Is Your Standard of Living Sustainable during Retirement? @bullet " " " " Opti Ruin Probabdltles, Asian Ons, and Life Annuities 1. Motivation

In this paper we compute the conditional and unconditional probability of ruin for an individual who wishes to consume a fixed periodic amount from an initial endowment invested in a portfolio earning a stochastic rate of return. The conditional probability of ruin is the probability that the net wealth becomes zero prior to the individual 's stochastic date of death. Unconditional is the probability that the wealth ever becomes zero. We solve this problem using insights from option pricing theory. Specifically, we show that the probability of ruin corresponds to the probability that a suitably parameterized Asian call option (a type of derivative security) will expire with value in-themoney. Under standard assumptions for the investment process, the unconditional probability of ruin is obtained analytically using well-known results leading to the Gamma distribution. The conditional probability of ruin is then approximated with moment-matching techniques using the same Gamma distribution. Finally, using realistic market values for equity and fixed-income investments, we apply our approximation to demonstrate that the conditional probability of ruin is minimized with a relatively high allocation to equity (the high-risk asset) until quite late in life. 1. Motivation Once an individual retires, lifetime consumption is funded by money saved and invested during the working part of the life cycle. The two classic finance problems for a retired individual are (1) What level of consumption can the individual enjoy from invested wealth, including investment earnings, without running out of money during his or her lifetime? and (2) How should the retirement fund be allocated to different investment assets? If the date of death and the rate of return are known with certainty, this problem is easily solved, but, of course, these assumptions are not realistic. In this paper we compute the conditional and unconditional probability of ruin for an individual (retiree) with a stochastic life span who is consuming a fixed real amount from a diversified investment portfolio. By the term conditional we mean the probability that the net-wealth process will hit zero while the individual is still alive, otherwise referred to as bankruptcy. By unconditional ] we mean the probability that the process will ever hit zero. The unconditional probability would be of interest to endowments or individuals with very strong bequest motives. In particular, ]Perhaps abusing conventions. IX. Is Your Standard of Living Sustainable during Retirement? 87 we tabulate the probability of ruin as an explicit function of the stochastic growth rate and volatility of the portfolio vis-a-vis the consumption rate. We view this research as an extension of the literature on ruin probabilities in insurance, such as the work by Pentikainen (1980) and Panjer (1986), as well as many others. The main distinction, of course, is that we focus on "ruin" from a personal perspective, and they do so on the company-firm level. In particular we assume that the (consumption) "claims" are deterministic and the (investment returns) "premiums" are stochastic. Interestingly, we demonstrate that the probability of ruin is equivalent to the probability that a suitably parameterized Asian call option--a type of path-dependent derivative security--will expire in-the-money. The actual price of this Asian call option can be interpreted as the cost of ensuring the retiree's prespecified standard of living, which is also analogous to the cost of an appropriately defined life annuity. Finally, we use a Gamma distribution approximation for life annuities with realistic market parameters for equity and fixed-income investments to demonstrate that the conditional probability of ruin and the implicit cost of insurance is minimized with a relatively high allocation to equity until quite late in life. This analytical approximation can be used to confirm earlier simulation-based studies by Milevsky, Ho, and Robinson (1997), which documented the effect of asset allocation on ruin probabilities. The essence of our approach is the actuarial intuition that the probability of ruin can be formulated as the probability that the stochastic present value---basically a life annuity or perpetuity--is greater than the initial wealth available to support the consumption. Thus, in our framework, an individual retires at age (x) with an initial wealth of Wo = w and a desired lifelong consumption stream of c real dollars per annum. In a deterministic world, with fixed time of death T and a fixed real interest rate r, the present value of the desired consumption stream is trivially calculated as S Te-'~ dt c(1 e -rr) PVT(c) = c ~ r (1) If the expression in Equation (1) is greater than the initial wealth w, the retiree does not have enough to support the desired consumption stream, and ruin occurs with probability one. Likewise, when T = ~o, Equation (1) becomes PV= (c) = c/r, which is the sum needed to fund a perpetuity of c dollars per annum. On the other hand, in a stochastic world, both the time of death and the rate of return on investment are stochastic. The stochastic analogue to the deterministic present value of consumption is the stochastic present value of lifetime consumption (SPV(c)) denoted by SPV~(c) = C~ro e-(~)' dt, (2) where the two sources of randomness, lF and/~,, are incorporated explicitly into the computation. The righthand side (r.h.s.) of Equation (2) is the actuarial definition of a life annuity under stochastic discounting. In addition, the r.h.s, of Equation (2) can be identified as the scaled payoff from an Asian put option (see Section 3 for more on this result). The higher the SPV, ceteris paribus, relative to the initial wealth-to-consumption ratio, the higher the probability of ruin. Once we have the probability density function (pdf) of the stochastic present value of lifetime consumption we can compute the probability that this quantity is greater than the initial level of wealth w. We denote this by Pr~ "~ve P(SPVt(c) > w) P(SPVr > w), (3) in ~ ~-C for the conditional case, and eruin " = P(SPV=(c) > w) = P(SPV= > ww), c for the unconditional case. The remainder of this paper is organized as follows. Section 2 introduces the investment and mortality dynamics, using the techniques of continuous time financial economics, and then derives an expression for the probability of ruin. Section 3 describes the connection and analogy between our problem and Asian options. Section 4 develops some techniques for computing the relevant probabilities using the Gamma distribution. Section 5 provides some numerical examples of the conditional and unconditional probability of ruin using realistic capital market and mortality parameters. Section 6 concludes the paper. 2. Investment and Mortality We start with the basic geometric Brownian motion (GBM) model of investment dynamics in which individual stocks (or asset classes) obey the stochastic differential equation (SDE) defined by dS] /Sj = ~tidt + (lidB~, (4) where B~ is a standard Brownian motion, ~t~ and t~ are the real (inflation-adjusted) mean and standard deviation of dSi/S i, and d(B ~, Bi), = 9~j is the correlation coefficient. An 88 Retirement Needs Framework investor (retiree) allocates and rebalances wealth among the universe of investment assets, provided by Equation (4), and consumes a fixed real amount c, per unit of time. By construction, the real net-wealth process will obey the SDE dWr = (BeW~ c)dt + ~pW~dB,, Wo = w, (5) where B, is a one-dimensional Brownian motion, c is the real fixed consumption rate, w is the initial level of wealth, and (lap, t~p) correspond to the portfolio mean and standard deviation as an implicit function of a static 2 asset allocation vector ct. Specifically, the scalar-valued mean return is ~tp= Ixa' = (~a,la~/, (6) and the scalar valued standard deviation (also known as volatility) of the portfolio is Op = \.,ot~ot = ~,,,'Z Z o~io,pijoj~, (7) i=l j=l where Ix is the vector of expected returns and ]~ is the variance-covariance matrix of the relevant assets in the market, all of which are lognormally distributed. The net-wealth process defined by Equation (5) has a drift coefficient ~p Wt c that may become negative if c is large enough relative to tap W,. This, in turn, implies that the process W, may eventually hit zero, in contrast to the classic geometric Brownian motion. Our intention is to compute the probability that W, will ever hit zero and compute the probability that W, will hit zero while 2A richer model would allow for dynamic portfolio strategies in which the investor can react to market conditions by optimizing asset allocation proportions to achieve greater utility over time. Indeed, a full theory of continuous time dynamic programming has been applied to investment-consumption problems by Samuelson (1969), Merton (1993), Richard (1975), and many others; see Karatzas and Shreve (1992), chapter 5.8, for further references. However, our intention is to simply (1) describe the analogy between the probability of ruin and Asian option pricing and (2) produce a reasonable, practical, and simple measure of sustainability as a function of consumption ratios and basic asset allocation proportions. Accordingly, we do not advocate that rational utility maximizing agents manage their portfolios (statically) so as to exclusively minimize the probability of bankruptcy. See Browne (1997) for a dynamic policy that does indeed minimize the unconditional probability of ruin in an infinite horizon framework. the investor is still alive. Naturally, the former quantity will be an upper bound for the latter. Lemma 1 The stochastic process Bit, defined by Equation (5), can be written (solved) explicitly as W, = H,[w cI~(Hs)-' ds ], (8) where the fundamental solution Hs is _ t a~)s +c~pB~ 1. H~ = exp[(l'tp (9) See Appendix for proof. 2.1 Mortality Function Following the actuarial literature and recent work on annuity pricing by Frees, Carriere, and Valdez (1996) we assume a Gompertz law for mortality. 3 In this model the probability of survival to age (x + t) conditional on survival at age (x) is denoted by tPx and defined equal to p(T > t I m, b, x) = ,p.~ exp{expI )E, ex./ /l } ,10, where m is the mode, b is the scale parameter, and ~F denotes the time-until-death random variable. For example, when the "mode" of life is m = 80 and the "scale" of life is b = 10, Equation (10) stipulates that the probability a 65-year-old, lives to age 85 is P(T _> 20 I 80, 10, 65) = 0.2404. The probability that a 75-year-old lives to age 85 is e(~ _> lO 180, 10, 75) =0.3527. (The chances of reaching age 85 increase the older you are.) The Gompertz model, with two free parameters, can be "fitted" to any mortality table, which we will do in Section 5. Substituting a value of t ---) 0o in Equation (10), with a finite value for m and b, results in exp {-oo } ~ 0, which confirms the natural boundary condition of human life (you can't live for ever). Likewise, a value of m ---) ~ in Equation (10) results in exp{0} ~ 1, Vt, which we call "the endowment" case. Therefore, the notation tPx can be used, without loss of generality, to include the unconditional (perpetuity) case as well. 3The probability of ruin, and the methodology we describe, can be applied using any analytic mortality law or mortality table, as will become evident in the next section. IX. Is Your Standard of Living Sustainable during Retirement? 89 2.2 Statement of Problem We would like to compute the probability that the netwealth process, defined by Equation (5), "hits" zero while the individual is still living and ever. Mathematically, p~t!~e.= p[ inf W, < O] n l l n " t o _ < / < _ J" (11)