A couple will retire in 40 years; they plan to spend about $31,000 a year in retirement, which should last about 20 years. They believe that they can earn 7% interest on retirement savings.
a. If they make annual payments into a savings plan, how much will they need to save each year? Assume the first payment comes in 1 year. (Do not round intermediate calculations. Round your answer to 2 decimal places.)
b. How would the answer to part (a) change if the couple also realize that in 15 years they will need to spend $61,000 on their child’s college education? (Do not round intermediate calculations. Round your answer to 2 decimal places.)
| a] | Amount required to be accumulated at the time of retirement = PV of the annuity of $31,000 to be received for 20 years, discounted at 7% = 31000*(1.07^20-1)/(0.07*1.07^20) = | $ 328,414.44 |
| The above amount is the FV of the annual deposits to be made for 40 years. Hence, the annual deposits = 328414.44*0.07/(1.07^40-1) = | $ 1,645.07 | |
| b] | PV [at t0] of 328414.44 = 328414.44/1.07^40 = | $ 21,931.64 |
| PV [at t0] of 61000 payable at t15 = 61000/1.07^15 = | $ 22,109.21 | |
| Total PV of amounts to be made up | $ 44,040.85 | |
| Amount to be deposited annually for 40 years = 44040.85*0.07*1.07^40/(1.07^40-1) = | $ 3,303.47 |
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