the second case we use carbon intensity. Figure 11.17 shows the Net-Zero Emissions by 2050 (NZE) scenario provided by the International Energy Agency<fnref n="28" />. This is a normative scenario based on a number of assumptions about the global energy sector. From this scenario, we can calculate the decarbonization path of the real economy and the different sectors. Figure 11.18 compares these with those used by the CTB and PAB benchmarks. It is clear that we are not comparing apples to apples. In fact, in the case of the real economy, carbon emissions $\mathcal{CE}(t)$ are assumed to follow the following trajectory:

$$\mathcal{CE}(t) = (1 - \mathcal{R}(t_0, t)) \mathcal{CE}(t_0)$$

while we have for the PAB and CTB pathways:

$$\mathcal{CI}(t) = (1 - \Delta\mathcal{R})^{t-t_0} (1 - \mathcal{R}^-) \mathcal{CI}(t_0)$$

where $\Delta\mathcal{R} = 7\%$ and $\mathcal{R}^-$ takes the values 30% (CTB) and 50% (PAB) respectively (TEG, 2019a). By construction, the reduction path expressed in terms of carbon intensity must be lower than the

<footnote n="28">²⁸This scenario has already been explored in Chapter 9 (see Figure 9.22 on page 919).</footnote>
<figure><img src="image_511.png" /><figcaption>Figure 11.17: CO₂ emissions by sector in the IEA NZE scenario (in GtCO₂e)</figcaption></figure>

Source: IEA (2021).

<figure><img src="image_512.png" /><figcaption>Figure 11.18: IEA, NZAOA, CTB and PAB decarbonization pathways</figcaption></figure>
reduction path expressed in terms of carbon emissions. This observation raises the question of the magnitude of the reduction rate. Let us assume that the base date is 2020. The Paris aligned benchmarks imply a reduction rate of 65% by 2025 and 75% by 2030 (Table 11.19). This is much higher than the reduction rates proposed by the International Energy Agency, which are about 15% and 40% by 2025 and 2030 respectively. By comparison, the net-zero frameworks for asset owners propose a reduction rate of about 30% by 2025 and 50% by 2030. The NZAOA curve corresponds to this average asset owner trajectory.

<table><caption>Table 11.19: IEA, NZAOA, CTB and PAB decarbonization rates (baseline = 2020)</caption><thead><tr><th>Year</th><th>CTB</th><th>PAB</th><th>NZE</th><th>NZAOA</th></tr><tr><th><i>R</i><sup>-</sup></th><th>30%</th><th>50%</th><th>IEA</th><th>Average</th></tr><tr><th><i>ΔR</i></th><th>7%</th><th>7%</th><th>Scenario</th><th>Scenario</th></tr></thead><tbody><tr><td>2020</td><td>30.0%</td><td>50.0%</td><td>0.0%</td><td>10.0%</td></tr><tr><td>2021</td><td>34.9%</td><td>53.5%</td><td>1.7%</td><td>14.0%</td></tr><tr><td>2022</td><td>39.5%</td><td>56.8%</td><td>3.9%</td><td>18.0%</td></tr><tr><td>2023</td><td>43.7%</td><td>59.8%</td><td>6.7%</td><td>22.0%</td></tr><tr><td>2024</td><td>47.6%</td><td>62.6%</td><td>9.9%</td><td>26.0%</td></tr><tr><td>2025</td><td>51.3%</td><td>65.2%</td><td>13.6%</td><td>30.0%</td></tr><tr><td>2026</td><td>54.7%</td><td>67.7%</td><td>17.8%</td><td>34.0%</td></tr><tr><td>2027</td><td>57.9%</td><td>69.9%</td><td>22.3%</td><td>38.0%</td></tr><tr><td>2028</td><td>60.8%</td><td>72.0%</td><td>27.2%</td><td>42.0%</td></tr><tr><td>2029</td><td>63.6%</td><td>74.0%</td><td>32.1%</td><td>46.0%</td></tr><tr><td>2030</td><td>66.1%</td><td>75.8%</td><td>37.1%</td><td>50.0%</td></tr><tr><td>-2035</td><td>76.4%</td><td>83.2%</td><td>60.2%</td><td>70.3%</td></tr><tr><td>2040</td><td>83.6%</td><td>88.3%</td><td>77.2%</td><td>89.6%</td></tr><tr><td>2045</td><td>88.6%</td><td>91.9%</td><td>87.6%</td><td>95.2%</td></tr><tr><td>2050</td><td>92.1%</td><td>94.3%</td><td>94.6%</td><td>100.0%</td></tr></tbody></table>

Source: Ben Slimane et al. (2023b).

The previous analysis considered a global path for the entire economy. However, Figure 11.17 shows that not all sectors are the same. In particular, three major sectors are affected (buildings, electricity and transportation), while some sectors are “hard-to-abate” such as materials, steel, cement, petrochemicals, etc. Therefore, a net-zero investment policy must focus on these sectors, which means that we must not spend too much effort on some sectors, such as health care or communication services. There is also a sequencing of decarbonization across sectors as shown in Figure 11.19. The order is as follows:

Electricity > Buildings > Transport > Industry

**The carbon budget approach** A different approach has been suggested by Bolton et al. (2022). The underlying idea is to consider the definition of a net-zero emissions scenario. In fact, a NZE scenario can be defined by a decarbonization pathway that satisfies the following constraints:

$$ \begin{cases} \mathbf{CB}(t_0, 2050) \leq \mathbf{CB}^+ \text{ GtCO}_2\text{e} \\ \mathbf{CE}(2050) \approx 0 \text{ GtCO}_2\text{e} \end{cases} $$

where $t_0$ is the base date and $\mathbf{CB}^+$ is the maximum carbon budget. If we look at the SR15 results from IPCC (2018), we can set $t_0 = 2019$ and $\mathbf{CB}^+ = 580 \text{ GtCO}_2\text{e}$, meaning that there is a 50%
<figure><img src="image_513.png" /><figcaption>Figure 11.19: Sectoral decarbonization pathways</figcaption></figure>

probability of limiting the global warning to 1.5°C. If we want to increase this probability, we can replace the maximum carbon budget $CB^+$ with a lower number, e.g., 420 GtCO$_2$e for a 66% probability and 300 GtCO$_2$e for an 83% probability. Over the years, the budget constraint moves, especially if the economy's decarbonization path is not met.

Let us assume that carbon dioxide emissions follow the pathways of the CTB/PAB:

$$ \mathcal{CE}(t) = (1 - \Delta\mathcal{R})^{t-t_0} (1 - \mathcal{R}^{-}) \mathcal{CE}(t_0) $$

Using Equation (9.7) on page 916, we obtain:

$$ \mathcal{CB}(t_0, t) = \left( \frac{(1 - \Delta\mathcal{R})^{t-t_0} - 1}{\ln(1 - \Delta\mathcal{R})} \right) (1 - \mathcal{R}^{-}) \mathcal{CE}(t_0) \quad (11.13) $$

By considering several values of $\mathcal{R}^{-}$ and $\Delta\mathcal{R}$, and assuming that $\mathcal{CE}$ (2020) = 36 GtCO$_2$e, we obtain the figures given in Table 11.20. For instance, the carbon budget $CB$ (2020, 2050) is equal to 308 GtCO$_2$e if $\mathcal{R}^{-} = 30\%$ and $\Delta\mathcal{R} = 7\%$.

Recall that carbon intensity $\mathcal{CI}(t)$ is defined as the ratio of carbon emissions $\mathcal{CE}(t)$ and the normalization variable $Y(t)$:

$$ \mathcal{CI}(t) = \frac{\mathcal{CE}(t)}{Y(t)} $$

Let $\mathcal{RCI}(t_0, t)$ and $\mathcal{RC\mathcal{E}}(t_0, t)$ be the reduction rates of carbon intensity and emissions between $t_0$ and $t$. We have the following relationship:

$$ \mathcal{RCI}(t_0, t) = \frac{\mathcal{CI}(t_0) - \mathcal{CI}(t)}{\mathcal{CI}(t_0)} = \frac{g_Y(t_0, t) + \mathcal{RC\mathcal{E}}(t_0, t)}{1 + g_Y(t_0, t)} $$
<table><caption>Table 11.20: Carbon budget CB (2020, 2050) (in GtCO₂e) when defining the decarbonization pathway of carbon emissions</caption><thead><tr><th><i>R</i><sup>-</sup></th><th>0%</th><th>10%</th><th>20%</th><th>30%</th><th>50%</th><th>75%</th></tr></thead><tbody><tr><th rowspan="6">Δ<i>R</i></th><th>5%</th><td>551</td><td>496</td><td>441</td><td>386</td><td>276</td><td>138</td></tr><tr><th>6%</th><td>491</td><td>442</td><td>393</td><td>344</td><td>245</td><td>123</td></tr><tr><th>7%</th><td>440</td><td>396</td><td>352</td><td>308</td><td>220</td><td>110</td></tr><tr><th>8%</th><td>396</td><td>357</td><td>317</td><td>277</td><td>198</td><td>99</td></tr><tr><th>9%</th><td>359</td><td>323</td><td>287</td><td>251</td><td>180</td><td>90</td></tr><tr><th>10%</th><td>327</td><td>294</td><td>262</td><td>229</td><td>164</td><td>82</td></tr></tbody></table>

where $g_Y(t_0, t)$ is the growth rate of the normalization variable. Assuming that $g_Y(t_0, t) \ge 0$, we can show that the rate of reduction in carbon intensity is always greater than the rate of reduction in carbon emissions. In most cases, we assume that the annual growth rate of the normalization variable is constant: $Y(t) = (1 + g_Y)Y(t-1)$. We deduce that the compound growth rate is equal to $g_Y(t_0, t) = (1 + g_Y)^{t-t_0} - 1$.

Since we have $\mathcal{CE}(t) = Y(t)\mathcal{CI}(t)$, we obtain:

$$ \begin{aligned} \mathcal{CB}(t_0, t) &= \mathcal{CE}(t_0) \int_{t_0}^{t} (1 + g_Y(t_0, s))(1 - \mathcal{RCI}(t_0, s)) \, ds \\ &= (t - t_0) \mathcal{CE}(t_0) + \mathcal{CE}(t_0) \underbrace{\int_{t_0}^{t} g_Y(t_0, s) \, ds}_{\mathcal{CB}_1(t_0,t)} - \underbrace{\int_{t_0}^{t} g_Y(t_0, s) \, ds}_{\mathcal{CB}_2(t_0,t)} \\ &= \underbrace{\mathcal{CE}(t_0) \int_{t_0}^{t} (1 + g_Y(t_0, s)) \mathcal{RCI}(t_0, s) \, ds}_{\mathcal{CB}_3(t_0,t)} \end{aligned} $$

We can divide the carbon budget into three components. The first component $\mathcal{CB}_1(t_0, t)$ corresponds to the total carbon emissions if nothing is done. The second component $\mathcal{CB}_2(t_0, t)$ corresponds to the additional carbon budget if the carbon intensity remains unchanged. The third component $\mathcal{CB}_3(t_0, t)$ is the carbon budget removed by the intensity reduction. If we assume that the annual growth rate of $Y(t)$ is constant and use the PAB/CTB formula for the intensity decarbonization pathway, we get:

$$ \mathcal{CB}(t_0, t) = \frac{(1 + g_Y)^{t-t_0} (1 - \Delta \mathcal{RCI})^{t-t_0} - 1}{\ln(1 + g_Y) + \ln(1 - \Delta \mathcal{RCI})} (1 - \mathcal{R}_{\mathcal{CI}}^{-}) \mathcal{CE}(t_0) \quad (11.14) $$

Table 11.21 shows the effect of $g_Y$, $\mathcal{R}_{\mathcal{CI}}^{-}$ and $\Delta \mathcal{RCI}$ on the calculation of $\mathcal{CB}$ (2020, 2050) when $\mathcal{CE}$ (2020) = 36 GtCO₂e. If $g_Y = 0\%$, we get the previous results given in Table 11.20. When $g_Y > 0$, the estimated carbon budget is higher. For example, if $\mathcal{R}_{\mathcal{CI}}^{-} = 30\%$ and $\Delta \mathcal{RCI} = 7\%$, the carbon budget is equal to 540 GtCO₂e if $g_Y = 5\%$ while it is equal to 308 GtCO₂e if $g_Y = 0\%$. Therefore, we need to use a more aggressive decarbonization pathway for carbon intensity than for carbon emissions.

To satisfy a given carbon budget $\mathcal{CB}(t_0, t)$, we can calibrate a decarbonization pathway in terms of carbon emissions or in terms of carbon intensity. In the first case, we can approximate the integral by the finite sum:

$$ \mathcal{CB}(t_0, t) \approx \sum_{t_i=t_0+1}^{t} \mathcal{CE}(t_i) $$
<table><caption>Table 11.21: Carbon budget CB (2020, 2050) (in GtCO₂e) when defining the decarbonization pathway of carbon intensity</caption><thead><tr><th>R&#773;<sub>CI</sub></th><th>0%</th><th>10%</th><th>20%</th><th>30%</th><th>50%</th><th>75%</th><th>0%</th><th>10%</th><th>20%</th><th>30%</th><th>50%</th><th>75%</th></tr></thead><tbody><tr><td></td><td colspan="5">g<sub>Y</sub> = 1%</td><td colspan="5">g<sub>Y</sub> = 3%</td></tr><tr><td rowspan="5">ΔR<sub>CI</sub></td><td>5%</td><td>619</td><td>557</td><td>495</td><td>433</td><td>309</td><td>155</td><td>793</td><td>714</td><td>635</td><td>555</td><td>397</td><td>198</td></tr><tr><td>6%</td><td>547</td><td>493</td><td>438</td><td>383</td><td>274</td><td>137</td><td>691</td><td>622</td><td>553</td><td>484</td><td>346</td><td>173</td></tr><tr><td>7%</td><td>487</td><td>438</td><td>390</td><td>341</td><td>244</td><td>122</td><td>607</td><td>546</td><td>485</td><td>425</td><td>303</td><td>152</td></tr><tr><td>8%</td><td>436</td><td>392</td><td>349</td><td>305</td><td>218</td><td>109</td><td>536</td><td>482</td><td>429</td><td>375</td><td>268</td><td>134</td></tr><tr><td>9%</td><td>393</td><td>353</td><td>314</td><td>275</td><td>196</td><td>98</td><td>476</td><td>429</td><td>381</td><td>333</td><td>238</td><td>119</td></tr><tr><td>10%</td><td>356</td><td>320</td><td>285</td><td>249</td><td>178</td><td>89</td><td>426</td><td>383</td><td>341</td><td>298</td><td>213</td><td>107</td></tr><tr><td></td><td colspan="5">g<sub>Y</sub> = 5%</td><td colspan="5">g<sub>Y</sub> = 10%</td></tr><tr><td rowspan="5">ΔR<sub>CI</sub></td><td>5%</td><td>1040</td><td>936</td><td>832</td><td>728</td><td>520</td><td>260</td><td>2245</td><td>2021</td><td>1796</td><td>1572</td><td>1123</td><td>561</td></tr><tr><td>6%</td><td>893</td><td>804</td><td>715</td><td>625</td><td>447</td><td>223</td><td>1859</td><td>1673</td><td>1487</td><td>1301</td><td>930</td><td>465</td></tr><tr><td>7%</td><td>772</td><td>695</td><td>618</td><td>540</td><td>386</td><td>193</td><td>1549</td><td>1394</td><td>1239</td><td>1084</td><td>774</td><td>387</td></tr><tr><td>8%</td><td>672</td><td>605</td><td>538</td><td>470</td><td>336</td><td>168</td><td>1299</td><td>1169</td><td>1039</td><td>909</td><td>649</td><td>325</td></tr><tr><td>9%</td><td>589</td><td>530</td><td>471</td><td>412</td><td>295</td><td>147</td><td>1096</td><td>987</td><td>877</td><td>767</td><td>548</td><td>274</td></tr><tr><td>10%</td><td>520</td><td>468</td><td>416</td><td>364</td><td>260</td><td>130</td><td>932</td><td>839</td><td>746</td><td>653</td><td>466</td><td>233</td></tr></tbody></table>

and choose a path {$\mathcal{C}\mathcal{E}(t_0+1), \dots, \mathcal{C}\mathcal{E}(t)$} that satisfies the carbon budget. We can also assume that $\mathcal{C}\mathcal{E}(t) = (1 - \Delta\mathcal{R}_{\mathcal{C}\mathcal{E}})^{t-t_0} (1 - \mathcal{R}_{\mathcal{C}\mathcal{E}}) \mathcal{C}\mathcal{E}(t_0)$ and calibrate the parameters $(\mathcal{R}_{\mathcal{C}\mathcal{E}}, \Delta\mathcal{R}_{\mathcal{C}\mathcal{E}})$ such that they satisfy the inequality:

$$ \frac{(1 - \Delta\mathcal{R}_{\mathcal{C}\mathcal{E}})^{t-t_0} - 1}{\ln(1 - \Delta\mathcal{R}_{\mathcal{C}\mathcal{E}})} (1 - \mathcal{R}_{\mathcal{C}\mathcal{E}}) \mathcal{C}\mathcal{E}(t_0) \le \mathcal{CB}(t_0, t) $$

We remark that:

$$ \frac{(1 - \Delta\mathcal{R}_{\mathcal{C}\mathcal{E}})^{t-t_0} - 1}{\ln(1 - \Delta\mathcal{R}_{\mathcal{C}\mathcal{E}})} \le \frac{1}{1 - \mathcal{R}_{\mathcal{C}\mathcal{E}}} \frac{\mathcal{CB}(t_0, t)}{\mathcal{C}\mathcal{E}(t_0)} \quad (11.15) $$

This means that the solution does not depend on the absolute value of the carbon budget, but on the ratio of the carbon budget to current emissions. In the second case, we assume that $g_Y$ is given and calibrate the parameters $(\mathcal{R}_{\mathcal{C}\mathcal{E}}, \Delta\mathcal{R}_{\mathcal{C}\mathcal{E}})$ in the same way:

$$ \frac{(1 - \Delta\mathcal{R}_{\mathcal{C}\mathcal{E}})^{t-t_0} - 1}{\ln(1 + g_Y) + \ln(1 - \Delta\mathcal{R}_{\mathcal{C}\mathcal{E}})} \le \frac{1}{(1 + g_Y)^{t-t_0} (1 - \mathcal{R}_{\mathcal{C}\mathcal{E}})} \frac{\mathcal{CB}(t_0, t)}{\mathcal{C}\mathcal{E}(t_0)} \quad (11.16) $$

**Example 58** We want to reduce the carbon emissions by 400% from $t_0 = 2020$ to $t = 2050$. We assume that $\mathcal{R}_{\mathcal{C}\mathcal{E}} = \mathcal{R}_{\mathcal{C}\mathcal{I}} = 0$ and $g_Y = 3\%$.

We have:

$$ \frac{(t - t_0) \mathcal{C}\mathcal{E}(t_0)}{\mathcal{CB}(t_0, t)} = 400\% \Leftrightarrow \frac{\mathcal{CB}(t_0, t)}{\mathcal{C}\mathcal{E}(t_0)} = \frac{30}{4} = 7.5 $$

because $t - t_0$ is equal to 30 years. The numerical solution to Equation (11.15) is $\Delta\mathcal{R}_{\mathcal{C}\mathcal{E}} = 5.88\%$. It follows that:

$$ \begin{aligned} (1 - \Delta\mathcal{R}_{\mathcal{C}\mathcal{E}}) = (1 + g_Y)(1 - \Delta\mathcal{R}_{\mathcal{C}\mathcal{I}}) &\Leftrightarrow \Delta\mathcal{R}_{\mathcal{C}\mathcal{I}} = 1 - \left(\frac{1 - \Delta\mathcal{R}_{\mathcal{C}\mathcal{E}}}{1 + g_Y}\right) \\ &\Leftrightarrow \Delta\mathcal{R}_{\mathcal{C}\mathcal{I}} = 1 - \left(\frac{1 - 5.88\%}{1 + 3\%}\right) = 8.62\% \end{aligned} $$

We check that this is the solution of Equation (11.16).
## Dynamic decarbonization and portfolio alignment

**General framework** While the decarbonization problem finds an optimal portfolio $w^*(\mathcal{R})$ with respect to a given reduction rate $\mathcal{R}$, the alignment problem defines an optimal portfolio $w^*(t)$ with respect to a given date $t$. Therefore, this second problem can be seen as a special case of the first problem, where we use the mapping function between the date $t$ and the reduction rate $\mathcal{R}$. The inequality constraint $\mathcal{CI}(w) \le (1 - \mathcal{R}) \mathcal{CI}(b)$ becomes:

$$ \mathcal{CI}(t, w) \le (1 - \mathcal{R}(t_0, t)) \mathcal{CI}(t_0, b(t_0)) \qquad (11.17) $$

where $t_0$ is the base year, $\mathcal{R}(t_0, t)$ is the decarbonization pathway of the NZE scenario and $\mathcal{CI}(t_0, b(t_0))$ is the carbon intensity of the benchmark at time $t_0$. Portfolio alignment is the process of aligning the decarbonization rate of the portfolio $w^*(t)$ with respect to the decarbonization pathway of the net-zero scenario. We have the following properties:

* Decarbonizing the aligned portfolio becomes easier over time as the benchmark decarbonizes itself:

$$ \mathcal{CI}(t, b(t)) \ll \mathcal{CI}(t_0, b(t_0)) \quad \text{for } t > t_0 $$

* Decarbonizing the aligned portfolio becomes more difficult over time as the benchmark carbonizes itself:

$$ \mathcal{CI}(t, b(t)) \gg \mathcal{CI}(t_0, b(t_0)) \quad \text{for } t > t_0 $$

* The aligned portfolio matches the benchmark portfolio if the benchmark is sufficiently decarbonized:

$$ \mathcal{CI}(t, b(t)) \le (1 - \mathcal{R}(t_0, t)) \mathcal{CI}(t_0, b(t_0)) $$

Since we have $\mathcal{CI}(t, b(t)) = \sum_{i=1}^{n} \mathcal{CI}_i(t) b_i(t)$, the decarbonization of a net-zero investment process is strongly influenced by two factors: changes in benchmark weights and changes in the carbon intensity of assets. In fact, we can imagine that the decarbonization process will become easier over time because the market capitalization of green assets will grow faster than the market capitalization of brown assets and/or because the global decarbonization of the world is well established and on the right track.

**Remark 141** If we consider a carbon budget/emissions approach, we change the inequality constraint (11.17) by $\mathcal{CE}(t, w) \le (1 - \mathcal{R}(t_0, t)) \mathcal{CE}(t_0, b(t_0))$ or $\mathcal{CB}(t_0, t, w) \le \mathcal{CB}^*(t_0, t)$ where $\mathcal{CB}^*(t_0, t)$ is the carbon budget target at time $t$. In addition, we saw in the previous section that we can calibrate an intensity-based scenario from an emissions- or budget-based scenario.

**Equity portfolios** Suppose the objective is to replicate a benchmark and to align the portfolio with a given scenario. The optimization problem becomes:

$$ w^*(t) = \arg \min \frac{1}{2} (w - b(t))^\top \Sigma(t) (w - b(t)) \qquad (11.18) $$

$$ \text{s.t.} \quad \begin{cases} \mathcal{CI}(t, w) \le (1 - \mathcal{R}(t_0, t)) \mathcal{CI}(t_0, b(t_0)) \\ w \in \Omega_0 \cap \Omega \end{cases} $$

where $\Omega_0 = \mathcal{C}_0 = \{w : \mathbf{1}_n^\top w = 1, \mathbf{0}_n \le w \le \mathbf{1}_n\}$ defines the long-only constraint and $\Omega$ is the set of additional constraints. Note that the benchmark $b(t)$, the covariance matrix $\Sigma(t)$, and the carbon intensity $\mathcal{CI}(t, x)$ are functions of time $t$. This means that the data is updated each time we rebalance the portfolio$^{29}$.

<footnote>$^{29}$For example, at time $t+1$, the optimization problem depends on the data available at that time, not the data available at time $t$ in the past.</footnote>
**Example 59** We consider Example 56. We want to align the portfolio with respect to the CTB scenario. To compute the optimal portfolio $w^*(t)$ where $t = t_0+h$ and $h = 0, 1, 2, ...$ years, we assume that the benchmark $b(t)$, the covariance matrix $\Sigma(t)$, and the vector $CI(t)$ of carbon intensities do not change over time.

First, we compute the mapping function between the time $t$ and the decarbonization rate $R(t, t)$:

$$R(t_0, t) = 1 - (1 - 30\%) \times (1 - 7\%)^h$$

We get $R(t_0, t_0) = 30\%$, $R(t_0, t_0 + 1) = 34.90\%$, $R(t_0, t_0 + 2) = 39.46\%$, and so on. Second, we solve the optimization problem (11.18) for the different values of time $t$. The results are shown in Table 11.22.

<table><caption>Table 11.22: Equity portfolio alignment (Example 59)</caption><thead><tr><th>t</th><th>b(t<sub>0</sub>)</th><th>t<sub>0</sub></th><th>t<sub>0</sub> + 1</th><th>t<sub>0</sub> + 2</th><th>t<sub>0</sub> + 3</th><th>t<sub>0</sub> + 4</th><th>t<sub>0</sub> + 5</th><th>t<sub>0</sub> + 10</th></tr></thead><tbody><tr><td>w<sup>*</sup><sub>1</sub></td><td>20.00</td><td>21.86</td><td>22.21</td><td>22.54</td><td>22.84</td><td>23.02</td><td>22.92</td><td>8.81</td></tr><tr><td>w<sup>*</sup><sub>2</sub></td><td>19.00</td><td>18.70</td><td>18.41</td><td>18.15</td><td>17.90</td><td>17.58</td><td>17.04</td><td>0.00</td></tr><tr><td>w<sup>*</sup><sub>3</sub></td><td>17.00</td><td>8.06</td><td>5.69</td><td>3.48</td><td>1.43</td><td>0.00</td><td>0.00</td><td>0.00</td></tr><tr><td>w<sup>*</sup><sub>4</sub></td><td>13.00</td><td>8.74</td><td>7.66</td><td>6.65</td><td>5.72</td><td>4.56</td><td>2.70</td><td>0.00</td></tr><tr><td>w<sup>*</sup><sub>5</sub></td><td>12.00</td><td>13.07</td><td>13.29</td><td>13.51</td><td>13.70</td><td>13.91</td><td>14.18</td><td>21.22</td></tr><tr><td>w<sup>*</sup><sub>6</sub></td><td>8.00</td><td>22.57</td><td>25.59</td><td>28.39</td><td>31.00</td><td>33.39</td><td>35.54</td><td>62.31</td></tr><tr><td>w<sup>*</sup><sub>7</sub></td><td>6.00</td><td>7.00</td><td>7.15</td><td>7.29</td><td>7.42</td><td>7.53</td><td>7.63</td><td>7.66</td></tr><tr><td>w<sup>*</sup><sub>8</sub></td><td>5.00</td><td>0.00</td><td>0.00</td><td>0.00</td><td>0.00</td><td>0.00</td><td>0.00</td><td>0.00</td></tr><tr><td>σ(w<sup>*</sup> | b(t))</td><td>0.01</td><td>104.10</td><td>126.22</td><td>147.14</td><td>166.79</td><td>185.24</td><td>203.51</td><td>352.42</td></tr><tr><td>CI(t, w)</td><td>160.57</td><td>112.40</td><td>104.53</td><td>97.22</td><td>90.41</td><td>84.08</td><td>78.20</td><td>54.40</td></tr><tr><td>R(w | b(t<sub>0</sub>))</td><td>0.00</td><td>30.00</td><td>34.90</td><td>39.46</td><td>43.70</td><td>47.64</td><td>51.30</td><td>66.12</td></tr></tbody></table>

The reduction rate and weights are expressed in %, while the tracking error volatility is measured in bps.

Instead of using the threshold approach, we can implement the order statistic or the naive approach when performing portfolio alignment. These approaches make sense because net-zero investing is an exclusion process, as shown by Barahhou et al. (2022). Examples of portfolio alignment based on order-statistic and naive approaches can be found in Jondeau et al. (2021), Bolton et al. (2022), and Ben Slimane et al. (2023b).

**Bond portfolios** For bonds, the tracking error volatility is replaced by the active risk function proposed by Barahhou et al. (2022):

$$ \mathcal{D}(w | b) = \varphi \underbrace{\sum_{s=1}^{n_{\text{Sector}}} \left| \sum_{i \in s} (w_i - b_i) \text{DTS}_i \right|}_{\text{DTS component}} + \underbrace{\frac{1}{2} \sum_{i \in b} |w_i - b_i|}_{\text{AS component}} + \underbrace{1_{\Omega_{\text{MD}}}(w)}_{\text{MD component}} \quad (11.19) $$

where $\text{DTS}_i$ and $\text{MD}_i$ are the duration-times-spread and modified duration factors, $\Omega_{\text{MD}} = \{w : \sum_{i=1}^n (w_i - b_i) \text{MD}_i = 0\}$ and $1_{\Omega}(w)$ is the convex indicator function. The optimization problem becomes then:

$$ \begin{array}{l} w^*(t) = \arg \min \mathcal{D}(w | b(t)) \qquad (11.20) \\ \text{s.t.} \quad \begin{cases} \mathcal{CI}(t, w) \le (1 - \mathcal{R}(t_0, t)) \mathcal{CI}(t_0, b(t_0)) \\ w \in \Omega_0 \cap \Omega \end{cases} \end{array} $$
Again, the benchmark $b(t)$ and the risk factors (modified duration, DTS, etc.) are functions of time $t$. To solve this mathematical problem, we transform it into a linear programming problem and use standard LP algorithms.

**Example 60** We consider Example 57. We want to align the portfolio with respect to the CTB scenario. To compute the optimal portfolio $w^*(t)$ where $t = t_0+h$ and $h = 0, 1, 2, ...$ years, we assume that the benchmark, the modified duration and the duration-times-spread factors do not change over time.

The corresponding LP problem is<fnref n="30" />:

$$x^* = \arg \min c^\top x \qquad (11.21) \\ \text{s.t.} \quad \begin{cases} Ax = B \\ Cx \le D \\ x^- \le x \le x^+ \end{cases}$$

where $x = (w, \tau_w, \tau_{DTS})$ is a $18 \times 1$ vector. The $18 \times 1$ vector $c$ is equal to $(\mathbf{0}_8, \frac{1}{2}\mathbf{1}_8, \varphi\mathbf{1}_2)$. The equality constraint includes the convex indicator function $1_{\Omega_{MD}}(w)$ and is defined by:

$$Ax = B \Leftrightarrow \begin{pmatrix} \mathbf{1}_8^\top & \mathbf{0}_8^\top & \mathbf{0}_2^\top \\ \mathbf{MD}^\top & \mathbf{0}_8^\top & \mathbf{0}_2^\top \end{pmatrix} x = \begin{pmatrix} 1 \\ 5.476 \end{pmatrix}$$

The inequality constraints are:

$$Cx \le D \Leftrightarrow \begin{pmatrix} I_8 & -I_8 & \mathbf{0}_{8,2} \\ -I_8 & -I_8 & \mathbf{0}_{8,2} \\ C_{DTS} & \mathbf{0}_{2,8} & -I_2 \\ -C_{DTS} & \mathbf{0}_{2,8} & -I_2 \\ \mathcal{CI}(t)^\top & \mathbf{0}_{1,8} & 0 \end{pmatrix} x \le \begin{pmatrix} b \\ -b \\ 192.68 \\ 108.37 \\ -192.68 \\ -108.37 \\ 160.574 \times (1 - \mathcal{R}(t_0, t)) \end{pmatrix}$$

where:

$$C_{DTS} = \begin{pmatrix} 100 & 0 & 575 & 436 & 0 & 0 & 0 & 365 \\ 0 & 155 & 0 & 0 & 159 & 145 & 804 & 0 \end{pmatrix}$$

Finally, the bounds are $x^- = \mathbf{0}_{18}$ and $x^+ = \infty \cdot \mathbf{1}_{18}$. The solutions are shown in Table 11.23.

## Defining a net-zero investment policy

**General framework** As explained by Barahhou et al. (2022) and Ben Slimane et al. (2023b), net-zero investment policies must address two dimensions: portfolio alignment and financing the transition. Therefore, the set of constraints to be applied must include this second dimension:

$$\Omega = \Omega_{\text{alignment}} \cap \Omega_{\text{transition}}$$

where:

$$\Omega_{\text{alignment}} = \{w : \mathcal{CI}(t, w) \le (1 - \mathcal{R}(t_0, t)) \mathcal{CI}(t_0, b(t_0))\}$$

<footnote n="30">³⁰We use the LP framework and the transformation problem (11.8) defined on page 1056.</footnote>
<table><caption>Table 11.23: Bond portfolio alignment (Example 60)</caption><thead><tr><th>t</th><th>b(t<sub>0</sub>)</th><th>t<sub>0</sub></th><th>t<sub>0</sub> + 1</th><th>t<sub>0</sub> + 2</th><th>t<sub>0</sub> + 3</th><th>t<sub>0</sub> + 4</th><th>t<sub>0</sub> + 5</th><th>t<sub>0</sub> + 10</th></tr></thead><tbody><tr><td>w<sub>1</sub><sup>*</sup></td><td>20.00</td><td>20.00</td><td>20.00</td><td>20.00</td><td>13.98</td><td>17.64</td><td>16.02</td><td>5.02</td></tr><tr><td>w<sub>2</sub><sup>*</sup></td><td>19.00</td><td>13.99</td><td>17.79</td><td>19.00</td><td>19.00</td><td>19.00</td><td>19.00</td><td>19.00</td></tr><tr><td>w<sub>3</sub><sup>*</sup></td><td>17.00</td><td>25.43</td><td>20.96</td><td>17.78</td><td>17.00</td><td>13.64</td><td>11.65</td><td>4.61</td></tr><tr><td>w<sub>4</sub><sup>*</sup></td><td>13.00</td><td>0.00</td><td>0.00</td><td>0.00</td><td>0.00</td><td>0.00</td><td>0.00</td><td>0.00</td></tr><tr><td>w<sub>5</sub><sup>*</sup></td><td>12.00</td><td>28.97</td><td>30.71</td><td>35.84</td><td>43.52</td><td>48.80</td><td>53.33</td><td>71.37</td></tr><tr><td>w<sub>6</sub><sup>*</sup></td><td>8.00</td><td>8.00</td><td>8.00</td><td>5.67</td><td>6.46</td><td>0.92</td><td>0.00</td><td>0.00</td></tr><tr><td>w<sub>7</sub><sup>*</sup></td><td>6.00</td><td>3.61</td><td>2.53</td><td>1.70</td><td>0.04</td><td>0.00</td><td>0.00</td><td>0.00</td></tr><tr><td>w<sub>8</sub><sup>*</sup></td><td>5.00</td><td>0.00</td><td>0.00</td><td>0.00</td><td>0.00</td><td>0.00</td><td>0.00</td><td>0.00</td></tr><tr><td>AS (w)</td><td>0.00</td><td>25.40</td><td>22.68</td><td>24.62</td><td>31.52</td><td>36.80</td><td>41.33</td><td>59.37</td></tr><tr><td>MD (w)</td><td>5.48</td><td>5.48</td><td>5.48</td><td>5.48</td><td>5.48</td><td>5.48</td><td>5.48</td><td>5.48</td></tr><tr><td>DTS (w)</td><td>301.05</td><td>274.61</td><td>248.91</td><td>230.60</td><td>220.10</td><td>204.46</td><td>197.26</td><td>174.46</td></tr><tr><td>D (w | b)</td><td>0.00</td><td>0.39</td><td>0.49</td><td>0.60</td><td>0.72</td><td>0.85</td><td>0.99</td><td>1.57</td></tr><tr><td>CI (w)</td><td>160.57</td><td>112.40</td><td>104.53</td><td>97.22</td><td>90.41</td><td>84.08</td><td>78.20</td><td>54.40</td></tr><tr><td>R (w | b)</td><td>0.00</td><td>30.00</td><td>34.90</td><td>39.46</td><td>43.70</td><td>47.64</td><td>51.30</td><td>66.12</td></tr></tbody></table>

The reduction rate, weights, and active share metrics are expressed in %, the MD metrics are measured in years, and the DTS metrics are calculated in bps.

While specifying $\Omega_{\text{alignment}}$ is straightforward, specifying $\Omega_{\text{transition}}$ is more complex. Indeed, the goal of a net-zero investment policy is to participate in the transformation to a low-carbon economy. As explained in Chapter 15 of IPCC (2022), this transformation involves large programs of capital reallocation for the low-carbon transition. However, ensuring efficient capital allocation in line with climate targets is not an easy task. Therefore, the number of factors to define net-zero investments can be very large. Barahhou et al. (2022) considers three of them, which are:

1. The self-decarbonization of the portfolio;

2. The greenness of the portfolio;

3. The exclusion of net-zero “enemies” (free riders).

Therefore, we have:

$$ \Omega_{\text{transition}} = \Omega_{\text{self-decarbonization}} \cap \Omega_{\text{greenness}} \cap \Omega_{\text{exclusion}} $$

**Self-decarbonization and endogeneity of the decarbonization pathway** In the context of a net-zero scenario, portfolio alignment is a dynamic approach to portfolio decarbonization. Most investors have solved this problem by considering a time-varying rate of carbon footprint reduction. In this case, the portfolio is periodically rebalanced to match the decarbonization pathway. This is what we have done in the previous paragraph. However, the decarbonization dimension of net-zero investing cannot be summarized by a sequence of decarbonization rates or a sequence of portfolio rebalancing. In fact, if net-zero investing consists of building successive independent portfolios, there is no mechanism that respects the endogenous aspect of the decarbonization pathway. In particular, if the time-varying decarbonization is due only to the rebalancing process, it is clear that the portfolio cannot claim to be net-zero. In fact, the endogenous aspect of the decarbonization pathway implies a self-decarbonization of the portfolio. Consider an example to illustrate the concept of self-decarbonization. Suppose the decarbonization rate at the beginning of year $t$ is 30%. For
the next year $t + 1$, the goal is to achieve a decarbonization rate of 35%. Two extreme cases are considered below. In the first panel on the left, effective decarbonization of the portfolio at the end of the year is 25%, which means that the carbon footprint of the portfolio has increased. In this case, we need to rebalance the portfolio to reach the 35% level at the beginning of year $t + 1$. This is the bad case because the self-decarbonization of the portfolio is zero. In the third panel on the right, the decarbonization rate of the portfolio is higher than 35% at the end of the year, which means that we do not need to rebalance the portfolio. This is the good case because the portfolio has decarbonized itself, meaning that the carbon footprint of the issuers in the portfolio is following a trajectory consistent with the net-zero scenario.

<table><thead><tr><th></th><th>Bad case</th><th>Mixed case</th><th>Good case</th></tr></thead><tbody><tr><td>Effective decarbonization</td><td></td><td></td><td></td></tr><tr><td>at the beginning of the year <i>t</i></td><td>30%</td><td>30%</td><td>30%</td></tr><tr><td>at the end of the year <i>t</i></td><td>25%</td><td>33%</td><td>36%</td></tr><tr><td>Self-decarbonization</td><td>0%</td><td>3%</td><td>6%</td></tr><tr><td>Relabancing requirement</td><td>10%</td><td>2%</td><td>0%</td></tr></tbody></table>

The previous example illustrates that we can always follow a decarbonization path by rebalancing the portfolio if it is composed of liquid assets, but this does not mean that the investment process is a net-zero investment policy. In particular, there are some financial businesses where it is difficult to rebalance the portfolio because the assets are not liquid, such as a portfolio of private equities, a portfolio of car insurance policies, or a portfolio of loans.

We therefore need to introduce an incentive mechanism to achieve a minimum level of self-decarbonization. The objective of the temperature ratings is precisely to assess an issuer's ability to adapt to a carbon emissions scenario. Implied temperature ratings can be seen as a synthetic scoring system based on the PAC framework (Le Guenedal et al., 2022), which measures the (past) participation, ambition and credibility of the issuer<fnref n="31" />. As a temperature rating system is often perceived as a black box, we may consider a simplified approach that is more transparent. For example, we can use net-zero targets that have been approved and validated by a third party. Using a linear interpolation model, we can calculate the annual self-decarbonization rate of issuers and derive the self-decarbonization level of portfolios. This simple approach is limited for two reasons. First, the data is not homogeneous, as target dates and scopes may differ. Second, self-decarbonization cannot be calculated for issuers without a net-zero commitment or validation. Another approach is to focus on the first pillar, which is participation. In fact, participation is a technical term used to identify past self-decarbonization. This explains why carbon trends and carbon momentum measures are very important metrics for a net-zero investor. This is a way to introduce a dynamic approach to carbon footprints and move beyond current levels, which are a poor estimate of the issuer's finish line and an even poorer estimate of how quickly the issuer will get there.

Following Barahhou et al. (2022), we can specify the self-decarbonization constraint as follows:

$$ \Omega_{\text{self-decarbonization}} = \{w : \mathcal{CM}(t, w) \leq \mathcal{CM}^*(t)\} $$

where $\mathcal{CM}(t, w)$ is the carbon momentum of the portfolio $w$ at time $t$ and $\mathcal{CM}^*(t)$ is the self-decarbonization minimum threshold. For example, if $\mathcal{CM}^*(t) = -3\%$, we expect to reduce the carbon footprint of the portfolio by 3% next year if the observed trend continues. By construction, carbon momentum is inherently a backward-looking approach to self-decarbonization that can be complemented by more forward-looking measures of self-decarbonization.

<footnote n="31">³¹See Section 9.3.2 on page 920.</footnote>
**Green footprint** The second factor in the transition dimension is the greenness of the portfolio. In fact, to achieve a low-carbon economy by 2050, we need to reduce the carbon footprint, but we also need to improve the green intensity of the economy. This means that net-zero investing is not just a carbon footprint issue, it is also a green footprint issue, and these two concepts are different. The goal of the second factor is then to finance the transition to a low-carbon economy, or in other words, to reallocate capital investment to green activities. The greenness constraint can be written as follows:

$$ \Omega_{\text{greenness}} = \{w : \mathcal{G}\mathcal{I}(t, w) \geq \mathcal{G}\mathcal{I}^*(t)\} $$

where $\mathcal{G}\mathcal{I}(t, w)$ is the green intensity of the portfolio $w$ at time $t$ and $\mathcal{G}\mathcal{I}^*(t)$ is the minimum threshold. In general, the absolute measure $\mathcal{G}\mathcal{I}^*(t)$ is expressed as a relative value with respect to the benchmark:

$$ \mathcal{G}\mathcal{I}^*(t) = (1 + \mathcal{G}) \mathcal{G}\mathcal{I}(t, b(t)) $$

where $\mathcal{G}$ is the minimum growth value. For example, if $\mathcal{G} = 100\%$, we want to improve the green footprint of the benchmark so that the green intensity of the portfolio is at least twice the green intensity of the benchmark$^{32}$.

**Remark 142** Another approach is to use a greenness pathway $\mathcal{G}\mathcal{I}^*(t) = (1 + \mathcal{G}(t_0, t)) \mathcal{G}\mathcal{I}(t, b(t_0))$. In this case, $\Omega_{\text{greenness}}$ and $\Omega_{\text{alignment}}$ are two symmetric constraints.

The choice of green intensity is critical. In theory, we want to improve the future green footprint of the economy. Therefore, an appropriate measure would be a forward-looking metric. The ideal candidate is green capex because it measures current green investment. It is therefore a proxy for future green revenues. However, the scarcity and current robustness of green capex data is an obstacle. As a result, most investors prefer to use green revenue share.

**Net-zero exclusion policy** Any net-zero investment portfolio must include an exclusion policy, which may include sectors and/or issuers. Most asset owners and managers have a coal exclusion policy when implementing net-zero. Some may also include the fossil fuel sector. These sector exclusions are related to ESG exclusion strategies. As with ESG investing, investors also create an exclusion list of issuers. In some cases, the net-zero exclusion list is the same as the ESG exclusion list, but in most cases, the net-zero exclusion list is specific. The criteria can be the issuer's carbon emissions or intensity relative to the sector, but the most popular metric is the temperature score. For example, an issuer will often be excluded if its temperature score is above a cap threshold, such as 4 or 5 degrees Celsius. The rationale is to exclude issuers that are not willing to participate in the net-zero journey. In the same spirit as the temperature score, Barahhou et al. (2022) suggests excluding issuers whose carbon momentum is greater than a threshold $\mathcal{C}\mathcal{M}^+$:

$$ \Omega_{\text{exclusion}} = \{w : \mathcal{C}\mathcal{M}_i \geq \mathcal{C}\mathcal{M}^+ \Rightarrow w_i = 0\} $$

Again, the underlying rationale is to measure the willingness of issuers to follow a net-zero scenario, and to exclude issuers who have not played the game in the past. For example, if $\mathcal{C}\mathcal{M}^+ = 10\%$, we exclude issuers that have recently increased their carbon footprint by more than 10%. On page 928 we have seen that the proportion of excluded issuers can be high, especially if we consider carbon emissions.

**Remark 143** To manage the self-decarbonization of the portfolio or to define the exclusion list, it is better to use the long-term carbon momentum than the short-term carbon momentum.

<footnote>$^{32}$ $\mathcal{G} = 100\%$ is the standard measure in the ETF market.</footnote>
**Equity portfolios** The optimization problem (11.18) on page 1089 becomes:

$$w^*(t) = \arg\min \frac{1}{2} (w - b(t))^\top \Sigma(t) (w - b(t)) \qquad (11.22)$$

$$\text{s.t.} \quad \begin{cases} \mathcal{CI}(t, w) \le (1 - \mathcal{R}(t_0, t)) \mathcal{CI}(t_0, b(t_0)) & \leftarrow \text{Alignment} \\ \mathcal{CM}(t, w) \le \mathcal{CM}^*(t) & \leftarrow \text{Self-decarbonization} \\ \mathcal{GI}(t, w) \ge (1 + \mathcal{G}) \mathcal{GI}(t, b(t)) & \leftarrow \text{Greenness} \\ 0 \le w_i \le 1 \{\mathcal{CM}_i(t) \le \mathcal{CM}^+\} & \leftarrow \text{Exclusion} \\ w \in \Omega_0 \cap \Omega & \leftarrow \text{Other constraints} \end{cases}$$

We deduce that the quadratic form is $Q = \Sigma(t)$, $R = \Sigma(t)b(t)$, $A = \mathbf{1}_n^\top$, $B = 1$, $w^- = \mathbf{0}_n$, $w^+ = \mathbf{1}\{\mathcal{CM}(t) \le \mathcal{CM}^+\}$ and:

$$Cw \le D \Leftrightarrow \begin{pmatrix} \mathcal{CI}(t)^\top \\ \mathcal{CM}(t)^\top \\ -\mathcal{GI}(t)^\top \end{pmatrix} w \le \begin{pmatrix} (1 - \mathcal{R}(t_0, t)) \mathcal{CI}(t_0, b(t_0)) \\ \mathcal{CM}^*(t) \\ -(1 + \mathcal{G}) \mathcal{GI}(t, b(t)) \end{pmatrix}$$

Here, we assume that the carbon momentum function is a linear function:

$$\mathcal{CM}(t, w) = w^\top \mathcal{CM}(t) = \sum_{i=1}^{n} w_i \mathcal{CM}_i(t)$$

where $\mathcal{CM}(t) = (\mathcal{CM}_1(t), \dots, \mathcal{CM}_n(t))$ is the carbon momentum vector.

**Remark 144** In the previous optimization problem, the carbon momentum constraint can be replaced by a temperature constraint $TS(t, w) \le TS^*(t)$ where $TS(t, w) = w^\top TS(t) = \sum_{i=1}^n w_i TS_i(t)$ and $TS_i(t)$ is the temperature score of issuer $i$.

Note that the aggregation of the carbon momentum or temperature score at the portfolio level uses the weighted average approach, as does the WACI formulation for carbon intensity. If we use long-term carbon momentum estimated with a linear trend model, we can perform an exact calculation of the carbon momentum at the portfolio level. Using the results in Box 11.2, we have:

$$\begin{align*} \mathcal{CM}(t, w) \le \mathcal{CM}^*(t) &\Leftrightarrow \sum_{i=1}^{n} \tilde{w}_i \mathcal{CM}_i(t) \le \mathcal{CM}^*(t) \\ &\Leftrightarrow \sum_{i=1}^{n} \frac{w_i \mathcal{CI}_i(t)}{\sum_{j=1}^{n} w_j \mathcal{CI}_j(t)} \mathcal{CM}_i(t) \le \mathcal{CM}^*(t) \\ &\Leftrightarrow \sum_{i=1}^{n} w_i \mathcal{CI}_i(t) \mathcal{CM}_i(t) \le \sum_{j=1}^{n} w_j \mathcal{CI}_j(t) \mathcal{CM}^*(t) \\ &\Leftrightarrow \sum_{i=1}^{n} w_i \mathcal{CI}_i(t) (\mathcal{CM}_i(t) - \mathcal{CM}^*(t)) \le 0 \\ &\Leftrightarrow \sum_{i=1}^{n} w_i \zeta_i \le 0 \end{align*}$$

where:

$$\zeta_i = \mathcal{CI}_i(t) (\mathcal{CM}_i(t) - \mathcal{CM}^*(t))$$
Box 11.2: Carbon momentum aggregation at the portfolio level

We recall that $\mathcal{CM}_i^{Long}(t) = \frac{\hat{\beta}_{i,1}(t)}{\mathcal{CI}_i(t)}$ where $i$ is the issuer, $\mathcal{CI}_i(t)$ is the carbon intensity, and $\hat{\beta}_{i,1}(t)$ is the slope of the linear trend model:

$$ \hat{\mathcal{CI}}_i(t) = \hat{\beta}_{i,0}(t) + \hat{\beta}_{i,1}(t) (t - t_0) $$

The carbon intensity of the portfolio is given by its weighted average: $\mathcal{CI}(t, w) = \sum_{i=1}^{n} w_i \mathcal{CI}_i(t)$. This follows:

$$ \begin{aligned} \hat{\mathcal{CI}}(t, w) &= \sum_{i=1}^{n} w_i \hat{\mathcal{CI}}_i(t) \\ &= \sum_{i=1}^{n} w_i \hat{\beta}_{i,0}(t) + \underbrace{\sum_{i=1}^{n} w_i \hat{\beta}_{i,1}(t) (t - t_0)}_{\hat{\beta}_1(t,w)} \end{aligned} $$

where $\hat{\beta}_1(t, w) = \sum_{i=1}^{n} w_i \hat{\beta}_{i,1}(t)$. We deduce that:

$$ \begin{aligned} \mathcal{CM}^{Long}(t, w) &= \frac{\hat{\beta}_1(t, w)}{\mathcal{CI}(t, w)} \\ &= \frac{\sum_{i=1}^{n} w_i \hat{\beta}_{i,1}(t)}{\sum_{i=1}^{n} w_i \mathcal{CI}_i(t)} \\ &= \frac{\sum_{i=1}^{n} w_i \mathcal{CI}_i(t) \mathcal{CM}_i^{Long}(t)}{\sum_{i=1}^{n} w_i \mathcal{CI}_i(t)} \\ &= \sum_{i=1}^{n} \tilde{w}_i \mathcal{CM}_i^{Long}(t) \end{aligned} $$

where the adjusted weight $\tilde{w}_i$ is equal to:

$$ \tilde{w}_i = \frac{w_i \mathcal{CI}_i(t)}{\sum_{j=1}^{n} w_j \mathcal{CI}_j(t)} $$

We see that $\mathcal{CM}^{Long}(t, w) \neq \sum_{i=1}^{n} w_i \mathcal{CM}_i^{Long}(t)$. This aggregation method is also valid at the sector level by using the weight of each issuer in its respective sector.

The formulation of the QP problem remains the same, but the inequality constraints are changed:

$$ Cw \leq D \Leftrightarrow \begin{pmatrix} \mathcal{CI}(t)^{\top} \\ \zeta^{\top} \\ -\mathcal{GI}(t)^{\top} \end{pmatrix} w \leq \begin{pmatrix} (1 - \mathcal{R}(t_0, t)) \mathcal{CI}(t_0, b(t_0)) \\ 0 \\ -(1 + \mathcal{G}) \mathcal{GI}(t, b(t)) \end{pmatrix} $$

where $\zeta = (\zeta_1, \dots, \zeta_n)$.

If we define the carbon momentum by considering the log-linear trend model, we get $\mathcal{CM}_i^{Long}(t) = \hat{\gamma}_{i,1}(t)$. We cannot find an analytically exact formula $\hat{\gamma}_{i,1}(t)$ for the portfolio. Therefore, we use the weighted average approach:

$$ \mathcal{CM}(t, w) = \sum_{i=1}^{n} w_i \hat{\gamma}_{i,1}(t) $$