The Future of Pension Plan Funding London School of Economics UBS Pensions Research Programme Paper: Optimal Portfolio Allocation for Pension Funds in the Presence of Background Risk by David McCarthy (Imperial College) and David Miles (Morgan Stanley) Discussant: Francisco Gomes London Business School

The pension fund has a deterministic finite horizon: Goal of the Paper This paper studies the optimal asset allocation for a defined benefit pension fund. The pension fund has a deterministic finite horizon: Terminal date at which its assets are converted into annuities that are used to pay off its liabilities. The fund chooses an investment strategy in every period before the terminal date: It can invest in equities and an “optimally-chosen” portfolio of short and long-term bonds.

Important: The pension fund’s liabilities are known in advance. Model Set-up Important: The pension fund’s liabilities are known in advance. The fund is closed to new employees. There is no accrual of new pension liabilities. Important: The fund only manages its initial funds, since it receives no additional income. There are no additional contributions from existing employees.

Shocks to survival probabilities. Model Set-up (cont.) This set-up significantly simplifies the problem, since there are only two sources of uncertainty: Asset returns. Shocks to survival probabilities. Concern: Ignoring other sources of uncertainty significantly decreases the degree of “background risk” faced by the pension fund: Potential large positive bias in the optimal equity allocation.

Model Set-up (cont.) Future worker contributions are like riskless asset holdings. In that case you get the Merton problem with a riskless income stream: The contributions act as an implicit bond holding, and thus ignoring them will biased the optimal equity investment downwards. So we have two important missing features that might affect the results (they are opposing biases but unlikely to cancel out each other).

Model Set-up (cont.) Firms usually extract fund surpluses gradually, by taking “contribution holidays”: this might also affect the optimal investment incentives of trustees.

The fund’s trustees are assumed to maximize: Objective Function The fund’s trustees are assumed to maximize: CRRA power utility preferences. Defined over the fund’s final asset position (after potential adjustments for deficits or surplus). Is this the right objective function? Study observed trustee behaviour. Ask trustees.

Treatment of Deficits and Surpluses If the fund’s assets (at the terminal date) exceed its liabilities then: The sponsor will (potentially) collect a fraction (t) of the surplus. If the fund’s assets (at the terminal date) fail to meet its liabilities then: The sponsor will cover any deficits with probability p. If the sponsor defaults (probability 1-p), the insurer will cover a fraction (s) of the deficit. Multiple cases are consider for t, p and s. Good!

Non-convexity in the asset allocation decision The paper discusses in detail the non-convexities in the optimal allocation just before the terminal date. These are theoretically interesting but probability not very important in practice. They don’t exist for t=T-10 and probably in other periods as well: the objective function is made concave by the probabilities. So this would likely disappear as well if we were to add uncertainty in the terminal period, the size of the contributions, the size of the liabilities, etc.

Numerical Solution and Related OR Literature The non-convexities in the objective function make this a very hard problem to solve: Computationally intensive. Suggestion: look at the OR literature on “asset-liability management”: Scenario optimization tools will probably be faster. This OR literature might have studied similar types of problems before.

Conclusion: This is a very nice paper! Summary Conclusion: This is a very nice paper! The authors are tackling a very interesting and important problem in a realistic model. Main suggestion: Think of ways to try to incorporate even more realism into the analysis (“clever” short-cuts or alternative solution techniques).