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HOW BOND PRICES VARY WITH INTEREST RATES

As interest rates change, so do bond prices. For example, suppose that investors demanded an interest rate of 6 percent on 3-year Treasury bonds. What would be the price of the Treasury 6s of 2002? Just repeat the last calculation with a discount rate of r = .06:

2 You may have noticed that the semiannually compounded interest rate on the bond is also the bond s APR, although this term is not generally used by bond investors. To find the effective rate, we can use a formula that we presented earlier:

where m is the number of payments each year. In the case of our Treasury bond

3 Why is the present value a bit higher in this case? Because now we recognize that half the annual coupon payment is received only 6 months into the year, rather than at year end. Because part of the coupon income is received earlier, its present value is higher.

Thus when the interest rate is the same as the coupon rate (6 percent in our example), the bond sells for its face value. We first valued the Treasury bond with an interest rate of 5.6 percent, which is lower than the coupon rate. In that case the price of the bond was higher than its face value.

We then valued it using an interest rate that is equal to the coupon and found that bond price equaled face value. You have probably already guessed that when the cash flows are discounted at a rate that is higher than the bond s coupon rate, the bond is worth less than its face value. The following example confirms that this is the case.

Bond Prices and Interest Rates

Investors will pay $1,000 for a 6 percent, 3-year Treasury bond, when the interest rate is 6 percent. Suppose that the interest rate is higher than the coupon rate at (say) 15 percent. Now what is the value of the bond? Simple! We just repeat our initial calculation The bond sells for 79.45 percent of face value but with r = .15:

We conclude that when the market interest rate exceeds the coupon rate, bonds sell for less than face value. When the market interest rate is below the coupon rate, bonds sell for more than face value.

YIELD TO MATURITY VERSUS CURRENT YIELD

Suppose you are considering the purchase of a 3-year bond with a coupon rate of 10 percent. Your investment adviser quotes a price for the bond. How do you calculate the rate of return the bond offers?

For bonds priced at face value the answer is easy. The rate of return is the coupon rate. We can check this by setting out the cash flows on your investment:

Notice that in each year you earn 10 percent on your money ($100/$1,000). In the final year you also get back your original investment of $1,000. Therefore, your total return is 10 percent, the same as the coupon rate.

Now suppose that the market price of the 3-year bond is $1,136.16. Your cash flows are as follows:

What s the rate of return now? Notice that you are paying out $1,136.16 and receiving an annual income of $100. So your income as a proportion of the initial outlay is $100/$1,136.16 = .088, or 8.8 percent. This is sometimes called the bond s current yield.

However, total return depends on both interest income and any capital gains or losses. A current yield of 8.8 percent may sound attractive only until you realize that the bond s price must fall. The price today is $1,136.16, but when the bond matures 3 years from now, the bond will sell for its face value, or $1,000. A price decline (i.e., a capital loss) of $136.16 is guaranteed, so the overall return over the next 3 years must be less than the 8.8 percent current yield.

Let us generalize. A bond that is priced above its face value is said to sell at a premium. Investors who buy a bond at a premium face a capital loss over the life of the bond, so the return on these bonds is always less than the bond s current yield. A bond priced below face value sells at a discount. Investors in discount bonds face a capital gain over the life of the bond; the return on these bonds is greater than the current yield:

Because it focuses only on current income and ignores prospective price increases or decreases, the current yield mismeasures the bond s total rate of return. It overstates the return of premium bonds and understates that of discount bonds.

We need a measure of return that takes account of both current yield and the change in a bond s value over its life. The standard measure is called yield to maturity. The yield to maturity is the answer to the following question: At what interest rate would the bond be correctly priced?

The yield to maturity is defined as the discount rate that makes the present value of the bond s payments equal to its price.

If you can buy the 3-year bond at face value, the yield to maturity is the coupon rate, 10 percent. We can confirm this by noting that when we discount the cash flows at 10 percent, the present value of the bond is equal to its $1,000 face value:

But if you have to buy the 3-year bond for $1,136.16, the yield to maturity is only 5 percent. At that discount rate, the bond s present value equals its actual market price, $1,136.16:

CURRENT YIELD Annual coupon payments divided by bond price.

YIELD TO MATURITY Interest rate for which the present value of the bond s payments equals the price.