Question

Bank offers following type of saving accounts requiring initial deposit of PLN 1,000,000: * account A - compunding interest account offering 8% pa under quarterly compunding (incl. interest accrued quarterly) opened for 3 years, * account B - simple interest account opened for 3 years. What should be the rate offered for account B so that it would give the same amount of total interest (income) after saving account is closed?

Answer 1

Account A :

Initial deposit amount = PL 1000,000

Interest rate = 8% p.a. Compounding quarterly

Deposit term = 3 years

First of all we need to make the frequency of interest rate compounding yearly since deposit is made for 3 periods of each period of 1 year

We know that effective annual interest rate is:

(1+APR/n)ⁿ -1

Here n = number of times interest rate is Compounding

Hence effective annual rate is:

= (1+0.08/4)⁴-1

= (1+0.02)⁴ -1

= 1.08243216 -1

=0.08243216

= 8.243216%

We know that interest amount in case of Compounding interest :

Deposit x[ (1+r)ⁿ - 1]

Here r = compounding interest rate  

n = number of periods for which deposit is made.

Hence interest amount is:

= 1,000,000 x [(1+0.08243216)³ -1)]

= 1,000,000 x (1.2682417946-1)

= 1,000,000x 0.2682417946

=$268241.79

We know that interest amount at simple interest rate r is calculated as follows:

Deposit amount x r x number of years

We want to calculate a simple interest rate that gives the same amount of interest as we are getting in interest rate 8% p.a. Compounding quarterly.

i.e. deposit amount x simple interest rate x number of years = 268241.79

i.e. 1,000,000 x simple interest rate x 3 = 268241.79

i.e. simple interest rate = 2468241.79/(1,000,000x3)

i.e. simple interest rate = 2468241.79/3,000,000

= 0.08941393

i.e. simple interest rate = 8.941393%

i.em simple interest rate = 8.94% (rounded off to two decimal places)

Hence if account B offer 8.94% simple interest, it will give the same amount of total interest as the account A will give when after 3 years account is closed.