Based on modern portfolio theory (Bodie et al.
2021), the CB considers the average expected yields of green
\(\mu _G\) and non-green bonds
\(\mu _N\), their average volatility (i.e.,, the standard deviation of their returns), given respectively by
\(\sigma _G, \sigma _N > 0\), and the covariance between the two types of corporate bonds
\(\sigma _{G,N}\).
9 The covariance
\(\sigma _{G,N}\) is related to the correlation coefficient
\(r_{G,N}=\frac{\sigma _{G,N}}{\sigma _G \, \sigma _N}\), which, to be economically meaningful, must range between
\(-1\) (i.e., perfect negative correlation) and
\(+1\) (i.e., perfect positive correlation). Thus, we impose that:
$$\begin{aligned} -1 \le \frac{\sigma _{G,N}}{\sigma _G \, \sigma _N} \le 1 \end{aligned}$$
(4)
According to the capital asset pricing model (CAPM), the CB portfolio’s expected yield
\(\mu _P (x)\) is a convex combination of the individual yields, where the weights are the share of green bonds
\(x\in (0,1)\) and non-green bonds
\(1-x\) (i.e., the complementary part) in the CB portfolio/market:
$$\begin{aligned} \mu _P (x)&= x \, \mu _G + (1-x) \, \mu _N \end{aligned}$$
(5)
Substituting the inverse supply functions of green (
2c) and non-green bonds (
3c) into Eq. (
5), and defining the CB portfolio’s expected variance
\(\sigma _P^2 (x)\), based on the volatility (i.e., standard deviation)
\(\sigma _i>0, i=G,N\), and the covariance
\(\sigma _{G,N}\) of the individual type of onds, we obtain:
$$\begin{aligned} {\left\{ \begin{array}{ll} \mu _P (x) = \frac{\alpha }{B_T} + \frac{\beta }{B_T} \\ \sigma ^2_P (x) = x^2 \, \sigma ^2_G + \left( 1-x\right) ^2 \, \sigma ^2_N + 2 \, x \, \left( 1-x\right) \, \sigma _{G,N} \end{array}\right. } \end{aligned}$$
(6)
The system of equations in (
6) determines a tuple of points, i.e., the expected yield and expected variance of the portfolio, in relation to share
x. It describes the mean-variance trade-off that the CB faces for all the possible combinations/allocations of green (
x) and non-green (
\(1-x\)) bonds.
10 Consequently, corporate bonds come in a variety of risk-reward levels depending on the issuing company’s creditworthiness. While the CB prefers assets that have the highest expected return, it also seeks to minimize uncertainty about corporate bonds’ future return. We assume that the CB chooses the combination of green and non-green bonds with the optimal risk-reward level and thus, the portfolio allocation that offers the maximum return-to-risk ratio, i.e., the optimal portfolio
\(x^*\) in the CAPM. The CB risk-averse preference function in a
neutral monetary policy setup can be formalized as a capital allocation line defined by the following (
7):
$$\begin{aligned} \mu _P (x) = r_F + S_P \, \sigma _P (x) \end{aligned}$$
(7)
The CB maximizes the portfolio return
\(\mu _P(x)\) for a given portfolio risk
\(\sigma _P(x)\), where
\(S_P\) is the Sharpe ratio or reward-to-risk ratio (Sharpe
1971), and
\(r_F \ge 0\) is the equivalent risk-free asset (i.e., the yield associated to a risk-free asset, for example a short-term U.S. treasury bond). Equation (
7) shows the trade-off between the expected portfolio return
\(\mu _P (x)\) and its volatility
\(\sigma _P (x)\) and thus defines the risk-aversion preference of the CB. The CB is willing to hold a riskier portfolio if and only if it guarantees a higher average return reflected in
\(S_P\). Therefore, the CB maximizes the reward-to-risk ratio
\(S_P\) given the constraints in (
6) by determining the share
x that maximizes the Sharpe ratio of a portfolio that is on the envelope of the Markowitz bullet (Markowitz
1952)
11:
$$\begin{aligned} \begin{aligned} \max _{x} \quad S_P&= \frac{\mu _P(x)-r_F}{\sigma _P(x)} \quad \text {s.t.} \\&\text {constraints in } (6) \end{aligned} \end{aligned}$$
(8)
Note that
\(\mu _P(x) \ge r_F\) in (
8) requires that:
$$\begin{aligned} \frac{\alpha + \beta }{B_T} \ge r_F \end{aligned}$$
(9)
From the Sharpe ratio condition (
8), it is also required that
\(\sigma ^2_P (x) > 0\) in (
6). It must therefore hold that:
$$\begin{aligned} \sigma _{G,N} > -\frac{x \sigma _G^2}{2 (1-x)} - \frac{(1-x) \sigma _N^2}{2 x} \end{aligned}$$
(10)
The problem in (
8) can be reduced to solving the unconstrained maximization problem
$$\begin{aligned} \max _{x} \quad \frac{\frac{\alpha }{B_T} + \frac{\beta }{B_T} - r_F}{\sqrt{x_G^2 \, \sigma ^2_G + \left( 1-x\right) ^2 \, \sigma ^2_N + 2 \, x \, \left( 1-x\right) \, \sigma _{G,N}}} \end{aligned}$$
(11)
The solutions to problem (
11) returns the optimal shares of green and non-green corporate bonds in the CB portfolio and in the market, and is given by:
$$\begin{aligned} x^*&= \frac{\sigma ^2_N - \sigma _{G,N}}{\sigma ^2_G + \sigma ^2_N - 2 \, \sigma _{G,N}} \end{aligned}$$
(12a)
$$\begin{aligned} 1-x^*&= \frac{\sigma ^2_G - \sigma _{G,N}}{\sigma ^2_G + \sigma ^2_N - 2 \, \sigma _{G,N}} \end{aligned}$$
(12b)
From condition (
4) and using (
12a), (
12b)
\(\in (0,1)\), it must hold:
$$\begin{aligned}&\sigma _N^2 > \sigma _{G,N} \end{aligned}$$
(13a)
$$\begin{aligned}&\sigma _G^2 > \sigma _{G,N} \end{aligned}$$
(13b)
In the following, we define the derivatives of the optimal shares (
12a), (
12b) with respect to the model parameters:
$$\begin{aligned} \frac{\partial x^*}{\partial \sigma ^2_N}&= \frac{\sigma ^2_G-\sigma _{G,N}}{\left( \sigma ^2_G-2 \sigma _{G,N}+\sigma ^2_N\right) ^2}>0 \end{aligned}$$
(14a)
$$\begin{aligned} \frac{\partial x^*}{\partial \sigma ^2_G}&= \frac{\sigma _{G,N}-\sigma ^2_N}{\left( \sigma ^2_G-2 \sigma _{G,N}+\sigma ^2_N\right) ^2}<0 \end{aligned}$$
(14b)
$$\begin{aligned} \frac{\partial x^*}{\partial \sigma _{G,N}}&= \frac{\sigma ^2_N-\sigma ^2_G}{\left( \sigma ^2_G-2 \sigma _{G,N}+\sigma ^2_N\right) ^2} \gtreqless 0 \end{aligned}$$
(14c)
$$\begin{aligned} \frac{\partial ^2 x^*}{\partial \sigma _{N}^2\partial \sigma _{G}^2}&=\frac{\sigma ^2_N-\sigma ^2_G}{\left( \sigma ^2 _G-2\sigma _{G,N}+\sigma ^2 _N\right) {}^3} \gtreqless 0 \end{aligned}$$
(14d)
$$\begin{aligned} \frac{\partial ^2 x^*}{\partial \sigma ^2_{G}\partial \sigma _{G,N}}&=\frac{2 \sigma _{G,N}+\sigma _G^2-3 \sigma ^2 _N}{\left( \sigma _G^2-2\sigma _{G,N}+\sigma _N^2\right) {}^3} \gtreqless 0 \end{aligned}$$
(14e)
$$\begin{aligned} \frac{\partial ^2 x^*}{\partial \sigma _{N}^2\partial \sigma _{G,N}}&=-\frac{2 \sigma _{G,N}-3 \sigma _G^2+\sigma _N^2}{\left( \sigma _G^2-2\sigma _{G,N}+\sigma _N^2\right) {}^3} \gtreqless 0 \end{aligned}$$
(14f)
As expected, an increase of the variance (i.e., financial risk) reduces the optimal share of the correspondent corporate bond in the CB portfolio, while the effect of the covariance on
\(x^*\) can be positive, negative or null, depending on the difference of the two variances.