Bonds are financial instruments that feature regular interest payments and the refund of the original purchase price at maturity. They can be bought and sold on the open market. Determining the market price of a bond requires calculating the present value of the interest and maturity payments. The interest rate used as a discount factor in the present value calculation can be the spot rate or yield to maturity.

## Bond Components

The purchase price of a bond is known as the par value. The interest payments are known as coupons, and they are calculated by multiplying the par value by the bond return and dividing the interest payments over the given coupon frequency. A 20-year bond with a $1,000 par value and a return of 6.0 percent with semiannual coupons will pay $30 in interest every 6 months ($1,000 x 0.06 / 2). At the end of the bond’s 20-year period, you receive the refund of the original $1,000 purchase price, known as the maturity value.

## Market Price

To determine the bond’s market price at any given point, calculate the present value of all the remaining coupon payments added to the present value of the maturity value. If the $1,000 bond with 6.0 percent return and semiannual coupons had 10 years remaining, you would calculate the present value of the remaining 20 coupons of $30 (10 years x semiannually), plus the present value of the $1,000 maturity in 10 years.

## Yield to Maturity

The yield to maturity is the interest rate used over the entire remaining period of the bond to determine the present value of the coupons and the maturity value. It represents the average investment return the bond will generate over the remaining term. For example, with a yield to maturity of 8.0 percent the market price of the bond would be:

PV of 20 $30 coupons at 8.0 percent + PV of $1,000 maturity in 10 years:

$410.50 + $463.19 = $873.69

By paying a lower price than the original $1,000, you increase the yield to maturity on the bond from 6.0 percent to 8.0 percent.

## Spot Rate

The spot rate is similar to the yield to maturity in that it is used to determine the fair market price of the bond. However, the spot rate differs from the yield to maturity in that it can vary from one period to the next as fluctuations in interest rates over the remaining bond period are anticipated. The spot rate can be a truer measure of the bond’s fair market price if interest rates are believed to rise or fall over the coming years. The spot rate can be any rate for any time period in the calculation of the bond price. You may use current rates for a fixed period and then a different rate for the remaining years. For example, you may use an 8.0 percent rate for the first five years and then a 10.0 percent rate for the last five years if interest rates are anticipated to rise. You would then calculate the fair market price of the bond as:

PV of $30 coupons years 1 through 5 at 8.0 percent

plus

PV of $30 coupons years 6 through 10 at 10.0 percent

plus

PV of $1,000 maturity value in 10 years at 8.0 percent years for 1 through 5 and 10.0 percent for years 6 through 10.

\= $244.26 + $158.57 + $422.59 = $825.42

The price using the spot rate is lower because interest rates are anticipated to rise in five years.

### References

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