comparing the interest rates

Question

John needs to borrow $500,000 for three years. Which of the following option is more beneficial for him?
(i) 3.2 % simple interest rate
(ii) 1.6% compound interest rate when it is compounded semi annually

Answer 1

Yousaf, you need to compare how much John is going to pay under either alternative:

 

1) 3.2% simple interest rate.    What John pays to borrow $500,000 for 3 years is equal to 

     $500,000*3.2%*3 = $48,000

 

2) 1.6% compound interest rate, compounded semi-annually.

Note that 1.6% is not the annual interest rate applied, but is already given to you as the compounded interest rate applied semi-annually. Otherwise it is clearly better to use option 2 as you pay half the interest rate than in option 1!  

In this case, one pays interest rates every 6 months. More specifically, one pays 1.6% every 6 months. In a period of 3 years, there are 6 periods of 6 months each.  So, John pays:

      $500,000*[(1+1.6%)^2*3 -1] = $49,961.45.

 

The conclusion is that John is better off paying a simple interest rate as it pays less over the assigned period of 3 years.

Answer 2

If I am understanding the problem, the interest rate in option ii is 1.6% ANNUALLY compounded every 6 months.  This is usually the convention when stating compound interest rates - the rate quoted is the yearly rate, and not the rate for one period.  So you divide the rate by the number of periods per year to find the rate per period.

 

If John borrows at the simple rate of 3.2% per year, assuming it's not compounded yearly, then over three years he will pay 9.6% interest or .096x500,000 = 48,000 dollars interest.

 

If john borrows at 1.6% compounded semi-annually, then he will get charged .8% semi-annually compounded semi-annually.  The number of compounding periods will be 6. That will come to 500000(1+.008)^6 - 500000 = $24,485.15 

 

So John  will save $23,514.84 by selecting borrowing option (ii).