<table><caption>Table B.5: Computation of the alpha return due to the ESG constraint</caption><thead><tr><th>Asset</th><th>β<sub>i</sub>(w<sub>m</sub>)</th><th>μ̃<sub>i</sub>(w<sub>m</sub>)<br>(in %)</th><th>μ̃<sub>i</sub>(w<sub>m</sub>) - r<br>(in %)</th><th>α<sub>i</sub><br>(in bps)</th></tr></thead><tbody><tr><td>1</td><td>0.1660</td><td>1.7228</td><td>0.7228</td><td>-22.28</td></tr><tr><td>2</td><td>0.4321</td><td>2.8813</td><td>1.8813</td><td>-38.13</td></tr><tr><td>3</td><td>0.7518</td><td>4.2733</td><td>3.2733</td><td>-77.33</td></tr><tr><td>4</td><td>0.8494</td><td>4.6984</td><td>3.6984</td><td>80.16</td></tr><tr><td>5</td><td>1.2395</td><td>6.3967</td><td>5.3967</td><td>110.33</td></tr><tr><td>6</td><td>1.9955</td><td>9.6885</td><td>8.6885</td><td>131.15</td></tr></tbody></table>

We deduce that:

$$ 
\begin{aligned}
\alpha(w_{\text{ew}}) &= \mu(w_{\text{ew}}) - \tilde{\mu}(w_{\text{ew}}) \\
&= 5.25\% - 4.9435\% \\
&= 30.65 \text{ bps}
\end{aligned}
$$

We verify that:

$$ \alpha(w_{\text{ew}}) = \sum_{i=1}^{6} w_{\text{ew},i} \alpha_i = \frac{\sum_{i=1}^{6} \alpha_i}{6} = 30.65 \text{ bps} $$

The equally-weighted portfolio has a positive alpha. The main reason is that its ESG score is lower than the ESG score of the market portfolio:

$$ \mathbf{S}(w_{\text{ew}}) = -0.33 \ll \mathbf{S}(w_m) = 0.50 $$

3. The objective of the investor is twice. He would like to manage the tracking error risk of his portfolio with respect to the benchmark $b = (15\%, 20\%, 19\%, 14\%, 15\%, 17\%)$ and have a better ESG score than the benchmark. Nevertheless, this investor faces a long-only constraint because he cannot leverage his portfolio and he cannot also be short on the assets.

(a) We have:

$$ \mathbf{S}(b) = \sum_{i=1}^{6} b_i \mathbf{S}_i = -0.1620 $$

(b) We have:

$$ 
\left\{ 
\begin{array}{l}
\mathbf{S}(w | b) = (w - b)^{\top} \mathbf{S} = 0.0470 \\
\mu(w | b) = (w - b)^{\top} \mu = -0.5 \text{ bps} \\
\sigma(w | b) = \sqrt{(w - b)^{\top} \Sigma (w - b)} = 2.8423\% \\
\text{IR}(w | b) = \frac{\mu(w | b)}{\sigma(w | b)} = -0.0018
\end{array}
\right.
$$

The portfolio $w$ is not optimal since it improves the ESG score of the benchmark, but its information ratio is negative. Nevertheless, the expected excess return is close to zero (less than -1 bps).

(c) We have $\mathbf{S}(w | b) = 0.1305$, $\mu(w | b) = 29.5 \text{ bps}$, $\sigma(w | b) = 2.4949\%$ and $\text{IR}(w | b) = 0.1182$. The portfolio $w$ has then a positive expected excess return of 29.5 bps and a tracking error volatility of 2.4949%. Moreover, it has a better ESG score than the benchmark since its excess ESG score is equal to 0.1305.
(d) The optimization problem is:

$$ w^* = \arg \min \frac{1}{2} \sigma^2 (w | b) - \gamma \mathbf{S} (w | b) \\ \text{s.t.} \quad \begin{cases} \mathbf{1}_6^\top w = 1 \\ \mathbf{0}_6 \le w \le \mathbf{1}_6 \end{cases} $$

(e) The objective function is equal to:

$$ \begin{align*} (*) &= \frac{1}{2}\sigma^2(w | b) - \gamma \mathbf{S}(w | b) \\ &= \frac{1}{2}(w - b)^{\top}\Sigma(w - b) - \gamma(w - b)^{\top}\mathbf{S} \\ &= \frac{1}{2}w^{\top}\Sigma w - w^{\top}(\Sigma b + \gamma\mathbf{S}) + \underbrace{(\gamma b^{\top}\mathbf{S} + \frac{1}{2}b^{\top}\Sigma b)}_{\text{does not depend on } w} \end{align*} $$

We remind that the form of the QP problem is:

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

We deduce that $Q = \Sigma$, $R = \Sigma b + \gamma \mathbf{S}$, $A = \mathbf{1}_6^\top$, $B = 1$, $w^- = \mathbf{0}_6$ and $w^+ = \mathbf{1}_6$.

(f) We solve the QP problem for several values of $\gamma \in [0, 5\%]$ and obtain Figure B.10.

(g) Using the QP numerical algorithm, we compute the optimal value $\sigma(w | b)$ for $\gamma = 0$ and $\gamma = 5\%$. Then, we apply the bisection algorithm to find the optimal value $\gamma^*$ such that:

$$ \sigma(w | b) = \sigma^* $$

We obtain the results given in Table B.6.

<table><caption>Table B.6: Solution of the $\sigma$-problem</caption><thead><tr><th>Target σ<sup>*</sup></th><th>0</th><th>1%</th><th>2%</th><th>3%</th><th>4%</th></tr></thead><tbody><tr><td>γ<sup>*</sup> (in bps)</td><td>0.000</td><td>4.338</td><td>8.677</td><td>13.015</td><td>18.524</td></tr><tr><td rowspan="7">w<sup>*</sup> (in %)</td><td>15.000</td><td>15.175</td><td>15.350</td><td>15.525</td><td>14.921</td></tr><tr><td>20.000</td><td>21.446</td><td>22.892</td><td>24.338</td><td>25.385</td></tr><tr><td>19.000</td><td>23.084</td><td>27.167</td><td>31.251</td><td>35.589</td></tr><tr><td>14.000</td><td>9.588</td><td>5.176</td><td>0.763</td><td>0.000</td></tr><tr><td>15.000</td><td>12.656</td><td>10.311</td><td>7.967</td><td>3.555</td></tr><tr><td>17.000</td><td>18.052</td><td>19.104</td><td>20.156</td><td>20.550</td></tr><tr><td>S(w<sup>*</sup> | b)</td><td>0.000</td><td>0.230</td><td>0.461</td><td>0.691</td><td>0.915</td></tr></tbody></table>

(h) Using the QP numerical algorithm, we compute the optimal value $\mathbf{S}(w | b)$ for $\gamma = 0$ and $\gamma = 5\%$. Then, we apply the bisection algorithm to find the optimal value $\gamma^*$ such that:

$$ \mathbf{S}(w | b) = \mathbf{S}^* $$
<figure><img src="image_550.png" /><figcaption>Figure B.10: Efficient frontier of tracking a benchmark with an ESG score objective</figcaption></figure>

An alternative approach consists in solving the following optimization problem:

$$ 
\begin{array}{l} 
w^* = \arg \min \frac{1}{2} \sigma^2 (w | b) \\ 
\text{s.t.} \quad \left\{ \begin{array}{l} 
\mathbf{1}_6^\top w = 1 \\ 
\mathbf{S}(w | b) = \mathbf{S}^* \\ 
\mathbf{0}_6 \le w \le \mathbf{1}_6 
\end{array} \right. 
\end{array} 
$$

It is easy to show that the QP problem is given by $Q = \Sigma$, $R = \Sigma b$, $A = (\begin{pmatrix} \mathbf{1}_6^\top \\ \mathbf{S}^\top \end{pmatrix})$, $B = (\begin{pmatrix} 1 \\ \mathbf{S}^* + \mathbf{S}^\top b \end{pmatrix})$, $w^- = \mathbf{0}_6$ and $w^+ = \mathbf{1}_6$. Finally, we obtain the results given in Table B.7.

<table><caption>Table B.7: Solution of the S-problem</caption><thead><tr><th>Target <b>S</b><sup>*</sup></th><th>0</th><th>0.1</th><th>0.2</th><th>0.3</th><th>0.4</th></tr></thead><tbody><tr><td>γ<sup>*</sup> (in bps)</td><td>0.000</td><td>1.882</td><td>3.764</td><td>5.646</td><td>7.528</td></tr><tr><td rowspan="8">w<sup>*</sup> (in %)</td><td>15.000</td><td>15.076</td><td>15.152</td><td>15.228</td><td>15.304</td></tr><tr><td>20.000</td><td>20.627</td><td>21.255</td><td>21.882</td><td>22.509</td></tr><tr><td>19.000</td><td>20.772</td><td>22.544</td><td>24.315</td><td>26.087</td></tr><tr><td>14.000</td><td>12.086</td><td>10.171</td><td>8.257</td><td>6.343</td></tr><tr><td>15.000</td><td>13.983</td><td>12.966</td><td>11.949</td><td>10.932</td></tr><tr><td>17.000</td><td>17.456</td><td>17.913</td><td>18.369</td><td>18.825</td></tr><tr><td>σ(w<sup>*</sup> | b) (in %)</td><td>0.000</td><td>0.434</td><td>0.868</td><td>1.301</td><td>1.735</td></tr></tbody></table>
(i) For the best-in-class strategy, the optimization problem becomes:

$$ w^* = \arg\min_{\mathbf{s}} \frac{1}{2}\sigma^2(\mathbf{w} | \mathbf{b}) - \gamma\mathbf{S}(\mathbf{w} | \mathbf{b}) \\ \text{s.t.} \quad \begin{cases} \mathbf{1}_6^\top \mathbf{w} = 1 \\ w_4 = w_5 = w_6 = 0 \\ \mathbf{0}_6 \le \mathbf{w} \le \mathbf{1}_6 \end{cases} $$

The QP form is defined by $Q = \Sigma$, $R = \Sigma b + \gamma\mathbf{S}$, $A = \mathbf{1}_6^\top$, $B = 1$, $w^- = \mathbf{0}_6$ and $w^+ = (\begin{matrix} \mathbf{1}_3 \\ \mathbf{0}_3 \end{matrix})$. For the worst-in-class strategy, the optimization problem becomes:

$$ w^* = \arg\min_{\mathbf{s}} \frac{1}{2}\sigma^2(\mathbf{w} | \mathbf{b}) - \gamma\mathbf{S}(\mathbf{w} | \mathbf{b}) \\ \text{s.t.} \quad \begin{cases} \mathbf{1}_6^\top \mathbf{w} = 1 \\ w_6 = 0 \\ \mathbf{0}_6 \le \mathbf{w} \le \mathbf{1}_6 \end{cases} $$

The QP form is defined by $Q = \Sigma$, $R = \Sigma b + \gamma\mathbf{S}$, $A = \mathbf{1}_6^\top$, $B = 1$, $w^- = \mathbf{0}_6$ and $w^+ = (\begin{matrix} \mathbf{1}_5 \\ \mathbf{0} \end{matrix})$. The efficient frontiers are reported in Figure B.11. The exclusion strategy has less impact than the selection strategy. This last one implies a high tracking error risk.

<figure><img src="image_551.png" /><figcaption>Figure B.11: Comparison of the efficient frontiers (ESG integration, best-in-class selection and worst-in-class exclusion)</figcaption></figure>
(j) We solve the first problem of Question 3.i with $\gamma = 0$ and obtain $\sigma(w | b) \ge 11.17\%$. When $\sigma(w^* | b) = 11.17\%$, the corresponding optimal portfolio is:

$$w^* = \begin{pmatrix} 16.31\% \\ 34.17\% \\ 49.52\% \\ 0\% \\ 0\% \\ 0\% \end{pmatrix}$$

**Remark 155** The impact of ESG scores on optimized portfolios depends on their relationship with expected returns, volatilities, correlations, beta coefficients, etc. In the previous exercise, the results are explained because the best-in-class assets are those with the lowest expected returns and beta coefficients while the worst-in-class assets are those with the highest expected returns and beta coefficients. For instance, we obtain a high tracking error risk for the best-in-class selection strategy, because the best-in-class assets have low volatilities and correlations with respect to worst-in-class assets, implying that it is difficult to replicate these last assets with the other assets.
B.1.4 Calculating the prevalence of undernourishment (Exercise 5.5.1)

1. (a) We have:

$$ \ln D = \ln X - \ln R \sim \mathcal{N}(\mu_d, \sigma_d^2) $$

where:

$$ \mu_d = \mathbb{E}[\ln X - \ln R] = \mathbb{E}[\ln X] - \mathbb{E}[\ln R] = \mu_x - \mu_r $$

and:

$$ \sigma_d^2 = \operatorname{var}(\ln X - \ln R) $$

$$ = \operatorname{var}(\ln X) + \operatorname{var}(\ln R) - 2 \operatorname{cov}(\ln X, \ln R) $$

$$ = \sigma_x^2 + \sigma_r^2 - 2\rho\sigma_x\sigma_r $$

Thus, we deduce that $D = X/R$ is a log-normal random variable: $D \sim \mathcal{LN}(\mu_d, \sigma_d^2)$. It follows that:

$$ \mathrm{PoU}^* = \mathrm{Pr}\{X < R\} $$

$$ = \mathrm{Pr}\{D < 1\} $$

$$ = \mathrm{Pr}\{\ln D < 0\} $$

$$ = \Phi\left(-\frac{\mu_d}{\sigma_d}\right) $$

$$ = \Phi\left(-\frac{\mu_x - \mu_r}{\sqrt{\sigma_x^2 + \sigma_r^2 - 2\rho\sigma_x\sigma_r}}\right) $$

(b) The probability density functions of $X$ and $R$ are given in Figure B.12.

(c) Figure B.13 illustrates the relationship between the correlation and the prevalence of undernourishment. We observe a decreasing function between $\rho$ and $\mathrm{PoU}^*$, because the standard deviation of $D$ is a decreasing function of the correlation between $X$ and $R$ and $\mu_x > \mu_r$. Conversely, if $\mu_x < \mu_r$, the relationship between $\rho$ and $\mathrm{PoU}^*$ becomes increasing.

(d) We have:

$$ \mathrm{PoU} = \mathrm{Pr}\{X < r_L\} = \mathrm{Pr}\{\ln X < \ln r_L\} = \Phi\left(\frac{\ln r_L - \mu_x}{\sigma_x}\right) $$

The relationship between $r_L$ and $\mathrm{PoU}$ is shown in Figure B.14.

(e) We deduce that:

$$ \begin{align*} \mathrm{PoU} = \mathrm{PoU}^* &\Leftrightarrow \Phi\left(\frac{\ln r_L^* - \mu_x}{\sigma_x}\right) = \Phi\left(-\frac{\mu_x - \mu_r}{\sqrt{\sigma_x^2 + \sigma_r^2 - 2\rho_{x,r}\sigma_x\sigma_r}}\right) \\ &\Leftrightarrow \frac{\ln r_L^* - \mu_x}{\sigma_x} = -\frac{\mu_x - \mu_r}{\sqrt{\sigma_x^2 + \sigma_r^2 - 2\rho_{x,r}\sigma_x\sigma_r}} \end{align*} $$

which implies:

$$ r_L^* = \exp\left(\mu_x - \frac{(\mu_x - \mu_r)\sigma_x}{\sqrt{\sigma_x^2 + \sigma_r^2 - 2\rho_{x,r}\sigma_x\sigma_r}}\right) $$

The calibration of $r_L^*$ is shown in Figure B.15.
<figure><img src="image_552.png" /><figcaption>Figure B.13: Relationship between the correlation $\rho$ and the prevalence of undernourishment PoU*</figcaption></figure>

<figure><img src="image_553.png" /></figure>
<figure><img src="image_554.png" /><figcaption>Figure B.15: Calibration of $r_L^*$</figcaption></figure>

<figure><img src="image_555.png" /></figure>
2. (a) We have:

$$\mu(X) = e^{\mu_x + \frac{1}{2}\sigma_x^2}$$

and:

$$\sigma^2(X) = e^{2\mu_x + \sigma_x^2} (e^{\sigma_x^2} - 1) = \mu^2(X) (e^{\sigma_x^2} - 1)$$

(b) We deduce that:

$$CV(X) = \frac{\sigma(X)}{\mu(X)} = \sqrt{e^{\sigma_x^2} - 1}$$

(c) We have:

$$CV(X) = \sqrt{e^{\sigma_x^2} - 1} \Leftrightarrow e^{\sigma_x^2} = CV^2(X) + 1 \\ \Leftrightarrow \sigma_x = \sqrt{\ln(CV^2(X) + 1)}$$

and:

$$\mu(X) = e^{\mu_x + \frac{1}{2}\sigma_x^2} \Leftrightarrow \mu_x = \ln \mu(X) - \frac{1}{2}\sigma_x^2 \\ \Leftrightarrow \mu_x = \ln \mu(X) - \frac{1}{2}\ln(CV^2(X) + 1) \\ \Leftrightarrow \mu_x = \ln \mu(X) - \ln \sqrt{CV^2(X) + 1} \\ \Leftrightarrow \mu_x = \ln \frac{\mu(X)}{\sqrt{CV^2(X) + 1}}$$

(d) The food available for human consumption is equal to:

$$Q_1 = 349 000 + 1 025 - 40 000 - (-1 000) - 45 000 - 9 000 = 257 025 \text{ tonnes}$$

$$Q_2 = 50 000 + 9 010 - 1 000 - 6 000 - 500 - 500 = 51 010 \text{ tonnes}$$

We deduce that the ADEC value is:

$$\mu(X) = \frac{257 025 \times 10^3 \times 3 500 + 51 010 \times 10^3 \times 500}{365 \times 10^6} = 2 534.50 \text{ kcal/capita/day}$$

(e) Let $x_j^y$ be the average dietary energy consumption for the jth household expenditure decile. We have:

$$\sigma (X | Y) = \sqrt{\sum_{j=1}^{10} f_j (x_i^y - \bar{x}y)^2} = 533.4906$$

where $f_j = 10\%$ is the frequency of each decile group and $\bar{x}^y = \sum_{j=1}^{10} f_j x_i^y$ is the mean. We verify that $\bar{x}^y = 2 534.50 = \mu(X)$. It follows that:

$$CV(X | Y) = \frac{\sigma (X | Y)}{\mu (X | Y)} = \frac{533.4906}{2 534.50} = 0.2105$$

We deduce that:

$$CV(X) = \sqrt{CV^2(X | Y) + CV^2(X | R)} = \sqrt{0.2105^2 + 0.20^2} = 0.2904$$
(f) We have:

$$\sigma_x = \sqrt{\ln(0.2904^2 + 1)} = 0.2845$$

and:

$$\mu_x = \ln \frac{2534.50}{\sqrt{0.2904^2 + 1}} = 7.7973$$

The prevalence of undernourishment is equal to:

$$\text{PoU} = \Phi \left( \frac{1850 - 7.7973}{0.2845} \right) = 16.75\%$$

Finally, the number of undernourished people is around 167 500:

$$\text{NoU} = N \cdot \text{PoU} = 10^6 \times 16.75\% = 167\,451$$

3. (a) We have:

$$\text{weight}_{\text{age,sex}}^{*} = \text{BMI}_{\text{age,sex}}(p) \cdot \text{height}_{\text{age,sex}}^{2}$$

For example, we have:

$$\begin{align*} \text{weight}_{-3\text{yr,female}}^{*} &= \text{BMI}_{-3\text{yr,female}} (5\%) \cdot \text{height}_{-3\text{yr,female}}^{2} \\ &= 15.5 \times 0.80^{2} \\ &= 9.92 \text{ kg} \end{align*}$$

We obtain the following results:

<table><thead><tr><th>Age</th><th>-3</th><th>3-10</th><th>10-18</th><th>18-30</th><th>30-60</th><th>60+</th></tr></thead><tbody><tr><td>Female</td><td>9.92</td><td>22.32</td><td>40.84</td><td>44.80</td><td>44.80</td><td>44.80</td></tr><tr><td>Male</td><td>12.00</td><td>23.83</td><td>42.44</td><td>54.73</td><td>54.73</td><td>54.73</td></tr></tbody></table>

This means that a 40-year-old woman is considered undernourished if she weighs less than 44.8 kg.

(b) The basal metabolic rate (BMR) is given by the Schofield equation:

$$\text{BMR}_{\text{age,sex}} = \alpha_{\text{age,sex}} + \beta_{\text{age,sex}} \cdot \text{weight}_{\text{age,sex}}^{*}$$

For example, we have:

$$\begin{align*} \text{BMR}_{-3\text{yr,female}} &= \alpha_{-3\text{yr,female}} + \beta_{-3\text{yr,female}} \cdot \text{weight}_{-3\text{yr,female}}^{*} \\ &= -31.1 + 58.317 \times 9.92 \\ &= 547.40 \, \text{kcal/capita/day} \end{align*}$$

We obtain the following results:

<table><thead><tr><th>Age</th><th>-3</th><th>3-10</th><th>10-18</th><th>18-30</th><th>30-60</th><th>60+</th></tr></thead><tbody><tr><td>Female</td><td>547</td><td>939</td><td>1239</td><td>1150</td><td>1210</td><td>1065</td></tr><tr><td>Male</td><td>684</td><td>1045</td><td>1409</td><td>1516</td><td>1501</td><td>1229</td></tr></tbody></table>
(c) The minimum dietary energy requirement is equal to the physical activity level times the basal metabolic rate:

$$MDER_{age,sex} = PAL_{age,sex} \cdot BMR_{age,sex}$$

We obtain the following results:

<table><thead><tr><th>Age</th><th>-3</th><th>3-10</th><th>10-18</th><th>18-30</th><th>30-60</th><th>60+</th></tr></thead><tbody><tr><td>Female</td><td>848</td><td>1 456</td><td>1 921</td><td>1 783</td><td>1 875</td><td>1 651</td></tr><tr><td>Male</td><td>1 060</td><td>1 620</td><td>2 184</td><td>2 350</td><td>2 326</td><td>1 904</td></tr></tbody></table>

This means that a 40-year-old woman is considered undernourished if her dietary energy consumption is less than 1 875 Calories per day.

(d) The minimum dietary energy requirement of the population is the weighted average of the different MDER values:

$$MDER = \sum_{sex} \sum_{age} f_{age,sex} \cdot MDER_{age,sex}$$

where $f_{age,sex}$ is the frequency of the group in the population. Finally, we obtain:

$$MDER = 1 849.90 \text{ kcal/capita/day}$$

4. (a) FD indicates how many calories would be needed to ensure that undernourished would be eliminated if properly distributed.

(b) We have:

$$ 
\begin{aligned}
\text{FD} &= \int_{x<r_L} (\bar{r}-x) f_x(x) \, dx \\
&= \int_0^{r_L} \bar{r} f_x(x) \, dx - \int_0^{r_L} x f_x(x) \, dx \\
&= \bar{r} \int_0^{r_L} f_x(x) \, dx - \frac{\int_0^{r_L} x f_x(x) \, dx}{\int_0^{r_L} f_x(x) \, dx} \int_0^{r_L} f_x(x) \, dx \\
&= (\bar{r} - \mathbb{E}[X | X < r_L]) \cdot \text{Pr}\{X < r_L\} \\
&= \text{PoU} \cdot (\bar{r} - \mathbb{E}[X | X < r_L])
\end{aligned}
$$

The depth of the food deficit is the product of the prevalence of undernourishment and the difference between the average dietary energy requirement and the average dietary energy consumption, conditional on consumption being below the minimum requirement. Another expression is:

$$\text{FD} = \text{PoU} \cdot \mathbb{E}[(\bar{r} - X)_+ | X < r_L]$$

because $\bar{r} \ge r_L$. $\mathbb{E}[(\bar{r} - X)_+ | X < r_L]$ is the expected shortfall of food security.

(c) Following Roncalli (2020a, page 319), we introduce the notation $\Phi_c(x) = \Phi((x - \mu_x)/\sigma_x)$, and we calculate the conditional moment $\mu'_m(X) = \mathbb{E}[X^m | X < r_L]$ for $m \ge 1$ by using the change of variable $y = \ln x$:

$$ 
\begin{aligned}
\mu'_m(X) &= \frac{1}{\Phi_c(\ln r_L)} \int_0^{r_L} \frac{x^m}{x\sigma_x\sqrt{2\pi}} \exp\left(-\frac{1}{2}\left(\frac{\ln x - \mu_x}{\sigma_x}\right)^2\right) dx \\
&= \frac{1}{\Phi_c(\ln r_L)} \int_{-\infty}^{\ln r_L} \frac{1}{\sigma_x\sqrt{2\pi}} \exp\left(-\frac{1}{2}\left(\frac{y - \mu_x}{\sigma_x}\right)^2 + my\right) dy
\end{aligned}
$$
We have:

$$ 
\begin{aligned}
-\frac{1}{2} \left( \frac{y - \mu_x}{\sigma_x} \right)^2 + my &= -\frac{1}{2} \left( \frac{y^2 - 2y(\mu_x + m\sigma_x^2) + \mu_x^2}{\sigma_x^2} \right) \\
&= -\frac{1}{2} \left( \frac{y - (\mu_x + m\sigma_x^2)}{\sigma_x} \right)^2 + \left( m\mu_x + \frac{1}{2}m^2\sigma_x^2 \right)
\end{aligned}
$$

We deduce that:

$$ \mu'_{m}(X) = \frac{\exp(m\mu_{x} + m^{2}\sigma_{x}^{2}/2)}{\Phi_{c}(\ln r_{L})} \int_{-\infty}^{\ln r_{L}} \frac{1}{\sigma_{x}\sqrt{2\pi}} \exp\left(-\frac{1}{2}\left(\frac{y - (\mu_{x} + m\sigma_{x}^{2})}{\sigma_{x}}\right)^{2}\right) dy $$

Using the change of variable $z = \frac{y - (\mu_x + m\sigma_x^2)}{\sigma_x}$, it follows that:

$$ \mu'_{m}(X) = \frac{\exp(m\mu_{x} + m^{2}\sigma_{x}^{2}/2)}{\Phi_{c}(\ln r_{L})} \int_{-\infty}^{z_{L}} \frac{1}{\sqrt{2\pi}} \exp\left(-\frac{1}{2}z^{2}\right) dz $$

where:

$$ z_L = \frac{\ln r_L - (\mu_x + m\sigma_x^2)}{\sigma_x} $$

Finally, we obtain:

$$ \mu'_{m}(X) = \frac{\Phi_{c}(\ln r_{L} - m\sigma_{x}^{2})}{\Phi_{c}(\ln r_{L})} \exp\left(m\mu_{x} + \frac{1}{2}m^{2}\sigma_{x}^{2}\right) $$

and:

$$ \mathbb{E}[X | X < r_L] = \mu'_1(X) = \frac{\Phi\left(\frac{\ln r_L - \mu_x - \sigma_x^2}{\sigma_x}\right)}{\Phi\left(\frac{\ln r_L - \mu_x}{\sigma_x}\right)} \exp\left(\mu_x + \frac{1}{2}\sigma_x^2\right) $$

The analytical expression of the depth of the food deficit is:

$$ FD = \bar{r}\Phi\left(\frac{\ln r_L - \mu_x}{\sigma_x}\right) - e^{\mu_x + \frac{1}{2}\sigma_x^2}\Phi\left(\frac{\ln r_L - \mu_x - \sigma_x^2}{\sigma_x}\right) $$

because:

$$ \text{PoU} = \text{Pr}\{X < r_L\} = \Phi\left(\frac{\ln r_L - \mu_x}{\sigma_x}\right) $$

(d) The depth of the food deficit is equal to:

$$ 
\begin{aligned}
FD &= 2500 \times \Phi\left(\frac{\ln 1850 - 7.7973}{0.2845}\right) - \\
&\quad \exp\left(7.7973 + \frac{1}{2}0.2845^2\right) \times \Phi\left(\frac{\ln 1850 - 7.7973 - 0.2845^2}{0.2845}\right) \\
&= 150.2968 \text{ kcal/capita/year}
\end{aligned}
$$
B.1.5 Calculating the species-area relationship using the theory of island bio-
geography (Exercise 5.5.2)

1. (a) We have:

$$
\begin{align*}
f'(x) &= a e^{b(x/c)^d} b d \left(\frac{x}{c}\right)^{d-1} \frac{1}{c} \\
&= \frac{a b d}{c} \left(\frac{x}{c}\right)^{d-1} e^{b(x/c)^d}
\end{align*}
$$

and:

$$
\begin{align*}
f''(x) &= \frac{abd}{c}(d-1)\left(\frac{x}{c}\right)^{d-2}\frac{1}{c}e^{b(x/c)^d} + \frac{abd}{c}\left(\frac{x}{c}\right)^{d-1}e^{b(x/c)^d}bd\left(\frac{x}{c}\right)^{d-1}\frac{1}{c} \\
&= \frac{abd(d-1)}{c^2}\left(\frac{x}{c}\right)^{d-2}e^{b(x/c)^d} + \frac{ab^2d^2}{c^2}\left(\frac{x}{c}\right)^{2d-2}e^{b(x/c)^d} \\
&= \frac{abd}{c^2}\left(\frac{x}{c}\right)^{d-2}\left(d-1+bd\left(\frac{x}{c}\right)^d\right)e^{b(x/c)^d}
\end{align*}
$$

Since *a*, *c* and *d* are positive, we deduce that $\frac{x}{c} \ge 0$, $e^{b(x/c)^d} \ge 0$ and:

$f'(x) \ge 0 \Leftrightarrow b \ge 0$

$f(x)$ is an increasing function of $x$ if the parameter $b$ is positive, otherwise $f(x)$ is a decreasing function of $x$. We have:

$$
f''(x) \ge 0 \Leftrightarrow g(x; b) = \underbrace{b(d-1)}_{-/+} + \underbrace{b^2 d \left(\frac{x}{c}\right)^d}_{+} \ge 0
$$

because we have:

$0 \le b^2 d \left(\frac{x}{c}\right)^d \le b^2 d$

We consider two cases:

i. $b = -1$

If $d \le 1$, then $g(x; -1) = 1 - d + d \left(\frac{x}{c}\right)^d$, $f''(x) \ge 0$ and the function is convex. If $d > 1$, the function is neither convex nor concave.

ii. $b = 1$

If $d \ge 1$, then $g(x; 1) = (d-1) + d \left(\frac{x}{c}\right)^d$, $f''(x) \ge 0$ and the function is convex. If $d \le \frac{1}{2}$, then $g(0; 1) = d-1 < 0$, $g(c; 1) = 2d-1 \le 0$, $f''(x) \le 0$ and the function is concave. If $d \in (\frac{1}{2}, 1)$, the function is neither convex nor concave.

(b) We have<fnref n="2" />:

$$
\frac{\partial \lambda(t)}{\partial S(t)} = - \frac{\beta_1 \lambda_0^2}{(1 - e^{-1}) S_s} \left(\frac{S(t)}{S_s}\right)^{\beta_1 \lambda_0 - 1} e^{-\left(\frac{S(t)}{S_s}\right)^{\beta_1 \lambda_0}} \le 0
$$

with:

$$
\lambda(t_0) = \lambda_0 \frac{e^{-0} - e^{-1}}{1 - e^{-1}} = \lambda_0
$$

<footnote n="2">²We set $a = \frac{\lambda_0}{1 - e^{-1}}$, $b = -1$, $c = S_s$ and $d = \beta_1 \lambda_0$.</footnote>
and:

$$ \lambda(t_s) = \lambda_0 \frac{e^{-1} - e^{-1}}{1 - e^{-1}} = 0 $$

The second-order derivative is:

$$ \frac{\partial^2 \lambda(t)}{\partial S(t)^2} = - \frac{\beta_1 \lambda_0^2}{(1 - e^{-1}) S_s^2} \left(\frac{S(t)}{S_s}\right)^{\beta_1 \lambda_0 - 2} \left(\beta_1 \lambda_0 - 1 - \beta_1 \lambda_0 \left(\frac{S(t)}{S_s}\right)^{\beta_1 \lambda_0}\right) e^{-\left(\frac{S(t)}{S_s}\right)^{\beta_1 \lambda_0}} $$

$\lambda(t)$ is convex if and only if $\beta_1\lambda_0 \le 1$. The function $\lambda(t)$ is plotted in Figure B.16. We verify that it is decreasing and convex for the first three sets of parameters because $\beta_1\lambda_0 \le 1$. For the last set of parameters, it is decreasing but not convex.

<figure><img src="image_556.png" /><figcaption>Figure B.16: Immigrate rate function $\lambda(t)$</figcaption></figure>

(c) We have<fnref n="3" />:

$$ \frac{\partial \mu^{\text{long}}(t)}{\partial S(t)} = \frac{\beta_2 \mu_s^2}{(e^{-1} - 1) S_s} \left(\frac{S(t)}{S_s}\right)^{\beta_2 \mu_s - 1} e^{\left(\frac{S(t)}{S_s}\right)^{\beta_2 \mu_s}} \geq 0 $$

with:

$$ \mu^{\text{long}}(t_0) = \mu_s \frac{e^0 - 1}{e^1 - 1} = 0 $$

and:

$$ \mu^{\text{long}}(t_s) = \mu_s \frac{e^1 - 1}{e^1 - 1} = \mu_s $$

<footnote n="3">³We set $a = \frac{\mu_s}{e^{-1} - 1}$, $b = 1$, $c = S_s$ and $d = \beta_2\mu_s$.</footnote>
The second-order derivative is:

$$ \frac{\partial^2 \mu^{\text{long}}(t)}{\partial S(t)^2} = \frac{\beta_2 \mu_s^2}{(e^{-1} - 1) S_s^2} \left(\frac{S(t)}{S_s}\right)^{\beta_2 \mu_s - 2} \left(\beta_2 \mu_s - 1 + \beta_2 \mu_s \left(\frac{S(t)}{S_s}\right)^{\beta_2 \mu_s}\right) e^{\left(\frac{S(t)}{S_s}\right)^{\beta_2 \mu_s}} $$

$\mu^{\text{long}}(t)$ is convex if and only if $\beta_2\mu_s \ge 1$ and concave if and only if $\beta_2\mu_s \le \frac{1}{2}$. The function $\mu^{\text{long}}(t)$ is plotted in Figure B.17. We verify that it is increasing and concave for the first two sets of parameters because $\beta_2\mu_s \le \frac{1}{2}$. For the last two sets of parameters, it is increasing and convex.

<figure><img src="image_557.png" /><figcaption>Figure B.17: Long-term extinction rate function $\mu^{\text{long}}(t)$</figcaption></figure>

(d) $\mu^{\text{short}}(t)$ is the product of two positive and decreasing functions: $\lambda(t)$ and $\beta_3\lambda(t) e^{-\beta_4 S(t)}$. We deduce that it is decreasing. We also have:

$$ \mu^{\text{short}}(t_0) = \beta_3 \lambda_0 e^{-\beta_4 S_0} \le \beta_3 \lambda_0 $$

and:

$$ \mu^{\text{short}}(t_s) = \beta_3 \lambda_s e^{-\beta_4 S_s} = 0 $$

The function $\mu^{\text{short}}(t)$ is plotted in Figure B.18.

(e) We have:

$$ \mu(t_0) = \mu^{\text{short}}(t_0) + \mu^{\text{long}}(t_0) = \mu^{\text{short}}(t_0) + 0 = \mu^{\text{short}}(t_0) $$

and:

$$ \mu(t_s) = \mu^{\text{short}}(t_s) + \mu^{\text{long}}(t_s) = 0 + \mu_s = \mu_s $$
<figure><img src="image_558.png" /><figcaption>Figure B.19: Extinction rate function $\mu(t)$</figcaption></figure>

<figure><img src="image_559.png" /></figure>
Figure B.19 illustrates the function $\mu(t)$. Since the extinction rate is the sum of two functions, $\mu^{\text{short}}(t)$ which is decreasing, and $\mu^{\text{long}}(t)$ which is increasing, $\mu(t)$ is not monotonic and can exhibit several shapes. However, for reasonable values of the parameters, we observe that $\mu(t)$ first decreases and then increases. This behavior arises because $\mu^{\text{short}}(t_0) > \mu^{\text{long}}(t_0)$ and $\mu^{\text{short}}(t_s) < \mu^{\text{long}}(t_s)$. Consequently, when $S(t)$ is low and close to zero, the short-term extinction rate dominates the long-term extinction rate, leading to a general decrease in $\mu(t)$. In contrast, when $S(t)$ is high and approaches the saturation state $S_s$, the long-term extinction rate becomes dominant, implying that $\mu(t)$ generally increases for high values of species richness. In summary, the extinction rate $\mu(t)$ has a U-shaped relationship with $S(t)$.

(f) An equilibrium occurs if and only if $\delta(t) = 0$ or $\lambda(t) = \mu(t)$. We deduce that such an equilibrium $S^*$ is achieved if the following equation has at least one solution:

$$ \lambda_0 \frac{e^{-\left(\frac{S^*}{S_s}\right)^{\beta_1 \lambda_0}} - e^{-1}}{1 - e^{-1}} \left(1 - \beta_3 e^{-\beta_4 S^*}\right) = \mu_s \frac{e^{\left(\frac{S^*}{S_s}\right)^{\beta_2 \mu_s}} - 1}{e^1 - 1} \quad (B.1) $$

Figure B.20 shows the functions $\lambda(t)$ and $\mu(t)$. Solving Equation (B.1) gives $S^* = 96.472$. This equilibrium is stable, because $\frac{dS(t)}{dt} < 0$ when $S(t) > S^*$ and $\frac{dS(t)}{dt} > 0$ when $S(t) < S^*$. This implies that any perturbation from $S^*$ will return to the steady state $S^*$.

(g) In Table B.8, we give the direction of variation of the functions $\lambda(t)$, $\mu^{\text{short}}(t)$, $\mu^{\text{long}}(t)$ and $\mu(t)$. We can then derive the table of variation of the net diversification rate $\delta(t) = \lambda(t) - \mu(t)$. When $\beta_3 = 0$, $\mu^{\text{short}}(t) = 0$ and $\delta(t)$ is a decreasing function with $\delta(t_0) = \delta_0 = \lambda_0 > 0$ and $\delta(t_s) = -\mu_s < 0$. We conclude that there is only one equilibrium. When $0 < \beta_3 \le 1$, $\delta(t)$ increases up to a threshold $S^*$ and then decreases. We have $\delta_0 = \lambda_0(1 - \beta_3) > 0$ and $\delta(t_s) = -\mu_s < 0$. We obtain the table of variation of the net diversification rate given in Table B.9. When $\beta_3 > 1$, $\delta(t)$ has the same behavior as the previous case, but $\delta_0 = \lambda_0(1 - \beta_3) < 0$ and we obtain the table of variation of the net diversification rate given in Table B.10. We deduce that there are two equilibria $S_1^*$ and $S_2^*$, with $S_1^* \le S_2^*$. The equilibrium $S_1^*$ is unstable because $\frac{dS(t)}{dt} < 0$ when $S(t) < S_1^*$ and $\frac{dS(t)}{dt} > 0$ when $S(t) > S_1^*$. Only $S_2^*$ is a stable steady state. Furthermore, the net diversification rate is negative when $S(t) < S_1^*$, which means that $S(t + dt) < S(t)$ and $S(t)$ tends to the equilibrium $S_0^* = 0$.

(h) The simulations of the species richness process are presented in Figures B.22 and B.23. In the one-equilibrium case, the process $S(t)$ converges to the equilibrium $S^* = 96.47$, regardless of the initial state $S_0$. In the three-equilibrium case, the process $S(t)$ converges to either $S_0^* = 0$ or $S_2^* = 97.35$, depending on the initial state $S_0$. The equilibrium $S_1^* = 27.65$ is reached only if $S_0 = S_1^*$. Otherwise, we get the equilibrium $S_0^* = 0$ if $S_0 < S_1^*$, and the equilibrium $S_2^* = 97.35$ if $S_0 > S_1^*$.

2. (a) We have:

$$ \frac{\partial \mu(t)}{\partial \beta_2} = \frac{\mu_s^2}{e^1 - 1} e^{\left(\frac{S(t)}{S_s}\right)^{\beta_2 \mu_s}} \left(\frac{S(t)}{S_s}\right)^{\beta_2 \mu_s} \ln\left(\frac{S(t)}{S_s}\right) \le 0 $$

This means that the relationship $\mu(t)$ shifts downward as $\beta_2$ increases. Figure B.24 illustrates the equilibrium for the three values of $\beta_2$. From this, we deduce that if $\beta'_2 \ge \beta_2$, then $S^*(\beta'_2) \ge S^*(\beta_2)$.
<table><caption>Table B.8: Table of variation for the immigration and extinction rates</caption><thead><tr><th>S</th><th>0</th><th>S*</th><th>S<sub>s</sub></th></tr></thead><tbody><tr><td>λ(t)</td><td>λ<sub>0</sub></td><td></td><td>0</td></tr><tr><td>μ<sup>short</sup>(t)</td><td>β<sub>3</sub>λ<sub>0</sub></td><td></td><td>0</td></tr><tr><td>μ<sup>long</sup>(t)</td><td>0</td><td>μ<sub>s</sub></td><td></td></tr><tr><td>μ(t)<br>(β<sub>3</sub> = 0)</td><td>0</td><td>μ<sub>s</sub></td><td></td></tr><tr><td>μ(t)<br>(β<sub>3</sub> > 0)</td><td>β<sub>3</sub>λ<sub>0</sub></td><td>μ<sup>*</sup></td><td>μ<sub>s</sub></td></tr></tbody></table>

<table><caption>Table B.9: Table of variation for the net diversification rate δ(t)</caption><thead><tr><th>S</th><th>0</th><th>S<sup>*</sup></th><th>S<sup>*</sup></th><th>S<sub>s</sub></th></tr></thead><tbody><tr><td>λ(t) - μ(t)<br>(0 < β<sub>3</sub> ≤ 1)</td><td>δ<sub>0</sub></td><td>δ<sup>*</sup></td><td>0</td><td>-μ<sub>s</sub></td></tr></tbody></table>

<table><caption>Table B.10: Table of variation for the net diversification rate δ(t)</caption><thead><tr><th>S</th><th>0</th><th>S<sup>*</sup><sub>1</sub></th><th>S<sup>*</sup></th><th>S<sup>*</sup><sub>2</sub></th><th>S<sub>s</sub></th></tr></thead><tbody><tr><td>λ(t) - μ(t)<br>(β<sub>3</sub> > 1)</td><td>δ<sub>0</sub></td><td>0</td><td>δ<sup>*</sup></td><td>0</td><td>-μ<sub>s</sub></td></tr></tbody></table>
<figure><img src="image_560.png" /><figcaption>Figure B.21: The three-equilibrium case</figcaption></figure>

<figure><img src="image_561.png" /></figure>
<figure><img src="image_562.png" /><figcaption>Figure B.23: Simulation of the species richness process under a three-equilibrium scenario</figcaption></figure>

<figure><img src="image_563.png" /></figure>
<figure><img src="image_564.png" /><figcaption>Figure B.24: How the equilibrium shifts with the parameter $\beta_2$</figcaption></figure>

(b) The relationship between $\beta_2$ and $S^*$ is shown in Figure B.25.

(c) We have:

$$ \frac{\partial \mu(t)}{\partial \mu_s} = \frac{e^{\left(\frac{S(t)}{S_s}\right)^{\beta_2 \mu_s}} - 1}{e^1 - 1} + \frac{\beta_2 \mu_s}{e^1 - 1} e^{\left(\frac{S(t)}{S_s}\right)^{\beta_2 \mu_s}} \left(\frac{S(t)}{S_s}\right)^{\beta_2 \mu_s} \ln\left(\frac{S(t)}{S_s}\right) $$

Using the default parameter values and $S(t) = 100$, we find that $\partial_{\mu_s}\mu(t) = 0.045$ when $\mu_s = 1$ and $\partial_{\mu_s}\mu(t) = -0.047$ when $\mu_s = 5$. Since the derivative can be either positive or negative, we cannot determine the effect of $\mu_s$ on the equilibrium $S^*$.

(d) The relationship between $\mu_s$ and $S^*$ is shown in Figure B.26.

(e) We can assume that the relationship between $A$ and $\beta_2$ is increasing. In this case, we obtain the species-area relationship. We can also assume that the relationship between $A$ and $\mu_s$ is increasing. In this case, we retrieve the species-area relationship only if $\mu_s$ is greater than a threshold.

(f) We obtain the following estimated values for the parameters:

<table><thead><tr><td>Parameter</td><td>Power</td><td>Exponential</td><td>Kobayashi</td></tr></thead><tbody><tr><td>c</td><td>74.513</td><td>82.098</td><td>73.539</td></tr><tr><td>z</td><td>0.562</td><td>45.199</td><td>1.797</td></tr></tbody></table>

The fitted models are displayed in Figure B.27. We observe that the fitted curves closely align with the equilibrium curve predicted by the theory of island biogeography.
<figure><img src="image_565.png" /><figcaption>Figure B.26: Relationship between $\mu_s$ and $S^*$</figcaption></figure>

<figure><img src="image_566.png" /></figure>
<figure><img src="image_567.png" /><figcaption>Figure B.27: Species-area relationship (TIB, power, exponential and Kobayashi models)</figcaption></figure>

B.1.6 Species abundance models (Exercise 5.5.3)

1. (a) We have:

$$ \tilde{S}_a = \sum_{i=1}^{S} \tilde{S}_i $$

Assuming that the random variables $\tilde{S}_1, \dots, \tilde{S}_S$ are independent, $\tilde{S}_a$ follows a Poisson binomial distribution:

$$ \tilde{S}_a \sim \mathcal{P}\mathcal{B}(p_1, \dots, p_S) $$

The probability mass function is given by:

$$ \mathrm{Pr} \{ \tilde{S}_a = k \} = \sum_{\mathcal{E} \in \mathcal{F}_k} \prod_{i \in \mathcal{E}} p_i \prod_{j \in \mathcal{E}^c} (1 - p_j) $$

where $\mathcal{F}_k$ is the set of all subsets of $k$ integers that can be selected from $\{1, \dots, n\}$ and $\mathcal{E}^c$ denotes the complement of the subset $\mathcal{E}$. From this, we deduce the expected value:

$$ S_a = \mathbb{E}[\tilde{S}_a] = \mathbb{E}\left[\sum_{i=1}^{S} \tilde{S}_i\right] = \sum_{i=1}^{S} p_i $$

Given $p = (20\%, 30\%, 10\%, 65\%, 10\%)$, we obtain the following probability mass function for $\tilde{S}_a$:

<table><thead><tr><th>k</th><th>0</th><th>1</th><th>2</th><th>3</th><th>4</th><th>5</th></tr></thead><tbody><tr><td>Pr{S&#x0303;<sub>a</sub> = k}</td><td>15.88%</td><td>43.79%</td><td>30.85%</td><td>8.48%</td><td>0.97%</td><td>0.04%</td></tr></tbody></table>

The expected number of species is $S_a = 1.35$.
(b) Let $\pi$ be the probability that an individual occupies the subarea a. We have:

$$ \pi = \frac{a}{A} $$

For example, if $A = 10 \text{ km}^2$, the occupancy probability is 20% when $a = 2 \text{ km}^2$. Let $\tilde{N}_i$ be the random variable indicating the number of individuals of species i present in the subarea a. $\tilde{N}_i$ follows a binomial distribution with parameters $n_i$ (the total number of individuals of species i) and $\pi$. Thus, we have:

$$ \Pr\{\tilde{N}_i = k\} = C_{n_i}^k \pi^k (1-\pi)^{n_i-k} $$

The probability $p_i$ of observing the species i on the subarea is then:

$$ \begin{aligned} p_i &= \Pr\{\tilde{S}_i = 1\} &= \Pr\{\tilde{N}_i > 0\} \\ &= 1 - \Pr\{N_i = 0\} \\ &= 1 - (1 - \pi)^{n_i} \\ &= 1 - \left(1 - \frac{a}{A}\right)^{n_i} \end{aligned} $$

Consequently, $\tilde{S}_a$ follows then a Poisson binomial distribution:

$$ \tilde{S}_a \sim \mathcal{P}\mathcal{B}\left(p_i = 1 - \left(1 - \frac{a}{A}\right)^{n_i}\right) $$

It follows that:

$$ \begin{aligned} S_a &= \sum_{i=1}^{S} p_i \\ &= \sum_{i=1}^{S} \left(1 - \left(1 - \frac{a}{A}\right)^{n_i}\right) \\ &= S - \sum_{i=1}^{S} \left(1 - \frac{a}{A}\right)^{n_i} \end{aligned} $$

Given $(n_1, \dots, n_5) = (10, 30, 25, 66, 8)$, $A = 10 \text{ km}^2$, and $a = 2 \text{ km}^2$, $\pi = 20\%$ and we obtain the following results:

<table><thead><tr><th>k or i</th><th>0</th><th>1</th><th>2</th><th>3</th><th>4</th><th>5</th></tr></thead><tbody><tr><td>p<sub>i</sub></td><td></td><td>89.26%</td><td>59.04%</td><td>99.62%</td><td>73.79%</td><td>83.22%</td></tr><tr><td>Pr{S&#x0303;<sub>a</sub> = k}</td><td>0.00%</td><td>0.21%</td><td>3.45%</td><td>19.78%</td><td>44.32%</td><td>32.34%</td></tr></tbody></table>

The expected number of species is $S_a = 4.05$.

(c) i. If the distribution is homogenous across species ($n_i = \frac{n}{S}$), we have:

$$ p_i = 1 - \left(1 - \frac{a}{A}\right)^{\frac{n}{S}} $$

$\tilde{S}_a$ follows then a binomial distribution:

$$ \tilde{S}_a \sim \mathcal{B}\left(S, p = 1 - \left(1 - \frac{a}{A}\right)^{\frac{n}{S}}\right) $$
The expected number of species is then equal to:

$$S_a = Sp = S \left( 1 - \left( 1 - \frac{a}{A} \right)^{\frac{n}{S}} \right)$$

ii. If the distribution is the most uneven ($n_i = 1$ for $i < S$ and $n_S = n - S + 1$), we have:

$$ 
\begin{aligned}
S_a &= S - \sum_{i=1}^{S-1} \left(1 - \frac{a}{A}\right)^{1} - \left(1 - \frac{a}{A}\right)^{n-S+1} \\
&= S - (S-1) \left(1 - \frac{a}{A}\right) - \left(1 - \frac{a}{A}\right)^{n-S+1} \\
&= 1 + (S-1) \frac{a}{A} - \left(1 - \frac{a}{A}\right)^{n-S+1}
\end{aligned}
$$

iii. We have:

$$ 
\begin{aligned}
S_a &= S - \sum_{i=1}^{s} \left(1 - \frac{a}{A}\right)^{1} - \sum_{i=s+1}^{S} \left(1 - \frac{a}{A}\right)^{n_{s+1}} \\
&= S - s \left(1 - \frac{a}{A}\right) - (S - s) \left(1 - \frac{a}{A}\right)^{n_{s+1}} \\
&= s \frac{a}{A} + (S - s) \left(1 - \left(1 - \frac{a}{A}\right)^{n_{s+1}}\right) \\
&= s \frac{a}{A} + (S - s) \left(1 - \left(1 - \frac{a}{A}\right)^{\frac{n-s}{S-s}}\right)
\end{aligned}
$$

If $s = 0$, we retrieve the most even model:

$$S_a = S \left( 1 - \left( 1 - \frac{a}{A} \right)^{\frac{n}{S}} \right)$$

If $s = S - 1$, we retrieve the most uneven model:

$$S_a = (S-1) \frac{a}{A} + \left( 1 - \left( 1 - \frac{a}{A} \right)^{\frac{n-S+1}{S-S+1}} \right) = 1 + (S-1) \frac{a}{A} - \left( 1 - \frac{a}{A} \right)^{n-S+1}$$

iv. We have:

$$ 
\begin{aligned}
S_a &= \sum_{i=1}^{S} \left( 1 - \left( 1 - \frac{a}{A} \right) \left( 1 + \frac{n_i a}{\kappa_i A} \right)^{-\kappa_i} \right) \\
&= S - \left( 1 - \frac{a}{A} \right) \sum_{i=1}^{S} \left( 1 + \frac{n_i a}{\kappa_i A} \right)^{-\kappa_i}
\end{aligned}
$$

v. We can approximate the harmonic sum by:

$$\sum_{k=i}^{S} \frac{1}{k} \approx \int_{i}^{S} \frac{1}{x} dx = \ln \left( \frac{S}{i} \right)$$

We deduce that:

$$n_i = \frac{n}{S} \sum_{k=i}^{S} \frac{1}{k} \approx \frac{n}{S} \ln \left( \frac{S}{i} \right)$$
We have:

$$ \left(1 - \frac{a}{A}\right)^{n_i} = e^{(n_i \ln(1 - \frac{a}{A}))} = \exp\left(\frac{n}{S} \ln\left(\frac{S}{i}\right) \ln\left(1 - \frac{a}{A}\right)\right) = \left(\frac{i}{S}\right)^{-\frac{n}{S} \ln(1 - \frac{a}{A})} $$

and:

$$ \sum_{i=1}^{S} \left(1 - \frac{a}{A}\right)^{n_i} = \sum_{i=1}^{S} \left(\frac{i}{S}\right)^{-\frac{n}{S} \ln\left(1 - \frac{a}{A}\right)} \approx \int_{0}^{S} \left(\frac{x}{S}\right)^{-\frac{n}{S} \ln\left(1 - \frac{a}{A}\right)} dx $$

Using the change of variable $u = \frac{x}{S}$, we obtain:

$$ \begin{aligned} \int_{0}^{S} \left(\frac{x}{S}\right)^{-\frac{n}{S} \ln\left(1 - \frac{a}{A}\right)} dx &= S \int_{0}^{1} u^{-\frac{n}{S} \ln\left(1 - \frac{a}{A}\right)} du \\ &= S \left[ \frac{u^{-\frac{n}{S} \ln\left(1 - \frac{a}{A}\right) + 1}}{-\frac{n}{S} \ln\left(1 - \frac{a}{A}\right) + 1} \right]_{0}^{1} \\ &= \frac{S}{-\frac{n}{S} \ln\left(1 - \frac{a}{A}\right) + 1} \\ &= \frac{S^2}{S - n \ln\left(1 - \frac{a}{A}\right)} \end{aligned} $$

Since we have:

$$ S_a = S - \sum_{i=1}^{S} \left(1 - \frac{a}{A}\right)^{n_i} \approx S - \frac{S^2}{S - n \ln\left(1 - \frac{a}{A}\right)} = \frac{-Sn \ln\left(1 - \frac{a}{A}\right)}{S - n \ln\left(1 - \frac{a}{A}\right)} $$

we conclude that:

$$ S_a = \frac{S \ln \left( 1 - \frac{a}{A} \right)}{\ln \left( 1 - \frac{a}{A} \right) - \frac{S}{n}} $$

(d) We have:

$$ S_a = \sum_{i=1}^{S} p_i = \sum_{j} \sum_{i \in j} p_i = \sum_{j} s(j) p_j $$

where $p_j$ is the probability that a species with abundance $j$ is present in the subarea $a$. Since $p_j = 1 - \left(1 - \frac{a}{A}\right)^j$, we get:

$$ S_a = \sum_{j} s(j) \left(1 - \left(1 - \frac{a}{A}\right)^j\right) = S - \sum_{j} s(j) \left(1 - \frac{a}{A}\right)^j $$

where $S = \sum_j s(j)$ is the total number of species. To calculate $S_a$ for the log-series distribution, we use a preliminary result: $\sum_{j=1}^{\infty} x^j/j$ is the series expansion for $-\ln(1-x)$ when $|x| < 1$. We have:

$$ S = \sum_{j=1}^{\infty} s(j) = \alpha \sum_{j=1}^{\infty} \frac{x^j}{j} = -\alpha \ln(1-x) $$
and:

$$ \sum_{j=1}^{\infty} s(j) \left(1 - \frac{a}{A}\right)^j = \alpha \sum_{j=1}^{\infty} \frac{x^j}{j} \left(1 - \frac{a}{A}\right)^j = -\alpha \ln \left(1 - x \left(1 - \frac{a}{A}\right)\right) $$

We deduce that:

$$ \begin{aligned} S_a &= -\alpha \ln(1-x) + \alpha \ln\left(1-x\left(1-\frac{a}{A}\right)\right) \\ &= \alpha \ln\left(\frac{1-x\left(1-\frac{a}{A}\right)}{1-x}\right) \\ &= \alpha \ln\left(1+\frac{x}{1-x}\frac{a}{A}\right) \end{aligned} $$

2. (a) Species i is locally endemic to subarea $a \subseteq A$ if all the individuals of this species are found in a. Let $E_i \sim \mathcal{B}(\tilde{p}_i)$ be the Bernoulli random variable that takes the value 1 if species i is locally endemic to a. We have:

$$ \tilde{p}_i = \Pr\{E_i = 1\} = \prod_{k=1}^{n_i} \left(\frac{a}{A}\right) = \left(\frac{a}{A}\right)^{n_i} $$

(b) The expected number $E_a$ of locally endemic species is equal to:

$$ E_a = \mathbb{E}[E_1 + E_2 + \dots + E_S] = \sum_{i=1}^{S} \mathbb{E}[E_i] = \sum_{i=1}^{S} \tilde{p}_i = \sum_{i=1}^{S} \left(\frac{a}{A}\right)^{n_i} $$

(c) Since we have $S_a = S - \sum_{i=1}^{S} \left(1 - \frac{a}{A}\right)^{n_i}$ and $E_a = \sum_{i=1}^{S} \left(\frac{a}{A}\right)^{n_i}$, we deduce that:

$$ S_{A-a} = S - \sum_{i=1}^{S} \left(1 - \frac{A-a}{A}\right)^{n_i} = S - \sum_{i=1}^{S} \left(\frac{a}{A}\right)^{n_i} $$

and:

$$ E_{A-a} = \sum_{i=1}^{S} \left(\frac{A-a}{A}\right)^{n_i} = \sum_{i=1}^{S} \left(1 - \frac{a}{A}\right)^{n_i} $$

It follows that:

$$ S_{A-a} + E_a = S - \sum_{i=1}^{S} \left(\frac{a}{A}\right)^{n_i} + \sum_{i=1}^{S} \left(\frac{a}{A}\right)^{n_i} = S $$

(d) We have:

$$ T_a = S_a + E_a = S - \sum_{i=1}^{S} \left(1 - \frac{a}{A}\right)^{n_i} + \sum_{i=1}^{S} \left(\frac{a}{A}\right)^{n_i} $$

It follows that:

$$ \begin{aligned} \frac{\partial T_a}{\partial a} &= -\sum_{i=1}^{S} n_i \left(1 - \frac{a}{A}\right)^{n_i-1} \left(-\frac{1}{A}\right) + \sum_{i=1}^{S} n_i \left(\frac{a}{A}\right)^{n_i-1} \left(\frac{1}{A}\right) \\ &= \frac{1}{A} \left( \sum_{i=1}^{S} n_i \left(1 - \frac{a}{A}\right)^{n_i-1} + \sum_{i=1}^{S} n_i \left(\frac{a}{A}\right)^{n_i-1} \right) \\ &\ge 0 \end{aligned} $$
This implies that $T_a$ is an increasing function of $a$. Moreover, we have:

$$T_{\emptyset} = S_{\emptyset} + E_{\emptyset} = S - \sum_{i=1}^{S} \left(1 - \frac{0}{A}\right)^{n_i} + \sum_{i=1}^{S} \left(\frac{0}{A}\right)^{n_i} = S - S + 0 = 0$$

and:

$$T_A = S_A + E_A = S - \sum_{i=1}^{S} \left(1 - \frac{A}{A}\right)^{n_i} + \sum_{i=1}^{S} \left(\frac{A}{A}\right)^{n_i} = S - 0 + S = 2S$$

We deduce that:

$$0 \leq S_a + E_a \leq 2S$$

### B.1.7 Valuation of risks to life and health (Exercise 5.5.4)

1. Using Bayes theorem, we have:

$$S(u|t) = \Pr\{\tau > u | \tau > t\} = \frac{\Pr\{\tau > u, \tau > t\}}{\Pr\{\tau > t\}} = \frac{S(u)}{S(t)}$$

2. Since $dS(t) = \partial S(t) dt = -f(t) dt$, we can express the expected survival time as:

$$\mathbb{E}[\tau] = \int_{0}^{\infty} tf(t) dt = - \int_{0}^{\infty} t dS(t)$$

Using integration by parts, we get:

$$\mathbb{E}[\tau] = -[tS(t)]_{0}^{\infty} + \int_{0}^{\infty} S(t) dt = 0 + \int_{0}^{\infty} S(t) dt = \int_{0}^{\infty} S(t) dt$$

We deduce the conditional life expectancy for an individual alive at time $t$:

$$LE(t) = \mathbb{E}[\tau | \tau > t] = \frac{\mathbb{E}[1\{\tau > t\} \cdot \tau]}{\Pr\{\tau > t\}} = \frac{\int_{t}^{\infty} S(u) du}{S(t)} = \int_{t}^{\infty} S(u | t) du$$

3. The expected present value of a payoff, taking into account the future lifetime, is given by:

$$\mathbb{E}[X; t, \varrho] = \int_{t}^{\infty} e^{-\varrho(u-t)} S(u|t) X(u) du = \mathbb{E}[\delta(u) X(u)] \quad (B.2)$$

where $\delta(u) = B(t, u) S(u|t)$ is the discount factor under uncertain lifetime and $B(t, u) = e^{-\varrho(u-t)}$ is the standard discount factor at time $u$. Therefore, we have:

$$\delta(u) = e^{-\varrho(u-t)} \Pr\{\tau > u | \tau > t\}$$

Equation (B.2) is the classical formula for the present value when the discount rate accounts for uncertainty about the individual's lifetime.

4. DLE($t; \varrho$) is the mathematical expectation of the discounted survival time, given that the survival time exceeds $t$:

$$\text{DLE}(t; \varrho) = \mathbb{E}[e^{-\varrho t} | \tau > t]$$
Using integration by parts with $u = e^{-\varrho t}t$ and $v' = f(t)$, we can express the discounted life expectancy at $t = 0$ as:

$$ 
\begin{aligned}
\text{DLE}(0; \varrho) &= \int_{0}^{\infty} e^{-\varrho t} tf(t) \, dt \\
&= [-e^{-\varrho t} t \mathbf{S}(t)]_{0}^{\infty} + \int_{0}^{\infty} (-\varrho e^{-\varrho t} t + e^{-\varrho t}) \mathbf{S}(t) \, dt \\
&= \int_{0}^{\infty} e^{-\varrho t} \mathbf{S}(t) \, dt - \varrho \int_{0}^{\infty} e^{-\varrho t} t \mathbf{S}(t) \, dt
\end{aligned}
$$

because $u' = -\varrho e^{-\varrho t}t + e^{-\varrho t}$ and $v = -\mathbf{S}(t)$. We deduce that:

$$ \text{DLE}(0; \varrho) = \text{LE}(0; \varrho) - \varrho \underbrace{\int_{0}^{\infty} e^{-\varrho t} t \mathbf{S}(t) \, dt}_{\ge 0} $$

and:

$$ \text{LE}(0; \varrho) \ge \text{DLE}(0; \varrho) = \int_{0}^{\infty} e^{-\varrho t} tf(t) \, dt $$

The generalization to the case $t \ne 0$ is straightforward. Using the change of variable $x = u - t$, we obtain $\text{DLE}(t; \varrho) = \text{LE}(t; \varrho) - \varrho \int_{0}^{\infty} e^{-\varrho x} x \mathbf{S}(t+x) \, dx$. Equality is achieved if and only if $\varrho = 0$.

5. Since $\mathbf{S}(t) = e^{-\lambda t}$, we have:

$$ 
\begin{aligned}
\text{LE}(t; \varrho) &= \frac{1}{e^{-\lambda t}} \int_{t}^{\infty} e^{-\varrho(u-t)} e^{-\lambda u} \, du \\
&= \int_{t}^{\infty} e^{-(\varrho+\lambda)(u-t)} \, du \\
&= \left[ \frac{e^{-(\varrho+\lambda)(u-t)}}{(\varrho+\lambda)} \right]_{t}^{\infty} \\
&= \frac{1}{\varrho+\lambda}
\end{aligned}
$$

and:

$$ 
\begin{aligned}
\text{DLE}(t; \varrho) &= \int_{t}^{\infty} e^{-\varrho(u-t)} (u-t) f(u|t) \, du \\
&= \int_{t}^{\infty} e^{-\varrho(u-t)} (u-t) \lambda e^{-\lambda(u-t)} \, du \\
&= \int_{0}^{\infty} \lambda e^{-(\varrho+\lambda)s} \, ds \\
&= \left[ -\lambda s \frac{e^{-(\varrho+\lambda)s}}{(\varrho+\lambda)} \right]_{0}^{\infty} + \lambda \int_{0}^{\infty} \frac{e^{-(\varrho+\lambda)s}}{(\varrho+\lambda)} \, ds \\
&= \lambda \left[ -\frac{e^{-(\varrho+\lambda)s}}{(\varrho+\lambda)^2} \right]_{0}^{\infty} \\
&= \frac{\lambda}{(\varrho+\lambda)^2}
\end{aligned}
$$
We verify that:

$$ \mathrm{LE}(t; \varrho) = \left(1 + \frac{\varrho}{\lambda}\right) \mathrm{DLE}(t; \varrho) \geq \mathrm{DLE}(t; \varrho) $$

Moreover, we note that $LE(t; \varrho)$ does not depend on time $t$ because exponential survival times satisfy the property of lack of memory.

6. (a) In the case of exponential survival time, we have:

$$ \mathbb{E}[\tau] = \frac{1}{\lambda} $$

It follows that:

$$ \lambda = \frac{1}{\mathbb{E}[\tau]} = \frac{1}{70} = 0.01429 $$

For the Weibull distribution, we have:

$$ \mathbb{E}[\tau] = a\Gamma\left(1 + \frac{1}{b}\right) $$

We deduce that:

$$ a = \frac{\mathbb{E}[\tau]}{\Gamma\left(1 + \frac{1}{b}\right)} = \frac{70}{\Gamma\left(1 + \frac{1}{5}\right)} = 76.23871 $$

The survival functions $S(t)$ and $S(u | t = 50)$ are shown in Figure B.28. It is evident that the exponential distribution is not suitable for modeling human lifetimes, as the probability of dying before reaching 50 years of age is approximately 50%. In contrast, the Weibull distribution provides a more realistic model for human lifetime.

(b) We have:

$$ \mathrm{LE}(0; 3\%) = \frac{1}{\varrho + \lambda} = \frac{1}{3\% + 1.43\%} = 22.58 \text{ years} $$

and:

$$ \mathrm{DLE}(0; 3\%) = \frac{\lambda}{(\varrho + \lambda)^2} = \frac{1.43\%}{(3\% + 1.43\%)^2} = 7.28 \text{ years} $$

The function $DLE(t; \varrho)$ discounts lifetime too rapidly and is not a suitable approach for calculating present values while accounting for future lifetime.

(c) Figure B.29 shows the results. We observe that $LE(t; \varrho)$ is a decreasing function with respect to both time $t$ and the discount rate $\varrho$. Using a standard discount rate of 3%, a life expectancy of 70 years corresponds to a discounted life expectancy of approximately 30 years at birth and 20 years at age 40.

7. (a) We have:

$$ \begin{align*} \mathrm{QALE}(t; \varrho) &= \frac{1}{\mathbf{S}(t)} \int_t^\infty e^{-\varrho(u-t)} \mathbf{S}(u) Q(t) \, du \\ &= \left( \frac{1}{\mathbf{S}(t)} \int_t^\infty e^{-\varrho(u-t)} \mathbf{S}(u) \, du \right) \cdot Q(t) \\ &= \mathrm{LE}(t; \varrho) \cdot Q(t) \end{align*} $$

The quality-adjusted life expectancy is the product of the discounted life expectancy and the average quality of life weight.
<figure><img src="image_568.png" /><figcaption>Figure B.29: Discounted lifetime expectancy LE (t; ϱ)</figcaption></figure>

<figure><img src="image_569.png" /></figure>
(b) Since $Q(t) \le 1$, we deduce that:

$$0 \le e^{-\varrho(u-t)} \frac{\mathbf{S}(u)}{\mathbf{S}(t)} Q(u) \le e^{-\varrho(u-t)} \frac{\mathbf{S}(u)}{\mathbf{S}(t)}$$

and:

$$\int_t^\infty e^{-\varrho(u-t)} \frac{\mathbf{S}(u)}{\mathbf{S}(t)} Q(u) \, du \le \int_t^\infty e^{-\varrho(u-t)} \frac{\mathbf{S}(u)}{\mathbf{S}(t)} du$$

We conclude that $\text{QALE}(t; \varrho) \le \text{LE}(t; \varrho)$.

(c) $Q_1(t)$ is a piecewise linear function representing three phases of life. Before $t_1$, the quality of life is equal to 1. Between $t_1$ and $t_2$, the quality of life decreases linearly, reaching 1 - $\kappa_1$ at $t = t_2$. The rate of decrease per year in this phase is $\kappa_1/(t_2 - t_1)$. In the third phase, between $t_2$ and $t_3$, the quality of life continues to decline linearly at a rate of $\kappa_2/(t_3 - t_2)$. Figure B.30 shows the functions $\text{LE}(t; \varrho)$, $\text{QALE}_1(t; \varrho)$ calculated with $Q_1(t)$, and $\text{QALE}_2(t; \varrho)$ calculated with $Q_2(t)$. The difference between $\text{LE}(t; \varrho)$ and $\text{QALE}_1(t; \varrho)$ is small, especially when $t \le t_1$. It is more pronounced for large values of $t$ because quality of life is strongly affected when people are old. In contrast, $\text{QALE}_2(t; \varrho)$ is much lower than $\text{LE}(t; \varrho)$, because the quality of life decreases exponentially 2% per year.

<figure><img src="image_570.png" /><figcaption>Figure B.30: Discounted lifetime expectancy $\text{LE}(t; \varrho)$</figcaption></figure>

8. (a) We have:

$$\text{VSL}(t) = \frac{v}{\Delta L(t)} = \frac{v}{\mathcal{R}(t) \cdot \Delta t}$$

We deduce that:

$$\text{VSLY}(t) = \frac{\text{VSL}(t) \cdot \Delta L}{\Delta \text{LE}(t)} = \frac{\text{VSL}(t)}{\text{LE}(t; \varrho)}$$
and:

$$VQALY(t) = \frac{VSL(t) \cdot \Delta L}{\Delta QALE(t)} = \frac{VSL(t)}{QALE(t; \varrho)}$$

(b) Since $\Delta L(t) = \mathcal{R}(t) = \mathcal{I}(t)$ because $\Delta t = 1$ year, the associated payoff is:

$$X(t) = VSL(t) \cdot \mathcal{R}(t) = VSL(t) \cdot \Delta L(t)$$

It follows that:

$$X(t) = VSLY(t) \cdot \Delta LE(t) = VQALY(t) \cdot \Delta QALE(t)$$

(c) If $VSL(t)$ and $\mathcal{I}(t)$ are constant, we get:

$$ \begin{aligned} \mathcal{G}(t; \varrho) &= VSL \cdot \left( \frac{1}{\mathbf{S}(t)} \int_t^\infty e^{-\varrho(u-t)} \mathbf{S}(u) \, du \right) \cdot \mathcal{I}(t) \\ &= VSL \cdot \mathcal{L}E(t; \varrho) \cdot \Delta L \end{aligned} \qquad (B.3) $$

The economic gain is the value of a statistical life multiplied by the discounted life expectancy and the expected number of lives saved.

(d) In the case of air pollution, $\mathcal{I}(t)$ is negative and the economic cost becomes:

$$C(t; \varrho) = -\mathcal{G}(t; \varrho) = VSL \cdot \mathcal{L}E(t; \varrho) \cdot \Delta L$$

where $\Delta L$ is the expected number of deaths due to air pollution. If we compare with the formula:

$$C = VSLY \cdot YLL$$

we can deduce that the years of life lost (YLL) is equal to:

$$YLL = \Delta L \cdot \mathcal{L}E(t; \varrho)$$

In this model, years of life lost is the product of the expected number of deaths and the discounted life expectancy. The traditional formula is:

$$YLL = \Delta L \cdot \Delta \tau$$

where $\Delta \tau$ is the difference between life expectancy without air pollution and life expectancy with air pollution. Therefore, we assume that:

$$\Delta \tau = \mathcal{L}E(t; \varrho) = \frac{1}{\mathbf{S}(t)} \int_t^\infty e^{-\varrho(u-t)} \mathbf{S}(u) \, du$$

This means that we take the discount rate into account when calculating years of life lost.

(e) Results are reported in Table B.11. By construction, the life expectancy of an individual at age 0 is 70 years. Using a discount rate of 3%, the discounted life expectancy is 28.73 years. At age 40, the remaining life expectancy is 31.49 years, so the average lifetime of an individual alive at age 40 is 71.49 years. This corresponds to a discounted life expectancy of 19.11 years. At age 80, the remaining life expectancy is 8.74 years, so the average lifetime of an individual alive at age 80 is 88.74 years. The impact of the quality-of-life weight is shown in the second row. The estimated value of a statistical life is calculated as:

$$VSL(t) = \frac{\$1\,000}{1/10\,000} = \$10\,mn$$
<table><caption>Table B.11: Calculation of the economic gain $G(t; \varrho)$</caption><thead><tr><th rowspan="2">Discount rate<br>Age</th><th colspan="3">&varrho; = 0%</th><th colspan="3">&varrho; = 3%</th><th rowspan="2">Unit</th></tr><tr><th>Birth</th><th>40 years</th><th>80 years</th><th>0</th><th>40 years</th><th>80 years</th></tr></thead><tbody><tr><td>LE (t; &varrho;)</td><td>70.00</td><td>31.49</td><td>8.74</td><td>28.73</td><td>19.11</td><td>7.19</td><td rowspan="2">years</td></tr><tr><td>QALE (t; &varrho;)</td><td>68.13</td><td>29.54</td><td>6.50</td><td>28.52</td><td>18.39</td><td>5.38</td></tr><tr><td>VSL (t)</td><td colspan="6">10 000</td><td></td></tr><tr><td>VSLY (t)</td><td>143</td><td>318</td><td>1 144</td><td>348</td><td>523</td><td>1 390</td><td>$1 000</td></tr><tr><td>VQALY (t)</td><td>147</td><td>339</td><td>1 538</td><td>351</td><td>544</td><td>1 858</td><td></td></tr><tr><td>VSL (t)</td><td>10 500</td><td>4 723</td><td>1 312</td><td>4 310</td><td>2 867</td><td>1 079</td><td></td></tr><tr><td>VSLY (t)</td><td colspan="6">150</td><td>$ 1000</td></tr><tr><td>VQALY (t)</td><td>154</td><td>160</td><td>202</td><td>151</td><td>156</td><td>200</td><td></td></tr><tr><td>G (t; &varrho;)</td><td>36 750</td><td>7 437</td><td>573</td><td>6 191</td><td>2 740</td><td>388</td><td>$</td></tr></tbody></table>

We can then compute the value of a statistical life year. Assuming an age of 40 years and a discount rate of 3%, we obtain:

$$VSLY (40) = \frac{VSL (40)}{LE (40; 3\%)} = \frac{\$10 mn}{19.11} = \$523 206$$

and:

$$VQALY (40) = \frac{VSL (40)}{QALE (40; 3\%)} = \frac{\$10 mn}{18.39} = \$543 817$$

The assumption that VSL(t) is constant is not realistic. For instance, the value per statistical life year for a newborn is $142 857, while for an 80-year-old individual it is $1 390 448. On the other hand, the assumption that VSLY(t) is constant is more realistic, even if it does not fully match reality. In fact, we can expect VSLY(40) ≥ VSLY(0) and VSLY(40) ≥ VSLY(80). Finally, we calculate the economic gain using Equation (B.3). For example, assuming an age of 40 years and a discount rate of 3%, the economic gain is:

$$G (40; 3\%) = \$2 866 938 \times 19.11 \times \frac{5}{100 000} = \$2 740$$

We observe that the economic gain is greater for newborns than for older individuals.
# B.2 Exercises related to climate risk

## B.2.1 Compensating and equivalent variation (Exercise 10.6.1)

1. The optimization problem of the consumer is:

$$ \begin{array}{l} \{x_1^*, x_2^*\} = \arg\max \mathcal{U}(x_1, x_2) \\ \text{s.t.} \quad p_1x_1 + p_2x_2 = y \end{array} $$

The Lagrange function is:

$$ \mathcal{L}(x_1, x_2; \lambda) = x_1^\alpha x_2^\beta - \lambda (p_1x_1 + p_2x_2 - y) $$

The first-order conditions are:

$$ \begin{cases} \partial_{x_1} \mathcal{L}(x_1, x_2; \lambda) = \alpha x_1^{\alpha-1} x_2^{\beta} - \lambda p_1 = 0 \\ \partial_{x_2} \mathcal{L}(x_1, x_2; \lambda) = \beta x_1^{\alpha} x_2^{\beta-1} - \lambda p_2 = 0 \\ \partial_{\lambda} \mathcal{L}(x_1, x_2; \lambda) = y - p_1 x_1 - p_2 x_2 = 0 \end{cases} $$

It follows that:

$$ \frac{\alpha}{\beta} \frac{x_2}{x_1} = \frac{p_1}{p_2} \Leftrightarrow x_2 = \frac{\beta p_1}{\alpha p_2} x_1 $$

and:

$$ p_1x_1 + p_2 \left( \frac{\beta p_1}{\alpha p_2} x_1 \right) = y \Leftrightarrow p_1x_1 \left( 1 + \frac{\beta}{\alpha} \right) = y $$

We deduce that:

$$ \left\{ \begin{array}{l} x_1^* = \frac{\alpha}{\alpha + \beta} \left( \frac{y}{p_1} \right) \\ x_2^* = \frac{\beta}{\alpha + \beta} \left( \frac{y}{p_2} \right) \end{array} \right. $$

The optimal utility is:

$$ u^* = \mathcal{U}(x_1^*, x_2^*) = \frac{\alpha^\alpha \beta^\beta}{(\alpha + \beta)^{\alpha+\beta}} \left( \frac{y^{\alpha+\beta}}{p_1^\alpha p_2^\beta} \right) $$

2. The optimization problem becomes:

$$ \begin{array}{l} \{x_1^*, x_2^*\} = \arg\min p_1x_1 + p_2x_2 \\ \text{s.t.} \quad \mathcal{U}(x_1, x_2) = u \end{array} $$

The Lagrange function is:

$$ \mathcal{L}(x_1, x_2; \lambda) = p_1x_1 + p_2x_2 - \lambda (x_1^\alpha x_2^\beta - u) $$

The first-order conditions are:

$$ \begin{cases} \partial_{x_1} \mathcal{L}(x_1, x_2; \lambda) = p_1 - \lambda \alpha x_1^{\alpha-1} x_2^{\beta} = 0 \\ \partial_{x_2} \mathcal{L}(x_1, x_2; \lambda) = p_2 - \lambda \beta x_1^{\alpha} x_2^{\beta-1} = 0 \\ \partial_{\lambda} \mathcal{L}(x_1, x_2; \lambda) = u - x_1^{\alpha} x_2^{\beta} = 0 \end{cases} $$

It follows that:

$$ \frac{p_1}{p_2} = \frac{\alpha}{\beta} \frac{x_2}{x_1} \Leftrightarrow x_2 = \frac{\beta p_1}{\alpha p_2} x_1 $$
and:

$$x_1^\alpha \left( \frac{\beta p_1}{\alpha p_2} x_1 \right)^\beta = u \Leftrightarrow x_1^{\alpha+\beta} \left( \frac{\beta p_1}{\alpha p_2} \right)^\beta = u$$

We deduce that:

$$\left\{ \begin{array}{l} x_1^* = u \frac{1}{\alpha + \beta} \left( \frac{\alpha p_2}{\beta p_1} \right) \frac{\beta}{\alpha + \beta} \\ x_2^* = u \frac{1}{\alpha + \beta} \left( \frac{\beta p_1}{\alpha p_2} \right) \frac{\alpha}{\alpha + \beta} \end{array} \right.$$

The minimum income to achieve utility $u$ is:

$$\begin{align*} y^* &= p_1 x_1^* + p_2 x_2^* \\ &= u^{\frac{1}{\alpha+\beta}} \left( p_1 \left( \frac{\alpha p_2}{\beta p_1} \right)^{\frac{\beta}{\alpha+\beta}} + p_2 \left( \frac{\beta p_1}{\alpha p_2} \right)^{\frac{\alpha}{\alpha+\beta}} \right) \\ &= u^{\frac{1}{\alpha+\beta}} \left( \frac{\frac{\alpha}{\alpha+\beta} p_1^{\frac{\alpha}{\alpha+\beta}} p_2^{\frac{\beta}{\alpha+\beta}}}{\beta^{\frac{\beta}{\alpha+\beta}}} + \frac{\frac{\beta}{\alpha+\beta} p_1^{\frac{\alpha}{\alpha+\beta}} p_2^{\frac{\beta}{\alpha+\beta}}}{\alpha^{\frac{\alpha}{\alpha+\beta}}} \right) \\ &= u^{\frac{1}{\alpha+\beta}} \left( \frac{\frac{\alpha}{\alpha+\beta} \frac{\alpha}{\alpha+\beta} + \frac{\beta}{\alpha+\beta} \frac{\beta}{\alpha+\beta}}{\frac{\alpha}{\alpha+\beta} \frac{\beta}{\alpha+\beta}} \right) p_1^{\frac{\alpha}{\alpha+\beta}} p_2^{\frac{\beta}{\alpha+\beta}} \\ &= u^{\frac{1}{\alpha+\beta}} \left( \frac{\frac{\alpha}{\alpha+\beta} \frac{\alpha}{\alpha+\beta} + \frac{\beta}{\alpha+\beta} \frac{\beta}{\alpha+\beta}}{\frac{\alpha}{\alpha+\beta} \frac{\beta}{\alpha+\beta}} \right) p_1^{\frac{\alpha}{\alpha+\beta}} p_2^{\frac{\beta}{\alpha+\beta}} \\ &= u^{\frac{1}{\alpha+\beta}} \left( \frac{\frac{\alpha}{\alpha+\beta}}{\frac{\alpha}{\alpha+\beta} \frac{\beta}{\alpha+\beta}} \right) p_1^{\frac{\alpha}{\alpha+\beta}} p_2^{\frac{\beta}{\alpha+\beta}} \end{align*}$$

We finally get:

$$y^* = (\alpha + \beta) u^{\frac{1}{\alpha + \beta}} \left( \frac{p_1^{\alpha} p_2^{\beta}}{\alpha^{\alpha} \beta^{\beta}} \right)^{\frac{1}{\alpha + \beta}}$$

3. The Marshallian demand functions $D_1(p_1, p_2, y)$ and $D_2(p_1, p_2, y)$ depend on the budget constraint and income $y$, while the Hicksian demand functions $D_1(p_1, p_2, u)$ and $D_2(p_1, p_2, u)$ depend on the utility level $u$:

<table><thead><tr><th rowspan="2">Demand</th><th colspan="2">α + β ≠ 1</th><th colspan="2">α + β = 1</th></tr><tr><th>Marshallian</th><th>Hicksian</th><th>Marshallian</th><th>Hicksian</th></tr></thead><tbody><tr><td>x<sub>1</sub></td><td>α / (α + β) (y / p<sub>1</sub>)</td><td>1 / (α + β) (αp<sub>2</sub> / βp<sub>1</sub>) / (α + β)</td><td>α y / p<sub>1</sub></td><td>u (αp<sub>2</sub> / βp<sub>1</sub>)<sup>β</sup></td></tr><tr><td>x<sub>2</sub></td><td>β / (α + β) (y / p<sub>2</sub>)</td><td>1 / (α + β) (βp<sub>1</sub> / αp<sub>2</sub>) / (α + β)</td><td>β y / p<sub>2</sub></td><td>u (βp<sub>1</sub> / αp<sub>2</sub>)<sup>α</sup></td></tr><tr><td>u</td><td>α<sup>α</sup>β<sup>β</sup> / (α + β)<sup>α+β</sup> (y<sup>α+β</sup> / p<sub>1</sub><sup>α</sup>p<sub>2</sub><sup>β</sup>)</td><td>u</td><td>α<sup>α</sup>β<sup>β</sup> / p<sub>1</sub><sup>α</sup>p<sub>2</sub><sup>β</sup></td><td>u</td></tr><tr><td>y</td><td>y</td><td>(α + β) (u p<sub>1</sub><sup>α</sup>p<sub>2</sub><sup>β</sup> / α<sup>α</sup>β<sup>β</sup>)</td><td>y</td><td>u p<sub>1</sub><sup>α</sup>p<sub>2</sub><sup>β</sup> / α<sup>α</sup>β<sup>β</sup></td></tr></tbody></table>
The key difference lies in what is held constant. Marshallian demand considers both the income and substitution effects of a price change when income is fixed and utility changes. In contrast, Hicksian demand only considers the substitution effect because the consumer remains on the same indifference curve while income adjusts.

4. Since we have:

$$u^* = \frac{\alpha^\alpha \beta^\beta}{(\alpha + \beta)^{\alpha+\beta} p_1^\alpha p_2^\beta} y^{\alpha+\beta}$$

and:

$$(u^*)^{\frac{1}{\alpha+\beta}} = \frac{y}{\alpha+\beta} \left( \frac{\alpha^\alpha \beta^\beta}{p_1^\alpha p_2^\beta} \right)^{\frac{1}{\alpha+\beta}}$$

we deduce that:

$$\tilde{x}_1^* = (u^*)^{\frac{1}{\alpha+\beta}} \left( \frac{\alpha p_2}{\beta \tilde{p}_1} \right)^{\frac{\beta}{\alpha+\beta}} = \frac{\alpha}{\alpha+\beta} \left( \frac{y}{p_1} \right) \left( \frac{p_1}{\tilde{p}_1} \right)^{\frac{\beta}{\alpha+\beta}}$$

and:

$$\tilde{x}_2^* = (u^*)^{\frac{1}{\alpha+\beta}} \left( \frac{\beta \tilde{p}_1}{\alpha p_2} \right)^{\frac{\alpha}{\alpha+\beta}} = \frac{\beta}{(\alpha+\beta)} \left( \frac{y}{p_2} \right) \left( \frac{\tilde{p}_1}{p_1} \right)^{\frac{\alpha}{\alpha+\beta}}$$

The demand functions depend on a weighted geometric mean of the old and new prices, where the weights are the preference parameters $\alpha$ and $\beta$. It follows that:

$$\tilde{y}^* = y \left( \frac{\tilde{p}_1}{p_1} \right)^{\frac{\alpha}{\alpha+\beta}}$$

The compensating variation is the amount of money that must be taken away from (or given to) consumers after a price decrease (or increase), respectively, to maintain the same level of utility. It is calculated as follows:

$$CV = y - \tilde{y}^* = y \left( 1 - \left( \frac{\tilde{p}_1}{p_1} \right)^{\frac{\alpha}{\alpha+\beta}} \right)$$

5. We have:

$$\left\{ \begin{array}{l} \bar{x}_1^* = \frac{\alpha}{(\alpha + \beta)} \left( \frac{y}{\bar{p}_1} \right) \\ \bar{x}_2^* = \frac{\beta}{(\alpha + \beta)} \left( \frac{y}{p_2} \right) \end{array} \right.$$

and:

$$\bar{u}^* = \frac{\alpha^\alpha \beta^\beta}{(\alpha + \beta)^{\alpha+\beta}} \left( \frac{y^{\alpha+\beta}}{\bar{p}_1^\alpha p_2^\beta} \right)$$

We deduce that:

$$\bar{x}_1^* = \frac{\alpha}{\alpha + \beta} \left( \frac{y}{\bar{p}_1} \right) \left( \frac{\bar{p}_1}{p_1} \right)^{\frac{\beta}{\alpha + \beta}}$$
<figure><img src="image_571.png" /><figcaption>Figure B.32: CV and EV when the price of good 1 increases</figcaption></figure>

<figure><img src="image_572.png" /></figure>
and:

$$ \bar{x}_2^* = \frac{\beta}{(\alpha + \beta)} \left( \frac{y}{p_2} \right) \left( \frac{p_1}{\bar{p}_1} \right) \frac{\alpha}{\alpha + \beta} $$

It follows that:

$$ \bar{y}^* = y \left( \frac{p_1}{\bar{p}_1} \right) \frac{\alpha}{\alpha + \beta} $$

The equivalent variation is the amount of money that, if given to the consumer before the price change (or taken away from them), would leave them as well off as they would be after the price change. It is calculated as follows:

$$ \text{EV} = \bar{y}^* - y = y \left( \left( \frac{p_1}{\bar{p}_1} \right) \frac{\alpha}{\alpha + \beta} - 1 \right) $$

6. Using the price system $(p_1, p_2)$, we obtain the optimal consumption bundle $x_1^* = 1.8$, $x_2^* = 1.2$. This solution corresponds to point A in Figure B.31. It represents the tangency between the budget line with income $y = 3$ and the indifference curve at utility level $u^* = 1.5305$ (blue curves). Under the alternative price system $(\tilde{p}_1, \tilde{p}_2)$, the optimal bundle becomes $x_1^* = 2.4$, $x_2^* = 1.2$, corresponding to point B in Figure B.31. This point represents the tangency between the new budget line (with income $y = 3$) and the higher indifference curve with utility level $u^* = 1.8189$ (red curves). To determine the compensating variation, we shift the red budget line until it becomes tangent to the indifference curve corresponding to utility level $u^* = 1.5305$ (green curves). This yields point C, with coordinates $\tilde{x}_1^* = 2.0195$ and $\tilde{x}_2^* = 1.0098$. The associated budget line corresponds to an income of $\bar{y}^* = 2.5244$. We deduce that $CV = 3 - 2.5244 = 0.4756$. To determine the equivalent variation, we shift the blue budget line until it becomes tangent to the indifference curve corresponding to utility level $u^* = 1.8189$ (violet curves). This gives point D, with coordinates $\bar{x}_1^* = 2.1391$ and $\bar{x}_2^* = 1.4261$. The associated income level is $\bar{y}^* = 3.5652$. We deduce that $EV = 3.5652 - 3 = 0.5652$.

7. Using the price system $(p_1, p_2)$, we obtain the optimal consumption bundle $x_1^* = 1.5$, $x_2^* = 1.5$. This solution corresponds to point A in Figure B.32. It represents the tangency between the budget line with income $y = 3$ and the indifference curve at utility level $u^* = 1.5$ (blue curves). Under the alternative price system $(\tilde{p}_1, \tilde{p}_2)$, the optimal bundle becomes $x_1^* = 0.8571$, $x_2^* = 1.5$, corresponding to point B in Figure B.32. This point represents the tangency between the new budget line (with income $y = 3$) and the lower indifference curve with utility level $u^* = 1.1339$ (red curves). To determine the compensating variation, we shift the red budget line until it becomes tangent to the indifference curve corresponding to utility level $u^* = 1.5$ (green curves). This yields point C, with coordinates $\tilde{x}_1^* = 1.1339$ and $\tilde{x}_2^* = 1.9843$. The associated budget line corresponds to an income of $\bar{y}^* = 3.9686$. We deduce that $CV = 3 - 3.9686 = -0.9686$. To determine the equivalent variation, we shift the blue budget line until it becomes tangent to the indifference curve corresponding to utility level $u^* = 1.1339$ (violet curves). This gives point D, with coordinates $\bar{x}_1^* = 1.1339$ and $\bar{x}_2^* = 1.1339$. The associated income level is $\bar{y}^* = 2.2678$. We deduce that $EV = 2.2678 - 3 = -0.7322$.
B.2.2 Equity and bond portfolio optimization with green preferences (Exercise 11.4.1)

1. (a) We have:

$$CI_{i,j} = \frac{CE_{i,j}}{Y_i}$$

For instance, if we consider the 8th issuer, we have<fnref n="4" />:

$$CI_{8,1} = \frac{CE_{8,1}}{Y_8} = \frac{5}{25} = 0.20 \text{ tCO}_2\text{e/}$ mn

$$CI_{8,2} = \frac{CE_{8,2}}{Y_8} = \frac{64}{25} = 2.56 \text{ tCO}_2\text{e/}$ mn

$$CI_{8,3} = \frac{CE_{8,3}}{Y_8} = \frac{199}{25} = 7.96 \text{ tCO}_2\text{e/}$ mn

Since we have:

<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>CE<sub>i,1</sub></td><td>75</td><td>5 000</td><td>720</td><td>50</td><td>2 500</td><td>25</td><td>30 000</td><td>5</td></tr><tr><td>CE<sub>i,2</sub></td><td>75</td><td>5 000</td><td>1 030</td><td>350</td><td>4 500</td><td>5</td><td>2000</td><td>64</td></tr><tr><td>CE<sub>i,3</sub></td><td>24 000</td><td>15 000</td><td>1 210</td><td>550</td><td>500</td><td>187</td><td>30 000</td><td>199</td></tr><tr><td>Y<sub>i</sub></td><td>300</td><td>328</td><td>125</td><td>100</td><td>200</td><td>102</td><td>107</td><td>25</td></tr></tbody></table>

we obtain:

<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>CI<sub>i,1</sub></td><td>0.25</td><td>15.24</td><td>5.76</td><td>0.50</td><td>12.50</td><td>0.25</td><td>280.37</td><td>0.20</td></tr><tr><td>CI<sub>i,2</sub></td><td>0.25</td><td>15.24</td><td>8.24</td><td>3.50</td><td>22.50</td><td>0.05</td><td>18.69</td><td>2.56</td></tr><tr><td>CI<sub>i,3</sub></td><td>80.00</td><td>45.73</td><td>9.68</td><td>5.50</td><td>2.50</td><td>1.83</td><td>280.37</td><td>7.96</td></tr></tbody></table>

(b) We have:

$$CI_{i,1-2} = \frac{CE_{i,1} + CE_{i,2}}{Y_i} = CI_{i,1} + CI_{i,2}$$

and:

$$CI_{i,1-3} = CI_{i,1} + CI_{i,2} + CI_{i,3}$$

We deduce that:

<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>CI<sub>i,1</sub></td><td>0.25</td><td>15.24</td><td>5.76</td><td>0.50</td><td>12.50</td><td>0.25</td><td>280.37</td><td>0.20</td></tr><tr><td>CI<sub>i,1-2</sub></td><td>0.50</td><td>30.49</td><td>14.00</td><td>4.00</td><td>35.00</td><td>0.29</td><td>299.07</td><td>2.76</td></tr><tr><td>CI<sub>i,1-3</sub></td><td>80.50</td><td>76.22</td><td>23.68</td><td>9.50</td><td>37.50</td><td>2.12</td><td>579.44</td><td>10.72</td></tr></tbody></table>

(c) We have:

$$ \begin{aligned} CI(b) &= \sum_{i=1}^{8} b_i CI_i \\ &= 0.22 \times 80.50 + 0.19 \times 76.2195 + 0.17 \times 23.68 + 0.13 \times 9.50 + \\ & \quad 0.11 \times 37.50 + 0.08 \times 2.1275 + 0.06 \times 579.4393 + 0.04 \times 10.72 \\ &= 76.9427 \text{ tCO}_2\text{e/}$ mn \end{aligned} $$

<footnote n="4">⁴Because 1 ktCO₂e/$ bn = 1 tCO₂e/$ mn.</footnote>
(d) i. We have:

$$b_i = \frac{MC_i}{\sum_{k=1}^{8} MC_k}$$

and $\sum_{k=1}^{8} MC_k = \$10$ bn. We deduce that:

$$MC_i = 10 \times b_i$$

We obtain the following values of market capitalization expressed in $ bn:

<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>MC<sub>i</sub></td><td>2 200</td><td>1 900</td><td>1 700</td><td>1 300</td><td>1 100</td><td>800</td><td>600</td><td>400</td></tr></tbody></table>

ii. Let $W$ be the wealth invested in the benchmark portfolio $b$. The wealth invested in asset $i$ is equal to $b_i W$. We deduce that the ownership ratio is equal to:

$$\varpi_i = \frac{b_i W}{MC_i} = \frac{b_i W}{b_i \sum_{k=1}^{n} MC_k} = \frac{W}{\sum_{k=1}^{n} MC_k}$$

When we invest in a capitalization-weighted portfolio, the ownership ratio is the same for all the assets. In our case, we have:

$$\varpi_i = \frac{1}{10 \times 1000} = 0.01\%$$

The ownership ratio is equal to 1 basis point.

iii. Using the financed emissions approach, the carbon emissions of our investment is equal to:

$$\begin{aligned} \mathcal{CE} (\$1 \text{ bn}) &= 0.01\% \times (75 + 75 + 24\,000) + \\ & 0.01\% \times (5\,000 + 5\,000 + 15\,000) + \\ & \dots + \\ & 0.01\% \times (5 + 64 + 199) \\ &= 12.3045 \text{ ktCO}_2\text{e} \end{aligned}$$

iv. We compute the revenues of our investment:

$$Y (\$1 \text{ bn}) = 0.01\% \sum_{i=1}^{8} Y_i = \$0.1287 \text{ bn}$$

We deduce that the exact carbon intensity is equal to:

$$\mathcal{CI} (\$1 \text{ bn}) = \frac{\mathcal{CE} (\$1 \text{ bn})}{Y (\$1 \text{ bn})} = \frac{12.3045}{0.1287} = 95.6061 \text{ tCO}_2\text{e}/\$ \text{ mn}$$

We notice that the WACI of the benchmark underestimates the exact carbon intensity of our investment by 19.5%:

$$76.9427 < 95.6061$$
2. (a) The covariance matrix $\Sigma = (\Sigma_{i,j})$ is defined by:

$$ \Sigma_{i,j} = \rho_{i,j} \sigma_i \sigma_j $$

We obtain the following numerical values (expressed in bps):

$$ \Sigma = \begin{pmatrix} 484.0 & 352.0 & 385.0 & 237.6 & 616.0 & 253.0 & 200.2 & 382.8 \\ 352.0 & 400.0 & 375.0 & 234.0 & 400.0 & 276.0 & 130.0 & 377.0 \\ 385.0 & 375.0 & 625.0 & 360.0 & 700.0 & 402.5 & 227.5 & 507.5 \\ 237.6 & 234.0 & 360.0 & 324.0 & 612.0 & 331.2 & 175.5 & 391.5 \\ 616.0 & 400.0 & 700.0 & 612.0 & 1600.0 & 552.0 & 416.0 & 754.0 \\ 253.0 & 276.0 & 402.5 & 331.2 & 552.0 & 529.0 & 149.5 & 466.9 \\ 200.2 & 130.0 & 227.5 & 175.5 & 416.0 & 149.5 & 169.0 & 226.2 \\ 382.8 & 377.0 & 507.5 & 391.5 & 754.0 & 466.9 & 226.2 & 841.0 \end{pmatrix} $$

(b) The tracking error variance of portfolio $w$ with respect to benchmark $b$ is equal to:

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

The carbon intensity constraint has the following expression:

$$ \sum_{i=1}^{8} w_i \mathcal{C} \mathcal{I}_i \leq (1 - \mathcal{R}) \mathcal{C} \mathcal{I}(b) $$

where $\mathcal{R}$ is the reduction rate and $\mathcal{C} \mathcal{I}(b)$ is the carbon intensity of the benchmark. Let $\mathcal{C} \mathcal{I}^* = (1 - \mathcal{R}) \mathcal{C} \mathcal{I}(b)$ be the target value of the carbon footprint. The optimization problem is then:

$$ \begin{array}{l} w^* = \arg \min \frac{1}{2} \sigma^2 (w | b) \\ \text{s.t.} \quad \begin{cases} \sum_{i=1}^{8} w_i \mathcal{C} \mathcal{I}_i \leq \mathcal{C} \mathcal{I}^* \\ \sum_{i=1}^{8} w_i = 1 \\ 0 \leq w_i \leq 1 \end{cases} \end{array} $$

We add the second and third constraints in order to obtain a long-only portfolio.

(c) The objective function is equal to:

$$ f(w) = \frac{1}{2} \sigma^2 (w | b) = \frac{1}{2} (w - b)^{\top} \Sigma (w - b) = \frac{1}{2} w^{\top} \Sigma w - w^{\top} \Sigma b + \frac{1}{2} b^{\top} \Sigma b $$

while the matrix form of the carbon intensity constraint is:

$$ \mathcal{C} \mathcal{I}^{\top} w \leq \mathcal{C} \mathcal{I}^{*} $$

where $\mathcal{C} \mathcal{I} = (\mathcal{C} \mathcal{I}_1, \dots, \mathcal{C} \mathcal{I}_8)$ is the column vector of carbon intensities. Since $b^{\top} \Sigma b$ is a constant and does not depend on $w$, we can cast the previous optimization problem into a QP problem:

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

We have $Q = \Sigma$, $R = \Sigma b$, $A = \mathbf{1}_8^\top$, $B = 1$, $C = \mathcal{C} \mathcal{I}^\top$, $D = \mathcal{C} \mathcal{I}^*$, $w^- = \mathbf{0}_8$ and $w^+ = \mathbf{1}_8$.
(d) We have:

$$ 
\begin{aligned}
\mathcal{CI}(b) &= 0.22 \times 0.50 + 0.19 \times 30.4878 + \dots + 0.04 \times 2.76 \\
&= 30.7305 \text{ tCO}_2\text{e/}$ mn
\end{aligned}
$$

We deduce that:

$$ \mathcal{CI}^* = (1 - \mathcal{R}) \mathcal{CI}(b) = 0.80 \times 30.7305 = 24.5844 \text{ tCO}_2\text{e/}$ mn

Therefore, the inequality constraint of the QP problem is:

$$ ( 0.50 \ 30.49 \ 14.00 \ 4.00 \ 35.00 \ 0.29 \ 299.07 \ 2.76 ) \begin{pmatrix} w_1 \\ w_2 \\ \vdots \\ w_7 \\ w_8 \end{pmatrix} \le 24.5844 $$

We obtain the following optimal solution:

$$ w^* = \begin{pmatrix} 23.4961\% \\ 17.8129\% \\ 17.1278\% \\ 15.4643\% \\ 10.4037\% \\ 7.5903\% \\ 4.0946\% \\ 4.0104\% \end{pmatrix} $$

The minimum tracking error volatility $\sigma(w^* | b)$ is equal to 15.37 bps.

(e) In Table B.12, we report the optimal solution $w^*$ (expressed in %) of the optimization problem for different values of $\mathcal{R}$. We also indicate the carbon intensity of the portfolio (in tCO$_2$e/$mn) and the tracking error volatility (in bps). For instance, if $\mathcal{R}$ is set to 50%, the weights of assets #1, #3, #4 and #8 increase whereas the weights of assets #2, #5, #6 and #7 decrease. The carbon intensity of this portfolio is equal to 15.3653 tCO$_2$e/$mn. The tracking error volatility is below 40 bps, which is relatively low.

(f) In this case, the inequality constraint $Cw \le D$ is defined by:

$$ C = \mathcal{CI}_{1-3}^{\top} = \begin{pmatrix} 80.5000 \\ 76.2195 \\ 23.6800 \\ 9.5000 \\ 37.5000 \\ 2.1275 \\ 579.4393 \\ 10.7200 \end{pmatrix}^{\top} $$

and:

$$ D = (1 - \mathcal{R}) \times 76.9427 $$

We obtain the results given in Table B.13.
<table><caption>Table B.12: Solution of the equity optimization problem (scope $\mathcal{SC}_{1-2}$)</caption><thead><tr><th><i>R</i></th><th>0%</th><th>20%</th><th>30%</th><th>50%</th><th>70%</th></tr></thead><tbody><tr><td>w<sub>1</sub></td><td>22.0000</td><td>23.4961</td><td>24.2441</td><td>25.7402</td><td>30.4117</td></tr><tr><td>w<sub>2</sub></td><td>19.0000</td><td>17.8129</td><td>17.2194</td><td>16.0323</td><td>9.8310</td></tr><tr><td>w<sub>3</sub></td><td>17.0000</td><td>17.1278</td><td>17.1917</td><td>17.3194</td><td>17.8348</td></tr><tr><td>w<sub>4</sub></td><td>13.0000</td><td>15.4643</td><td>16.6964</td><td>19.1606</td><td>23.3934</td></tr><tr><td>w<sub>5</sub></td><td>11.0000</td><td>10.4037</td><td>10.1055</td><td>9.5091</td><td>7.1088</td></tr><tr><td>w<sub>6</sub></td><td>8.0000</td><td>7.5903</td><td>7.3854</td><td>6.9757</td><td>6.7329</td></tr><tr><td>w<sub>7</sub></td><td>6.0000</td><td>4.0946</td><td>3.1418</td><td>1.2364</td><td>0.0000</td></tr><tr><td>w<sub>8</sub></td><td>4.0000</td><td>4.0104</td><td>4.0157</td><td>4.0261</td><td>4.6874</td></tr><tr><td><i>CI</i>(w)</td><td>30.7305</td><td>24.5844</td><td>21.5114</td><td>15.3653</td><td>9.2192</td></tr><tr><td><i>σ</i>(w | b)</td><td>0.00</td><td>15.37</td><td>23.05</td><td>38.42</td><td>72.45</td></tr></tbody></table>

<table><caption>Table B.13: Solution of the equity optimization problem (scope $\mathcal{SC}_{1-3}$)</caption><thead><tr><th><i>R</i></th><th>0%</th><th>20%</th><th>30%</th><th>50%</th><th>70%</th></tr></thead><tbody><tr><td>w<sub>1</sub></td><td>22.0000</td><td>23.9666</td><td>24.9499</td><td>26.4870</td><td>13.6749</td></tr><tr><td>w<sub>2</sub></td><td>19.0000</td><td>17.4410</td><td>16.6615</td><td>8.8001</td><td>0.0000</td></tr><tr><td>w<sub>3</sub></td><td>17.0000</td><td>17.1988</td><td>17.2981</td><td>19.4253</td><td>24.1464</td></tr><tr><td>w<sub>4</sub></td><td>13.0000</td><td>16.5034</td><td>18.2552</td><td>25.8926</td><td>41.0535</td></tr><tr><td>w<sub>5</sub></td><td>11.0000</td><td>10.2049</td><td>9.8073</td><td>7.1330</td><td>3.5676</td></tr><tr><td>w<sub>6</sub></td><td>8.0000</td><td>7.4169</td><td>7.1254</td><td>7.0659</td><td>8.8851</td></tr><tr><td>w<sub>7</sub></td><td>6.0000</td><td>3.2641</td><td>1.8961</td><td>0.0000</td><td>0.0000</td></tr><tr><td>w<sub>8</sub></td><td>4.0000</td><td>4.0043</td><td>4.0065</td><td>5.1961</td><td>8.6725</td></tr><tr><td><i>CI</i>(w)</td><td>76.9427</td><td>61.5541</td><td>53.8599</td><td>38.4713</td><td>23.0828</td></tr><tr><td><i>σ</i>(w | b)</td><td>0.00</td><td>21.99</td><td>32.99</td><td>104.81</td><td>414.48</td></tr></tbody></table>

(g) In Figure B.33, we report the relationship between the reduction rate $\mathcal{R}$ and the tracking error volatility $\sigma(w | b)$. The choice of the scope has little impact when $\mathcal{R} \le 45\%$. Then, we notice a high increase when we consider the scope $1+2+3$. The portfolio's weights are given in Figure B.34. For assets #1 and #3, the behavior is divergent when we compare scopes $1+2$ and $1+2+3$.

3. (a) We have:

$$ 
\begin{aligned}
\text{MD}(b) &= \sum_{i=1}^{n} b_i \text{MD}_i \\
&= 0.22 \times 3.56 + 0.19 \times 7.48 + \dots + 0.04 \times 7.96 \\
&= 5.96 \text{ years}
\end{aligned}
$$

and:

$$ 
\begin{aligned}
\text{DTS}(b) &= \sum_{i=1}^{n} b_i \text{DTS}_i \\
&= 0.22 \times 103 + 0.19 \times 155 + \dots + 0.04 \times 155 \\
&= 210.73 \text{ bps}
\end{aligned}
$$
<figure><img src="image_573.png" /><figcaption>Figure B.34: Impact of the scope on the portfolio allocation (in %)</figcaption></figure>

<figure><img src="image_574.png" /></figure>
(b) We have:

$$
\left\{
\begin{array}{l}
\text{MD}(w_{\text{ew}}) = 6.20 \text{ years} \\
\text{DTS}(w_{\text{ew}}) = 228.50 \text{ bps} \\
\sigma_{\text{AS}}(w_{\text{ew}} | b) = 17.03\% \\
\sigma_{\text{MD}}(w_{\text{ew}} | b) = 1.00 \text{ years} \\
\sigma_{\text{DTS}}(w_{\text{ew}} | b) = 36.19 \text{ bps}
\end{array}
\right.
$$

(c) We have:

$$
\begin{align*}
\mathcal{R}_{\text{AS}}(w | b) ={}& (w_1 - 0.22)^2 + (w_2 - 0.19)^2 + (w_3 - 0.17)^2 + (w_4 - 0.13)^2 + \\
& (w_5 - 0.11)^2 + (w_6 - 0.08)^2 + (w_7 - 0.06)^2 + (w_8 - 0.04)^2
\end{align*}
$$

The objective function is then:

$$
f(w) = \frac{1}{2} \mathcal{R}_{\text{AS}}(w | b)
$$

The optimal solution is equal to:

$$
w^* = (17.30\%, 17.41\%, 20.95\%, 14.41\%, 10.02\%, 11.09\%, 0\%, 8.81\%)
$$

The risk metrics are:

$$
\left\{
\begin{array}{l}
\text{MD}(w^*) = 5.96 \text{ years} \\
\text{DTS}(w^*) = 210.73 \text{ bps} \\
\sigma_{\text{AS}}(w^* | b) = 10.57\% \\
\sigma_{\text{MD}}(w^* | b) = 0.43 \text{ years} \\
\sigma_{\text{DTS}}(w^* | b) = 15.21 \text{ bps}
\end{array}
\right.
$$

(d) We have^5:

$$
\begin{align*}
\mathcal{R}_{\text{MD}}(w | b) &= \left( \sum_{i=1,3,4,6} (w_i - b_i) \text{MD}_i \right)^2 + \left( \sum_{i=2,5,7,8} (w_i - b_i) \text{MD}_i \right)^2 \\
&= \left( \sum_{i=1,3,4,6} w_i \text{MD}_i - \text{MD}_1^* \right)^2 + \left( \sum_{i=2,5,7,8} w_i \text{MD}_i - \text{MD}_2^* \right)^2 \\
&= (3.56w_1 + 6.54w_3 + 10.23w_4 + 2.30w_6 - 3.4089)^2 + \\
&\quad (7.48w_2 + 2.40w_5 + 9.12w_7 + 7.96w_8 - 2.5508)^2
\end{align*}
$$

The objective function is then:

$$
f(w) = \frac{\varphi_{\text{AS}}}{2} \mathcal{R}_{\text{AS}}(w | b) + \frac{\varphi_{\text{MD}}}{2} \mathcal{R}_{\text{MD}}(w | b)
$$

The optimal solution is equal to:

$$
w^* = (16.31\%, 18.44\%, 17.70\%, 13.82\%, 11.67\%, 11.18\%, 0\%, 10.88\%)
$$

The risk metrics are:

$$
\left\{
\begin{array}{l}
\text{MD}(w^*) = 5.93 \text{ years} \\
\text{DTS}(w^*) = 210.73 \text{ bps} \\
\sigma_{\text{AS}}(w^* | b) = 11.30\% \\
\sigma_{\text{MD}}(w^* | b) = 0.03 \text{ years} \\
\sigma_{\text{DTS}}(w^* | b) = 3.70 \text{ bps}
\end{array}
\right.
$$

<footnote n="5">⁵We verify that 3.4089 + 2.5508 = 5.9597 years.</footnote>
(e) We have$^6$:

$$
\begin{align*}
\mathcal{R}_{\text{DTS}}(w | b) &= \left( \sum_{i=1,3,4,6} (w_i - b_i) \text{DTS}_i \right)^2 + \left( \sum_{i=2,5,7,8} (w_i - b_i) \text{DTS}_i \right)^2 \\
&= (103w_1 + 75w_3 + 796w_4 + 45w_6 - 142.49)^2 + \\
&\quad (155w_2 + 89w_5 + 320w_7 + 245w_8 - 68.24)^2
\end{align*}
$$

The objective function is then:

$$
f(w) = \frac{\varphi_{\text{AS}}}{2} \mathcal{R}_{\text{AS}}(w | b) + \frac{\varphi_{\text{MD}}}{2} \mathcal{R}_{\text{MD}}(w | b) + \frac{\varphi_{\text{DTS}}}{2} \mathcal{R}_{\text{DTS}}(w | b)
$$

The optimal solution is equal to:

$$
w^* = (16.98\%, 17.21\%, 18.26\%, 13.45\%, 12.10\%, 9.46\%, 0\%, 12.55\%)
$$

The risk metrics are:

$$
\left\{
\begin{array}{l}
\text{MD}(w^*) = 5.97 \text{ years} \\
\text{DTS}(w^*) = 210.68 \text{ bps} \\
\sigma_{\text{AS}}(w^* | b) = 11.94\% \\
\sigma_{\text{MD}}(w^* | b) = 0.03 \text{ years} \\
\sigma_{\text{DTS}}(w^* | b) = 0.06 \text{ bps}
\end{array}
\right.
$$

(f) We summarize the results in Table B.14.

<table><caption>Table B.14: Solution of the bond optimization problem (scope **SC**<sub>1-3</sub>)</caption>
  <thead>
    <tr>
      <th>Problem</th>
      <th>Benchmark</th>
      <th>3.(c)</th>
      <th>3.(d)</th>
      <th>3.(e)</th>
    </tr>
  </thead>
  <tbody>
    <tr>
      <td>w<sub>1</sub></td>
      <td>22.0000</td>
      <td>17.3049</td>
      <td>16.3102</td>
      <td>16.9797</td>
    </tr>
    <tr>
      <td>w<sub>2</sub></td>
      <td>19.0000</td>
      <td>17.4119</td>
      <td>18.4420</td>
      <td>17.2101</td>
    </tr>
    <tr>
      <td>w<sub>3</sub></td>
      <td>17.0000</td>
      <td>20.9523</td>
      <td>17.6993</td>
      <td>18.2582</td>
    </tr>
    <tr>
      <td>w<sub>4</sub></td>
      <td>13.0000</td>
      <td>14.4113</td>
      <td>13.8195</td>
      <td>13.4494</td>
    </tr>
    <tr>
      <td>w<sub>5</sub></td>
      <td>11.0000</td>
      <td>10.0239</td>
      <td>11.6729</td>
      <td>12.1008</td>
    </tr>
    <tr>
      <td>w<sub>6</sub></td>
      <td>8.0000</td>
      <td>11.0881</td>
      <td>11.1792</td>
      <td>9.4553</td>
    </tr>
    <tr>
      <td>w<sub>7</sub></td>
      <td>6.0000</td>
      <td>0.0000</td>
      <td>0.0000</td>
      <td>0.0000</td>
    </tr>
    <tr>
      <td>w<sub>8</sub></td>
      <td>4.0000</td>
      <td>8.8075</td>
      <td>10.8769</td>
      <td>12.5464</td>
    </tr>
    <tr>
      <td>MD&#773;(w)</td>
      <td>5.9597</td>
      <td>5.9597</td>
      <td>5.9344</td>
      <td>5.9683</td>
    </tr>
    <tr>
      <td>DTS&#773;(w)</td>
      <td>210.7300</td>
      <td>210.7300</td>
      <td>210.7300</td>
      <td>210.6791</td>
    </tr>
    <tr>
      <td>&sigma;<sub>AS</sub> (w | b)</td>
      <td>0.0000</td>
      <td>10.5726</td>
      <td>11.3004</td>
      <td>11.9400</td>
    </tr>
    <tr>
      <td>&sigma;<sub>MD</sub> (w | b)</td>
      <td>0.0000</td>
      <td>0.4338</td>
      <td>0.0254</td>
      <td>0.0308</td>
    </tr>
    <tr>
      <td>&sigma;<sub>DTS</sub> (w | b)</td>
      <td>0.0000</td>
      <td>15.2056</td>
      <td>3.7018</td>
      <td>0.0561</td>
    </tr>
    <tr>
      <td><b>CI</b>(w)</td>
      <td>76.9427</td>
      <td>38.4713</td>
      <td>38.4713</td>
      <td>38.4713</td>
    </tr>
  </tbody>
</table>

(g) The goal is to write the objective function into a quadratic function:

$$
\begin{align*}
f(w) &= \frac{\varphi_{\text{AS}}}{2} \mathcal{R}_{\text{AS}}(w | b) + \frac{\varphi_{\text{MD}}}{2} \mathcal{R}_{\text{MD}}(w | b) + \frac{\varphi_{\text{DTS}}}{2} \mathcal{R}_{\text{DTS}}(w | b) \\
&= \frac{1}{2} w^{\top} Q(b) w - w^{\top} R(b) + c(b)
\end{align*}
$$

<footnote>$^6$We verify that 142.49 + 68.24 = 210.73 bps.</footnote>
where:

$$ \mathcal{R}_{\text{AS}}(w | b) = (w_1 - 0.22)^2 + (w_2 - 0.19)^2 + (w_3 - 0.17)^2 + (w_4 - 0.13)^2 + (w_5 - 0.11)^2 + (w_6 - 0.08)^2 + (w_7 - 0.06)^2 + (w_8 - 0.04)^2 $$

$$ \mathcal{R}_{\text{MD}}(w | b) = (3.56w_1 + 6.54w_3 + 10.23w_4 + 2.30w_6 - 3.4089)^2 + (7.48w_2 + 2.40w_5 + 9.12w_7 + 7.96w_8 - 2.5508)^2 $$

$$ \mathcal{R}_{\text{DTS}}(w | b) = (103w_1 + 75w_3 + 796w_4 + 45w_6 - 142.49)^2 + (155w_2 + 89w_5 + 320w_7 + 245w_8 - 68.24)^2 $$

We use the analytical approach which is described in Section 11.1.3 on pages 1053-1058. Moreover, we rearrange the universe such that the first fourth assets belong to the first sector and the last fourth assets belong to the second sector. In this case, we have:

$$ w = \left( \underbrace{w_1, w_3, w_4, w_6}_{\mathcal{S}ector_1}, \underbrace{w_2, w_5, w_7, w_8}_{\mathcal{S}ector_2} \right) $$

The matrix $Q(b)$ is block-diagonal:

$$ Q(b) = \begin{pmatrix} Q_1 & \mathbf{0}_{4,4} \\ \mathbf{0}_{4,4} & Q_2 \end{pmatrix} $$

where the matrices $Q_1$ and $Q_2$ are equal to:

$$ Q_1 = \begin{pmatrix} 11\,025.8400 & 8\,307.0600 & 82\,898.4700 & 4\,839.7000 \\ 8\,307.0600 & 6\,794.2900 & 61\,372.6050 & 3\,751.0500 \\ 82\,898.4700 & 61\,372.6050 & 636\,332.3225 & 36\,408.2250 \\ 4\,839.7000 & 3\,751.0500 & 36\,408.2250 & 2\,257.2500 \end{pmatrix} $$

and:

$$ Q_2 = \begin{pmatrix} 25\,523.7600 & 14\,243.8000 & 51\,305.4400 & 39\,463.5200 \\ 14\,243.8000 & 8\,165.0000 & 29\,027.2000 & 22\,282.6000 \\ 51\,305.4400 & 29\,027.2000 & 104\,579.3600 & 80\,214.8800 \\ 39\,463.5200 & 22\,282.6000 & 80\,214.8800 & 61\,709.0400 \end{pmatrix} $$

The vector $R(b)$ is defined as follows:

$$ R(b) = \begin{pmatrix} 15\,001.8621 \\ 11\,261.1051 \\ 114\,306.8662 \\ 6\,616.0617 \\ 11\,073.1996 \\ 6\,237.4080 \\ 22\,424.3824 \\ 17\,230.4092 \end{pmatrix} $$

Finally, the value of $c(b)$ is equal to:

$$ c(b) = 12\,714.3386 $$
Using a QP solver, we obtain the following numerical solution:

$$ \begin{pmatrix} w_1 \\ w_3 \\ w_4 \\ w_6 \\ w_2 \\ w_5 \\ w_7 \\ w_8 \end{pmatrix} = \begin{pmatrix} 16.9796 \\ 18.2582 \\ 13.4494 \\ 9.4553 \\ 17.2102 \\ 12.1009 \\ 0.0000 \\ 12.5464 \end{pmatrix} \times 10^{-2} $$

We observe some small differences (after the fifth digit) because the QP solver is more efficient than a traditional nonlinear solver.

4. (a) The optimization problem is:

$$ \begin{array}{l} w^* = \arg\min_{\mathbf{D}} \mathbf{D}(w | b) \\ \text{s.t.} \quad \begin{cases} \mathbf{1}_8^\top w = 1 \\ \mathbf{C}\mathbf{I}^\top w \le (1 - \mathbf{R}) \mathbf{C}\mathbf{I} (b) \\ \mathbf{0}_8 \le w \le \mathbf{1}_8 \end{cases} \end{array} $$

(b) We use the absolute value trick and obtain the following optimization problem:

$$ w^* = \arg\min \frac{1}{2} \varphi_{\text{AS}} \sum_{i=1}^{8} \tau_{i,w} + \varphi_{\text{MD}} \sum_{j=1}^{2} \tau_{j,\text{MD}} + \varphi_{\text{DTS}} \sum_{j=1}^{2} \tau_{j,\text{DTS}} $$

$$ \text{s.t.} \quad \begin{cases} \mathbf{1}_8^\top w = 1 \\ \mathbf{0}_8 \le w \le \mathbf{1}_8 \\ \mathbf{C}\mathbf{I}^\top w \le (1 - \mathbf{R}) \mathbf{C}\mathbf{I} (b) \\ |w_i - b_i| \le \tau_{i,w} \\ \sum_{i \in \mathbf{S} \text{ector}_j} (w_i - b_i) \text{MD}_i \le \tau_{j,\text{MD}} \\ \sum_{i \in \mathbf{S} \text{ector}_j} (w_i - b_i) \text{DTS}_i \le \tau_{j,\text{DTS}} \\ \tau_{i,w} \ge 0, \tau_{j,\text{MD}} \ge 0, \tau_{j,\text{DTS}} \ge 0 \end{cases} $$

We can now formulate this problem as a standard LP problem:

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

where $x$ is the $20 \times 1$ vector defined as follows:

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

The $20 \times 1$ vector $c$ is equal to:

$$ c = \begin{pmatrix} \mathbf{0}_8 \\ \frac{1}{2} \varphi_{\text{AS}} \mathbf{1}_8 \\ \varphi_{\text{MD}} \mathbf{1}_2 \\ \varphi_{\text{DTS}} \mathbf{1}_2 \end{pmatrix} $$
The equality constraint is defined by $A = (\mathbf{1}_8^\top \ \mathbf{0}_8^\top \ \mathbf{0}_2^\top \ \mathbf{0}_2^\top)$ and $B = 1$. The bounds are $x^- = \mathbf{0}_{20}$ and $x^+ = \infty \cdot \mathbf{1}_{20}$. For the inequality constraint, we have$^7$:

$$ C x \leq D \Leftrightarrow \left( \begin{array}{c} \begin{array}{cccc} I_8 & -I_8 & \mathbf{0}_{8,2} & \mathbf{0}_{8,2} \\ -I_8 & -I_8 & \mathbf{0}_{8,2} & \mathbf{0}_{8,2} \\ C_{\text{MD}} & \mathbf{0}_{2,8} & -I_2 & \mathbf{0}_{2,2} \\ -C_{\text{MD}} & \mathbf{0}_{2,8} & -I_2 & \mathbf{0}_{2,2} \\ C_{\text{DTS}} & \mathbf{0}_{2,8} & \mathbf{0}_{2,2} & -I_2 \\ -C_{\text{DTS}} & \mathbf{0}_{2,8} & \mathbf{0}_{2,2} & -I_2 \\ \mathbf{CI}^\top & \mathbf{0}_{1,8} & 0 & 0 \end{array} \\ x \leq \begin{array}{c} b \\ -b \\ \text{MD}^* \\ -\text{MD}^* \\ \text{DTS}^* \\ -\text{DTS}^* \\ (1 - \mathbf{R}) \mathbf{CI} (b) \end{array} \end{array} \right) $$

where:

$$ C_{\text{MD}} = \begin{pmatrix} 3.56 & 0.00 & 6.54 & 10.23 & 0.00 & 2.30 & 0.00 & 0.00 \\ 0.00 & 7.48 & 0.00 & 0.00 & 2.40 & 0.00 & 9.12 & 7.96 \end{pmatrix} $$

and:

$$ C_{\text{DTS}} = \begin{pmatrix} 103 & 0 & 75 & 796 & 0 & 45 & 0 & 0 \\ 0 & 155 & 0 & 0 & 89 & 0 & 320 & 245 \end{pmatrix} $$

The 2 × 1 vectors MD$^*$ and DTS$^*$ are respectively equal to (3.4089, 2.5508) and (142.49, 68.24).

(c) We obtain the following solution:

$$ w^* = (18.7360, 15.8657, 17.8575, 13.2589, 11, 9.4622, 0, 13.8196) \times 10^{-2} $$

$$ \tau_w^* = (3.2640, 3.1343, 0.8575, 0.2589, 0, 1.4622, 6, 9.8196) \times 10^{-2} $$

$$ \tau_{\text{MD}} = (0, 0) $$

$$ \tau_{\text{DTS}} = (0, 0) $$

In Table B.15, we compare the two solutions$^8$. They are very close. In fact, we notice that the LP solution matches perfectly the MD and DTS constraints, but has a higher AS risk $\sigma_{\text{AS}}(w | b)$. If we note the two solutions $w^*(\mathcal{L}_1)$ and $w^*(\mathcal{L}_2)$, we have:

$$ \begin{cases} \mathcal{R}(w^*(\mathcal{L}_2) | b) = 1.4524 < \mathcal{R}(w^*(\mathcal{L}_1) | b) = 1.5584 \\ \mathcal{D}(w^*(\mathcal{L}_2) | b) = 13.9366 > \mathcal{D}(w^*(\mathcal{L}_1) | b) = 12.3982 \end{cases} $$

There is a trade-off between the $\mathcal{L}_1$- and $\mathcal{L}_2$-norm risk measures. This is why we cannot say that one solution dominates the other.

<footnote>$^7C$ is a 25 × 8 matrix and $D$ is a 25 × 1 vector.</footnote>

<footnote>$^8$The units are the following: % for the weights $w_i$, and the active share metrics $\sigma_{\text{AS}}(w | b)$ and $\mathcal{D}_{\text{AS}}(w | b)$; years for the modified duration metrics MD($w$), $\sigma_{\text{MD}}(w | b)$ and $\mathcal{D}_{\text{MD}}(w | b)$; bps for the duration-times-spread metrics DTS($w$), $\sigma_{\text{DTS}}(w | b)$ and $\mathcal{D}_{\text{DTS}}(w | b)$; tCO$_2$e/$\$ mn for the carbon intensity DTS($w$).</footnote>
<table><caption>Table B.15: Solution of the bond optimization problem (scope $SC_{1-3}$)</caption><thead><tr><th>Problem</th><th>Benchmark</th><th>3.(e)</th><th>4.(c)</th></tr></thead><tbody><tr><td>w<sub>1</sub></td><td>22.0000</td><td>16.9796</td><td>18.7360</td></tr><tr><td>w<sub>2</sub></td><td>19.0000</td><td>17.2102</td><td>15.8657</td></tr><tr><td>w<sub>3</sub></td><td>17.0000</td><td>18.2582</td><td>17.8575</td></tr><tr><td>w<sub>4</sub></td><td>13.0000</td><td>13.4494</td><td>13.2589</td></tr><tr><td>w<sub>5</sub></td><td>11.0000</td><td>12.1009</td><td>11.0000</td></tr><tr><td>w<sub>6</sub></td><td>8.0000</td><td>9.4553</td><td>9.4622</td></tr><tr><td>w<sub>7</sub></td><td>6.0000</td><td>0.0000</td><td>0.0000</td></tr><tr><td>w<sub>8</sub></td><td>4.0000</td><td>12.5464</td><td>13.8196</td></tr><tr><td>MD(w)</td><td>5.9597</td><td>5.9683</td><td>5.9597</td></tr><tr><td>DTS(w)</td><td>210.7300</td><td>210.6791</td><td>210.7300</td></tr><tr><td>σ<sub>AS</sub>(w | b)</td><td>0.0000</td><td>11.9400</td><td>12.4837</td></tr><tr><td>σ<sub>MD</sub>(w | b)</td><td>0.0000</td><td>0.0308</td><td>0.0000</td></tr><tr><td>σ<sub>DTS</sub>(w | b)</td><td>0.0000</td><td>0.0561</td><td>0.0000</td></tr><tr><td>D<sub>AS</sub>(w | b)</td><td>0.0000</td><td>25.6203</td><td>24.7964</td></tr><tr><td>D<sub>MD</sub>(w | b)</td><td>0.0000</td><td>0.0426</td><td>0.0000</td></tr><tr><td>D<sub>DTS</sub>(w | b)</td><td>0.0000</td><td>0.0608</td><td>0.0000</td></tr><tr><td>CI(w)</td><td>76.9427</td><td>38.4713</td><td>38.4713</td></tr></tbody></table>
Bibliography

ABOUMAHBOUB, T., AUER, C., BAUER, N., ..., and UECKERDT, F. (2020). *REMIND — REgional Model of INvestments and Development*. Technical Documentation, Version 2.1.0.

ABRAMS, P. A. (2009). When Does Greater Mortality Increase Population Size? The Long History and Diverse Mechanisms underlying the Hydra Effect. *Ecology Letters*, 12(5), pp. 462-474.

ACHAKULWISUT, P., ERICKSON, P., GUIVARCH, C., SCHAEFFER, R., BRUTSCHIN, E., and PYE, S. (2023). Global Fossil Fuel Reduction Pathways under Different Climate Mitigation Strategies and Ambitions. *Nature Communications*, 14(1), 5425, 15 pages.

ADAR, Z., and GRIFFIN, J. M. (1976). Uncertainty and the Choice of Pollution Control Instruments. *Journal of Environmental Economics and Management*, 3(3), pp. 178-188.

ADMATI, A. R., and PFLEIDERER, P. (2009). The “Wall Street Walk” and Shareholder Activism: Exit as a Form of Voice. *Review of Financial Studies*, 22(7), pp. 2645-2685.

AGRAWAL, A., and HOCKERTS, K. (2021). Impact Investing: Review and Research Agenda. *Journal of Small Business & Entrepreneurship*, 33(2), pp. 153-181.

AIZEN, M. A., AGUIAR, S., BIESMEIJER, J. C., ..., and SEYMOUR, C. L. (2019). Global Agricultural Productivity is Threatened by Increasing Pollinator Dependence without a Parallel Increase in Crop Diversification. *Global change biology*, 25(10), pp. 3516-3527.

AIZEN, M. A., GARIBALDI, L. A., CUNNINGHAM, S. A., and KLEIN, A. M. (2009). How Much Does Agriculture Depend on Pollinators? Lessons from Long-term Trends in Crop Production. *Annals of Botany*, 103(9), pp. 1579-1588.

ÅKERMAN, M. (2003). What Does ‘Natural Capital’ Do? The Role of Metaphor in Economic Understanding of the Environment. *Environmental Values*, 12(4), pp. 431-448.

ALBUQUERQUE, R., KOSKINEN, Y., and ZHANG, C. (2019). Corporate Social Responsibility and Firm Risk: Theory and Empirical Evidence. *Management Science*, 65(10), pp. 4451-4469.

ALESINA, A., ÖZLER, S., ROUBINI, N., and SWAGEL, P. (1996). Political Instability and Economic Growth. *Journal of Economic Growth*, 1, pp. 189-211.

ALESSI, L., and BATTISTON, S. (2022). Two Sides of the Same Coin: Green Taxonomy Alignment versus Transition Risk in Financial Portfolios. *International Review of Financial Analysis*, 102319.

ALGEO, T. J., and SHEN, J. (2024). Theory and Classification of Mass Extinction Causation. *National Science Review*, 11, nwad237, 21 pages.
ALKEMADE, R., VAN OORSCHOT, M., MILES, L., NELLEMANN, C., BAKKENES, M., and TEN BRINK, B. (2009). GLOBIO3: A Framework to Investigate Options for Reducing Global Terrestrial Biodiversity Loss. *Ecosystems*, 12, pp. 374-390.

ALLEN, M., AXELSSON, K., CALDECOTT, B., HALE, T., HEPBURN, C., HICKEY, C., MITCHELL-LARSON, E., MALHI, Y., OTTO, F., SEDDON, N., and SMITH, S. (2020). The Oxford Principles for Net Zero Aligned Carbon Offsetting. *Report*, September, 13 pages.

ALLEY, R. B., ANANDAKRISHNAN, S., CHRISTIANSON, K., HORGAN, H. J., MUTO, A., PARIZEK, B. R., POLLARD, D., and WALKER, R. T. (2015). Oceanic Forcing of Ice-Sheet Retreat: West Antarctica and More. *Annual Review of Earth and Planetary Sciences*, 43, pp. 207-231.

ALMOND, R. E. A., GROOTEN, M., JUFFE BIGNOLI, D., and PETERSEN, T. (Eds) (2022). Living Planet Report 2022 — Building a Nature-positive Society. *Report*, WWF.

ALTMAN, E. I. (1968). Financial Ratios, Discriminant Analysis and the Prediction of Corporate Bankruptcy. *Journal of Finance*, 23(4), pp. 589-609.

ANDERSON, S. J., KUBISZEWSKI, I., and SUTTON, P. C. (2024). The Ecological Economics of Light Pollution: Impacts on Ecosystem Service Value. *Remote Sensing*, 16(14), 2591, 11 pages.

ANDERSSON, M., BOLTON, P., and SAMAMA, F. (2016). Hedging Climate Risk. *Financial Analysts Journal*, 72(3), pp. 13-32.

ANTHOFF, D., and TOL, R. S. J. (2014). *The Climate Framework for Uncertainty, Negotiation and Distribution (Fund)*. Technical Description, Version 3.9.

ANTRÀS, P., CHOR, D., FALLY, T., and HILLBERRY, R. (2012). Measuring the Upstreamness of Production and Trade Flows. *American Economic Review*, 102(3), pp. 412-416.

APERGIS, N., POUFINAS, T., and ANTONOPOULOS, A. (2022). ESG Scores and Cost of Debt. *Energy Economics*, 112, 106186.

APPEL, I. R., GORMLEY, T. A., and KEIM, D. B. (2016). Passive Investors, Not Passive Owners. *Journal of Financial Economics*, 121(1), pp. 111-141.

ARCHER, D., and PIERREHUMBERT, R. (Eds) (2011). *The Warming Papers: The Scientific Foundation for the Climate Change Forecast*. John Wiley & Sons.

ARDITI, R., and GINZBURG, L. R. (1989). Coupling in Predator-prey Dynamics: Ratio-dependence. *Journal of Theoretical Biology*, 139(3), pp. 311-326.

ARLIDGE, W. N. S., BULL, J. W., ADDISON, P. F. E., ..., and MILNER-GULLAND, E. J. (2018). A Global Mitigation Hierarchy for Nature Conservation. *BioScience*, 68(5), pp. 336-347.

ARMSTRONG, R. A., and McGEHEE, R. (1980). Competitive Exclusion. *American Naturalist*, 115(2), pp. 151-170.

ARMSTRONG MCKAY, D. I., STAAL, A., ABRAMS, J. F., ..., and LENTON, T. M. (2022). Exceeding 1.5°C Global Warming Could Trigger Multiple Climate Tipping Points. *Science*, 377(6611), eabn7950, 11 pages.
ARRHENIUS, S. (1896). On the Influence of Carbonic Acid in the Air upon the Temperature of the Ground. *The London, Edinburgh, and Dublin Philosophical Magazine and Journal of Science*, 41(251), pp. 237-276.

ARRHENIUS, S. (1908). *Worlds in the Making: The Evolution of the Universe*. Harper.

ARRHENIUS, O. (1921). Species and Area. *Journal of Ecology*, 9(1), pp. 95-99.

ARROW, K. J. (1950). A Difficulty in the Concept of Social Welfare. *Journal of Political Economy*, 58(4), pp. 328-346.

ATKINSON, G., BRAATHEN, N. A., GROOM, B., and MOURATO, S. (2018). *Cost-Benefit Analysis and the Environment: Further Developments and Policy Use*. OECD Publishing, June, 458 pages.

ATTIG, N., EL GHOUL, S., GUEDHAMI, O., and SUH, J. (2013). Corporate Social Responsibility and Credit Ratings. *Journal of Business Ethics*, 117, pp. 679-694.

ATTRIDGE, S., and ENGEN, L. (2019). Blended Finance in the Poorest Countries: The Need for a Better Approach. *ODI Report*, Overseas Development Institute, April, 76 pages.

ATZ, U., VAN HOLT, T., LIU, Z. Z., and BRUNO, C. C. (2022). Does Sustainability Generate Better Financial Performance? Review, Meta-analysis, and Propositions. *Journal of Sustainable Finance & Investment*, 13(1), pp. 802-825.

Autorité de Contrôle Prudentiel et de Résolution (2021). A First Assessment of Financial Risks Stemming from Climate Change: The Main Results of the 2020 Climate Pilot Exercise. *Analyses et synthèses*, 122-2021, 4 May 2021.

AVRAMOV, D., CHENG, S., LIOUI, A., and TARELLI, A. (2022). Sustainable Investing with ESG Rating Uncertainty. *Journal of Financial Economics*, 145(2), pp. 642-664.

AZAR, J., DURO, M., KADACH, I., and ORMAZABAL, G. (2021). The Big Three and Corporate Carbon Emissions around the World. *Journal of Financial Economics*, 142(2), pp. 674-696.

BAILON, M., BOR, A. M., and REDÍN, J. G. (2024). *Biodiversity Measurement Approaches — A Practitioner's Guide for Financial Institutions*. November, 86 pages.

BAKER, M., BERGSTRESSER, D., SERAFEIM, G., and WURGLER, J. (2022). The Pricing and Ownership of US Green Bonds, *Annual Review of Financial Economics*, 14, pp. 415-437.

BALITZKY, S., and MOSSON, N. (2024). Impact Investing — Do SDG Funds Fulfil their Promises?. *ESMA TRV Risk Analysis*, February, 17 pages.

BAMBACH, R. K. (2006). Phanerozoic Biodiversity Mass Extinctions. *Annual Review of Earth Planetary Sciences*, 34(1), pp. 127-155.

BANDEIRA, B., JAMET, J.-L., JAMET, D., and GINOUX, J.-M. (2013). Mathematical Convergences of Biodiversity Indices. *Ecological Indicators*, 29, pp. 522-528.

Bank of England (2022). Results of the 2021 Climate Biennial Exploratory Scenario (CBES). *Report*, 24 May 2022.

BARAHHOU, I., BEN SLIMANE, M., OULID AZOUZ, N., and RONCALLI, T. (2022). Net Zero Investment Portfolios — Part 1. The Comprehensive Integrated Approach. *SSRN*, 4283998.
BARAHHOU, I., FERREIRA, P., and MAALEJ, Y. (2023). A Framework to Align Sovereign Bond Portfolios with Net Zero Trajectories. SSRN, 4515462.

BARBER, B. M., MORSE, A., and YASUDA, A. (2021). Impact Investing. *Journal of Financial Economics*, 139(1), pp. 162-185.

BARKO, T., CREMERS, M., and RENNEBOOG, L. (2022). Shareholder Engagement on Environmental, Social, and Governance Performance. *Journal of Business Ethics*, 180(2), pp. 777-812.

BARLOW, J., LENNOX, G. D., FERREIRA, J., ..., and GARDNER, T. A. (2016). Anthropogenic Disturbance in Tropical Forests Can Double Biodiversity Loss from Deforestation. *Nature*, 535(7610), pp. 144-147.

BARNETT, M. L., and SALOMON, R. M. (2006). Beyond Dichotomy: The Curvilinear Relationship between Social Responsibility and Financial Performance. *Strategic Management Journal*, 27(11), pp. 1101-1122.

BARNOSKY, A. D., MATZKE, N., TOMIYA, S., ..., and FERRER, E. A. (2011). Has the Earth's Sixth Mass Extinction Already Arrived?. *Nature*, 471, pp. 51-57.

Basel Committee on Banking Supervision (2022). Principles for the Effective Management and Supervision of Climate-related Financial Risks. Guidelines, 15 June 2022, 15 pages.

Basel Committee on Banking Supervision (2023). Disclosure of Climate-related Financial Risks. Consultative Document, 29 November 2023, 38 pages.

BATHIANY, S., NOTZ, D., MAURITSEN, T., RAEDEL, G., and BROVKIN, V. (2016). On the Potential for Abrupt Arctic Winter Sea Ice Loss. *Journal of Climate*, 29(7), pp. 2703-2719.

BATISTA, M., and KARAWIA, A. (2009). The Use of the Sherman-Morrison-Woodbury Formula to Solve Cyclic Block Tri-diagonal and Cyclic Block Penta-diagonal Linear Systems of Equations. *Applied Mathematics and Computation*, 210(2), pp. 558-563.

BAUER, N., MOURATIADOU, I., LUDERER, G., ..., and KRIEGLER, E. (2016). Global Fossil Energy Markets and Climate Change Mitigation — An Analysis with REMIND. *Climatic Change*, 136(1), pp. 69-82.

BAUER, R., and HANN, D. (2010). Corporate Environmental Management and Credit Risk. SSRN, 1660470.

BAUER, R., KOEDIJK, K., and OTTEN, R. (2005). International Evidence on Ethical Mutual Fund Performance and Investment Style. *Journal of Banking & Finance*, 29(7), pp. 1751-1767.

BAUMOL, W. J., and SIDAK, J. G. (1994). Stranded Costs. *Harvard Journal of Law and Public Policy*, 18(3), pp. 835-849.

BEABOUT, G. R., and SCHMIESING, K. E. (2003). Socially Responsible Investing: An Application of Catholic Social Thought. *Logos: A Journal of Catholic Thought and Culture*, 6(1), pp. 63-99.

BEAUGRAND, G., KLÉPARSKI, L., LUCZAK, C., GOBERVILLE, E., and KIRBY, R. R. (2024). A Niche-based Theory of Island Biogeography. *Ecology and Evolution*, 14(6), e11540, 14 pages.

BEBCHUK, L. A., and HIRST, S. (2019). The Specter of the Giant Three. NBER, 25914.
BECKER, M. G., MARTIN, F., and WALTER, A. (2022). The Power of ESG Transparency: The Effect of the New SFDR Sustainability Labels on Mutual Funds and Individual Investors. *Finance Research Letters*, 47 (Part B), 102708.

BEGON, M., & TOWNSEND, C. R. (2021). *Ecology: From Individuals to Ecosystems*. Fifth edition, John Wiley & Sons, 864 pages.

BEKJAROVSKI, F., and BRIÈRE, M. (2018). Shareholder Activism: Why Should Investors Care?. *Amundi Discussion Paper*, 30.

BEN AMAR, W., GOMES, M., KHURSHEED, H., and MARSAT, S. (2022). Climate Change Exposure and Internal Carbon Pricing Adoption. *Business Strategy and the Environment*, 31(7), pp. 2854-2870.

BEN DOR, A., GUAN, J., KELLEHER, A., LAURETIG, A., PRECLAW, R., and ZENG, X. (2022). ESG and Alternative Data: Capturing Corporates’ Sustainability-Related Activities with Job Postings. *Journal of Financial Data Science*, 4(1), pp. 130-144.

BEN SLIMANE, M. (2021). Bond Index Tracking with Genetic Algorithms. *ResearchGate*, 350756323.

BEN SLIMANE, M., BRARD, E., LE GUENEDAL, T., RONCALLI, T., and SEKINE, T. (2019a). ESG Investing in Fixed Income: It's Time to Cross the Rubicon. *SSRN*, 3683477.

BEN SLIMANE, M., DA FONSECA, D., and MAHTANI, V. (2020). Facts and Fantasies about the Green Bond Premium. *ResearchGate*, 348650560.

BEN SLIMANE, M., and DE JONG, M. (2017). Bond Liquidity Scores. *Journal of Fixed Income*, 27(1), pp. 77-82.

BEN SLIMANE, M., LE GUENEDAL, T., RONCALLI, T., and SEKINE, T. (2019b). ESG Investing in Corporate Bonds: Mind the Gap. *SSRN*, 3683472.

BEN SLIMANE, M., LUCIUS, D., RONCALLI, T., and XU, J. (2023b). Net Zero Investment Portfolios — Part 2. The Core-Satellite Approach. *SSRN*, 4611418.

BEN SLIMANE, M., RONCALLI, T., and SEMET, R. (2023a). Green vs. Social Bond Premium. *SSRN*, 4448651.

BEN SLIMANE, M., and MENCHAOUI, G. (2023). Bond Portfolio Optimisation and Mixed Integer Programming. *ResearchGate*, 376687891.

BENGIS, R. G., LEIGHTON, F. A., FISCHER, J. R., ARTOIS, M., MÖRNER, T., and TATE, C. M. (2004). The Role of Wildlife in Emerging and Re-emerging Zoonoses. *Revue Scientifique et Technique (International Office of Epizootics)*, 23(2), pp. 497-512.

BENNANI, L., LE GUENEDAL, T., LEPETIT, F., LY, L., MORTIER, V., RONCALLI, T., and SEKINE, T. (2018). How ESG Investing Has Impacted the Asset Pricing in the Equity Market. *SSRN*, 3316862.

BENTO, N., and GIANFRATE, G. (2020). Determinants of Internal Carbon Pricing. *Energy Policy*, 143, 111499, 8 pages.

BENTO, N., GIANFRATE, G., and ALDY, J. E. (2021). National Climate Policies and Corporate Internal Carbon Pricing. *Energy Journal*, 42(5), pp. 89-100.
BENTON, M. J. (1995). Diversification and extinction in the history of life. *Science*, 268(5207), pp. 52-58.

BENTON, M. J. (2015). *When Life Nearly Died: The Greatest Mass Extinction of All Time*. Thames & Hudson, 336 pages.

BERG, F., KÖLBEL, J. F., and RIGOBON, R. (2022). Aggregate Confusion: The Divergence of ESG Ratings. *Review of Finance*, 26(6), pp. 1315-1344.

BERGER, L., SPEARE, R., DASZAK, P., ..., and PARKES, H. (1998). Chytridiomycosis Causes Amphibian Mortality Associated with Population Declines in the Rain Forests of Australia and Central America. *Proceedings of the National Academy of Sciences*, 95(15), pp. 9031-9036.

BERK, J., and VAN BINSBERGEN, J. H. (2024). The Impact of Impact Investing. *Journal of Financial Economics*, forthcoming.

BERNARD, J., BOIRAL, O., GUILLAUMIE, L., and BROTHERTON, M. C. (2023). Does Proxy Voting Really Promote Corporate Sustainability?. *Corporate Governance: An International Review*, 31(3), pp. 445-463.

BERNSTEIN, A. S., ANDO, A. W., LOCH-TEMZELIDES, T., ..., and DOBSON, A. P. (2022). The Costs and Benefits of Primary Prevention of Zoonotic Pandemics. *Science Advances*, 8(5), eabl4183, 13 pages.

BERRADA, T., ENGELHARDT, L., GIBSON, R., and KRUEGER, P. (2022). The Economics of Sustainability Linked Bonds. *Swiss Finance Institute Research Paper*, 22-26.

BETTS, M. G., WOLF, C., RIPPLE, W. J., ..., and LEVI, T. (2017). Global Forest Loss Disproportionately Erodes Biodiversity in Intact Landscapes. *Nature*, 547(7664), pp. 441-444.

BICKNELL, J. E., STRUEBIG, M. J., and DAVIES, Z. G. (2015). Reconciling Timber Extraction with Biodiversity Conservation in Tropical Forests using Reduced-impact Logging. *Journal of Applied Ecology*, 52(2), pp. 379-388.

BILLIO, M., COSTOLA, M., HRISTOVA, I., LATINO, C., and PELIZZON, L. (2021). Inside the ESG Ratings: (Dis)agreement and Performance. *Corporate Social Responsibility and Environmental Management*, 28(5), pp. 1426-1445.

BINGLER, J. A., KRAUS, M., LEIPPOLD, M., and WEBERSINKE, N. (2022). Cheap Talk and Cherry-Picking: What ClimateBert has to Say on Corporate Climate Risk Disclosures. *Finance Research Letters*, 47(B), 102776, June.

BLACK, F. (1972). Capital Market Equilibrium with Restricted Borrowing. *Journal of Business*, 45(3), pp. 444-455.

BLACK, F. and LITTERMAN, R. B. (1991). Asset Allocation: Combining Investor Views with Market Equilibrium. *Journal of Fixed Income*, 1(2), pp. 7-18.

BLACK, F. and LITTERMAN, R. B. (1992). Global Portfolio Optimization. *Financial Analysts Journal*, 48(5), pp. 28-43.

BLEHERT, D. S., HICKS, A. C., BEHR, M., ..., and STONE, W. B. (2009). Bat White-Nose Syndrome: An Emerging Fungal Pathogen?. *Science*, 323(5911), pp. 227.
Board of Governors of the Federal Reserve System (2011). Supervisory Guidance of Model Risk Management. Joint Publication with the Office of the Comptroller of the Currency. *Supervision and Regulation Letters*, SR 11-7, April.

Board of Governors of the Federal Reserve System (2021). Interagency Statement on Model Risk Management for Bank Systems Supporting Bank Secrecy Act/Anti-Money Laundering Compliance. Joint Publication with the Federal Deposit Insurance Corporation and the Office of the Comptroller of the Currency. *Interagency Statement*, April.

BOBBINK, R., HICKS, K., GALLOWAY, J., ..., and DE VRIES, W. (2010). Global Assessment of Nitrogen Deposition Effects on Terrestrial Plant Diversity: A Synthesis. *Ecological Applications*, 20(1), pp. 30-59.

BOFFO, R., and PATALANO, R. (2020). ESG Investing: Practices, Progress and Challenges. *Report*, OECD Paris.

BÖHRINGER, C. (2003). The Kyoto Protocol: A Review and Perspectives. *Oxford Review of Economic Policy*, 19(3), pp. 451-466.

BOLIN, B., and ERIKSSON, E. (1958). Changes in the Carbon Dioxide Content of the Atmosphere and Sea due to Fossil Fuel Combustion. In Bolin, B. (Ed.), *The Atmosphere and the Sea in Motion: Scientific Contributions to the Rossby Memorial Volume*, Rockefeller Institute Press, pp. 130-142.

BOLTON, P., and KACPERCZYK, M. (2021). Do Investors Care about Carbon Risk?. *Journal of Financial Economics*, 142(2), pp. 517-549.

BOLTON, P., KACPERCZYK, M., and SAMAMA, F. (2022). Net-zero Carbon Portfolio Alignment. *Financial Analysts Journal*, 78(2), pp. 19-33.

BOLTON, P., MUSCA, X., and SAMAMA, F. (2020). Global Public-Private Investment Partnerships: A Financing Innovation with Positive Social Impact. *Journal of Applied Corporate Finance*, 32(2), pp. 31-41.

BOS, K., and GUPTA, J. (2019). Stranded Assets and Stranded Resources: Implications for Climate Change Mitigation and Global Sustainable Development. *Energy Research & Social Science*, 56, 101215, 15 pages.

BOSETTI, V., CARRARO, C., GALEOTTI, M., MASSETTI, E., and TAVONI, M. (2006). A World Induced Technical Change Hybrid Model. *Energy Journal*, 27, Special Issue #2, pp. 13-37.

BOSMANS, P., and DE MARIZ, F. (2023). The Blue Bond Market: A Catalyst for Ocean and Water Financing. *Journal of Risk and Financial Management*, 16(3), 184, 48 pages.

Boston Consulting Group (2023a). The Voluntary Carbon Market: 2022 Insights and Trends. *Report*, January, 21 pages.

Boston Consulting Group (2023b). Sustainability in Private Equity, 2023. *Report*, October, 32 pages.

BOUYÉ, E., and MENVILLE, D. (2021). The Convergence of Sovereign Environmental, Social and Governance Ratings. *Policy Research Working Paper*, World Bank, 9583.

BOWEN, H. R. (1953). *Social Responsibilities of the Businessman*. Harper & Row.

BOWKER, G. C. (2000). Biodiversity Datadiversity. *Social Studies of Science*, 30(5), pp. 643-683.
BOYD, S., PARIKH, N., CHU, E., PELEATO, B., and ECKSTEIN, J. (2010). Distributed Optimization and Statistical Learning via the Alternating Direction Method of Multipliers. *Foundations and Trends® in Machine learning*, 3(1), pp. 1-122.

BRANDER, M., ASCUI, F., SCOTT, V., and TETT, S. (2021). Carbon Accounting for Negative Emissions Technologies. *Climate Policy*, 21(5), pp. 699-717.

BREIDERT, C. (2006). *Estimation of Willingness-to-pay: Theory, Measurement, Application*. Springer.

BRIÈRE, M., KEIP, M., and LE BERTHE, T. (2022). Artificial Intelligence for Sustainable Finance: Why it may Help. *Amundi Institute Research Paper*, October 2022.

BRIÈRE, M., POUGET, S., and URECHE-RANGAU, L. (2020). BlackRock vs Norway Fund at Shareholder Meetings: Institutional Investors' Votes on Corporate Externalities. In Bank for International Settlements (Ed.), *Evolving Practices in Public Investment Management*, Proceedings of the Seventh Public Investors Conference, pp. 81-108.

BRIÈRE, M., and RAMELLI, S. (2021). Green Sentiment, Stock Returns, and Corporate Behavior. SSRN, 3850923.

BRITO-RAMOS, S., CORTEZ, M. C., and SILVA, F. (2024a). Do Sustainability Signals Diverge? An Analysis of Labeling Schemes for Socially Responsible Investments. *Business & Society*, forthcoming.

BRITO-RAMOS, S., CORTEZ, M. C., COVACHEV, S., and SILVA, F. (2024b). In Labels we Trust? The Influence of Sustainability Labels in Mutual Fund Flows. SSRN, 4748816.

BROCCARDO, E., HART, O. D., and ZINGALES, L. (2022). Exit versus Voice. *Journal of Political Economy*, 130(12), pp. 3101-3145.

BROECKER, W. S. (1975). Climatic Change: Are we on the Brink of a Pronounced Global Warming?. *Science*, 189(4201), pp. 460-463.

BROWN, J. H. (1999). The Legacy of Robert MacArthur: From Geographical Ecology to Macroecology. *Journal of Mammalogy*, 80(2), pp. 333-344.

BRUMMITT, C. D., BARNETT, G., and D'SOUZA, R. M. (2015). Coupled Catastrophes: Sudden Shifts Cascade and Hop among Interdependent Systems. *Journal of The Royal Society Interface*, 12(112), 20150712, 12 pages.

BUDYKO, M. I. (1969). The Effect of Solar Radiation Variations on the Climate of the Earth. *Tellus*, 21(5), pp. 611-619.

BUGG-LEVINE, A., and EMERSON, J. (2011). *Impact Investing: Transforming How We Make Money While Making a Difference*. John Wiley & Sons, 310 pages.

BURIVALOVA, Z., PURNOMO, ORNDORFF, S., TRUSKINGER, A., ROE, P., and GAME, E. T. (2021). The Sound of Logging: Tropical Forest Soundscape Before, During, and After Selective Timber Extraction. *Biological Conservation*, 254, 108812, 6 pages.

BURKE, M., DRISCOLL, A., LOBELL, D. B., and ERMON, S. (2021). Using Satellite Imagery to Understand and Promote Sustainable Development. *Science*, 371(6535), March.
BUSCH, T., BRUCE-CLARK, P., DERWALL, J., ECCLES, R., HEBB, T., HOEPNER, A., KLEIN, C., KRUEGER, P., PAETZOLD, F., SCHOLTENS, B. and WEBER, O. (2021). Impact Investments: A Call for (Re)Orientation. *SN Business & Economics*, 1, pp. 1-13.

BUSCH, T., PRUESSNER, E., OULTON, W., PALINSKA, A., and GARRAULT, P. (2024). Methodology for Eurosif Market Studies on Sustainability-related Investments. *Eurosif Report*, February, 30 pages.

BUTCHART, S. H. M., RESIT AKÇAKAYA, H., CHANSON, J., ..., and HILTON-TAYLOR, C. (2007). Improvements to the Red List Index. *PloS One*, 2(1), e140.

CALABRESE, E. J., and MATTSON, M. P. (2017). How Does Hormesis Impact Biology, Toxicology, and Medicine?. *npj Aging and Mechanisms of Disease*, 3(13), 8 pages.

CALDECOTT, B. (2017). Introduction to Special Issue: Stranded Assets and the Environment. *Journal of Sustainable Finance & Investment*, 7(1), pp. 1-13.

CALDECOTT, B., DERICKS, G., PFEIFFER, A., and ASTUDILLO, P. (2017). *Stranded Assets: The Transition to a Low Carbon Economy — Overview for the Insurance Industry*. Lloyd's Innovation Series, Emerging Risk Report 2017, 33 pages.

CALDECOTT, B., DERICKS, G., TULLOCH, D. J., KRUITWAGEN, L., and KOK, I. (2016a). Stranded Assets and Thermal Coal in Japan: An Analysis of Environment-related Risk Exposure. *University of Oxford Working Paper*, May, 107 pages.

CALDECOTT, B., HARNETT, E., COJOIANU, T., KOK, I., and PFEIFFER, A. (2016b). *Stranded Assets: A Climate Risk Challenge*. Inter-American Development Bank, Report, 113 pages.

CALDECOTT, B., HOWARTH, N., and MCSHARRY, P. (2013). *Stranded Assets in Agriculture: Protecting Value from Environment-related Risks*. University of Oxford, Smith School of Enterprise and the Environment, Report, 109 pages.

CALEL, R., and STAINFORTH, D. A. (2017a). On the Physics of Three Integrated Assessment Models. *Bulletin of the American Meteorological Society*, 98(6), pp. 1199-1216.

CALEL, R., and STAINFORTH, D. A. (2017b). On the Physics of Three Integrated Assessment Models: Supplement. *Internet Supplementary Materials*, https://doi.org/10.1175/BAMS-D-16-0034.1.

CALLENDAR, G. S. (1938). The Artificial Production of Carbon Dioxide and its Influence on Temperature. *Quarterly Journal of the Royal Meteorological Society*, 64(275), pp. 223-240.

CALVIN, K., PATEL, P., CLARKE, L., ..., and WISE, M. (2019). GCAM v5.1: Representing the Linkages between Energy, Water, Land, Climate, and Economic Systems. *Geoscientific Model Development*, 12, 677-698.

CAPELLE-BLANCARD, G., CRIFO, P., DIAYE, M. A., SCHOLTENS, B., and OUEGHLISSI, R. (2019). Sovereign Bond Yield Spreads and Sustainability: An Empirical Analysis of OECD Countries. *Journal of Banking & Finance*, 98, pp. 156-169.

CAPELLE-BLANCARD, G., and MONJON, S. (2014). The Performance of Socially Responsible Funds: Does the Screening Process Matter?. *European Financial Management*, 20(3), pp. 494-520.

Carbon Tracker Initiative (2011). *Unburnable Carbon — Are the World's Financial Markets Carrying A Carbon Bubble?*. Report, January, 36 pages.
Carbon Tracker Initiative (2013). *Unburnable Carbon 2013 — Wasted Capital and Stranded Assets*. Report, April, 40 pages.

CARBONE, L., NERGADZE, S. G., MAGNANI, E., ..., and GIULOTTO, E. (2006). Evolutionary Movement of Centromeres in Horse, Donkey, and Zebra. *Genomics*, 87(6), pp. 777-782.

CARDINALE, B. J., DUFFY, J. E., GONZALEZ, A., ..., and NAEEM, S. (2012). Biodiversity Loss and its Impact on Humanity. *Nature*, 486(7401), pp. 59-67.

CAREY, M. (2010). *In the Shadow of Melting Glaciers: Climate Change and Andean Society*. Oxford University Press.

CARL, J., and FEDOR, D. (2016). Tracking Global Carbon Revenues: A Survey of Carbon Taxes Versus Cap-and-trade in the Real World. *Energy Policy*, 96, pp. 50-77.

CARNEY, M. (2015). Breaking the Tragedy of the Horizon — Climate Change and Financial Stability. *Speech given at Lloyd's of London*, 29 September 2015.

CARNEY, M. (2019). Fifty Shades of Green. *Finance and Development*, 56(4), pp. 12-15.

CARROLL, A. B. (1999). Corporate Social Responsibility: Evolution of a Definitional Construct. *Business & Society*, 38(3), pp. 268-295.

CARROLL, A. B., and SHABANA, K. M. (2010). The Business Case for Corporate Social Responsibility: A Review of Concepts, Research and Practice. *International Journal of Management Reviews*, 12(1), pp. 85-105.

CARSON, R. (1962). *Silent Spring*. Houghton Mifflin Harcourt Publishing Company.

CASE, T. J., and GILPIN, M. E. (1974). Interference Competition and Niche Theory. *Proceedings of the National Academy of Sciences*, 71(8), pp. 3073-3077.

CATLING, D. C., and ZAHNLE, K. J. (2020). The Archean Atmosphere. *Science Advances*, 6(9), eaax1420, pp. 1-16.

CAWTHORN, D. M., and HOFFMAN, L. C. (2015). The Bushmeat and Food Security Nexus: A Global Account of the Contributions, Conundrums and Ethical Collisions. *Food Research International*, 76, pp. 906-925.

CAZZOLLA GATTI, R., REICH, P. B., GAMARRA, J. G. P., ..., and LIANG, J. (2022). The Number of Tree Species on Earth. *Proceedings of the National Academy of Sciences*, 119(6), e2115329119, 11 pages.

CDP (2021). Putting a Price on Carbon: The State of Internal Carbon Pricing by Corporates Globally. *Report*, 24 pages.

CDP (2022). Accounting of Scope 2 Emissions — CDP Climate Change Questionnaire. *Technical Note*, Version 9.0, March.

CDP (2023). 2023 CDP Database. October.

CEBALLOS, G., EHRLICH, P. R., BARNOSKY, A. D., GARCÍA, A., PRINGLE, R. M., and PALMER, T. M. (2015). Accelerated Modern Human-induced Species Losses: Entering the Sixth Mass Extinction. *Science Advances*, 1(5), e1400253, 5 pages.
CEBALLOS, G., EHRLICH, P. R., and DIRZO, R. (2017). Biological Annihilation via the Ongoing Sixth Mass Extinction Signaled by Vertebrate Population Losses and Declines. *Proceedings of the National Academy of Sciences*, 114(30), pp. E6089-E6096.

CEDERGREEN, N., RITZ, C., and STREIBIG, J. C. (2005). Improved Empirical Models Describing Hormesis. *Environmental Toxicology and Chemistry: An International Journal*, 24(12), pp. 3166-3172.

CHABER, A. L., MOLONEY, G. K., RENAULT, V., ..., and GAUBERT, P. (2023). Examining the International Bushmeat Traffic in Belgium: A Threat to Conservation and Public Health. *One Health*, 17, 100605, 8 pages.

CHAFIK, L., HOLLIDAY, N. P., BACON, S., and ROSSBY, T. (2022). Irminger Sea is the Center of Action for Subpolar AMOC Variability. *Geophysical Research Letters*, 49(17), e2022GL099133, 11 pages.

CHAMBERS, D., DIMSON, E., and QUIGLEY, E. (2020). To Divest or to Engage? A Case Study of Investor Responses to Climate Activism. *Journal of Investing*, 29(2), pp. 10-20.

CHAO, A. (1984). Nonparametric Estimation of the Number of Classes in a Population. *Scandinavian Journal of Statistics*, 11(4), pp. 265-270.

CHAO, A. (1987). Estimating the Population Size for Capture-Recapture Data with Unequal Catchability. *Biometrics*, 43(4), pp. 783-791.

CHAPMAN, J., DEAN, M., ORTOLEVA, P., SNOWBERG, E., and CAMERER, C. (2017). Willingness to Pay and Willingness to Accept are Probably Less Correlated Than You Think. *NBER*, 23954.

CHARNAY, B., WOLF, E. T., MARTY, B., and FORGET, F. (2020). Is the Faint Young Sun Problem for Earth Solved?. *Space Science Reviews*, 216(90), pp. 1-29.

CHARNEY, J. G., ARAKAWA, A., BAKER, D. J., BOLIN, B., DICKINSON, R. E., GOODY, R. M., LEITH, C. E., STOMMEL, H. M., and WUNSCH, C. I. (1979). *Carbon Dioxide and Climate: A Scientific Assessment*. US National Academy of Sciences, 22 pages.

CHARNEY, J. G., FJÖRTOFT, R., and VON NEUMANN, J. (1950). Numerical Integration of the Barotropic Vorticity Equation. *Tellus*, 2(4), pp. 237-254.

CHATTERJI, A. K., DURAND, R., LEVINE, D. I., and TOUBOUL, S. (2016). Do Ratings of Firms Converge? Implications for Managers, Investors and Strategy Researchers. *Strategic Management Journal*, 37(8), pp. 1597-1614.

CHAZDON, R. L., BRANCALION, P. H. S., LAESTADIUS, L., ..., and WILSON, S. J. (2016). When is a Forest a Forest? Forest Concepts and Definitions in the Era of Forest and Landscape Restoration. *Ambio*, 45(5), pp. 538-550.

CHEN, T., DONG, H., and LIN, C. (2020). Institutional Shareholders and Corporate Social Responsibility. *Journal of Financial Economics*, 135(2), pp. 483-504.

CHEN, Y. H. H., LANDRY, E., and REILLY, J. M. (2023). An Economy-wide Framework for Assessing the Stranded Assets of Energy Production Sector under Climate Policies. *Climate Change Economics*, 14(01), 2350003.
CHEN, Z., VAN KHOA, L. D., TEOH, E. N., NAZIR, A., KARUPPIAH, E. K., and LAM, K. S. (2018). Machine Learning Techniques for Anti-money Laundering (AML) Solutions in Suspicious Transaction Detection: A Review. *Knowledge and Information Systems*, 57(2), pp. 245-285.

CHERIEF, A., SEKINE, T., and STAGNOL, L. (2025). A Novel Nature-based Risk Index: Application to Acute Risks and their Financial Materiality on Corporate Bonds. *Ecological Economics*, 228, 108427, 10 pages.

CHERTOW, M. R. (2000). The IPAT Equation and its Variants. *Journal of Industrial Ecology*, 4(4), pp. 13-29.

CHESSON, P. (1990). MacArthur's Consumer-Resource Model. *Theoretical Population Biology*, 37(1), pp. 26-38.

CHESSON, P. (2000). Mechanisms of Maintenance of Species Diversity. *Annual Review of Ecology and Systematics*, 31(1), pp. 343-366.

CHESTER, D., LYNCH, C., MERCURE, J. F., and JARVIS, A. (2024). Stranded Human and Produced Capital in a Net-zero Transition. *Environmental Research: Climate*, 3(4), 045012, 14 pages.

CHIAPPINI, H., MARINELLI, N., JALAL, R. N., and BIRINDELLI, G. (2023). Past, Present and Future of Impact Investing and Closely Related Financial Vehicles: A Literature Review. *Sustainability Accounting, Management and Policy Journal*, 14(7), pp. 232-257.

CHIN, L., COJOIANU, T., HOEPNER, A., MORENO, D., PATIL, R., and VU, A. (2023). NZAOA Evidence on the Paris Alignment of Members' Disclosure, Proxy Voting, and Fossil Fuel Bond Investing. *Climate Votes Report*, January, 114 pages.

CHIQUIER, S., PATRIZIO, P., BUI, M., SUNNY, N., and MAC DOWELL, N. (2022). A Comparative Analysis of the Efficiency, Timing, and Permanence of CO2 Removal Pathways. *Energy & Environmental Science*, 15(10), pp. 4389-4403.

CHOWDHRY, B., DAVIES, S. W., and WATERS, B. (2019). Investing For Impact. *Review of Financial Studies*, 32(3), pp. 864-904.

CLARK, C. M., and TILMAN, D. (2008). Loss of Plant Species After Chronic Low-level Nitrogen Deposition to Prairie Grasslands. *Nature*, 451(7179), pp. 712-715.

CLAUS, J., and THOMAS, J. (2001). Equity Premia as Low as Three Percent? Evidence from Analysts' Earnings Forecasts for Domestic and International Stock Markets. *Journal of Finance*, 56(5), pp. 1629-1666.

Climate Bonds Initiative (2019). Climate Bonds Standard Version 3.0. Guidelines, December, 34 pages.

Climate Bonds Initiative (2022). Sustainable Debt — Global State of the Market 2021. Report, April, 38 pages.

Climate Bonds Initiative (2023). Sustainable Debt — Global State of the Market 2022. Report, April, 32 pages.

Climate Bonds Initiative (2024b). Climate Bonds Standard Version 4.1. Guidelines, February, 71 pages.
Climate Bonds Initiative (2024b). Sustainability-Linked Bonds: Building a High-Quality Market. *Report*, March, 36 pages.

Climate Bonds Initiative (2024c). Sustainable Debt — Global State of the Market 2023. *Report*, May, 30 pages.

Climate Disclosure Standards Board (2021a). Application Guidance for Water-related Disclosures. *CDSB Framework*, August 2021.

Climate Disclosure Standards Board (2021b). Application Guidance for Biodiversity-related Disclosures. *CDSB Framework*, November 2021.

Climate Disclosure Standards Board (2021c). TCFD Good Practice Handbook. *Report*, November 2021.

CLINE, W. R. (1992). *The Economics of Global Warming*. Institute for International Economics.

COASE, R. H. (1960). The Problem of Social Cost. *Journal of Law & Economics*, 3, pp. 1-44.

COHEN, A. J., BRAUER, M., BURNETT, R., ..., and FOROUZANFAR, M. H. (2017). Estimates and 25-year Trends of the Global Burden of Disease Attributable to Ambient Air Pollution: An Analysis of Data from the Global Burden of Diseases Study 2015. *The Lancet*, 389(10082), pp. 1907-1918.

COINTE, B., and GUILLEMOT, H. (2023). A History of the 1.5°C Target. *WIREs Climate Change*, 14(3), e824, 11 pages.

COJOIANU, T., HOEPNER, A., LIN, Y., VAN DER MERVE, K., and VU, A. (2021). NZAOA Climate Voting Transparency and Benchmarking Report. *Climate Votes Report*, December, 75 pages.

COLEMAN, B. D. (1981). On Random Placement and Species-Area Relations. *Mathematical Biosciences*, 54(3-4), pp. 191-215.

COLMAN, R., and SODEN, B. J. (2021). Water Vapor and Lapse Rate Feedbacks in the Climate System. *Reviews of Modern Physics*, 93(4), 045002, 90 pages.

COLWELL, R. K., and CODDINGTON, J. A. (1994). Estimating Terrestrial Biodiversity Through Extrapolation. *Philosophical Transactions of the Royal Society of London Series B: Biological Sciences*, 345(1311), pp. 101-118.

COLWELL, R. K., MAO, C. X., and CHANG, J. (2004). Interpolating, Extrapolating, and Comparing Incidence-based Species Accumulation Curves. *Ecology*, 85(10), pp. 2717-2727.

Commissariat Général au Développement Durable (2020). L'empreinte carbone des Français reste stable. *Report*, 4 pages, January.

CONDIT, R., PITMAN, N., LEIGH Jr, E. G., ..., and HUBBELL, S. P. (2002). Beta-Diversity in Tropical Forest Trees. *Science*, 295(5555), pp. 666-669.

CONNOR, E. F., and MCCOY, E. D. (1979). The Statistics and Biology of the Species-Area Relationship. *The American Naturalist*, 113(6), pp. 791-833.

Convention on Biological Diversity (2022). *15/4 Kunming-Montreal Global Biodiversity Framework*. Conference of the Parties, 16 pages.
Convergence (2024). State of Blended Finance 2024. *Report*, April, 93 pages.

COOPER, E. W., and UZUN, H. (2015). Corporate Social Responsibility and the Cost of Debt. *Journal of Accounting & Finance*, 15(8), pp. 11-29.

COPPOLA, D., LAURITANO, C., PALMA ESPOSITO, F., RICCIO, G., RIZZO, C., and DE PASCALE, D. (2021). Fish Waste: From Problem to Valuable Resource. *Marine Drugs*, 19(2), 116, 39 pages.

COQUERET, G. (2022). *Perspectives in Sustainable Equity Investing*. CRC Press.

COQUERET, G., GIROUX, T., and ZERBIB, O. D. (2025). The Biodiversity Premium. *Ecological Economics*, 228, 108435, 14 pages.

CORTEZ, M. C., SILVA, F., and AREAL, N. (2009). The Performance of European Socially Responsible Funds. *Journal of Business Ethics*, 87(4), pp. 573-588.

COSTANZA, R., D'ARGE, R., DE GROOT, R., ..., and VAN DEN BELT, M. (1997). The Value of the World's Ecosystem Services and Natural Capital. *Nature*, 387(6630), pp. 253-260.

COSTANZA, R., and DALY, H. E. (1992). Natural Capital and Sustainable Development. *Conservation Biology*, 6(1), pp. 37-46.

COSTANZA, R., DE GROOT, R., SUTTON, P. C., ..., and TURNER, R. K. (2014). Changes in the Global Value of Ecosystem Services. *Global Environmental Change*, 26, pp. 152-158.

COWIE, R. H., BOUCHET, P., and FONTAINE, B. (2022). The Sixth Mass Extinction: Fact, Fiction or Speculation?. *Biological Reviews*, 97(2), pp. 640-663.

COX, G. M., HALVERSON, G. P., STEVENSON, R. K., ..., and MACDONALD, F. A. (2016). Continental Flood Basalt Weathering as a Trigger for Neoproterozoic Snowball Earth. *Earth and Planetary Science Letters*, 446, pp. 89-99.

CRAIG, H. (1961). Isotopic Variations in Meteoric Waters. *Science*, 133(3465), pp. 1702-1703.

CRIFO, P., DIAYE, M. A., and OUEGHLISSI, R. (2015). Measuring the Effect of Government ESG Performance on Sovereign Borrowing Cost. *Quarterly Review of Economics & Finance*, 66, pp. 13-20.

CRIPPA, M., GUIZZARDI, D., PAGANI, F., ..., and VIGNATI, E. (2023). GHG Emissions of All World Countries. *Publications Office of the European Union*, JRC134504, 268 pages.

CROWTHER, D., and ARAS, G. (2008). *Corporate Social Responsibility*. Bookboon.

CRUTZEN, P. J. and STOERMER, E. F. (2000). The Anthropocene. *IGBP Global Change News*, 41, pp. 17-18.

CUI, W., MARSLAND III, R., and MEHTA, P. (2024). Les Houches Lectures on Community Ecology: From Niche Theory to Statistical Mechanics. *arXiv*, 2403.05497.

CURTIS, P. G., SLAY, C. M., HARRIS, N. L., TYUKAVINA, A., and HANSEN, M. C. (2018). Classifying Drivers of Global Forest Loss. *Science*, 361(6407), pp. 1108-1111.

DAILY, G. C. (1997). Introduction: What Are Ecosystem Services. In Daily, G. C. (Ed.), *Nature's Services: Societal Dependence On Natural Ecosystems*, Island Press, Chapter 1, pp. 1-10.
DALY, H. E. (1968). On Economics as a Life Science. *Journal of Political Economy*, 76(3), pp. 392-406.

DANIEL, K. D., LITTERMAN, R. B., and WAGNER, G. (2016). Applying Asset Pricing Theory to Calibrate the Price of Climate Risk. *NBER*, 22795.

DANIEL, K. D., LITTERMAN, R. B., and WAGNER, G. (2019). Declining CO₂ Price Paths. *Proceedings of the National Academy of Sciences*, 116(42), pp. 20886-20891.

DANSGAARD, W. (1964). Stable Isotopes in Precipitation. *Tellus*, 16(4), pp. 436-468.

DAO, I., RONCALLI, T., and SEMET, R. (2024). An Introduction to Carbon Pricing: Carbon Tax, Cap & Trade, ETS and Internal Carbon Price. SSRN, 4940475, 65 pages.

DASGUPTA, P. (2021). *The Economics of Biodiversity: The Dasgupta Review*. London: HM Treasury, February, 610 pages.

DAUBANES, J. X., MITALI, S. F., and ROCHET, J-C. (2021). Why Do Firms Issue Green Bonds?. *Swiss Finance Institute Research Paper*, 21-97.

DAUMAS, L. (2024). Financial Stability, Stranded Assets and the Low-carbon Transition — A Critical Review of the Theoretical and Applied Literatures. *Journal of Economic Surveys*, 38(3), pp. 601-716.

DAVIES, S. W., and VAN WESEP, E. D. (2018). The Unintended Consequences of Divestment. *Journal of Financial Economics*, 128(3), pp. 558-575.

DAVIS, S. J., and CALDEIRA, K. (2010). Consumption-based Accounting of CO₂ Emissions. *Proceedings of the National Academy of Sciences*, 107(12), pp. 5687-5692.

DAVIS, A. P., CHADBURN, H., MOAT, J., O'SULLIVAN, R., HARGREAVES, S., and LUGHADHA, E. N. (2019). High Extinction Risk for Wild Coffee Species and Implications for Coffee Sector Sustainability. *Science Advances*, 5(1), eaav3473, 9 pages.

DAVIS, G. F., and KIM, E. H. (2007). Business Ties and Proxy Voting by Mutual Funds. *Journal of Financial Economics*, 85(2), pp. 552-570.

DE GROOT, R., BRANDER, L., VAN DER PLOEG, S., ..., and VAN BEUKERING, P. (2012). Global Estimates of the Value of Ecosystems and Their Services in Monetary Units. *Ecosystem Services*, 1(1), pp. 50-61.

DE PALMA, A., HOSKINS, A., GONZALEZ, R. E., ..., and PURVIS, A. (2021). Annual Changes in the Biodiversity Intactness Index in Tropical and Subtropical Forest Biomes, 2001–2012. *Scientific Reports*, 11(1), 20249, 13 pages.

DE QUEIROZ, K. (2007). Species Concepts and Species Delimitation. *Systematic biology*, 56(6), pp. 879-886.

DE RYCK, J., DRIESEN, K., VERHELST, J., and LAMMERANT, J. (2024). *Assessment of Biodiversity Measurement Approaches for Businesses and Financial Institutions*. Update Report 5 on behalf of the EU Business & Biodiversity Platform, October, 272 pages.
DE VOS, J. M., JOPPA, L. N., GITTLEMAN, J. L., STEPHENS, P. R., and PIMM, S. L. (2015). Estimating the Normal Background Rate of Species Extinction. *Conservation Biology*, 29(2), pp. 452-462.

DECHEZLEPRÊTRE, A., RIVERS, N., and STADLER, B. (2019). The Economic Cost of Air Pollution: Evidence from Europe. *OECD Economics Department Working Papers*, 1584, 62 pages.

DEL MONTE-LUNA, P., NAKAMURA, M., VICENTE, A., ..., and LLUCH-COTA, S. E. (2023). A Review of Recent and Future Marine Extinctions. *Cambridge Prisms: Extinction*, 1, e13, 9 pages.

DELMAS, R. J., ASCENCIO, J. M., and LEGRAND, M. (1980). Polar Ice Evidence that Atmospheric CO₂ 20,000 yr BP was 50% of Present. *Nature*, 284(5752), pp. 155-157.

DELMAS, M. A., and BURBANO, V. C. (2011). The Drivers of Greenwashing. *California Management Review*, 54(1), pp. 64-87.

DEMARTINI, A. (2020). Provision of Non-financial Data: Mapping of Stakeholders, Products and Services. *AMF Risks and Trend Mapping*, December.

DENGLER, J. (2009). Which Function Describes the Species-Area Relationship Best? A Review and Empirical Evaluation. *Journal of Biogeography*, 36(4), pp. 728-744.

DENNIG, F., BUDOLFSON, M. B., FLEURBAEY, M., SIEBERT, A., and SOCOLOW, R. H. (2015). Inequality, Climate Impacts on the Future Poor, and Carbon Prices. *Proceedings of the National Academy of Sciences*, 112(52), pp. 15827-15832.

DERWALL, J., and KOEDIJK, K. (2009). Socially Responsible Fixed-income Funds. *Journal of Business Finance & Accounting*, 36(1-2), pp. 210-229.

DERWALL, J., and VERWIJMEREN, P. (2007). Corporate Governance and the Cost of Equity Capital: Evidence from GMI's Governance Rating. *European Centre for Corporate Engagement Research Note*, 6(1), 11 pages.

DERYUGINA, T., MOORE, F. C., and TOL, R. S. J. (2021). Environmental Applications of the Coase Theorem. *Environmental Science & Policy*, 120, pp. 81-88.

DESCOMBES, P., WISZ, M. S., LEPRIEUR, F., ..., and PELLISSIER, L. (2015). Forecasted Coral Reef Decline in Marine Biodiversity Hotspots under Climate Change. *Global Change Biology*, 21(7), pp. 2479-2487.

DESNOS, B., LE GUENEDAL, T., MORAIS, P., and RONCALLI, T. (2023). From Climate Stress Testing to Climate Value-at-Risk: A Stochastic Approach. *SSRN*, 4497124, 182 pages.

DEUTZ, A., HEAL, G. M., NIU, R., ..., and TOBIN-DE LA PUENTE, J. (2020). *Financing Nature: Closing the Global Biodiversity Financing Gap*. The Paulson Institute, The Nature Conservancy, and the Cornell Atkinson Center for Sustainability, 262 pages.

DI FRANCO, E., PIERSON, P., DI IORIO, L., ..., and GUIDETTI, P. (2020). Effects of Marine Noise Pollution on Mediterranean Fishes and Invertebrates: A Review. *Marine Pollution Bulletin*, 159, 111450, 10 pages.

DIAGNE, C., LEROY, B., VAISSIÈRE, A. C., ..., and COURCHAMP, F. (2021). High and Rising Economic Costs of Biological Invasions Worldwide. *Nature*, 592(7855), pp. 571-576.
DIAZ, D. B., and MOORE, F. C. (2017a). Quantifying the Economic Risks of Climate Change. *Nature Climate Change*, 7(11), pp. 774-782.

DIAZ, D. B., and MOORE, F. C. (2017b). Quantifying the Economic Risks of Climate Change: Supplementary Information. Available at www.nature.com/articles/nclimate3411.

DICKS, L. V., BREEZE, T. D., NGO, H. T., ..., and POTTS, S. G. (2021). A Global-scale Expert Assessment of Drivers and Risks Associated With Pollinator Decline. *Nature Ecology & Evolution*, 5(10), pp. 1453-1461.

DIETRICH, J. P., BODIRSKY, B. L., HUMPENÖDER, F., ..., and POPP, A. (2019). MAgPIE 4 — A Modular Open-source Framework for Modeling Global Land Systems. *Geoscientific Model Development*, 12(4), pp. 1299-1317.

DIETZENBACHER, E., LOS, B., STEHRER, R., TIMMER, M. P., and DE VRIES, G. (2013). The Construction of World Input-Output Tables in the WIOD Project. *Economic Systems Research*, 25(1), pp. 71-98.

DIMSON, E., KARAKAŞ, O., and LI, X. (2015). Active Ownership. *Review of Financial Studies*, 28(12), pp. 3225-3268.

DIMSON, E., KARAKAŞ, O., and LI, X. (2021). Coordinated Engagements. SSRN, 3209072.

DIMSON, E., KREUTZER, I., LAKE, R., SJO, H., and STARKS, L. T. (2013). Responsible Investment and the Norwegian Government Pension Fund Global. *Strategy Council 2013*, Report, 34 pages, 11 November 2013.

DINERSTEIN, E., OLSON, D. M., JOSHI, A., ..., and SALEEM, M. (2017). An Ecoregion-based Approach to Protecting Half the Terrestrial Realm. *BioScience*, 67(6), pp. 534-545.

DIRZO, R., YOUNG, H. S., GALETTI, M., CEBALLOS, G., ISAAC, N. J. B., and COLLEN, B. (2014). Defaunation in the Anthropocene. *Science*, 345(6195), pp. 401-406.

DISLEY, E., GIACOMANTONIO, C., KRUTHOF, K., and SIM, M. (2016). The Payment by Results Social Impact Bond Pilot at HMP Peterborough: Final Process Evaluation Report. *RAND Europe Research Series*, February, 73 pages.

DISLEY, E., RUBIN, J., SCRAGGS, E., BURROWES, N., and CULLEY, D. (2011). Lessons Learned from the Planning and Early Implementation of the Social Impact Bond at HMP Peterborough. *RAND Europe Research series*, May, 86 pages.

DIZ-PITA, E., and OTERO-ESPINAR, M. V. (2021). Predator-prey Models: A Review of Some Recent Advances. *Mathematics*, 9(15), 1783, 34 pages.

DOMMENGET, D. (2022). An Introduction to Climate Dynamics. *Lectures Notes for the Undergraduate Course in Climate Dynamics*, 408 pages.

DREI, A., LE GUENEDAL, T., LEPETIT, F., MORTIER, V., RONCALLI, T. and SEKINE, T. (2019). ESG Investing in Recent Years: New Insights from Old Challenges. SSRN, 3683469.

DRÉZE, J., and STERN, N. (1987). The Theory of Cost-Benefit Analysis. In Auerbach, A. J., and Feldstein, M. (Eds), *Handbook of Public Economics II*, Elsevier, Chapter 14, pp. 909-989.

DRUCKER, P. F. (1954). *The Practice of Management*. Harper & Row.
DRUCKER, P. F. (1984). The New Meaning of Corporate Social Responsibility. *California Management Review*, 26(2), pp. 53-63.

DUARTE, C. M., AGUSTI, S., BARBIER, E. B., ..., and WORM, B. (2020). Rebuilding Marine Life. *Nature*, 580(7801), pp. 39-51.

DUCOULOMBIER, F. (2021). Understanding the Importance of Scope 3 Emissions and the Implications of Data Limitations. *Journal of Impact and ESG Investing*, 1(4), pp. 63-71.

DUDÁS, F., and NAFFA, H. (2020). The Predictive Role of Country-Level ESG Indicators. *Economy and Finance*, 7(4), pp. 441-453.

EASTON, P. D. (2004). PE Ratios, PEG Ratios, and Estimating the Implied Expected Rate of Return on Equity Capital. *Accounting Review*, 79(1), pp. 73-95.

ECCLES, R. G., LEE, L. E., and STROEHLE, J. C. (2020). The Social Origins of ESG: An Analysis of Innovest and KLD. *Organization & Environment*, 33(4), pp. 575-596.

ECCLES, R. G., SHANDAL, V., YOUNG, D., and MONTGOMERY, B. (2023). Private Equity Should Take the Lead in Sustainability. In Young, D., and Reeves, M. (Eds), *Sustainable Business Model Innovation*, De Gruyter, Chapter 21, pp. 203-214.

ECCLES, R. G., and STROEHLE, J. C. (2018). Exploring Social Origins in the Construction of Environmental, Social and Governance Measures. *SSRN*, 3212685.

Ecosystem Marketplace (2023a). All in on Climate: The Role of Carbon Credits in Corporate Climate Strategies. *Report*, October, 36 pages.

Ecosystem Marketplace (2023b). State of the Voluntary Carbon Markets 2023. *Insights Report*, November, 28 pages.

EDMANS, A. (2009). Blockholder Trading, Market Efficiency, and Managerial Myopia. *Journal of Finance*, 64(6), pp. 2481-2513.

EDMANS, A. (2011). Does the Stock Market fully Value Intangibles? Employee Satisfaction and Equity Prices. *Journal of Financial Economics*, 101(3), pp. 621-640.

EDMANS, A. (2014). Blockholders and Corporate Governance. *Annual Review of Financial Economics*, 6, pp. 23-50.

EDMANS, A., LEVIT, D., and SCHNEEMEIER, J. (2022). Socially Responsible Divestment. European Corporate Governance Institute, *Finance Working Paper*, 823, April.

EDWARDS, M. R., CUI, R., BINDL, M., ..., and ZHAO, A. (2022). Quantifying the Regional Stranded Asset Risks from New Coal Plants under 1.5°C. *Environmental Research Letters*, 17(2), 024029, 12 pages.

EHLERS, T., and PACKER, F. (2017). Green Bond Finance and Certification. *BIS Quarterly Review*, September, pp. 89-104.

EILERS, E. J., KREMEN, C., SMITH GREENLEAF, S., GARBER, A. K., and KLEIN, A. M. (2011). Contribution of Pollinator-mediated Crops to Nutrients in the Human Food Supply. *PLoS One*, 6(6), e21363, 6 pages.
EL GHOUL, S., GUEDHAMI, O., KWOK, C. C., and MISHRA, D. R. (2011). Does Corporate Social Responsibility Affect the Cost of Capital?. *Journal of Banking & Finance*, 35(9), pp. 2388-2406.

ELIWA, Y., ABOUD, A., and SALEH, A. (2021). ESG Practices and the Cost of Debt: Evidence from EU Countries. *Critical Perspectives on Accounting*, 79, 102097, 21 pages.

EMERSON, J. (2003). The Blended Value Proposition: Integrating Social and Financial Returns. *California Management Review*, 45(4), pp. 35-51.

ENCORE (2024). *ENCORE: Exploring Natural Capital Opportunities, Risks and Exposure*. www.encorenature.org.

Energy Transitions Commission (2020). Making Mission Possible: Delivering a Net-Zero Economy. *Report*, September, 89 pages.

Energy Transitions Commission (2021). Making Clean Electrification Possible: 30 Years to Electrify the Global Economy. *Report*, April, 108 pages.

Energy Transitions Commission (2023a). Financing the Transition: How to Make the Money Flow for a Net-Zero Economy. *Report*, March, 101 pages.

Energy Transitions Commission (2023b). Material and Resource Requirements for the Energy Transition. *Report*, July, 130 pages.

Energy Transitions Commission (2024). NDCs, NCQG, and Financing the Transition: Unlocking Flows for a Net-Zero Future. *Briefing Note*, October, 27 pages.

EPICA community members (2004). Eight Glacial Cycles from an Antarctic Ice Core. *Nature*, 429(6992), pp. 623-628.

EHRLICH, P. R. (1995). The Scale of the Human Enterprise and Biodiversity Loss. In Lawton, J.H., and May, R. M. (Eds), *Extinction Rates*, Oxford University Press, Chapter 14, pp. 214-226.

ESCHER, B. I., STAPLETON, H. M., and SCHYMANSKI, E. L. (2020). Tracking Complex Mixtures of Chemicals in our Changing Environment. *Science*, 367(6476), pp. 388-392.

European Banking Authority (2014). Regulatory Technical Standards on the Conditions for Assessing the Materiality of Extensions and Changes of Internal Approaches for Credit, Market and Operational Risk. *Technical Standards, Guidelines & Recommendations*, July.

European Banking Authority (2023). EBA Progress Report on Greenwashing Monitoring and Supervision. *Report*, 2023/16, 74 pages.

European Central Bank (2022). 2022 Climate Risk Stress Test. *Report*, July 2022.

European Commission (2022a). Targeted Consultation on the Functioning of the ESG Ratings Market in the European Union and on the Consideration of ESG Factors in Credit Ratings. *Consultation Document*, April 2022.

European Commission (2022b). Targeted Consultation on the Functioning of the ESG Ratings Market in the European Union and on the Consideration of ESG Factors in Credit Ratings. *Summary Report*, August 2022.
European Environment Agency (2024a). *The Costs to Health and the Environment from Industrial Air Pollution in Europe — 2024 Update*. Briefing 24/2023, January, 18 pages.

European Environment Agency (2024b). *Estimating the External Costs of Industrial Air Pollution — Trends 2012–2021*. Technical Note, January, 177 pages.

European Financial Reporting Advisory Group (2022). Draft European Sustainability Reporting Standards (Appendix II — CSRD Requirements for the Development of sustainability Reporting Standards and their Coverage by the draft ESRS). *Draft*, November, 20 pages.

European Fund and Asset Management Association (2023a). Fact Book 2023. *Trends in European Investment Funds*, 21st Edition, June, 326 pages.

European Fund and Asset Management Association (2023b). The SFDR Fund Market — State of Play. *Market Insights*, 12, June, 8 pages.

European Investment Bank (2021). Measuring the EIB Group's Impact — Methods and Studies. *Report*, October, 46 pages.

European Securities and Markets Authority (2022). ESMA's Technical Advice to the European Commission on Integrating Sustainability Risks and Factors in MiFID II. *Final Report*, September 2022.

European Securities and Markets Authority (2023). Final Report on Draft RTS on the Review of PAI and Financial Product Disclosures in the SFDR Delegated Regulation. *Report*, JC-2023-55, 4 December 2023, 221 pages.

European Securities and Markets Authority (2024). EU Alternative Investment Funds 2023. *Market Report*, ESMA50-524821-3095, 30 January 2024, 53 pages.

European Sustainable Investment Forum (2018). *2018 European SRI Study*. November.

European Sustainable Investment Forum (2021). *2021 Eurosif Report: Fostering Investor Impact — Placing it at the Heart of Sustainable Finance*. October, 45 pages.

EZE, M. O., GEORGE, S. C., and HOSE, G. C. (2021). Dose-Response Analysis of Diesel Fuel Phytotoxicity on Selected Plant Species. *Chemosphere*, 263, 128382, 10 pages.

FA, J. E., PERES, C. A., and MEEUWIG, J. (2002). Bushmeat Exploitation in Tropical Forests: An Intercontinental Comparison. *Conservation Biology*, 16(1), pp. 232-237.

FABOZZI, F. J., MA, K. C., and OLIPHANT, B. J. (2008). Sin Stock Returns. *Journal of Portfolio Management*, 35(1), pp. 82-94.

FAHRIG, L. (2003). Effects of Habitat Fragmentation on Biodiversity. *Annual Review of Ecology, Evolution, and Systematics*, 34(1), pp. 487-515.

FAHRIG, L. (2017). Ecological Responses to Habitat Fragmentation per se. *Annual Review of Ecology, Evolution, and Systematics*, 48(1), pp. 1-23.

FAHRIG, L., ARROYO-RODRÍGUEZ, V., BENNETT, J. R., ..., and WATLING, J. I. (2019). Is Habitat Fragmentation Bad for Biodiversity?. *Biological Conservation*, 230, pp. 179-186.
FAMA E. F., and FRENCH, K. R. (1993). Common Risk Factors in the Returns on Stocks and Bonds. *Journal of Financial Economics*, 33(1), pp. 3-56.

FATICA, S., PANZICA, R., and RANCAN, M. (2021). The Pricing of Green Bonds: Are Financial Institutions Special?. *Journal of Financial Stability*, 54, 100873.

FATTORINI, S., BORGES, P. A., DAPPORTO, L., and STRONA, G. (2017). What Can the Parameters of the Species-Area Relationship (SAR) Tell Us? Insights from Mediterranean Islands. *Journal of Biogeography*, 44(5), pp. 1018-1028.

FAULWASSER, T. J., KELLETT, C. M., and WELLER, S. R. (2018). *MPC-DICE: Model Predictive Control — Dynamic Integrated model of Climate and Economy*. Code source available at https://github.com/cmkellett/MPC-DICE.

FAWCETT, T. (2006). An Introduction to ROC Analysis. *Pattern Recognition Letters*, 27(8), pp. 861-874.

FEENSTRA, R. C., INKLAAR, R., and TIMMER, M. P. (2015). The Next Generation of the Penn World Table. *American Economic Review*, 105(10), pp. 3150-3182.

FENDER, I., MCMORROW, M., SAHAKYAN, V., and ZULAICA, O. (2019). Green Bonds: The Reserve Management Perspective. *BIS Quarterly Review*, September, pp. 49-53.

FEULNER, G. (2012). The Faint Young Sun Problem. *Reviews of Geophysics*, 50(2), RG2006, 29 pages.

FICHTNER, J., HEEMSKERK, E. M., and GARCIA-BERNARDO, J. (2017). Hidden Power of the Big Three? Passive Index Funds, Re-concentration of Corporate Ownership, and New Financial Risk. *Business and Politics*, 19(2), pp. 298-326.

FIGUEIREDO, L., KRAUSS, J., STEFFAN-DEWENTER, I., and SARMENTO CABRAL, J. (2019). Understanding Extinction Debts: Spatio-temporal Scales, Mechanisms and A Roadmap for Future Research. *Ecography*, 42(12), pp. 1973-1990.

Financial Stability Board (2015). Financial Reforms — Progress on the Work Plan for the Antalya Summit. *Chair's Letter to G20 Finance Ministers and Central Bank Governors*.

FINSTER, M., WOOD, M., and RAJA, S. N. (2005). The Apgar Score has Survived the Test of Time. *Journal of the American Society of Anesthesiologists*, 102(4), pp. 855-857.

FISHER, R. A., CORBET, A. S., and WILLIAMS, C. B. (1943). The Relation between the Number of Species and the Number of Individuals in a Random Sample of an Animal Population. *Journal of Animal Ecology*, 12(1), pp. 42-58.

FLAMMER, C. (2021). Corporate Green Bonds. *Journal of Financial Economics*, 142(2), pp. 499-516.

FLAMMER, C., GIROUX, T., and HEAL, G. M. (2024). Blended Finance. *NBER*, 32287.

FLAMMER, C., GIROUX, T., and HEAL, G. M. (2025). Biodiversity Finance. *Journal of Financial Economics*, 164, 103987, 15 pages.

FLETCHER Jr, R. J., DIDHAM, R. K., BANKS-LEITE, C., ..., and HADDAD, N. M. (2018). Is Habitat Fragmentation Good for Biodiversity?. *Biological Conservation*, 226, pp. 9-15.
FLÖRKE, M., KYNAST, E., BÄRLUND, I., EISNER, S., WIMMER, F., and ALCAMO, J. (2013). Domestic and Industrial Water Uses of the Past 60 Years as a Mirror of Socio-economic Development: A Global Simulation Study. *Global Environmental Change*, 23(1), pp. 144-156.

Food and Agriculture Organization of the United Nations (2019). *The State of the World's Biodiversity for Food and Agriculture*. FAO Commission on Genetic Resources for Food and Agriculture, prepared by Bélanger J., and Pilling D. (Eds), February, 572 pages.

Food and Agriculture Organization of the United Nations (2020). *Global Forest Resources Assessment 2020 — Key Findings*. 16 pages.

Food and Agriculture Organization of the United Nations (2022). *Blue Transformation — Roadmap 2022–2030: A Vision for FAO’s Work on Aquatic Food Systems*. 40 pages.

Food and Agriculture Organization of the United Nations (2023a). FRA 2025: Terms and Definitions. *Forest Resources Assessment Working Paper*, 194, 29 pages.

Food and Agriculture Organization of the United Nations (2023b). *World Food and Agriculture — Statistical Yearbook 2023*. November, 384 pages.

Food and Agriculture Organization of the United Nations (2024a). *The State of Food Security and Nutrition in the World 2024 – Financing to End Hunger, Food Insecurity and Malnutrition in all its Forms*. September, 316 pages.

Food and Agriculture Organization of the United Nations (2024b). *The State of World Fisheries and Aquaculture 2024 — Blue Transformation in action*. October, 264 pages.

FOSTER, G. L., ROYER, D. L., and LUNT, D. J. (2017). Future Climate Forcing Potentially without Precedent in the last 420 Million Years. *Nature Communications*, 8, 14845, 8 pages.

FOURIER, J. (1824). Remarques générales sur les températures du globe terrestre et des espaces planétaires. *Annales de Chimie et de Physique*, 27, pp. 136-167.

France Invest & Forum pour l'investissement responsable (2021). Impact Investing: A Demanding Definition For Listed and Non-listed Products. *Guidelines*, March, 19 pages.

FRANKIGNOUL, C., and HASSELMANN, K. (1977). Stochastic Climate Models, Part II Application to Sea-surface Temperature Anomalies and Thermocline Variability. *Tellus*, 29(4), pp. 289-305.

FRAZZINI, A., and PEDERSEN, L. H. (2014). Betting Against Beta. *Journal of Financial Economics*, 111(1), pp. 1-25.

FREEMAN, R. E. (2004). *Strategic Management: A Stakeholder Approach*. Pitman.

FREIRE-GONZÁLEZ, J. (2018). Environmental Taxation and the Double Dividend Hypothesis in CGE Modelling Literature: A Critical Review. *Journal of Policy Modeling*, 40(1), pp. 194-223.

Freshfields Bruckhaus Deringer (2005). A Legal Framework for the Integration of Environmental, Social and Governance Issues into Institutional Investment. *UNEP Report*, October.

FRETWELL, S. D. (1975). The Impact of Robert MacArthur on Ecology. *Annual Review of Ecology and Systematics*, 6, pp. 1-13.
FRICAUDET, M., PRAKASH, V., SOHM, S., SMITH, T., and REHMATULLA, N. (2024). Fossil Fuel Carrying Ships and the Risk of Stranded Assets. *Kühne Foundation Report*, June, 50 pages.

FRIEDE, G., BUSCH, T., and BASSEN, A. (2015). ESG and Financial Performance: Aggregated Evidence from more than 2000 Empirical Studies. *Journal of Sustainable Finance & Investment*, 5(4), pp. 210-233.

FRIEDERICH, D., KAACK, L. H., LUCCIONI, A., and STEFFEN, B. (2021). Automated Identification of Climate Risk Disclosures in Annual Corporate Reports. *arXiv*, 2108.01415.

FRIEDLINGSTEIN, P., O'SULLIVAN, M., JONES, M. W., ..., and ZHENG, B. (2022). Global Carbon Budget 2022. *Earth System Science Data*, 14(11), pp. 4811-4900.

FRIEDMAN, M. (1962). *Capitalism and Freedom*. University of Chicago Press.

FRIEDMAN, M. (1970). The Social Responsibility of Business Is to Increase Its Profits. *New York Times Magazine*, September 13, Section SM, pp. 17 (reproduced in Zimmerli, W. Ch., Holzinger, M., and Richter, K. (Eds). (2007), *Corporate Ethics and Corporate Governance*, Springer, pp. 173-178).

FRIEDRICH, O., NORRIS, R. D., and ERBACHER, J. (2012). Evolution of Middle to Late Cretaceous Oceans — A 55 my Record of Earth’s Temperature and Carbon Cycle. *Geology*, 40(2), pp. 107-110.

FUJIMORI, S., HASEGAWA, T., and MASUI, T. (2017). AIM/CGE V2.0: Basic Feature of the Model. In Fujimori, S., Kainuma, M., and Masui, T. (Eds), *Post-2020 Climate Action: Global and Asian Perspectives*, Springer, pp. 305-328.

FULLER, R., LANDRIGAN, P. J., BALAKRISHNAN, K., ..., and YAN, C. (2022). Pollution and Health: A Progress Update. *The Lancet Planetary Health*, 6(6), e535-e547.

FUSS, S., LAMB, W. F., CALLAGHAN, ..., and MINX, J. C. (2018). Negative Emissions — Part 2: Costs, Potentials and Side Effects. *Environmental Research Letters*, 13(6), 063002, 47 pages.

GABAY, D., and MERCIER, B. (1976). A Dual Algorithm for the Solution of Nonlinear Variational Problems via Finite Element Approximation. *Computers & Mathematics with Applications*, 2(1), pp. 17-40.

GAMBHIR, A., BUTNAR, I., LI, P. H., SMITH, P., and STRACHAN, N. (2019). A Review of Criticisms of Integrated Assessment Models and Proposed Approaches to Address These, Through the Lens of BECCS. *Energies*, 12(9), 1747.

GANTCHEV, N. (2013). The Costs of Shareholder Activism: Evidence from a Sequential Decision Model. *Journal of Financial Economics*, 107(3), pp. 610-631.

GANTCHEV, N., GREDIL, O. R., and JOTIKASTHIRA, C. (2019). Governance under the Gun: Spillover Effects of Hedge Fund Activism. *Review of Finance*, 23(6), pp. 1031-1068.

GARIBALDI, L. A., STEFFAN-DEWENTER, I., WINFREE, R., ..., and KLEIN, A. M. (2013). Wild Pollinators Enhance Fruit Set of Crops Regardless of Honey Bee Abundance. *Science*, 339(6127), pp. 1608-1611.

GASTON, K. J. (2010). Biodiversity. In Sodhi, N. S., and Ehrlich, P. R. (Eds), *Conservation Biology for All*, Oxford University Press, Chapter 2, pp. 27-44.
GASTON, K. J., and SPICER, J. I. (2004). *Biodiversity: An Introduction*. Second Edition, Wiley-Blackwell, 208 pages.

GATTI, L. V., BASSO, L. S., MILLER, J. B., ..., and NEVES, R. A. (2021). Amazonia as a Carbon Source Linked to Deforestation and Climate Change. *Nature*, 595(7867), pp. 388-393.

GEBHARDT, W. R., LEE, C. M., and SWAMINATHAN, B. (2001). Toward an Implied Cost of Capital. *Journal of Accounting Research*, 39(1), pp. 135-176.

GECZY, C., JEFFERS, J. S., MUSTO, D. K., and TUCKER, A. M. (2021). Contracts with (Social) Benefits: The Implementation of Impact Investing. *Journal of Financial Economics*, 142(2), pp. 697-718.

GEMECHU, E. D., BUTNAR, I., LLOP, M., and CASTELLS, F. (2014). Economic and Environmental effects of CO₂ Taxation: An Input-output Analysis for Spain. *Journal of Environmental Planning and Management*, 57(5), pp. 751-768.

GENEST, C., and MACKAY, J. (1986a). Copules Archimédiennes et Familles de Lois Bidimensionnelles dont les Marges sont données. *Canadian Journal of Statistics*, 14(2), pp. 145-159.

GENEST, C., and MACKAY, J. (1986b), The Joy of Copulas: Bivariate Distributions with Uniform Marginals, *American Statistician*, 40(4), pp. 280-283.

GENTLEMAN, W., LEISING, A., FROST, B., STROM, S., and MURRAY, J. (2003). Functional Responses for Zooplankton Feeding on Multiple Resources: A Review of Assumptions and Biological Dynamics. *Deep Sea Research Part II: Topical Studies in Oceanography*, 50(22-26), pp. 2847-2875.

GERARD, B. (2019). ESG and Socially Responsible Investment: A Critical Review. *Beta*, 33(1), pp. 61-83.

GHG Protocol (2004). The Greenhouse Gas Protocol: A Corporate Accounting and Reporting Standard. *Standards*.

GHG Protocol (2011). The Value Chain (Scope 3) Standard. *Standards*.

GHG Protocol (2013). Technical Guidance for Calculating Scope 3 Emissions: Supplement to the Corporate Value Chain (Scope 3) Accounting & Reporting Standard. *Standards*.

GHG Protocol (2015). GHG Protocol Scope 2 Guidance: An Amendment to the GHG Protocol Corporate Standard. *Standards*.

GHOSH, B., and KAR, T. K. (2013). Maximum Sustainable Yield and Species Extinction in a Prey-predator System: Some New Results. *Journal of Biological Physics*, 39, pp. 453-467.

GIANFRATE, G., and PERI, M. (2019). The Green Advantage: Exploring the Convenience of Issuing Green Bonds. *Journal of Cleaner Production*, 219, pp. 127-135.

GIANONCELLI, A., GAGGIOTTI, G., BOIARDI, P., and PICÓN MARTÍNEZ, A. (2019). 15 Years of Impact — Taking Stock and Looking Ahead. *EVPA Report*, 112 pages.

GIBBARD, P., WALKER, M., BAUER, A., ..., and RUDDIMAN, W. (2022). The Anthropocene as an Event, not an Epoch. *Journal of Quaternary Science*, 37(3), pp. 395-399.
GIGLIO, S., MAGGIORI, M., STROEBEL, J., TAN, Z., UTKUS, S., and XU, X. (2023a). Four Facts about ESG Beliefs and Investor Portfolios. *NBER*, 31114

GIGLIO, S., KUCHLER, T., STROEBEL, J., and ZENG, X. (2023b). Biodiversity Risk. *NBER*, 31137.

GILLAN, S. L., and STARKS, L. T. (2000). Corporate Governance Proposals and Shareholder Activism: The Role of Institutional Investors. *Journal of Financial Economics*, 57(2), pp. 275-305.

GILLINGHAM, K., and STOCK, J. H. (2018). The Cost of Reducing Greenhouse Gas Emissions. *Journal of Economic Perspectives*, 32(4), pp. 53-72.

GILPIN, M. E., CASE, T. J., and AYALA, F. J. (1976). θ-selection. *Mathematical Biosciences*, 32(1-2), pp. 131-139.

GINI, C. (1921). Measurement of Inequality of Incomes. *Economic Journal*, 31(121), pp. 124-126.

GIRONA, M. M., MORIN, H., GAUTHIER, S., and BERGERON, Y. (Eds) (2023). *Boreal Forests in the Face of Climate Change: Sustainable Management*. Springer Nature, Advances in Global Change Research, 14, 837 pages.

GLEASON, H. A. (1922). On the Relation Between Species and Area. *Ecology*, 3(2), pp. 158-162.

Global Development Network (2021). Measuring Impacts — The Experience of the EIB-GDN Programme. *Report*, 114 pages.

Global Impact Investing Network (2021). *Impact Investing: A Guide to this Dynamic Market*. January, 20 pages.

Global Sustainable Investment Alliance (2013). *2012 Global Sustainable Investment Review*. January, 46 pages.

Global Sustainable Investment Alliance (2015). *2014 Global Sustainable Investment Review*. February, 36 pages.

Global Sustainable Investment Alliance (2017). *2016 Global Sustainable Investment Review*. March, 31 pages.

Global Sustainable Investment Alliance (2019). *2018 Global Sustainable Investment Review*. April, 29 pages.

Global Sustainable Investment Alliance (2021). *2020 Global Sustainable Investment Review*. July, 32 pages.

Global Sustainable Investment Alliance (2023). *2022 Global Sustainable Investment Review*. November, 47 pages.

GLUSZEK, S., VIOLLAZ, J., MWINYIHALI, R., WIELAND, M., and GORE, M. L. (2021). Using Conservation Criminology to Understand the Role of Restaurants in the Urban Wild Meat Trade. *Conservation Science and Practice*, 3(5), e368, 13 pages.

GODE, D., and MOHANRAM, P. (2003). Inferring the Cost of Capital using the Ohlson-Juettner Model. *Review of Accounting Studies*, 8, pp. 399-431.

GOEDKOOP, M., HEIJUNGS, R., HUIJBREGTS, M., DE SCHRYVER, A., STRUIJS, J., and VAN ZELM, R. (2009). *ReCiPe 2008*. First edition, 132 pages.
GOLDREYER, E. F., AHMED, P., and DILTZ, J. D. (1999). The Performance of Socially Responsible Mutual Funds: Incorporating Sociopolitical Information in Portfolio Selection. *Managerial Finance*, 25(1), pp. 23-36.

GOLLIER, C. (2010). Ecological Discounting. *Journal of Economic Theory*, 145(2), pp. 812-829.

GOLLIER, C. (2013). *Pricing the Planet's Future: The Economics of Discounting in an Uncertain World*. Princeton University Press.

GOLLIER, C., and POUGET, S. (2022). Investment Strategies and Corporate Behaviour with Socially Responsible Investors: A Theory of Active Ownership. *Economica*, 89(356), pp. 997-1023.

GOLUB, G. H., and VAN LOAN, C. F. (2013). *Matrix Computations*. Fourth edition, Johns Hopkins University Press.

GOMBEER, S., NEBESSE, C., MUSABA, P., ..., and VERHEYEN, E. (2021). Exploring the Bushmeat Market in Brussels, Belgium: A Clandestine Luxury Business. *Biodiversity and Conservation*, 30, pp. 55-66.

GOMPERS, P., ISHII, J., and METRICK, A. (2003). Corporate Governance and Equity Prices. *Quarterly Journal of Economics*, 118(1), pp. 107-156.

GONÇALVES, T. C., DIAS, J., and BARROS, V. (2022). Sustainability Performance and the Cost of Capital. *International Journal of Financial Studies*, 10(3), 63, 32 pages.

GONZÁLEZ-MARTÍN, J., KRAAKMAN, N. J. R., PÉREZ, C., LEBRERO, R., and MUÑOZ, R. (2021). A State-of-the-art Review on Indoor Air Pollution and Strategies for Indoor Air Pollution Control. *Chemosphere*, 262, 128376, 16 pages.

GOOSSE, H., KAY, J. E., ARMOUR, K. C., ..., and VANCOPPENOLLE, M. (2018). Quantifying Climate Feedbacks in Polar Regions. *Nature Communications*, 9(1), 1919, 13 pages.

GORDON, H. S. (1954). The Economic Theory of a Common-Property Resource: The Fishery. *Journal of Political Economy*, 62(2), pp. 124-142.

GOTELLI, N. J., and COLWELL, R. K. (2001). Quantifying Biodiversity: Procedures and Pitfalls in the Measurement and Comparison of Species Richness. *Ecology Letters*, 4(4), pp. 379-391.

GOULDER, L. H. (1995). Environmental Taxation and the Double Dividend: A Reader's Guide. *International Tax and Public Finance*, 2, pp. 157-183.

GOULDER, L. H., and PARRY, I. W. H. (2008). Instrument Choice in Environmental Policy. *Review of Environmental Economics and Policy*, 2(2), pp. 152-174.

GOURIÉROUX, C. (1992). Courbes de Performance, de Sélection et de Discrimination. *Annales d'Économie et de Statistique*, 28(4), pp. 107-123.

GOURIÉROUX, C., and JASIAK, J. (2007). *The Econometrics of Individual Risk: Credit, Insurance, and Marketing*. Princeton University Press.

GOUTELLE, S., MAURIN, M., ROUGIER, F., ..., and MAIRE, P. (2008). The Hill Equation: A Review of Its Capabilities in Pharmacological Modelling. *Fundamental & Clinical Pharmacology*, 22(6), pp. 633-648.
GRAFF ZIVIN, J., and NEIDELL, M. (2013). Environment, Health, and Human Capital. *Journal of Economic Literature*, 51(3), pp. 689-730.

GRATCHEVA, E. M., EMERY, T., and WANG, D. (2020). Demystifying Sovereign ESG. *Report*, World Bank, Equitable Growth, Finance and Institutions Insight, December.

GRAY, A. (2019). The Ecology of Plant Extinction: Rates, Traits and Island Comparisons. *Oryx*, 53(3), pp. 424-428.

GREGORY, A., THARYAN, R., and WHITTAKER, J. (2014). Corporate Social Responsibility and Firm Value: Disaggregating the Effects on Cash Flow, Risk and Growth. *Journal of Business Ethics*, 124(4), pp. 633-657.

GREEN, J. L., and OSTLING, A. (2003). Endemics-Area Relationships: The Influence of Species Dominance and Spatial Aggregation. *Ecology*, 84(11), pp. 3090-3097.

GREWAL, J., SERAFEIM, G., and YOON, A. (2016). Shareholder Activism on Sustainability Issues. *SSRN*, 2805512.

GROSSMAN, B. R., and SHARPE, W. F. (1986). Financial Implications of South African Divestment. *Financial Analysts Journal*, 42(4), pp. 15-29.

GRUBB, M., WIENERS, C., and YANG, P. (2021). Modeling Myths: On DICE and Dynamic Realism in Integrated Assessment Models of Climate Change Mitigation. *WIREs Climate Change*, 12(3), e698.

GRUBER, J. (2005). *Public Finance and Public Policy*. Third edition published in 2011, Worth Publishers, 860 pages.

GUPTA, J., and VEGELIN, C. (2016). Sustainable Development Goals and Inclusive Development. *International Environmental Agreements: Politics, Law and Economics*, 16, pp. 433-448.

GUPTA, K. (2018). Environmental Sustainability and Implied Cost of Equity: International Evidence. *Journal of Business Ethics*, 147(2), pp. 343-365.

GUTIERREZ, M. S. (2008). *Economic Activity and Atmospheric Pollution in Spain: An Input-output Approach*. PhD Thesis, Universitat de Barcelona.

HADDAD, N. M., BRUDVIG, L. A., CLOBERT, J., ..., and TOWNSHEND, J. R. (2015). Habitat Fragmentation and its Lasting Impact on Earth's Ecosystems. *Science Advances*, 1(2), e1500052, 9 pages.

HALLEY, J. M., MONOKROUSOS, N., MAZARIS, A. D., NEWMARK, W. D., and VOKOU, D. (2016a). Dynamics of Extinction Debt Across Five Taxonomic Groups. *Nature Communications*, 7(1), 12283, 6 pages.

HALLEY, J. M., MONOKROUSOS, N., MAZARIS, A. D., NEWMARK, W. D., and VOKOU, D. (2016b). Supplementary Information. *Nature Communications*, 7(1), 12283, 34 pages.

HALLMANN, C. A., SORG, M., JONGEJANS, E., ..., and DE KROON, H. (2017). More than 75 Percent Decline over 27 Years in Total Flying Insect Biomass in Protected Areas. *PloS One*, 12(10), e0185809, 21 pages.
HAMMITT, J. K. (2023). Consistent Valuation of a Reduction in Mortality Risk using Values per Life, Life Year, and Quality-Adjusted Life Year. *Health Economics*, 32(9), pp. 1964-1981.

HAMMITT, J. K., and ROBINSON, L. A. (2011). The Income Elasticity of the Value per Statistical Life: Transferring Estimates Between High and Low Income Populations. *Journal of Benefit-Cost Analysis*, 2(1), 28 pages.

HAND, D., SUNDERJI, S., NOVA, N. and DE, I. (2021). Impact Investing Decision-making: Insights on Financial Performance. *GIIN Report*, January, 55 pages.

HAND, D., RINGEL, B., and DANEL, A. (2022). Sizing the Impact Investing Market: 2022. *GIIN Report*, October, 17 pages.

HANEMANN, W. M. (1991). Willingness to Pay and Willingness to Accept: How Much Can They Differ?. *American Economic Review*, 81(3), pp. 635-647.

HANEMANN, W. M. (2008). What is the Economic Cost of Climate Change?. *Working Paper*, University of California, Berkeley, 1071.

HÄNSEL, M. C., DRUPP, M. A., JOHANSSON, D. J., ..., and STERNER, T. (2020). Climate Economics Support for the UN Climate Targets. *Nature Climate Change*, 10(8), pp. 781-789.

HANSEN, J. E., LACIS, A., RIND, D., RUSSELL, G., STONE, P., FUNG, I., RUEDY, R., and LERNER, J. (1984). Climate Sensitivity: Analysis of Feedback Mechanisms. In Hansen, J. E., and Takahashi, T. (Eds), *Climate Processes and Climate Sensitivity*, 29, pp. 130-163.

HANSEN, J. E., SATO, M., RUSSELL, G., and KHARECHA, P. (2013a). Climate Sensitivity, Sea Level and Atmospheric Carbon Dioxide. *Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences*, 371(2001), 20120294, 31 pages.

HANSEN, M. C., POTAPOV, P. V., MOORE, R., ..., and TOWNSHEND, J. R. (2013b). High-Resolution Global Maps of 21st-century Forest Cover Change. *Science*, 342(6160), pp. 850-853.

HANSEN, T. A. (2022). Stranded Assets and Reduced Profits: Analyzing the Economic Underpinnings of the Fossil Fuel Industry's Resistance to Climate Stabilization. *Renewable and Sustainable Energy Reviews*, 158, 112144, 14 pages.

HANSKI, I., and OVASKAINEN, O. (2002). Extinction Debt at Extinction Threshold. *Conservation Biology*, 16(3), pp. 666-673.

HANSKI, I., and OVASKAINEN, O. (2003). Metapopulation Theory for Fragmented Landscapes. *Theoretical Population Biology*, 64(1), pp. 119-127.

HANSKI, I., ZURITA, G. A., BELLOCQ, M. I., and RYBICKI, J. (2013). Species-fragmented Area Relationship. *Proceedings of the National Academy of Sciences*, 110(31), pp. 12715-12720.

HARDIN, G. (1968). The Tragedy of the Commons. *Science*, 162(3859), pp. 1243-1248.

HARPANKAR, K. (2019). Internal Carbon Pricing: Rationale, Promise and Limitations. *Carbon Management*, 10(2), pp. 219-225.

HARTIGAN Jr, J. (2017). *Care of the Species: Races of Corn and the Science of Plant Biodiversity*. University of Minnesota Press, 376 pages.
HARTLEY, L., ALEXANDER, J., and HOLDEN, P. (2025). *Stranding: Modelling the UK's Exposure to At-Risk Fossil Fuel Assets*. UKSIF Special Report, 14 pages.

HARTMANN, D. L. (2016). *Global Physical Climatology*. Second edition, Elsevier Science.

HARVEY, A. C. (1990). *Forecasting, Structural Time Series Models and the Kalman Filter*. Cambridge University Press.

HASSELMANN, K. (1976). *Stochastic Climate Models, Part I. Theory*. *Tellus*, 28(6), pp. 473-485.

HASTIE, T., TIBSHIRANI, R., and FRIEDMAN, J.H. (2009). *The Elements of Statistical Learning*. Second edition, Springer.

HAVLÍK, P., VALIN, H., MOSNIER, A., ..., and HASEGAWA, T. (2018). *GLOBIOM Documentation*. International Institute for Applied Systems Analysis.

HAYS, J. D., IMBRIE, J., and SHACKLETON, N. J. (1976). *Variations in the Earth's Orbit: Pacemaker of the Ice Ages*. *Science*, 194(4270), pp. 1121-1132.

HE, F., and HUBBELL, S. P. (2011). *Species-Area Relationships Always Overestimate Extinction Rates from Habitat Loss*. *Nature*, 473(7347), pp. 368-371.

HE, F., and LEGENDRE, P. (2002). *Species Diversity Patterns Derived from Species-Area Models*. *Ecology*, 83(5), pp. 1185-1198.

HEEB, F., KÖLBEL, J. F., PAETZOLD, F., and ZEISBERGER, S. (2023). *Do Investors Care About Impact?*. *Review of Financial Studies*, 36(5), pp. 1737-1787.

HEIN, L., BAGSTAD, K. J., OBST, C., ..., and CAPARRÓS, A. (2020). *Progress in Natural Capital Accounting for Ecosystems*. *Science*, 367(6477), pp. 514-515.

HELD, I. M., and SODEN, B. J. (2006). *Robust Responses of the Hydrological Cycle to Global Warming*. *Journal of Climate*, 19(21), pp. 5686-5699.

HENDREN, N., and SPRUNG-KEYSER, B. (2020). *A Unified Welfare Analysis of Government Policies*. *Quarterly Journal of Economics*, 135(3), pp. 1209-1318.

HENRY, P. B., and MILLER, C. (2009). *Institutions Versus Policies: A Tale of Two Islands*. *American Economic Review*, 99(2), pp. 261-267.

HEPBURN, C. (2006). *Regulation by Prices, Quantities, or Both: A Review of Instrument Choice*. *Oxford Review of Economic Policy*, 22(2), pp. 226-247.

HERAS-SAIZARBITORIA, I., URBIETA, L., and BOIRAL, O. (2022). *Organizations' Engagement with Sustainable Development Goals: From Cherry-picking to SDG-washing?*. *Corporate Social Responsibility and Environmental Management*, 29(2), pp. 316-328.

HERSBACH, H., BELL, B., BERRISFORD, P., ..., and THÉPAUT, J. N. (2020). *The ERA5 Global Reanalysis*. *Quarterly Journal of the Royal Meteorological Society*, 146(730), pp. 1999-2049.

HEYWOOD, V. H. (1995). *The Global Biodiversity Assessment*. UNEP & Cambridge University Press, 1140 pages.

HICKS, J. R. (1939). *The Foundations of Welfare Economics*. *Economic Journal*, 49(196), pp. 696-712.
High-Level Expert Group on Sustainable Finance (2018). Financing a Sustainable European Economy. *Final Report*, January.

HILDERINK, M. H., and DE WINTER, I. I. (2021). No Need to Beat Around the Bushmeat — The Role of Wildlife Trade and Conservation Initiatives in the Emergence of Zoonotic Diseases. *Heliyon*, 7, e07692, 10 pages.

HINKLEY, D. V. (1969). On the Ratio of Two Correlated Normal Random Variables. *Biometrika*, 56(3), pp. 635-639.

HIRSCHMAN, A. O. (1970). *Exit, Voice, and Loyalty: Responses to Decline in Firms, Organizations, and States*. Harvard University Press.

HJORT, J., STRELETSKIY, D., DORÉ, G., WU, Q., BJELLA, K., and LUOTO, M. (2022). Impacts of Permafrost Degradation on Infrastructure. *Nature Reviews Earth & Environment*, 3(1), pp. 24-38.

HOEGH-GULDBERG, O. I., BEAL, D., CHAUDHRY, T., ..., and SMITS, M. (2015). Reviving the Ocean Economy: The Case for Action. *WWF Report*, April, 60 pages.

HOEL, M., and KARP, L. (2002). Taxes versus Quotas for a Stock Pollutant. *Resource and Energy Economics*, 24(4), pp. 367-384.

HOEL, M., and KVERNDOKK, S. (1996). Depletion of Fossil Fuels and the Impacts of Global Warming. *Resource and Energy Economics*, 18(2), pp. 115-136.

HOFFMAN, P. F., KAUFMAN, A. J., HALVERSON, G. P., and SCHRAG, D. P. (1998). A Neoproterozoic Snowball Earth. *Science*, 281(5381), pp. 1342-1346.

HOFFMAN, P. F., and SCHRAG, D. P. (2002). The Snowball Earth Hypothesis: Testing the Limits of Global Change. *Terra Nova*, 14(3), pp. 129-155.

HÖLKER, F., WOLTER, C., PERKIN, E. K., and TOCKNER, K. (2010). Light Pollution as a Biodiversity Threat. *Trends in Ecology & Evolution*, 25(12), pp. 681-682.

HOLLING, C. S. (1959a). The Components of Predation as Revealed by a Study of Small-mammal Predation of the European Pine Sawfly. *Canadian Entomologist*, 91(5), pp. 293-320.

HOLLING, C. S. (1959b). Some Characteristics of Simple Types of Predation and Parasitism. *Canadian Entomologist*, 91(7), pp. 385-398.

HONG, H., and KACPERCZYK, M. (2009). The Price of Sin: The Effects of Social Norms on Markets. *Journal of Financial Economics*, 93(1), pp. 15-36.

HOPE, C. (2011). The PAGE09 Integrated Assessment Model: A Technical Description. *Cambridge Judge Business School Working Paper*, 4.

HÖCHSTÄDTER, A. K., and SCHECK, B. (2015). What's in a Name: An Analysis of Impact Investing Understandings by Academics and Practitioners. *Journal of Business Ethics*, 132, pp. 449-475.

HOWARD, P. H., and STERNER, T. (2017). Few and not so Far Between: A Meta-analysis of Climate Damage Estimates. *Environmental and Resource Economics*, 68(1), pp. 197-225.

HÜBEL, B. (2022). Do Markets Value ESG Risks in Sovereign Credit Curves?. *Quarterly Review of Economics and Finance*, 85, pp. 134-148.
HUMPHREY, J. E., LEE, D. D., and SHEN, Y. (2012). Does it Cost to be Sustainable?. *Journal of Corporate Finance*, 18(3), pp. 626-639.

HUPPMANN, D., GIDDEN, M., FRICKO, ..., and KREY, V. (2019). The MESSAGEix Integrated Assessment Model and the ix Modeling Platform (ixmp): An Open Framework for Integrated and Cross-cutting Analysis of Energy, Climate, the Environment, and Sustainable Development. *Environmental Modelling & Software*, pp. 112, 143-156.

HURLBERT, S. H. (1971). The Nonconcept of Species Diversity: A Critique and Alternative Parameters. *Ecology*, 52(4), pp. 577-586.

HUTCHINSON, G. E. (1961). The Paradox of the Plankton. *American Naturalist*, 95(882), pp. 137-145.

HUTCHINSON, M. C., and LUCEY, B. (2024). A Bibliometric and Systemic Literature Review of Biodiversity Finance. *Finance Research Letters*, 105377, 7 pages.

Institute for Climate Economics (2025). Global Carbon Accounts 2025: Carbon Pricing Instruments and the Unlocked Potential of Carbon Revenues. *Report*, June, 15 pages.

Institute for Global Environmental Strategies (IGES) (2022). *Nationally Determined Contributions (NDC) Database Version 7.7*. Source: https://pub.iges.or.jp/pub/iges-ndc-database.

Interagency Working Group on Social Cost of Greenhouse Gases (2021). *Technical Support Document: Social Cost of Carbon, Methane, and Nitrous Oxide, Interim Estimates under Executive Order 13990*. United States Government, 2021.

Intergovernmental Panel on Climate Change (1999). Procedures for the Preparation, Review, Acceptance, Adoption, Approval and Publication of IPCC Reports. *Appendix A to the Principles Governing IPCC Work*, 29 pages.

Intergovernmental Panel on Climate Change (2006). 2006 IPCC Guidelines for National Greenhouse Gas Inventories. Prepared by the National Greenhouse Gas Inventories Programme, Eggleston H. S., Buendia L., Miwa K., Ngara T. and Tanabe K. (Eds).

Intergovernmental Panel on Climate Change (2007). Climate Change 2007: The Physical Science Basis — Contribution of Working Group I to the Fourth Assessment Report of the IPCC. *Report*.

Intergovernmental Panel on Climate Change (2013). Climate Change 2013: The Physical Science Basis — Contribution of Working Group I to the Fifth Assessment Report of the IPCC. *Report*.

Intergovernmental Panel on Climate Change (2014a). Climate Change 2014: Mitigation of Climate Change — Contribution of Working Group III to the Fifth Assessment Report of the IPCC. *Report*.

Intergovernmental Panel on Climate Change (2014b). Climate Change 2014: Synthesis Report — Contribution of Working Groups I, II and III to the Fifth Assessment Report of the IPCC. *Report*, 151 pages.

Intergovernmental Panel on Climate Change (2018). Global Warming of 1.5°C. *Special Report*, 616 pages.
Intergovernmental Panel on Climate Change (2019). *2019 Refinement to the 2006 IPCC Guidelines for National Greenhouse Gas Inventories*. Prepared by Calvo Buendia, E., Tanabe, K., Kranjc, A., Baasansuren, J., Fukuda, M., Ngarize, S., Osako, A., Pyrozhenko, Y., Shermanau, P. and Federici, S. (Eds).

Intergovernmental Panel on Climate Change (2021). *Climate Change 2021: The Physical Science Basis — Contribution of Working Group I to the Sixth Assessment Report of the IPCC*. *Report*.

Intergovernmental Panel on Climate Change (2022). *Climate Change 2022: Mitigation of Climate Change — Contribution of Working Group III to the Sixth Assessment Report of the IPCC*. *Report*.

Intergovernmental Science-Policy Platform on Biodiversity and Ecosystem Services (2016). *The Assessment Report of the Intergovernmental Science-Policy Platform on Biodiversity and Ecosystem Services on Pollinators, Pollination and Food Production*. IPBES secretariat, prepared by Potts, S.G., Imperatriz-Fonseca, V. L., and Ngo H. T. (Eds), 552 pages.

Intergovernmental Science-Policy Platform on Biodiversity and Ecosystem Services (2019). *Global Assessment Report on Biodiversity and Ecosystem Services of the Intergovernmental Science-Policy Platform on Biodiversity and Ecosystem Services*. IPBES secretariat, prepared by Brondízio, E. S., Settele, J., Díaz, S., and Ngo, H. T. (Eds), 1144 pages.

Intergovernmental Science-Policy Platform on Biodiversity and Ecosystem Services (2019). *Thematic Assessment Report on Invasive Alien Species and their Control*. IPBES secretariat, prepared by Roy, H. E., Pauchard, A., Stoett, P., and Renard Truong, T. (Eds.), 952 pages.

International Association of Insurance Supervisors (2021). *Supervision of Climate-related Risks in the Insurance Sector*. *Application Paper*, May, 37 pages.

International Association of Insurance Supervisors (2023). *Public Consultation on Climate Risk Supervisory Guidance — Part One*. *Consultation Document*, March, 12 pages.

International Capital Market Association (2021a). *Green Bond Principles — Voluntary Process Guidelines for Issuing Green Bonds*. *Guidelines*, June.

International Capital Market Association (2021b). *Social Bond Principles — Voluntary Process Guidelines for Issuing Social Bonds*. *Guidelines*, June.

International Capital Market Association (2022). *Green, Social and Sustainability Bonds: A High-Level Mapping to the Sustainable Development Goals*. *Report*, June.

International Capital Market Association (2023). *Bonds to Finance the Sustainable Blue Economy*. *Report*, September, 33 pages.

International Carbon Action Partnership (2023). *Emissions Trading Worldwide: 2023 ICAP Status Report*. *Status Report*, March, 218 pages.

International Energy Agency (2017). *Energy Technology Perspectives 2017: Catalysing Energy Technology Transformations*. *Flagship Report*, June, 441 pages.

International Energy Agency (2021). *Net Zero by 2050: A Roadmap for the Global Energy Sector*. *Report*, July, 223 pages.
International Energy Agency (2022a). The Role of Critical Minerals in Clean Energy Transitions. *Report*, March, 287 pages.

International Energy Agency (2022b). Direct Air Capture: A Key Technology for Net Zero. *Report*, April, 76 pages.

International Energy Agency (2023a). Tracking Clean Energy Progress 2023. *Report*, July.

International Energy Agency (2023b). Net Zero Roadmap: A Global Pathway to Keep the 1.5°C Goal in Reach — 2023 Update. *Report*, September, 226 pages (Revised version, November 2024).

International Energy Agency (2024a). Renewables 2024 — Analysis and Forecast to 2030. *Report*, October, 177 pages.

International Energy Agency (2024b). Renewable Energy Progress Tracker. *Report*, October.

International Energy Agency (2025). World Energy Outlook 2025. *Report*, October, ??? pages.

International Finance Corporation (2020). Green Bond Handbook: A Step-by-step Guide to Issuing a Green Bond. *Guidelines*.

International Finance Corporation (2022). Blue Finance — Guidance for Financing the Blue Economy, Building on the Green Bond Principles and the Green Loan Principles. *Guidelines*, January.

International Organization of Securities Commissions (2021). Environmental, Social and Governance (ESG) Ratings and Data Products Providers. *Report*, 23 November 2021, 56 pages.

International Organization of Securities Commissions (2023). Supervisory Practices to Address Greenwashing. *Report*, 4 December 2023, 94 pages.

International Sustainability Standards Board (2022a). IFRS S1 General Requirements for Disclosure of Sustainability-related Financial Information. *Exposure Draft*, March.

International Sustainability Standards Board (2022b). IFRS S2 Climate-related Disclosures. *Exposure Draft*, March.

ISRAEL, R. B., ROSENTHAL, J. S., and WEI, J. Z. (2001). Finding Generators for Markov Chains via Empirical Transition Matrices, with Applications to Credit Ratings. *Mathematical Finance*, 11(2), pp. 245-265.

ISS Governance (2022). Climate & Voting: 2021 Review and Global Trends. *Insights Report*, May.

IUCN (2012). *IUCN Red List Categories and Criteria*. Version 3.1, second edition, 32 pages.

IUCN (2022). *The IUCN Red List of Threatened Species*. Version 2022-1, www.iucnredlist.org.

IUCN (2024a). *The IUCN Red List of Threatened Species*. Version 2024-1, www.iucnredlist.org.

IUCN (2024b). *Summary Statistics*. Version 2024-1, www.iucnredlist.org.

IZZO, M. F., and MAGNANELLI, B. S. (2012). Does it Pay or Does Firm Pay? The relation Between CSR Performance and the Cost of Debt. *SSRN*, 1986131.

JACKSON, E. T. (2013). Interrogating the Theory of Change: Evaluating Impact Investing where It Matters Most. *Journal of Sustainable Finance & Investment*, 3(2), pp. 95-110.
JACKSON, J. B., KIRBY, M. X., BERGER, W. H., ..., and WARNER, R. R. (2001). Historical Overfishing and the Recent Collapse of Coastal Ecosystems. *Science*, 293(5530), pp. 629-637.

JAEGER, W. K. (2012). The Double Dividend Debate. In Milne, J. E., and Andersen, M. S. (Eds.), *Handbook of Research on Environmental Taxation*, Edward Elgar Publishing, Chapter 12, pp. 211-229.

JAHNKE, P. (2019). Ownership Concentration and Institutional Investors' Governance through Voice and Exit. *Business and Politics*, 21(3), pp. 327-350.

JARROW, R.A., LANDO, D., and TURNBULL, S.M. (1997). A Markov Model for the Term Structure of Credit Risk Spreads. *Review of Financial Studies*, 10(2), pp. 481-523.

JAULIN, T., LE BERTHE, T., DEIXONNE, F., and DE Vecchi, V. (2022). EU Sustainable Finance Action Plan: State of Play. *Amundi ESG Thema*, 11, 13 pages.

JEEVANJEE, N. (2023). Climate Sensitivity from Radiative-Convective Equilibrium: A Chalkboard Approach. *American Journal of Physics*, 91(9), pp. 731-745.

JENSEN, M. C. (1968). The Performance of Mutual Funds in the Period 1945-1964. *Journal of Finance*, 23(2), pp. 389-416.

JENSEN, M. C. (1993). The Modern Industrial Revolution, Exit, and the Failure of Internal Control Systems. *Journal of Finance*, 48(3), pp. 831-880.

JENSEN, M. C., and MECKLING, W. H. (1976). Theory of the Firm: Managerial Behavior, Agency Costs and Ownership Structure. *Journal of Financial Economics*, 3(4), pp. 305-360.

JEWELL, J., and CHERP, A. (2020). On the Political Feasibility of Climate Change Mitigation Pathways: Is it Too Late to Keep Warming Below 1.5°C?. *WIREs Climate Change*, 11(1), e621, 12 pages.

JHA, A., and COX, J. (2015). Corporate Social Responsibility and Social Capital. *Journal of Banking & Finance*, 60, pp. 252-270.

JOE, H. (1997). *Multivariate Models and Dependence Concepts*. Monographs on Statistics and Applied Probability, 73, Chapman & Hall.

JOHANSEN, D. F., and VESTVIK, R. A. (2020). The Cost of Saving our Ocean — Estimating the Funding Gap of Sustainable Development Goal 14. *Marine Policy*, 112, 103783, 12 pages.

JOHNSON, S. A., MOORMAN, T. C., and SORESCU, S. (2009). A Reexamination of Corporate Governance and Equity Prices. *Review of Financial Studies*, 22(11), pp. 4753-4786.

JOLLES, A. E. (2007). Population Biology of African Buffalo (*Syncerus caffer*) at Hluhluwe-iMfolozi Park, South Africa. *African Journal of Ecology*, 45(3), pp. 398-406.

JONDEAU, E., MOJON, B., and PEREIRA DA SILVA, L. A. (2021). Building Benchmark Portfolios with Decreasing Carbon Footprints. *Swiss Finance Institute Research Paper*, 21-91.

JONES, K. C., and DE VOOGT, P. (1999). Persistent Organic Pollutants (POPs): State of the Science. *Environmental Pollution*, 100(1-3), pp. 209-221.
JONES, M. W., PETERS, G. P., GASSER, T., ..., and LE QUÉRÉ, C. (2023). National Contributions to Climate Change due to Historical Emissions of Carbon Dioxide, Methane, and Nitrous Oxide since 1850. *Scientific Data*, 10(1), 23 pages.

JONES, T. M. (1980). Corporate Social Responsibility Revisited, Redefined. *California Management Review*, 22(3), pp. 59-67.

JOOS, F., ROTH, R., FUGLESTVEDT, J. S., ..., and WEAVER, A. J. (2013). Carbon Dioxide and Climate Impulse Response Functions for the Computation of Greenhouse Gas Metrics: A Multi-model Analysis. *Atmospheric Chemistry and Physics*, 13(5), pp. 2793-2825.

JOPPA, L. N., LUERS, A., WILLMOTT, E., FRIEDMANN, S. J., HAMBURG, S. P., and BROZE, R. (2021). Microsoft's Million-Tonne CO₂-Removal Purchase — Lessons for Net Zero. *Nature*, 597(7878), pp. 629-632.

JORION, P. (2003). Portfolio Optimization with Tracking-Error Constraints. *Financial Analysts Journal*, 59(5), pp. 70-82.

JOUZEL, J. (2013). A Brief History of Ice Core Science over the last 50 yr. *Climate of the Past*, 9(6), pp. 2525-2547.

JOUZEL, J., MASSON-DELMOTTE, V., CATTANI, O., ..., and WOLFF, E. W. (2007). Orbital and Millennial Antarctic Climate Variability over the Past 800,000 Years. *Science*, 317(5839), pp. 793-796.

KALDOR, N. (1939). Welfare Propositions of Economics and Interpersonal Comparisons of Utility. *Economic Journal*, 49(195), pp. 549-552.

KAPLAN, R. S., RAMANNA, K., and ROSTON, M. (2023). Accounting for Carbon Offsets — Establishing the Foundation for Carbon-trading Markets. *Harvard Business School Working Paper*, 23-050, 16 pages.

KAPRAUN, J., LATINO, C., SCHEINS, C., and SCHLAG, C. (2021). (In)-credibly Green: Which Bonds Trade at a Green Bond Premium?. *SSRN*, 3347337.

KAR, T. K., and GHOSH, B. (2013). Impacts of Maximum Sustainable Yield Policy to Prey-predator Systems. *Ecological Modelling*, 250, pp. 134-142.

KAUFMANN, D., KRAAY, A., and MASTRUZZI, M. (2010). The Worldwide Governance Indicators: Methodology and Analytical Issues. *World Bank Policy Research Working Paper*, 5430 (SSRN, 1682130).

KAVVATHAS, D. (2001). Estimating Credit Rating Transition Probabilities for Corporate Bonds. *SSRN*, 248421.

KAYA, Y., and YOKOBORI, K. (Eds) (1997). *Environment, Energy, and Economy: Strategies for Sustainability*. United Nations University Press, 393 pages.

KEELING, C. D. (1960). The Concentration and Isotopic Abundances of Carbon Dioxide in the Atmosphere. *Tellus*, 12(2), pp. 200-203.

KEELING, C. D., BACASTOW, R. B., BAINBRIDGE, A. E., EKDAHL Jr, C. A., GUENTHER, P. R., WATERMAN, L. S., and CHIN, J. F. S. (1976). Atmospheric Carbon Dioxide Variations at Mauna Loa Observatory, Hawaii. *Tellus*, 28(6), pp. 538-551.
KEELING, C. D. (1978). The Influence of Mauna Loa Observatory on the Development of Atmospheric CO₂ Research. In Miller, J (Ed.), *Mauna Loa Observatory: A 20th Anniversary Report*, National Oceanic and Atmospheric Administration Special Report, pp. 36-54.

KEELING, C. D., PIPER, S. C., BACASTOW, R. B., WAHLEN, M., WHORF, T. P., HEIMANN, M., and MEIJER, H. A. (2001). Exchanges of Atmospheric CO₂ and ¹³CO₂ with the Terrestrial Biosphere and Oceans from 1978 to 2000. *I. Global aspects. Scripps Institution of Oceanography, San Diego, SIO Reference Series*, 01-06, 28 pages.

KELLETT, C. M., FAULWASSER, T., and WELLER, S. R. (2016). *DICE-2013R-mc* — *A Mat-lab/CasADi Implementation of Vanilla DICE-2013R*. Code source available at https://github.com/cmkellett/DICE2013R-mc.

KELLETT, C. M., WELLER, S. R., FAULWASSER, T., GRÜNE, L., and SEMMLER, W. (2019). Feedback, Dynamics, and Optimal Control in Climate Economics. *Annual Reviews in Control*, 47, pp. 7-20.

KELLY, D. L., and KOLSTAD, C. D. (1999). Integrated Assessment Models for Climate Change Control. In Folmer, H., and Tietenberg, T. (Eds), *International Yearbook of Environmental and Resource Economics 1999/2000*, Edward Elgar, pp. 171-197.

KEMP, L., XU, C., DEPLEDGE, J., ..., and LENTON, T. M. (2022). Climate Endgame: Exploring Catastrophic Climate Change Scenarios. *Proceedings of the National Academy of Sciences*, 119(34), e2108146119, 9 pages.

KHALED, R., ALI, H., and MOHAMED, E. K. (2021). The Sustainable Development Goals and Corporate Sustainability Performance: Mapping, Extent and Determinants. *Journal of Cleaner Production*, 311, 127599.

KHANCHEL, I., and LASSOUED, N. (2022). ESG Disclosure and the Cost of Capital: Is there a Ratcheting Effect over Time?. *Sustainability*, 14(15), 9237, 19 pages.

KIEHL, J. T., and TRENBERTH, K. E. (1997). Earth's Annual Global Mean Energy Budget. *Bulletin of the American Meteorological Society*, 78(2), pp. 197-208.

KITCHEN, A. M., GESE, E. M., and SCHAUSTER, E. R. (1999). Resource Partitioning Between Coyotes and Swift Foxes: Space, Time, and Diet. *Canadian Journal of Zoology*, 77(10), pp. 1645-1656.

KITZES, J. (2013). An Introduction to Environmentally-extended Input-output Analysis. *Resources*, 2(4), pp. 489-503.

KLEIN, A. M., VAISSIÈRE, B. E., CANE, J. H., ..., and TSCHARNTKE, T. (2007). Importance of Pollinators in Changing Landscapes for World Crops. *Proceedings of the Royal Society B: Biological Sciences*, 274(1608), pp. 303-313.

KLEINBERG, R. L. (2020). The Global Warming Potential Misrepresents the Physics of Global Warming Thereby Misleading Policy Makers. *EarthArXiv*, November.

KLOSE, A. K., KARLE, V., WINKELMANN, R., and DONGES, J. F. (2020). Emergence of Cascading Dynamics in Interacting Tipping Elements of Ecology and Climate. *Royal Society Open Science*, 7(6), 200599, 18 pages.
KOEHLER, D. A., and BERTOCCI, B. (2016). *Stranded Assets: What Lies Beneath*. UBS Asset Management, 19 pages.

KOFOWOROLA, O., DODD, N., BOYANO, A., KONSTANTAS, A., WOLF, O., ALESSI, L., and OSSOLA, E. (2019). Development of EU Ecolabel Criteria for Retail Financial Products — Technical Report 2.0. *Joint Research Centre Technical Report*, December, 119 pages.

KÖLBEL, J. F., LEIPPOLD, M., RILLAERTS, J., and WANG, Q. (2022). Ask BERT: How Regulatory Disclosure of Transition and Physical Climate Risks Affects the CDS Term Structure. *Swiss Finance Institute Research Paper*, 21-19.

KOLBERT, E. (2014). *The Sixth Extinction: An Unnatural History*. Henry Holt and Company, 319 pages.

KÖPPL, A., and SCHRATZENSTALLER, M. (2023). Carbon Taxation: A Review of the Empirical Literature. *Journal of Economic Surveys*, 37(4), pp. 1353-1388.

KOSTANT, P. C. (1999). Exit, Voice and Loyalty in the Course of Corporate Governance and Counsel's Changing Role. *Journal of Socio-Economics*, 28(3), pp. 203-246.

KONSTANTAS, A., FARACA, G. DODD, N., KOFOWOROLA, O., BOYANO, A., WOLF, O., ALESSI, L., and OSSOLA, E. (2019). Development of EU Ecolabel Criteria for Retail Financial Products — Technical Report 4.0. *Joint Research Centre Technical Report*, March, 141 pages.

KREMEN, C. (2005). Managing Ecosystem Services: What Do We Need to Know about Their Ecology? *Ecology letters*, 8(5), pp. 468-479.

KRUEGER, P., SAUTNER, Z., and STARKS, L. T. (2020). The Importance of Climate Risks for Institutional Investors. *Review of Financial Studies*, 33(3), pp. 1067-1111.

KRUSE, H., KIRKEMO, A. M., and HANDELAND, K. (2004). Wildlife as Source of Zoonotic Infections. *Emerging Infectious Diseases*, 10(12), pp. 2067-2072.

KUIPERS, K. J. J., MAY, R., and VERONES, F. (2021). Considering Habitat Conversion and Fragmentation in Characterisation Factors for Land-use Impacts on Vertebrate Species Richness. *Science of the Total Environment*, 801, 149737, 10 pages.

KURTZ, L. (1999). Investment in Minority-Owned Media: A Social Investor's Perspective. *Federal Communications Law Journal*, 51(3), pp. 681-691.

KUUSSAARI, M., BOMMARCO, R., HEIKKINEN, R. K., ..., and STEFFAN-DEWENTER, I. (2009). Extinction Debt: A Challenge for Biodiversity Conservation. *Trends in Ecology & Evolution*, 24(10), pp. 564-571.

KVENVOLDEN, K. A. (1988). Methane Hydrates and Global Climate. *Global Biogeochemical Cycles*, 2(3), pp. 221-229.

LAFORTUNE, G., FULLER, G., MORENO, J., SCHMIDT-TRAUB, G., and KROLL, C. (2018). SDG Index and Dashboards: Detailed Methodological Paper. *Supplementary Materials*, September, 56 pages.

LAMB, W. F., WIEDMANN, T., PONGRATZ, J., ..., and MINX, J. (2021). A Review of Trends and Drivers of Greenhouse Gas Emissions by Sector from 1990 to 2018. *Environmental Research Letters*, 16(7), 073005.
LANDRIGAN, P. J., FULLER, R., ACOSTA, N. J. R., ..., and ZHONG, M. (2018). The *Lancet* Commission on Pollution and Health. *The Lancet*, 391(10119), pp. 462-512.

LANGWAY, C. C. (2008). The History of Early Polar Ice Cores. *Cold Regions Science and Technology*, 52(2), pp. 101-117.

LAPOLA, D. M., PINHO, P., BARLOW, J., ..., and WALKER, W. S. (2023). The Drivers and Impacts of Amazon Forest Degradation. *Science*, 379(6630), eabp8622, 11 pages.

LARCKER, D. F., TAYAN, B., and COPLAND, J. R. (2018). The Big Thumb on the Scale: An Overview of the Proxy Advisory Industry. *Harvard Law School Forum on Corporate Governance*, 9 pages, 14 June 2018.

LARCKER, D. F., and WATTS, E. M. (2020). Where's the Greenium?. *Journal of Accounting and Economics*, 69(2-3), 101312.

LARSEN, B. B., MILLER, E. C., RHODES, M. K., and WIENS, J. J. (2017). Inordinate Fondness Multiplied and Redistributed: The Number of Species on Earth and the New Pie of Life. *Quarterly Review of Biology*, 92(3), pp. 229-265.

LE GUENEDAL, T. (2019). Economic Modeling of Climate Risks. SSRN, 3693661.

LE GUENEDAL, T., DROBINSKI, P., and TANKOV, P. (2021). Measuring and Pricing Cyclone-Related Physical Risk Under Changing Climate. SSRN, 3850673.

LE GUENEDAL, T., DROBINSKI, P., and TANKOV, P. (2022). Cyclone Generation Algorithm Including a Thermodynamic Module for Integrated National Damage Assessment (CATHERINA 1.0) Compatible with Coupled Model Intercomparison Project (CMIP) Climate Data. *Geoscientific Model Development*, 15(21), pp. 8001-8039.

LE GUENEDAL, T., LOMBARD, F., RONCALLI, T., and SEKINE, T. (2022). Net Zero Carbon Metrics. SSRN, 4033686.

LE GUENEDAL, T., and RONCALLI, T. (2022). Portfolio Construction with Climate Risk Measures. In Jurchenko, E. (Ed.), *Climate Investing: New Strategies and Implementation Challenges*, Wiley, December 2022, pp. 49-86.

LEAKEY, R. E., and LEWIN, R. (1995). *The Sixth Extinction: Patterns of Life and the Future of Humankind*. Doubleday, 271 pages.

LEBRETON, L., SLAT, B., FERRARI, F., ..., and REISSER, J. (2018). Evidence that the Great Pacific Garbage Patch is Rapidly Accumulating Plastic. *Scientific Reports*, 8(1), pp. 1-15.

LEGESSE, F., DEGEFA, S., and SOROMESSA, T. (2022). Valuation Methods in Ecosystem Services: A Meta-analysis. *World Journal of Forest Research*, 1(1), pp. 1-12.

LEITE, P., and CORTEZ, M. C. (2016). The Performance of European Socially Responsible Fixed-Income Funds. SSRN, 2726094.

LELIEVELD, J., EVANS, J. S., FNAIS, M., GIANNADAKI, D., and POZZER, A. (2015). The Contribution of Outdoor Air Pollution Sources to Premature Mortality on a Global Scale. *Nature*, 525(7569), pp. 367-371.