The Foundation
5.4 Bond Pricing Relationships
Concept Check 5.3
1 What are the two distinct types of cash flows associated with a straight bond?
2 What is the meaning of default risk and call risk?
3 What are the three adjustments needed to convert the formula for an annual-pay bond into a semiannual-pay bond?
4 Why does a zero coupon bond typically sell at a discount to its par value?
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This inverse relationship should not be surprising because a bond’s value is the present value of the bond’s future cash flows, and present values have an inverse relationship with interest rates. What becomes important to bond investors is the sensitivity of a bond’s price to changes in interest rates, commonly referred to as interest rate risk or price volatility of a bond. We take a closer look at interest rate risk later in this section. Example 5.7 illustrates the price changes for a bond as the required rate of return (market yield) changes.
Example 5.7 Changes in Required Returns (Yields)
Returning to Example 5.5, Potomac Corporation has a 10-year, $1,000 par, 10 percent semiannual-pay bond. What is the value of the bond to an investor with a required rate of return of 8 percent, 10 percent, and 12 percent?
Solution: Use the BA II PLUS® calculator to determine the value of this bond. This semiannual-pay bond requires doubling the number of periods (n = 2 × 10 = 20), reducing the coupon by half (PMT = 100/2 = 50), and reducing the required rate of return by half (I/Y = 8/2 = 4) and keying in:
Required rate Inputs Output
of return (%)
8 20 N; 4 I/YR; 50 +/− PMT; FV +/− 1,000; CPT PV 1,135.90 10 20 N; 5 I/YR; 50 +/− PMT; FV +/− 1,000; CPT PV 1,000.00 12 20 N; 6 I/YR; 50 +/− PMT; FV +/− 1,000; CPT PV 885.30
The results show several important bond pricing properties. One property is that if the required rate of return equals the stated coupon rate, the bond price, Po, will equal the maturity or par value, M. This relationship holds regardless of the remaining term to matu- rity. Using the financial calculator, we can easily verify this relationship by entering different values for N and checking the price with the PV key. In our example, when both the required rate of return and the coupon rate are 10 percent, the price of the bond is $1,000.
Bond Pricing Property #2 If the required rate of return is equal to the coupon rate, the price of a bond will be equal to its par value.
The logic behind this bond pricing fundamental is simple. Compare the purchase of the
$1,000 bond to a $1,000 deposit at a bank. Assume the bank pays an annual interest rate of 10 percent, compounded semi-annually. You can withdraw the $50 in interest that you earn every six months and still maintain your $1,000 principal amount. You could do this for 20 years (withdrawing the $50 interest twice per year), and then withdraw your original
$1,000 at the end of the 20th year. The time diagram for this would be identical to the time diagram for our bond in Example 5.5. Thus, the amount an investor would pay for a bond with the same future payments would logically be the same as the $1,000 “deposit.”
Another bond pricing property is that if the required rate of return increases, the bond price decreases. This relationship holds regardless of the remaining term to maturity. At any time during the bond’s life, if the required return exceeds the bond’s coupon rate, the bond will sell at a discount, which equals M −P0. In our example, the required rate of return of 12 percent is greater than the coupon rate of 10 percent. As a result, the bond sells for
$885.30, which is a $114.70 discount below the par value of $1,000. Investors are unwilling to pay par value because the bond pays less coupon interest than newly issued bonds with similar characteristics.
Bond Pricing Property #3 If the required return is greater than the coupon rate, the price of a bond will be less than par value. The bond sells at a discount.
Still another bond pricing property is if the required rate of return decreases, then the bond price will increase. Such a bond sells at a premium, which equals Po− M. In this example, if the required rate of return decreases from 10 percent to 8 percent, the bond price will increase. Since the required rate of return is less than the coupon rate, the bond will sell at $1,135.90, which is a $135.90 premium above the par value of $1,000. Investors are willing to pay more than par value for this bond because of its higher coupon interest payments.
Bond Pricing Property #4 When the required return is less than the coupon rate, the price of a bond will be greater than par value. The bond sells at a premium.
Price-yield Curve
A price-yield curve is a plot of a bond’s required rate of return (market yield) to its correspond- ing price. As Figure 5.6 illustrates, the price-yield curve is not a straight-line, but instead is convex. This price-yield curve for option-free bonds has an important property: as bond yields rise, their prices will fall but at a decreasing rate; as bond yields fall, their prices will rise, but at an increasing rate. This property, known as positive convexity, means that bond prices go up faster than they go down.
Bond Pricing Property #5 As yields change for option-free bonds, bond prices go up faster than they go down.
To illustrate this property, review the results from Example 5.7. A 2 percentage point increase in yield (from 10 percent to 12 percent) led to a $114.70 decrease in price (from
124 THEFOUNDATION
Figure 5.6 Price-yield curve
Yield Price
$1,000.00 to $885.30); while a 2 percentage point decrease in yield (from 10 percent to 8 percent) led to a $135.90 increase in price (from $1,000.00 to $1,135.90). These results show that bond prices increase faster than they go down with the same change in yield.
Practical Financial Tip 5.5
The extent to which bond prices move depends both on the direction of change in market interest rates and the magnitude of such change: the greater the moves in interest rates, the greater the swings in bond prices.
Maturity and Price Convergence
The final bond pricing property that we discuss relates the price of a bond to its remaining term to maturity. Regardless of its required yield, the price of a bond will converge toward par value as maturity approaches.
Bond Pricing Property #6 As the maturity date of a bond approaches, the price of the bond will approach par value.
Figure 5.7 shows that, holding the bond’s required rate of return constant, the price of a bond selling at a premium will decline gradually over its remaining term to maturity. Simi- larly, the price of bond selling at a discount will gradually increase. If the required rate of return of a bond does not remain constant, the prices will still converge to par value, but not as gradually as depicted in Figure 5.7.
Par value Premium bond
Discount bond
Maturity Approaching date
maturity
Holding the bond’s required return constant Market value
Figure 5.7 Time path of a bond
Table 5.1 Price convergence of bonds at different maturities
Required rate of return (yield)
Time to maturity (years) 8% 10% 12%
10 $1,135.90 $1,000.00 $ 885.30
5 1,081.11 1,000.00 926.40
4 1,067.33 1,000.00 937.90
3 1,052.42 1,000.00 950.83
2 1,036.30 1,000.00 965.35
1 1,018.86 1,000.00 981.67
0 1,000.00 1,000.00 1,000.00
Example 5.8 Maturity and Price Convergence
Returning to Example 5.7, what happens to the bond price of Potomac Corporation’s 10-year, $1,000 par, 10 percent semiannual-pay bond as it approaches maturity if an investor requires an 8 percent, 10 percent, and 12 percent yield?
Solution: Table 5.1 shows how the prices for the premium and discount bonds con- verge to the maturity value of $1,000 over time.
126 THEFOUNDATION
Based on the expected price appreciation for discount bonds shown in Figure 5.7 and Table 5.1, can we logically conclude that investors will prefer discount bonds to premium bonds? Ignoring differences in the taxation of long-term capital gains and ordinary income, investors would not have a preference. Recall that a bond sells at a discount from par value when the coupon rate is less than the bond’s required rate of return (bond pricing property
#3). Thus, discount bonds pay interest income that is less than what is required for the bond to sell at par value. In order for an investor to earn the required rate of return on a discount bond, the expected price appreciation must augment the income yield (the return earned from the interest payments) so that the investor can expect to earn the required rate of return.
The opposite occurs for a premium bond. These bonds pay more interest income than investors require. As a result, such bonds sell at a premium to par value. The decrease in price over the bond’s life will reduce the higher income yield offered on premium bonds such that an investor can expect to earn the required rate of return. The upshot of this argument is that both discount and premium bonds will offer investors an expected rate of return equal to the required rate of return.
Concept Check 5.4
1 Should bond prices rise or fall as the general level of interest rates in the economy rise?
2 If a bond has a coupon rate that exceeds its required rate of return, should the bond sell at a discount or a premium? Why?
3 What does the term positive convexity mean?
4 What happens to the price of a discount and a premium bond as it approaches maturity?