# Definition of compound interest

The textbook by John Hull (9/e) describes "quarterly compounding" on page 80:

If the interest rate is measured with annual compounding, the bank’s statement that the interest rate is 10% means that \$100 grows to

\$100 x 1.1 = \$110

at the end of 1 year...When the interest rate is measured with quarterly compounding, the bank’s statement [that the interest rate is 10%] means that 2.5% is earned every 3 months, with the interest being reinvested. The \$100 then grows to

\$100 x (1.025)^4 = \$110.38

at the end of 1 year

Then in example 4.2 on p. 82, he gives the following calculation which seems at odds with the definition of quarterly compounding (it doesn't reinvest the interest):

Suppose that a lender quotes the interest rate on loans as 8% per annum with continuous compounding, and that interest is actually paid quarterly. ..[T]he equivalent rate with quarterly compounding is

4 x (e^(0.08/4) - 1) = 0.0808

or 8.08% per annum. This means that on a \$1,000 loan, interest payments of \$20.20 would be required each quarter.

Given the earlier definition of quarterly compounding, at a rate of 8.08% with quarterly compounding I would expect my balance at the end of one year to be

\$1000 x (1.0202)^4 = \$1083.28

However, my balance is actually \$1000 + \$20.20*4 = \$1080.80. Then, using the earlier definition of quarterly compounding, my actual quarterly compounded rate is

\$1000 x (1 + R/4)^4 = \$1080.80 => R = 7.85%

which is not 8.08% as described.

Is this example indeed inconsistent with his definition, or am I missing something?

• The third example is continuous compounding, not quarterly. – Pete Becker Jun 24 '19 at 13:05

Is this example indeed inconsistent with his definition, or am I missing something?

It sounds like you have an 8.08% APY (annual percentage yield), not 8.08% APR (annual percentage rate).

With APY they have already taken compounding into account, which is convenient for comparison purposes and also looks better in advertisements than the lower APR. With \$1,000 at 8.08% APY it's just \$1,000 * 0.0808 = \$80.80 divided by however many periods they are paying it out.

If you pay off all the interest as it becomes due, then the amount borrowed/owed doesn't change. If the amount borrowed doesn't changed, then neither does the interest due.

The same would be the case if you withdrew the interest from a savings account. Even though the interest would compound if you didn't withdraw it, so long as you withdraw it as it is earned, you will never earn interest on interest.

There are many different ways of adding interest. For example, you could calculate it daily, but only add it quarterly. It could even be added quarterly, but only compounded yearly - that would give equal interest payments for each quarter.

You would have to look very closely at your contract to see which compounding method they are applying.

• "It could even be added quarterly, but only compounded yearly - that would give equal interest payments for each quarter." Equal payment is what the example ends up with despite quarterly compounding. Should we not have different payments as the interest is reinvested in accordance with quarterly compounding? – bcf Jun 23 '19 at 21:21

If you apply the formula for continuous compounding, you'll see the amount of interest owed on the debt increasing in the four quarters by (rounding to the nearest cent)

``````20.20
20.61
21.03
21.45
-----
83.29
``````

which is consistent with what you expect, but this is for the case that interest compounds all year long and the borrower makes no payments.

However, the second quoted example describes the borrower paying an interest-only payment on a loan every quarter. So, with an initial balance of 1000.00 and daily compounding, at the end of the first quarter the borrower owes 1020.20. At this time the borrower makes an interest-only payment of 20.20, so there is no further compounding on that interest.

The second quarter begins with a balance of 1000.00 and interest begins accruing on that balance with daily compounding, resulting in the same 20.20 in interest for the quarter, and similarly for the subsequent quarters.

\$1,000 at 8% compounded daily is \$1,083.28 at the end of a year. When it's paid is irrelevant.

If you are being charged a different amount on something ("my balance" ?), check your contract for the terms.

• Doesn’t “compounded daily” effectively mean it’s paid daily? Otherwise, it could only compound after it has been paid? Not quite sure what you mean by “When it’s paid is irrelevant.” statement. – Peter K. Jun 23 '19 at 21:14
• @Bob Baerker Indeed, I expect \$1,083.28 at the end of the year; however I end up with only \$1,080.80, which as I show implies only a 7.85% rate (quarterly compounded) – bcf Jun 23 '19 at 21:23
• Peter K, You might be too young to have had a passbook savings account that compounded daily but paid monthly. @bcf - I'm confused by your confusion. 7.85% rate compounded quarterly is \$1,080.84 and 8% compounded daily is \$1,083.28 – Bob Baerker Jun 23 '19 at 21:35
• @BobBaerker The interest rate is quoted as 8.08% with quarterly compounding, which means (I thought) in each quarter (since payments are also made quarterly) my balance would increase to \$1000*(1.0202) after Q1 payment, 1000*(1.0202)^2 after Q2 payment, 1000*(1.0202)^3 after Q3 payment, and 1000*(1.0202)^4 = \$1,083.28 after Q4 (and final) payment. However, my balance after the final payment is only \$1000 + \$20.20*4 = \$1080.80. The reasoning to get to the \$20.20 quarterly payments (and thus the \$1080.80 total) is my confusion. – bcf Jun 23 '19 at 22:28
• @BobBaerker You were too young to understand it, perhaps? What was happening in those was that the interest was calculated daily, but compounded monthly. I was taught that compounding means earning interest on interest, which can't happen if money isn't added? Plus, I have no idea what relevance your comment `7.85% rate compounded quarterly is \$1,080.84 and 8% compounded daily is \$1,083.28` has to the discussion? – Peter K. Jun 23 '19 at 22:52