A Sustainable Spending Rate without Simulation

Table 2 reciprocal gamma approximation for ruin

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Table 2. Reciprocal Gamma Approximation for Ruin Probability vs. Exact Results Using Correct Mortality Table Real Annual Spending per $100 of Nest Egg Retirement Age Median Age at Death Hazard Rate, λ $2.0 $3.0 $4.0 $5.0 $6.0 $7.0 $8.0 $9.0 $10.0 NA Infinity 0.00% Approx.: 15.1% 30.0% 45.1% 58.4% 69.4% 77.9% 84.4% 89.1% 92.5% Exact.: 15.1 30.0 45.1 58.4 69.4 77.9 84.4 89.1 92.5 Diff.: 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 50 78.1 2.47 Approx.: 4.27 10.27 18.0 26.8 35.8 44.6 52.8 60.3 66.9 Exact.: 3.04 9.10 17.8 27.7 37.8 47.2 55.5 62.6 68.5 Diff.: 1.2 1.2 0.3 –0.9 –2.0 –2.6 –2.7 –2.3 –1.6 55 83.0 2.48 Approx.: 4.26 10.23 18.0 26.7 35.7 44.5 52.7 60.2 66.8 Exact.: 2.83 8.95 18.0 28.7 39.6 49.9 59.0 66.7 73.0 Diff.: 1.4 1.3 0.0 –2.0 –3.9 –5.4 –6.3 –6.5 –6.3 60 83.4 2.96 Approx.: 3.48 8.54 15.3 23.1 31.4 39.7 47.6 55.0 61.7 Exact.: 1.82 6.36 13.7 22.9 32.9 42.7 51.7 59.6 66.4 Diff.: 1.7 2.2 1.6 0.2 –1.5 –3.0 –4.1 –4.6 –4.6 65 83.9 3.67 Approx.: 2.64 6.68 12.27 18.9 26.2 33.7 41.1 48.3 54.9 Exact.: 1.02 4.03 9.43 16.8 25.3 34.1 42.7 50.5 57.4 Diff.: 1.6 2.7 2.8 2.1 0.9 –0.4 –1.5 –2.2 –2.5 70 84.6 4.75 Approx.: 1.61 4.73 8.95 14.2 20.1 26.5 33.0 39.5 45.8 Exact.: 0.48 2.20 5.71 11.0 17.6 24.9 32.4 39.6 46.4 Diff.: 1.3 2.5 3.2 3.2 2.6 1.6 0.6 –0.1 –0.6 75 85.7 6.48 Approx.: 1.07 2.90 5.69 9.32 13.6 18.5 23.6 29.0 34.4 Exact.: 0.18 0.98 2.89 6.10 10.5 15.8 21.7 27.7 33.7 Diff.: 0.9 1.9 2.8 3.2 3.1 2.6 1.9 1.2 0.7 80 87.4 9.37 Approx.: 0.52 1.47 3.00 5.10 7.71 10.8 14.2 18.0 21.9 Exact.: 0.05 0.34 1.16 2.76 5.20 8.43 12.3 16.6 21.1 Diff.: 0.5 1.1 1.8 2.3 2.5 2.3 1.9 1.4 0.8 NA = not applicable. Notes : Mean arithmetic portfolio return = 7 percent; standard deviation of return = 20 percent; mean geometric portfolio return = 5 percent. Differences may not be exact because of rounding.
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Financial Analysts Journal 96 pubs .org ©2005, CFA Institute To understand the intuition behind the num- bers, recall that the mean or expected value of the SPV of $1 of real spending is 1/( μ σ 2 + λ ), where μ and σ are the investment parameters and λ is the mortality rate parameter induced by a given median remaining lifetime. For a 65-year-old of either sex, the median remaining lifetime is 18.9 years (83.9 median age of death in Table 2 minus actual age of 65) according to the RP-2000 Society of Actuaries mortality table. To obtain the 50 per- cent probability point with an exponential distri- bution, we solve for e –18.9 λ = 0.5, which leads to λ = ln(2)/18.9 = 0.0367 as the implied rate of mortal- ity. The mean value of the SPV for μ = 7 percent and σ = 20 percent works out to 1/(0.07 – 0.04 + 0.0367), which is an average of $15 for the SPV per dollar of desired consumption. Thus, if the retiree intends to spend $90,000 a year, it should come as no surprise that a nest egg of only 11 times this amount is barely sustainable on average. Note that the expected value of the SPV decreases in μ and λ and increases in σ . Higher mean is good, higher volatility is bad, and the benefit of a higher mor- tality rate comes from reducing the length of time over which the withdrawals are taken. Effects of Investment Strategies We are not entering the debate about what are the “right” values for return expectations because our work makes no contribution to answering that important but contentious question. But we can use our model to show the effect of various portfolio composition and return assumptions. The portfolio in Table 2 is an all-equity portfolio with mean return of 7 percent and volatility of 20 percent. As expected, if the mean return is higher or the vola- tility lower, with all else held constant, the sustain- ability improves, and vice versa. What happens, however, if we change both parameters in the same
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