A bond's coupon rate is equal to its yield to maturity if its purchase price is equal to its par value. The par value of a bond is its face value, or the stated value of the bond at the time of issuance, as determined by the issuing entity. Most bonds have par values of $100 or $1,000.

The par value of a bond does not dictate its market price, however. Instead, the market or selling price of a bond is influenced by a number of factors in addition to its par. These factors include the bond's coupon rate, maturity date, prevailing interest rates and the availability of more lucrative bonds.

Defining the Coupon Rate, Maturity Date and Market Value of Bonds

The coupon rate of a bond is its interest rate, or the amount of money it pays the bondholder each year, expressed as a percentage of its par value. A bond with a $1,000 par value and coupon rate of 5% pays $50 in interest each year until maturity.

Suppose you purchase an IBM Corp. bond with a $1,000 face value, and it is issued with semi-annual payments of $10. To calculate the bond's coupon rate, divide the total annual interest payments by the face value. In this case, the total annual interest payment equals $10 x 2 = $20. The annual coupon rate for IBM bond is therefore equal to $20 ÷ $1000 = 2%.

The coupons are fixed; no matter what price the bond trades for, the interest payments always equal $20 per year. So if interest rates went up, driving down the price of IBM's bond to $980, the 2% coupon on the bond will remain unchanged.

A bond's maturity date is simply the date on which the bondholder receives repayment for his investment. At maturity, the issuing entity must pay the bondholder the par value of the bond, regardless of its current market value. This means that if an investor purchases a five-year $1,000 bond for $800, they collect $1,000 at the end of five years in addition to any coupon payments they received during that time.

The market value of bonds has a negative correlation with prevailing interest rates. As interest rates go up, the price of pre-existing bonds goes down. As rates decline, current bonds with higher rates become more valuable. 

For example, if a company issues a $1,000 bond with a 4% interest rate, but the government subsequently raises the minimum interest rate to 5%, then any new bonds being issued have higher coupon payments than the company's initial 4% bond. To entice investors to purchase the bond despite its lower coupon payments, the company has to sell the bond at less than its par value, which is called a discount. If interest rates were to drop to 3%, the pre-existing 4% bond sells for more than its par value, which is called a premium.

Since the market price of bonds is so changeable, it is possible to make a profit in addition to that generated by coupon payments by purchasing bonds at a discount. The yield to maturity of a bond is the rate of return generated by a bond after accounting for its market price, expressed as a percentage of its par value. Considered a more accurate estimate of a bond's profitability than other yield calculations, the yield to maturity of a bond incorporates the gain or loss created by the difference between the bond's purchase price and its par value.

Comparing Bond Coupon Rates and Yields

The coupon rate is often different from the yield. A bond's yield is more accurately thought of as the effective rate of return based on the actual market value of the bond. At face value, the coupon rate and yield equal each other. If you sell your IBM Corp. bond at a $100 premium, the bond's yield is now equal to $20 / $1,100 = 1.82%. Assuming interest rates increased and the price of your bond fell to $980, your yield from selling the bond at a discount will be $20 / $980 = 2.04%  Thus, yield and price are inversely related.

Because coupon payments are not the only source of bond profits, the yield to maturity calculation incorporates the potential gains or losses generated by variations in market price. If an investor purchases a bond for its par value, the yield to maturity is equal to the coupon rate. If the investor purchases the bond at a discount, its yield to maturity is always higher than its coupon rate. Conversely, a bond purchased at a premium always has a yield to maturity that is lower than its coupon rate.

Yield to maturity approximates the average return of the bond over its remaining term. A single discount rate is applied to all future interest payments to create a present value roughly equivalent to the price of the bond. The entire calculation takes into account the coupon rate; current price of the bond; difference between price and face value; and time until maturity. Along with the spot rate, yield to maturity is one of the most important figures in bond valuation.

When a Bond's Yield to Maturity Equals Its Coupon Rate

If a bond is purchased at par, its yield to maturity is thus equal to its coupon rate, because the initial investment is offset entirely by repayment of the bond at maturity, leaving only the fixed coupon payments as profit. If a bond is purchased at a discount, then the yield to maturity is always higher than the coupon rate. If it is purchased at a premium, the yield to maturity is always lower.