A portfolio approach to climate investments: CAPM and endogenous risk

Abstract

Is there a role for investments in climate change mitigation despite low expected return? We use a model of intertemporal expected utility maximisation to analyse this question. Similar to the capital asset pricing model (CAPM) the rate of return depends on the correlation of risk between the return on investments in climate change mitigation and the market portfolio, but in contrast to the classical CAPM we admit the fact that economic and environmental systems are jointly determined, implying that environmental risk is endogenous. Therefore, investments in climate change mitigation may reduce risk via self-protection and self-insurance. If risk reduction is accounted for in cost–benefit evaluations, climate investments may be justified despite low expected return. These aspects of climate investments are not, however, communicated via standard cost–benefit analyses of climate policy. Optimal climate policy may therefore be more ambitious than previously considered.

Appendix

This Appendix explains the derivations of the main equations presented in the text, i.e. Eqs. 3, 5, 13 and 14, respectively.

Derivation of Equation (3)

For convenience, first recall the third condition of Eq. 2:

$$ E\left[u^{\prime}\left(\tilde{C}_2\right)\left(\tilde{R}^{N}-R^{0} \right)\right]+\int u\left(C_2\right)\hbox{d}F_{K^{N}}\left(C_2;K^{N},\xi\right)=0.$$

Then, using integration by parts over C ₂, the last term of Eq. 2 can be written

$$ \eqalign{ \int u (C_2)\hbox{d}F_{K^{N}}(C_2;K^{N},\xi)=& \ u(\infty)F_{K^{N}}(\infty) -u({-\infty})F_{K^{N}}(-\infty)\cr & \quad -\int u^{\prime}(C_2)F_{K^{N}}(C_2;K^{N},\xi)\hbox{d}C_2\cr =& -\int u^{\prime}(C_2)F_{K^N}(C_2;K^N,\xi)\hbox{d}C_2 }<!endaligned> $$

since F _K (−∞) =  F _K (∞) =  0 and we assume that u(C ₂) is bounded above and below or, more generally, ${\lim_{C_2\to-\infty}u(C_2)F_K (C_2) = \lim_{C_2\to\infty} u(C_2 )F_{K} (C_2) = 0}$. Now we can rewrite Eq. 2:

$$ E\left[u^{\prime}\left(\tilde{C}_{2}\right)\left(\tilde{R}^N -R^{0}\right) \right] - \int u^{\prime}(C_2)F_{K^N} (C_2;K^N,\xi)\hbox{d}C_2 =0.$$

(15)

By applying the covariance identity and rearranging, we get

$$ E\tilde{R}^{N} = R^{0} -\frac{\hbox{cov}(u^{\prime}(\tilde{C}_2),\tilde{R}^{N})}{Eu^{\prime}(\tilde{C}_2)}- \frac{\int u^{\prime}(C_2)F_{K^N}(C_2;K^{N},\xi)\hbox{d}C_2}{Eu^{\prime}(\tilde{C}_2)}.$$

(16)

Then we make use of the Stein–Rubinstein lemma (Stein 1973; Rubinstein 1976), which requires that joint distribution of ${\tilde{R}^{N}}$ and ${\tilde{C}_2}$ is bivariate normal and u(·) is twice differential. Invoking the result on the numerator of the first fraction of Eq. 16, such that ${\hbox{cov}(u^{\prime}(\tilde{C}_2), \tilde{R}^{N}) = Eu^{\prime\prime}(\tilde{C}_2) \hbox{cov}(\tilde{C}_2, \tilde{R}^N)}$ we can rewrite Eq. 16 as

$$ E\tilde{R}^N=R^{0}+\left(-\frac{Eu^{\prime\prime}(\tilde{C}_2)} {Eu^{\prime}(\tilde{C}_{2})}\right)\hbox{cov}(\tilde{C}_2,\tilde{R}^N)-\frac{\int u^{\prime}(C_2)F_{K^{N}}(C_2;K^{N},\xi)\hbox{d}C_2}{Eu^{\prime}({\tilde{C}_2})}.$$

(17)

Returning to the second condition of Eq. 2, ${E\left\lfloor u^{\prime} \left(\tilde{C}_2\right)\left(\tilde{R}^{M}-R^{0}\right)\right\rfloor=0}$, and rearranging it, we get

$$ E\tilde{R}^{M}=R^{0}-\frac{\hbox{cov}(u^{\prime}(\tilde{C}_2),\tilde{R}^M)}{Eu^{\prime}(\tilde{C}_{2})}.$$

(18)

Invoking again the Stein–Rubinstein lemma and recalling the assumption of perfect correlation between ${\tilde{C}_2}$ and ${\tilde{R}^{M}}$, we can write Eq. 18 as

$$ -\frac{Eu^{\prime\prime}(\tilde{C}_2)}{Eu^{\prime}(\tilde{C}_2)} =\frac{E\tilde{R}^{M}-R^{0}}{\hbox{var}(\tilde{C}_2)}.$$

(19)

Equation 19 allows us to rewrite Eq. 17 to obtain Eq. 3 in the text.

Derivation of Equation (5)

We have

$${F(C_2; K^{N}) =\Phi\left({C_2-\mu({}{K^N}/{}{\mu}_0)\over{\sigma}}\right)}$$

and

$$\eqalign{{F_{K} (C_2;K^{N})} =-{\mu^{\prime}/\mu_0\over\sigma}\Phi^{\prime} \left({C_2-\mu(K^N)/\mu_0\over\sigma}\right)\cr =-{\mu^{\prime}/\mu_0\over\sigma}\phi\left({C_2-\mu(K^N)/\mu_0\over\sigma}\right)=-\mu^{\prime}/\mu_0f(C_2)\cdot}$$

ϕ is the standard normal density function. Consequently,

$$ {\int u^{\prime}(C_2)F_{K^N}(C_2;K^{N})\hbox{d}C_2\over Eu^{\prime}(\tilde{C_2})} ={\int-u^{\prime}(C_2)\mu^{\prime}/\mu_0f(C_2)\hbox{d}C_2\over Eu^{\prime}(\tilde{C}_2)}=-{\mu^{\prime}Eu^{\prime}(\tilde{C}_2)\over\mu_0 Eu^{\prime}(\tilde{C}_2)}=-{\mu^{\prime}\over{\mu}_0}.$$

(20)

By definition μ=EC̃₂. We assume μ₀ is chosen such that it equals actual period two␣expected consumption, i.e., μ₀=μ. Inserting this in equation (20) now yields equation (5).

Derivation of Equation (13)

The production function of this model is not constant returns to scale and will therefore generate profit income. We assume that profit is related to investment of K ^M and define

$$ \begin{aligned} {\tilde R}^N&=\frac{\hbox{d}{\tilde Y}}{\hbox{d}K^N}+1=\frac{\theta_2\eta}{T}{\tilde D}Y^*+1\\ {\tilde R}^M&=\frac{\tilde Y}{K^M}-\frac{\hbox{d}{\tilde Y}}{\hbox{d}K^N}\frac{K^N}{K^M}+1. \end{aligned}\\ $$

(21)

In words, K ^N gets its marginal return and K ^M gets the rest. With these assumptions we obtain

$$ \begin{aligned} \tilde C_2&=Y^0+\tilde Y+K^M+K^N\\ =&R^0{K^0}+\frac{\tilde Y}{K^M}K^M+K^M+K^N\\ =&R^0{K^0}+\left(\frac{\tilde Y}{K^M}-\frac{\hbox{d}\tilde Y}{\hbox{d}K^N}\frac{K^N}{K^M}\right)K^M+\frac{\hbox{d}\tilde Y}{\hbox{d}K^N}{K^N+K^M+K^N}\\ =&R^0{K^0}+\left(\frac{\tilde Y}{K^M}-\frac{\hbox{d}\tilde Y}{\hbox{d}K^N}\frac{K^N}{K^M}+1\right)K^M+\left(\frac{\hbox{d}\tilde Y}{\hbox{d}K^N}+1\right)K^N\\ =&R^0{K^0}+R^M{K^M}+R^N{K^N}\\ =&(W-C_1)({R^0}x^0+{R^M}x^M+{R^N}x^N).\\ \end{aligned}\\ $$

(22)

Derivation of Equation (14)

Since ${\tilde\theta_1\sim N(\bar\theta_1(K^N),\sigma_{\theta})}$ and ${{\tilde C}_2=(1-\tilde\theta_1{T^{\theta_2}})Y^*+Y^0+K^M+K^N}$ (equations 8, 9, 12) it follows that ${{\tilde C}_2}$ is normal and we may apply Eq. 5:

$$ E{\tilde R}^N=R^0+\beta(E{\tilde R}^M-R^0)-E{\tilde C'}_2/E{\tilde C}_2,\quad \hbox{where}\ \beta=\frac{\hbox{cov}({\tilde C}_2,R^N)}{\hbox{var}({\tilde C}_2)}. $$

First, we develop a simpler expression for β. With ${{\tilde C}_2=Y^0+\tilde Y+K^M+K^N}$ (Eq. 12) we have

$$ \frac{\hbox{cov}({\tilde C}_2,R^N)}{\hbox{var}({\tilde C}_2)}=\frac{\hbox{cov}({\tilde Y},R^N)}{\hbox{var}({\tilde Y})}.$$

(23)

From Eq. 21 we know that ${{\tilde R}^N=({\theta_2}{\eta}/T)\tilde DY^*+1,}$ which from Eq. 9 is ${{\tilde R}^N=({\theta_2}{\eta}/T)(Y^*-\tilde Y)+1}$. Inserting in Eq. 23 gives us

$$ \frac{\hbox{cov}({\tilde Y},R^N)}{\hbox{var}({\tilde Y})}=\frac{\hbox{cov}\left({\tilde Y},\frac{\theta_2{\eta}}{T}({Y^*}-\tilde Y)+1\right)}{\hbox{var}({\tilde Y})}=\frac{-\frac{\theta_2{\eta}}{T}\hbox{var}(\tilde Y)}{\hbox{var}(\tilde Y)}=\frac{\theta_2{\eta}}{T}.$$

(24)

Then we develop an expression for ${E{\tilde C'}_2}$. By definition ${E{\tilde C}_2=\int C_2\hbox{d}F_{C_2}(C_2;K^N,\xi)}$. Since, as stated, ${{\tilde C}_2=(1-\tilde\theta_1T^{\theta_2})Y^*+Y^0+K^M+K^N}$, it follows that

$$ \eqalign{ E{\tilde C'}_2=& \int\frac{\hbox{d}[(1-\theta_1(K^N)T^{\theta_2})Y^*+Y^0+K^M+K^N]}{\hbox{d}K^N}\hbox{d}F(C_2;K^N,\xi)\cr =& \int-\theta'_1{T^{\theta_2}Y^*}\hbox{d}F(C_2;K^N,\xi)=\frac{-E\tilde\theta'_1}{E\tilde\theta_1}E\tilde\theta_1{T^{\theta_2}}Y^*=-\frac{\bar\theta'_1}{\bar\theta_1}E{\tilde D}Y^*.\cr }<!endaligned> $$

(25)

From Eq. 25 we get

$$ -\frac{E{\tilde C'}_2}{E{\tilde C}_2}=\frac{\bar\theta'_1}{\bar\theta_1}\frac{Y^*}{{E{\tilde C}_2}}E\tilde D.$$

(26)

Assume that ${E{\tilde C}_2\approx E\tilde Y}$. This is an assumption in the spirit of the model since it only disregards the somewhat artificial storage technology Y ⁰ and initial wealth K ^M and K ^N, consumption of which is an artefact of the two-period setup. Now

$$ \frac{\bar\theta'_1}{\bar\theta_1}\frac{Y^*}{{E{\tilde C}_2}}E\tilde D=\frac{E\tilde D}{1-E\tilde D}\frac{\bar\theta'}{\bar\theta_1}\approx E\tilde D\frac{\bar\theta'}{\bar\theta_1}.$$

(27)

Expression (27) underestimates the effect to the extent that ${Y^* > E{\tilde C}_2}$. Combining results we have

$$ E{\tilde R}^N=R^0+\beta(E{\tilde R}^M-R^0)-E{\tilde C'}_2=R^0-\frac{{\theta_2}\eta}{T}({E\tilde R}^M-R^0)-E\tilde D\frac{-\bar\theta'_1}{\bar\theta_1}.$$

(28)