Implied Life Credits: A Note on Developing an Index for Life Annuities

Implied Life Credits: A Note on Developing an Index for Life Annuities In this paper the authors employ the concept of self-annuitization to develop a …nancial index for tracking the time series behavior of life annuity payouts. Aside from increasing awareness of the bene…ts of voluntary annuitization, there are a number of practical uses for such an index such as benchmarking relative competitiveness and analyzing the optimal age at which to annuitize. From a technical point of view the proposed index goes beyond a (trivial) cross-sectional average of life annuity payouts o¤ered by di¤erent insurance companies. Rather, the index is de…ned equal to the internal rate of return (IRR) that an individual would have to earn on their portfolio if they chose to self-annuitize, instead of purchasing a life annuity. We de…ne this IRR –which is based on the current term structure of annuity payouts –as the implied life credits at a given age and for a given deferral period. The implied life credit value solves a non-linear equation that we proceed to approximate quadratically and for which we obtain a relatively simple and intuitive expression. We then suggest age 65 –with a ten year period certain –compared against the same annuity at age 75, as the standard benchmark for the index. As an illustration of the concept we calibrate the index to a comprehensive time-series of weekly (Canadian) life annuity quotes for the years 2000 to 2003. During this period the implied life credits varied from 5.53% to 6.81% for males and 5.11% to 6.40% for females and is shown to be highly correlated with the yield on a 10-year Government of Canada bond. JEL Classi…cation: H55, G12, G22, Keywords: Pensions, Financial Index, Insurance, Investments, Self-Annuitization 1 BACKGROUND AND MOTIVATION There have been a number of recent papers in the pension and insurance literature that have explored the risk and return properties of a …nancial strategy known as self-annuitization during the retirement years. Self-annuitization is a consumption and investment strategy that attempts to closely mimic the payout from a life annuity while allocating assets to minimize the probability of lifetime ruin. This strategy might not be optimal within a classical life-cycle model with no bequest motives, as originally demonstrated by Yaari (1965) and recently extended by Brown, Davido¤ and Diamond (2003). But the popularity and interest in this ‘annuity alternative’continues to grow, especially given some of the imperfections in the life annuity market, as pointed out by Yagi and Nishigaki (1993). In this paper the authors employ the concept of self-annuitization to develop a …nancial index for tracking the time series behavior of life annuity payouts. Aside from increasing awareness of the bene…ts of voluntary annuitization, there are a number of practical uses for this type of index such as benchmarking competitiveness and analyzing the optimal age at which to annuitize. From a technical point of view, the proposed index goes beyond a (trivial) cross-sectional average of life annuity payouts o¤ered by di¤erent insurance companies. Rather, the index is de…ned equal to the internal rate of return (IRR) that an individual would have to earn on their portfolio, if they chose to self-annuitize, instead of purchasing a life annuity at a given age. We de…ne this IRR –which is based on the current term structure of annuity payouts –as the implied life credits (ILC) at a given age and for a given deferral period. The (unique) implied life credit value solves a non-linear equation that is at the core of the paper. We approximate this equation to arrive at a relatively simple and intuitive expression for the ILC –which is the root of a quadratic equation –that can be easily computed on a pocket calculator. We then suggest age 65 –with a ten year period certain –compared against the same annuity at age 75, as the standard benchmark for the index. This age range appears to be the most common one at which these decisions are made. As an illustration of the concept we calibrate the index formula and implied life credits to a comprehensive time-series of weekly (Canadian) life annuity quotes for the years 2000 to 2003. During this period, the implied life credits varied from 5.53% to 6.81% for males, and 5.11% to 6.40% for females, and is