John Adams is the CEO of a nursing home in San Jose. He is now 50 years old and

Question

John Adams is the CEO of a nursing home in San Jose.  He is now 50 years old and plans to retire in 10 years.  He expects to live for 25 years after he retires, that is, until he is 85.  He wants a fixed retirement income that has the same purchasing power at the time he retires as $40,000 has today (he realizes that the real value of his retirement income will decline year by year after he retires).  His retirement income will begin the day he retires, 10 years from today, and he will then get 24 additional annual payments.      Inflation is expected to be 5 percent per year for 10 years (ignore inflation after John retires); he currently has $100,000 saved up, and he expects to earn a return on his savings of 8 percent per year, annual compounding.  To the nearest dollar, how much must he save during each of the next 10 years (with deposits being made at the end of each year) to meet his retirement goal?  (Hint: the inflation rate 5 percent per year is used only to calculate desired retirement income.

Explanation / Answer

Value of today's 40,000 in 10 years = 40,000*(1+inflation rate)^10 = 40,000*(1+5%)^10 = 65,155.79

So John gets $65,155.79 10 years from now, and another 24 payments of $65,155.79 each

Present value of this annuity stream of 24 payments of 65,155.79 = annuity amount*(1-1/(1+discount rate)^n)/discount rate = 65,155.79*(1-1/(1+8%)^24)/8% = 686,009.56

Adding the 65,155.79 he gets 10 years from now, the total present value of the 25 payments he gets = 65,155.79 + 686,009.56 = 751,165.35

This is the value of the 25 years of retirement income as of 10 years from now.

So the future value of the 100,000 he has saved up today plus the 10 more years of savings should be equal to $751,165.35

Let John save $X each year for the next 10 years.

Future value of this stream of payments at the end of 10 years = annuity amount*((1+discount rate)^n-1)/discount rate = X*((1+8%)^10-1)/8% = 14.48656*X

So total future value of the savings = 100,000*(1+8%)^10+14.48656*X = 215,892.5+14.48656*X

This should equal to the value calculated earlier, i.e. 751,165.35

So 215,892.5+14.48656*X = 751,165.35

Solving, we get X = 36,949.61

So John has to save $36,949.61 each year for the next 10 years to realize his goal.

Hope this helped ! Let me know in case of any queries.

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