The Impact of Stochastic Interest and Mortality Rates on Ruin Probability and Annuitization Decisions Faced by Retirees

This dissertation focuses on two issues in retirement planning. The first issue, annuitization problem, provides insight on how interest rates may affect annuitization decisions for retirees under an all-or-nothing framework. The second issue, ruin probability, studies the probability for a retired individual who might run out of money, under a fixed consumption strategy before the end of his/her life under stochastic hazard rates. These two financial problems have been very important in personal finance for both retirees and financial advisors throughout the world, especially in the developed countries as the baby boom generation nears retirement. They are the direct results of both longevity risk and demise of Defined Benefit (DB) pension plans.

The existing literature of the annuitization problem, such as Richard (1975), concludes that it is always optimal to annuitize with no bequest motives under a constant interest rate. To see the effect of stochastic interest rates on the annuitization decisions under a constrained consumption strategy without bequest motives, we present two life cycle models. They investigate the optimal annuitization strategy for a retired individual whose objective is to maximize his/her lifetime utility under a variety of institutional restrictions, in an all-or-nothing framework. The individual is required to annuitize all his/her wealth in a lump sum at some time at retirement. The first life cycle model we have presented assumes full consumption after annuity purchasing. A free boundary exists in this case upon the assumption of constant spread between the expected return of the risky asset and the riskless interest rate. The second life cycle model applies the optimal consumption strategy after annuitization, and numerical analysis shows that it is always optimal to annuitize no matter what the current interest rate is. This conclusion is based on the assumption of constant risk premium, no loads and no bequest motives.

Historical data show that mortality rates for human beings behave stochastically. Motivated by this, we study the ruin probability for a retired individual who withdraws $1 per annum with various initial wealth for log-normal mortality with constant drift and volatility, which is a special form of the most widely accepted Lee-Carter model. This problem is converted to a Partial Differential Equation (PDE) and solved numerically by the Alternative Direction Implicit (ADI) method. For any given initial wealth, ruin probability can be obtained for various initial hazard rates. The correlation between the wealth process and the mortality process slightly affects the ruin probability at time zero.