### DEFINITION of Interest-On-Interest

Interest-on-interest is the interest that is earned when interest payments are reinvested. Interest-on-interest is primarily used in the context of bonds which have their coupon payments assumed to be re-invested at some interest rate and held until the bond is sold or matures.

Interest-on-interest is also called compound interest.

### BREAKING DOWN Interest-On-Interest

An example of a financial security that pays investors interest-on-interest is the U.S. Savings bond, issued by a governmental body to raise funds from the public to fund its capital projects and other operations necessary to manage the economy. The savings bonds are zero-coupon bonds that do not pay interest until they are redeemed or until maturity date. 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 maturity date is issued electronically to the bondholder’s designated bank account.

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. A Treasury bond is an example of a debt security that pays simple interest. For example, 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 Treasury 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 the value of the bond. Using our example above, the first interest earned on the 10-year bond is $250. For the second period, interest will be calculated on the increased value of the bond. In this case, 2.5% x ($10,000 + $250) = 2.5% x $10,250 = $256.25 is the interest earned for the second compounding period. Therefore, 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. Each interest earned is added back to the principal value for which the next interest is calculated on.

Interest-on-interest can be calculated using this formula: P [(1 + i)^{n} – 1]

Where P = principal value

n = number of compounding periods

An investor who holds this 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 is higher than a bond that pays simple interest which will earn $5,000 (calculated as $500 x 10 years, or $250 x 20 compounding periods) over the life of the bond. When calculating interest-on-interest, 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.

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