Question

Problem 2.2 Effective interest rate Given: The nominal interest rate is 7%. You wish to know the difference in the frequency of compounding Find: The effective (annual) interest rate if the nominal interest rate of 7% is compounded (a) quarterly, (b) monthly, (c) weekly, (d) daily, and (e) continuously. Solution:

Answer 1

The formula for the effective interest rate is:

effective interest rate = (1 + i / n) ^ n - 1

i = 7%

1) compounded quarterly

1 year = 4 quarters so n = 4

eir = (1 + 7% / 4) ^ 4 - 1

eir = (1 + 1.75%) ^ 4 - 1

eir = (1.0175) ^ 4 - 1

eir = 1.071859 - 1

eir = 0.071859 or 7.1859%

2) compounded monthly

1 year = 12 months so n = 12

eir = (1 + 7% / 12) ^ 12 - 1

eir = (1 + 0.005833) ^ 12 - 1

eir = (1.005833) ^ 12 - 1

eir = 1.07229 - 1

eir = 0.07229 or 7.229%

3) compounded weekly

1 year = 52 weeks so n = 52

eir = (1 + 7% / 52) ^ 52 - 1

eir = (1 + 0.001346) ^ 52 - 1

eir = (1.001346) ^ 52 - 1

eir = 1.072458 - 1

eir = 0.072458 or 7.2458%

4) compounded daily

1 year = 365 days or n = 365

eir = (1 + 7% / 365) ^ 365 - 1

eir = (1 + 0.000192) ^ 365 - 1

eir = (1.000192) ^ 365 - 1

eir = 1.072501 - 1

eir = 0.072501 or 7.2501%