Carbon-Penalised Portfolio Insurance Strategies in a Stochastic Factor Model with Partial Information

Abstract

Given the increasing importance of environmental, social and governance (ESG) factors, particularly carbon emissions, we investigate optimal proportional portfolio insurance (PPI) strategies accounting for carbon footprint reduction. PPI strategies enable investors to mitigate downside risk while retaining the potential for upside gains. This paper aims to determine the multiplier of the PPI strategy to maximise the expected utility of the terminal cushion, where the terminal cushion is penalised proportionally to the realised volatility of stocks issued by firms operating in carbon-intensive sectors. We model the risky assets' dynamics using geometric Brownian motions whose drift rates are modulated by an unobservable common stochastic factor to capture market-specific or economy-wide state variables that are typically not directly observable. Using classical stochastic filtering theory, we formulate a suitable optimization problem and solve it for CRRA utility function. We characterise optimal carbon penalised PPI strategies and optimal value functions under full and partial information and quantify the loss of utility due incomplete information. Finally, we carry a numerical analysis showing that the proposed strategy reduces carbon emission intensity without compromising financial performance.

Figures (14)

• Figure 6.1: True trajectory of the common stochastic factor $Y$ (solid blue line) and trajectory of its filtered estimate $\Gamma$ (dashed magenta line).
• Figure 6.2: Bar charts displaying the optimal exposure to the $i$-th stock of the carbon-penalised PPI strategy at $t=0$ for different levels of $\delta$ and $\varepsilon$.
• Figure 6.3: Optimal multiplier $\bar{m}^\star_0$ (left panel) and optimal exposure to the risk-free asset $S^0$ (right panel) as a function of carbon aversion $\varepsilon$. The optimal PPI strategy's exposure to $S^0$ is given by $1-\mathbf{1}^\top\mathbf{\bar{E}}^\star_t$ for every $t\in[0,T]$.
• Figure 6.4: Simulated paths of the carbon-penalised PPI strategy’s optimal exposures to $\mathbf{S}$. Parameters of $\mathbf{S}$ and $Y$ are reported in Table \ref{['tab:model_params']}. PPI strategy parameters: $\delta=1$, $\varepsilon=1$, $V_0=1$, $\mathrm{PL}=1$ and $T=5$ years.
• Figure 6.5: Tornado plots displaying the percentage change in the expected terminal value (left panels) and in the variance (right panels) of the optimal strategy at maturity, following a $20$% one-at-a-time perturbation of the market parameters relative to the baseline values reported in Table \ref{['tab:model_params']}. Orange and blue bars correspond to downward and upward perturbations, respectively. Results are shown under partial information for risk aversion $\delta = 1$ and carbon aversion $\varepsilon = 1$.

Theorems & Definitions (38)

• Remark 2.1
• Remark 3.1
• Definition 4.1
• Theorem 4.2: Verification Theorem
• proof
• Theorem 4.3
• proof
• Proposition 4.4
• proof
• Remark 4.5