## An Example Bond Pricing

We discussed earlier an 8% coupon, 30-year maturity bond with par value of \$1,000 paying 60 semiannual coupon payments of \$40 each. Suppose that the interest rate is 8% annually, or r = 4% per six-month period. Then the value of the bond can be written as

= \$40 X Annuity factor (4%, 60) + \$1,000 X PV factor (4%, 60)

It is easy to confirm that the present value of the bond's 60 semiannual coupon payments of \$40 each is \$904.94 and that the \$1,000 final payment of par value has a present value of \$95.06, for a total bond value of \$1,000. You can either calculate the value directly from equation 14.2, perform these calculations on any financial calculator,4 or use a set of present value tables.

In this example, the coupon rate equals yield to maturity, and the bond price equals par value. If the interest rate were not equal to the bond's coupon rate, the bond would not sell at par value. For example, if the interest rate were to rise to 10% (5% per six months), the bond's price would fall by \$189.29 to \$810.71, as follows:

\$40 X Annuity factor(5%, 60) + \$1,000 X PV factor(5%, 60)

At a higher interest rate, the present value of the payments to be received by the bondholder is lower. Therefore, the bond price will fall as market interest rates rise. This illustrates a crucial general rule in bond valuation. When interest rates rise, bond prices must fall because the present value of the bond's payments are obtained by discounting at a higher interest rate.

Figure 14.3 shows the price of the 30-year, 8% coupon bond for a range of interest rates, including 8%, at which the bond sells at par, and 10%, at which it sells for \$810.71. The negative slope illustrates the inverse relationship between prices and yields. Note also from the figure (and from Table 14.2) that the shape of the curve implies that an increase in the

3 Here is a quick derivation of the formula for the present value of an annuity. An annuity lasting T periods can be viewed as a equivalent to a perpetuity whose first payment comes at the end of the current period less another perpetuity whose first payment comes at the end of the (T + 1)st period. The immediate perpetuity net of the delayed perpetuity provides exactly T payments. We know that the value of a \$1 per period perpetuity is \$1/r. Therefore, the present value of the delayed perpetuity is \$1/r discounted

for T additional periods, or - X —-—. The present value of the annuity is the present value of the first perpetuity minus the

present value of the delayed perpetuity, or - I 1--1 ).

4 On your financial calculator, you would enter the following inputs: n (number of periods) = 60; FV (face or future value) = 1000; PMT (payment each period) = 40; i (per period interest rate) = 4%; then you would compute the price of the bond (COMP PV or CPT PV). You should find that the price is \$1,000. Actually, most calculators will display the result as minus \$1000. This is because most (but not all) calculators treat the initial purchase price of the bond as a cash outflow.

CHAPTER 14 Bond Prices and Yields

Figure 14.3 The inverse relationship between bond prices and yields. Price of an 8% coupon bond with 30-year maturity making semiannual payments.

Bond price

Bond price

Interest rate

Interest rate

Table 14.2 Bond Prices at Different Interest Rates (8% coupon bond, coupons paid semiannually)

Bond Price at Given Market Interest Rate

Bond Price at Given Market Interest Rate

Table 14.2 Bond Prices at Different Interest Rates (8% coupon bond, coupons paid semiannually)

 Time to Maturity 4% 6% 8% 10% 12% 1 year 1,038.83 1,029.13 1,000.00 981.41 963.33 10 years 1,327.03 1,148.77 1,000.00 875.35 770.60 20 years 1,547.11 1,231.15 1,000.00 828.41 699.07 30 years 1,695.22 1,276.76 1,000.00 810.71 676.77

interest rate results in a price decline that is smaller than the price gain resulting from a decrease of equal magnitude in the interest rate. This property of bond prices is called convexity because of the convex shape of the bond price curve. This curvature reflects the fact that progressive increases in the interest rate result in progressively smaller reductions in the bond price.5 Therefore, the price curve becomes flatter at higher interest rates. We return to the issue of convexity in Chapter 16.

CONCEPT CHECK ^ QUESTION 2

Calculate the price of the bond for a market interest rate of 3% per half year. Compare the capital gains for the interest rate decline to the losses incurred when the rate increases to 5%.

Corporate bonds typically are issued at par value. This means that the underwriters of the bond issue (the firms that market the bonds to the public for the issuing corporation) must choose a coupon rate that very closely approximates market yields. In a primary issue of bonds, the underwriters attempt to sell the newly issued bonds directly to their customers. If the coupon rate is inadequate, investors will not pay par value for the bonds.

After the bonds are issued, bondholders may buy or sell bonds in secondary markets, such as the one operated by the New York Stock Exchange or the over-the-counter market, where most bonds trade. In these secondary markets, bond prices move in accordance with market forces. The bond prices fluctuate inversely with the market interest rate.

5 The progressively smaller impact of interest increases results from the fact that at higher rates the bond is worth less. Therefore, an additional increase in rates operates on a smaller initial base, resulting in a smaller price reduction.

### PART IV Fixed-Income Securities

The inverse relationship between price and yield is a central feature of fixed-income securities. Interest rate fluctuations represent the main source of risk in the fixed-income market, and we devote considerable attention in Chapter 16 to assessing the sensitivity of bond prices to market yields. For now, however, it is sufficient to highlight one key factor that determines that sensitivity, namely, the maturity of the bond.

A general rule in evaluating bond price risk is that, keeping all other factors the same, the longer the maturity of the bond, the greater the sensitivity of price to fluctuations in the interest rate. For example, consider Table 14.2, which presents the price of an 8% coupon bond at different market yields and times to maturity. For any departure of the interest rate from 8% (the rate at which the bond sells at par value), the change in the bond price is smaller for shorter times to maturity.

This makes sense. If you buy the bond at par with an 8% coupon rate, and market rates subsequently rise, then you suffer a loss: You have tied up your money earning 8% when alternative investments offer higher returns. This is reflected in a capital loss on the bond— a fall in its market price. The longer the period for which your money is tied up, the greater the loss, and correspondingly the greater the drop in the bond price. In Table 14.2, the row for one-year maturity bonds shows little price sensitivity—that is, with only one year's earnings at stake, changes in interest rates are not too threatening. But for 30-year maturity bonds, interest rate swings have a large impact on bond prices.

This is why short-term Treasury securities such as T-bills are considered to be the safest. They are free not only of default risk, but also largely of price risk attributable to interest rate volatility.

We have noted that the current yield of a bond measures only the cash income provided by the bond as a percentage of bond price and ignores any prospective capital gains or losses. We would like a measure of rate of return that accounts for both current income and the price increase or decrease over the bond's life. The yield to maturity is the standard measure of the total rate of return of the bond over its life. However, it is far from perfect, and we will explore several variations of this measure.

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