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# comparing the interest rates

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

### 2 Answers by Expert Tutors

Maurizio T. | Statistics Ph.D and CFA charterholder with a true passion to teach.Statistics Ph.D and CFA charterholder wi...
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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.

I looked up the term "compound interest rate" online, and the response is:

In my view, it's not reasonable to interpret the problem this way.
Kenneth G. | Experienced Tutor of Mathematics and StatisticsExperienced Tutor of Mathematics and Sta...
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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).

Kenneth, it is quite possible that they meant that and that is the standard approach. But the 1.6% is not qualified as an "annual nominal interest rate". The lack of "annual" leaves the door open to interpretations. Of course, comparing an annual interest rate of 1.6% vs. one of 3.2% (albeit simply compounded) does not even require calculations to tell which one is the best option of the two for this lucky John. Even if one compounded the 1.6% continuously, you would only pay \$500,000*e.016*3 = \$524,585.33.
i.e. only \$24,585.33 of interest vs. a whooping \$48,000 choosing the first option.

Maurizio,
I agree with you. It's not a bad thing that you posted a different interpretation of the problem. It illustrates that there can be alternate interpretations of a mathematical problem. Students need to understand that there is not necessarily one right answer in Mathematics, and that there can be different interpretations of a math problem.

By stating that my interpretation was incorrect (which it might have been if the problem had been worded better) you didn't allow for my interpretation and didn't check to make sure the term “compound interest rate” was well defined, which is why I posted my comment to your solution.

Kenneth,
I do not believe I wrote that your interpretation is incorrect. In fact, I even wrote that "it is quite possible that they meant that"; i.e., what you wrote, and added that, in general, it would be the standard approach as a matter of fact. I just pointed out that the 1.6% lacks the qualification of "annual" which is usually required. I apologize if I gave you that impression.

Ultimately, there is no doubt that the wording of the problem is ambiguous and open to interpretations. As a student, I would have clarified with my teacher/professor/instructor what is the meaning of that 1.6%.
I agree.  Thanks for the great discussion!