How long would it take $1,000 to double if it were invested in a bank that pays 6 percent per year? How long would it take if the rate were 10 percent?

FINDING THE NUMBER OF YEARS

We sometimes need to know how long it will take to accumulate a given sum of money, given our beginning funds and the rate we will earn on those funds. For example, suppose we believe that we could retire comfortably if we had $1 mil- lion, and we want to find how long it will take us to have $1 million, assuming we now have $500,000 invested at 4.5 percent. We cannot use a simple formula— the situation is like that with interest rates. We could set up a formula that uses logarithms, but calculators and spreadsheets can find N very quickly. Here’s the calculator setup:

Enter I/YR 4.5, PV 500000, PMT 0, and FV 1000000. Then, when we press the N key, we get the answer, 15.7473 years. If you plug N 15.7473 into the FV formula, you can prove that this is indeed the correct number of years: FV PV(1 I)N $500,000(1.045)15.7473 $1,000,000 We would also get N 15.7473 with a spreadsheet.

• How long would it take $1,000 to double if it were invested in a bank that pays 6 percent per year? How long would it take if the rate were 10 percent? (11.9 years; 7.27 years)
  
• Microsoft’s 2004 earnings per share were $1.04, and its growth rate during the prior 10 years was 24.1 percent per year. If that growth rate were maintained, how long would it take for Microsoft’s EPS to double? (3.21 years)