Question

3. A family friend is planning her retirement from work in the U.S. She is 61 years old right now (time 0) and has the choice of taking her Social Security (a public pension program) in time 1, in time 5 or in time 9 according to the following schedule:

I. Early retirement (Age 62 exactly) $1,200 per month for life

II. Regular retirement (Age 66 exactly) $1,800 per month for life

III. Delayed retirement (Age 70 exactly) $2,500 per month for life

If her expected life expectancy is 88 years old (exactly), what are the present values of the choices? (Assume r = 4% (annual))

Answer 1

Answer I:

Early retirement (Age 62 exactly) $1,200 per month for life

Her expected life expectancy is = 88 years

Number of months = (88 - 62) * 12 = 312

Monthly interest = 4% / 12

Present value at age 62 = PV (rate, nper, pmt, fv, type) = PV (4%/12, 312, -1200, 0, 0) = $232,536.05726

Present Value at age 61 = PV = FV / (1 + Periodic Interest rate) ^{Number of periods} = 232536.05726 / (1 + 4%/12) ¹² = $223,433.05

Present values of the choice I (now at age 61) = $223,433.05

Answer II:

Regular retirement (Age 66 exactly) $1,800 per month for life

Number of months = (88 - 66) * 12 = 264

Present value at age 66 = PV (rate, nper, pmt, fv, type) = PV (4%/12, 264, -1800, 0, 0) = $315689.2078

Number of months (age 61 to age 66) = 5 * 12 = 60

Present Value at age 61 = PV = FV / (1 + Periodic Interest rate) ^{Number of periods} =315689.2078 / (1 + 4%/12) ⁶⁰ = $258,550.44

Present values of the choice II (now at age 61) = $258,550.44

Answer III:

Delayed retirement (Age 70 exactly) $2,500 per month for life

Number of months = (88 - 70) * 12 = 216

Present value at age 70 = PV (rate, nper, pmt, fv, type) = PV (4%/12, 216, -2500, 0, 0) = $384498.4403

Number of months (age 61 to age 70) = 9 * 12 = 108

Present Value at age 61 = PV = FV / (1 + Periodic Interest rate) ^{Number of periods} = 384498.4403 / (1 + 4%/12) ¹⁰⁸ = $268,416.10

Present values of the choice III (now at age 61) = $268,416.10

Hence:

Present values of the choice I (now at age 61) = $223,433.05

Present values of the choice II (now at age 61) = $258,550.44

Present values of the choice III (now at age 61) = $268,416.10