Question

What is the nominal annual rate of interest compounded quarterly if a loan of $ 25,000 is repaid in seven years by payments of ​$2000 made at the end of every six​ months?

Answer 1

Loan amount = $25,000
Semiannual payment = $2,000
Number of payments = 14 (7 years)

Let semiannual interest rate be i%

$25,000 = $2,000 * PVIFA(i%, 14)

Using financial calculator:
N = 14
PV = 25000
PMT = -2000
FV = 0

I = 1.5485%

Semiannual interest rate = 1.5485%

Effective annual rate = (1 + Semiannual interest rate)^2 - 1
Effective annual rate = (1 + 0.015485)^2 - 1
Effective annual rate = 1.03121 - 1
Effective annual rate = 0.03121 or 3.121%

Quarterly interest rate = (1 + Effective annual rate)^(1/4) - 1
Quarterly interest rate = (1 + 0.03121)^(1/4) - 1
Quarterly interest rate = 1.007713 - 1
Quarterly interest rate = 0.007713 or 0.7713%

Annual interest rate = 4 * Quarterly interest rate
Annual interest rate = 4 * 0.7713%
Annual interest rate = 3.085% or 3.09%

Therefore, nominal annual interest rate is 3.09% compounded quarterly