Dana Reyes: Lump Sum Versus a Fixed Pension Annuity
Dana Reyes, 58, is retiring from a utility and must choose between a lump sum of $320,000 paid immediately and a fixed annuity of $2,100 per month for 20 years, with the first payment one month from now. The annuity has no cost-of-living adjustment and no survivor benefit. Dana is in good health, and her family history suggests a life expectancy beyond 85. She plans to invest the lump sum in a balanced portfolio. Her financial adviser uses a nominal annual discount rate of 5.4%, compounded monthly. The plan's actuary notes that the annuity is backed by the employer, while the lump sum would be exposed to market risk. Dana also has the option to delay taking Social Security, which would increase her benefit by 7% per year of delay. She wants a recommendation that considers both the arithmetic and the risks she would bear.
1. Compute the present value of the annuity at the adviser's discount rate and compare it with the lump sum. Show the annuity factor and the monthly rate.
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Monthly rate = 5.4%/12 = 0.45%. Number of payments = 20 × 12 = 240. PV = $2,100 × [1 − (1.0045)^−240]/0.0045. (1.0045)^240 ≈ 2.9378, so the annuity factor is (1 − 0.3404)/0.0045 ≈ 146.58. PV ≈ $307,800. The lump sum is $320,000, about $12,200 larger. On a pure expected-value basis at the adviser's discount rate, the lump sum wins. But the difference is only about 4%, which is within the uncertainty of the discount rate and mortality assumptions. The comparison also assumes the lump sum is invested at 5.4% net of fees and taxes, and that Dana actually lives to receive all 240 payments. Those are not equivalent risks.
2. At what monthly discount rate are the two options equivalent? What does that rate represent, and how should Dana's health and risk aversion affect the choice?
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Solve for the monthly rate that equates $320,000 to $2,100 × annuity factor(240). The required factor is $320,000/$2,100 = 152.38. At 0.41% monthly the factor is about 152.5; at 0.412% it is about 152.2. Break-even is roughly 0.411% monthly, or 4.94% annually. That rate is the implicit return on the annuity if she lives the full 20 years. If her alternative investment can earn only 4.8%, the annuity is worth about $323,600 and beats the lump sum. The break-even rate is below the adviser's 5.4%, so the recommendation is sensitive to the assumed return. Given her good health and family longevity, the annuity's longevity protection has extra value; a risk-averse retiree should weight the guaranteed income and employer backing, while recognizing credit risk and the lack of survivor benefits.
3. How should expected inflation of 2.5% and the lack of a COLA change the comparison? Be explicit about whether you should double-count inflation by adjusting both the payments and the discount rate.
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Do not double-count expected inflation. The 5.4% nominal discount rate already embeds the market's expected inflation. Discounting fixed nominal payments at a nominal rate is correct. The real issue is unexpected inflation and the absence of a COLA: $2,100 at 2.5% inflation buys only about $1,281 in today's dollars after 20 years. That decline is a transfer of inflation risk to Dana, not a correction to the PV. A useful sensitivity is to compare the annuity with a lump sum invested in TIPS or a balanced portfolio that may hedge inflation. If she strongly fears inflation, the lump sum gains appeal despite its slightly lower PV. If she fears outliving her assets, the annuity's mortality pooling is valuable. The decision should combine PV with risk allocation, not treat the 2.5% as a second discount.