Question

2.48(c)

2.4 8 You have $50 000 to invest in the stock market and have sought the expertise of Adam, an experienced colleague who is willing to advise you, for a fee. Adam informs you he has found a one-year investment that provides 15 percent interest, compounded monthly (a) What is the effective annual interest rate based on a 15 percent nominal annual rate and monthly compounding? (b) Adam says he will make the investment for a modest fee of 2 percent of the investment's value one year from now. If you invest the $50 000 today, how much will you have at the end of one year (before Adam's fee)? (c) What is the effective annual interest rate of this investment, including Adam's fee?

Answer 1

(a)

Effective interest rate (EAR) = [1 + (r/N)]^N - 1, where N: Number of compounding periods

(a) r = 15% = 0.15, N = 12

EAR = [1 + (0.15/12)]¹² - 1 = (1 + 0.0125)¹² - 1 = (1.0125)¹² - 1 = 1.1608 - 1 = 0.1608 = 16.08%

(b) Monthly nominal interest rate = 15%/12 = 1.25% and number of months in 1 year = 12

Value after 1 year = $50,000 x F/P(1.25%, 12) = $50,000 x (1.0125)¹² = $50,000 x 1.1608 = $58,040

(c) Adam's fee = $58,040 x 2% = $1,160.8

Net value received = $(58,040 - 1,160.8) = $56,879.2

Annual nominal interest rate (r) = ($56,879.2 / $50,000) - 1 = 1.1376 - 1 = 0.1376 = 13.76%

r = 13.76% = 0.1376, N = 12

EAR = [1 + (0.1376/12)]¹² - 1 = (1 + 0.0115)¹² - 1 = (1.0115)¹² - 1 = 1.1466 - 1 = 0.1466 = 14.66%