In Table 2.28, we observe that the initial empirical distribution of ratings is:

$$ \hat{\pi}^{(0)} = (3.683\%, 11.242\%, 18.317\%, 23.983\%, 21.517\%, 17.483\%, 3.775\%) $$

We conclude that the long-term dynamics of the Markov chain has dramatically change the initial probability distribution. In Table 2.28, we also observe that the final distribution of ratings after one year is:

$$ \hat{\pi}^{(1)} = (3.942\%, 12.708\%, 19.817\%, 23.133\%, 21.458\%, 15.290\%, 3.650\%) $$

We reiterate that the Kolmogorov equation applied to the distribution<sup>73</sup> $\pi^{(n)}$ is given by $\pi^{(n+1)} = P^{\top}\pi^{(n)}$. In particular, we verify that $\hat{\pi}^{(1)} = \hat{P}^{\top}\hat{\pi}^{(0)}$ where $\hat{P} = (\hat{p}_{i,j})$. In Figure 2.36, we have reported the dynamics of $\pi^{(n)}$ with $\pi^{(0)} = \hat{\pi}^{(0)}$. We conclude that the distribution of the score $S(t)$ is not stationary.

<figure><img src="image_61.png" /><figcaption>Figure 2.36: Dynamics of the probability distribution $\pi^{(n)}$ (migration matrix #1)</figcaption></figure>

## Continuous-time modeling

**Markov generator** We now consider the case $t \in \mathbb{R}_+$. We note $P(s;t)$ the transition matrix defined as follows:

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

Assuming that the Markov chain is time-homogenous, we have $P(t) = P(0;t)$. Jarrow et al. (1997) introduce the generator matrix $\Lambda = (\lambda_{i,j})$ where $\lambda_{i,j} \ge 0$ for all $i \ne j$ and $\lambda_{i,i} = -\sum_{j \ne i}^{K} \lambda_{i,j}$. In this case, the transition matrix satisfies the following relationship:

$$ P(t) = \exp(t\Lambda) \qquad (2.6) $$

<footnote n="73"><sup>73</sup>We have $\pi_k^{(n)} = \Pr\{\mathcal{R}(n) = R_k\}$.</footnote>
where $\exp(A)$ is the matrix exponential of $A$. Let us give a probabilistic interpretation of $\Lambda$. If we assume that the probability of jumping from rating $R_i$ to rating $R_j$ in a short time period $\Delta t$ is proportional to $\Delta t$, we have $p(t, i; t + \Delta t, j) = \lambda_{i,j}\Delta t$. The matrix form of this equation is $P(t; t + \Delta t) = \Lambda \Delta t$. We deduce that:

$$P(t + \Delta t) = P(t) P(t; t + \Delta t) = P(t) \Lambda \Delta t$$

and:

$$dP(t) = P(t) \Lambda dt$$

Because we have $\exp(0) = I$, we obtain the solution $P(t) = \exp(t\Lambda)$. We then interpret $\lambda_{i,j}$ as the instantaneous transition rate of jumping from rating $R_i$ to rating $R_j$.

**Remark 27** In Appendix A.1.1, we present the matrix exponential function and its mathematical properties. In particular, we have $e^{A+B} = e^A e^B$ and $e^{A(s+t)} = e^A e^{At}$ where A and B are two square matrices such that $AB = BA$ and $s$ and $t$ are two real numbers.

**Example 8** We consider a rating system with three states: A (good rating), B (average rating) and C (bad rating). The Markov generator is equal to:

$$\Lambda = \begin{pmatrix} -0.30 & 0.20 & 0.10 \\ 0.15 & -0.40 & 0.25 \\ 0.10 & 0.15 & -0.25 \end{pmatrix}$$

The one-year transition probability matrix is equal to:

$$P(1) = e^{\Lambda} = \begin{pmatrix} 75.63\% & 14.84\% & 9.53\% \\ 11.63\% & 69.50\% & 18.87\% \\ 8.52\% & 11.73\% & 79.75\% \end{pmatrix}$$

For the two-year maturity, we get:

$$P(2) = e^{2\Lambda} = \begin{pmatrix} 59.74\% & 22.65\% & 17.61\% \\ 18.49\% & 52.24\% & 29.27\% \\ 14.60\% & 18.76\% & 66.63\% \end{pmatrix}$$

We verify that $P(2) = P(1) \cdot P(1)$. This derives from the property of the matrix exponential:

$$P(t) = e^{t\Lambda} = (e^{\Lambda})^t = P(1)^t$$

The continuous-time framework allows to calculate transition matrices for non-integer maturities, which do not correspond to full years. For instance, the one-month transition probability matrix of the previous example is equal to:

$$P\left(\frac{1}{12}\right) = e^{\frac{1}{12}\Lambda} = \begin{pmatrix} 97.54\% & 1.62\% & 0.83\% \\ 1.22\% & 96.74\% & 2.03\% \\ 0.82\% & 1.22\% & 97.95\% \end{pmatrix}$$
Box 2.8: Estimation of the Markov generator

One of the issues with the continuous-time framework is to estimate the Markov generator $\Lambda$. One solution consists in using the empirical transition matrix $\hat{P}(t)$, which have been calculated for a given time horizon $t$. In this case, the estimate $\hat{\Lambda}$ must satisfy the relationship $\hat{P}(t) = \exp(t\hat{\Lambda})$. We deduce that:

$$ \hat{\Lambda} = \frac{1}{t} \ln (\hat{P}(t)) $$

where $\ln A$ is the matrix logarithm of A. However, the matrix $\hat{\Lambda}$ cannot verify the Markov conditions $\hat{\lambda}_{i,j} \ge 0$ for all $i \neq j$ and $\sum_{j=1}^K \lambda_{i,j} = 0$. Therefore, Israel et al. (2001) propose two estimators to obtain a valid generator:

1. the first approach consists in adding the negative values back into the diagonal values:

$$ \left\{ \begin{array}{ll} \bar{\lambda}_{i,j} = \max(\hat{\lambda}_{i,j}, 0) & i \neq j \\ \bar{\lambda}_{i,i} = \hat{\lambda}_{i,i} + \sum_{j \neq i} \min(\hat{\lambda}_{i,j}, 0) & \end{array} \right. $$

2. in the second method, we carry forward the negative values on the matrix entries which have the correct sign:

$$ \left\{ \begin{array}{ll} G_i = |\hat{\lambda}_{i,i}| + \sum_{j \neq i} \max(\hat{\lambda}_{i,j}, 0) & \\ B_i = \sum_{j \neq i} \max(-\hat{\lambda}_{i,j}, 0) & \\ \tilde{\lambda}_{i,j} = \begin{cases} 0 & \text{if } i \neq j \text{ and } \hat{\lambda}_{i,j} < 0 \\ \frac{\hat{\lambda}_{i,j} - B_i}{G_i} & \text{if } G_i > 0 \\ \hat{\lambda}_{i,j} & \text{if } G_i = 0 \end{cases} & \end{array} \right. $$

**Markov property** The Markov property refers to the lack of memory of stochastic processes. This implies that the probability distribution of future states of the process conditional on both past and present values depends only upon the present state. Therefore, given the present, the future does not depend on the past. In order to better understand the implications of this property, we consider the following example with three companies:

<table><thead><tr><th>t - 2</th><th>t - 1</th><th>t</th><th>t + 1</th></tr></thead><tbody><tr><td>AAA</td><td>→ BBB</td><td>→ BBB</td><td>→ ?</td></tr><tr><td>BBB</td><td>→ BBB</td><td>→ BBB</td><td>→ ?</td></tr><tr><td>BB</td><td>→ BB</td><td>→ BBB</td><td>→ ?</td></tr></tbody></table>

Today, the three companies are rated BBB. We would like to predict the ESG rating of those companies at time $t + 1$. If the ESG ratings are Markovian, these entities are equivalent and have the same conditional probabilities to become AAA, AA, etc. Otherwise, this means that the conditional probabilities depend on the past trajectory. In this case, we have:

$$ \mathrm{Pr}\{\mathcal{R}_{c_1}(t+1) = R_j | \mathcal{R}_{c_1}(t) = R_i\} \neq \mathrm{Pr}\{\mathcal{R}_{c_2}(t+1) = R_j | \mathcal{R}_{c_2}(t) = R_i\} $$

for two different companies $c_1$ and $c_2$. In our example, the firms have different past trajectories. They don't have the same transition matrix if the rating process has not the Markov property.
To verify the Markov property, we compute the matrix $\Lambda' = \ln(P)$ and measure whether $\Lambda'$ is a Markov generator or not. Using the rating migration matrix #1, we obtain the results given in Table 2.29. We notice that $\ln P$ is not a Makov generator since 11 off-diagonal elements are not positive. Using the first method of Israel et al. (2001) described in Box 2.8, we transform this matrix into a Markov generator<sup>74</sup> $\bar{\Lambda}$ (Table 2.30). We recompute the one-year transition matrix $\bar{P}(1) = \exp(\bar{\Lambda})$ and observe some small differences with the original transition matrix (see Table 2.31 vs. Table 2.22).

<table><caption>Table 2.29: Non-Markov generator $\Lambda' = \ln(P)$ of the migration matrix #1 (in %)</caption><thead><tr><th></th><th>AAA</th><th>AA</th><th>A</th><th>BBB</th><th>BB</th><th>B</th><th>CCC</th></tr></thead><tbody><tr><th>AAA</th><td>-7.663</td><td>6.427</td><td>0.542</td><td>0.466</td><td>0.245</td><td>-0.016</td><td>-0.000</td></tr><tr><th>AA</th><td>4.770</td><td>-20.604</td><td>15.451</td><td>-0.001</td><td>0.318</td><td>0.066</td><td>-0.001</td></tr><tr><th>A</th><td>-0.267</td><td>20.259</td><td>-35.172</td><td>14.953</td><td>0.152</td><td>0.083</td><td>-0.008</td></tr><tr><th>BBB</th><td>0.102</td><td>-1.051</td><td>28.263</td><td>-40.366</td><td>13.100</td><td>-0.128</td><td>0.080</td></tr><tr><th>BB</th><td>0.032</td><td>0.307</td><td>-1.762</td><td>28.351</td><td>-37.889</td><td>10.832</td><td>0.129</td></tr><tr><th>B</th><td>-0.005</td><td>-0.008</td><td>0.503</td><td>-2.240</td><td>30.227</td><td>-31.482</td><td>3.006</td></tr><tr><th>CCC</th><td>0.000</td><td>-0.024</td><td>0.194</td><td>0.469</td><td>0.365</td><td>16.806</td><td>-17.810</td></tr></tbody></table>

<table><caption>Table 2.30: Markov generator of the migration matrix #1 (in %)</caption><thead><tr><th></th><th>AAA</th><th>AA</th><th>A</th><th>BBB</th><th>BB</th><th>B</th><th>CCC</th></tr></thead><tbody><tr><th>AAA</th><td>-7.679</td><td>6.427</td><td>0.542</td><td>0.466</td><td>0.245</td><td>0.000</td><td>0.000</td></tr><tr><th>AA</th><td>4.770</td><td>-20.606</td><td>15.451</td><td>0.000</td><td>0.318</td><td>0.066</td><td>0.000</td></tr><tr><th>A</th><td>0.000</td><td>20.259</td><td>-35.447</td><td>14.953</td><td>0.152</td><td>0.083</td><td>0.000</td></tr><tr><th>BBB</th><td>0.102</td><td>0.000</td><td>28.263</td><td>-41.545</td><td>13.100</td><td>0.000</td><td>0.080</td></tr><tr><th>BB</th><td>0.032</td><td>0.307</td><td>0.000</td><td>38.351</td><td>-39.651</td><td>10.832</td><td>0.129</td></tr><tr><th>B</th><td>0.000</td><td>0.000</td><td>0.503</td><td>0.000</td><td>30.227</td><td>-33.735</td><td>3.006</td></tr><tr><th>CCC</th><td>0.000</td><td>0.000</td><td>0.194</td><td>0.469</td><td>0.365</td><td>16.806</td><td>-17.834</td></tr></tbody></table>

<table><caption>Table 2.31: ESG migration Markov matrix #1 (one-year transition probability in %)</caption><thead><tr><th></th><th>AAA</th><th>AA</th><th>A</th><th>BBB</th><th>BB</th><th>B</th><th>CCC</th></tr></thead><tbody><tr><th>AAA</th><td>92.75</td><td>5.66</td><td>0.90</td><td>0.45</td><td>0.23</td><td>0.01</td><td>0.00</td></tr><tr><th>AA</th><td>4.17</td><td>82.73</td><td>11.85</td><td>0.89</td><td>0.30</td><td>0.07</td><td>0.00</td></tr><tr><th>A</th><td>0.40</td><td>15.51</td><td>72.79</td><td>10.39</td><td>0.81</td><td>0.10</td><td>0.01</td></tr><tr><th>BBB</th><td>0.12</td><td>2.11</td><td>19.60</td><td>68.69</td><td>8.91</td><td>0.50</td><td>0.07</td></tr><tr><th>BB</th><td>0.04</td><td>0.43</td><td>2.79</td><td>19.25</td><td>69.65</td><td>7.61</td><td>0.23</td></tr><tr><th>B</th><td>0.01</td><td>0.09</td><td>0.65</td><td>2.98</td><td>21.21</td><td>72.71</td><td>2.35</td></tr><tr><th>CCC</th><td>0.00</td><td>0.02</td><td>0.25</td><td>0.58</td><td>2.19</td><td>13.09</td><td>83.87</td></tr></tbody></table>

<footnote n="74"><sup>74</sup>The matrix $\bar{\Lambda}$ is the best Markov generator that minimize the $L_1$-norm distance to P.</footnote>
**Dynamic analysis** We have now all the tools to conduct a dynamic analysis of the ESG rating system. There is tremendous potential. For instance, we compute the probability to reach the states A with the following formula:

$$ \pi_k(t, \mathcal{A}) = \Pr\{\mathcal{R}(t) \in \mathcal{A} | \mathcal{R}(0) = k\} = \sum_{j \in \mathcal{A}} \mathbf{e}_k^{\top} e^{t\Lambda} \mathbf{e}_j^{\top} $$

Some examples are given in Figure 2.37. We can also use the continuous-time framework to investigate the probability density function of conditional events, the probability over a given interval, the m-order derivative of time functions, etc. We use the properties $\partial_t \exp(\Lambda t) = \Lambda \exp(\Lambda t)$, $\partial_t^m \exp(\Lambda t) = \Lambda^m \exp(\Lambda t)$ and$^{75} \int_0^t e^{\Lambda s} ds = (e^{\Lambda t} - I_K) \Lambda^{-1}$. For example, we have:

$$ \pi_k^{(m)}(t, \mathcal{A}) := \frac{\partial \pi_k(t, \mathcal{A})}{\partial t^m} = \sum_{j \in \mathcal{A}} \mathbf{e}_k^{\top} \Lambda^m e^{t\Lambda} \mathbf{e}_j^{\top} $$

$\pi_k^{(1)}(t, \mathcal{A})$ may be interpreted as a “time density function”. In Figure 2.37, we report $\pi_k(t, \text{AAA})$, $\pi_k^{(1)}(t, \text{AAA})$, $\pi_k(t, \text{CCC})$ and $\pi_k^{(1)}(t, \text{CCC})$. We observe the strange behavior of the CCC rating towards the AAA rating.

<figure><img src="image_62.png" /><figcaption>Figure 2.37: Probability $\pi_k(t, \mathcal{A})$ to reach A at time t (migration matrix #1)</figcaption></figure>

**Remark 28** The previous analysis can be used to check consistency of ratings. In particular, the fact that ratings satisfy ordering preferences implies that we must generally observe a monotone behavior of quantities that are a non-decreasing and concave function of ratings.

<footnote>$^{75}$For more general integrals of type $\int_0^t e^{\Lambda s} Qf(s) ds$, we use the numerical algorithms developed by Van Loan (1978).</footnote>
<figure><img src="image_63.png" /><figcaption>Box 2.9: Computing statistical moments with continuous-time Markov chains</figcaption></figure>

The distribution $\pi(t)$ follows the Kolmogorov equation:

$$ \frac{d\pi(t)}{dt} = \Lambda\pi(t) $$

It follows that $\pi(t) = e^{\Lambda t}\pi(0)$. Let $Y(t) = \phi(\mathcal{R}(t))$ be a random variable that depends on the ratings. We have:

$$ \mu(t) = \sum_{k=1}^{K} \phi(R_k) \pi_k(t) $$

and:

$$ \sigma^2(t) = \sum_{k=1}^{K} (\phi(R_k) - \mu(t))^2 \pi_k(t) $$
### 2.3.4 Comparison with credit ratings

The modeling of credit ratings is similar to this of ESG ratings, but there is one important difference. The states include the default state. This means that $R_K$ is the absorbing state, implying that any entity which has reached this state remains in this state. In this case, $p_{i,K}$ is the one-period default probability of rating $R_i$ and we have $p_{K,K} = 1$. An example of credit migration matrix is given in Table 2.32. It is the S&P one-year transition probability matrix for corporate bonds estimated by Kavvathas (2001) for the period 1960–1998. More recent credit migration matrices<fnref n="76" /> are given in Table 2.33.

<table><caption>Table 2.32: Example of credit migration matrix (one-year probability transition in %)</caption><thead><tr><th></th><th>AAA</th><th>AA</th><th>A</th><th>BBB</th><th>BB</th><th>B</th><th>CCC</th><th>D</th></tr></thead><tbody><tr><th>AAA</th><td>92.82</td><td>6.50</td><td>0.56</td><td>0.06</td><td>0.06</td><td>0.00</td><td>0.00</td><td>0.00</td></tr><tr><th>AA</th><td>0.63</td><td>91.87</td><td>6.64</td><td>0.65</td><td>0.06</td><td>0.11</td><td>0.04</td><td>0.00</td></tr><tr><th>A</th><td>0.08</td><td>2.26</td><td>91.66</td><td>5.11</td><td>0.61</td><td>0.23</td><td>0.01</td><td>0.04</td></tr><tr><th>BBB</th><td>0.05</td><td>0.27</td><td>5.84</td><td>87.74</td><td>4.74</td><td>0.98</td><td>0.16</td><td>0.22</td></tr><tr><th>BB</th><td>0.04</td><td>0.11</td><td>0.64</td><td>7.85</td><td>81.14</td><td>8.27</td><td>0.89</td><td>1.06</td></tr><tr><th>B</th><td>0.00</td><td>0.11</td><td>0.30</td><td>0.42</td><td>6.75</td><td>83.07</td><td>3.86</td><td>5.49</td></tr><tr><th>CCC</th><td>0.19</td><td>0.00</td><td>0.38</td><td>0.75</td><td>2.44</td><td>12.03</td><td>60.71</td><td>23.50</td></tr><tr><th>D</th><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>100.00</td></tr></tbody></table>

Source: Kavvathas (2001).

<table><caption>Table 2.33: Credit migration matrix in % (Moody's, 1983–2021)</caption><thead><tr><th></th><th>Aaa</th><th>Aa</th><th>A</th><th>Baa</th><th>Ba</th><th>B</th><th>Caa</th><th>W</th><th>D</th></tr></thead><tbody><tr><th colspan="10">Sovereign issuers</th></tr><tr><th>Aaa</th><td>96.99</td><td>2.87</td><td>0.03</td><td>0.08</td><td>0.00</td><td>0.00</td><td>0.00</td><td>0.03</td><td>0.00</td></tr><tr><th>Aa</th><td>2.73</td><td>93.52</td><td>2.56</td><td>0.62</td><td>0.09</td><td>0.00</td><td>0.00</td><td>0.47</td><td>0.00</td></tr><tr><th>A</th><td>0.00</td><td>3.60</td><td>92.17</td><td>3.19</td><td>0.98</td><td>0.06</td><td>0.00</td><td>0.00</td><td>0.00</td></tr><tr><th>Baa</th><td>0.00</td><td>0.00</td><td>5.43</td><td>89.17</td><td>4.98</td><td>0.39</td><td>0.03</td><td>0.00</td><td>0.00</td></tr><tr><th>Ba</th><td>0.00</td><td>0.00</td><td>0.00</td><td>6.91</td><td>85.72</td><td>6.53</td><td>0.29</td><td>0.10</td><td>0.44</td></tr><tr><th>B</th><td>0.00</td><td>0.00</td><td>0.00</td><td>0.00</td><td>4.31</td><td>88.49</td><td>4.50</td><td>0.26</td><td>2.43</td></tr><tr><th>Caa</th><td>0.00</td><td>0.00</td><td>0.00</td><td>0.00</td><td>0.06</td><td>13.60</td><td>73.24</td><td>0.75</td><td>12.35</td></tr><tr><th colspan="10">Corporates issuers</th></tr><tr><th>Aaa</th><td>87.16</td><td>8.05</td><td>0.45</td><td>0.08</td><td>0.03</td><td>0.00</td><td>0.00</td><td>4.23</td><td>0.00</td></tr><tr><th>Aa</th><td>0.70</td><td>85.02</td><td>8.57</td><td>0.42</td><td>0.06</td><td>0.04</td><td>0.02</td><td>5.17</td><td>0.02</td></tr><tr><th>A</th><td>0.05</td><td>2.44</td><td>86.84</td><td>5.15</td><td>0.45</td><td>0.10</td><td>0.04</td><td>4.88</td><td>0.05</td></tr><tr><th>Baa</th><td>0.02</td><td>0.12</td><td>3.73</td><td>86.43</td><td>3.42</td><td>0.65</td><td>0.16</td><td>5.31</td><td>0.15</td></tr><tr><th>Ba</th><td>0.00</td><td>0.03</td><td>0.38</td><td>6.02</td><td>75.95</td><td>7.19</td><td>0.86</td><td>8.78</td><td>0.77</td></tr><tr><th>B</th><td>0.01</td><td>0.03</td><td>0.12</td><td>0.42</td><td>4.73</td><td>73.61</td><td>7.34</td><td>10.79</td><td>2.95</td></tr><tr><th>Caa</th><td>0.00</td><td>0.01</td><td>0.02</td><td>0.07</td><td>0.26</td><td>5.58</td><td>70.41</td><td>14.82</td><td>8.83</td></tr></tbody></table>

Source: Moody's (2020).

<footnote n="76">⁷⁶W means that the issuer has required to stop the rating (withdrawn).</footnote>
Since there are few research on ESG ratings, credit migration matrices can be used as a bench-
mark to compare the two rating systems. For that, we consider the trace statistics:

$$
\lambda(t) = \frac{\operatorname{trace}(e^{t\Lambda})}{K}
$$

It is the average of the diagonal transition probabilities. It measures the average probability to remain in its state<sup>77</sup>. Results are reported in Figure 2.39. Even if the two ESG rating systems used here are fictitious examples, we generally conclude that ESG rating systems are less stable than credit rating systems, and the time horizon of ESG ratings for prediction is shorter.

<figure><img src="image_64.png" /><figcaption>Figure 2.39: Trace statistics of credit and ESG migration matrices</figcaption></figure>

2.4 Exercises

2.4.1 Score normalization when the features are independent

We study the behavior of the score defined as:

$$
\mathcal{S} = \frac{X_1 + X_2 + \cdots + X_m}{m}
$$

where $X_1, \dots, X_m$ is a sequence of iid uniform random variables, whose distribution is $\mathbf{F}$.

1. We consider the case $m = 2$ and $X_j \sim U_{[0,1]}$. The score is a weighted average of $X_1$ and $X_2$:

$$
\mathcal{S} = \omega X_1 + (1 - \omega) X_2
$$

where $\omega \in [0, 1]$.

<footnote n="77"><sup>77</sup>For a credit migration matrix, we consider all the states except the absorbing state. In this case, we have
$$
\lim_{t \to \infty} \lambda(t) = 0.
$$</footnote>
(a) Let $s \in [0, 1]$. Find the points of intersection between the curve $x_2 = (s - \omega X_1) / (1 - \omega)$ and the unit square. Discuss the different cases.

(b) For each case, compute the area $A(s)$ defined as:

$$A(s) = \iint_{\Omega(s)} dx_1 dx_2$$

where $\Omega(s) = \{(x_1, x_2) \in [0, 1]^2 : \omega x_1 + (1 - \omega) x_2 \le s\}$. Deduce the cumulative distribution function $G$ of the score.

(c) Compute the density function $g$.

(d) Find $G$ and $g$ when $X_j \sim U_{[a,b]}$ where $b > a$.

2. We consider the case $m = 3$ and $X_j \sim U_{[0,1]}$. The volume $V(s)$ is equal to:

$$V(s) = \iiint_{\Omega(s)} dx_1 dx_2 dx_3$$

where $\Omega(s) = \{(x_1, x_2, x_3) \in [0, 1]^3 : x_1 + x_2 + x_3 \le s\}$.

(a) Compute the volume $V(s)$ when $0 \le s \le 1$.

(b) Compute the volume difference $V(s) - V(1)$ when $1 \le s \le 2$.

(c) Compute the volume difference $V(s) - V(2)$ when $1 \le s \le 2$.

(d) Deduce the cumulative distribution function $G$ of the score.

(e) Compute the density function $g$.

3. We consider that $X_j \sim U_{[0,1]}$ and $m \ge 1$. We note $G_m(s)$ the probability $\Pr\{S \le s\}$.

(a) Give the expression of $G_m(s)$ and the associate density function $g_m(s)$.

(b) We assume that $X_j \sim U_{[a,b]}$ where $b > a$. Deduce the expressions of the density and distribution functions of the score $S$.

4. We assume that $X_j \sim G(\alpha_j, \beta)$ where $\alpha_j > 0$ and $\beta > 0$.

(a) Compute the cumulative distribution function $G$ of the score.

(b) Deduce the density function $g$.

(c) Compute the mean and the variance of $S$.

(d) We assume that $\alpha_j = 2$ and $\beta = 2$.

i. Draw the functions $E[S]$ and $\text{var}(S)$ with respect to the number $m$ of features.

ii. Find the value $m^+(p, \varepsilon)$ such that:

$$m^+(p, \varepsilon) = \inf m : \Pr\{2 - \varepsilon \le S \le 2 - \varepsilon\} \le p$$

for the pairs $(p, \varepsilon) = (99\%, 5\%)$.

iii. Draw the function $m^+(p, \varepsilon)$ with respect to $p$ when $\varepsilon = 1\%$.

iv. Draw the function $m^+(p, \varepsilon)$ with respect to $\varepsilon$ when $p = 99.99\%$.
## 2.4.2 Probability distribution of an ESG score

1. We consider an investment universe of 8 issuers with the following ESG scores:

<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></tr></thead><tbody><tr><td>(E)</td><td>-2.80</td><td>-1.80</td><td>-1.75</td><td>0.60</td><td>0.75</td><td>1.30</td><td>1.90</td><td>2.70</td></tr><tr><td>(S)</td><td>-1.70</td><td>-1.90</td><td>0.75</td><td>-1.60</td><td>1.85</td><td>1.05</td><td>0.90</td><td>0.70</td></tr><tr><td>(G)</td><td>0.30</td><td>-0.70</td><td>-2.75</td><td>2.60</td><td>0.45</td><td>2.35</td><td>2.20</td><td>1.70</td></tr></tbody></table>

(a) Calculate the ESG score of the issuers if we assume the following weighting scheme: 40% for (E), 40% for (S) and 20% for (G).

(b) Calculate the ESG score of the equally-weighted portfolio $x_{ew}$.

2. We assume that the ESG scores are iid and follow a standard Gaussian distribution:

$$ \mathbf{S}_i \sim \mathcal{N}(0, 1) $$

(a) We note $x_{ew}^{(n)}$ the equally-weighted portfolio composed of $n$ issuers. Calculate the distribution of the ESG score $\mathbf{S}(x_{ew}^{(n)})$ of the portfolio $x_{ew}^{(n)}$.

(b) What is the ESG score of a well-diversified portfolio?

(c) We note $T \sim \mathbf{F}_\alpha$ where $\mathbf{F}_\alpha(t) = t^\alpha$, $t \in [0, 1]$ and $\alpha \ge 0$. Draw the graph of the probability density function $f_\alpha(t)$ when $\alpha$ is respectively equal to 0.5, 1.5, 2.5 and 70. What do you notice?

(d) We assume that the weights of the portfolio$^{78}$ $x = (x_1, \dots, x_n)$ follow a power-law distribution $\mathbf{F}_\alpha$:

$$ x_i \sim cT_i $$

where $T_i \sim \mathbf{F}_\alpha$ are iid random variables and $c$ is a normalization constant. Explain how to simulate the portfolio weights $x = (x_1, \dots, x_n)$. Represent one simulation of the portfolio $x$ for the previous values of $\alpha$. Comment on these results. Deduce the relationship between the Herfindahl index $H_\alpha(x)$ of the portfolio weights $x$ and the parameter $\alpha$.

(e) We assume that the weight $x_i$ and the ESG score $\mathbf{S}_i$ of the issuer $i$ are independent. How to simulate the portfolio's score $\mathbf{S}(x)$? Using 50 000 replications, estimate the probability distribution function of $\mathbf{S}(x)$ by the Monte Carlo method. Comment on these results.

(f) We now assume that the weight $x_i$ and the ESG score $\mathbf{S}_i$ of the issuer $i$ are positively correlated. More precisely, the dependence function between $x_i$ and $\mathbf{S}_i$ is the Normal copula function with parameter $\rho$. Show that this is also the copula function between $T_i$ and $\mathbf{S}_i$. Deduce an algorithm to simulate $\mathbf{S}(x)$.

(g) Using 50 000 replications, estimate the probability distribution function of $\mathbf{S}(x)$ by the Monte Carlo method when the correlation parameter $\rho$ is set to 50%. Comment on these results.

(h) Estimate the relationship between the correlation parameter $\rho$ and the expected ESG score $\mathbb{E}[\mathbf{S}(x)]$ of the portfolio $x$. Comment on these results.

(i) How are the previous results related to the size bias of ESG scoring?

<footnote>$^{78}$We use $n = 50$ in the rest of the exercise.</footnote>
3. Let $\mathbf{S}$ be the ESG score of the issuer. We assume that the ESG score follows a standard Gaussian distribution:

$$ \mathbf{S} \sim \mathcal{N}(0, 1) $$

The ESG score $\mathbf{S}$ is also converted into an ESG rating $\mathcal{R}$, which can take the values$^{79}$ A, B, C and D.

(a) We assume that the breakpoints of the rating system are -1.5, 0 and +1.5. Compute the frequencies of the ratings.

(b) We would like to build a rating system such that each category has the same frequency. Find the mapping function.

(c) We would like to build a rating system such that the frequency of the median ratings B and C is 40% and the frequency of the extreme ratings A and D is 10%. Find the mapping function.

4. Let $\mathbf{S}(t)$ be the ESG score of the issuer at time $t$. The ESG scoring system is evaluated every month. The index time $t$ corresponds to the current month, whereas the previous month is $t-1$. We assume that:

i. The ESG score at time $t-1$ follows a standard Gaussian distribution:

$$ \mathbf{S}(t-1) \sim \mathcal{N}(0, 1) $$

ii. The variation of the ESG score is Gaussian between two months:

$$ \Delta \mathbf{S}(t) = \mathbf{S}(t) - \mathbf{S}(t-1) \sim \mathcal{N}(0, \sigma^2) $$

iii. The ESG score $\mathbf{S}(t-1)$ and the variation $\Delta \mathbf{S}(t)$ are independent.

The ESG score $\mathbf{S}(t)$ is converted into an ESG rating $\mathcal{R}(t)$, which can take following grades:

$$ \mathcal{R}_1 \prec \mathcal{R}_2 \prec \dots \prec \mathcal{R}_k \prec \dots \prec \mathcal{R}_{K-1} \prec \mathcal{R}_K $$

We assume that the breakpoints of the rating system are $(s_1, s_2, \dots, s_{K-1})$. We also note $s_0 = -\infty$ and $s_K = +\infty$.

(a) Compute the bivariate probability distribution of the random vector $(\mathbf{S}(t-1), \Delta \mathbf{S}(t))$.

(b) Compute the bivariate distribution of the random vector $(\mathbf{S}(t-1), \mathbf{S}(t))$.

(c) Compute the probability $p_k = \text{Pr}\{\mathcal{R}(t-1) = \mathcal{R}_k\}$.

(d) Compute the joint probability $\text{Pr}\{\mathcal{R}(t) = \mathcal{R}_k, \mathcal{R}(t-1) = \mathcal{R}_j\}$.

(e) Compute the transition probability $p_{j,k} = \text{Pr}\{\mathcal{R}(t) = \mathcal{R}_k | \mathcal{R}(t-1) = \mathcal{R}_j\}$.

(f) Compute the monthly turnover $\mathcal{T}(\mathcal{R}_k)$ of the ESG rating $\mathcal{R}_k$.

(g) Compute the monthly turnover $\mathcal{T}(\mathcal{R}_1, \dots, \mathcal{R}_K)$ of the ESG rating system.

(h) For each rating system given in Questions 3.a, 3.b and 3.c, compute the corresponding migration matrix and the monthly turnover of the rating system if we assume that $\sigma$ is equal to 10%. What is the best ESG rating system if we would like to control the turnover of ESG ratings?

<footnote>$^{79}$A is the best rating and D is the worst rating.</footnote>
(i) Draw the relationship between the parameter $\sigma$ and the turnover $T(\mathcal{R}_1, \dots, \mathcal{R}_K)$ for the three ESG rating systems.

(j) We consider a uniform ESG rating system where:

$$ \Pr\{\mathcal{R}(t-1) = \mathcal{R}_k\} = \frac{1}{K} $$

Draw the relationship between the number of notches $K$ and the turnover $T(\mathcal{R}_1, \dots, \mathcal{R}_K)$ when the parameter $\sigma$ takes the values 5%, 10% and 25%.

(k) Why is an ESG rating system different than a credit rating system? What do you conclude from the previous analysis? What is the issue of ESG exclusion policy and negative screening?
# Chapter 3

## Impact of ESG Investing on Asset Prices and Portfolio Returns

The question of ESG performance is on everyone's lips. This question is linked to several other questions, which can be summarized as follows: What is the impact of ESG on corporate financial performance? What is the impact of ESG investing on risk premia? How does ESG screening affect portfolio returns? Is there a difference between ESG investing and climate investing? In fact, we can multiply the questions because the term *ESG performance* covers different topics, and we need to be more precise when we talk about it. First, we need to distinguish between operational performance, social performance, accounting performance, market performance, etc. For example, it is not the same thing to measure performance based on financial statements (balance sheet and income statement) or share price evolution. Second, we can measure performance from an investor's or issuer's perspective. The third ambiguity concerns the nature of the financial assets. Are they stocks or bonds? Because we know that fixed-income and equity markets react differently. Another important source of discrepancy is the choice of financial instruments. We can compare the performance of securities, mutual funds, asset owners or backtests. For example, simulated performance must be different from live performance. The fifth issue is the investment universe and sample. We can imagine that the impact of ESG is different from one region to another, from one sector to another, one period to another, etc. Finally, if we focus on the financial performance of ESG strategies, the last question is the implementation of the portfolio strategy. Are we talking about an exclusion, selection, integration or momentum strategy? Are we talking about active or passive management? Moreover, since ESG scores vary widely from one rating agency to another, we are not sure that we are capturing the performance of the ESG market, but perhaps some idiosyncratic patterns. There are therefore many factors to consider, and it is no coincidence that there are many academic studies with different conclusions. It is impossible to cite all of them, even the most famous. They are described in meta-analyses, e.g., Orlitzky et al. (2003), Margolis et al. (2009), Friede et al. (2015), Atz et al. (2022) and Coqueret (2022). Rather than delve into all these empirical studies, we take a different approach. In fact, fifteen or twenty years ago, ESG investing did not exist or was so marginal. Moreover, ESG data is certainly not robust or relevant before 2010. Therefore, it is better to focus on theoretical research when analyzing the performance of ESG investing. This first section is mainly based on the work of Pástor et al. (2021) and Pedersen et al. (2021). It will help us understand when, where and why ESG investing may underperform or outperform business-as-usual investing. The second section is devoted to empirical studies, but we make a selection to illustrate the theoretical results and concepts defined in the first section. Finally, the third section examines the impact of ESG on the cost of capital.
## 3.1 Theoretical models

Before discussing the impact of ESG on the theory of risk premia and security selection, we summarize the main results of modern portfolio theory as presented in Roncalli (2013).

### 3.1.1 An introduction to modern portfolio theory

The concept of the market portfolio has a long history, dating back to the seminal work of Markowitz (1952). He showed that an efficient portfolio is the portfolio that maximizes the expected return for a given level of risk. Markowitz concluded that there is not just one optimal portfolio, but a set of optimal portfolios called the efficient frontier. By studying the liquidity preference, Tobin (1958) showed that the efficient frontier becomes a straight line in the presence of a risk-free asset. In this case, optimal portfolios correspond to a combination of the risk-free asset and a particular efficient portfolio named the tangency portfolio. Sharpe (1964) summarized the results of Markowitz and Tobin as follows: “the process of investment choice can be broken down into two phases: first, the choice of a unique optimum combination of risky assets<fnref n="1" />; and second, a separate choice concerning the allocation of funds between such a combination and a single riskless asset.” This two-step process is today known as the mutual fund separation theorem. In this seminal research paper, Sharpe developed the CAPM theory and emphasized the relationship between the risk premium of the asset (the difference between the expected return and the risk-free rate) and its beta (the systematic risk with respect to the tangency portfolio). Assuming that the market is at equilibrium, he showed that asset prices are such that the tangency portfolio is the market portfolio, which is composed of all risky assets in proportion to their market capitalization.

#### The efficient frontier

**The optimization problem** Seventy years ago, Markowitz introduced the concept of the efficient frontier. Consider a universe of $n$ assets. Let $w = (w_1, ..., w_n)$ be the vector of weights in the portfolio. We assume that the portfolio is fully invested, so $\sum_{i=1}^n w_i = \mathbf{1}_n^\top w = 1$. We denote $R = (R_1, ..., R_n)$ the vector of asset returns where $R_i$ is the return of asset $i$. The portfolio return is then $R(w) = \sum_{i=1}^n w_i R_i = w^\top R$. Let $\mu = \mathbb{E}[R]$ and $\Sigma = \mathbb{E}[(R - \mu)(R - \mu)^\top]$ be the vector of expected returns and the covariance matrix of asset returns. The expected return $\mu(w) := \mathbb{E}[R(w)]$ of the portfolio is equal to:

$$ \mu(w) = \mathbb{E}[w^\top R] = w^\top \mathbb{E}[R] = w^\top \mu $$

while its variance $\sigma^2(w) := \text{var}(R(w))$ is given by:

$$ \begin{aligned} \sigma^2(w) &= \mathbb{E}[(R(w) - \mu(w))(R(w) - \mu(w))^\top] \\ &= \mathbb{E}[w^\top(R - \mu)(R - \mu)^\top w] \\ &= w^\top \Sigma w \end{aligned} $$

We can then formulate the investor's financial problem as follows:

1. Maximize the expected return of the portfolio under a volatility constraint ($\sigma$-problem):

$$ \max \mu(w) \quad \text{s.t.} \quad \sigma(w) \le \sigma^* \qquad (3.1) $$

<footnote n="1">¹It is precisely the tangency portfolio.</footnote>
2. Or minimize the volatility of the portfolio under a return constraint ($\mu$-problem):

$$ \min \sigma(w) \quad \text{s.t.} \quad \mu(w) \geq \mu^* \qquad (3.2) $$

By considering all the portfolios belonging to the simplex set defined by {$w \in [0, 1]^n : \mathbf{1}_n^\top w = 1$}, we can compute the expected return and volatility bounds of the portfolios: $\mu^- \leq \mu(w) \leq \mu^+$ and $\sigma^- \leq \sigma(w) \leq \sigma^+$. There is also a solution to the first problem if $\sigma^* \geq \sigma^-$. The second problem has a solution if $\mu^* \leq \mu^+$. When these two conditions are verified, the inequality constraints become $\sigma(w) = \min(\sigma^*, \sigma^+)$ and $\mu(w) = \max(\mu^-, \mu^*)$.

The key idea of Markowitz (1956) was to transform the original nonlinear optimization problem (3.1) into a quadratic optimization problem which is easier to solve numerically. To do this, he introduced the mean-variance (or quadratic) utility function:

$$ \mathcal{U}(w) := \mathbb{E}[R(w)] - \frac{\bar{\gamma}}{2} \mathrm{var}(R(w)) = w^{\top}\mu - \frac{\bar{\gamma}}{2}w^{\top}\Sigma w $$

where $\bar{\gamma}$ is the absolute risk-aversion parameter. We get the following problem:

$$ \begin{array}{l} w^*(\bar{\gamma}) = \arg\max \left\{ \mathcal{U}(w) = w^\top\mu - \frac{\bar{\gamma}}{2}w^\top\Sigma w \right\} \qquad (3.3) \\ \text{s.t.} \quad \mathbf{1}_n^\top w = 1 \end{array} $$

If $\bar{\gamma} = 0$, the optimized portfolio is the one that maximizes the expected return and we have $\mu(w^*(0)) = \mu^+$. If $\bar{\gamma} = \infty$, the risk-aversion parameter is maximum, we obtain the global minimum variance (GMV) portfolio:

$$ \begin{array}{l} w^*(\infty) = \arg\min \frac{1}{2}w^\top\Sigma w \\ \text{s.t.} \quad \mathbf{1}_n^\top w = 1 \end{array} $$

and we have $\sigma(w^*(\infty)) = \sigma^-$. In practice, we formulate the optimization problem (3.3) as follows:

$$ \begin{array}{l} w^*(\gamma) = \arg\min \frac{1}{2}w^\top\Sigma w - \gamma w^\top\mu \qquad (3.4) \\ \text{s.t.} \quad \mathbf{1}_n^\top w = 1 \end{array} $$

where $\gamma = \bar{\gamma}^{-1}$ is the inverse of the risk-aversion parameter and is called the risk-tolerance coefficient. The reason is that this new formulation is a standard quadratic programming (QP) problem². From a numerical point of view, it is therefore better to use Problem (3.4). In this case, the minimum variance portfolio corresponds to $\gamma = 0$. The set of solutions {$w^*(\gamma), \gamma \geq 0$} corresponds to mean-variance optimized (MVO) portfolios.

**Example 9** Consider an investment universe with five assets. Their expected returns are equal to 5%, 7%, 6%, 10% and 8% while their volatilities are equal to 18%, 20%, 22%, 25% and 30%. The correlation matrix of the asset returns is given by the following matrix:

$$ \mathbb{C} = \begin{pmatrix} 100\% & & & & \\ 70\% & 100\% & & & \\ 20\% & 30\% & 100\% & & \\ -30\% & 20\% & 10\% & 100\% & \\ 0\% & 0\% & 0\% & 0\% & 100\% \end{pmatrix} $$
<figure><img src="image_65.png" /><figcaption>Figure 3.1: Efficient frontier (Example 9)</figcaption></figure>

In Figure 3.1, we report the efficient frontier {$\sigma(w^*(\gamma)), \mu(w^*(\gamma))$} by considering several values<fnref n="3" /> of $\gamma \in [-0.5, 1]$. We see that the optimized portfolios significantly improve the risk/return profile with respect to the five assets represented by the cross symbol. Some special solutions are given in Table 3.1. The portfolio weights, its return and its volatility are expressed in %. For instance, the GMV portfolio is obtained with $\gamma = 0$. The solution is (66.35%, -28.52%, 15.31%, 34.85%, 12.02%) and it is not possible to find a portfolio whose volatility is less than 10.40%.

<table><caption>Table 3.1: Solution of the Markowitz optimization problem (in %)</caption><thead><tr><th>γ</th><th>0.00</th><th>0.10</th><th>0.20</th><th>0.50</th><th>1.00</th><th>5.00</th></tr></thead><tbody><tr><th>w<sup>*</sup><sub>1</sub>(γ)</th><td>66.35</td><td>58.25</td><td>50.14</td><td>25.84</td><td>-14.67</td><td>-338.72</td></tr><tr><th>w<sup>*</sup><sub>2</sub>(γ)</th><td>-28.52</td><td>-22.67</td><td>-16.82</td><td>0.74</td><td>30.00</td><td>264.12</td></tr><tr><th>w<sup>*</sup><sub>3</sub>(γ)</th><td>15.31</td><td>13.30</td><td>11.30</td><td>5.28</td><td>-4.74</td><td>-84.93</td></tr><tr><th>w<sup>*</sup><sub>4</sub>(γ)</th><td>34.85</td><td>37.65</td><td>40.44</td><td>48.82</td><td>62.78</td><td>174.50</td></tr><tr><th>w<sup>*</sup><sub>5</sub>(γ)</th><td>12.02</td><td>13.48</td><td>14.94</td><td>19.32</td><td>26.62</td><td>85.03</td></tr><tr><th>μ(w<sup>*</sup>(γ))</th><td>6.69</td><td>6.97</td><td>7.25</td><td>8.09</td><td>9.49</td><td>20.71</td></tr><tr><th>σ(w<sup>*</sup>(γ))</th><td>10.40</td><td>10.53</td><td>10.93</td><td>13.35</td><td>19.71</td><td>84.38</td></tr></tbody></table>

Solving the $\mu$-problem or the $\sigma$-problem is equivalent to finding the optimal value of $\gamma$ such that $\mu(w^*(\gamma)) = \mu^*$ or $\sigma(w^*(\gamma)) = \sigma^*$. We know that the functions $\mu(w^*(\gamma))$ and $\sigma(w^*(\gamma))$ are increasing in $\gamma$ and bounded. The optimal value of $\gamma$ can then be easily computed using the bisection

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

<footnote n="3">³If $\gamma < 0$, $w^*(\gamma)$ is not a MVO portfolio because it has a lower expected return than the GMV portfolio with higher volatility. In fact, Problem (3.4) defines the convex hull {$\mu(w), \sigma(w)$} of all possible portfolios {$w : 1_n^T w = 1$}.</footnote>
algorithm described on page 1139. This is the approach used in practice, because it benefits from the numerical efficiency of quadratic programming solvers<fnref n="4" />. For instance, if we want a portfolio with $\sigma^* = 15\%$, we know that $\gamma \in [0.5, 1]$. The optimal solution $w^*$ is (14.06%, 9.25%, 2.37%, 52.88%, 21.44%). The bisection algorithm returns $\gamma = 0.6455$. In this case, we get $\mu(w^*(\gamma)) = 8.50\%$. Now consider a $\mu$-problem with $\mu^* = 9\%$. We find $\gamma = 0.8252$, $w^* = (-0.50\%, 19.77\%, -1.23\%, 57.90\%, 24.07)$ and $\sigma(w^*(\gamma)) = 17.30\%$.

**Adding some constraints** The Lagrange function of the optimization problem (3.4) is equal to:

$$ \mathcal{L}(w; \lambda_0) = \frac{1}{2} w^\top \Sigma w - \gamma w^\top \mu + \lambda_0 (\mathbf{1}_n^\top w - 1) $$

where $\lambda_0$ is the Lagrange coefficient associated with the constraint $\mathbf{1}_n^\top w = 1$. The solution $w^*$ verifies the following first-order conditions:

$$ \left\{ \begin{array}{l} \partial_w \mathcal{L}(w; \lambda_0) = \Sigma w - \gamma \mu + \lambda_0 \mathbf{1}_n = \mathbf{0}_n \\ \partial_{\lambda_0} \mathcal{L}(w; \lambda_0) = \mathbf{1}_n^\top w - 1 = 0 \end{array} \right. $$

We obtain $w = \Sigma^{-1}(\gamma\mu - \lambda_0\mathbf{1}_n)$. Since $\mathbf{1}_n^\top w - 1 = 0$, we have $\gamma\mathbf{1}_n^\top\Sigma^{-1}\mu - \lambda_0\mathbf{1}_n^\top\Sigma^{-1}\mathbf{1}_n = 1$. It follows that:

$$ \lambda_0 = \frac{\gamma \mathbf{1}_n^\top \Sigma^{-1} \mu - 1}{\mathbf{1}_n^\top \Sigma^{-1} \mathbf{1}_n} $$

The solution is then:

$$ \begin{aligned} w^*(\gamma) &= \frac{\Sigma^{-1}\mathbf{1}_n}{\mathbf{1}_n^\top\Sigma^{-1}\mathbf{1}_n} + \gamma \frac{(\mathbf{1}_n^\top\Sigma^{-1}\mathbf{1}_n)\Sigma^{-1}\mu - (\mathbf{1}_n^\top\Sigma^{-1}\mu)\Sigma^{-1}\mathbf{1}_n}{\mathbf{1}_n^\top\Sigma^{-1}\mathbf{1}_n} \\ &= w_{\text{gmv}} + \gamma w_{\text{lsp}} \end{aligned} $$

where $w_{\text{gmv}} = (\Sigma^{-1}\mathbf{1}_n) / (\mathbf{1}_n^\top\Sigma^{-1}\mathbf{1}_n)$ is the global minimum variance portfolio and $w_{\text{lsp}}$ is a long/short cash-neutral portfolio<fnref n="5" /> such that $\mathbf{1}_n^\top w_{\text{lsp}} = 0$. We deduce that a QP solver is not needed to find the solution of the optimization problem (3.4). For example, analytic calculus gives $w_{\text{gmv}} = (66.35\%, -28.52\%, 15.31\%, 34.85\%, 12.02\%)$ and $w_{\text{lsp}} = (-81.01\%, 58.53\%, -20.05\%, 27.93\%, 14.60\%)$. Using the numerical results in Table 3.1, we verify that the equation $w^*(\gamma) = w_{\text{gmv}} + \gamma w_{\text{lsp}}$ is satisfied. However, these solutions are not realistic because they correspond to leveraged long/short portfolios, but most investors cannot have short positions. In addition, short selling can only be implemented for a few assets that are very liquid and highly tradable. Otherwise, the cost of short selling is enormous. Therefore, in practice, portfolio optimization takes other constraints into account:

$$ \begin{array}{l} w^*(\gamma) = \arg \min \frac{1}{2} w^\top \Sigma w - \gamma w^\top \mu \qquad (3.5) \\ \text{s.t.} \quad \left\{ \begin{array}{l} \mathbf{1}_n^\top w = 1 \\ w \in \Omega \end{array} \right. \end{array} $$

where $w \in \Omega$ is the set of restrictions. The most common constraints are certainly the no short-selling restriction and the asset bounds. In the first case, $w_i \ge 0$ and $\Omega = [0, 1]^n$. The second constraint imposes $w_i \le w^+$ to ensure that the portfolio is not concentrated in a small number of assets.

<footnote n="4">⁴From a numerical point of view, it is generally better to solve several QP problems than one nonlinear optimization problem.</footnote>

<footnote n="5">⁵We have $\mathbf{1}_n^\top w^*(\gamma) = 1 \Leftrightarrow \mathbf{1}_n^\top w_{\text{gmv}} + \gamma \mathbf{1}_n^\top w_{\text{lsp}} = 1 \Leftrightarrow \mathbf{1}_n^\top w_{\text{lsp}} = 0$ because $\mathbf{1}_n^\top w_{\text{gmv}} = 1$.</footnote>
Let us introduce some constraints in Example 9. In Figure 3.2, we have plotted two constrained efficient frontiers, the first by imposing no short selling and the second by imposing that the weights are between 0% and 33%. We verify that investment constraints can significantly reduce opportunity arbitrage.

<figure><img src="image_66.png" /><figcaption>Figure 3.2: Impact of constraints on the efficient frontier (Example 9)</figcaption></figure>

## The tangency portfolio

**Two-fund separation theorem** We recall that in Markowitz’s view, there is a set of optimized portfolios. However, Tobin (1958) showed that one optimized portfolio dominates all others if there is a risk-free asset. Consider a combination of the risk-free asset and a portfolio w. We denote the return on the risk-free asset by r. We have$^6$:

$$R(\tilde{w}) = (1 - \alpha)r + \alpha R(w)$$

where $\tilde{w} = (\alpha w, 1 - \alpha)$ is a vector of dimension $(n + 1)$ and $\alpha \ge 0$ is the fraction of wealth invested in the risky portfolio. It follows that:

$$\mu(\tilde{w}) = (1 - \alpha)r + \alpha\mu(w) = r + \alpha(\mu(w) - r)$$

and:

$$\sigma^2(\tilde{w}) = \alpha^2\sigma^2(w)$$

We deduce that:

$$\mu(\tilde{w}) = r + \frac{(\mu(w) - r)}{\sigma(w)}\sigma(\tilde{w}) \qquad (3.6)$$

<footnote>$^6$We have n + 1 assets in the universe, where the first n assets correspond to the previous risky assets and the last asset is the risk-free asset.</footnote>
It is the equation of a linear function between the expected return and the volatility of the combined portfolio $\tilde{w}$. In Figure 3.3, we plot the previous (unconstrained) efficient frontier. The dashed line corresponds to the combination between the risk-free asset ($r$ equals 3%) and the optimized portfolio $w^*$. Nevertheless this combination is suboptimal because it is dominated by other combinations. We notice that one straight line dominates all other straight lines and the efficient frontier. This line is tangent to the efficient frontier and is called the capital market line. It implies that an optimized risky portfolio dominates all other risky portfolios, namely the tangency (or tangent) portfolio. We denote it by $w^*$.

<figure><img src="image_67.png" /><figcaption>Figure 3.3: Capital market line (Example 9)</figcaption></figure>

Let $\text{SR}(w|r)$ be the Sharpe ratio of portfolio $w$:

$$ \text{SR}(w|r) = \frac{\mu(w) - r}{\sigma(w)} $$

Notice that we can write Equation (3.6) as follows:

$$ \frac{\mu(\tilde{w}) - r}{\sigma(\tilde{w})} = \frac{\mu(w) - r}{\sigma(w)} \Leftrightarrow \text{SR}(\tilde{w} | r) = \text{SR}(w | r) $$

We deduce that the tangency portfolio is the one that maximizes the angle $\theta(w)$ or equivalently $\tan\theta(w)$ which is equal to the Sharpe ratio. The tangency portfolio is also the risky portfolio corresponding to the maximum Sharpe ratio. We also note that any portfolio belonging to the capital market line has the same Sharpe ratio. If we look at our example with $r = 3\%$, the composition of the tangency portfolio $w^*$ is (42.57%, -11.35%, 9.43%, 43.05%, 16.30%) and we have $\mu(w^*) = 7.51\%$, $\sigma(w^*) = 11.50\%$, $\text{SR}(w^*|r) = 0.39$ and $\theta(w^*) = 21.40$ degrees.
**Augmented optimization problem** If the risk-free asset belongs to the investment universe, the optimization problem becomes:

$$ \tilde{w}^*(\gamma) = \arg\min \frac{1}{2} \tilde{w}^\top \tilde{\Sigma} \tilde{w} - \gamma \tilde{w}^\top \tilde{\mu} \quad (3.7) \\ \text{s.t.} \quad \begin{cases} \mathbf{1}_n^\top \tilde{w} = 1 \\ \tilde{w} \in \Omega \end{cases} $$

where $\tilde{w} = (w, w_r)$ is the augmented allocation vector of dimension $n+1$. It follows that:

$$ \tilde{\Sigma} = \begin{pmatrix} \Sigma & \mathbf{0}_n \\ \mathbf{0}_n^\top & 0 \end{pmatrix} \quad \text{and} \quad \tilde{\mu} = \begin{pmatrix} \mu \\ r \end{pmatrix} $$

In the case where $\Omega = \mathbb{R}^{n+1}$, Roncalli (2013, pages 13-14) showed that the optimal solution is equal to:

$$ \tilde{w}^*(\gamma) = \underbrace{\alpha \begin{pmatrix} w^* \\ 0 \end{pmatrix}}_{\text{risky assets}} + \underbrace{(1 - \alpha) \begin{pmatrix} \mathbf{0}_n \\ 1 \end{pmatrix}}_{\text{risk-free asset}} $$

where $w^*$ is the tangency portfolio:

$$ w^* = \frac{\Sigma^{-1}(\mu - r\mathbf{1}_n)}{\mathbf{1}_n^\top \Sigma^{-1} (\mu - r\mathbf{1}_n)} $$

and the proportion of risky assets is equal to $\alpha = \gamma \mathbf{1}_n^\top \Sigma^{-1} (\mu - r\mathbf{1}_n)$. It follows that the risk-tolerance coefficient associated with the tangency portfolio is given by:

$$ \gamma(w^*) = \frac{1}{\mathbf{1}_n^\top \Sigma^{-1} (\mu - r\mathbf{1}_n)} $$

When $\alpha \neq 1$, the weights $\tilde{w}^*(\gamma)$ of the optimal portfolio are proportional to the weights $w^*$ of the tangency portfolio, while the wealth invested in the risk-free asset is the complement $(1 - \alpha)$ to obtain a total exposure of 100%. We then retrieve the two-fund separation theorem.

## Market equilibrium and CAPM

**Risk premium and beta** Based on the results of Markowitz and Tobin, Sharpe (1964) developed the capital asset pricing model (CAPM). Let $w^*$ be the tangency portfolio. On the efficient frontier, we have seen that every portfolio $w$ satisfies the capital market line:

$$ \mu(w) = r + \frac{\sigma(w)}{\sigma(w^*)} (\mu(w^*) - r) $$

We consider a portfolio $x$ with a fraction $\omega$ invested in the asset $i$ and a fraction $(1 - \omega)$ invested in the tangency portfolio $w^*$. We have$^7$ $\mu(x) = \omega\mu_i + (1 - \omega)\mu(w^*)$ and $\sigma^2(x) = \omega^2\sigma_i^2 + (1 - \omega)^2\sigma^2(w^*) + 2\omega(1 - \omega)\rho(\mathbf{e}_i, w^*)\sigma_i\sigma(w^*)$. It follows that:

$$ \frac{\partial \mu(x)}{\partial \sigma(x)} = \frac{\mu_i - \mu(w^*)}{(\omega\sigma_i^2 + (\omega - 1)\sigma^2(w^*) + (1 - 2\omega)\rho(\mathbf{e}_i, w^*)\sigma_i\sigma(w^*))\sigma^{-1}(x)} $$

<footnote>$^7$e_i is the unit vector with 1 in the i'th position and 0 elsewhere. It then corresponds to the portfolio fully invested in asset i.</footnote>
When $\omega = 0$, the portfolio $x$ is the tangency portfolio $w^*$ and the previous derivative is equal to the Sharpe ratio SR($w^* | r$):

$$ \lim_{\omega \to 0} \frac{\partial \mu(x)}{\partial \sigma(x)} = \tan \theta(w^*) = \frac{\mu(w^*) - r}{\sigma(w^*)} $$

We deduce that:

$$ \frac{(\mu_i - \mu(w^*)) \sigma(w^*)}{\rho(\mathbf{e}_i, w^*) \sigma_i \sigma(w^*) - \sigma^2(w^*)} = \frac{\mu(w^*) - r}{\sigma(w^*)} $$

which is equivalent to:

$$ \pi_i := \mu_i - r = \beta_i (\mu(w^*) - r) \quad (3.8) $$

where $\pi_i$ is the risk premium of the asset $i$ and:

$$ \beta_i = \frac{\rho(\mathbf{e}_i, w^*) \sigma_i}{\sigma(w^*)} = \frac{\operatorname{cov}(R_i, R(w^*))}{\operatorname{var}(R(w^*))} \quad (3.9) $$

The coefficient $\beta_i$ is the ratio of the covariance between the return of asset $i$ and the return of the tangency portfolio to the variance of the tangency portfolio return. Equation (3.8) tells us that the risk premium of asset $i$ is equal to its beta times the excess return of the tangency portfolio. It is easy to show that this relationship holds for any portfolio $w$ and not just for the assets that make up the tangency portfolio.

Box 3.1: Computation of the beta coefficient

Let $R_{i,t}$ and $R_t(w)$ be the returns of asset $i$ and portfolio $w$ at time $t$. We consider the linear regression:

$$ R_{i,t} = \alpha_i + \beta_i R_t(w) + \varepsilon_{i,t} $$

where $\varepsilon_{i,t}$ is a white noise process. The OLS coefficient $\hat{\beta}_i$ is an estimate of the beta $\beta_i$ of the asset $i$. We can generalize this approach to estimate the beta of a portfolio $x$ with respect to another portfolio $w$. We have:

$$ R_t(x) = \alpha + \beta R_t(w) + \varepsilon_t $$

Another way to calculate beta is to use the following relationship:

$$ \beta(x | w) = \frac{\sigma(x, w)}{\sigma^2(w)} = \frac{x^\top \Sigma w}{w^\top \Sigma w} $$

We deduce that the expression for the beta of asset $i$ is also:

$$ \beta_i = \beta(\mathbf{e}_i | w) = \frac{\mathbf{e}_i^\top \Sigma w}{w^\top \Sigma w} = \frac{(\Sigma w)_i}{w^\top \Sigma w} $$

It follows that the beta of a portfolio is the weighted average of the betas of the assets that make up the portfolio:

$$ \beta(x | w) = \frac{x^\top \Sigma w}{w^\top \Sigma w} = x^\top \frac{\Sigma w}{w^\top \Sigma w} = \sum_{i=1}^{n} x_i \beta_i $$

The relationship (3.8) is very important and highlights the role of the beta coefficient. However, this result is not the only main finding of Sharpe (1964). In his article, Sharpe also showed that
when the market is at equilibrium, the prices of assets are such that the tangency portfolio $w^*$ is the market portfolio $w_m$ (or the market-cap portfolio). With this result, the characterization of the tangency portfolio does not depend on assumptions about expected returns, volatilities, and correlations.

In the case of Example 9, we have seen that the composition of the tangency portfolio is $w^* = (42.57\%, -11.35\%, 9.43\%, 43.05\%, 16.30\%)$. Since its expected return is $\mu(w^*) = 7.51\%$ and $r = 3\%$, we deduce that the market risk premium is 4.51%. In Table 3.2, we report the beta of each asset and two portfolios: the equally weighted (EW) portfolio $w_{ew}$ and the GMV portfolio $w_{gmv}$. We compute the associated expected return $\mu(w) = w^\top\mu$ and the risk premium explained by the tangency portfolio $\pi(w|w^*) = \beta(w|w^*)(\mu(w^*) - r)$. We check the relationship $\pi(w|w^*) = \mu(w) - r$. For instance, the beta of the first asset is 0.444 and we have $0.444 \times 4.51\% = 2\%$, which is also the difference between 5% and 3%. For the EW portfolio, the risk premium is equal to $0.932 \times 4.51\% = 4.20\%$. We also check that it is equal to the difference between the expected return of 7.20% and the risk-free rate of 3%.

<table><caption>Table 3.2: Computation of beta and risk premium (Example 9)</caption><thead><tr><th>Portfolio</th><th>μ(w)</th><th>μ(w) - r</th><th>β(w | w*)</th><th>π(w | w*)</th></tr></thead><tbody><tr><td>e<sub>1</sub></td><td>5.00%</td><td>2.00%</td><td>0.444</td><td>2.00%</td></tr><tr><td>e<sub>2</sub></td><td>7.00%</td><td>4.00%</td><td>0.887</td><td>4.00%</td></tr><tr><td>e<sub>3</sub></td><td>6.00%</td><td>3.00%</td><td>0.665</td><td>3.00%</td></tr><tr><td>e<sub>4</sub></td><td>10.00%</td><td>7.00%</td><td>1.553</td><td>7.00%</td></tr><tr><td>e<sub>5</sub></td><td>8.00%</td><td>5.00%</td><td>1.109</td><td>5.00%</td></tr><tr><td>w<sub>ew</sub></td><td>7.20%</td><td>4.20%</td><td>0.932</td><td>4.20%</td></tr><tr><td>w<sub>gmv</sub></td><td>6.69%</td><td>3.69%</td><td>0.817</td><td>3.69%</td></tr></tbody></table>

**Risk premium and alpha return** Jensen (1968) analyzed the performance of active management by using the following regression model:

$$R_{j,t} - r = \alpha_j + \beta_j (R_t(w_m) - r) + \varepsilon_{j,t}$$

where $R_{j,t}$ is the return of the mutual fund j at time t, $R_t(w_m)$ is the return of the market portfolio and $\varepsilon_{j,t}$ is an idiosyncratic risk. If the mutual fund outperforms the market portfolio, the assumption $\alpha_j > 0$ is not rejected. However, Jensen rejected this assumption for most mutual funds and concluded that active management does not generate alpha. More generally, alpha is defined as the difference between the risk premium $\pi(w)$ of portfolio w and the beta$^8$ $\beta(w)$ of the portfolio times the market risk premium $\pi_m$:

$$ \alpha = (\mu(w) - r) - \beta(w | w_m)(\mu(w_m) - r) \\ = \pi(w) - \beta(w)\pi_m $$

If we now impose a no short-selling constraint by using a lower bound $x_i \ge 0$, the tangency portfolio becomes $w^* = (33.62\%, 0\%, 8.79\%, 40.65\%, 16.95\%)$ in our previous example. We check that the portfolio has no short exposure to the second asset. Since we have $\mu(w^*) = 7.63\%$ and $r = 3\%$, we deduce that the market risk premium is 4.63%. It is higher than in the unconstrained case. We report the beta and the risk premium in Table 3.3. We notice that the equality $\mu(w) - r = \beta(w)(\mu_m - r)$ is

<footnote>$^8$The notation $\beta(w)$ means that the beta is calculated with respect to the market portfolio.</footnote>
Box 3.2: Computing the implied risk premia of investors

Consider the optimization problem$^a$:

$$ 
\begin{array}{l} 
w^* = \arg\min \frac{1}{2} w^\top \Sigma w - \gamma w^\top (\mu - r \mathbf{1}_n) \\ 
\text{s.t.} \quad \begin{cases} \mathbf{1}_n^\top w = 1 \\ w \in \Omega \end{cases} 
\end{array} 
$$

If we omit the constraints, the solution is $w^* = \gamma \Sigma^{-1} (\mu - r \mathbf{1}_n)$. In the Markowitz model, the unknown variable is the vector $w$ of weights. We now assume that the investor has a current asset allocation $w_0$. By construction, $w_0$ is the optimal portfolio for this investor, otherwise he will change his investment policy. We deduce that$^b$:

$$ w_0 = \gamma \Sigma^{-1} (\mu - r \mathbf{1}_n) \Leftrightarrow \tilde{\pi} = \mu - r \mathbf{1}_n = \frac{1}{\gamma} \Sigma w_0 \quad (3.10) $$

We can interpret $\tilde{\pi}$ as the vector of risk premia consistent with the portfolio $w_0$ (Black and Litterman, 1991, 1992). $\tilde{\pi}$ is then the risk premium priced by the portfolio manager. Calculating $\tilde{\pi}$ requires specifying the investor's risk tolerance. Let us assume that the investor targets a Sharpe ratio $\text{SR}(w_0 | r)$ for his portfolio. We deduce that:

$$ \text{SR}(w_0 | r) = \frac{\mu(w_0) - r}{\sigma(w_0)} = \frac{w_0^\top (\mu - r \mathbf{1}_n)}{\sqrt{w_0^\top \Sigma w_0}} = \frac{1}{\gamma} \sqrt{w_0^\top \Sigma w_0} $$

Finally, we obtain:

$$ \tilde{\pi} = \text{SR}(w_0 | r) \cdot \frac{\Sigma w_0}{\sqrt{w_0^\top \Sigma w_0}} $$

Let us consider Example 9. We assume that the current allocation $w_0$ is equal to (35%, 25%, 15%, 15%, 10%). The volatility of the portfolio is then equal to $\sigma(x_0) = 12.52\%$. The objective of the portfolio manager is to target a Sharpe ratio of 0.25. The implied risk tolerance is $\gamma = 0.50$ and the implied risk premia are $\tilde{\pi} = (3.36\%, 4.45\%, 2.83\%, 1.59\%, 1.80\%)$.

not always satisfied. This is especially true for the second asset, which has a negative alpha of 49 bps. We know that the true risk premium of this asset is 4%. Nevertheless, investors are constrained and cannot short this asset. Theoretically, the optimal demand for this asset must be negative. Because of the lower bound $x_i \ge 0$, the market demand is higher than the expected demand derived from the CAPM. Therefore, there is a upward pressure on the price of this asset due to the lack of arbitrage. The risk premium perceived by the market is then higher, creating a negative alpha because the price is overvalued. Since the equally-weighted portfolio is long this asset, it also has a negative alpha. This is not the case for the GMV portfolio, which is short this asset (its weight is $-28.52\%$ — see Table 3.1 on page 140).

<footnote>$^a$We notice that the excess expected return is equal to $w^\top (\mu - r \mathbf{1}_n) = w^\top \mu - r$. Adding the risk-free rate has no effect on the mean-variance utility function.</footnote>

<footnote>$^b$From this equation we also deduce the following relationship: $\tilde{\pi}(w_0) = \gamma^{-1}\sigma^2(w_0)$.</footnote>
<table><caption>Table 3.3: Computation of alpha return (Example 9)</caption><thead><tr><th>Portfolio</th><th>μ(w)</th><th>μ(w) - r</th><th>β(w | w*)</th><th>π(w | w*)</th><th>α(w | w*)</th></tr></thead><tbody><tr><td>e<sub>1</sub></td><td>5.00%</td><td>2.00%</td><td>0.432</td><td>2.00%</td><td>0.00%</td></tr><tr><td>e<sub>2</sub></td><td>7.00%</td><td>4.00%</td><td>0.970</td><td>4.49%</td><td>-0.49%</td></tr><tr><td>e<sub>3</sub></td><td>6.00%</td><td>3.00%</td><td>0.648</td><td>3.00%</td><td>0.00%</td></tr><tr><td>e<sub>4</sub></td><td>10.00%</td><td>7.00%</td><td>1.512</td><td>7.00%</td><td>0.00%</td></tr><tr><td>e<sub>5</sub></td><td>8.00%</td><td>5.00%</td><td>1.080</td><td>5.00%</td><td>0.00%</td></tr><tr><td>w<sub>ew</sub></td><td>7.20%</td><td>4.20%</td><td>0.929</td><td>4.30%</td><td>-0.10%</td></tr><tr><td>w<sub>gmv</sub></td><td>6.69%</td><td>3.69%</td><td>0.766</td><td>3.55%</td><td>0.14%</td></tr></tbody></table>

The previous analysis can be applied to a more general framework. There are two main ex- planations for alpha generation. The first concerns the assumptions of the CAPM. In particular, this model assumes that investors are not constrained in terms of leverage, short selling, transaction costs, etc. In practice, investors are highly constrained, especially large institutional investors. Since they cannot leverage their portfolios, they will not use the tangency portfolio. They will prefer a portfolio with a lower Sharpe ratio but a higher expected return. This explains why the demand for high beta assets is greater than the demand predicted by the CAPM. Therefore, we observe a positive alpha return for low beta assets and a negative alpha return for high beta assets (Black, 1972; Frazzini and Pedersen, 2014). The second explanation is the existence of other risk factors that are not priced by the CAPM (Ross, 1976). The development of factor investing and alternative risk premia in the aftermath of the 2008 global financial crisis is related to this issue. If investors use systematic strategies with the same approach and these strategies are very popular, they can affect asset prices (Roncalli, 2017). In both cases, alpha generation is rooted in the imbalance between supply and demand and the dynamics of investment flows.

## Portfolio optimization in the presence of a benchmark

**Utility function revisited** The Markowitz approach to portfolio optimization assumes that the investor has a mean-variance utility function without reference to a given investment policy. We now extend the optimization problem when a strategic asset allocation imposes a benchmark represented by a portfolio $b$. The tracking error between the active portfolio $w$ and its benchmark $b$ is the difference between the return of the portfolio and the return of the benchmark:

$$ \epsilon = R(w) - R(b) = \sum_{i=1}^{n} w_i R_i - \sum_{i=1}^{n} b_i R_i = w^{\top} R - b^{\top} R = (w - b)^{\top} R $$

The tracking error $\epsilon$ is a stochastic random variable. The expected excess return is equal to:

$$ \mu(w | b) := \mathbb{E}[\epsilon] = (w - b)^{\top} \mu $$

whereas the volatility of the tracking error is defined as:

$$ \sigma(w | b) := \sigma(\epsilon) = \sqrt{(w - b)^{\top} \Sigma (w - b)} $$

The investor's objective is then to maximize the expected tracking error, subject to a constraint on the tracking error volatility:

$$ \begin{array}{l} w^* = \arg \max \mu(w | b) \qquad (3.11) \\ \text{s.t.} \quad \begin{cases} \mathbf{1}_n^\top x = 1 \\ \sigma(w | b) \le \sigma^* \end{cases} \end{array} $$
Like the Markowitz problem, we transform this $\sigma$-problem into a $\gamma$-problem:

$$w^*(\gamma) = \arg\min f(w | b)$$

where:

$$ 
\begin{aligned}
f(w | b) &= \frac{1}{2}\sigma^2(w | b) - \gamma\mu(w | b) \\
&= \frac{1}{2}(w - b)^{\top}\Sigma(w - b) - \gamma(w - b)^{\top}\mu \\
&= \frac{1}{2}w^{\top}\Sigma w - w^{\top}(\gamma\mu + \Sigma b) + \underbrace{\frac{1}{2}b^{\top}\Sigma b + \gamma b^{\top}\mu}_{\text{constant}}
\end{aligned}
$$

Again, we recognize a quadratic programming problem. The efficient frontier is then the parametric curve $(\sigma(w^*(\gamma) | b), \mu(w^*(\gamma) | b))$ with $\gamma \ge 0$.

**Remark 29** Using Equation (3.10), we find that $w^\top(\gamma\mu + \Sigma b) = 2\gamma w^\top(\frac{\pi + \tilde{\pi}}{2}) - \gamma$ where $\tilde{\pi}$ is the implied risk premia associated with the benchmark $b$. We obtain a Markowitz problem where the vector of expected returns is replaced by an average of the true and implied risk premia.

**Example 10** Consider an investment universe with four assets. Their expected returns are 5%, 6.5%, 8% and 6.5% while their volatilities are 15%, 20%, 25% and 30%. The correlation matrix of the asset returns is given by the following matrix:

$$C = \begin{pmatrix} 100\% & & & \\ 10\% & 100\% & & \\ 40\% & 70\% & 100\% & \\ 50\% & 40\% & 80\% & 100\% \end{pmatrix}$$

We consider Example 10 with the benchmark $b = (60\%, 40\%, 20\%, -20\%)$. In Figure 3.4, we have represented the corresponding efficient frontier. We verify that it is a straight line if there is no constraint (Roll, 1992). If we impose that $w_i \ge -10\%$, the efficient frontier is shifted to the right. For the third case, we assume that the weights are between a lower bound and an upper bound: $w_i^- \le w_i \le w_i^+$ with $w_i^+ = 50\%$. For the first three assets, the lower bound $w_i^-$ is set to 0, while for the fourth asset it is $-20\%$.

**Information ratio** To compare the performance of different portfolios, a better measure than the Sharpe ratio is the information ratio which is defined as follows:

$$IR(w | b) = \frac{\mu(w | b)}{\sigma(w | b)} = \frac{(w - b)^{\top} \mu}{\sqrt{(w - b)^{\top} \Sigma (w - b)}}$$

If we consider a combination of the benchmark $b$ and the active portfolio $w$, the composition of the portfolio is:

$$x = (1 - \alpha)b + \alpha w$$

where $\alpha \ge 0$ is the proportion of wealth invested in the portfolio $w$. It follows that:

$$\mu(x | b) = (x - b)^{\top} \mu = \alpha \mu(w | b)$$
<figure><img src="image_68.png" /><figcaption>Figure 3.5: Tangency portfolio with respect to a benchmark (Example 10)</figcaption></figure>

<figure><img src="image_69.png" /></figure>
and:

$$ \sigma^2 (x | b) = (x - b)^{\top} \Sigma (x - b) = \alpha^2 \sigma^2 (w | b) $$

We deduce that:

$$ \mu (x | b) = \text{IR}(w | b) \sigma (x | b) $$

It is the equation of a linear function between the tracking error volatility and the expected tracking error of the portfolio $x$. It implies that the efficient frontier is a straight line:

> "If the manager is measured solely in terms of excess return performance, he or she should pick a point on the upper part of this efficient frontier. For instance, the manager may have a utility function that balances expected value added against tracking error volatility. Note that because the efficient set consists of a straight line, the maximal Sharpe ratio is not a usable criterion for portfolio allocation" (Jorion, 2003, page 172).

If we add some other constraints to the portfolio optimization problem (3.11), the efficient frontier is no longer a straight line. In this case, one optimized portfolio dominates all other portfolios. It is the portfolio that belongs to the efficient frontier and the straight line that is tangent to the efficient frontier. It is also the portfolio that maximizes the information ratio.

Let us look at the previous efficient frontier when we impose lower and upper bounds (third case). When we combine it with the benchmark, we get the straight line shown in Figure 3.5 and the tangency portfolio $w^*$ is equal to (46.56%, 33.49%, 39.95%, -20.00%).

### 3.1.2 ESG risk premium

We now analyze the impact of ESG investing on the CAPM. However, it is important to reiterate that the risk premium is the expected excess return that investors earn because they are exposed to systematic risk. Therefore, we need to distinguish between expected (or required) returns and historical (or realized) returns. Moreover, it is not very clear whether the risk premium is the long-term performance or a specific requirement of investors. This is the difference between the unconstrained risk premium $\pi_i$ and the implied risk premium $\tilde{\pi}_i$.

#### The Pástor-Stambaugh-Taylor model

In this section, we present the model developed by Pástor et al. (2021) (hereafter, PST model). It is a direct extension of the CAPM and has the advantage of emphasizing many intuitively stylized facts.

**Model settings** Pástor et al. (2021) consider an investment universe of $n$ assets corresponding to the stocks of $n$ firms. They assume that the asset excess returns $\tilde{R} = R - r = (\tilde{R}_1, \dots, \tilde{R}_n)$ are normally distributed — $\tilde{R} \sim N(\pi, \Sigma)$, and the firms produce social impact. Each firm has an ESG characteristic $G_i$, which is positive for *esg-friendly* (or *green*) firms and negative for *esg-unfriendly* (or *brown*) firms. This means that $G_i > 0$ induces positive social impacts, while $G_i < 0$ induces negative externalities on society. Consider an economy with a continuum of agents ($j = 1, 2, \dots, \infty$). We note $w_{i,j}$ the fraction of wealth invested by agent $j$ in stock $i$, and $w_j = (w_{1,j}, \dots, w_{n,j})$ the allocation vector of agent $j$. The relationship between the initial and final wealth $W_j$ and $\tilde{W}_j$ is given by:

$$ \tilde{W}_j = (1 + r + w_j^{\top} \tilde{R}) W_j $$
Pástor et al. (2021) assume that the economic agent j has an exponential CARA utility function:

$$ \mathcal{U}(\tilde{W}_j, w_j) = -\exp(-\bar{\gamma}_j \tilde{W}_j - w_j^\top b_j W_j) $$

where $\bar{\gamma}_j$ is the absolute risk-aversion and $b_j = \varphi_j \mathcal{G}$ is the vector of non-pecuniary benefits depending on the green intensity $\mathcal{G}$ and the ESG preference coefficient $\varphi_j \ge 0$ of the economic agent.

**Optimal portfolio** The expected utility is equal to:

$$ \begin{aligned} \mathbb{E}[\mathcal{U}(\tilde{W}_j, w_j)] &= \mathbb{E}[-\exp(-\bar{\gamma}_j \tilde{W}_j - w_j^\top b_j W_j)] \\ &= \mathbb{E}[-\exp(-\bar{\gamma}_j (1+r + w_j^\top \tilde{R}) W_j - w_j^\top b_j W_j)] \\ &= -e^{-\bar{\gamma}_j (1+r) W_j} \mathbb{E}[\exp(-\bar{\gamma}_j w_j^\top W_j (\tilde{R} + \bar{\gamma}_j^{-1} b_j))] \\ &= e^{-\bar{\Gamma}_j (1+r)} \mathbb{E}[\exp(-\bar{\Gamma}_j w_j^\top (\tilde{R} + \bar{\gamma}_j^{-1} b_j))] \end{aligned} \quad (3.12) $$

where $\bar{\Gamma}_j = \bar{\gamma}_j W_j$ is the nominal risk aversion. We notice that $\tilde{R} + \bar{\gamma}_j^{-1} b_j \sim \mathcal{N}(\pi + \bar{\gamma}_j^{-1} b_j, \Sigma)$ and:

$$ -\bar{\Gamma}_j w_j^\top (\tilde{R} + \bar{\Gamma}_j^{-1} b_j) \sim \mathcal{N}(-\bar{\Gamma}_j w_j^\top (\pi + \bar{\gamma}_j^{-1} b_j), \bar{\Gamma}_j^2 w_j^\top \Sigma w_j) $$

Using the mathematical expectation formula of the log-normal distribution<fnref n="9" />, we deduce that:

$$ \mathbb{E}[\mathcal{U}(\tilde{W}_j, w_j)] = e^{-\bar{\Gamma}_j(1+r)} \exp \left( -\bar{\Gamma}_j w_j^\top (\pi + \bar{\gamma}_j^{-1} b_j) + \frac{1}{2} \bar{\Gamma}_j^2 w_j^\top \Sigma w_j \right) $$

The first-order condition is equal to:

$$ -\bar{\Gamma}_j (\pi + \bar{\gamma}_j^{-1} b_j) + \bar{\Gamma}_j^2 \Sigma w_j = \mathbf{0}_n $$

Finally, Pástor et al. (2021) conclude that the optimal portfolio is:

$$ w_j^* = \Gamma_j \Sigma^{-1} (\pi + \gamma_j b_j) \quad (3.13) $$

where $\Gamma_j = \bar{\Gamma}_j^{-1}$ and $\gamma_j = \bar{\gamma}_j^{-1}$ are the relative nominal and unitary risk-tolerance coefficients. This is the unconstrained optimal portfolio where asset returns include the green sentiment $\gamma_j b_j = \gamma_j \varphi_j \mathcal{G}$.

**Remark 30** We assume that $W_j = 1$. Since we have $\mathbf{1}_n^\top w_j = 1$, $w_j^\top r = r$ and $\bar{\Gamma}_j = \bar{\gamma}_j$, we deduce that:

$$ \begin{aligned} -\ln \mathbb{E}[\mathcal{U}(\tilde{W}_j, w_j)] &= \bar{\gamma}_j (1+r) + \bar{\gamma}_j w_j^\top (\pi + \bar{\gamma}_j^{-1} b_j) - \frac{1}{2} \bar{\gamma}_j^2 w_j^\top \Sigma w_j \\ &\propto w_j^\top (\pi + r \mathbf{1}_n + \bar{\gamma}_j^{-1} b_j) - \frac{1}{2} \bar{\gamma}_j w_j^\top \Sigma w_j \\ &= w_j^\top (\mu + \bar{\gamma}_j^{-1} b_j) - \frac{1}{2} \bar{\gamma}_j w_j^\top \Sigma w_j \end{aligned} $$

Maximizing expected utility is then equivalent to solving the classical Markowitz QP problem:

$$ \begin{aligned} w_j^*(\gamma_j) &= \arg \min \frac{1}{2} w_j^\top \Sigma w_j - \gamma_j w_j^\top \mu' \\ \text{s.t.} \quad & \mathbf{1}_n^\top w_j = 1 \end{aligned} $$

where $\gamma_j = \bar{\gamma}_j^{-1}$ is the relative risk tolerance and $\mu' = \mu + \gamma_j b_j$ is the vector of modified expected returns reflecting the ESG sentiment of the economic agent regarding the social impact of firms.

<footnote n="9">⁹See Appendix A.2.1 on page 1153.</footnote>
**Example 11** We consider a universe of *n* risky assets, where *n* is an even number. The risk-free rate *r* is set at 3%. We assume that the Sharpe ratio of these assets is the same and equal to 20%. The volatility of asset *i* is equal to $\sigma_i = 0.10 + 0.20 e^{-n^{-1}[0.5i]}$. The correlation between asset returns is constant: $C = C_n(\rho)$. The social impact of firms is given by the vector **G**. If **G** is not specified, it is equal to the cyclic vector (+1%, -1%, +1%, ..., +1%, -1%). This implies that half of the firms (green firms) have a positive social impact while the others (brown firms) have a negative impact.

We consider the case where $n = 6$ and $\rho = 25\%$, and we assume that we cannot be short the assets. We calibrate the risk-tolerance parameter $\gamma$ so that the long-only optimized portfolio of the non-ESG investor has a volatility of 20%. We find $\gamma = 1.5456$ and obtain the results reported in Table 3.4. We verify that the optimized portfolio depends on the ESG preference coefficient $\phi$. We consider a second set of ESG characteristics: $\mathcal{G} = (10\%, 5\%, 2\%, 3\%, 25\%, 30\%)$. Since $\mathcal{G}_i > 0$, we can assume that this investment universe has been filtered to retain only the best-in-class issuers and to implement an ESG selection strategy. Again, we measure the impact of $\phi$ on the optimized portfolios. In Figure 3.6, we show the efficient frontier when the investment universe consists of 20 assets. We verify that the expected returns of the efficient frontier are reduced when ESG preferences are taken into account, and this reduction depends on the ESG preference coefficient $\phi$. We also find that all of these efficient frontiers start at the same point, since the global minimum variance portfolio is not affected by the investor's ESG preferences.

<table><caption>Table 3.4: Mean-variance optimized portfolios with ESG preferences (Example 11, $n = 6$, $\rho = 25\%$)</caption><thead><tr><th rowspan="2">&phi;</th><th colspan="4">G = (1%, -1%, 1%, -1%, 1%, -1%)</th><th colspan="4">G = (10%, 5%, 2%, 3%, 25%, 30%)</th></tr><tr><th>0.00%</th><th>1.00%</th><th>5.00%</th><th>50.00%</th><th>0.00%</th><th>0.50%</th><th>1.00%</th><th>2.00%</th></tr></thead><tbody><tr><td>w<sub>1</sub><sup>*</sup></td><td>44.97%</td><td>48.87%</td><td>58.65%</td><td>67.48%</td><td>44.97%</td><td>46.83%</td><td>28.69%</td><td>0.00%</td></tr><tr><td>w<sub>2</sub><sup>*</sup></td><td>44.97%</td><td>41.06%</td><td>19.60%</td><td>0.00%</td><td>44.97%</td><td>37.06%</td><td>9.17%</td><td>0.00%</td></tr><tr><td>w<sub>3</sub><sup>*</sup></td><td>5.03%</td><td>9.82%</td><td>21.75%</td><td>32.52%</td><td>5.03%</td><td>0.00%</td><td>0.00%</td><td>0.00%</td></tr><tr><td>w<sub>4</sub><sup>*</sup></td><td>5.03%</td><td>0.25%</td><td>0.00%</td><td>0.00%</td><td>5.03%</td><td>0.00%</td><td>0.00%</td><td>0.00%</td></tr><tr><td>w<sub>5</sub><sup>*</sup></td><td>0.00%</td><td>0.00%</td><td>0.00%</td><td>0.00%</td><td>0.00%</td><td>0.83%</td><td>16.62%</td><td>21.09%</td></tr><tr><td>w<sub>6</sub><sup>*</sup></td><td>0.00%</td><td>0.00%</td><td>0.00%</td><td>0.00%</td><td>0.00%</td><td>15.28%</td><td>45.53%</td><td>78.91%</td></tr><tr><td>&mu; (w<sup>*</sup>)</td><td>8.33%</td><td>8.33%</td><td>8.27%</td><td>8.22%</td><td>8.33%</td><td>8.23%</td><td>7.79%</td><td>7.43%</td></tr><tr><td>&sigma; (w<sup>*</sup>)</td><td>20.00%</td><td>20.09%</td><td>20.07%</td><td>21.56%</td><td>20.00%</td><td>19.33%</td><td>16.70%</td><td>19.17%</td></tr><tr><td>SR (w<sup>*</sup> | r)</td><td>0.27</td><td>0.27</td><td>0.26</td><td>0.24</td><td>0.27</td><td>0.27</td><td>0.29</td><td>0.23</td></tr></tbody></table>

**Remark 31** In this numerical example, the impact of ESG preferences is small because the assets have similar financial characteristics: the same Sharpe ratio and the same cross-correlation values. This explains why the optimized portfolios are different, but their Sharpe ratios are very close. Nevertheless, the expected return is always lower when implementing an ESG strategy<fnref n="10" />.

**Risk premium** The total market wealth $W$ is equal to $\int W_j \, dj$. Let $\omega_j = W_j/W$ be the market share of economic agent $j$. The amount $W_{i,j}$ invested in stock $i$ is equal to $W_{i,j} = w_{i,j}^* W_j = w_{i,j}^* \omega_j W$. The total dollar amount invested in stock $i$ is then equal to

$$W_i = \int_j W_{i,j} \, dj = \int_j w_{i,j}^* \omega_j W \, dj$$

<footnote n="10">¹⁰In the following, we will see that this is not always the case, as it depends on the sign of the aggregate ESG preference $w_m^* \mathcal{G}$ where $w_m$ is the market portfolio.</footnote>
<figure><img src="image_70.png" /><figcaption>Figure 3.6: Efficient frontier with ESG preferences (Example 11, n = 20, ρ = 25%)</figcaption></figure>

Let $w_m = (w_{1,m}, \dots, w_{n,m})$ be the market portfolio. We have:

$$w_{i,m} = \frac{W_i}{W} = \int_j w_{i,j}^* \omega_j \, dj$$

and $\int_j \omega_j \, dj = 1$. Pástor et al. (2021) deduce that the market clearing condition satisfies:

$$\begin{align*} w_m &= \int_j \omega_j w_j^* \, dj \\ &= \int_j \omega_j \Gamma_j \Sigma^{-1} (\pi + \gamma_j b_j) \, dj \\ &= \int_j \omega_j \Gamma_j \Sigma^{-1} (\pi + \gamma_j \varphi_j \mathcal{G}) \, dj \\ &= \left( \int_j \Gamma_j \omega_j \, dj \right) \Sigma^{-1} \pi + \left( \int_j \omega_j \Gamma_j \psi_j \, dj \right) \Sigma^{-1} \mathcal{G} \end{align*}$$

where $\psi_j = \gamma_j \varphi_j$. Let $\Gamma_m = \int_j \Gamma_j \omega_j \, dj$ and $\psi_m = \Gamma_m^{-1} \left( \int_j \omega_j \Gamma_j \psi_j \, dj \right)$ be the average risk tolerance and the weighted average of the ESG preferences. The expression for the market portfolio is then equal to:

$$w_m = \Gamma_m \Sigma^{-1} \pi + \Gamma_m \psi_m \Sigma^{-1} \mathcal{G}$$

We deduce that the asset risk premia are equal to:

$$\pi = \frac{1}{\Gamma_m} \Sigma w_m - \psi_m \mathcal{G}$$
while the market risk premium is defined as:

$$
\begin{align*}
\pi_m &= w_m^\top \pi \\
&= \frac{1}{\Gamma_m} w_m^\top \Sigma w_m - \psi_m w_m^\top \mathcal{G} \\
&= \frac{1}{\Gamma_m} \sigma_m^2 - \psi_m \mathcal{G}_m
\end{align*}
$$

where $\sigma_m = \sqrt{w_m^\top \Sigma w_m}$ and $\mathcal{G}_m = w_m^\top \mathcal{G}$ are the volatility and greenness (or greenness) of the market portfolio. On page 147 (see footnote b.), we have seen that $\pi_m^{\text{capm}} = \Gamma_m^{-1} \sigma_m^2$ is the risk premium derived from the CAPM. Since $\Gamma_m \ge 0$ and $\psi_m \ge 0$, Pástor et al. (2021) notice that:

* The risk premium including ESG sentiment is lower than the CAPM risk premium when the market's ESG intensity is positive:

$$
\mathcal{G}_m > 0 \implies \pi_m \le \pi_m^{\text{capm}}
$$

* It is greater than the CAPM risk premium when the market ESG intensity is negative:

$$
\mathcal{G}_m < 0 \implies \pi_m \ge \pi_m^{\text{capm}}
$$

* The gap $\Delta\pi_m^{\text{esg}} := |\pi_m - \pi_m^{\text{capm}}|$ is an increasing function of market ESG sentiment $\psi_m$:

$$
\psi_m \nearrow \Rightarrow \Delta\pi_m^{\text{esg}} \nearrow
$$

If we assume that $\mathcal{G}_m \approx 0$, we have $\Gamma_m = \sigma_m^2/\pi_m$ and:

$$
\pi = \beta \pi_m - \psi_m \mathcal{G} \tag{3.14}
$$

because $\beta(w_m) = (w_m^\top \Sigma w_m)^{-1} \Sigma w_m$ is the vector of asset betas with respect to the market portfolio. This is the main result of Pástor et al. (2021). It follows that the alpha of asset $i$ is equal to:

$$
\alpha_i = \pi_i - \beta_i \pi_m = -\psi_m \mathcal{G}_i
$$

Pástor et al. (2021) conclude that when $\psi_m > 0$, “green stocks have negative alphas and brown stocks have positive alphas. Moreover, greener stocks have lower alphas.”

**Example 12** We consider Example 11. The market consists of two long-only investors ($j = 1, 2$): a non-ESG investor ($\varphi_1 = 0$) and an ESG investor ($\varphi_2 > 0$). We assume that they have the same risk tolerance $\gamma$. We note $W_1$ and $W_2$ their financial wealth, which is fully invested in the risky assets. We assume that $W_1 = W_2 = 1$.

If the market is at equilibrium, we need to compute the market portfolio. If there is no short-
selling constraint, we have seen that the weights of the tangency portfolio are equal to:

$$
w^* = \frac{\Sigma^{-1} (\mu - r \mathbf{1}_n)}{\mathbf{1}_n^\top \Sigma^{-1} (\mu - r \mathbf{1}_n)}
$$

We obtain $w^* = (15.04\%, 15.04\%, 16.65\%, 16.65\%, 18.31\%, 18.31\%)$. Since there is no short position, this is the market portfolio<fnref n="11" /> without ESG preferences. It follows that the optimal portfolio $w_1^*$ of the first economic agent is equal to $w^*$. Then, we deduce the risk-tolerance coefficient of this agent:

$$
\gamma_1 = \frac{1}{\mathbf{1}_n^\top \Sigma^{-1} (\mu - r \mathbf{1}_n)} = 0.4558
$$

<footnote n="11">¹¹Otherwise, we must solve the Markowitz QP problem with the constraints $\mathbf{1}_n^\top w = 1$ and $w_i \ge 0$.</footnote>
We can now compute the optimal portfolio of the second economic agent by assuming that $\gamma_2 = \gamma_1$ and considering the following optimization problem:

$$ w_2^* = \arg \min \frac{1}{2} w^\top \Sigma w - \gamma_2 w^\top (\mu + \gamma_2 \varphi_2 \mathcal{G}) \\ \text{s.t.} \quad \begin{cases} \mathbf{1}_n^\top w = 1 \\ w \ge \mathbf{0}_n \end{cases} $$

It is important to use the QP program and not the analytical formula, because the ESG-tilted returns $\mu' = \mu + \gamma_2\varphi_2\mathcal{G}$ can be very different from the asset returns $\mu$. In this example, the long-only market portfolio is equal to the long/short tangency portfolio because we assume a uniform correlation of 25% and a constant Sharpe ratio of 20%. The ESG preference $\varphi_2$ combined with the greenness vector $\mathcal{G}$ can dramatically change the Sharpe ratio of the assets when calculated with the ESG-tilted returns. We obtain $w_2^* = (18.86\%, 11.22\%, 21.33\%, 11.97\%, 23.96\%, 12.65\%)$. The market portfolio is then equal to:

$$ \begin{aligned} w_m &= \frac{W_1}{W} w_1^* + \frac{W_2}{W} w_2^* \\ &= (1 - \omega^{\text{esg}}) w_1^* + \omega^{\text{esg}} w_2^* \end{aligned} $$

where $W = W_1 + W_2$ and $\omega^{\text{esg}}$ is the wealth share of ESG investors. If $W_1 = W_2 = 1$, we obtain $w_m = (16.95\%, 13.13\%, 18.99\%, 14.31\%, 21.13\%, 15.48\%)$, $\mu_m = 7.86\%$ and $\sigma_m = 14.93\%$. It follows that the beta values are $\beta = (1.15, 1.05, 1.04, 0.95, 0.95, 0.86)$. We deduce that the risk premia are $\pi = (5.58\%, 5.12\%, 5.06\%, 4.61\%, 4.62\%, 4.17\%)$. Finally, we conclude that the alpha vector expressed in bps is $\alpha = (-19.09, 26.19, -19.43, 25.84, -19.72, 25.55)$. A summary of these results is given in Table 3.5.

<table><caption>Table 3.5: Computation of alpha returns (Example 12, $n = 6$, $\rho = 25\%$)</caption><thead><tr><th rowspan="2">i</th><th colspan="3">Portfolio w<sub>1</sub><sup>*</sup></th><th colspan="4">Portfolio w<sub>2</sub><sup>*</sup></th><th colspan="4">Portfolio w<sub>m</sub></th></tr><tr><th>w<sub>i</sub><br>(in %)</th><th>β<sub>i</sub></th><th>π<sub>i</sub><br>(in %)</th><th>w<sub>i</sub><br>(in %)</th><th>β<sub>i</sub></th><th>π<sub>i</sub><br>(in %)</th><th>α<sub>i</sub><br>(in bps)</th><th>w<sub>i</sub><br>(in %)</th><th>β<sub>i</sub></th><th>π<sub>i</sub><br>(in %)</th><th>α<sub>i</sub><br>(in bps)</th></tr></thead><tbody><tr><td>1</td><td>15.04</td><td>1.11</td><td>5.39</td><td>18.86</td><td>1.17</td><td>5.69</td><td>-30</td><td>16.95</td><td>1.15</td><td>5.58</td><td>-19</td></tr><tr><td>2</td><td>15.04</td><td>1.11</td><td>5.39</td><td>11.22</td><td>0.99</td><td>4.80</td><td>58</td><td>13.13</td><td>1.05</td><td>5.12</td><td>26</td></tr><tr><td>3</td><td>16.65</td><td>1.00</td><td>4.87</td><td>21.33</td><td>1.07</td><td>5.18</td><td>-32</td><td>18.99</td><td>1.04</td><td>5.06</td><td>-19</td></tr><tr><td>4</td><td>16.65</td><td>1.00</td><td>4.87</td><td>11.97</td><td>0.88</td><td>4.30</td><td>57</td><td>14.31</td><td>0.95</td><td>4.61</td><td>26</td></tr><tr><td>5</td><td>18.31</td><td>0.91</td><td>4.43</td><td>23.96</td><td>0.98</td><td>4.76</td><td>-33</td><td>21.13</td><td>0.95</td><td>4.62</td><td>-20</td></tr><tr><td>6</td><td>18.31</td><td>0.91</td><td>4.43</td><td>12.65</td><td>0.80</td><td>3.87</td><td>56</td><td>15.48</td><td>0.86</td><td>4.17</td><td>26</td></tr></tbody></table>

**Remark 32** In Figure 3.7, we show the evolution of the alpha return $\alpha_i$ with respect to the market share $\omega^{\text{esg}}$ of ESG investors. It increases in absolute value because the deviation of the market portfolio including ESG preferences increases with $\omega^{\text{esg}}$. We note that $\alpha_1 \approx \alpha_3 \approx \alpha_5$ and $\alpha_2 \approx \alpha_4 \approx \alpha_6$ because of the specification of the exercise problem. When the Sharpe ratio of the assets is different and the correlation is not uniform, the alpha returns are more diffuse.

**Interpretation of the results** As mentioned earlier, we need to distinguish between expected returns and realized returns. From a theoretical point of view, there is a scientific consensus that the risk premium of brown assets is positive, which implies that the risk premium of green assets
<figure><img src="image_71.png" /><figcaption>Figure 3.7: Evolution of alpha return relative to market share of ESG investors (Example 12, n = 6, ρ = 25%)</figcaption></figure>

is negative (Zerbib, 2019; Ben Slimane et al., 2020; Bolton and Kacperczyk, 2021). This is because there is a systematic market risk when investing in brown assets due to several factors, including carbon pricing, regulatory, reputational, asset stranding and climate hedging risks. Moreover, it is clear that high demand for green assets from ESG investors lowers their expected returns. However, we must be careful because the positive expected excess return of brown assets does not necessarily imply that the performance of green assets is lower than the performance of brown assets:

> "In equilibrium, green assets have low expected returns because investors enjoy holding them and because green assets hedge climate risk. Green assets nevertheless outperform when positive shocks hit the ESG factor, which captures shifts in customers' tastes for green products and investors' tastes for green holdings." (Pástor et al., 2021).

The important word in this quote is *equilibrium*, which means that green assets have low expected returns in the long run. In this case, investors need to earn an additional return to compensate for the risk they take by investing in brown assets. However, in the short run, when the market is not at equilibrium, green assets may outperform brown assets, especially when we observe a supply/demand imbalance.

We might ask, what does equilibrium mean? In fact, it refers to a certain long-term period. To quantify the long run more precisely, let us consider the one-factor risk model:

$$R_i - r = \alpha_i + \beta_i (R_m - r) + \varepsilon_i$$
where $R_m \sim \mathcal{N}(\mu_m, \sigma_m^2)$ is the stochastic market return, $\varepsilon_i \sim \mathcal{N}(0, \tilde{\sigma}_i^2)$ is the idiosyncratic risk and $\varepsilon_i \perp \varepsilon_j$. It follows that $(R_i, R_j)$ follows a bivariate Gaussian distribution:

$$ \begin{pmatrix} R_i \\ R_j \end{pmatrix} = \mathcal{N} \left( \left( \begin{pmatrix} \mu_i \\ \mu_j \end{pmatrix} \right), \begin{pmatrix} \sigma_i^2 & \sigma_{i,j} \\ \sigma_{i,j} & \sigma_j^2 \end{pmatrix} \right) $$

where $\mu_i = r + \alpha_i + \beta_i (\mu_m - r)$, $\sigma_i^2 = \beta_i^2 \sigma_m^2 + \tilde{\sigma}_i^2$ and $\sigma_{i,j} = \beta_i \beta_j \sigma_m^2$. We deduce that $R_i - R_j = \mathcal{N}(\mu_{i-j}, \sigma_{i-j}^2)$ where:

$$ \mu_{i-j} = (\alpha_i - \alpha_j) + (\beta_i - \beta_j) (\mu_m - r) $$

and:

$$ \sigma_{i-j}^2 = (\beta_i - \beta_j)^2 \sigma_m^2 + \tilde{\sigma}_i^2 + \tilde{\sigma}_j^2 $$

Let us assume that the two assets have the same systematic risk: $\beta_i = \beta_j$. We obtain:

$$ R_i - R_j = \mathcal{N}(\alpha_i - \alpha_j, \tilde{\sigma}_i^2 + \tilde{\sigma}_j^2) $$

In the standard CAPM, the alpha returns are equal to zero and we deduce that:

$$ \Pr\{R_i < R_j\} = \frac{1}{2} $$

If two assets have the same systematic risk, the probability that one will underperform the other is 50%. Now let us take ESG preferences into account by assuming that asset i is the green asset and asset j is the brown asset. The one-year underperformance probability becomes:

$$ p_u(\Delta\alpha) = \Pr\{R_i < R_j\} = \Phi\left(\frac{\alpha_j - \alpha_i}{\sqrt{\tilde{\sigma}_i^2 + \tilde{\sigma}_j^2}}\right) > \frac{1}{2} $$

because we have $\Delta\alpha = \alpha_j - \alpha_i > 0$. We can extend this formula to a holding period longer than one year. If we assume that the dynamics of asset returns are Brownian motions, we get:

$$ p_u(\Delta\alpha, t) = \Phi\left(\frac{(\alpha_j - \alpha_i)\sqrt{t}}{\sqrt{\tilde{\sigma}_i^2 + \tilde{\sigma}_j^2}}\right) $$

where $t$ is the holding period. Using this formula, we can find the holding period to achieve a given underperformance probability $p_u$:

$$ t(\Delta\alpha, p_u) = \frac{(\tilde{\sigma}_i^2 + \tilde{\sigma}_j^2)}{(\alpha_j - \alpha_i)^2} \Phi^{-1}(p_u)^2 $$

In Figure 3.8, we show the relationship between $\Delta\alpha$ and the underperformance probability $p_u(\Delta\alpha, t)$ for several values of the holding period$^{12}$. For plausible values of $\Delta\alpha$ (less than 200 bps), we see that the probability for a one-year holding period is less than 55%. It increases to 70% for a ten-year period, which is not very high. It follows that the values of $t(\Delta\alpha, p_u)$ are very high at the asset level. Let us now consider the same exercise at the portfolio level. We consider an equally-weighted portfolio of 500 green assets and 500 brown assets. Results are given in the bottom/right panel. If the alpha difference is 40 bps, the probability of underperformance is 90% over two years.

<footnote>12We assume a typical value of 10% for the idiosyncratic volatility: $\tilde{\sigma}_i = \tilde{\sigma}_j = 10\%$.</footnote>
<figure><img src="image_72.png" /><figcaption>Figure 3.8: Impact of alpha returns on underperformance probability</figcaption></figure>

**Remark 33** All these results show that the term equilibrium refers to long holding periods. At the asset level, alpha returns require at least ten years to observe a significant difference. At the portfolio level, a three-year holding period is required for alpha returns to materialize.

One might ask whether the PST model accounts for the ESG risk premium or whether it is better suited to assess the green risk premium. In fact, the sustainable characteristic of the firm $i$ is measured by a non-random metric $G_i$. For instance, $G_i$ may correspond to the carbon footprint or the green intensity<fnref n="13" /> of the firm. If we apply this model with ESG characteristics, $G_i$ is the ESG score $S_i$ of the firm. In this case, the assumption that all investors have the same view on the ESG score is a strong hypothesis. In particular, we have already seen that there is a large divergence between ESG scoring models (Berg et al., 2022). In this context, the original formulation of the PST model is certainly better suited to assess the climate risk premium than the ESG risk premium.

## Extension of the model

**ESG uncertainty** The previous problem is addressed by Avramov et al. (2022) (hereafter, ACLT model), who analyze the impact of ESG score uncertainty on the ESG risk premium. They assume that ESG scores are stochastic and may be correlated with asset excess returns:

$$ \left( \begin{array}{c} \tilde{R} \\ \mathbf{S} \end{array} \right) \sim \mathcal{N} \left( \left( \begin{array}{c} \pi \\ \mu_s \end{array} \right), \left( \begin{array}{cc} \Sigma & \Sigma_{\pi,s} \\ \Sigma_{s,\pi} & \Sigma_s \end{array} \right) \right) $$

where $\tilde{R}$ and $\mathbf{S}$ are the random vectors of excess returns and ESG scores, respectively. It follows that $\tilde{b}_j = \varphi_j \mathbf{S}$ is stochastic and not constant. Using Equation (3.12), we deduce that the expected

<footnote n="13">¹³Measured, for example, by the green revenue share.</footnote>
utility of the economic agent j is equal to:

$$ \mathbb{E}[\mathcal{U}(\tilde{W}_j, w_j)] = e^{-\bar{\Gamma}_j(1+r)} \mathbb{E}[\exp(-\bar{\Gamma}_j w_j^\top (\tilde{R} + \psi_j \mathcal{S}))] $$

where $\psi_j = \gamma_j \varphi_j$. Since we have $\tilde{R} + \psi_j \mathcal{S} \sim \mathcal{N}(\tilde{\mu}_j, \tilde{\Sigma}_j)$ where $\tilde{\mu}_j = \pi + \psi_j \mu_s$ and $\tilde{\Sigma}_j = \Sigma + \psi_j^2 \Sigma_s + 2\psi_j \Sigma_{\pi,s}$, a new expression for the expected utility is:

$$ \mathbb{E}[\mathcal{U}(\tilde{W}_j, w_j)] = e^{-\bar{\Gamma}_j(1+r)} \exp \left( -\bar{\Gamma}_j w_j^\top \tilde{\mu}_j + \frac{1}{2} \bar{\Gamma}_j^2 w_j^\top \tilde{\Sigma}_j w_j \right) $$

The first-order condition is equal to $-\bar{\Gamma}_j \tilde{\mu}_j + \bar{\Gamma}_j^2 \tilde{\Sigma}_j w_j = 0$, which means that the optimal portfolio is:

$$ w_j^* = \bar{\Gamma}_j \tilde{\Sigma}_j^{-1} \tilde{\mu}_j = \bar{\Gamma}_j \tilde{\Sigma}_j^{-1} (\pi + \psi_j \mu_s) $$

This is exactly the same expression as in Equation (3.13) where the asset covariance matrix $\Sigma$ is replaced by the augmented covariance matrix $\tilde{\Sigma}_j$ and the greenness vector $\mathcal{G}$ is equal to the vector $\mu_s$ of expected ESG scores. Avramov et al. (2022) introduce the matrix $\Omega_j = \tilde{\Sigma}_j^{-1} - \Sigma^{-1}$ and rewrite the optimal solution as follows:

$$ w_j^* = \underbrace{\bar{\Gamma}_j \Sigma^{-1} (\pi + \psi_j \mu_s)}_{\text{PST solution}} + \underbrace{\bar{\Gamma}_j^{-1} \Omega_j (\pi + \psi_j \mu_s)}_{\text{ESG uncertainty}} $$

Thus, the optimal portfolio consists of two components. The first is the optimal portfolio of the PST model. The second component is another portfolio due to the uncertainty in the ESG scores. The two portfolios have the same expression, except that the second portfolio depends on the matrix $\Omega_j$ rather than the covariance matrix $\Sigma$. Using market clearing conditions, Avramov et al. (2022) derive the CAPM-like relationship:

* If there is no ESG uncertainty ($\mathcal{S} = \mu_s$ and $\Sigma_s = 0_{n,n}$), the vector of risk premia is given by:

$$ \begin{aligned} \pi^{\text{esg}} &= \beta \pi_m - \psi_m (\mu_s - \beta \bar{\mathcal{S}}_m) \\ &= \pi^{\text{capm}} - \psi_m (\mu_s - \beta \bar{\mathcal{S}}_m) \end{aligned} $$

where:

$$ \left\{ \begin{aligned} \pi_m &= \frac{1}{\Gamma_m} \sigma_m^2 - \psi_m \bar{\mathcal{S}}_m \\ \sigma_m^2 &= w_m^\top \Sigma w_m \\ \beta &= \frac{\Sigma w_m}{w_m^\top \Sigma w_m} \\ \bar{\mathcal{S}}_m &= w_m^\top \mu_s \end{aligned} \right. $$

Here, $\pi_m$, $\sigma_m$ and $\bar{\mathcal{S}}_m$ are the risk premium, volatility and ESG score of the market portfolio $w_m$, and $\beta$ is the vector of beta coefficients. $\Gamma_m$ and $\psi_m$ are the aggregate risk tolerance and ESG preference across the investors:

$$ \left\{ \begin{aligned} \Gamma_m &= \frac{\int_j \omega_j \Gamma_j \, dj}{\int_j \omega_j \, dj} = \int_j \omega_j \Gamma_j \, dj \\ \psi_m &= \frac{\int_j \omega_j \Gamma_j \psi_j \, dj}{\int_j \omega_j \Gamma_j \, dj} = \frac{\int_j \omega_j \Gamma_j \psi_j \, dj}{\Gamma_m} \end{aligned} \right. $$

The authors retrieve the formula (3.14) when the market is ESG neutral<fnref n="14" />.

<footnote n="14">¹⁴Pástor et al. (2021) assume that $\mathcal{G}_m = 0$ and find that $\pi = \beta\pi_m - \psi_m\mathcal{G}$.</footnote>
• If there is uncertainty in the ESG scores ($\mathbf{S} \neq \mu_s$ and $\Sigma_s \neq \mathbf{0}_{n,n}$), the vector of risk premia will be:

$$ 
\begin{aligned}
\check{\pi}^{\text{esg}} &= \check{\beta}\check{\pi}_m - \psi_m (\check{\mu}_s - \check{\beta}\check{\mathbf{S}}_m) \\
&= \beta\pi_m + (\check{\beta} - \beta)\pi_m - \psi_m (\check{\mu}_s - \check{\beta}\check{\mathbf{S}}_m)
\end{aligned}
$$

where $\beta$ and $\psi_m$ are the previously defined values and:

$$ 
\left\{
\begin{aligned}
\check{\pi}_m &= \frac{1}{\Gamma_m} \check{\sigma}_m^2 - \psi_m \check{\mathbf{S}}_m \\
\check{\sigma}_m^2 &= w_m^\top \check{\mathbf{S}}_m w_m \\
\check{\beta} &= \frac{\check{\mathbf{S}}_m w_m}{w_m^\top \check{\mathbf{S}}_m w_m} \\
\check{\mu}_s &= \frac{\Psi_m \mu_s}{\psi_m} \\
\check{\mathbf{S}}_m &= w_m^\top \check{\mu}_s
\end{aligned}
\right.
$$

Here, $\check{\pi}_m$, $\check{\sigma}_m$ and $\check{\mathbf{S}}_m$ are the risk premium, volatility and ESG score of the market portfolio $w_m$, $\check{\beta}$ is the vector of effective beta coefficients, and $\check{\mu}_s$ is the vector of modified average ESG scores. These quantities depend on $\check{\mathbf{S}}_m$ and $\Psi_m$:

$$ 
\left\{
\begin{aligned}
\check{\mathbf{S}}_m &= \left( \frac{\int_j \omega_j \Gamma_j \check{\mathbf{S}}_j^{-1} \mathrm{d}j}{\Gamma_m} \right)^{-1} \\
\Psi_m &= \left( \int_j \omega_j \Gamma_j \check{\mathbf{S}}_j^{-1} \mathrm{d}j \right)^{-1} \int_j \omega_j \Gamma_j \check{\mathbf{S}}_j^{-1} \psi_j \mathrm{d}j
\end{aligned}
\right.
$$

The relationship $\check{\pi}^{\text{esg}} = \check{\beta}\check{\pi}_m - \psi_m (\check{\mu}_s - \check{\beta}\check{\mathbf{S}}_m)$ obtained with ESG uncertainty is very close to the equilibrium formula $\pi^{\text{esg}} = \beta\pi_m - \psi_m (\mu_s - \beta\check{\mathbf{S}}_m)$ obtained without ESG uncertainty. In fact, ESG uncertainty changes the risk perception of investors. Therefore, the ESG-tilted covariance matrix $\check{\mathbf{S}}_j = \Sigma + \psi_j^2 \Sigma_s + 2\psi_j \Sigma_{\pi,s}$ is no longer equal to the asset covariance matrix $\Sigma$. It impacts the quantities related to the market portfolio.

To better understand the impact of ESG uncertainty on alpha returns, Avramov et al. (2022) consider the special case where agents have homogeneous preferences ($\check{\gamma}_j = \check{\gamma}, \varphi_j = \varphi$) and the same wealth ($W_j = 1$), the covariance matrix of ESG scores is diagonal ($\Sigma_s = \text{diag}(\sigma_{1,s}^2, \dots, \sigma_{n,s}^2)$) and returns are independent of ESG scores ($\Sigma_{\pi,s} = \mathbf{0}_{n,n}$). They deduce that $\Gamma_m = \gamma, \psi_m = \psi = \gamma\varphi, \check{\mathbf{S}}_m = \Sigma + \psi^2 \Sigma_s, \Psi_m = \psi I_n, \check{\sigma}_m^2 = \sigma_m^2 + \psi^2 \sigma_s^2, \sigma_m^2 = w_m^\top \Sigma w_m, \sigma_s^2 = w_m^\top \Sigma_s w_m, \check{\mu}_s = \mu_s$ and $\check{\mathbf{S}}_m = w_m^\top \mu_s$. The vector of the effective beta is equal to:

$$ 
\begin{aligned}
\check{\beta} &= \frac{\sigma_m^2}{\check{\sigma}_m^2} \beta + \psi^2 \frac{\sigma_s^2}{\check{\sigma}_m^2} \beta_s \\
&= \beta + \psi^2 \frac{\sigma_s^2}{\check{\sigma}_m^2} (\beta_s - \beta)
\end{aligned}
$$

where:

$$ \beta_s = \frac{\Sigma_s w_m}{\sigma_s^2} $$
Avramov et al. (2022) find that:

$$
\begin{align*}
\check{\alpha}^{\text{esg}} &= \check{\pi}^{\text{esg}} - \beta \pi_m \\
&= \psi^2 \frac{\sigma_s^2}{\check{\sigma}_m^2} (\beta_s - \beta) \pi_m - \psi_m \left( \mu_s - \left( \beta + \psi^2 \frac{\sigma_s^2}{\check{\sigma}_m^2} (\beta_s - \beta) \right) \check{\mathcal{S}}_m \right) \\
&= \psi^2 \frac{\sigma_s^2}{\check{\sigma}_m^2} (\beta_s - \beta) \left( \pi_m + \psi_m \check{\mathcal{S}}_m \right) - \psi_m \left( \mu_s - \beta \check{\mathcal{S}}_m \right)
\end{align*}
$$

If we consider the asset *i*, we obtain:

$$
\beta_{i,s} = w_{i,m} \frac{\sigma_{i,s}^2}{\sigma_s^2}
$$

$\beta_{i,s}$ increases with the volatility $\sigma_{i,s}$ of the score $\mathbf{S}_i$. We deduce that:

$$
\frac{\partial \check{\alpha}_i^{\text{esg}}}{\partial \sigma_{i,s}} > 0
$$

This implies that alpha increases with ESG uncertainty. The authors also examine the impact of ESG uncertainty on demand and test the model using the standard deviations from ESG rating agencies as a proxy for ESG uncertainty. Their conclusion is as follows:

"In equilibrium, the market premium increases and demand for stocks declines under ESG uncertainty. In addition, the CAPM alpha and effective beta both rise with ESG uncertainty and the negative ESG-alpha relation weakens." Avramov et al. (2022, page 642).

**Example 13** Consider an investment universe with four assets. Their expected returns are 5%, 6%, 7% and 8%, while their volatilities are 15%, 20%, 30% and 30%. The correlation matrix of asset returns is given by the following matrix:

$$
\mathbb{C} = \begin{pmatrix}
100\% & & & \\
10\% & 100\% & & \\
40\% & 60\% & 100\% & \\
50\% & 40\% & 80\% & 100\%
\end{pmatrix}
$$

The risk-free return is set to 2%. The average ESG scores are +3%, -2%, +1% and -1%, respectively, while the standard deviation of ESG score is the same for all assets and is 20%. We assume that the long-only investor's ESG preference $\phi$ is equal to 0.50 while his risk tolerance corresponds to the market risk tolerance.

We begin by calculating the market risk tolerance $\gamma_m = (\mathbf{1}_n^\top \Sigma^{-1} (\mu - r\mathbf{1}_n))^{-1} = 0.4654$ and the market portfolio weights:

$$
w_m = \gamma_m \Sigma^{-1} (\mu - r \mathbf{1}_n) = \begin{pmatrix} 50.08\% \\ 49.98\% \\ -22.52\% \\ 23.47\% \end{pmatrix}
$$

Then we consider the following optimization problem:

$$
\begin{array}{ll}
w^*(\psi, \mu_s, \Sigma_s) = & \arg\min \frac{1}{2} w^\top (\Sigma + \psi^2 \Sigma_s) w - \gamma_m (\mu + \psi \mu_s) \\
\text{s.t.} \quad \left\{ \begin{array}{l} \mathbf{1}_n^\top w = 1 \\ w \ge \mathbf{0}_n \end{array} \right. &
\end{array}
$$
where $\Sigma_s = 4\% \times I$, $\mu_s = (3\%, -2\%, 1\%, -1\%)$ and $\psi = \gamma_m\varphi = 0.4954 \times 0.50 = 0.2327$. To decompose the risk premium of the long-only ESG investor, we compute the long-only portfolio $w^*(0, \mathbf{0}_n, \mathbf{0}_{n,n})$, the long-only portfolio without ESG uncertainty $w^*(\psi, \mu_s, \mathbf{0}_{n,n})$ and the long-only portfolio with ESG uncertainty $w^*(\psi, \mu_s, \Sigma_s)$. For each portfolio, we compute the beta coefficient $\beta(w^*(\psi, \mu_s, \Sigma_s) | w_m)$ with respect to the market portfolio. We derive the risk premium $\pi(w^*(\psi, \mu_s, \Sigma_s)) = \beta(w^*(\psi, \mu_s, \Sigma_s) | w_m) \cdot \pi_m$ where $\pi_m = \mu(x_m) - r$. We get the following decomposition:

$$ \pi(w^*(\psi, \mu_s, \Sigma_s)) = \underbrace{\pi(w_m)}_{\text{market risk premium}} + \underbrace{\pi(w^*(0, \mathbf{0}_n, \mathbf{0}_{n,n})) - \pi(w_m)}_{\text{long-only alpha}} + \underbrace{\pi(w^*(\psi, \mu_s, \mathbf{0}_{n,n})) - \pi(w^*(0, \mathbf{0}_n, \mathbf{0}_{n,n}))}_{\text{ESG score alpha}} + \underbrace{\pi(w^*(\psi, \mu_s, \Sigma_s)) - \pi(w^*(\psi, \mu_s, \mathbf{0}_{n,n}))}_{\text{ESG uncertainty alpha}} $$

The optimal weights and risk statistics are given in Tables 3.6 and 3.7. We have:

$$ \pi(w^*(\psi, \mu_s, \Sigma_s)) = 3.74\% - 0.96 \text{ bps} - 19.80 \text{ bps} + 5.41 \text{ bps} = 3.50\% $$

In this example, the cost of the long-only constraint is -0.96 bps, the ESG score alpha is -19.80 bps, while the ESG uncertainty alpha is 5.41 bps.

<table><caption>Table 3.6: Weights of optimized portfolios (Example 13)</caption><thead><tr><th>Asset</th><th>w<sub>m</sub></th><th>w<sup>*</sup>(0, 0<sub>n</sub>, 0<sub>n,n</sub>)</th><th>w<sup>*</sup>(ψ, μ<sub>s</sub>, 0<sub>n,n</sub>)</th><th>w<sup>*</sup>(ψ, μ<sub>s</sub>, Σ<sub>s</sub>)</th></tr></thead><tbody><tr><td>#1</td><td>50.08%</td><td>52.00%</td><td>64.03%</td><td>61.87%</td></tr><tr><td>#2</td><td>48.98%</td><td>39.65%</td><td>31.51%</td><td>32.05%</td></tr><tr><td>#3</td><td>-22.52%</td><td>0.00%</td><td>0.00%</td><td>0.00%</td></tr><tr><td>#4</td><td>23.47%</td><td>8.35%</td><td>4.47%</td><td>6.09%</td></tr></tbody></table>

<table><caption>Table 3.7: Risk statistics of optimized portfolios (Example 13)</caption><thead><tr><th>Portfolio</th><th>μ(w)</th><th>σ(w)</th><th>β(w | w<sub>m</sub>)</th><th>π(w)</th><th>S(w)</th></tr></thead><tbody><tr><td>w<sub>m</sub></td><td>5.74%</td><td>13.20%</td><td>1.0000</td><td>3.74%</td><td>0.06%</td></tr><tr><td>w<sup>*</sup>(0, 0<sub>n</sub>, 0<sub>n,n</sub>)</td><td>5.65%</td><td>13.33%</td><td>0.9743</td><td>3.65%</td><td>0.68%</td></tr><tr><td>w<sup>*</sup>(ψ, μ<sub>s</sub>, 0<sub>n,n</sub>)</td><td>5.45%</td><td>12.86%</td><td>0.9214</td><td>3.45%</td><td>1.25%</td></tr><tr><td>w<sup>*</sup>(ψ, μ<sub>s</sub>, Σ<sub>s</sub>)</td><td>5.50%</td><td>12.99%</td><td>0.9358</td><td>3.50%</td><td>1.15%</td></tr></tbody></table>

**Risk factor model** In the capital asset pricing model, the asset return $R_i$ satisfies the one-factor risk model:

$$ R_i - r = \alpha_i + \beta_i (R_m - r) + \varepsilon_i $$

where $\alpha_i = 0$, $\beta_i$ is the CAPM beta of asset $i$, $R_m \sim \mathcal{N}(\mu_m, \sigma_m^2)$ is the market return, $\varepsilon_i \sim \mathcal{N}(0, \sigma_i^2)$ is the residual. Moreover, we have $\varepsilon_i \perp R_m$ and $\varepsilon_i \perp \varepsilon_j$. In matrix form, we obtain:

$$ R = r + \beta (R_m - r) + \varepsilon $$
where $R \sim \mathcal{N}(\mu, \Sigma)$, $\varepsilon = (\varepsilon_1, \dots, \varepsilon_n) \sim \mathcal{N}(\mathbf{0}_n, D)$ and $D = \text{diag}(\tilde{\sigma}_1^2, \dots, \tilde{\sigma}_n^2)$. We deduce that:

$$ \mu = \mathbb{E}[R] = r + \beta(\mu_m - r) $$

or:

$$ \pi = \mu - r = \beta\pi_m $$

It follows that:

$$ R - \mu = \beta(R_m - \mu_m) + \varepsilon $$

Therefore, the expression of the covariance matrix is:

$$ \begin{aligned} \Sigma &= \mathbb{E}[(R - \mu)(R - \mu)^{\top}] \\ &= \mathbb{E}[(\beta(R_m - \mu_m) + \varepsilon)(\beta(R_m - \mu_m) + \varepsilon)^{\top}] \\ &= \mathbb{E}[\beta(R_m - \mu_m)(R_m - \mu_m)\beta^{\top} + \varepsilon\varepsilon^{\top} + 2\beta(R_m - \mu_m)\varepsilon^{\top}] \\ &= \sigma_m^2\beta\beta^{\top} + D \end{aligned} $$

When we introduce ESG preferences, we obtain a two-factor model:

$$ R = r + \beta(R_m - r) + \beta_{\text{esg}}R_{\text{esg}} + \varepsilon $$

where $R \sim \mathcal{N}(\mu, \Sigma)$, $R_m \sim \mathcal{N}(\mu_m, \sigma_m^2)$ is the market return, $\varepsilon = (\varepsilon_1, \dots, \varepsilon_n) \sim \mathcal{N}(\mathbf{0}_n, D)$ and $D = \text{diag}(\tilde{\sigma}_1^2, \dots, \tilde{\sigma}_n^2)$. Here, $R_{\text{esg}} \sim \mathcal{N}(\mu_{\text{esg}}, \sigma_{\text{esg}}^2)$ is the return of the ESG portfolio $w_{\text{esg}} \propto \Sigma^{-1}\mu_s$, which is a zero-beta strategy (Pástor et al., 2021):

$$ \beta^{\top}w_{\text{esg}} = 0 $$

This is why we assume that $R_{\text{esg}} \perp R_m$ and $R_{\text{esg}} \perp \varepsilon$. We deduce that:

$$ \mu = r + \beta(\mu_m - r) + \beta_{\text{esg}}\mu_{\text{esg}} $$

or:

$$ \pi = \beta\pi_m + \beta_{\text{esg}}\mu_{\text{esg}} $$

For the covariance matrix, we obtain:

$$ \Sigma = \sigma_m^2\beta\beta^{\top} + \sigma_{\text{esg}}^2\beta_{\text{esg}}\beta_{\text{esg}}^{\top} + D $$

Even if the ESG portfolio has a zero return ($\mu_{\text{esg}} = 0$), we see that it can have a big impact on the structure of the covariance matrix.

**Remark 34** The CAPM and ESG coefficients $\beta_i$ and $\beta_{i,\text{esg}}$ do not have the same status. In fact, we generally assume that $\beta_i \ge 0$. Otherwise, the asset risk premium is negative: $\pi_i = \beta_i\pi_m$. Moreover, the CAPM average beta $\bar{\beta}$ is close to 1 because the beta of the market portfolio is equal to 1:

$$ \begin{aligned} \beta(w_m) = 1 &\Leftrightarrow \frac{w_m^{\top}\Sigma w_m}{w_m^{\top}\Sigma w_m} = 1 \\ &\Leftrightarrow w_m^{\top}\frac{\Sigma w_m}{w_m^{\top}\Sigma w_m} = 1 \\ &\Leftrightarrow \sum_{i=1}^{n} w_{i,m}\beta_i = 1 \end{aligned} $$
In practice, we assume that $\beta_i \in [0, 3]$ in the stock market. This is not the case for ESG beta because the ESG factor is a zero-beta long/short portfolio. This means that $\beta_{i,esg}$ can be positive or negative. In practice, the ESG factor¹⁵ is constructed so that $\beta_{i,esg} \in [-1, 1]$.

To measure the impact of ESG on the covariance matrix, we compare the one-factor and two-factor models. The expression for the asset variance is respectively $\sigma_i^2$ (capm) = $\sigma_m^2\beta_i^2 + \tilde{\sigma}_i^2$ for the one-factor model and $\sigma_i^2$ (esg) = $\sigma_m^2\beta_i^2 + \sigma_{esg}^2\beta_{i,esg}^2 + \tilde{\sigma}_i^2$ for the two-factor model. We deduce that:

$$ \sigma_i^2(\text{esg}) - \sigma_i^2(\text{capm}) = \sigma_{esg}^2\beta_{i,esg}^2 \geq 0 $$

Theoretically, the introduction of the ESG factor increases asset volatility. This is because a new risk is priced in by the market. From a practical point of view, the impact may be lower:

$$ \sigma_i^2(\text{esg}) - \sigma_i^2(\text{capm}) \leq \sigma_{esg}^2\beta_{i,esg}^2 $$

because the idiosyncratic volatility can be reduced in the two-factor model. In fact, we can assume that the ESG factor may capture some of the CAPM residual risk. If we focus on the correlation, we obtain:

$$ \rho_{i,j}(\text{esg}) = \frac{\sigma_m^2\beta_i\beta_j + \sigma_{esg}^2\beta_{i,esg}\beta_{j,esg}}{\sigma_i(\text{esg})\sigma_j(\text{esg})} $$

and:

$$ \rho_{i,j}(\text{esg}) - \rho_{i,j}(\text{capm}) = \underbrace{\left( \frac{1}{\sigma_i(\text{esg})\sigma_j(\text{esg})} - \frac{1}{\sigma_i(\text{capm})\sigma_j(\text{capm})} \right)}_{\text{negative}} \sigma_m^2\beta_i\beta_j + \underbrace{\frac{\sigma_{esg}^2\beta_{i,esg}\beta_{j,esg}}{\sigma_i(\text{esg})\sigma_j(\text{esg})}}_{\text{not signed}} $$

We have two effects:

1. As asset volatilities increase, the contribution of the CAPM covariance factor $\beta_i\beta_j$ in the two-factor model decreases.

2. The second component depends on the sign of the two ESG beta coefficients. If $\beta_{i,esg}$ and $\beta_{j,esg}$ are both positive or negative, the contribution is positive, otherwise it is negative. This means that the ESG factor increases the correlation between ESG-friendly (or green) assets. The correlation also increases between ESG-unfriendly (or brown) assets. However, the correlation between a green asset and a brown asset decreases.

**Example 14** We consider an investment universe consisting of five assets. Their market betas are 0.9, 0.8, 1.2, 0.7 and 1.3, respectively, while their specific volatilities are 4%, 12%, 5%, 8% and 5%. The volatility of the market portfolio is 25%. As for the ESG factor, we have $\sigma_{esg} = 10\%$, while the ESG sensitivity values are set to -0.5, 0.7, 0.2, 0.9 and -0.3.

The covariance matrices are reported in Tables 3.8 and 3.9. We verify that the asset volatilities have increased with the two-factor model: $\sigma_i(\text{esg}) > \sigma_i(\text{capm})$. We notice that most of correlations have decreased except for the cross-correlation $\rho_{2,4}$ between the second and fourth assets. These assets have the largest ESG sensitivities: $\beta_{2,esg} = 0.7$ and $\beta_{4,esg} = 0.9$. The cross-correlation $\rho_{1,5}$ is not reduced even through we have $\beta_{1,esg} = -0.5$ and $\beta_{5,esg} = -0.3$. In fact, the product $\beta_{1,esg}\beta_{5,esg}$
<table><caption>Table 3.8: CAPM covariance matrix (Example 14)</caption><thead><tr><th>Asset</th><th>σ<sub>i</sub> (in %)</th><th colspan="6">ρ<sub>i,j</sub> (in %)</th></tr></thead><tbody><tr><td>#1</td><td>22.85</td><td>100.00</td><td>84.43</td><td>97.12</td><td>89.54</td><td>97.31</td></tr><tr><td>#2</td><td>23.32</td><td>84.43</td><td>100.00</td><td>84.58</td><td>77.99</td><td>84.75</td></tr><tr><td>#3</td><td>30.41</td><td>97.12</td><td>84.58</td><td>100.00</td><td>89.71</td><td>97.49</td></tr><tr><td>#4</td><td>19.24</td><td>89.54</td><td>77.99</td><td>89.71</td><td>100.00</td><td>89.89</td></tr><tr><td>#5</td><td>32.88</td><td>97.31</td><td>84.75</td><td>97.49</td><td>89.89</td><td>100.00</td></tr></tbody></table>

<table><caption>Table 3.9: Two-factor covariance matrix (Example 14)</caption><thead><tr><th>Asset</th><th>σ<sub>i</sub> (in %)</th><th colspan="6">ρ<sub>i,j</sub> (in %)</th></tr></thead><tbody><tr><td>#1</td><td>23.39</td><td>100.00</td><td>72.85</td><td>93.27</td><td>70.18</td><td>96.61</td></tr><tr><td>#2</td><td>24.35</td><td>72.85</td><td>100.00</td><td>82.72</td><td>79.84</td><td>78.23</td></tr><tr><td>#3</td><td>30.48</td><td>93.27</td><td>82.72</td><td>100.00</td><td>83.87</td><td>96.28</td></tr><tr><td>#4</td><td>21.24</td><td>70.18</td><td>79.84</td><td>83.87</td><td>100.00</td><td>77.24</td></tr><tr><td>#5</td><td>33.02</td><td>96.61</td><td>78.23</td><td>96.28</td><td>77.24</td><td>100.00</td></tr></tbody></table>

is no large enough to compensate for the increase in volatilities: $\sigma_1$ (esg) - $\sigma_1$ (capm) = 54 bps and $\sigma_5$ (esg) - $\sigma_5$ (capm) = 14 bps.

We consider the portfolio $w = \vartheta\Sigma^{-1}\eta$ where $\eta$ is a $n \times 1$ vector and $\vartheta = 1/(\mathbf{1}_n^\top\Sigma^{-1}\eta)$ is the scalar such that $\mathbf{1}_n^\top w = 1$. Using results in Box 3.3, we deduce that:

$$ 
\begin{aligned}
w &= \vartheta D^{-1}\eta - \vartheta M^{-1}\eta \\
&= \vartheta D^{-1}\eta - \vartheta\omega_1\tilde{\beta}\tilde{\beta}^\top\eta - \vartheta\omega_2\tilde{\beta}_{\text{esg}}\tilde{\beta}_{\text{esg}}^\top\eta + \vartheta\omega_3(\tilde{\beta}_{\text{esg}}\tilde{\beta}^\top + \tilde{\beta}\tilde{\beta}_{\text{esg}}^\top)\eta \\
&= \vartheta\left(D^{-1} - \omega_1\tilde{\beta}\tilde{\beta}^\top\right)\eta + \vartheta\left(\omega_3\tilde{\beta}_{\text{esg}}\tilde{\beta}^\top + \omega_2\tilde{\beta}\tilde{\beta}_{\text{esg}}^\top - \omega_1\tilde{\beta}\tilde{\beta}^\top\right)\eta
\end{aligned} 
\quad (3.15) 
$$

Therefore, we can derive analytical formulas for GMV ($\eta = \mathbf{1}_n$), MVO ($\eta = \gamma\mu$) and tangency ($\eta = \mu - r\mathbf{1}_n$) portfolios. For instance, if we look at the minimum variance portfolio, Roncalli et al. (2020) show that:

$$ \omega_{i,gmv} = \frac{\sigma^2(\omega_{gmv})}{\tilde{\sigma}_i^2} \max \left( 1 - \frac{\beta_i}{\beta^*} - \frac{\beta_{i,esg}^*}{\beta_{esg}^*}, c \right) $$

where $c = -\infty$ if there is no constraint and $c = 0$ if there is no short selling. Since the mean of the beta coefficients is close to one, $\beta^*$ is positive, implying that the asset weight is a decreasing function of the asset beta. Therefore, the minimum variance portfolio is a low beta strategy. The effect of the ESG factor is more complex because the mean of the ESG beta coefficients is close to zero, implying that the threshold $\beta_{esg}^*$ can be positive or negative. We conclude that the asset weight can be a decreasing or increasing function of the ESG betas.

**Example 15** Consider an investment universe with five assets. Their market betas are 0.7, 0.8, 0.9, 1.2 and 1.5, respectively, while their specific volatilities are 10%, 8%, 3%, 5% and 4%. The volatility of the market portfolio is 25%. As for the ESG factor, we have $\sigma_{esg} = 10\%$, while the ESG sensitivity values are set to -0.5, -0.7, -0.5, 0.9 and 1.3.

<footnote n="15">¹⁵Because the ESG factor is long/short, we can always scale its volatility by leveraging or deleveraging.</footnote>
Box 3.3: One-factor and two-factor precision matrices

In portfolio optimization, several variables (weights, risk premium, etc.) are expressed in terms of the inverse of the covariance matrix, which is called the precision matrix. In the case of the one-factor model, we apply the Sherman-Morrison-Woodbury formula<sup>a</sup> with A = D and u = v = σ<sub>m</sub>β, and we obtain:

$$ \Sigma^{-1} = D^{-1} - \frac{\sigma_m^2}{1 + \sigma_m^2 \tilde{\beta}^\top \beta} \tilde{\beta} \tilde{\beta}^\top $$

where $\tilde{\beta}_i = \beta_i / \tilde{\sigma}_i^2$.

For the two-factor model, we use the generalized Sherman-Morrison-Woodbury formula with A = D, u<sub>1</sub> = v<sub>1</sub> = σ<sub>m</sub>β and u<sub>2</sub> = v<sub>2</sub> = σ<sub>esg</sub>β<sub>esg</sub>. It follows that the inverse of the covariance matrix is equal to:

$$ \Sigma^{-1} = D^{-1} - D^{-1} U S^{-1} V^\top D^{-1} $$

where $U = V = (\sigma_m\beta \quad \sigma_{esg}\beta_{esg})$ and:

$$ S = \begin{pmatrix} 1 + \sigma_m^2 \beta^\top D^{-1} \beta & \sigma_m \sigma_{esg} \beta^\top D^{-1} \beta_{esg} \\ \sigma_m \sigma_{esg} \beta^\top D^{-1} \beta_{esg} & 1 + \sigma_{esg}^2 \beta_{esg}^\top D^{-1} \beta_{esg} \end{pmatrix} $$

Roncalli et al. (2020) show that:

$$ \Sigma^{-1} = D^{-1} - M^{-1} $$

where:

$$ M^{-1} = \omega_1 \tilde{\beta} \tilde{\beta}^{\top} + \omega_2 \tilde{\beta}_{esg} \tilde{\beta}_{esg}^{\top} - \omega_3 (\tilde{\beta}_{esg} \tilde{\beta}^{\top} + \tilde{\beta} \tilde{\beta}_{esg}^{\top}) $$

and:

$$ \left\{ \begin{array}{l} \tilde{\beta}_i = \beta_i / \tilde{\sigma}_i^2 \\ \tilde{\beta}_{i,esg} = \beta_{i,esg} / \tilde{\sigma}_i^2 \\ \omega_0 = 1 + \sigma_m^2 \tilde{\beta}^\top \beta + \sigma_{esg}^2 \tilde{\beta}_{esg}^\top \beta_{esg} + \sigma_m^2 \sigma_{esg}^2 \left( (\tilde{\beta}^\top \beta) (\tilde{\beta}_{esg}^\top \beta_{esg}) - (\tilde{\beta}^\top \beta_{esg})^2 \right) \\ \omega_1 = \omega_0^{-1} \sigma_m^2 (1 + \sigma_{esg}^2 \tilde{\beta}_{esg}^\top \beta_{esg}) \\ \omega_2 = \omega_0^{-1} \sigma_{esg}^2 (1 + \sigma_m^2 \tilde{\beta}^\top \beta) \\ \omega_3 = \omega_0^{-1} \sigma_m^2 \sigma_{esg}^2 (\tilde{\beta}^\top \beta_{esg}) \end{array} \right. $$

Tables 3.10 and 3.11 show the weights of the GMV and long-only MV portfolios and compare the allocation between the one-factor and two-factor models. If we consider Example 14, adding the ESG factor increases (resp. decreases) the weights of assets with negative (resp. positive) values of β<sub>i,esg</sub>. The reason lies in the fact that the threshold β<sub>esg</sub><sup>*</sup> is positive. In the case of Example 15, β<sub>esg</sub><sup>*</sup> is equal to −3.5677 for the GMV portfolio and −7.5752 for the long-only MV portfolio. The relationship between β<sub>i,esg</sub> and ω<sub>i,gmv</sub> becomes more complex. Indeed, the long exposure condition is β<sub>i</sub>/β<sup>*</sup> + β<sub>i,esg</sub>/β<sub>esg</sub><sup>*</sup> ≤ 1. If β<sub>i</sub> ≤ β<sup>*</sup>, ω<sub>i,gmv</sub> may be positive if β<sub>i,esg</sub> is greater than the bound

<footnote><sup>a</sup>See Appendix A.1.1 on page 1138.</footnote>
$\beta_{\text{esg}}^*(1 - \beta_i/\beta^*)$, which is negative. Therefore, both positive and negative values of $\beta_{i,\text{esg}}$ can lead to a long exposure. If $\beta_i \ge \beta^*$, the bound is positive and only an asset with a positive ESG sensitivity has a positive weight.

<table><caption>Table 3.10: Minimum variance portfolios (Example 14)</caption><thead><tr><th rowspan="2">Asset</th><th rowspan="2">β<sub>i</sub></th><th rowspan="2">β<sub>i,esg</sub></th><th colspan="2">One-factor</th><th colspan="2">Two-factor</th></tr><tr><th>GMV</th><th>MV</th><th>GMV</th><th>MV</th></tr></thead><tbody><tr><td>#1</td><td>0.90</td><td>-0.50</td><td>147.33%</td><td>0.00%</td><td>166.55%</td><td>33.54%</td></tr><tr><td>#2</td><td>0.80</td><td>0.70</td><td>24.67%</td><td>9.45%</td><td>21.37%</td><td>1.46%</td></tr><tr><td>#3</td><td>1.20</td><td>0.20</td><td>-49.19%</td><td>0.00%</td><td>-58.80%</td><td>0.00%</td></tr><tr><td>#4</td><td>0.70</td><td>0.90</td><td>74.20%</td><td>90.55%</td><td>65.06%</td><td>64.99%</td></tr><tr><td>#5</td><td>1.30</td><td>-0.30</td><td>-97.01%</td><td>0.00%</td><td>-94.18%</td><td>0.00%</td></tr><tr><td colspan="2">σ(w)</td><td></td><td>11.45%</td><td>19.19%</td><td>11.54%</td><td>20.40%</td></tr><tr><td colspan="2">β(w)</td><td></td><td>0.1913</td><td>0.7095</td><td>0.1954</td><td>0.7686</td></tr><tr><td colspan="2">β<sub>esg</sub>(w)</td><td></td><td>0.2965</td><td>0.8811</td><td>0.0674</td><td>0.4274</td></tr><tr><td colspan="2">β<sup>*</sup></td><td></td><td>1.0972</td><td>0.8307</td><td>1.0906</td><td>0.8667</td></tr><tr><td colspan="2">β<sub>esg</sub><sup>*</sup></td><td></td><td></td><td></td><td>19.7724</td><td>9.7394</td></tr></tbody></table>

<table><caption>Table 3.11: Minimum variance portfolios (Example 15)</caption><thead><tr><th rowspan="2">Asset</th><th rowspan="2">β<sub>i</sub></th><th rowspan="2">β<sub>i,esg</sub></th><th colspan="2">One-factor</th><th colspan="2">Two-factor</th></tr><tr><th>GMV</th><th>MV</th><th>GMV</th><th>MV</th></tr></thead><tbody><tr><td>#1</td><td>0.70</td><td>-0.50</td><td>26.21%</td><td>66.96%</td><td>57.34%</td><td>73.46%</td></tr><tr><td>#2</td><td>0.80</td><td>-0.70</td><td>32.17%</td><td>33.04%</td><td>19.57%</td><td>26.54%</td></tr><tr><td>#3</td><td>0.90</td><td>-0.50</td><td>166.32%</td><td>0.00%</td><td>10.31%</td><td>0.00%</td></tr><tr><td>#4</td><td>1.20</td><td>0.90</td><td>-7.55%</td><td>0.00%</td><td>130.35%</td><td>0.00%</td></tr><tr><td>#5</td><td>1.50</td><td>1.30</td><td>-117.15%</td><td>0.00%</td><td>-117.58%</td><td>0.00%</td></tr><tr><td colspan="2">σ(w)</td><td></td><td>8.10%</td><td>19.69%</td><td>17.21%</td><td>20.47%</td></tr><tr><td colspan="2">β(w)</td><td></td><td>0.0899</td><td>0.7330</td><td>0.4513</td><td>0.7265</td></tr><tr><td colspan="2">β<sub>esg</sub>(w)</td><td></td><td>-2.7786</td><td>-0.5661</td><td>-0.8306</td><td>-0.5531</td></tr><tr><td colspan="2">β<sup>*</sup></td><td></td><td>1.1664</td><td>0.8462</td><td>1.0505</td><td>0.9227</td></tr><tr><td colspan="2">β<sub>esg</sub><sup>*</sup></td><td></td><td></td><td></td><td>-3.5677</td><td>-7.5752</td></tr></tbody></table>

**Remark 35** The previous examples illustrate that the global minimum variance portfolio can have a positive or negative ESG beta. In fact, it depends on the correlation between the CAPM betas and the ESG betas. In general, the GMV portfolio has a positive ESG beta if there is a negative correlation between $\beta_i$ and $\beta_{i,esg}$.

If market risk and ESG factors are uncorrelated, we can assume that$^{16}$ $\beta^\top\beta_{esg} \approx 0$. If we consider mean-variance optimized portfolios, Equation (3.15) becomes:

$$w = \vartheta (D^{-1} - \omega_1 \tilde{\beta} \tilde{\beta}^{\top} - \omega_2 \tilde{\beta}_{esg} \tilde{\beta}_{esg}^{\top}) \mu$$

<footnote>$^{16}$Otherwise, it means that green assets are generally associated with high beta assets if $\beta^\top\beta_{esg} \gg 0$ or low beta assets if $\beta^\top\beta_{esg} \ll 0$.</footnote>
We deduce that:

$$w_i \propto \omega_\eta \frac{\mu_i}{\tilde{\sigma}_i^2} - \omega_\beta \frac{\beta_i}{\tilde{\sigma}_i^2} - \omega_{\beta_{\text{esg}}} \frac{\beta_{i, \text{esg}}}{\tilde{\sigma}_i^2}$$

where:

$$\left\{ \begin{array}{l} \omega_\eta = 1 + \sigma_m^2 \tilde{\beta}^\top \beta + \sigma_{\text{esg}}^2 \tilde{\beta}^\top \beta_{\text{esg}} \beta_{\text{esg}} \ge 0 \\ \omega_\beta = \sigma_m^2 \left( 1 + \sigma_{\text{esg}}^2 \tilde{\beta}^\top \beta_{\text{esg}} \beta_{\text{esg}} \right) \sum_{j=1}^n \frac{\beta_j \mu_j}{\tilde{\sigma}_j^2} \ge 0 \\ \omega_{\beta_{\text{esg}}} = \sigma_{\text{esg}}^2 \left( 1 + \sigma_m^2 \tilde{\beta}^\top \beta \right) \sum_{j=1}^n \frac{\beta_{j, \text{esg}} \mu_j}{\tilde{\sigma}_j^2} \le 0 \end{array} \right.$$

We deduce that $w_i$ is an increasing function of $\mu_i$ and a decreasing function of $\beta_i$ and $\tilde{\sigma}_i$. Like the minimum variance portfolio, $w_i$ can be a decreasing or increasing function of $\beta_{i, \text{esg}}$ because $\omega_{\beta_{\text{esg}}}$ can be positive or negative.

### 3.1.3 ESG efficient frontier

Pedersen et al. (2021) propose an extension of the Markowitz optimization model to include ESG preferences (hereafter, the PFP model). Although the model settings are similar, the PFP model differs slightly from the PST model by focusing more on the efficient frontier.

#### Model settings

The investment universe consists of $n$ assets. We have $\tilde{R} = R - r \sim \mathcal{N}(\pi, \Sigma)$. The assets have an ESG score given by $\mathbf{S} = (\mathbf{S}_1, \dots, \mathbf{S}_n)$. Let $w = (w_1, \dots, w_n)$ be the investor's portfolio. His initial wealth is $W$ whereas his terminal wealth is given by $\tilde{W} = (1 + r + w^\top \tilde{R}) W$. The model uses the mean-variance utility function, which is tilted by the ESG score of the portfolio:

$$\begin{aligned} \mathcal{U}(\tilde{W}, w) &= \mathbb{E}[\tilde{W}] - \frac{\bar{\gamma}}{2} \text{var}(\tilde{W}) + \zeta(\mathbf{S}(w)) W \\ &= \left(1 + r + w^\top \pi - \frac{\bar{\gamma}}{2} w^\top \Sigma w + \zeta(w^\top \mathbf{S})\right) W \end{aligned}$$

where $\zeta$ is a function that depends on the investor. Optimizing the utility function is equivalent to finding the mean-variance-esg optimized portfolio:

$$\begin{aligned} w^* &= \arg\max_{\mathbf{S}} w^\top \pi - \frac{\bar{\gamma}}{2} w^\top \Sigma w + \zeta(w^\top \mathbf{S}) \\ \text{s.t.} \quad &\mathbf{1}_n^\top w = 1 \end{aligned}$$

Let $\sigma(w) = \sqrt{w^\top \Sigma w}$ and $\mathbf{S}(w) = w^\top \mathbf{S}$. The optimization problem can be decomposed as follows:

$$w^* = \arg \left\{ \max_{\bar{\mathbf{S}}} \left\{ \max_{\bar{\sigma}} \left\{ \max_{w} \left\{ f(w; \pi, \Sigma, \mathbf{S}) \text{ s.t. } w \in \Omega(\bar{\sigma}, \bar{\mathbf{S}}) \right\} \right\} \right\} \right\} \quad (3.16)$$

where:

$$f(w; \pi, \Sigma, \mathbf{S}) = w^\top \pi - \frac{\bar{\gamma}}{2} \sigma^2(w) + \zeta(\mathbf{S}(w))$$

and:

$$\Omega = \left\{ w \in \mathbb{R}^n : \mathbf{1}_n^\top w = 1, \sigma(w) = \bar{\sigma}, \mathbf{S}(w) = \bar{\mathbf{S}} \right\}$$
## The optimal portfolio

We consider the first optimization subproblem, which is a $\sigma - \mathbf{S}$ problem:

$$ w^*(\bar{\sigma}, \bar{\mathbf{S}}) = \arg\max_{\mathbf{S}} w^\top \pi \\ \text{s.t.} \quad \begin{cases} \mathbf{1}_n^\top w = 1 \\ \sigma(w) = \sqrt{w^\top \Sigma w} = \bar{\sigma} \\ \mathbf{S}(w) = w^\top \mathbf{S} = \bar{\mathbf{S}} \end{cases} $$

Pedersen et al. (2021) rewrite the last two equations as $w^\top \Sigma w - \bar{\sigma}^2 = 0$ and $w^\top (\mathbf{S} - \bar{\mathbf{S}}\mathbf{1}_n) = 0$ because<fnref n="17" /> $\mathbf{1}_n^\top w = 1$. Therefore, the Lagrange function is:

$$ \mathcal{L}(w; \lambda_1, \lambda_2) = w^\top \pi + \lambda_1 (w^\top \Sigma w - \bar{\sigma}^2) + \lambda_2 (w^\top (\mathbf{S} - \bar{\mathbf{S}}\mathbf{1}_n)) $$

The first-order condition is:

$$ \frac{\partial \mathcal{L}(w; \lambda_1, \lambda_2)}{\partial w} = \pi + 2\lambda_1 \Sigma w + \lambda_2 (\mathbf{S} - \bar{\mathbf{S}}\mathbf{1}_n) = \mathbf{0}_n $$

We deduce that the optimal portfolio is given by:

$$ w = -\frac{1}{2\lambda_1} \Sigma^{-1} (\pi + \lambda_2 (\mathbf{S} - \bar{\mathbf{S}}\mathbf{1}_n)) $$

The second constraint $w^\top (\mathbf{S} - \bar{\mathbf{S}}\mathbf{1}_n) = 0$ implies that:

$$ \begin{align*} (*) &\Leftrightarrow (\mathbf{S} - \bar{\mathbf{S}}\mathbf{1}_n)^\top \frac{1}{2\lambda_1} \Sigma^{-1} (\pi + \lambda_2 (\mathbf{S} - \bar{\mathbf{S}}\mathbf{1}_n)) = 0 \\ &\Leftrightarrow \lambda_2 = -\frac{(\mathbf{S} - \bar{\mathbf{S}}\mathbf{1}_n)^\top \Sigma^{-1} \pi}{(\mathbf{S} - \bar{\mathbf{S}}\mathbf{1}_n)^\top \Sigma^{-1} (\mathbf{S} - \bar{\mathbf{S}}\mathbf{1}_n)} \\ &\Leftrightarrow \lambda_2 = \frac{\bar{\mathbf{S}} (\mathbf{1}_n^\top \Sigma^{-1} \pi) - \mathbf{S}^\top \Sigma^{-1} \pi}{\mathbf{S}^\top \Sigma^{-1} \mathbf{S} - 2\bar{\mathbf{S}} (\mathbf{1}_n^\top \Sigma^{-1} \mathbf{S}) + \bar{\mathbf{S}}^2 (\mathbf{1}_n^\top \Sigma^{-1} \mathbf{1}_n)} \\ &\Leftrightarrow \lambda_2 = \frac{C_{1,\pi} \bar{\mathbf{S}} - C_{s,\pi}}{C_{s,s} - 2C_{1,s} \bar{\mathbf{S}} + C_{1,1} \bar{\mathbf{S}}^2} \end{align*} $$

where $C_{x,y}$ is the compact notation for $x^\top \Sigma^{-1} y - C_{1,\pi} = \mathbf{1}_n^\top \Sigma^{-1} \pi$, $C_{s,\pi} = \mathbf{S}^\top \Sigma^{-1} \pi$, $C_{s,s} = \mathbf{S}^\top \Sigma^{-1} \mathbf{S}$, $C_{1,s} = \mathbf{1}_n^\top \Sigma^{-1} \mathbf{S}$ and $C_{1,1} = \mathbf{1}_n^\top \Sigma^{-1} \mathbf{1}_n$. Using the first constraint $w^\top \Sigma w - \bar{\sigma}^2 = 0$, we deduce that:

$$ \begin{align*} \bar{\sigma}^2 &= -\frac{1}{2\lambda_1} w^\top \Sigma \Sigma^{-1} (\pi + \lambda_2 (\mathbf{S} - \bar{\mathbf{S}}\mathbf{1}_n)) \\ &= -\frac{1}{2\lambda_1} (w^\top \pi + \lambda_2 w^\top (\mathbf{S} - \bar{\mathbf{S}}\mathbf{1}_n)) \\ &= -\frac{1}{2\lambda_1} w^\top \pi \\ &= \frac{1}{4\lambda_1^2} \pi^\top \Sigma^{-1} (\pi + \lambda_2 (\mathbf{S} - \bar{\mathbf{S}}\mathbf{1}_n)) \end{align*} $$

<footnote n="17">¹⁷This last constraint $\mathbf{1}_n^\top w = 1$ is not used in the sequel, implying that the fraction of wealth invested in the risk-free asset is equal to $w_r = 1 - \mathbf{1}_n^\top w$.</footnote>
The first Lagrange coefficient is then equal to:

$$ 
\begin{aligned}
\lambda_1 &= -\frac{1}{2\bar{\sigma}} \sqrt{\pi^\top \Sigma^{-1} \pi + \lambda_2 (\pi^\top \Sigma^{-1} \mathbf{S} - \bar{\mathbf{S}} (\pi^\top \Sigma^{-1} \mathbf{1}_n))} \\
&= -\frac{1}{2\bar{\sigma}} \sqrt{C_{\pi,\pi} - \frac{(C_{1,\pi} \bar{\mathbf{S}} - C_{s,\pi})^2}{C_{s,s} - 2C_{1,s} \bar{\mathbf{S}} + C_{1,1} \bar{\mathbf{S}}^2}}
\end{aligned}
$$

where $C_{\pi,\pi} = \pi^\top \Sigma^{-1} \pi$. Pedersen et al. (2021) notice that the optimal portfolio is the product of the volatility $\bar{\sigma}$ and the vector $\varrho(\bar{\mathbf{S}})$:

$$ 
\begin{aligned}
w^*(\bar{\sigma}, \bar{\mathbf{S}}) &= -\frac{1}{2\lambda_1} \Sigma^{-1} (\pi + \lambda_2 (\mathbf{S} - \bar{\mathbf{S}} \mathbf{1}_n)) \\
&= \bar{\sigma} \varrho(\bar{\mathbf{S}})
\end{aligned}
$$

where:

$$ \varrho(\bar{\mathbf{S}}) = \frac{1}{\lambda_1} \Sigma^{-1} (\pi + \lambda_2 (\mathbf{S} - \bar{\mathbf{S}} \mathbf{1}_n)) $$

and:

$$ \lambda_1' = \sqrt{C_{\pi,\pi} - \frac{(C_{1,\pi} \bar{\mathbf{S}} - C_{s,\pi})^2}{C_{s,s} - 2C_{1,s} \bar{\mathbf{S}} + C_{1,1} \bar{\mathbf{S}}^2}} $$

**Example 16** We consider an investment universe with four assets. Their expected returns are 6%, 7%, 8% and 10%, respectively, while their volatilities are 15%, 20%, 25% and 30%. The correlation matrix of the asset returns is given by the following matrix:

$$ C = \begin{pmatrix} 100\% & & & \\ & 20\% & 100\% & \\ & & 30\% & 50\% & 100\% \\ & & & 40\% & 60\% & 70\% & 100\% \end{pmatrix} $$

The risk-free rate is set to 2%. The ESG score vector is $\mathbf{S} = (3\%, 2\%, -2\%, -3\%)$.

We obtain $C_{1,\pi} = 2.4864$, $C_{s,\pi} = 0.0425$, $C_{s,s} = 0.1274$, $C_{1,s} = 1.9801$, $C_{1,1} = 64.1106$ and $C_{\pi,\pi} = 0.1193$. If we target $\bar{\sigma} = 20\%$ and $\bar{\mathbf{S}} = 1\%$, we deduce that $\lambda_1 = -0.8514$ and $\lambda_2 = -0.1870$. The optimal portfolio is then:

$$ w^*(\bar{\sigma}, \bar{\mathbf{S}}) = \begin{pmatrix} 59.31\% \\ 29.52\% \\ 21.76\% \\ 20.72\% \end{pmatrix} $$

It follows that the portfolio is leveraged since we have $w_r = 1 - \mathbf{1}_n^\top w = -31.31\%$. We verify that $\sqrt{w^*(\bar{\sigma}, \bar{\mathbf{S}})^\top \Sigma w^*(\bar{\sigma}, \bar{\mathbf{S}})} = 20\%$ and $(w^*(\bar{\sigma}, \bar{\mathbf{S}})^\top \mathbf{S}) / (\mathbf{1}_n^\top w^*(\bar{\sigma}, \bar{\mathbf{S}})) = 1\%$. We also notice that:

$$ \varrho(\bar{\mathbf{S}}) = \begin{pmatrix} 2.9657 \\ 1.4759 \\ 1.0881 \\ 1.0358 \end{pmatrix} $$

and verify that $w^*(\bar{\sigma}, \bar{\mathbf{S}}) = \bar{\sigma} \varrho(\bar{\mathbf{S}})$. The portfolio is then leveraged when $\bar{\sigma} \ge 1 / (\mathbf{1}_n^\top \varrho(\bar{\mathbf{S}})) = 17.75\%$.
## The Sharpe ratio of the optimal portfolio

Let us rewrite the first-order condition as:

$$ 
\begin{align*}
(*) \quad & \Leftrightarrow \pi + 2\lambda_1 \Sigma w + \lambda_2 (\mathbf{S} - \bar{\mathbf{S}}\mathbf{1}_n) = \mathbf{0}_n \\
& \Leftrightarrow w^\top \pi + 2\lambda_1 w^\top \Sigma w + \lambda_2 w^\top (\mathbf{S} - \bar{\mathbf{S}}\mathbf{1}_n) = \mathbf{0}_n \\
& \Leftrightarrow w^\top \pi + 2\lambda_1 \bar{\sigma}^2 = \mathbf{0}_n \\
& \Leftrightarrow \lambda_1 = -\frac{1}{2} \frac{w^\top \pi}{\bar{\sigma}^2} \\
& \Leftrightarrow \lambda_1 = -\frac{1}{2} \frac{\text{SR}(w | r)}{\bar{\sigma}}
\end{align*}
$$

We deduce that the Sharpe ratio of the optimal portfolio $w^*(\bar{\sigma}, \bar{\mathbf{S}})$ is equal to:

$$ \text{SR}(w^*(\bar{\sigma}, \bar{\mathbf{S}}) | r) = \sqrt{C_{\pi, \pi} - \frac{(C_{1, \pi} \bar{\mathbf{S}} - C_{s, \pi})^2}{C_{s, s} - 2C_{1, s} \bar{\mathbf{S}} + C_{1, 1} \bar{\mathbf{S}}^2}} = \text{SR}(\bar{\mathbf{S}} | \pi, \Sigma, \mathbf{S}) $$

Therefore, it depends on the asset parameters $\pi, \Sigma, \mathbf{S}$, the investor's ESG objective $\bar{\mathbf{S}}$, but not on the volatility target $\bar{\sigma}$.

Using Example 16, we deduce that the Sharpe ratio of the optimal portfolio $w^*(20\%, 1\%)$ is equal to 0.3406. More generally, we verify that $\text{SR}(w^*(\bar{\sigma}, \bar{\mathbf{S}}) | r)$ does not depend on the value of $\bar{\sigma}$. For instance, we have $\text{SR}(w^*(\bar{\sigma}, -3\%) | r) = 0.2724$, $\text{SR}(w^*(\bar{\sigma}, -2\%) | r) = 0.2875$, $\text{SR}(w^*(\bar{\sigma}, -1\%) | r) = 0.3052$, $\text{SR}(w^*(\bar{\sigma}, 0\%) | r) = 0.3242$, $\text{SR}(w^*(\bar{\sigma}, 1\%) | r) = 0.3406$, $\text{SR}(w^*(\bar{\sigma}, 2\%) | r) = 0.3443$, and $\text{SR}(w^*(\bar{\sigma}, 3\%) | r) = 0.3221$. In Figure 3.9, we show the relationship between the target value $\bar{\mathbf{S}}$ and the Sharpe ratio $\text{SR}(w^*(\bar{\sigma}, \bar{\mathbf{S}}) | r)$.

## The ESG-SR frontier

Since the objective function is equal to:

$$ 
\begin{align*}
f(w^*(\bar{\sigma}, \bar{\mathbf{S}}); \pi, \Sigma, \mathbf{S}) &= \left( \frac{w^*(\bar{\sigma}, \bar{\mathbf{S}})^\top \pi}{\bar{\sigma}} \right) \bar{\sigma} - \frac{\bar{\gamma}}{2} \bar{\sigma}^2 + \zeta(\bar{\mathbf{S}}) \\
&= \text{SR}(\bar{\mathbf{S}} | \pi, \Sigma, \mathbf{S}) \bar{\sigma} - \frac{\bar{\gamma}}{2} \bar{\sigma}^2 + \zeta(\bar{\mathbf{S}})
\end{align*}
$$

the $\sigma$-problem becomes:

$$ \max_{\bar{\sigma}} \left\{ \max_{w} \left\{ f(w; \pi, \Sigma, \mathbf{S}) \text{ s.t. } w \in \Omega(\bar{\sigma}, \bar{\mathbf{S}}) \right\} \right\} = \max_{\bar{\sigma}} \left\{ \text{SR}(\bar{\mathbf{S}} | \pi, \Sigma, \mathbf{S}) \bar{\sigma} - \frac{\bar{\gamma}}{2} \bar{\sigma}^2 + \zeta(\bar{\mathbf{S}}) \right\} $$

The first-order condition is $\text{SR}(\bar{\mathbf{S}} | \pi, \Sigma, \mathbf{S}) - \bar{\gamma}\bar{\sigma} = 0$ or $\bar{\sigma} = \bar{\gamma}^{-1} \text{SR}(\bar{\mathbf{S}} | \pi, \Sigma, \mathbf{S})$, and we have:

$$ 
\begin{align*}
f(w^*(\bar{\sigma}, \bar{\mathbf{S}}); \pi, \Sigma, \mathbf{S}) &= \bar{\gamma}^{-1} \text{SR}^2(\bar{\mathbf{S}} | \pi, \Sigma, \mathbf{S}) - \frac{1}{2}\bar{\gamma}^{-1} \text{SR}^2(\bar{\mathbf{S}} | \pi, \Sigma, \mathbf{S}) + \zeta(\bar{\mathbf{S}}) \\
&= \frac{1}{2}\bar{\gamma}^{-1} (\text{SR}^2(\bar{\mathbf{S}} | \pi, \Sigma, \mathbf{S}) + 2\bar{\gamma}\zeta(\bar{\mathbf{S}}))
\end{align*}
$$

We conclude that the $\mathbf{S}$-problem becomes:

$$ \mathbf{S}^* = \arg \max_{\bar{\mathbf{S}}} \left\{ \text{SR}^2(\bar{\mathbf{S}} | \pi, \Sigma, \mathbf{S}) + 2\bar{\gamma}\zeta(\bar{\mathbf{S}}) \right\} \quad (3.17) $$
<figure><img src="image_73.png" /><figcaption>Figure 3.9: Relationship between $\bar{S}$ and SR ($\bar{S} | \pi, \Sigma$)</figcaption></figure>

and the optimal portfolio is:

$$w^* = w^*(\sigma^*, \mathbf{S}^*)$$

where $\mathbf{S}^*$ is the solution of the $\mathbf{S}$-problem and $\sigma^* = \bar{\gamma}^{-1} \text{SR}(\mathbf{S}^* | \pi, \Sigma, \mathbf{S})$. Pedersen et al. (2021) distinguish three groups of investors:

* Type-U or ESG-unaware investors have no ESG preferences and do not use the information of ESG scores;

* Type-A or ESG-aware investors have no ESG preferences, but use ESG scores to update their views on risk premia;

* Type-M or ESG-motivated investors have ESG preferences, which means that they would like to have a high ESG score.

Type-U investors hold the same portfolio, which is the standard tangency portfolio calculated without the information of ESG scores:

$$w_U^* = \frac{\Sigma^{-1}\pi}{1_n^\top \Sigma^{-1}\pi}$$

Type-A investors choose the optimal portfolio with the highest Sharpe ratio. This is equivalent to setting $\zeta(s) = 0$ in Equation (3.17) and we note $\mathbf{S}_A^*$ the optimal ESG score. Finally, type-M investors choose an optimal portfolio on the ESG-SR efficient frontier that has an ESG score greater than the optimal ESG score: $\mathbf{S}_M^* \ge \mathbf{S}_A^*$. In this case, we have $\text{SR}(\mathbf{S}_M^* | \pi, \Sigma, \mathbf{S}) \le \text{SR}(\mathbf{S}_A^* | \pi, \Sigma, \mathbf{S})$. Therefore, type-M investors reduce their Sharpe ratio to achieve a better ESG score. While the optimal portfolio is the same for all type-A investors, it is different for two type-M investors who do not have the same risk-aversion coefficient $\bar{\gamma}$ and the same ESG utility function $\zeta(s)$.
<figure><img src="image_74.png" /><figcaption>Figure 3.11: Optimal portfolio for type-A investors (Example 16)</figcaption></figure>

<figure><img src="image_75.png" /></figure>
<figure><img src="image_76.png" /><figcaption>Figure 3.13: Optimal portfolio for type-M investors when $\zeta(s) = 0.2\sqrt{\max(s, 0)}$ (Example 16)</figcaption></figure>

<figure><img src="image_77.png" /></figure>
We consider Example 16. We compute the optimal portfolio for type-U investors. In this case, the previous analysis is not necessarily since the optimal portfolio is the traditional tangency portfolio. In the case of type-A investors, we need to find the portfolio that corresponds to the maximum Sharpe ratio of the ESG-SR efficient frontier (Figure 3.11). For type-M investors, we first calculate the function $\xi(\bar{\mathbf{S}})$:

$$ \xi(\bar{\mathbf{S}}) = \mathrm{SR}^2(\bar{\mathbf{S}} | \pi, \Sigma, \mathbf{S}) + 2\bar{\gamma}\zeta(\bar{\mathbf{S}}) $$

The optimal portfolio then corresponds to the optimal ESG score that maximizes $\xi(\bar{\mathbf{S}})$. Two examples are provided in Figures 3.12 and 3.13. Results are summarized in Table 3.12. For instance, if $\bar{\gamma} = 1.5$ and $\zeta(s) = 0.2\sqrt{\max(s, 0)}$, the optimal portfolio is $w_M^* = (107.2\%, 66.0\%, 6.5\%, -7.9\%)$ and its Sharpe ratio is 0.332. This portfolio has an ESG score of 2.7%.

<table><caption>Table 3.12: Optimal portfolios (Example 16)</caption><thead><tr><th rowspan="3">Statistics</th><th rowspan="3">Type-U</th><th rowspan="3">Type-A</th><th colspan="6">Type-M</th></tr><tr><th colspan="3">ζ(s) = s</th><th colspan="3">ζ(s) = 0.2√max(s, 0)</th></tr><tr><th>0.500</th><th>1.000</th><th>1.500</th><th>0.500</th><th>1.000</th><th>1.500</th></tr></thead><tbody><tr><td>γ̃</td><td></td><td></td><td></td><td></td><td></td><td></td><td></td></tr><tr><td>S(w*)</td><td>0.017</td><td>0.017</td><td>0.023</td><td>0.028</td><td>0.034</td><td>0.021</td><td>0.024</td><td>0.027</td></tr><tr><td>σ(w*)</td><td>0.139</td><td>0.100</td><td>0.682</td><td>0.329</td><td>0.203</td><td>0.687</td><td>0.339</td><td>0.221</td></tr><tr><td>SR(w*|r)</td><td>0.345</td><td>0.345</td><td>0.341</td><td>0.329</td><td>0.305</td><td>0.343</td><td>0.339</td><td>0.332</td></tr><tr><td>w<sub>1</sub>*</td><td>-0.524</td><td>-0.378</td><td>-3.028</td><td>-1.623</td><td>-1.090</td><td>-2.900</td><td>-1.542</td><td>-1.072</td></tr><tr><td>w<sub>2</sub>*</td><td>0.289</td><td>0.208</td><td>1.786</td><td>1.009</td><td>0.718</td><td>1.673</td><td>0.919</td><td>0.660</td></tr><tr><td>w<sub>3</sub>*</td><td>0.120</td><td>0.086</td><td>0.383</td><td>0.073</td><td>-0.056</td><td>0.464</td><td>0.169</td><td>0.065</td></tr><tr><td>w<sub>4</sub>*</td><td>0.067</td><td>0.048</td><td>-0.012</td><td>-0.144</td><td>-0.178</td><td>0.106</td><td>-0.035</td><td>-0.079</td></tr><tr><td>w<sub>r</sub>*</td><td>-0.000</td><td>-0.280</td><td>-4.184</td><td>-1.562</td><td>-0.574</td><td>-4.143</td><td>-1.596</td><td>-0.718</td></tr></tbody></table>

## Impact on asset returns

Pedersen et al. (2021) use the previous framework to analyze the dynamics of asset prices. They show that the impact of ESG depends strongly on the relative proportions of the three types of investors. Let $\omega^U$, $\omega^A$ and $\omega^M$ be the wealth shares of type-U, type-A and type-M investors, respectively. The authors assume that the security dividend payoff is given by the vector $v = (v_1, \dots, v_n)$ and depends on the ESG scores:

$$ \mathbb{E}[v | \mathbf{S}] = \hat{\mu} + \theta(\mathbf{S} - \mathbf{S}_m) $$

where $\mathbf{S}_m$ is the ESG score of the market portfolio and the parameter $\theta$ determines how informative ESG scores are of future profits. In particular, $\theta = 0$ if ESG scores are not informative. Otherwise, we can expect $\theta > 0$, implying that firms with better ESG scores are more profitable on average. Pedersen et al. (2021) derive the following propositions:

* If $\omega^U = 1$ and $\omega^A = \omega^M = 0$, then the unconditional expected returns are given by the CAPM:

$$ \mathbb{E}[R_i] - r = \beta_i (\mathbb{E}[R_m] - r) $$

but conditional expected returns depend on the ESG scores:

$$ \mathbb{E}[R_i | \mathbf{S}] - r = \beta_i (\mathbb{E}[R_m] - r) + \theta \frac{\mathbf{S}_i - \mathbf{S}_m}{P_i} $$

where $P_i$ is the price of asset i. Two assets with the same beta do not necessarily have the same conditional risk premium.
* If $\omega^A = 1$ and $\omega^U = \omega^M = 0$, then the informational value of ESG scores is fully incorporated into asset prices, and we have:

$$ \mathbb{E}[R_i | \mathbf{S}] - r = \tilde{\beta}_i (\mathbb{E}[R_m | \mathbf{S}] - r) $$

where $\tilde{\beta}_i$ is the ESG-adjusted beta coefficient.

* If $\omega^M = 1$ and $\omega^U = \omega^A = 0$, then the conditional expected return is given by:

$$ \mathbb{E}[R_i | \mathbf{S}] - r = \tilde{\beta}_i (\mathbb{E}[R_m | \mathbf{S}] - r) + \lambda_2 (\mathbf{S}_i - \mathbf{S}_m) $$

The best case for an ESG investor is $\omega^U = 1$ and $\omega^A = \omega^M = 0$ when all other investors are ESG-unaware. The adjustment of market prices then depends on the growth of type-A and type-M investors. More generally, negative and/or positive alpha returns are explained by asymmetric information, supply/demand imbalances and trading motivations. Therefore, there is no obvious conclusion:

> "If all types of investors exist, then several things can happen. If a security has a higher ESG score, then, everything else equal, its expected return can be higher or lower. A higher ESG score increases the demand for the stock from type-M investors, leading to a higher price and, therefore, a lower required return [...] Companies with poor ESG scores that are down-weighted by type-M investors will have lower prices and higher cost of capital. [...] Furthermore, the force that can increase the expected return is that the higher ESG could be a favorable signal of firm fundamentals, and if many type-U investors ignore this, the fundamental signal perhaps would not be fully reflected in the price [...] A future increase in ESG investing would lead to higher prices for high-ESG stocks [...]. If these flows are unexpected (or not fully captured in the price for other reasons), then high-ESG stocks would experience a return boost during the period of this repricing of ESG. If these flows are expected, then expected returns should not be affected." (Pedersen et al., 2021).

In this context, it is difficult to predict whether ESG investments will outperform or underperform in the short run, as it depends on many factors. In particular, the PST and PFP models use the efficient market hypothesis (EMH), which implies that asset prices must reflect all available information. For instance, as seen above, one consequence of EMH is that expected returns will not be affected if ESG investment flows are expected. In the real world, this type of assumption is difficult to verify because we know that asset prices do not react instantaneously. Assuming that the dynamics of asset prices depend only on unexpected events in the short run also limits the validity of the theoretical analysis. In the end, asset prices are driven by trading orders, regardless of the real motivations of investors. These motivations may be rational or irrational, related to fundamental or extra-financial information, etc. Moreover, the PST and PFP models consider a specific trading strategy that mimics the ESG integration strategy as defined on page 42. The previous results do not necessarily hold when we consider<fnref n="18" /> a worst-in-class exclusion strategy, a best-in-class selection strategy, or an ESG momentum strategy. For all these reasons, the performance of ESG investing remains a hotly debated topic in the investment community. Nevertheless, these models are very useful because they provide a normative framework and help make sense of the chaos of empirical results.

<footnote n="18">¹⁸See for instance Zerbib (2022).</footnote>
## 3.2 Empirical results

As noted above, the body of empirical research on the performance of ESG investing is impressive. However, there is no obvious consensus because there are so many factors to consider. First, ESG investing has evolved over the last thirteen years. The data are not the same — most of them didn't exist ten years ago — the practice of ESG scoring has definitely changed over time, the use of ESG considerations is new to many investors, etc. Backtesting an ESG strategy over a long history does not make sense. Second, we cannot assume that the relationship between ESG and performance is static (positive or negative). Rather, we must accept that the relationship between ESG and performance is dynamic. Sometimes ESG can create performance, and sometimes it does not. It was the case in the past, it will be the case in the future. This is because the relationship depends primarily on the investment and trading flows of investors. Third, the performance of ESG investing depends on portfolio implementation. This is not the same as considering an exclusion filter, adding an ESG score to an existing asset picking model, implementing a selection screen, etc. Finally, the relationship is different because it depends on the country, asset class, security universe, ESG definition, etc. Let us illustrate this with a few examples. When we talk about ESG performance, are we talking about the global ESG score or one of the pillars E, S, and G? Are we talking about specific securities such as green bonds? Are we talking about American, European, Japanese or EM assets? As the questions can be multiplied endlessly, let us focus more on the why rather than the whether. Why has ESG investing created or destroyed value for a given investment universe over a given period of time?

### 3.2.1 Equity markets

The relationship between ESG and performance has been extensively studied in equity markets. According to Coqueret (2022, Sections 4.2-4.5, pages 51-66), we can classify them into four categories: (1) ESG improves performance, (2) ESG does not impact performance, (3) ESG is financially detrimental and (4) it depends on many factors. According to Friede et al. (2015), the first category dominates the others:

> "[...] The results show that the business case for ESG investing is empirically very well founded. Roughly 90% of studies find a nonnegative ESG-CFP relation. More importantly, the large majority of studies reports positive findings. We highlight that the positive ESG impact on CFP appears stable over time. Promising results are obtained when differentiating for portfolio and non-portfolio studies, regions, and young asset classes for ESG investing such as emerging markets, corporate bonds, and green real estate." (Friede et al., 2015, page 2010).

In fact, their results are not obvious to accept because the concept of corporate financial performance covers many dimensions and is not limited to financial performance in the stock market. For instance, CFP can also refer to the cost of capital (El Ghoul et al., 2011). In addition, a large proportion of these studies focus on the G pillar (Gompers et al., 2003) or use some proxy variables other than ESG scores or ratings (Edmans, 2011). We can also find many studies whose conclusion is more neutral or negative (Barnett and Salomon, 2006; Fabozzi et al., 2008; Hong and Kacperczyk, 2009; Johnson et al., 2009; Capelle-Blancard and Monjon, 2014; Matos, 2020).

Since these various publications, a consensus has emerged among professionals. Like other investment styles, ESG investing has its ups and downs, and the relationship between ESG and performance is not straightforward and depends on many factors. Understanding these factors is the key challenge for investors, rather than having a set of strong predetermined beliefs.
## Simulated results

Below we summarize the results obtained by Bennani et al. (2018) and Drei et al. (2019), who analyzed the impact of ESG on three equity portfolio management approaches: active management, passive management, and factor investing.

**Sorted portfolios** Bennani et al. (2018) use the Amundi scoring system. For each company and each date, they access the global ESG score and its three components (E, S and G). The scores are normalized sector by sector to obtain a z-score shape, which means that they have a range roughly between -3 and +3. This also means that the scores are sector-neutral and are distributed as a standard Gaussian probability distribution.

<caption>Box 3.4: The method of characteristic-sorted portfolios</caption>

Portfolio sorting was popularized by Fama and French (1993) to test the impact of characteristics in asset pricing and to identify profitable investment strategies. The idea is to sort individual assets into portfolios with respect to a given variable. If each portfolio has roughly the same number of constituents and differs only in the level of the sorting variable, then differences in performance can be attributed to the impact of the sorting variable. Typically, each portfolio is equally or value weighted to maximize diversification. In the univariate case, the most popular approach is the quintile method, where the breakpoints for the sorting variable correspond to the 20th, 40th, 60th and 80th percentiles.

<table><caption>Table 3.A: An illustrative example</caption><thead><tr><th>Asset</th><th>S<sub>i</sub></th><th>Rank</th><th>Q<sub>i</sub></th><th>Weight</th></tr></thead><tbody><tr><td>#1</td><td>-0.3</td><td>6</td><td>Q<sub>3</sub></td><td>+50%</td></tr><tr><td>#2</td><td>0.2</td><td>5</td><td>Q<sub>3</sub></td><td>+50%</td></tr><tr><td>#3</td><td>-1.0</td><td>7</td><td>Q<sub>4</sub></td><td>+50%</td></tr><tr><td>#4</td><td>1.5</td><td>3</td><td>Q<sub>2</sub></td><td>+50%</td></tr><tr><td>#5</td><td>-2.9</td><td>10</td><td>Q<sub>5</sub></td><td>+50%</td></tr><tr><td>#6</td><td>0.8</td><td>4</td><td>Q<sub>2</sub></td><td>+50%</td></tr><tr><td>#7</td><td>-1.4</td><td>8</td><td>Q<sub>4</sub></td><td>+50%</td></tr><tr><td>#8</td><td>2.3</td><td>2</td><td>Q<sub>1</sub></td><td>+50%</td></tr><tr><td>#9</td><td>2.8</td><td>1</td><td>Q<sub>1</sub></td><td>+50%</td></tr><tr><td>#10</td><td>-2.2</td><td>9</td><td>Q<sub>5</sub></td><td>+50%</td></tr></tbody></table>

Consider the example above, where the sorting variable is an ESG score. Since we have 10 assets, each sorted portfolio has two assets. Portfolio $Q_1$ corresponds to the highest scores, while Portfolio $Q_5$ corresponds to the lowest scores. We obtain $Q_1 = (#8, #9)$, $Q_2 = (#4, #6)$, $Q_3 = (#1, #2)$, $Q_4 = (#3, #7)$ and $Q_5 = (#5, #10)$.

To construct the active management strategy, the authors use the sorting portfolio method. Each quarter, they rank the stocks according to their ESG score, and create five quintile portfolios<fnref n="19" />. Portfolio $Q_1$ corresponds to the 20% best-ranked stocks while Portfolio $Q_5$ corresponds to the 20% worst-rated stocks. The selected stocks are then equally weighted and each portfolio is invested on the first trading day of the quarter and held for three months. Quarterly rebalancing is performed to limit turnover.

<footnote n="19">¹⁹Given a universe of stocks, each portfolio then consists of 20% of assets.</footnote>
<figure><img src="image_78.png" /><figcaption>Figure 3.14: Annualized return of ESG-sorted portfolios (MSCI North America, global score)</figcaption></figure>

Source: Bennani et al. (2018).

They consider five investment universes using the following MSCI indexes: North America, EMU, Europe-ex-EMU, Japan and World. For each universe and each quintile portfolio, they calculate gross performance excluding transaction costs. Analyzing the results, the authors observe a break during the study period 2010–2017. Typically, the first half of the period is less favorable for ESG screening than the second half. In Figure 3.14, we report their results for North American stocks. Over the period 2010–2013, Portfolio $Q_1$ has a gross return of 14.6%, while Portfolio $Q_5$ has a gross return of 17.8%. We observe an increasing function between return and quintile. During this period, the best-in-class stocks underperformed worst-in-class stocks. The story is different if we focus on the period 2014–2017. Portfolio $Q_1$ shows a performance of 13.0%, while Portfolio $Q_5$ shows a performance of 9.4%. Clearly, the best-in-class stocks outperformed the worst-in-class stocks during this second period. Looking at individual pillars, Bennani et al. (2018) obtained very similar results in Figure 3.15. **E**, **S** and **G** stock picking has a negative impact on performance between 2010 and 2013, while the impact of **E**, **S** and **G** stock picking on performance is positive between 2014 and 2017. During the 2014–2017 period, environmental screening produces the best result, followed by governance scoring. However, for the governance component, the performance difference between Portfolios $Q_1$, $Q_2$, $Q_3$ and $Q_4$ is not significant. Only Portfolio $Q_5$ significantly underperforms, meaning that the worst-rated stocks are penalized, but the best-rated stocks are not necessarily rewarded.

These results clearly show that active ESG management was penalized during the period 2010–2013, while it generated excess performance between 2014 and 2017. In the case of the Eurozone, the conclusion is the same for the global ESG score, and its three components. For example, Portfolio $Q_1$ generated a return of 8.6%, while Portfolio $Q_5$ generated a return of 10.0% between 2010 and
<figure><img src="image_79.png" /><figcaption>Figure 3.15: Annualized return of ESG-sorted portfolios (MSCI North America, individual pillars)</figcaption></figure>

Source: Bennani et al. (2018).

2013 (Figure 3.16). On the contrary, the performance for the period 2014–2017 was 14.7% and 7.5% for Portfolios $Q_1$ and $Q_5$, respectively. Thus, the first period is characterized by a U-shape, while, in the second period, the best-in-class stocks significantly outperformed the worst-in-class stocks. Note that the performance difference mainly affects Portfolios $Q_1$ and $Q_5$, but not Portfolios $Q_2$, $Q_3$ and $Q_4$, implying that worst-in-class stocks are penalized and best-in-class stocks are rewarded. Looking at each individual pillar, governance is the most discriminating component (Figure 3.17). The difference between the $Q_1$ and $Q_5$ portfolios exceeds 7% in the last period. For the **E** score, we observe a U-shaped behavior between 2010 and 2013. Since 2014, the relationship between the quintile portfolios and their returns is clearly decreasing. It is less impressive than for the **G** score, but it affects all portfolios<fnref n="20" />. The integration of the social pillar is the least convincing.

For the other investment universes, the results are more heterogeneous. In the case of the Europe-ex-EMU universe, the ESG integration is country specific, which means that the performance is highly dependent on the overweight or underweight of each country. For example, the **G** screen largely overweights UK stocks when considering a $Q_1 – Q_5$ long/short portfolio. On the contrary, **E** or **S** screens favor Swedish stocks. The case of Japan is puzzling. In fact, ESG screening was less favorable in the period 2014–2017. If we consider the universe of the MSCI World index, the results are similar to those obtained for North America and the Eurozone. These different results are summarized in Table 3.13, where we have reported the impact of ESG screening on the returns of sorted portfolios. Again, the results illustrate the contrast between the two periods. In summary, Bennani et al. (2018) concluded that the relationship between performance and ESG is

<footnote n="20">²⁰For the **G** score, the difference mainly concerns the $Q_1$ and $Q_5$ portfolios, and less so the median portfolios.</footnote>
<figure><img src="image_80.png" /><figcaption>Figure 3.16: Annualized return of ESG-sorted portfolios (MSCI EMU, global score)</figcaption></figure>

Source: Bennani et al. (2018).

<figure><img src="image_81.png" /><figcaption>Figure 3.17: Annualized return of ESG-sorted portfolios (MSCI EMU, individual pillars)</figcaption></figure>

Source: Bennani et al. (2018).
time-varying and depends on several factors, particularly region and ESG pillar. They also found that some investment universes have ESG country biases, which implies that the relationship between performance and ESG cannot be analyzed. This is the case for the MSCI Europe-ex-EMU index, but such bias is not excluded for the MSCI World index.

<table><caption>Table 3.13: Impact of ESG screening on sorted portfolio returns (2010–2017)</caption><thead><tr><th>Period</th><th>Pillar</th><th>North America</th><th>EMU</th><th>Europe-ex-EMU</th><th>Japan</th><th>World</th></tr></thead><tbody><tr><td rowspan="4">2010–2013</td><td>ESG</td><td>--</td><td>-</td><td>0</td><td>+</td><td>0</td></tr><tr><td>E</td><td>-</td><td>0</td><td>+</td><td>-</td><td>0</td></tr><tr><td>S</td><td>-</td><td>-</td><td>0</td><td>-</td><td>-</td></tr><tr><td>G</td><td>-</td><td>0</td><td>+</td><td>0</td><td>+</td></tr><tr><td rowspan="4">2014–2017</td><td>ESG</td><td>++</td><td>++</td><td>0</td><td>-</td><td>+</td></tr><tr><td>E</td><td>++</td><td>++</td><td>-</td><td>+</td><td>++</td></tr><tr><td>S</td><td>+</td><td>+</td><td>0</td><td>0</td><td>+</td></tr><tr><td>G</td><td>+</td><td>++</td><td>0</td><td>+</td><td>++</td></tr></tbody></table>

Source: Bennani et al. (2018).

The study by Drei et al. (2019) is an update of the analysis of Bennani et al. (2018) considering the recent period 2018–2019. They use exactly the same data, the same investment universes and the same methodology. Their main results are reported in Figures 3.18 and 3.19.

Box 3.5: Computing the performance of long/short portfolios

Let $w_{Long}$ and $w_{Short}$ be the long and short portfolios, respectively. We note $R_t(w)$ the annualized return of Portfolio $w$ between $t-1$ and $t$. The performance of the long/short portfolio $w_{Long} - w_{Short}$ satisfies the following definition:

$$ (1 + R_t(w_{Long})) = (1 + \alpha_t(w_{Long} - w_{Short})) (1 + R_t(w_{Short})) $$

where $\alpha_t(w_{Long} - w_{Short})$ is the alpha return of $w_{Long} - w_{Short}$. We deduce that:

$$ \alpha_t (w_{Long} - w_{Short}) = \frac{R_t (w_{Long}) - R_t (w_{Short})}{1 + R_t (w_{Short})} \quad (3.18) $$

Looking at the $Q_1 - Q_5$ long-short portfolios in North America (Figure 3.18) and the Eurozone (Figure 3.19), we can see the evolution of ESG integration and its pillars in both markets. In the 2010–2013 period, sustainable investors were penalized, as shown by the negative return of the $Q_1 - Q_5$ long-short portfolios. In the 2014–2017 period, after the radical break in ESG integration, ESG investing gained momentum and delivered positive returns across all pillars on both sides of the Atlantic. However, after eight years of parallel development, Drei et al. (2019) observed a contradictory trend in ESG investing between North America and the Eurozone between 2018 and 2019. Indeed, the latter period is characterized by a squeeze in alpha returns on all dimensions in North America, and even a loss on the E pillar. This loss is important because it is the first long-short portfolio with a negative return since the 2014 ESG turning point in these two investment universes. Moreover, we observe a decline in the performance of the S and G pillars during the period 2018–2019. Looking at the global ESG score, its performance remains positive but is divided
<figure><img src="image_82.png" /><figcaption>Figure 3.18: Annualized return of long/short $Q_1 - Q_5$ sorted portfolios (MSCI North America)</figcaption></figure>

Source: Drei et al. (2019).

<figure><img src="image_83.png" /><figcaption>Figure 3.19: Annualized return of long/short $Q_1 - Q_5$ sorted portfolios (MSCI EMU)</figcaption></figure>

Source: Drei et al. (2019).
by a factor of six compared to the period 2014–2017. On the Eurozone side, the verdict is more positive. All long-short portfolio returns are positive. In the period 2018–2019, the E and S pillars generate even stronger returns compared to the previous period. The decline in the G long-short portfolio return can be partly attributed to a mean-reversion effect after an extraordinary period of impressive performance<fnref n="21" />. Drei et al. (2019) concluded that the period 2018–2019 is in line with the previous period for the Eurozone investment universe as the two periods post an annualized return around 6% in the case of the global ESG score.

How can these different results be explained? Bennani et al. (2018) assumed that two main ef-
fects contributed to ESG performance from 2014 to 2017: the selection effect of ESG screening and
the demand effect of ESG assets. The selection effect refers to the direct impact of extra-financial
information on stock prices. By taking into account other risk dimensions, the ESG investor can
select companies that are better managed from a social, environmental and governance perspective,
or avoid companies with extra-financial weaknesses. The idea is that sooner or later these extra-
financial risks will have a financial impact on the company’s performance. The second effect is
related to the balance between supply and demand. In fact, a stock price is the equilibrium between
supply and demand for that stock. Bennani et al. (2018) found that ESG investment flows that have
been observed since 2014 have largely contributed to the good performance of ESG investments over
the period 2014–2017, while the contribution of the selection effect is marginal. How to explain the
2014 break? In November 2013, the Norwegian Sovereign Wealth Fund adopted a new responsible
investment policy (Dimson et al., 2013). At about the same time, we observe a strong mobilization
of the largest European institutional investors (APG, PGGM, ERAFP, FRR, etc.), which are mas-
sively invested in European and American equities<fnref n="22" />. The good performance of ESG investing in
the period 2014–2017 is mainly explained by the portfolio rebalancing of these tier-one European
institutional investors. The 2018–2019 period is different. Indeed, this first mobilization has been
followed by a further mobilization of mid-sized (or tier-two) European institutional investors, while
the involvement of US investors remains weak. Nevertheless, this second wave of investors has a
low exposure to North American equities. The transatlantic divide observed between 2018 and 2019
is then mainly explained by the strategic asset allocation of these tier-two institutional investors.
They rebalanced their portfolios, but the trading operations mainly concerned European stocks and
not American stocks. A first explanation for the American setback can then be found in these
engagement differences between European and American investors. In addition to these two effects
选言的 (selection and supply/demand balance), Drei et al. (2019) suggest that a third factor may contribute
to ESG performance: the political and regulatory environment. The poor performance of the (E)
pillar in the US may be explained by the announced withdrawal of the United States from the Paris
Climate Agreement and some changes at the US Environmental Protection Agency (EPA). More
generally, another justification for the transatlantic divide could be the public policy of the Trump
administration in terms of its ESG roadmap.

**Remark 36** The original idea of the Amundi Investment Institute studies (Bennani et al., 2018; Drei et al., 2019) was to frequently update the empirical relationship between ESG and performance. However, the first study showed that there are country biases that are difficult to control for. Therefore, the second study focused on the investment universe of the MSCI North America and EMU indexes and considered the empirical relationship between ESG and performance to be monotonic. For instance, if the relationship is positive, we must observe $R_t(Q_1) \ge R_t(Q_2) \ge R_t(Q_3) \ge R_t(Q_4) \ge R_t(Q_5)$, while a negative relationship implies $R_t(Q_1) \le R_t(Q_2) \le R_t(Q_3) \le R_t(Q_4) \le R_t(Q_5)$.

<footnote n="21">²¹In fact, the annualized return was 7.9% between 2014 and 2017, compared to just 1.3% for the period 2018–2019.</footnote>

<footnote n="22">²²For instance, the Norwegian Sovereign Wealth Fund had an exposure to US equities greater than that of the three largest US pension funds (CalPERS, CalSTRS and NYSCRF).</footnote>
<figure><img src="image_84.png" /><figcaption>Figure 3.20: The monotonic assumption of the ESG-performance relationship</figcaption></figure>

More generally, there are some patterns we need to observe in order to interpret the results. Some of them are shown in Figure 3.20. For example, the implementation of an ESG exclusion strategy has a return-based rationale if the relationship is positive or if portfolio Q₅ underperforms the other quintile portfolios. Similarly, the implementation of an ESG selection strategy has a return-based rationale if the relationship is positive or if portfolio Q₁ outperforms the other quintile portfolios. However, Drei et al. (2019) noticed that most of the monotonous relationships observed during the period 2010–2017 were no longer valid between 2018 and 2019. For instance, they found that Rₜ (Q₁) ≥ Rₜ (Q₅) for the global ESG score in the Eurozone, but they also found that Rₜ (Q₄) ≥ Rₜ (Q₁), which is a puzzling result. This ranking disorder goes beyond the binary outcome in where Q₁ ≻ Q₅ holds or does not. Drei et al. (2019) argue that this puzzle marks the emergence of new ESG investment strategies. The Q₁ – Q₅ approach is representative of a static view of ESG scores, where the best-in-class stocks remain best-in-class and the worst-in-class stocks remain worst-in-class, while playing the intermediate quintiles, especially the fourth quintile, seems to be associated with the ESG momentum strategy (Figure 3.21) and a dynamic view of ESG scores. In the period 2018–2019, ESG strategies have become more complex, which may explain the ranking disorder. This finding is consistent with the results reported by GSIA (2019). In its 2018 investment review, the organization documents that the most common way to participate in sustainable investing (as measured by assets under management allocated to each strategy) is to implement negative screening, but this approach is closely followed by ESG integration and corporate engagement strategies. Similarly, Eurosif (2018) found similar results a year earlier, stating that “the main strategy is exclusion, but in the last two years the growth rate of this strategy slowed down. In contrast, best-in-class and ESG integration have had a high growth rate.” Investment strategies based on the dynamics of ESG ratings do not clearly correspond to negative or positive screening, but are more related to ESG integration. In this approach, an improvement in an extra-financial criterion can lead to portfolio rebalancing, just as we observe
for financial metrics. The convergence between the extra-financial approach and traditional security analysis certainly increases the focus on the dynamics and momentum of ESG ratings, and not just their static level. In this context, the analysis of the relationship between the performance of ESG investments and the static level of ESG scores is certainly outdated.

<figure><img src="image_85.png" /><figcaption>Figure 3.21: How to play ESG momentum?</figcaption></figure>

**Optimized portfolios** Many institutional investors implement ESG policies through passive management. In this case, they use two techniques: exclusion and optimization. The first approach consists of reducing the universe of the stock index by excluding the worst-rated stocks and then applying a capitalization-weighted scheme to construct the investment portfolio. The second approach is to improve the score of the investment portfolio with respect to the score of the benchmark portfolio, while controlling the tracking error risk. The first solution can be approximated using the second method, which means that optimized portfolios can be used to simulate the performance of passive ESG management (Bennani et al., 2018; Drei et al., 2019).

**Remark 37** Let us compare the ESG-optimized approach with the ESG-sorting method. We note $\mathbf{F}$ the probability distribution of the score $\mathbf{S}$. Portfolio $Q_1$ corresponds to best-in-class stocks $\{i \in Q_1 \Leftrightarrow \mathbf{S}_i \ge \mathbf{F}^{-1}(80\%)\}$, while Portfolio $Q_5$ corresponds to worst-in-class stocks $\{i \in Q_5 \Leftrightarrow \mathbf{S}_i \le \mathbf{F}^{-1}(20\%)\}$. Also the weights are uniform: $w_i(Q_j) = (5n)^{-1}$ where n is the total number of assets in the investment universe. In the case of the ESG-optimized approach, we have $w^*(\gamma) = \Sigma^{-1}(\gamma\mathbf{S} + \Sigma b)$, which implies that $w_i^*(\gamma) = b_i + \gamma(\Sigma^{-1}\mathbf{S})_i$. Therefore, the benchmark weights are tilted by the inverse of the covariance matrix times the vector of ESG scores. For example, assuming the covariance matrix is diagonal and $\mathbf{S}_i \sim N(0, 1)$, we obtain:

$$w_i^*(\gamma) = b_i + \gamma \frac{\mathbf{S}_i}{\sigma_i^2}$$

A positive score increases the benchmark weight, while $\gamma$ controls the discrepancy. If $\gamma = 0$, the optimal portfolio is the benchmark. If $\gamma$ tends to $+\infty$, the optimal weight is proportional to the score divided by the variance. When we add the long-only constraint, the optimization problem selects the stocks such that the ratio $\mathbf{S}_i/\sigma_i^2$ is greater than a threshold that depends on the parameter $\gamma$. We can then calculate the value of $\gamma$ to retrieve the stock selection<fnref n="23" /> given by portfolios $Q_1$, $Q_1 + Q_2$, etc. Therefore ESG-optimized and ESG-sorting approaches are related and generally produce similar results.

<footnote n="23">²³See Exercise 3.4.1 on page 227.</footnote>
Box 3.6: ESG-optimized portfolios

Let b be the benchmark, **S** the vector of ESG scores and Σ the covariance matrix. We consider the following optimization problem:

$$w^*(\gamma) = \arg\min \frac{1}{2}\sigma^2(w|b) - \gamma\mathbf{S}(w|b)$$

where $\sigma^2(w|b) = (w-b)^{\top}\Sigma(w-b)$ and $\mathbf{S}(w|b)$ are the ex-ante tracking error variance and ESG excess score of portfolio w relative to benchmark b. Since we have:

$$\mathbf{S}(w|b) = (w-b)^{\top}\mathbf{S} = \mathbf{S}(w) - \mathbf{S}(b)$$

we get the following optimization function:

$$w^*(\gamma) = \arg\min \frac{1}{2}w^\top\Sigma w - w^\top(\gamma\mathbf{S} + \Sigma b)$$

The ESG-variance efficient frontier is defined by the parametric curve $(\sigma^2(w^*(\gamma)|b), \mathbf{S}(w^*(\gamma)|b))$ with $\gamma \ge 0$. The QP form is given by $Q = \Sigma$ and $R = \gamma\mathbf{S} + \Sigma b$. If we target an ESG excess score, for instance $\mathbf{S}(w|b) \ge \Delta\mathbf{S}^*$, we set $\gamma = 0$ and add the inequality constraint $\mathbf{S}^\top w \ge \mathbf{S}(b) + \Delta\mathbf{S}^*$. The QP form is given by $Q = \Sigma$, $R = \Sigma b$, $C = -\mathbf{S}^\top$ and $D = -(\mathbf{S}(b) + \Delta\mathbf{S}^*)$. If we use the traditional long-only constraint $(\mathbf{1}_n^\top w = 1$ and $w_i \ge 0)$, we have $A = \mathbf{1}_n^\top$, $B = 1$ and $w^- = \mathbf{0}_n$.

Figure 3.22 shows the ESG-variance efficient frontier estimated by Bennani et al. (2018). It represents the relationship between excess score and tracking error volatility for the MSCI World universe. For example, improving the score<fnref n="24" /> of the index portfolio by 0.5 implies accepting a tracking error of 32 bps on average, and an excess score of 1.0 leads to a tracking error of 85 bps. Using risk attribution analysis, the authors also show that the governance pillar generates more tracking error than the environmental and social pillars<fnref n="25" />. These results imply that ESG passive management requires taking on significant tracking error risk relative to capitalization-weighted benchmarks.

Figure 3.24 shows the performance of ESG-optimized portfolios with respect to excess score. We see that the integration of ESG into passive management reduced the performance between 2010 and 2013, while it improved the annualized return between 2014 and 2017. For instance, an excess score of 1.0 resulted in an excess return of -34 bps in the first period and +45 bps in the second period. We also note that the relationship between excess score and excess return is not necessarily monotonic. For instance, targeting an excess score of 1.5 instead of 1.0 reduces the excess return from 45 bps to 19 bps in the second period. This is most likely due to the diversification effect. In fact, by increasing the excess score, we reduce the number of positions in the invested portfolios. There is a threshold at which the gains from ESG screening are offset by the losses from diversification reduction. Looking at the individual pillars, Bennani et al. (2018) draw the same conclusions that they have found for active management. For the MSCI World universe, all pillars destroyed value between 2010 and 2013, except for the environmental pillar, for which the results are neutral or slightly positive. This is particularly true for the governance pillar, whose underperformance is

<footnote n="24">²⁴We recall that these studies use z-scores, which means that the range is between -3 and +3.</footnote>

<footnote n="25">²⁵On average, optimized portfolios with the **G** score have 50% more tracking error than those with **E** and **S** scores (see Figure 3.23).</footnote>
<figure><img src="image_86.png" /><figcaption>Figure 3.22: Efficient frontier of ESG-optimized portfolios (MSCI World, 2010–2017, global score)</figcaption></figure>

Source: Bennani et al. (2018).

<figure><img src="image_87.png" /><figcaption>Figure 3.23: Efficient frontier of ESG-optimized portfolios (MSCI World, 2010–2017, individual pillars)</figcaption></figure>

Source: Bennani et al. (2018).
<figure><img src="image_88.png" /><figcaption>Figure 3.24: Annualized excess return of ESG-optimized portfolios (MSCI World, 2010–2017, global score)</figcaption></figure>

Source: Bennani et al. (2018).

<figure><img src="image_89.png" /><figcaption>Figure 3.25: Annualized excess return of ESG-optimized portfolios (MSCI World, 2010–2013, individual pillars)</figcaption></figure>

Source: Bennani et al. (2018).
<figure><img src="image_90.png" /><figcaption>Figure 3.26: Annualized excess return of ESG-optimized portfolios (MSCI World, 2014–2017, individual pillars)</figcaption></figure>

Source: Bennani et al. (2018).

<figure><img src="image_91.png" /><figcaption>Figure 3.27: Annualized excess return in bps of ESG-optimized portfolios (MSCI North America and EMU, 2010–2017)</figcaption></figure>

Source: Bennani et al. (2018).
about two to three times greater than the underperformance of the overall ESG score (Figure 3.25). For the period 2014–2017, the story changes. Every score generates outperformance, except for the social pillar (Figure 3.26). If we look at the North America and Europe investment universes<fnref n="26" />, the performance of the optimized portfolios is in line with the performance of the stock-picking portfolios (Figure 3.27). Over the period 2010–2013, only the E score would have generated outperformance in Europe. In this region, the authors found that the performance of the ESG score was also neutral when targeting low tracking error risk (less than 60 bps) or low excess score (less than 0.8). In all other cases, we observe a negative excess return, especially in North America. Between 2014 and 2017, we obtain opposite results. All scores generate outperformance, except for the social pillar. The results are more significant in Europe than in North America. To sum up, the two big winners between 2014 and 2017 were the environmental pillar in North America and the governance pillar in Europe.

The updated study by Drei et al. (2019) has confirmed most of the results found by Bennani et al. (2018), in particular the trade-off between excess score and tracking error risk, and the reversal phenomenon of the ESG-performance relationship, which is negative when a high excess score is targeted. This reversal phenomenon is most likely due to the diversification effect. Indeed, by increasing the excess score, we reduce the number of positions held in the managed portfolio. There is therefore a threshold at which the gains from ESG screening are offset by the losses from reduced diversification. Since the relationship between quintile portfolios and performance is not monotonic, Drei et al. (2019) find that the performance of ESG-optimized portfolios is less impressive between 2018 and 2019 than between 2014 and 2017. Therefore, they observe a decrease in the maximum excess return. Focusing on the Eurozone investment universe, where the loss of diversification is reached more quickly than in North America, they conclude that “optimized portfolios generate worse results (except for the social pillar)” and, more generally, that “risk-return profiles are less interesting than before.” Therefore, the dynamic view of ESG investing implies that the performance of ESG-optimized portfolios will not necessarily be in line with the performance of $Q_1 - Q_5$ sorted portfolios due to the impact of the other sorted portfolios, particularly the fourth quintile portfolio.

**A new risk factor?** Previously, we saw that the long/short $Q_1 - Q_5$ strategy generated positive alpha between 2014 and 2019, while ESG investing has penalized ESG investors between 2010 and 2013. When we talk about alpha generation, we generally refer to factor investing. In fact, factor investing makes the difference between the financial performance of systematic factors and the financial performance of specific factors. Put another way, factor investing makes the difference between alpha and beta returns. In factor investing, the beta (or systematic) factors correspond to the common risk factors that explain a significant portion of the cross-section of stock returns. As ESG changes the landscape of asset management, we may ask whether ESG has become a new risk factor that needs to be integrated into a factor investing framework, or whether it remains an alpha strategy. To answer this question, Roncalli (2020b) uses the single-factor model:

$$R_{i,t} = \alpha_{i,j} + \beta_{i,j} \mathcal{F}_{j,t} + \varepsilon_{i,t}$$

where $R_{i,t}$ is the return of stock i at time t, $\mathcal{F}_{j,t}$ is the value of the jth common risk factor at time t and $\varepsilon_{i,t}$ is the idiosyncratic risk. The coefficients $\alpha_{i,j}$ and $\beta_{i,j}$ are estimated using ordinary least squares. For each stock, we compute the coefficient of determination:

$$\mathcal{R}_{i,j}^2 = 1 - \frac{\text{var}(\varepsilon_{i,t})}{\text{var}(R_{i,t})}$$

<footnote n="26">²⁶Bennani et al. (2018) have merged Eurozone and Europe-ex-EMU stocks into the same investment universe because tracking error optimization forces the portfolio to be more or less country-neutral.</footnote>
We then calculate the average proportion of return variance explained by the common factor: $\mathfrak{R}_j^2 = \frac{1}{n} \sum_{i=1}^n \mathfrak{R}_{i,j}^2$. We consider the standard factors derived from a factor investing framework: size, value, momentum, low-volatility and quality. These factors $\mathcal{F}_{j,t}$ are constructed using the Fama-French method of sorted portfolios. Contrary to the academic literature, a long-only framework is used, which is the common approach of institutional investors. This means that the factors correspond to $Q_1$ portfolios or best-in-class stocks. We also consider the traditional market factor, which corresponds to the capitalization-weighted portfolio. All analyses use weekly returns. Results are shown in Table 3.14. We read these figures as follows: between 2010 and 2013, the market risk factor explains 40.8% of the dispersion of North American stock returns, this figure is 39.3% if we consider the size factor, and so one. We find that ESG is a strong contender as a stand-alone factor, competing with the market risk factor. On average, since 2014, the market risk factor explains 28.6% of the cross-section variance, while the ESG factor has an explanatory power of 27.4% in North America. In the Eurozone, these figures are 36.3% and 35.3%, respectively. Moreover, ESG has more explanatory power than the other risk factors in both North America and the Eurozone over the two periods: 2010–2013 and 2014–2019.

<table><caption>Table 3.14: Results of cross-section regression with long-only risk factors (single-factor linear regression model, average $\mathfrak{R}^2$)</caption><thead><tr><th rowspan="2">Factor</th><th colspan="2">North America</th><th colspan="2">Eurozone</th></tr><tr><th>2010-2013</th><th>2014-2019</th><th>2010-2013</th><th>2014-2019</th></tr></thead><tbody><tr><td>Market</td><td>40.8%</td><td>28.6%</td><td>42.8%</td><td>36.3%</td></tr><tr><td>Size</td><td>39.3%</td><td>26.1%</td><td>37.1%</td><td>23.3%</td></tr><tr><td>Value</td><td>38.9%</td><td>26.7%</td><td>41.6%</td><td>33.6%</td></tr><tr><td>Momentum</td><td>39.6%</td><td>26.3%</td><td>40.8%</td><td>34.1%</td></tr><tr><td>Low-volatility</td><td>35.8%</td><td>25.1%</td><td>38.7%</td><td>33.4%</td></tr><tr><td>Quality</td><td>39.1%</td><td>26.6%</td><td>42.4%</td><td>34.6%</td></tr><tr><td>ESG</td><td>40.1%</td><td>27.4%</td><td>42.6%</td><td>35.3%</td></tr></tbody></table>

Source: Roncalli (2020b).

We now consider a multi-factor model:

$$R_{i,t} = \alpha_i + \sum_{j=1}^{m} \beta_{i,j} \mathcal{F}_{j,t} + \varepsilon_{i,t}$$

where *m* is the number of risk factors. In this approach, we compare the CAPM, the standard five-factor model based on size, value, momentum, low-volatility and quality risk factors, and the six-factor model, which consists of adding the ESG factor to the universe of the five alternative risk factors. In Table 3.15, we verify that the five-factor model increases the proportion of systematic risk relative to the CAPM. For example, the CAPM and the 5F model explain 28.6% and 38.4%, respectively, of the cross-section variance in North America in the second period. Adding the ESG factor has a small impact between 2014 and 2019: 39.7% versus 38.4% in North America and 45.8% versus 45.0% in the Eurozone. This means that the ESG factor does not significantly improve the five-factor model. However, if we apply statistical significance tests to the six-factor model, we find that ESG is statistically significant in the Eurozone, but not in North America. We can conclude that ESG may be a risk factor in the Eurozone but not in North America.
<table><caption>Table 3.15: Results of cross-section regression with long-only risk factors (multi-factor linear regression model, average $\mathbb{R}^2$)</caption><thead><tr><th rowspan="2">Model</th><th colspan="2">North America</th><th colspan="2">Eurozone</th></tr><tr><th>2010-2013</th><th>2014-2019</th><th>2010-2013</th><th>2014-2019</th></tr></thead><tbody><tr><td>CAPM</td><td>40.8%</td><td>28.6%</td><td>42.8%</td><td>36.3%</td></tr><tr><td>5F model</td><td>46.1%</td><td>38.4%</td><td>49.5%</td><td>45.0%</td></tr><tr><td>6F model (5F + ESG)</td><td>46.7%</td><td>39.7%</td><td>50.1%</td><td>45.8%</td></tr></tbody></table>

Source: Roncalli (2020b).

The previous results may be disturbing. Indeed, cross-section regressions show that ESG is a very good single factor, but the added value of ESG in a multi-factor framework is limited. The difference between the two approaches is the cross-correlation between the risk factors included in the cross-section multi-factor regression. To better understand these results, Roncalli (2020b) considers a factor picking (or factor selection) approach. This approach is similar to the multi-factor approach, but a lasso penalized regression is used instead of the traditional least squares regression:

$$ \{\hat{\alpha}_i, \hat{\beta}_{i,1}, \dots, \hat{\beta}_{i,m}\} = \arg \min \left\{ \frac{1}{2} \mathrm{var}(\varepsilon_{i,t}) + \lambda \|\beta_i\|_1 \right\} $$

The advantage is that we can control the factor intensity of the multi-factor portfolio. This gives us a factor selection procedure. Starting with a low factor intensity ($\lambda \approx \infty$), we can determine which risk factor is the most important. We then increase the factor intensity to establish an order among the risk factors. When the factor intensity reaches 100% ($\lambda = 0$), we get the same results as before with the linear regression. The results are shown in Figures 3.28 and 3.29 for the period 2014–2019. In North America, we see that quality is the first selected factor, followed by ESG, momentum, value and finally low-volatility. Therefore, ESG is the second factor in North America. Thus, ESG should be an important factor when constructing a multi-factor portfolio. However, we observe that as we increase the factor intensity, the ESG beta first increases and then decreases. When the factor intensity reaches 100%, ESG represents a low exposure. Therefore, part of the ESG exposure has been replaced by exposure to other risk factors. This means that ESG has a high contribution in a low-diversified portfolio, but is somewhat redundant in an already well-diversified portfolio. In the case of the Eurozone, the situation is different. ESG is the first factor selected and remains an important factor even if we increase the factor intensity. For example, it is more important than momentum and low-volatility.

These different results (single-factor, multi-factor and factor picking) show that ESG investing remains an alpha strategy in North America. It may have generated outperformance, but the ESG risk factor cannot explain the dispersion of stock returns better than the standard five-factor risk model. This means that adding ESG to a multi-factor portfolio that is already well diversified adds very little value. This is clearly the definition of an alpha strategy. On the contrary, we find that ESG is a significant factor in a Eurozone multi-factor portfolio. We can then improve the diversification of multi-factor portfolios by including an ESG risk factor. Thus, in the Eurozone, an ESG strategy seems to be more of a beta strategy than an alpha strategy.

**Remark 38** These last observations can be linked to the development of factor investing, for example low-volatility and quality risk factors (Roncalli, 2017). Low-volatility strategies have been around for many years, but they really took off in the asset management industry between 2003 and 2004, following the dot.com bubble. Initially, low-volatility strategies were viewed as alpha strategies. After the 2008 Global Financial Crisis, they were massively implemented and thus became beta strategies.
<figure><img src="image_92.png" /><figcaption>Figure 3.28: Factor picking (MSCI North America, 2014–2019, global score)</figcaption></figure>

Source: Roncalli (2020b).

<figure><img src="image_93.png" /><figcaption>Figure 3.29: Factor picking (MSCI EMU, 2014–2019, global score)</figcaption></figure>

Source: Roncalli (2020b).
The case of the quality anomaly is similar. This shows that there is no clear boundary between alpha and beta. When an alpha strategy is massively invested, it has enough impact on the structure of asset prices to become a risk factor. The alpha/beta status of ESG strategies is related to investment flows. Indeed, an alpha strategy becomes a common market risk factor when it represents a significant portion of investment portfolios and explains the cross-section dispersion of asset returns. This may explain why ESG is more of a risk factor in the Eurozone than in North America.

<table><caption>Table 3.16: Performance of ESG equity indexes (MSCI World, 2010–2023)</caption><thead><tr><th rowspan="2">Year</th><th colspan="3">Return (in %)</th><th colspan="2">Alpha (in bps)</th></tr><tr><th>CW</th><th>ESG</th><th>SRI</th><th>ESG</th><th>SRI</th></tr></thead><tbody><tr><td>2010</td><td>11.8</td><td>10.7</td><td>10.6</td><td>-109</td><td>-114</td></tr><tr><td>2011</td><td>-5.5</td><td>-5.4</td><td>-5.5</td><td>12</td><td>2</td></tr><tr><td>2012</td><td>15.8</td><td>14.5</td><td>13.2</td><td>-135</td><td>-258</td></tr><tr><td>2013</td><td>26.7</td><td>27.6</td><td>27.4</td><td>89</td><td>71</td></tr><tr><td>2014</td><td>4.9</td><td>4.9</td><td>3.9</td><td>-6</td><td>-102</td></tr><tr><td>2015</td><td>-0.9</td><td>-1.1</td><td>-1.6</td><td>-23</td><td>-71</td></tr><tr><td>2016</td><td>7.5</td><td>7.3</td><td>7.7</td><td>-26</td><td>18</td></tr><tr><td>2017</td><td>22.4</td><td>21.0</td><td>23.6</td><td>-142</td><td>124</td></tr><tr><td>2018</td><td>-8.7</td><td>-7.8</td><td>-6.7</td><td>94</td><td>199</td></tr><tr><td>2019</td><td>27.7</td><td>28.2</td><td>29.8</td><td>48</td><td>209</td></tr><tr><td>2020</td><td>15.9</td><td>15.3</td><td>19.9</td><td>-61</td><td>396</td></tr><tr><td>2021</td><td>21.8</td><td>24.7</td><td>27.0</td><td>288</td><td>523</td></tr><tr><td>2022</td><td>-18.1</td><td>-19.6</td><td>-22.5</td><td>-143</td><td>-436</td></tr><tr><td>2023</td><td>23.8</td><td>25.4</td><td>27.8</td><td>161</td><td>404</td></tr><tr><td>3Y</td><td>7.3</td><td>8.0</td><td>8.0</td><td>67</td><td>70</td></tr><tr><td>5Y</td><td>12.8</td><td>13.2</td><td>14.4</td><td>39</td><td>157</td></tr><tr><td>7Y</td><td>10.7</td><td>11.0</td><td>12.3</td><td>25</td><td>161</td></tr><tr><td>10Y</td><td>8.6</td><td>8.7</td><td>9.5</td><td>11</td><td>93</td></tr></tbody></table>

## Equity indexes

Another way to illustrate the time-varying nature of ESG investment performance is to analyze the annualized returns of equity indexes. In Table 3.16, we compare the MSCI World capitalization-weighted index (CW) with the MSCI World ESG Leaders and SRI indexes<fnref n="27" />. For each index, we report the annualized return in % and also calculate the alpha in bps relative to the CW parent index. For instance, the 2022 return was -18.1% for the MSCI World index (CW), -19.6% for the MSCI World ESG Leaders index and -22.5% for the MSCI World SRI index. The alpha of the ESG and SRI indexes is therefore negative, at -143 and -436 bps, respectively. Over the past fourteen years, the benchmark index has outperformed the ESG index eight times and the SRI index only five times. The 3Y, 5Y, 7Y and 10Y annualized returns are higher for the ESG and SRI indexes than for the CW index. These results clearly show that ESG investing can generate alpha in some periods. Moreover, the relative performance depends on the construction of the ESG index. In fact, we do not observe the same patterns between ESG Leaders and SRI indexes.

<footnote n="27">²⁷"MSCI ESG Leaders indexes target companies that have the highest ESG-rated performance in each sector of the parent index. MSCI SRI indexes are designed to track the performance of companies with high ESG ratings. They use a best-in-class selection approach to target the top 25% companies in each sector according to their MSCI ESG Ratings" (www.msci.com/our-solutions/indexes/esg-indexes).</footnote>
In Table 3.17, we confirm that the performance of ESG investments depends on several factors. For instance, the region has a significant impact. The MSCI EMU SRI Index has generated positive alpha every year between 2015 and 2021, with an average of 393 bps per year. For the MSCI EM ESG Leaders Index, the period of euphoria is between 2010 and 2017 with an average alpha of 423 bps per year. The choice of ESG scoring model is another factor. In fact, if we look at the S&P 500 ESG Index, it has a positive alpha in 2014, 2020 and 2022, while the MSCI USA ESG Leaders Index has a negative alpha in these years (19 vs. -49 bps in 2014, 138 vs. -251 bps in 2020 and 43 vs. -73 bps in 2022).

<figure><img src="image_94.png" /><figcaption>Figure 3.30: Alpha return of ESG equity indexes (in bps)</figcaption></figure>

Source: MSCI, Factset & Author's calculations.

In Figure 3.30, we have plotted the distribution of alpha returns of ESG equity indexes over time. This illustrates perfectly that “ESG investing has its good times and its bad times.” While ESG indexes massively underperform CW indexes in 2022, they also massively outperform in 2021. The underperformance in 2022 is mainly due to the Russian war in Ukraine and the energy shock. In 2023, seven of the ten ESG indexes have positive alpha. The average alpha of the ten ESG indexes is 1.81% in 2020, 3.81% in 2021, -1.84% in 2022 and 1.21% in 2023. From 2010 to 2013, the average alpha is 82 bps per year. We retrieve the two-period results obtained by Bennani et al. (2018), where the average annual alpha is only 12 bps from 2010 to 2015 and 129 bps from 2016 to 2023. Curiously, the alpha is not negative in the first period. Moreover, the average alpha is negative only for the years 2012 and 2022, suggesting that ESG investing has had more good times than bad over the past fourteen years.
<table><caption>Table 3.17: Performance of ESG equity indexes</caption><thead><tr><th rowspan="2">Year</th><th colspan="3">MSCI USA</th><th colspan="3">MSCI EMU</th><th colspan="3">MSCI EM</th><th colspan="3">S&P 500</th></tr><tr><th>CW</th><th>Return ESG</th><th>SRI</th><th>Alpha ESG</th><th>CW</th><th>Return ESG</th><th>SRI</th><th>Alpha ESG</th><th>CW</th><th>Return ESG</th><th>Alpha ESG</th><th>CW</th><th>Return ESG</th><th>Alpha ESG</th></tr></thead><tbody><tr><td>2010</td><td>14.8</td><td>11.0</td><td></td><td>-373</td><td>2.4</td><td>3.6</td><td>1.6</td><td>120</td><td>-83</td><td>18.9</td><td>25.5</td><td>665</td><td>14.4</td><td>13.9</td><td>-45</td></tr><tr><td>2011</td><td>1.4</td><td>0.6</td><td>0.1</td><td>-79</td><td>-14.9</td><td>-13.3</td><td>-14.9</td><td>162</td><td>-3</td><td>-18.4</td><td>-13.1</td><td>536</td><td>1.5</td><td>2.5</td><td>101</td></tr><tr><td>2012</td><td>15.3</td><td>12.6</td><td>10.2</td><td>-270</td><td>19.3</td><td>19.5</td><td>23.0</td><td>20</td><td>371</td><td>18.2</td><td>21.2</td><td>299</td><td>15.2</td><td>15.1</td><td>-15</td></tr><tr><td>2013</td><td>31.8</td><td>32.8</td><td>31.7</td><td>98</td><td>-6</td><td>23.4</td><td>20.4</td><td>23.1</td><td>-301</td><td>-27</td><td>-2.6</td><td>1.3</td><td>386</td><td>31.5</td><td>29.7</td><td>-182</td></tr><tr><td>2014</td><td>12.7</td><td>12.2</td><td>11.8</td><td>-49</td><td>-90</td><td>4.3</td><td>3.7</td><td>2.1</td><td>-57</td><td>-218</td><td>-2.2</td><td>4.8</td><td>703</td><td>13.0</td><td>13.2</td><td>19</td></tr><tr><td>2015</td><td>0.7</td><td>-2.0</td><td>-3.7</td><td>-264</td><td>-438</td><td>9.8</td><td>14.0</td><td>14.7</td><td>415</td><td>493</td><td>-14.9</td><td>-12.3</td><td>262</td><td>0.7</td><td>-0.2</td><td>-91</td></tr><tr><td>2016</td><td>10.9</td><td>11.0</td><td>12.6</td><td>10</td><td>174</td><td>4.4</td><td>7.8</td><td>9.4</td><td>345</td><td>498</td><td>11.2</td><td>13.4</td><td>221</td><td>11.2</td><td>11.8</td><td>53</td></tr><tr><td>2017</td><td>21.2</td><td>19.8</td><td>23.4</td><td>-137</td><td>221</td><td>12.5</td><td>12.4</td><td>14.0</td><td>-9</td><td>154</td><td>37.3</td><td>40.4</td><td>309</td><td>21.1</td><td>20.5</td><td>-62</td></tr><tr><td>2018</td><td>-5.0</td><td>-3.7</td><td>-1.9</td><td>137</td><td>314</td><td>-12.7</td><td>-10.2</td><td>-8.0</td><td>248</td><td>472</td><td>-14.6</td><td>-15.0</td><td>-42</td><td>-4.9</td><td>-4.5</td><td>40</td></tr><tr><td>2019</td><td>30.9</td><td>30.9</td><td>32.5</td><td>2</td><td>159</td><td>25.5</td><td>26.3</td><td>29.9</td><td>85</td><td>444</td><td>18.4</td><td>19.8</td><td>134</td><td>30.7</td><td>32.5</td><td>184</td></tr><tr><td>2020</td><td>20.7</td><td>18.2</td><td>25.8</td><td>-251</td><td>508</td><td>-1.0</td><td>0.3</td><td>0.0</td><td>127</td><td>100</td><td>18.3</td><td>20.1</td><td>178</td><td>17.8</td><td>19.1</td><td>138</td></tr><tr><td>2021</td><td>26.5</td><td>31.2</td><td>35.3</td><td>473</td><td>882</td><td>22.2</td><td>22.4</td><td>28.0</td><td>28</td><td>588</td><td>-2.5</td><td>-2.1</td><td>48</td><td>28.2</td><td>31.2</td><td>305</td></tr><tr><td>2022</td><td>-19.8</td><td>-20.6</td><td>-23.6</td><td>-73</td><td>-373</td><td>-12.5</td><td>-12.5</td><td>-16.0</td><td>-6</td><td>-349</td><td>-20.1</td><td>-22.8</td><td>-270</td><td>-18.5</td><td>-18.1</td><td>43</td></tr><tr><td>2023</td><td>26.5</td><td>28.5</td><td>32.8</td><td>205</td><td>630</td><td>18.8</td><td>16.7</td><td>19.7</td><td>-209</td><td>92</td><td>9.8</td><td>6.2</td><td>-367</td><td>25.7</td><td>27.4</td><td>170</td></tr><tr><td>3Y</td><td>8.7</td><td>10.2</td><td>11.2</td><td>159</td><td>251</td><td>8.3</td><td>7.7</td><td>8.8</td><td>-58</td><td>51</td><td>-5.1</td><td>-7.1</td><td>-199</td><td>9.5</td><td>11.1</td><td>156</td></tr><tr><td>5Y</td><td>15.2</td><td>15.7</td><td>18.0</td><td>53</td><td>284</td><td>9.5</td><td>9.6</td><td>10.8</td><td>7</td><td>130</td><td>3.7</td><td>2.9</td><td>-77</td><td>15.1</td><td>16.7</td><td>158</td></tr><tr><td>7Y</td><td>12.9</td><td>13.3</td><td>15.7</td><td>41</td><td>281</td><td>6.4</td><td>6.9</td><td>8.4</td><td>47</td><td>192</td><td>5.0</td><td>4.7</td><td>-30</td><td>12.8</td><td>13.9</td><td>109</td></tr><tr><td>10Y</td><td>11.4</td><td>11.3</td><td>12.9</td><td>-5</td><td>152</td><td>6.4</td><td>7.4</td><td>8.4</td><td>101</td><td>209</td><td>2.7</td><td>3.7</td><td>103</td><td>11.4</td><td>12.1</td><td>72</td></tr></tbody></table>

Source: MSCI, Factset & Author's calculations.