3. A family friend is planning her retirement from work in the U. S. She is 62 years old right now (time 0) and has the choice of taking her Social Security (a public pension program) according to the following schedule (first payment noted in parentheses): I. Early retirement at age 62 (month 0) $1,300 per month for life II. Regular retirement at age 67 (month 60) $1,800 per month for life III. Delayed retirement at age 70 (month 96) $2,300 per month for life If her expected life expectancy is 92 years old (exactly), what are the present values of the choices? (Assume r = 4% (annual))

the present value of option 1 is:

PV = monthly payment x annuity factor

monthly payment = $1,300

PV annuity factor, 360 periods, 0.3333% = 209.86361

PV = $1,300 x 209.86361 = $272,822.69

the PV₆₀ (at month 60) of option 2 is:

PV₆₀ = monthly payment x annuity factor

monthly payment = $1,800

PV annuity factor, 300 periods, 0.3333% = 189.53178

PV₆₀ = $1,800 x 189.53178 = $341,157.20

now we have to determine the value of this annuity today:

PV₀ = $341,157.20 / 1.04⁵ = $280,406.35

the PV₉₂ (at month 92) of option 3 is:

PV₉₂ = monthly payment x annuity factor

monthly payment = $2,300

PV annuity factor, 264 periods, 0.3333% = 175.449

PV₉₂ = $2,300 x 175.449 = $403,532.70

now we have to determine the value of this annuity today:

PV₀ = $403,532.70 / 1.04⁸ = $294,857.39

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