A Sustainable Spending Rate without Simulation

The problems with the growing number of these and

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The problems with the growing number of these and similar studies based on Monte Carlo simulations—which are intellectually motivated by the “game of life” simulations envisioned by Markowitz (1991)—are that they (1) are difficult to replicate, (2) conduct only a minimal number of simulations, and (3) provide little pedagogical intu- ition on the financial trade-off between retirement risk and return. 2 Arnott (2004, p. 6) claimed that “our industry pays scant attention to the concept of sustainable spending, which is key to effective strategic plan- ning for corporate pensions, public pensions, foundations, and endowments—even for individ- uals.” Financial advisors continue to test the sus- tainability of spending strategies, but the financial literature lacks a coherent modeling framework on which to base the discussion. Moshe Milevsky is associate professor of finance at the Schulich School of Business at York University, Toronto, and executive director of the Individual Finance and In- surance Decisions ( IFID) Centre, Toronto. Chris Robin- son is associate professor of finance at the Atkinson School of Administrative Studies, York University, Toronto. R
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Financial Analysts Journal 90 pubs .org ©2005, CFA Institute We provide an intuitive and consistent plan- ning model by deriving an analytic relationship between spending, aging, and sustainability in a random portfolio environment. We introduce the concept of stochastic present value (SPV) and an expression for the probability that an initial corpus or investment (nest egg) will be depleted under a fixed consumption rule when both rates of return and time until death are stochastic. And, in contrast to almost all other authors who have tackled this problem, we do not depend on Monte Carlo simu- lations or historical (bootstrap) studies. Instead, we base the analysis on the SPV and a continuous-time approximation under lognormal returns and expo- nential lifetimes. In the case of a foundation or endowment with an infinite horizon (perpetual consumption), this formula is exact. In the case of a random finite future lifetime (the situation of a retiree), the for- mula is based on moment-matching approxima- tions, which target the first and second moments of the “true” stochastic present value. The results are remarkably accurate when compared with more costly and time-consuming simulations. We provide numerical examples to demon- strate the versatility of the closed-form expression for the SPV in determining sustainable withdrawal rates and their respective probabilities. This for- mula, which can easily be implemented in Excel, produces results that are within the standard error of extensive Monte Carlo simulations. 3 The Retirement Finances Triangle The main qualitative contribution of this article can be understood by reference to the triangle in Figure 1 . It provides a graphical illustration of the relationships among the three most important fac- tors in retirement planning: spending rates, invest- ment asset allocation, and mortality (determined by gender and age). We link these three factors in
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