## What Is Interest-On-Interest?

Interest-on-interest, also referred to as 'compound interest', is the interest that is earned when interest payments are reinvested. Interest-on-interest is primarily used in the context of bonds, whose coupon payments are assumed to be re-invested and held until the bond is sold or matures.

### Key Takeaways

- Interest-on-interest, also referred to as 'compound interest', is the interest that is earned when interest payments are reinvested.
- It is primarily used in the context of bonds, whose coupon payments are assumed to be re-invested and held until sale or maturity.
- Interest-on-interest applies to the principal amount of the bond or loan and to any other interest that has previously accrued.
- Simple interest, on the other hand, is only charged on the original principal amount.

## Understanding Interest-On-Interest

An example of a financial security that pays investors interest-on-interest is the U.S. Savings bond, which is issued by a governmental body to raise funds from the public to fund its capital projects and other operations necessary to manage the economy.

These savings bonds are zero-coupon bonds that do not pay interest until they are redeemed or mature. The interest compounds semi-annually and accrues monthly every year for 30 years. Every six months, the monthly interest calculation is adjusted to include the accrued interest from the previous six months.

An investor who purchases the bond at the end of the month will still receive the interest accrued for the entire month since the Treasury only counts full months. Any interest paid at redemption or the maturity date is then issued electronically to the bondholder’s designated bank account.

## Interest-On-Interest vs. Simple Interest

Interest-on-interest differs from simple interest. While interest-on-interest applies to the principal amount of the bond or loan and to any other interest that has previously accrued, simple interest is only charged on the original principal amount.

### Examples of Interest-On-Interest vs. Simple Interest

Consider a bond issued with a $10,000 par value and 10 years to maturity. The interest rate on the bond is 5% and compounds semi-annually. If this bond was a simple interest-paying Treasury Bond (T-Bond) or conventional corporate bond, investors will receive (5%/2) x $10,000 = 2.5% x $10,000 = $250 each payment period. In sum, they would receive $500 per year in interest income. Notice how the interest only applies to the par value or principal amount.

On the other hand, if the bond was, say, a Series EE bond (a type of U.S. Savings bond), the interest calculated for a period is added to the interest earned and accumulated from prior periods. Since the savings bond does not pay interest until it matures, any interest earned is added back to the principal amount of the bond, increasing its value.

With interest-on-interest, each interest payment earned is added back to the principal value for which the next interest is calculated.

Using our example above, the first interest earned on the 10-year bond is $250. For the second period, interest will then be calculated on the increased value of the bond. In this case, the interest earned for the second compounding period is: 2.5% x ($10,000 + $250) = 2.5% x $10,250 = $256.25.

So, in the first year an investor holding this bond will earn $250 + $256.25 = $506.25. The third interest can be calculated as 2.5% x ($10,250 + 256.25) = $262.66, and so on.

## Calculating Interest-On-Interest

Interest-on-interest can be calculated using the following formula: P [(1 + i)n – 1]

Where P = principal value

i = nominal annual interest rate

n = number of compounding periods

If we use this formula on the example above, we can see that an investor who holds the bond until it matures after 10 years (or 20 payment periods) will earn:

Interest-on-interest = $10,000 x (1.025^{20} – 1)

= $10,000 x (1.6386 – 1)

= $10,000 x 0.638616

= $6,386.16

This figure comes in higher than the bond that pays simple interest. That particular bond would have earnt $5,000 instead (calculated as $500 x 10 years, or $250 x 20 compounding periods) over its lifespan.

For simplification, the interest rate used to calculate interest-on-interest can be the bond's yield at the time the coupon payment is made.

## Special Considerations

Interest-on-interest is an important consideration an investor must make when analyzing potential investments and forecasting an investment's total cash return.

When calculating interest-on-interest, it's important to remember that the number of compounding periods makes a significant difference. The basic rule is that the higher the number of compounding periods, the greater the amount of interest-on-interest.