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

This paper derives optimal proportional portfolio insurance strategies that penalize carbon emissions within a stochastic factor model under partial information, demonstrating through numerical analysis that such approaches can reduce carbon intensity without compromising financial performance.

Original authors: Katia Colaneri, Federico D'Amario, Daniele Mancinelli

Published 2026-03-25

📖 5 min read🧠 Deep dive

CC BY 4.0

Original authors: Katia Colaneri, Federico D'Amario, Daniele Mancinelli

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

Imagine you are the captain of a ship (your investment portfolio) sailing toward a distant island (your retirement or financial goal). You have a safety net: a lifeboat that guarantees you won't sink below a certain depth, no matter how rough the seas get. This is the core idea of Portfolio Insurance.

However, the ocean is changing. There are new storms coming—not just financial ones, but climate storms. Ships that burn dirty fuel (carbon-intensive "brown" companies) are at risk of being stranded by new regulations, shifting markets, or bad reputations.

This paper proposes a new navigation system for your captain. It's a strategy that doesn't just try to keep you afloat; it actively steers you away from the dirty ships, without sacrificing your speed or your ability to reach the island.

Here is the breakdown of their "Carbon-Penalised" strategy, explained simply:

1. The Safety Net (The Cushion)

In traditional portfolio insurance, you have a "floor" (the minimum money you must keep safe). The money you have above that floor is called the Cushion.

The Rule: You only take risks with the Cushion. If the Cushion gets too small, you move all your money into the safe lifeboat (cash).
The Innovation: The authors say, "Let's make the Cushion smaller if you are holding too many dirty ships." They add a Carbon Penalty. Think of this as a "tax" on your happiness (utility) if you hold too much carbon. The more carbon you hold, the less "satisfied" the strategy feels, even if the money is the same.

2. The Foggy Ocean (Partial Information)

Here is the tricky part. In the real world, you can't see the future weather perfectly. You know the wind is blowing, but you don't know exactly how strong the storm will be or why it's happening.

The Model: The authors assume there is a hidden "Weather Factor" (like a giant, invisible atmospheric pressure system) that drives the stock market.
The Problem: Investors can't see this factor directly. They only see the waves (stock prices).
The Solution: They use a Mathematical Foggy Glasses (Stochastic Filtering). This is a way of guessing the hidden weather based on the waves you can see. It's like a sailor looking at the color of the water and the movement of birds to guess if a storm is coming, even though they can't see the clouds yet.

3. The Smart Captain (The Strategy)

The paper solves a complex math puzzle to tell the captain exactly how to steer. The strategy has two main jobs:

Protect the Floor: Make sure you don't run out of money.
Avoid the Brown: Reduce exposure to dirty companies.

How does it work?
The strategy doesn't just ban "brown" stocks. That would be like saying, "No sailing allowed unless the ship is 100% solar." Instead, it says:

"If a dirty ship is moving incredibly fast and making a lot of money, we might still sail with it, but we will keep a very tight grip on the wheel. If a clean ship is moving slowly, we might sail with it more freely."

It balances Risk vs. Reward against Clean vs. Dirty. If a dirty stock is too risky (high volatility) or offers a low reward, the "Carbon Penalty" kicks in hard, and the strategy dumps it. If a dirty stock is a "diamond in the rough" (high return, low risk), the strategy might keep a small amount of it, but it will never be the main focus.

4. The "Foggy" vs. "Clear" Vision

The authors compared two captains:

Captain Clear: Can see the hidden weather factor perfectly.
Captain Foggy: Has to guess the weather using the "Foggy Glasses" (partial information).

The Surprising Result:
Usually, having perfect information is a huge advantage. But the paper found that when you add the Carbon Penalty, the gap between Captain Clear and Captain Foggy gets smaller!

Why? Because the Carbon Penalty forces both captains to be more cautious and conservative. When you are forced to be careful about the environment, you don't need to be a genius weather forecaster to do well. The "safety" of the carbon strategy compensates for the "blindness" of not knowing the future perfectly.

5. The Bottom Line

The paper proves that you can build a financial safety net that is greener without being poorer.

For the Investor: You get the same (or very similar) financial protection and growth potential.
For the Planet: You significantly reduce the amount of money flowing into high-carbon industries.
The Trade-off: You might miss out on the absolute highest possible gains from a risky, dirty stock, but you gain massive protection against the risk that those stocks will crash due to climate change.

In a Nutshell:
This paper gives investors a new set of sunglasses. These glasses don't just protect your eyes from the sun (financial risk); they also tint the view to make the dirty, smoggy parts of the ocean look less attractive, guiding you toward a cleaner, safer, and still-profitable journey.

1. Problem Statement

The paper addresses the challenge of integrating carbon risk into Proportional Portfolio Insurance (PPI) strategies within a continuous-time stochastic control framework.

Context: Institutional investors increasingly seek to protect capital (via PPI) while adhering to Environmental, Social, and Governance (ESG) mandates, specifically reducing exposure to carbon-intensive ("brown") assets.
The Gap: Existing literature often treats sustainability via exclusion rules or hard constraints. Few studies incorporate carbon risk directly into the objective function of a portfolio insurance strategy, particularly under partial information where key market drivers (like expected returns) are unobservable.
Objective: To derive optimal dynamic allocation rules that maximize the expected utility of a carbon-penalised terminal cushion, balancing downside protection, financial performance, and carbon exposure, while accounting for the fact that investors cannot directly observe the underlying economic factors driving asset returns.

2. Methodology

A. Market Model

The authors model a financial market with $n$ risky assets and one risk-free asset.

Asset Dynamics: Risky asset prices follow Geometric Brownian Motions (GBM). Crucially, the drift (expected return) is stochastic and unobservable, driven by a common latent factor $Y_t$ .
Latent Factor: The factor $Y_t$ follows an Ornstein-Uhlenbeck (OU) process, representing macro-financial states (e.g., business cycles, transition risks) that are mean-reverting but not directly observable.
Partial Information: The investor observes only asset prices ( $S_t$ ), not the factor $Y_t$ . The investor must estimate $Y_t$ using stochastic filtering theory (Kalman filtering), resulting in a filtered estimate $\Gamma_t = \mathbb{E}[Y_t | \mathcal{F}_t]$ and an estimation error variance $P_t$ .

B. The Carbon-Penalised PPI Strategy

Standard PPI: The portfolio value $V_t$ is split between a risk-free asset and a risky reference portfolio $X_t$ . The exposure to $X_t$ is proportional to the cushion $C_t = V_t - F_t$ (where $F_t$ is the guaranteed floor), scaled by a multiplier $m_t$ .
Carbon Penalty: The authors modify the terminal objective. Instead of maximizing the utility of the terminal cushion $C_T$ $C_{T}$ , they maximize the utility of a penalised cushion $\hat{C}_T$ $\hat{C}_{T}$ :
$\hat{C}_T = C_T \exp\left( -\frac{1}{2} \int_0^T m_s^2 \pi_s^\top (\Sigma_S \Sigma_S^\top \odot e) \pi_s \, ds \right)$
- Here, $e$ is a vector where entries are $0$ for green stocks and $\varepsilon > 0$ for brown stocks.
- The penalty term is proportional to the realised variance of the brown stocks held in the portfolio.
- Interpretation: This acts as an endogenous increase in risk aversion specifically toward carbon-intensive assets. It does not strictly exclude them; rather, it reduces their weight if their risk-adjusted return does not compensate for the carbon penalty.

C. Optimization Framework

The problem is solved using Dynamic Programming and the Hamilton-Jacobi-Bellman (HJB) equation under two scenarios:

Full Information: The investor observes $Y_t$ directly.
Partial Information: The investor observes only $S_t$ and uses the filtered estimate $\Gamma_t$ .

The authors assume a CRRA (Constant Relative Risk Aversion) utility function (and analyze the Logarithmic case as a limit). They employ a guess-and-verify approach, assuming the value function has an exponential-affine form in terms of the state variables (cushion and factor/estimate).

3. Key Contributions

Novel Objective Function: The paper introduces a carbon penalty directly into the utility function of a PPI strategy. Unlike exclusionary approaches, this allows for a trade-off where high-performing brown assets may still be held if their Sharpe ratio justifies the carbon penalty.
Partial Information with Filtering: The model explicitly addresses the unobservability of expected returns. By modeling the drift as a latent OU process, the authors derive explicit solutions using stochastic filtering, providing a realistic framework for institutional investors who rely on estimated parameters.
Analytical Solutions: The authors derive closed-form expressions for:
- The optimal control (multiplier and portfolio weights) under both full and partial information.
- The value function.
- The Loss of Utility and Efficiency (monetary value of information) caused by the inability to observe the latent factor.
Characterization of the "Learning Risk": In the partial information case, the optimal strategy includes an additional hedging demand term related to the estimation error variance ( $P_t$ ), representing a hedge against parameter uncertainty.

4. Key Results

A. Optimal Strategy Structure

Full Information: The optimal control $\theta^*$ (product of multiplier and weights) consists of a myopic demand (instantaneous mean-variance trade-off) and an intertemporal hedging demand (hedging against changes in the investment opportunity set driven by $Y_t$ ). Both are scaled by a matrix $\hat{\Theta}^{-1}$ , which incorporates both the risk aversion parameter $\delta$ and the carbon penalty $\varepsilon$ .
Partial Information: The structure remains similar, but the latent factor $Y_t$ is replaced by its filtered estimate $\Gamma_t$ . Crucially, the hedging component is modified by the conditional variance of the estimation error, $P_t$ .
Effect of Carbon Penalty ( $\varepsilon$ ):
- Increasing $\varepsilon$ reduces the exposure to brown stocks and increases exposure to green stocks.
- It effectively increases the risk aversion toward brown assets, lowering the optimal multiplier $m^*$ and shifting allocation toward the risk-free asset (increasing safety).
- The strategy is not a simple exclusion; if a brown stock has a sufficiently high Sharpe ratio, it may still be included, but at a reduced weight.

B. Numerical Analysis

Carbon Reduction: Numerical experiments show that increasing carbon aversion significantly reduces the variance of terminal wealth and improves downside protection (higher 5th percentile), with only marginal losses in expected return.
Scenario Sensitivity: The reduction in carbon exposure is most pronounced when brown stocks outperform green stocks (Scenario 3), demonstrating the strategy's ability to penalize high-carbon assets even when they are financially attractive.
Information Value:
- The Loss of Utility due to partial information is strictly positive but decreases as risk aversion ( $\delta$ ) or carbon aversion ( $\varepsilon$ ) increases.
- Counter-intuitive Finding: High levels of carbon penalisation actually improve the relative efficiency of the partially informed investor compared to the fully informed one. The penalty acts as a conservative constraint that narrows the performance gap caused by information asymmetry.
- Under high risk aversion, the "information premium" virtually vanishes, suggesting that prudent behavior offsets the disadvantage of not observing the latent factor.

5. Significance and Implications

For Practitioners: The paper provides a rigorous mathematical framework for asset managers to implement "Green PPI" products. It demonstrates that sustainability goals can be met without sacrificing the primary goal of capital protection.
For Regulators: The model aligns with regulatory requirements (e.g., Solvency II, ORSA) that mandate the assessment of climate-related transition risks. The carbon penalty can be interpreted as a prudential risk adjustment.
Theoretical Advancement: It bridges the gap between sustainable finance and portfolio insurance under partial information. It highlights that the "cost" of not knowing the true market state is lower for investors who are already constrained by strong sustainability preferences.
Policy Insight: The results suggest that internalizing carbon risk through utility penalties is a flexible mechanism that allows for dynamic rebalancing based on market conditions, rather than rigid exclusion, potentially leading to more robust and adaptive sustainable portfolios.

In summary, the paper demonstrates that carbon-penalised PPI strategies offer a mathematically tractable and practically viable method for institutional investors to manage the dual objectives of capital preservation and decarbonization, even in the face of significant market uncertainty and incomplete information.