Question

If you want to be a millionaire by age 70, and you start saving $50 a month at age 18 (with no previous savings), what expected rate of return do you need to reach your goal? A. Between 8% and 8.9% B. Between 9% and 9.9% C. Between 10% and 10.9% D. 11% or more

Answer 1

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Answer:

B. Between 9% and 9.9%

Explanation:

The equation for the sum of a series of monthly payments P earning interest rate r compounded monthly for t years is given by ...

A = P((1 +r/12)^(12t) -1)/(r/12)

The number of years in this case is ...

70 -18 = 52

We want to find r such that ...

1,000,000 = 50·12((1 +r/12)^(12·52) -1)/r

10,000/6 = ((1 +r/12)^624 -1)/r

We can try some values for r to see what may work.

r = 9%

((1 +.09/12)^624 -1)/.09 = 1165.60 . . . not quite enough

r = 10%

((1 +0.10/12)^624 -1)/0.10 = 176408 . . . > 1666.67

The required interest rate will be between 9% and 10%, option B.

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In general, finding the exact interest rate is an iterative process. A financial calculator can do it for you. The result is near 9.86%. The graph shows a solution using a graphing calculator.