Question

You are saving for retirement and you want to have $1,000,000 in 40 years. Suppose you earn an average return of 5% on all of your investments. How much do you need to set aside each year to reach your goal?

You want to blow all of your retirement savings from the previous problem in 20 years from retirement. You now earn only 2% on your investment returns. How much can you spend each year?

Answer #1

(a) Let the amount set aside each year be P

Number of years = n = 40 years

Future value after 40 years = 1000000

rate of returns = r = 5%

Hence, FV = P(1+r)^{n-1} +....+ P(1+r)^{2} +
P(1+r) + P = P[(1+r)^{n} -1]/r

=> 1000000 = P(1.05^{40} -1)/0.05

=> P = $8278.16

(b) All the investment is used up in 20 years

n = 20 years

rate of return = r = 2%

PV = 1000000

Let amount spent each year be P

=> PV = P/(1+r) + P/(1+r)^{2} +....+
P/(1+r)^{n} = P[1- (1+r)^{-n}]/r

=> 1000000 = P(1 - 1.02^{-20})/0.02

=> P = $61156.72

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