LG 6 6.4 Bond Valuation
LG 5 LG 6 6.4 Bond Valuation
The basic valuation equation can be customized for use in valuing specific securi- ties: bonds, common stock, and preferred stock. We describe bond valuation in this chapter, and valuation of common stock and preferred stock in Chapter 7.
BOND FUNDAMENTALS As noted earlier in this chapter, bonds are long-term debt instruments used by
business and government to raise large sums of money, typically from a diverse group of lenders. Most corporate bonds pay interest semiannually (every 6 months) at a stated coupon interest rate, have an initial maturity of 10 to 30 years, and have a par value, or face value, of $1,000 that must be repaid at maturity.
Example 6.7 3 Mills Company, a large defense contractor, on January 1, 2013, issued a 10% coupon interest rate, 10-year bond with a $1,000 par value that pays interest
annually. Investors who buy this bond receive the contractual right to two cash flows: (1) $100 annual interest (10% coupon interest rate * $1,000 par value) distributed at the end of each year and (2) the $1,000 par value at the end of the
PART 3
Valuation of Securities
We will use data for Mills’s bond issue to look at basic bond valuation. BASIC BOND VALUATION
The value of a bond is the present value of the payments its issuer is contractually obligated to make, from the current time until it matures. The basic model for the
value, B 0 , of a bond is given by Equation 6.5:
d+M*c (1 + r n d
B 0 I* ca (6.5)
t=1 (1 + r ) t
where
B 0 = value of the bond at time zero
I = annual interest paid in dollars n = number of years to maturity M = par value in dollars r d = required return on the bond
We can calculate bond value by using Equation 6.5 and a financial calculator or by using a spreadsheet.
Personal Finance Example 6.8 3 Tim Sanchez wishes to determine the current value of the Mills
Company bond. Assuming that interest on the Mills Company bond issue is paid annually and that the required return is equal to the bond’s coupon interest rate,
I = $100, r d = 10%, M = $1,000, and n = 10 years. The computations involved in finding the bond value are depicted graphi- cally on the following time line.
Time line for bond
End of Year
valuation (Mills 2013 2014 2015 2016 2017 2018 2019 2020 2021 2022 Company’s 10% coupon interest rate, 10-year
$100 $100 $100 $100 $100 $100 $100 $100 $100 $100 $1,000 maturity, $1,000 par,
January 1, 2013, issue date, paying annual interest, and required return of 10%)
CHAPTER 6
Interest Rates and Bond Valuation
Calculator Use Using the Mills Company’s inputs shown at the left, you should
Input Function
find the bond value to be exactly $1,000. Note that the calculated bond value is
10 N
10 I equal to its par value; this will always be the case when the required return is equal to the coupon interest rate. 3
100 PMT 1000
FV Spreadsheet Use The value of the Mills Company bond also can be calculated
as shown in the following Excel spreadsheet.
CPT
PV
Solution A B
BOND VALUE, ANNUAL INTEREST, REQUIRED 1 RETURN = COUPON INTEREST RATE 2 Annual interest payment
3 Coupon interest rate
4 Number of years to maturity
5 Par value
6 Bond value
Entry in Cell B6 is =PV(B3,B4,B2,B5,0) Note that Excel will return a negative $1000 as the price that must be paid to acquire this bond.
BOND VALUE BEHAVIOR In practice, the value of a bond in the marketplace is rarely equal to its par value.
In the bond data (see Table 6.2 on page 235), you can see that the prices of bonds often differ from their par values of 100 (100 percent of par, or $1,000). Some bonds are valued below par (current price below 100), and others are valued above par (current price above 100). A variety of forces in the economy, as well as the passage of time, tend to affect value. Although these external forces are in no way controlled by bond issuers or investors, it is useful to understand the impact that required return and time to maturity have on bond value.
Required Returns and Bond Values Whenever the required return on a bond differs from the bond’s coupon interest
rate, the bond’s value will differ from its par value. The required return is likely to differ from the coupon interest rate because either (1) economic conditions have changed, causing a shift in the basic cost of long-term funds; or (2) the firm’s risk has changed. Increases in the basic cost of long-term funds or in risk will raise the required return; decreases in the cost of funds or in risk will lower the required return.
3. Note that because bonds pay interest in arrears, the prices at which they are quoted and traded reflect their value plus any accrued interest. For example, a $1,000 par value, 10% coupon bond paying interest semiannually and having a calculated value of $900 would pay interest of $50 at the end of each 6-month period. If it is now 3 months since the beginning of the interest period, three-sixths of the $50 interest, or $25 (that is, 3/6 * $50), would be accrued. The bond would therefore be quoted at $925—its $900 value plus the $25 in accrued interest. For conven- ience, throughout this book, bond values will always be assumed to be calculated at the beginning of the interest
PART 3
Valuation of Securities
discount Regardless of the exact cause, what is important is the relationship between
The amount by which a bond
the required return and the coupon interest rate: When the required return is
sells at a value that is less than
greater than the coupon interest rate, the bond value, B 0 , will be less than its par
its par value.
value, M. In this case, the bond is said to sell at a discount, which will equal premium
M-B 0 . When the required return falls below the coupon interest rate, the bond
The amount by which a bond
value will be greater than par. In this situation, the bond is said to sell at a
sells at a value that is greater
premium, which will equal B 0 - M .
than its par value.
Example 6.9 3 The preceding example showed that when the required return equaled the coupon interest rate, the bond’s value equaled its $1,000 par value. If for the
same bond the required return were to rise to 12% or fall to 8%, its value in each
Input Function
case could be found using Equation 6.5 or as follows.
10 N
12 I Calculator Use Using the inputs shown at the left for the two different required
returns, you will find the value of the bond to be below or above par. At a 12%
100 PMT
FV required return, the bond would sell at a discount of $113.00 ($1,000 par value - $887.00 value). At the 8% required return, the bond would sell for a
CPT
premium of $134.20 ($1,134.20 value - $1,000 par value). The results of these
PV
calculations for Mills Company’s bond values are summarized in Table 6.6 and
Solution
graphically depicted in Figure 6.4. The inverse relationship between bond value
and required return is clearly shown in the figure. Spreadsheet Use The values for the Mills Company bond at required returns of
Input Function
12% and 8% also can be calculated as shown in the following Excel spreadsheet.
10 N
Once this spreadsheet has been configured you can compare bond values for any
8 I two required returns by simply changing the input values.
100 PMT 1000
FV CPT
A B C Solution
PV
BOND VALUE, ANNUAL INTEREST, REQUIRED
1 RETURN NOT EQUAL TO COUPON INTEREST RATE
2 Annual interest payment
3 Coupon interest rate
4 Annual required return
5 Number of years to maturity
6 Par value
7 Bond value
$1,134.20 Entry in Cell B7 is =PV(B4,B5,B2,B6,0)
Note that the bond trades at a discount (i.e., below par) because the bond’s coupon rate is below investors’
required return. Entry in Cell C7 is =PV(C4,C5,C2,C6,0)
Note that the bond trades at a premium because the bond’s coupon rate is above investors’
required return.
CHAPTER 6
Interest Rates and Bond Valuation
Bond Values for Various Required Returns (Mills Company’s 10% TA B L E 6 . 6 Coupon Interest Rate, 10-Year Maturity, $1,000 Par, January 1,
2013, Issue Date, Paying Annual Interest)
Required return, r d Bond value, B 0 Status
Par value
Premium
FIGURE 6.4 Bond Values and Required
Returns Bond values and required
returns (Mills Company’s
10% coupon interest rate,
10-year maturity, $1,000
par, January 1, 2013, issue
Premium 1,134
date, paying annual
Market Value of Bond,
Required Return, r d (%)
Time to Maturity and Bond Values Whenever the required return is different from the coupon interest rate, the
amount of time to maturity affects bond value. An additional factor is whether required returns are constant or change over the life of the bond.
Constant Required Returns When the required return is different from the coupon interest rate and is constant until maturity, the value of the bond will approach its par value as the passage of time moves the bond’s value closer to maturity. (Of course, when the required return equals the coupon interest rate, the bond’s value will remain at par until it matures.)
Example 6.10 3 Figure 6.5 depicts the behavior of the bond values calculated earlier and pre- sented in Table 6.6 for Mills Company’s 10% coupon interest rate bond paying
PART 3
Valuation of Securities
FIGURE 6.5 Time to Maturity and Bond
Values
Premium Bond, Required Return, r d = 8%
Relationship among time to 0 B 1,134
maturity, required returns,
and bond values (Mills
Company’s 10% coupon
Par-Value Bond, Required Return, r d = 10%
interest rate, 10-year
M maturity, $1,000 par,
January 1, 2013, issue
date, paying annual
interest) Discount Bond, Required Return, r d = 12%
Market Value of Bond,
Time to Maturity (years)
returns—12%, 10%, and 8%—is assumed to remain constant over the 10 years to the bond’s maturity. The bond’s value at both 12% and 8% approaches and ultimately equals the bond’s $1,000 par value at its maturity, as the discount (at 12%) or premium (at 8%) declines with the passage of time.
Changing Required Returns The chance that interest rates will change and interest rate risk
thereby change the required return and bond value is called interest rate risk. 4
The chance that interest rates
Bondholders are typically more concerned with rising interest rates because a rise
will change and thereby
in interest rates, and therefore in the required return, causes a decrease in bond change the required return and value. The shorter the amount of time until a bond’s maturity, the less responsive
bond value. Rising rates, which result in decreasing bond
is its market value to a given change in the required return. In other words, short values, are of greatest concern. maturities have less interest rate risk than long maturities when all other features
(coupon interest rate, par value, and interest payment frequency) are the same. This is because of the mathematics of time value; the present values of short-term cash flows change far less than the present values of longer-term cash flows in response to a given change in the discount rate (required return).
Example 6.11 3 The effect of changing required returns on bonds with differing maturities can be illustrated by using Mills Company’s bond and Figure 6.5. If the required return
rises from 10% to 12% when the bond has 8 years to maturity (see the dashed line at 8 years), the bond’s value decreases from $1,000 to $901—a 9.9% decrease. If the same change in required return had occurred with only 3 years to
4. A more robust measure of a bond’s response to interest rate changes is duration. Duration measures the sensitivity of a bond’s prices to changing interest rates. It incorporates both the interest rate (coupon rate) and the time to maturity into a single statistic. Duration is simply a weighted average of the maturity of the present values of all the contractual cash flows yet to be paid by the bond. Duration is stated in years, so a bond with a 5-year duration will decrease in value by 5 percent if interest rates rise by 1 percent or will increase in value by 5 percent if interest rates
CHAPTER 6
Interest Rates and Bond Valuation
maturity (see the dashed line at 3 years), the bond’s value would have dropped to just $952—only a 4.8% decrease. Similar types of responses can be seen for the change in bond value associated with decreases in required returns. The shorter the time to maturity, the less the impact on bond value caused by a given change in the required return.
YIELD TO MATURITY (YTM) When investors evaluate bonds, they commonly consider yield to maturity
(YTM). This is the compound annual rate of return earned on a debt security purchased on a given day and held to maturity. (The measure assumes, of course, that the issuer makes all scheduled interest and principal payments as promised.) 5 The yield to maturity on a bond with a current price equal to its par value (that
is, B 0 = M ) will always equal the coupon interest rate. When the bond value dif- fers from par, the yield to maturity will differ from the coupon interest rate. Assuming that interest is paid annually, the yield to maturity on a bond can
be found by solving Equation 6.5 for r d . In other words, the current value, the annual interest, the par value, and the number of years to maturity are known, and the required return must be found. The required return is the bond’s yield to maturity. The YTM can be found by using a financial calculator, by using an Excel spreadsheet, or by trial and error. The calculator provides accurate YTM values with minimum effort.
Personal Finance Example 6.12 3 Earl Washington wishes to find the YTM on Mills Company’s bond. The bond currently sells for $1,080, has a 10% coupon
interest rate and $1,000 par value, pays interest annually, and has 10 years to maturity.
Input Function
0 in this case) or the future values (
Calculator Use Most calculators require either the present value (B
10 N
I and M in this case) to be input as negative numbers to
–1080 PV
calculate yield to maturity. That approach is employed here. Using the inputs
100 PMT
shown at the left, you should find the YTM to be 8.766%.
1000 FV CPT
Spreadsheet Use The yield to maturity of Mills Company’s bond also can be
I calculated as shown in the following Excel spreadsheet. First, enter all the bond’s cash flows. Notice that you begin with the bond’s price as an outflow (a negative
Solution
number). In other words, an investor has to pay the price up front to receive the
cash flows over the next 10 years. Next, use Excel’s internal rate of return func- tion. This function calculates the discount rate that makes the present value of a series of cash flows equal to zero. In this case, when the present value of all cash flows is zero, the present value of the inflows (interest and principal) equals the present value of the outflows (the bond’s initial price). In other words, the internal rate of return function is giving us the bond’s YTM, the discount rate that equates the bond’s price to the present value of its cash flows.
5. Many bonds have a call feature, which means they may not reach maturity if the issuer, after a specified time period, calls them back. Because the call feature typically cannot be exercised until a specific future date, investors often calculate the yield to call (YTC). The yield to call represents the rate of return that investors earn if they buy a callable bond at a specific price and hold it until it is called back and they receive the call price, which would be set
PART 3
Valuation of Securities
1 YIELD TO MATURITY, ANNUAL INTEREST
2 Year
Cash Flow 3 0 ($1,080) 4 1 $100 5 2 $100 6 3 $100 7 4 $100 8 5 $100 9 6 $100 10 7 $100 11 8 $100 12 9 $100
Entry in Cell B14 is =IRR(B3:B13)
SEMIANNUAL INTEREST AND BOND VALUES The procedure used to value bonds paying interest semiannually is similar to that
shown in Chapter 5 for compounding interest more frequently than annually, except that here we need to find present value instead of future value. It involves
1. Converting annual interest,
I, to semiannual interest by dividing I by 2.
2. Converting the number of years to maturity, n, to the number of 6-month periods to maturity by multiplying n by 2.
3. Converting the required stated (rather than effective) 6 annual return for similar-risk bonds that also pay semiannual interest from an annual rate, r d ,
to a semiannual rate by dividing r d by 2. Substituting these three changes into Equation 6.5 yields
6. As we noted in Chapter 5, the effective annual rate of interest, EAR, for stated interest rate r, when interest is paid semiannually ( m = 2), can be found by using Equation 5.17:
EAR =
a1 + r
For example, a bond with a 12% required stated annual return, r d , that pays semiannual interest would have an
effective annual rate of
1 = 1.1236 - 1 = 0.1236 = 12.36% Because most bonds pay semiannual interest at semiannual rates equal to 50 percent of the stated annual rate, their
EAR =
a1 + 0.12
249 Example 6.13 3 Assuming that the Mills Company bond pays interest semiannually and that the
CHAPTER 6
Interest Rates and Bond Valuation
required stated annual return, r d , is 12% for similar-risk bonds that also pay semiannual interest, substituting these values into Equation 6.6 yields
0.12 t S
$1,000 * C
0.12 20 S
2 b Calculator Use In using a calculator to find bond value when interest is paid
a1 + 2 b a1 +
Input Function
semiannually, we must double the number of periods and divide both the required
20 N
6 I stated annual return and the annual interest by 2. For the Mills Company bond,
we would use 20 periods (2 PMT *
10 years), a required return of 6% (12% 50 , 2), and an interest payment of $50 ($100 , 2). Using these inputs, you should find the
FV bond value with semiannual interest to be $885.30, as shown at the left.
CPT PV
Spreadsheet Use The value of the Mills Company bond paying semiannual interest at a required return of 12% also can be calculated as shown in the fol-
Solution
lowing Excel spreadsheet.
A B 1 BOND VALUE, SEMIANNUAL INTEREST 2 Semiannual interest payment
3 Semiannual required return
4 Number of periods to maturity
5 Par value
6 Bond value
Entry in Cell B6 is =PV(B3,B4,B2,B5,0) Note that Excel will produce a negative value for the
bond’s price
Comparing this result with the $887.00 value found earlier for annual com- pounding, we can see that the bond’s value is lower when semiannual interest is paid. This will always occur when the bond sells at a discount. For bonds selling at a premium, the opposite will occur: The value with semiannual interest will be greater than with annual interest.