Optimal pension asset allocation strategy when terminal utility is a function of

replacement ratio

Qing-Ping Ma

This paper considers the optimal asset allocation problem for defined-contribution

pension plan members whose terminal utility is a function of replacement ratio, i.e.

the pension-to-final wage ratio. When three asset types are available for investment,

the optimal portfolio composition, which is horizon dependent, includes investment in

both riskless and risky assets. The investment in risky assets has three components

to hedge wage risk, to speculate on risk premiums and to hedge for financial market

risk respectively.

When the terminal utility is a power function, closed form solution is derived for

the cases where there is no further contribution from wage incomes or there is no

nonhedgeable wage risk. The horizon dependence of optimal pension portfolio is

deterministic under assumptions of constant equity risk premium, constant interest rate

volatility and constant stock return volatility. The short-sale of wage replicating portfolio

also contributes to the horizon dependence of pension plan financial wealth (the sum of

pension portfolio and the short-sold wage replicating portfolio), and the effect is

stochastic due to the stochastic interest rate and stock return. Therefore, the optimal

asset allocation strategy in terms of financial wealth is “stochastic lifestyling”.

For the cases where wage incomes cannot be hedged due to non- hedgeable wage

risk, optimal asset proportions can be solved numerically by Monte Carlo simulation. The

optimal asset allocation is still horizon dependent. The proportions invested in bonds and

stocks are higher than those when wage replicating portfolio is used, hence more

shortsale of cash assets. The differences get smaller as retirement approaches.

Keywords : Optimal asset allocation; Defined-contribution pension plan; Annuity; Power

utility; Hamilton-Jacobi-Bellman equation.