💰 quantitative finance

Carbon-Sensitive Fund Construction and Hedging for Green Unit-Linked Life Insurance

This paper addresses the dual challenges of constructing a carbon-sensitive investment fund through an endogenous portfolio selection rule and developing a quadratic hedging strategy to minimize the variance of costs for unit-linked life insurance policies exposed to market, carbon, and mortality risks.

Original authors: Katia Colaneri, Alessandra Cretarola, Edoardo Lombardo, Daniele Mancinelli

Published 2026-05-27

📖 5 min read🧠 Deep dive

CC BY 4.0

Original authors: Katia Colaneri, Alessandra Cretarola, Edoardo Lombardo, 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 an insurance company selling a special kind of life insurance policy. Unlike traditional policies that just pay out a fixed amount, these "Unit-Linked" policies are like a hybrid: part life insurance, part investment fund. The money you get back depends entirely on how well a specific stock market portfolio performs.

Now, imagine the world is changing. People want their money to go to companies that are good for the planet (low carbon emissions), not just companies that make the most profit. This paper tackles two big problems that arise when an insurance company tries to sell these "Green" policies:

How do you build the "Green" investment fund?
How do you protect (hedge) the insurance company from losing money on these policies?

Here is a simple breakdown of how the authors solved these puzzles.

Part 1: Building the "Green" Fund (The Garden Analogy)

Usually, when investors want a "green" portfolio, they might just throw out all the "dirty" companies (like coal or oil) and only keep the "clean" ones. The authors say this is too rigid. It's like a gardener who refuses to plant any flower that isn't 100% organic, even if that flower is beautiful and hardy.

Instead, the authors propose a smarter way to build the garden: The Carbon Penalty.

The Analogy: Imagine every plant in your garden has a "pollution score." If a plant is very dirty, it gets a heavy penalty. But, if that dirty plant is also incredibly strong and produces a lot of fruit (high profit), the penalty might be worth it.
The Mechanism: The authors created a mathematical rule that "punishes" the final value of the portfolio if it holds too many dirty plants. The punishment gets worse the dirtier the plant is.
The Result: The computer doesn't just ban dirty plants. It calculates a balance. If a dirty plant is super profitable, it might still stay in the garden, but in smaller amounts. If a clean plant is profitable, it gets a bigger share. This creates a fund that is naturally "greener" without being blindly exclusionary. They tested this with real stock market data and found that even a small "penalty" significantly reduced the carbon footprint of the portfolio.

Part 2: Hedging the Risk (The Umbrella Analogy)

Now, the insurance company has this Green Fund and has sold policies based on it. They face three types of storms:

Market Risk: The stock market crashes.
Mortality Risk: People die sooner or later than expected.
Carbon Risk: The "greenness" of the companies changes (e.g., a company suddenly starts polluting more).

The Problem: You can buy an umbrella for the Market Risk (by trading stocks). But you cannot buy an umbrella for the other two. There is no "Carbon Stock" or "Death Stock" you can trade to perfectly protect yourself. This is called an incomplete market. You can't perfectly replicate the insurance payout using only standard stocks.

The Solution: Since you can't be perfect, the authors use a strategy called Risk Minimization.

The Analogy: Imagine you are trying to walk through a rainstorm without a perfect umbrella. You can't stop the rain (mortality and carbon risk), but you can adjust your path and hold your umbrella at the best angle to minimize how wet you get.
The Strategy: The authors developed a mathematical formula that tells the insurance company exactly how much of the Green Fund to hold at every single moment. This strategy doesn't promise to eliminate all losses, but it guarantees that the variance (the ups and downs) of the money they lose is as small as possible.

Part 3: The Simulation (The Weather Forecast)

To prove this works, the authors ran a massive computer simulation (like a super-advanced weather forecast).

They simulated 1,000 different policyholders over 20 years.
They compared three approaches:
1. No Hedging: Just sitting in the bank and hoping for the best (The "Duck" strategy).
2. Static Hedging: Picking a strategy at the start and never changing it (The "Set it and forget it" strategy).
3. Dynamic Hedging: Constantly adjusting the portfolio based on the math (The "Risk Minimization" strategy).

The Findings:

No Hedging was a disaster, with huge losses and wild swings.
Static Hedging was better but still had significant volatility.
Dynamic Hedging was the clear winner. It reduced the average loss to almost zero and made the results incredibly stable.

They also found that if you have a huge group of people (a portfolio of 1,000) rather than just one person, the "Mortality Risk" smooths out (because not everyone dies at the exact same time), making the hedging even more effective.

Summary

In short, this paper provides a blueprint for insurance companies to sell "Green" life insurance without going broke.

Build the fund by mathematically penalizing dirty companies rather than just banning them.
Protect the company by constantly adjusting their investments to minimize the "wetness" from the storms they can't control (death and carbon changes).

The result is a system that balances financial profit with environmental responsibility, while keeping the insurance company's risk under control.

Technical Summary: Carbon-Sensitive Fund Construction and Hedging for Green Unit-Linked Life Insurance

Problem Statement
The paper addresses the dual challenge faced by insurance companies issuing unit-linked life insurance policies with sustainability features. These contracts link policyholder benefits to an underlying investment fund that incorporates environmental criteria. The authors identify two distinct but interconnected problems:

Fund Construction: How to construct an investment fund that endogenously selects assets based on carbon intensity without relying on arbitrary pre-screening or fixed ESG scores.
Risk Management: How to hedge the liabilities of these policies in a market that is incomplete due to the coexistence of financial risk, mortality risk, and a newly introduced "carbon risk." Since carbon emissions and mortality are not spanned by traded assets, perfect replication of liabilities is impossible, necessitating strategies that minimize residual hedging costs.

Methodology
The authors propose a unified framework that separates the asset allocation problem from the liability hedging problem, though they are linked by the underlying fund dynamics.

Asset Side (Fund Construction):
- The authors model a continuous-time financial market with $d$ risky assets and a risk-free asset.
- They introduce a carbon-penalized optimization problem where the objective is to maximize the expected utility of terminal wealth.
- Carbon Penalization: Instead of imposing hard constraints, the model penalizes terminal wealth via a multiplicative exponential term. This term depends on the realized volatility of the assets and their carbon intensity ( $C_{i,t}$ ). The penalization coefficient $\alpha_i(t)$ reflects time-varying environmental preferences.
- Optimization: Using dynamic programming and the Hamilton-Jacobi-Bellman (HJB) equation, they derive an explicit optimal portfolio strategy. The solution shows that the optimal weights are adjusted by the carbon intensity and the penalization parameter, effectively increasing the "effective risk aversion" for high-carbon assets.
- Carbon Intensity Dynamics: The paper explores several stochastic models for the carbon intensity process, including a finite-state Markov chain, an Ornstein-Uhlenbeck (OU) process, an exponential OU process, and a Cox-Ingersoll-Ross (CIR) process.
Liability Side (Hedging):
- The authors extend the market to include mortality risk, assuming a deterministic hazard rate (Gompertz-Makeham law) and independence between financial, carbon, and mortality risks.
- They analyze unit-linked contracts for pure endowment, term insurance, and endowment insurance.
- Hedging Approach: Due to market incompleteness, they employ a quadratic hedging criterion (local risk minimization). They utilize the Föllmer–Schweizer decomposition to derive pseudo-optimal strategies that minimize the variance of the hedging costs in the $L^2$ -sense.
- The hedging strategy is derived under the minimal martingale measure, which adjusts the drift of traded financial risks while leaving non-traded carbon and mortality risks unchanged.
Numerical Implementation:
- For the static analysis, they use real market data (S&P 500 stocks) with fixed carbon intensity values.
- For dynamic simulations, they employ a multidimensional Ninomiya-Victoir weak second-order scheme to simulate the CIR carbon intensity process.
- To ensure computational efficiency, they utilize a variance-reduction technique based on the conditional Gaussian structure of the fund's logarithm.

Key Contributions and Results

Endogenous Carbon Integration: The paper proposes a portfolio selection rule where carbon risk is integrated endogenously through a penalization of terminal wealth. This allows for a flexible trade-off between financial performance and environmental impact, avoiding rigid exclusion rules.
Explicit Hedging Strategies: The authors derive explicit risk-minimizing hedging strategies for green unit-linked contracts in an incomplete market. These strategies account for the residual risks arising from non-traded carbon and mortality factors.
Empirical Findings (Static): Using S&P 500 data, the authors demonstrate that even moderate levels of carbon penalization ( $\alpha$ ) significantly reduce the carbon intensity index of the optimal portfolio. The results show that high-carbon assets are not automatically excluded; they are retained if their financial attractiveness (risk premium) sufficiently offsets the carbon penalty.
Simulation Findings (Dynamic):
- The dynamic simulation confirms that the model balances risk-return trade-offs with carbon risk. Assets with the lowest carbon intensity receive the highest weights, but high-return assets with higher carbon intensity are still included.
- Hedging Performance: Numerical analysis of hedging costs reveals that "no hedging" results in substantial losses and high variability. Static hedging (rebalancing only at $t=0$ ) improves upon no hedging but retains significant variability. The proposed continuous dynamic risk-minimizing strategy significantly reduces both the average loss (near zero) and the standard deviation of hedging costs.
- Diversification: The study confirms that while individual policy hedging costs are volatile, diversifying across a large portfolio of policies (e.g., 1000 policyholders) drastically reduces the dispersion of hedging costs due to the law of large numbers acting on mortality risk.

Significance and Claims
The paper claims to bridge a gap in the literature by providing a unified framework that combines carbon-sensitive portfolio construction with the hedging of insurance liabilities. Its primary significance lies in:

Actuarial-Sustainable Finance Intersection: It extends classical actuarial frameworks for unit-linked contracts by explicitly incorporating climate-related risk as an additional source of market incompleteness.
Tractability: It offers a tractable setting for assessing the interaction between sustainability considerations and insurance risk management, providing explicit formulas for optimal strategies under various carbon intensity dynamics.
Practical Relevance: The numerical results suggest that the proposed dynamic hedging approach is a viable method for insurers to manage the complex risk profile of green unit-linked products, achieving significant loss reduction compared to static or non-hedged approaches.

The authors maintain a modest tone, noting that their framework assumes independence between financial and mortality risks and that future work could explore dependencies (e.g., through common stochastic factors) to capture systemic shocks like pandemics or climate catastrophes. They also acknowledge that the model relies on specific assumptions regarding carbon intensity dynamics due to limited data availability.

Part 1: Building the "Green" Fund (The Garden Analogy)

Part 2: Hedging the Risk (The Umbrella Analogy)

Part 3: The Simulation (The Weather Forecast)

Summary

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