Question

You deposit $1,000 at the end of the year (k = 0) into an account that pays interest at a rate of 6% compounded annually. Two years after your deposit, the savings account interest rate changes to 12% nominal interest compounded monthly. Five years after your deposit, the savings account again changes its interest rate, this time the interest rate becomes 8% nominal interest compounded quarterly. Eight years after your deposit, the saving account changes its rate once more to 5% compounded annually. a. How much money should be in the savings account 16 years after the initial deposit, assuming no further changes in the account's interest rate? b. What interest rate, compounded annually, is equivalent to the interest pattern of the saving account in Part (a) over the entire 16 year period?

Answer 1

Deposit amount = $1,000

Interest rate for year 1-2 is 6% compounded annually

Interest rate for year 3-5 is 12% compounded monthly

Interest rate for year 6-8 is 8% compounded quarterly

Interest rate for year 9-16 is 5% compounded annually

Future value of compound interest is:

Amount Invested [1 + (Rate of interest / Compounding done annually)] ^{(Number of years * Compounding done annually)}

a) In two years, $1,000 invested becomes 1,000 (1+0.06)² = 1,123.6

In year 3-5, 1,123.6 becomes 1,123.6 [1 + (0.12/12)]³⁶ = 1,607.61

In year 6-8, 1,607.61 becomes 1,607.61 [1 + (0.08 / 4]¹² = 2,038.83

In year 9-16, 2038.83 becomes 2038.83 [1 + 0.05]⁸ = 3,012.28

Thus after 16 years, $1,000 deposited becomes $3,012.28

b) Assume there is single interest rate of X% per year.

3,012.8 = 1,000 [1 + X]¹⁶

3.0128^(1/16) = 1 + X

X = 7.13%