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Let's say a pension fund guarantees an annual return of at least 5% to their customers/investors, such that the investors face a payoff like the one of a call option (no downside). For this guarantee the investors pay a premium. From the fund's point of view, this guarantee is like a short put as they only face a downside; if the return is less than 5% they have to withdraw funds from a reserve account to cover the loss, while if the return is more than 5% the investors receive the excess return. For this short put the fund requires a premium. So, how should one perform a valuation of this short put, i.e. what would the fund have to pay to be completely hedged? In my view, this has to be the value of the minimum guaranteed return. However, as the underlying asset is the portfolio consisting of stocks, government bonds, corporate bonds, and investments in other funds, using option pricing theory (Black-Scholes, binomial pricing) is not so straightforward.

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I'm not sure if this truly belongs in quantitative finance, but as an actuary, I can't resist responding. The answer to your question literally fills thousands of pages of regulations, research papers, best-practice articles, and study materials.

Pension funds are VERY exotic options. They're not just puts. They're puts tied to mortality, employee behavior, and statutory law (early retirement, employment longevity, ERISA, etc.).

For instance, statutory law allows public plans to discount their expected future cashflows at the rate of return expected on their backing assets. Actuaries and investment analysts set these discount rates based on the basket of securities the pension holds. Private plans discount cashflows based on the AA corporate bond yield curve no matter the backing assets, but again, these bear a spread and risk premium over the risk free rate. The reason I bring this up is because pension liabilities (valued with a risk premium) will NEVER move with your risk-neutral hedging assets. You can get close if you match the duration of your pension obligations with a basket of AA bonds. That's called Portfolio Immunization, but even then, it's not exact, and it's not what you're talking about.

Ignoring the discount rate, you're on the right track with how to think about one type of risk present in pension plans: market risk. To value the embedded put option in a pension guarantee, you'd have to model out thousands of possible growth paths for your various investments with average growth set to your risk-free term structure using Monte Carlo scenario generation. Then, taking into account all the actuarial risks associated with pension plans under each scenario, discount the projected payout in scenarios where the pension assets cannot cover future withdrawals.

I would suggest looking into actuarial forums for further exploration.

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