Farmers Bank offers to lend you $50,000 at a nominal rate of 5.0%, simple interest, with interest paid quarterly. Merchants Bank offers to lend you the $50,000, but it will charge 6.0%, simple interest, with interest paid at the end of the year. What's the difference in the effective annual rates charged by the two banks?

The answer is 0.91%

Explanation:

Solution

Farmers Bank:

Lending Amount = $50,000

Nominal rate (APR) = 5.0%

Interest paid = Quarterly (4 periods in a year)

Thus

The effective annual rate (EAR) = (1 + APR/Number of compounding periods a year) ^ (number of compounding periods a year) - 1

= (1 + 5.0%/4) ^4 - 1

= (1 + 0.0125) ^4 - 1

= (1.0125) ^4 - 1

=1.05094533691406 - 1

= 0.5094533691406

= 5.0954%

Therefore the effective annual rate in farmer bank is 5.0954%

Merchants Bank:

Lending Amount = $50,000

Nominal rate (APR) = 6.0%

Interest paid = Annually (1 period in a year)

Thus

The effective annual rate (EAR) = (1 + APR/Number of compounding periods a year) ^ (number of compounding periods a year) - 1

= (1 + 6.0%/1) ^1 - 1

= (1+0.06) ^1 - 1

= (1.06) ^1 - 1

=1.06-1

=0.06 or 6.0000%

Therefore the effective annual rate of the Merchant bank is 6.000%

Now,

The difference between the annual rates=EAR merchant bank - EAR Farmers bank

=6.0000% - 5.0945%

=0.9055% or 0.91%

Therefore the difference between the effective annual rates charged by the two banks is 0.91%