Question

Suppose, Bank X offers you an account with a nominal rate of 3% with semiannual compounding. Bank Y has the same Effective annual rate as Bank X's effective rate, but interest will be compounded monthly. 1. Find EAR for Bank X; 2. Find Nominal rate for Bank Y. 3. Which Bank offers you a better deal on deposits?

Answer 1

Bank X

Nominal rate = 3% with semiannual compounding
Thus semiannual rate = 3%/2 = 1.5%
Thus if we deposit $1000, the bank will give $15 (1.5% of $1000 ) at the end of first 6 months period.
We now have $1000 + $15 = $ 1015.
At the end of the second 6 months period the bank will give $15.225 = (1.5% of $1000 + $15)

Thus effective interest = $15 + $15.225 = $30.225
Thus effective interest rate = ($30.225 / $1000) x 100
Thus effective interest rate = 3.0225%

Mathematically:

Effective interest Rate i = [1+ (r/m)]^m - 1
r = Nominal Interest Rate
m = No. of compounding periods per year.

i = [1+ (0.03/2)]^2 - 1
i = 3.0225%

Bank Y

EAR for Bank Y = EAR for Bank X = 3.0225%
Compounding Frequency = Monthly ; Thus m = 12
Thus using the above formula

Effective interest Rate i = [1+ (r/m)]^m - 1

3.0225% = [1+ (r/12)]^12 - 1
r = 3.1782%

Thus Nominal Interest Rate for Bank Y = 3.1782%

Comparing both of them we get that Bank Y is giving a better deal on deposit than Bank X in terms of both EAR & Nominal Interest Rate