CHAPTER 8 Bond Valuation Valuation and the Structure of Interest Rates
Before You Go On Questions and Answers Section 8.1 1.
What are the main differences between the bond markets and stock markets?
A corporate bond market is much larger than the stock market. The biggest investors in vestors in corporate bonds are mutual funds, life insurance companies, and pension funds, and given the size of these investors, the trades are conducted co nducted in much larger blocks block s than in the stock market. Also, while most stocks are traded in organized securities markets, most bond transactions take place through dealers in the OTC market.
2.
A bond has a 7 percent coupon rate, a face value of $1,000, and a maturity of four years. On a time line, lay out the cash flows for the bond.
The annual payments for the bond will be $70 ($1,000 x 7%); thus the time line for cash inflows would be as follows: 0
3.
1
_2
$70
$70
_______3
$70
_____4
$1,070 ($1,000 + $70)
Explain what a convertible bond is.
Convertible bonds are bonds that can be converted into shares of common stock at some predetermined ratio at the discretion of the bondholder. The convertible feature allows
the bondholder to take advantage advanta ge of the firm’s prosperity if the share prices rises above a certain value.
Section 8.2 1.
Explain conceptually how bonds are priced.
The current price of a bond is equal to the present value of o f all the cash flows that will be received from the investment. There are two sets of cash flows from a bond investment. First, there are the coupon payments to be received either annually or semiannually throughout the life of the bond. Second, there is the principal or face value of $1,000 that will be received when the bond matures. In order to find the price of the bond, bo nd, we must find the present value of the coupons and the present value of the face value. We do this by discounting the entire cash flow stream at the current market rate and adding them up. This gives us the current price of the bond. Recognize that the coupons represent an annuity and that we can use the equation for the present value of an annuity from Chapter 6 to calculate the present value of this cash flow stream.
2.
What is the compounding period for most bonds sold in the United States?
Most bonds sold in the United States pay interest semiannually, s emiannually, whereas European bonds typically only pay interest once a year.
3.
What are zero coupon bonds, and how are they priced?
Zero coupon bonds are debt instruments that do not pay coupon interest but promise a single payment (interest earned plus principal) paid at maturity. The price of a zero coupon bond can be calculated using the same equation as used for coupon bonds, but setting the coupon payments to zero. The resulting formula is as follows: PB = Fmn/(1 + i/m)mn Because zero coupon bonds bon ds offer the entire payment at maturity, for a given change in interest rates, their price fluctuates more than coupon bonds with a similar maturity.
Section 8.3 1.
Explain how bond yields are calculated.
A bond’s yield can be defined d efined as the interest rate that equates a bond’s bon d’s price to the present value of its interest payments and principal amount. The calculation of a bond’s yield, or its yield to maturity, takes into account the bond’s time to maturity, the coupon rate, and par.
Section 8.4 1.
What is interest rate risk?
Bond prices are negatively related to interest rate movements. As interest rates rise, bond prices fall, and vice versa. Interest rate risk risk simply recognizes the fact that bond prices fluctuate as interest rates change, and, if you sell a bond before maturity, you may ma y sell the bond for a price other than what you paid for it. The greater the fluctuation in bond prices due to changes in interest rates, the greater the interest rate risk.
2.
Explain why long-term bonds with zero coupons are riskier than short-term bonds that pay coupon interest.
According to bond theorems number two and three, for a given change in interest rates, longer-term bonds with low coupon rates have greater price changes than shorter-term bonds with higher coupon rates. Thus, long-term zero coupon bonds have greater interest rate risk greater price swings — than than short-term bonds that pay coupon payments. — greater
Section 8.5 1.
What are default risk premiums, and what do they measure?
Default risk premiums are the amount of return that investors must be paid to purchase a security that possesses default risk compared to a similar risk-free investment. Default risk premiums, at any point in time, represent compensation for the expected financial injury for owning a bond plus some additional premium for bearing risk.
2.
Describe the three most prominent bond rating systems.
Default risk premiums tend to increase during periods of economic decline de cline and to narrow during periods of economic expansion. This phenomenon is due to changes in investors’ willingness to own bonds with different credit ratings over the business cycle, the socalled flight to quality argument. Specifically, during periods of ex pansion when few defaults take place, investors are willing to invest in bond s with low credit quality to gain higher yields. In contrast, during tough economic times when many businesses fail, investors are concerned with safety. Accordingly, they adjust their portfolios to include more high-quality credits and sell off bonds with with low credit ratings. The three most
prominent credit rating agencies are Moody’s Investors Service (Moody’s), Standard & Poor’s (S&P) and Fitch. Exhibit 8.4 describes the corporate bond rating systems used by the three rating agencies.
3.
What are the key factors that most affect the level and shape of the yield curve?
The key factors that most affect the shape of the yield curve are the real rate of interest, the expected rate of inflation, and interest rate risk. If the future real rate of interest is expected to rise, it will result in an upward slope of the real rate of interest and consequently in an upward bias to the market yield curve. Similarly, increasing the expected rate of inflation will result in an upward-sloping yield curve, because long-term interest rates will contain a larger inflation premium than short-term interest rates. If these two variables are expected to decline in the future, the result will be a downward
bias to the yield curve. In contrast, the longer a bond’s maturity, the greater the bond’s interest rate risk. Thus, interest rate risk premium always adds an upward bi as to the slope of the yield curve, since the longer the maturity of a security, the greater its interest rate risk.
Self Study Problems 8.1
Calculate the price of a five-year bond that has a coupon of 6.5 percent paid annually. The current market rate is 5.75 percent.
Solution:
0
5.75%
1
2
3
4
5 Year
├───────┼────────┼───────┼────────┼───────┤ $65 PB
C1 1 i
C2 (1 i)
2
$65
(1 0.0575)1
$65 C3 (1 i)
3
$65 (1.0575)2
$65
C (1 i)
4
$65
$1,065
C4 F (1 i) 5
$65 (1.0575)3
$65 (1.0575)4
($65 $1, 000) (1.0575)5
$61.47 47 $58. $58.12 12 $54. $54.96 96 $51. $51.95 95 $805 $805.2 .28 8 $61.
$1, $1,031. 031.81 81
8.2
Bigbie Corp issued a five-year bond a year ago with a coupon of 8 percent. The bond pays interest semiannually. If the yield to maturity maturity on this bond is 9 percent, what is the price of the bond?
Solution:
0 9% 1
2
3
4
5
6
7
8 Semiannual Period
├───┼───┼───┼────┼───┼───┼───┼────┤ PB =? $40
PB
C1 / m (1 i / m )
1
$40
$40
C2 / m (1 i /
m)
2
$40
$40
C3 / m (1 i / m )
3
$40
..........
$40
$1,040
C8 F (1 i /
8
m)
$ 80 / 2 $40 $40 ($40 $1, 000) ........ (1 0.09 / 2)1 (1.045)2 (1.045)3 (1.045)8
$38.28 28 $36. $36.63 63 $35. $35.05 05 $33. $33.54 54 $32. $32.10 10 $30. $30.72 72 $29. $29.39 39 $731 $731.3 .31 1 $38.
$967.02
Alternatively, we can use the present value annuity factor from Chapter 6 (Equation 6.1) and the present value equation from Chapter 5 to solve for the price of the bond: 1 1 1 i m PB C i m
mn
F 1 i m n
mn
1 1 8 0.045 5 1 0.04 $4 0 0.045
$ 1, 00 0 8 1.0 45
$263.84 $263.84 $703.19 $703.19 $967.03
8.3
Rockwell Industries has a three-year bond outstanding that pays a 7.25 percent coupon and is currently priced at $913.88. What is the yield to maturity of this bond? Assume annual coupon payments.
Solution:
0
1
2
3
├───────┼────────┼───────┤ PB = $913.88
$72.50
$72.50
$1,072.50
Use the trial-and error approach error approach to solve for YTM. Since the bond is selling at a discount, we know that the yield to maturity is higher than the coupon rate. Try YTM = 10%. PB
C1 C2 C 3 F3 2 3 1 i 1 i 1 i $72. $72.50 50 $72. $72.50 50 $72. $72.50 50 $1,00 $1,000 0 2 3 1 .1 0 (1 .10 ) (1.1 0)
$65 $65.9 .91 1 $59 $59.9 .92 2 $80 $805. 5.79 79 $931.61 Try a higher rate, say YTM = 11%.
PB
C1 C2 C 3 F3 2 3 1 i 1 i 1 i $72. $72.50 50 $72. $72.50 50 $72. $72.50 50 $1,00 $1,000 0 2 3 1 .1 1 (1 .11) (1.11)
$65 $65.3 .32 2 $58 $58.8 .84 4 $78 $784. 4.20 20 $908.36
Since this is less than the price of the bond, bon d, we know that the YTM is between 10 and 11 percent and closer to 11 percent. Try YTM = 10.75%. PB
C1 C2 C3 F3 2 3 1 i 1 i 1 i $72.50 $72.50 $72.50 $1,000 ,000 2 1.1075 (1 (1.1075) (1.1075)3
$65.4 .46 6 $59.1 $59.11 1 $78 $789. 9.53 53 $65
$914.09 Alternatively, we can use the present value valu e annuity factor from Chapter 6 (Equation 6.1) and the present value equation from Chapter 5 to solve for the price of the bond:
1 1 (1 i ) n F PB C n i (1 i ) 1 1 3 (1.1075) $1,000 $913. $913.88 88 $72.5 $72.50 0 3 0 . 1 0 7 5 (1.1075) $177.94 94 $736. $736.15 15 $914.09 $177.
Thus, the YTM is approximately 10.75 percent. Using a financial calculator provided an exact YTM of 10.7594 percent.
8.4
Hindenberg, Inc., has a 10-year bond that is priced at $1,100.00. It has a coupon of 8 percent paid semiannually. What is the yield to maturity on this bond?
Solution:
0
1
2
3
4
5
6
19
├───┼────┼───┼───┼───┼────┼── $40
$40
$40
$40
$40
20
─┼────┤
$40
$40
$40 $1,000
The easiest way to calculate the yield to maturity is with a financial calculator. The inputs are as follows: Enter
20 N
Answer
i
40
−1,100
1,000
PMT
PV
FV
3.31
The answer we get is 3.31 percent, which is the semiannual interest rate. To obtain an annualized yield to maturity, we multiply this by b y two: YTM = 3.31% 2 YTM = 6.62%
8.5
Highland Corp., a U.S. company, has a five-year bond whose yield to maturity is 6.5 percent. The bond has no coupon payments. What is the price of this zero coupon bond? Solution:
You have the following information: YTM = 6.5% No coupon payments Most U.S. bonds pay interest semiannually. Thus m x n = 5 × 2 = 10 and i/2 = 0.065/2 = 0.0325. Using Equation 8.3, we obtain the following:
PB
Fmn
1 i
m
mn
$1,000 (1.0325)10
$726.27
Critical Thinking Questions 8.1
Because the conversion feature in a convertible bond is valuable to bondholders, convertible bond issues have lower coupon payments than otherwise similar bonds that are not convertible. Does this mean that a company can lower its cost of borrowing b orrowing by selling convertible debt? Explain.
No. While the interest (coupon) payments that the company must make are lower, the overall cost of borrowing is not. The reduction in the value of the interest payments pa yments is
offset by the value of the conversion co nversion feature. If the company’s stock price goes above the price implied by the conversion ratio, the existing stockholders must share some of their gains with the bondholders. Investors are going to require a return that compensates them for the risk that they are bearing. The only o nly difference with a convertible bond bon d is that some of that compensation comes in the form of the ability to benefit from appreciation in the
company’s stock price.
8.2
What economic conditions would prompt investors to take advantage of a bond’s convertibility feature?
A bond’s convertibility feature becomes attractive when the company’s compan y’s stock price rises above the bond’s price. This usually happens in times of economic expansion when the stock market is booming and interest rates are increasing, hence lowering the bond’s price.
8.3
We know that a vanilla bond that has a coupon rate which is below the market rate of interest will sell for a discount and that a vanilla bond b ond which has a coupon cou pon rate above the market rate of interest will sell for a premium. What kind of bon d will sell at its par value regardless of what happens to the market rate of o f interest?
A bond that pays a variable coupon rate that moves up and down with the market rate of interest. While corporate bonds in the U.S. do not have variable coupon payments, bank loans often have variable rates which adjust frequently enough enou gh so that the value of o f the loans remains relatively constant as interest rates move up and down over time.
8.4
Define yield to maturity. Why is it important?
Yield to maturity (YTM) is the rate of return earned by investors if they buy a bond today
at its market price and hold it to maturity. It is important because it represents the opportunity cost to the investor or the discount rate that makes the present value of the
bond’s cash flows (i.e., its coupons and its principal) equal to the market price. So, YTM is also referred to as the going market rate or the appropriate app ropriate discount rate for a bond’s cash flows. It is important to understand that any investor who buys b uys a bond and holds it to maturity will have a realized gain equal to the yield to maturity. If the investor sells before the maturity date, then realized gain will not be equal to the YTM, but will only be based on cash flows earned to that point. Similarly, for callable bonds, investors are are guaranteed a gain to the point in time when the bond is first called, but they cannot be assured of the yield to maturity because the issuer could call the bond before maturity!
8.5
Define interest rate risk . How can the CFOs C FOs manage this risk?
The change in a bond's prices caused by changes in interest rates is called interest rate risk .
In other words, we can measure the interest rate risk to a bond’s investor by measuring the percentage percentage change in the bond’s price caused caused by a 1 percent change in the market interest rates. The key to managing interest rate risk is to understand the relationships between interest rates, bond prices, the coupon rate, and the bond’s term to maturity. Portfolio managers need to understand that as interest rates rise bond prices decline, and it declines more for low-coupon bonds and longer-term bonds than for the others. In such a scenario, bond portfo portfolio lio manager managerss can reduce the size size and and maturity maturity of their their portfol portfolio io to reduce the impact of interest rate increases. When interest rates decline, bond prices increase and rise more for longer-term bonds and higher coupon bonds. At such times, CFOs can increase the size and maturity of their portfolios to take advantage of the inverse relationship between interest rates and bond prices.
8.6
Explain why bond prices and interest rates are negatively related. What is the role of the coupon rate and term-to-maturity in this relationship?
Bond prices and interest rates are negatively related because the market rate varies, while the coupon rate is constant over the life of the bond. Thus, as rates increase, demand and bond prices prices of existi existing ng bonds bonds decline, decline, while newer bonds with coupon rates at the current current rate rate are are in greater demand. o
For a given change in interest rates, longer-term bonds experience greater price changes ( price price volatility volatility) than shorter-term bonds. Longer-term bonds have more of their cash flows farther in the future, and their present value will be lower due to the compounding effect. In addition, the longer it takes for investors to receive the cash flows, the more uncertainty they have to deal with and hence the more price-volatile the bond will be.
o
Lower coupon bonds are more price volatile than higher coupon bonds. The same argument used above also explains this relationship. The lower the coupon on a bond, the the greater greater the proport proportion ion of cash flows that investors investors receive receive at maturity. maturity.
8.7
If rates are expected to increase, should investors look to long-term bonds or short-term securities? Explain.
As interest rates increase, bond prices decrease with longer-term bonds, ex periencing a bigger decline than shorter-term securities. securities. So, investors expecting an increase in interest rates should choose short-term securities over long-term securities and reduce their interest rate risk.
8.8
Explain what you would assume the yield curve would look like during economic expansion and why.
At the beginning of an economic expansion, the yield curve tends to be rather steep as the rates begin to rise once the demand for capital is beginning to pick up due to growing economic activity. The yield curve will retain its positive slope during the economic expansion, which reflects the investors’ expectations that the economy will grow in the future and that the inflation rates will also rise in the future.
8.9
An investor holds a 10-year bond paying a coupon of 9 percent. The yield to maturity of the bond is 7.8 percent. Would you expect the investor to be holding a par-value, premium, or discount bond? What if the yield to maturity was 10.2 percent? Explain.
Since the bond’s coupon of o f 9 percent is greater than the yield to maturity, the bond will be a premium bond. As market market rates rates of interest interest drop drop below below the the coupon coupon rate of the 9 percent percent bond, demand for the the bond bond increases increases,, driving driving up the price price of the bond bond above above face face value. value. If the yield to maturity is at 10.2 percent, then the bond is paying a lower coupon than the going market rate and will be less attractive to investors. The demand for the 9
percent percent bond will decline, decline, driving driving its price price below the face face value. value. This This will will be a discount bond.
8- year bond. If interest rate 8.10 a. Investor A holds a 10-year bond, while investor B has an 8-year increases by 1 percent, which investor will have the higher interest rate risk? Explain.
Since A holds the longer-term bond, he or she will face the higher h igher interest rate risk. Longer-term bonds are more price volatile than shorter-term sh orter-term bonds.
b. Investor A holds a 10-year bond paying 8 percent a year, while investor B also has a
10-year bond that pays a 6 percent coupon. Which investor will have the higher interest rate risk? Explain.
Investor B will have the higher interest rate risk since lower coupo n bonds have a higher interest rate risk than higher coupon bonds bon ds of the same maturity.
Questions and Problems
BASIC 8.1
w ith a coupon rate of 8 percent. The Bond price: BA Corp is issuing a 10-year bond with interest rate for similar bonds is currently 6 percent. Assuming annual payments, what is the value of the bond? LO 2
Solution:
Years to maturity = n = 10 Coupon rate = C = 8% Annual coupon = $1,000 × 0.08 = $80
Current market rate = i = 6% Present value of bond = PB 0 6% 1
2
3
4
5
6
10
├───┼────┼───┼───┼───┼────┼── $80
$80
$80
$80
$80
─────┤
$80
$80 $1,000
C1
PB
(1 i )
1
C2 (1 i )
2
C3 (1 i )
3
C10 F (1 i )10
1 1 1 1 10 (1 i ) n F $1,000 (1.06) C $80 n 10 i (1 i ) 0.06 (1.06) $588.81 $558.39 $1,147.20
8.2
Bond price: Pierre Dupont just received a cash gift from his grandfather. He plans to
invest in a five-year bond issued by Venice Corp. that pays an annual coupon of 5.5 percent. If the current market rate is 7.25 percent, what is the maximum amount Pierre should be willing to pay for this bond? LO 2
Solution:
0
7.25%
1
2
3
4
5
├───────┼────────┼───────┼────────┼───────┤ $55
$55
$55
$55
$1,055
Coupon rate = C = 5.5% Annual coupon = $1,000 × 0.055 = $55 Current market rate = i = 7.25% Present value of bond = PB 1 1 1 1 n 5 (1 i ) (1.0725) F $1,000 PB C $ 55 n 5 i (1 i ) 0.0725 (1.0725) $224.01 $704.72 $928.72
8.3
Bond price: Knight, Inc., has issued a three-year bond that pays a coupon of 6.10
percent. Coupon payments are made semiannually. Given the market rate of interest of 5.80 percent, what is the market value of the bond? LO 2
Solution:
Years to maturity = n = 3 Coupon rate = C = 6.1% Frequency of payment = m = 2 Semiannual coupon = $1,000 × (0.061/2) = $30.50 Current market rate = i = 5.8% Present value of bond = PB 0 5.8% 1
2
3
4
5
6
├───┼────┼───┼───┼───┼────┤ $30.50 $30.50 $30.50 $30.50 $30.50 $30.50 $1,000 1 1 1 2 n 1 i 6 1 F $1,000 (1.029) 2 PB C $ 30 . 50 2n 2 i 0.029 (1.029) 6 i 1 2 2 $165.77 $842.38 $1,008.15
8.4
Bond price: Regatta, Inc., has seven-year bonds outstanding that pay a 12 percent
coupon rate. Investors buying these bonds today can expect to earn a yield to maturity of 8.875 percent. What is the current value of these bonds? Assume annual coupon payments. LO 2
Solution:
Years to maturity = n = 7
Coupon rate = C = 12% Annual coupon = $1,000 x 0.12 = $120 Current market rate = i = 8.875% Present value of bond = PB 0
1
2
3
4
5
6
7
├───┼────┼───┼───┼───┼────┼───┤ $120
$120
$120
$120
$120
$120
$120 $1,000
1 1 1 1 n 7 (1 i ) (1.08875) F $1,000 PB C $ 120 n 7 i (1 i ) 0.08875 (1.08875) $606.50 $551.14 $1,157.94
8.5
Bond price: You are interested in investing in a five-year bond that pays 7.8 percent
coupon with interest to be received semiannually. Your You r required rate of return is 8.4 percent. What is the most you would be willing to pay for this bond? LO 2
Solution:
Years to maturity = n = 5 Coupon rate = C = 7.8% Frequency of payment = m = 2 Semi-annual coupon = $1,000 × (0.078/2) = $39.00 Current market rate = i = 8.4% Present value of bond = PB
0 8.4% 1
2
3
4
5
6
├───┼────┼───┼───┼───┼────┼── $39
$39
$39
$39
$39
$39
10
─────┤ $39 $1,000
1 1 2 n 1 i F 2 PB C 2 i 1 i2 2 $313.20 $662.71 $975.91
8.6
2n
1 1 (1.042)10 $1,000 $39 10 0.042 (1.042)
Zero coupon bonds: Diane Carter is interested in buying a five-year zero coupon bond
with a face value is $1,000. She understands that the market interest rate for similar investments is 9 percent. Assume annual coupon payments. What is the current value of this bond? LO 1, LO 2
Solution:
Years to maturity = n = 5 Coupon rate = C = 0% Current market rate = i = 9% 0
1
2
3
4
5
├───┼────┼───┼───┼───┤ $0
$0
$0
$0
$0 $1,000
PB
8.7
Fmn
1 i m
mn
$1,000 5
1.09
$649.93
Zero coupon bonds: Ten-year zero coupon bonds issued by the U.S. Treasury have a
face value of $1,000 and interest is compounded semiannually. If similar bonds in the market yield 10.5 percent, what is the value of these bonds? LO 1, LO 2
Solution:
Years to maturity = n = 10 Frequency of payment = m = 2
Coupon rate = C = 0% Current market rate = i = 10.5% 0
1
2
3
4
5
6
├───┼────┼───┼───┼───┼────┼── $0
$0
$0
$0
$0
$0
20
─────┤ $0 $1,000
PB
8.8
Fmn
1 m i
mn
$1,000
$359.38
1.052520
Zero coupon bonds: Northrop Real Estate Company is planning to fund a development
project by issuing 10-year zero coupon bonds with a face value of $1,000. Assuming semiannual compounding, what will be the price of these bonds if the appropriate discount rate is 14 percent? LO 1, LO 2
Solution:
Years to maturity = n = 10 Coupon rate = C = 0% Current market rate = i = 14% Assume semiannual coupon payments. 0
1
2
3
4
5
6
├───┼────┼───┼───┼───┼────┼── $0
$0
$0
$0
$0
$0
20
─────┤ $0 $1,000
PB
8.9
Fmn
1 m i
mn
$1,000
1.0720
$258.42
Yield to maturity: Ruth Hornsby is looking to invest in a three-year bond that makes
semiannual coupon payments at a rate of 5.875 percent. If these bonds have a market
price of $981.13, what yield to maturity and effective annual yield can she expect to earn? LO 3
Solution:
Years to maturity = n = 3 Coupon rate = C = 5.875% Frequency of payment = m = 2 Semi-annual coupon = $1,000 × (0.05875/2) = $29.375 Yield to maturity = i Present value of bond = PB = $981.13 Use the trial-and-error approach trial-and-error approach to solve for YTM. Since the bond is selling at a discount, we know that the yield to maturity is higher than the coupon rate. Try YTM = 6%. 1 1 2n i 1 2 PB C 2 i 2
F 2n 1 i 2
1 1 6 1 0 .0 3 $981.13 $981.13 $29.375 $29.375 0.03
$159.13 $159.13 $837.48 $837.48 $996.61 Try a higher rate, say YTM = 6.6%.
$1,000 6 1.03
1 1 2n i 1 2 PB C 2 i 2
F 2n 1 i 2
1 1 6 1 0.03 0.033 3 $981. $98 1.13 13 $29 $29.3 .375 75 0.033
$1,000 6 1.033
$157.56 $157.56 $823.00 $823.00 $980.56 The YTM is approximately 6.6 percent. Using a financial calculator provided an exact YTM of 6.58 percent Enter
6
$29.375 -$981.13 $1,000
N Answer
i%
PMT
PV
FV
6.58%
The effective annual yield can be computed as: EAY (1 Quoted rate m) m 1
1 0.06578 2 2 1 (1.0335) 2 1 0.06686 6.69%
8.10
Yield to maturity: Rudy Sandberg wants to invest in four-year bonds that are currently
priced at $868.43. These bonds have a coupon rate of 6 percent and make mak e semiannual coupon payments. What is the current market yield on this bond? LO 3
Solution:
Years to maturity = n = 4 Coupon rate = C = 6% Frequency of payment = m = 2 Semiannual coupon = $1,000 × (0.06/2) = $30
Yield to maturity = i Present value of bond = PB = $868.43 Use the trial-and-error approach to solve for YTM. Since the bond is selling at a discount, we know that the yield to maturity is higher than the coupon rate. Try YTM = 10%. 1 1 2n i 1 2 PB C 2 i 2
F 2n i 1 2
1 1 8 1 0 .05 $868 $868.4 .43 3 $30 $30 0.05
$1,000 8 1.05
$193.90 $676.84 $676.84 $193.90
$870.74 Try a higher rate, say YTM = 10.1%. 1 1 2n i 1 2 PB C 2 i 2
F 2n 1 i 2
1 1 8 0.0505 05 1 0.05 $1,000 8 $868 $868.4 .43 3 $30 $30 * 0.0505 1.0505 $193.51 $674.27 $674.27 $193.51
$867.77 The YTM is approximately 10.1 percent. Using a financial calculator provided an exact YTM of 10.08 percent. Enter
8 N
Answer
$30 i% 10.08%
PMT
-$868.43 PV
$1,000 FV
8.11
Realized yield: Josh Kavern bought 10-year, 12 percent coupon bonds issued by the U.S.
Treasury three years ago at $913.44. If he sells these bonds, which have a face value of $1,000, at the current price of $804.59, what is the realized return on these bonds? Assume similar coupon-paying bonds make annual coupon payments. LO 3
Solution:
Purchase price of bond = $913.44 Years investment held = n = 3 Coupon rate = C = 12% Frequency of payment = m = 1 Annual coupon = $1,000 × (0.12) = $120 Realized yield = i Selling price of bond = PB = $804.59 To compute the realized return, either the trial-and-error approach or the financial calculator can be used. Since the price has declined, market rates must have ha ve increased. So, the realized return is going to be less than the bond’s coupon. Try rates lower than the coupon rate. Try i = 10%. 1 1 (1 i ) n FV PB C n i (1 i ) 1 1 (1.10) 3 $804.59 $913.44 $120 3 0.10 (1.10) $298.42 $604.50 $902.92 Try a lower rate, i = 9.5%.
1 1 (1 i ) n FV PB C n i (1 i ) 1 1 3 (1.095) $804.59 $913.44 $120 3 0 . 095 (1.095) $301.07 $612.82 $913.89 The realized rate of return is approximately 9.5 percent. Using a financial calculator provided an exact yield of 9.52 percent. Enter
3 N
Answer
8.12
$120 i%
PMT
-$913.44 PV
$804.59 FV
9.52%
Realized yield: Four years ago, Lisa Stills bought six-year, 5.5 percent coupon bonds
issued by the Fairways Corp. for $947.68. If she sells these bonds at the current price of $894.52, what will be her realized yield on the bonds? Assume similar coupon-paying bonds make annual coupon payments. LO 3
Solution:
Purchase price of bond = $947.68 Years investment held = n = 4 Coupon rate = C = 5.5% Frequency of payment = m = 1 Annual coupon = $1,000 × (0.055) = $55 Realized yield = i Selling price of bond = PB = $894.52 To compute the realized return, either the trial-and-error approach or the financial calculator can be used. Since the price has declined, market rates must have ha ve increased.
So, the realized return is going to be less than the bond’s coupon. Try Tr y rates lower than the coupon rate. Try i = 5%. 1 1 (1 i ) n FV PB C n i (1 i ) 1 1 (1.05) 4 $894.52 $947.68 $55 4 0 . 05 (1.05) $195.03 $735.92 $930.95 Try a lower rate, i = 4.5%. 1 1 (1 i ) n FV PB C n i (1 i ) 1 1 4 (1.045) $894.52 $947.68 $55 4 0 . 045 (1.045) $197.31 $750.11 $947.42 The realized rate of return is approximately 4.5 percent. Using a financial calculator provided an exact yield of 4.49 percent. Enter
4 N
Answer
$55 i%
PMT
-$947.68 PV
$894.52 FV
4.49%
INTERMEDIATE 8.13
Bond price: The International Publishing Group is raising $10 million by issuing 15-year
bonds with a coupon rate of 8.5 percent. Coupon payments will be made annually.
Investors buying the bond currently will earn a yield to maturity of 8.5 percent. At what price will the bonds sell in the marketplace? Explain. LO 2
Solution:
Years to maturity = n = 15 Coupon rate = C = 8.5% Annual coupon = $1,000 × 0.085 = $85 Current market rate = i = 8.5% Present value of bond = PB 0
1
2
3
4
15
├───┼────┼───┼───┼─── $85
$85
$85
$85
─────┤ $85 $1,000
n = 7;
C = 8.5%;
i = YTM = 8.85%
1 1 1 1 n 15 (1 i ) (1.085) $1,000 F PB C $ 85 n 15 ( 1 ) 0 . 085 i i (1.085) $705.86 $294.14 $1,000.00 This answer should have been intuitive. Since the bond is paying a coupon equal to the going market rate of 8.5 percent, the bond should be selling at its par value of $1,000.
Enter
15 N
Answer
8.14
8.85% i%
$85 PMT
$1,000 PV
FV
-$1,000
Bond price: Pullman Corp issued 10-year bonds four years ago with a coupon rate of
9.375 percent, paid semiannually. At the time of issue, the bonds sold at par. pa r. Today, bonds of similar risk and maturity must must pay an annual coupon of 6.25 percent pe rcent to sell at par value. Assuming semi-annual coupon payments, what will be the current market price
of the firm’s bonds?
LO 2, LO 4
Solution:
Years to maturity = n = 6 Coupon rate = C = 9.375% Semiannual coupon = $1,000 × (0.09375/2) = $46.875 Current market rate = i = 6.25% Present value of bond = PB 0
1
2
3
4
12
├───┼────┼───┼───┼─── $46.875 $46.875………
─────┤ $46.875 $1,000
n = 6; m = 2; C = 9.375%; i = YTM = 6.25%
1 1 1 i 2 2n C PB 2 i 2
1 1 12 F $1, 000 (1.03125) $46.875 2n 12 1 i 2 0.03125 (1.03125)
$463.13 3.13 $691.25 $691.25 $1,154.38 $46 Enter
12
3.125%
N
i%
$46.875 PMT
Answer
8.15
$1,000 PV
FV
$1,154.38
co upon rate of 6.5 Bond price: Marshall Company is issuing eight-year bonds with a coupon percent and semiannual coupon payments. If the current market rate for similar bonds is 8 percent, what will be the bond price? If the company wants to raise $1.25 million, how many bonds does the firm have to sell? LO 2
Solution:
Years to maturity = n = 8 Coupon rate = C = 6.5%
Semiannual coupon = $1,000 × (0.065/2) = $32.50 Current market rate = i = 8% Present value of bond = PB
0 8% 1
2
3
4
16
├───┼────┼───┼───┼─── $32.50
─────┤
$32.50………..$32.50
$32.50 $1,000
1 1 2n i 1 F 2 PB C 2 i i 1 2 2 $378.70 $533.91 $912.61
2n
1 1 (1.04)16 $1,000 $32.50 16 0.04 (1.04)
To raise $1.25 million, the firm would have to sell: Number of bonds = $1,250,000 / $912.61 = 1,370 bond contracts Enter
16
4%
N
i%
$32.50 PMT
Answer
8.16
$1,000 PV
FV
-$912.61
Bond price: Rockne, Inc., has 15-year bonds that will mature in six years and pay an 8
percent coupon, interest being paid semiannually. If you paid $1036.65 today, and your required rate of return was 6.6 percent, did you pay the right price for the bond? b ond? LO 2, LO 4
Solution:
Years to maturity = n = 6 Coupon rate = C = 8% Semiannual coupon = $1,000 × (0.08/2) = $40 Current market rate = i = 6.6% Present value of bond = PB
0
1
2
3
12
├───────┼────────┼────────┼── $40
$40
─────────┤
$40
$40 $1,000
1 1 1 2n 1 i 12 (1.033) $1,000 1 F 2 PB C $ 40 2n 2 i 0 . 033 (1.033)12 i 1 2 2 $391.12 $677.32 $1,068.45
You paid less than what the bond bo nd is worth. That was a good price! Enter
12
3.3%
$40
N
i%
$1,000
PMT
PV
Answer
8.17
FV
-$1,068.45
Bond price: Nanotech, Inc., has a bond issue maturing in seven years that is paying a
coupon rate of 9.5 percent (semiannual payments). The company wants to retire a portion of the issue by buying the securities in the open market. If it can refinance at 8 percent, how much will Nanotech pay to buy back its current outstanding bonds? LO 2, LO 4
Solution:
Years to maturity = n = 7 Coupon rate = C = 9.5% Semi-annual coupon = $1,000 × (0.095/2) = $47.50 Current market rate = i = 8% Present value of bond = PB 0
1
2
3
├───────┼────────┼────────┼── $47.50
$47.50
$47.50
14
─────────┤ $47.50 $1,000
1 1 1 2n 1 i 14 1 F $1,000 (1.04) 2 PB C $ 47 . 50 2n 2 i 0.04 (1.04)14 i 1 2 2 $501.75 $577.48 $1,079.22
The firm will be willing to pay no more than $1,079.22 for their bond.
14
Enter
4%
N
$47.50
i%
$1,000
PMT
PV
Answer
8.18
FV
-$1,079.22
Zero coupon bonds: Kintel, Inc., wants to raise $1 million by issuing six-year zero
coupon bonds with a face value of $1,000. Its investment banker states that investors would use an 11.4 percent discount rate to value such bonds. At what price would these bonds sell in the marketplace? How many bonds would the firm have to issue to raise $1 million? Assume semiannual interest payments. LO 1, LO 2, LO 4
Solution:
Years to maturity = n = 6 Coupon rate = C = 0% Current market rate = i = 11.4% Assume semi-annual coupon payments. 0
1
2
3
4
5
6
├───┼────┼───┼───┼───┼────┼── $0
$0
$0
$0
$0
$0
12
─────┤ $0 $1,000
PB
Fmn
1 m i
mn
$1,000
1.05712
$514.16
At the price of $514.16, the firm needs to raise $1 million. To do so, the firm will have to issue:
Number of contracts
8.19
= $1,000,000 / $514.16 = 1,945 contracts
Zero coupon bonds: Rockinghouse Corp. plans to issue seven-year zero coupon bonds.
It has learned that these bonds will sell today at a price of $439.76. Assuming annual annu al coupon payments, what is the yield to maturity on these bonds? LO 1, LO 2, LO 4
Solution:
Years to maturity = n = 7 Coupon rate = C = 0% Current market rate = i Assume annual coupon payments. Present value of bond = PB = $439.76 0
1
2
3
4
5
6
7
├───┼────┼───┼───┼───┼────┼───┤ $0
$0
$0
$0
$0
$0
$0 $1,000
To solve for the YTM, a trial-and-error approach has to be used. Try YTM = 10%. PB
Fmn
1 i m
mn
$1,000
1.107
$439.76 $513.16 Try a higher rate, YTM = 12%. PB
Fmn
1 i m
mn
$1,000
1.127
$439.76 $452.35 Try YTM=12.5%. PB
Fmn
1 i m
mn
$1,000
1.1257
$439.76 $438.46 The YTM is approximately 12.5 percent.
7
Enter
$0
N Answer
8.20
i%
-$439.76 $1,000
PMT
PV
FV
12.453%
Yield to maturity: Electrolex, Inc., has four-year bonds outstanding that pay a coupon
rate of 6.6 percent and make coupon payments semiannually. If these bonds are currently selling at $914.89, what is the yield to maturity that an investor can expect to earn on these bonds? What is the effective annual yield? LO 3, LO 4
Solution:
Years to maturity = n = 4 Coupon rate = C = 6.6% Current market rate = i Semiannual coupon payments = $1,000 × (0.066/2) = $33 Present value of bond = PB = $914.89 0
1
2
3
├───────┼────────┼────────┼── $33
$33
$33
8
─────────┤ $33 $1,000
To solve for the YTM, a trial-and-error approach has to be used. Since this is a discount bond, the market rate should be higher than 6.6 percent. Try i = 8% or i/2 = 4%.
1 1 (1 i ) n Fn PB C n i (1 i ) 1 1 (1.04)8 $1,000 $914 $914.8 .89 9 $33 $33 8 0.04 (1.04) $222.1 .18 8 $730 $730.6 .69 9 $952 $952.8 .87 7 $222
Try a higher rate, i = 9%, i/2 = 4.5%.
1 1 (1 i ) n Fn PB C n i (1 i ) 1 1 8 (1.045) $1,000 $914 $914.8 .89 9 $33 $33 8 0 . 0 4 5 (1.045) $217. 7.66 66 $70 $703. 3.19 19 $92 $920.8 0.85 5 $21 Try a higher rate, i = 9.2%, i/2 = 4.6%. 1 1 (1 i ) n Fn PB C n i (1 i ) 1 1 (1.046)8 $1,000 $914 $914.8 .89 9 $33 $33 8 0 . 0 4 6 (1.046) $216. 6.78 78 $69 $697. 7.82 82 $914. $914.60 60 $21 The yield to maturity is approximately 9.2 percent. The effective annual yield can be computed as: EAY (1 Quoted rate m) m 1 2
1.046 1 0.0941 9.41% 8
Enter
$33
N
i%
Answer
-$914.89
$1,000
PV
FV
PMT
4.5954%
The effective annual yield can be computed as: EAY (1 Quoted rate m) m 1 2
1.045954 1 0.09399 9.4%
8.21
Yield to maturity: Serengeti Corp. has five-year bonds outstanding that pay a coupon of
8.8 percent. If these bonds are priced at $1,064.86, what is the yield to maturity on these bonds? Assume semiannual coupon payments. What is the effective annual yield? LO 3, LO 4
Solution:
Years to maturity = n = 5 Coupon rate = C = 8.8% Current market rate = i Semiannual coupon payments = $1,000 x (0.088/2) = $44 Present value of bond = PB = $1,064.86 0
1
2
3
├───────┼────────┼────────┼── $44
44
$44
10
─────────┤ $44 $1,000
To solve for the YTM, a trial-and-error approach has to be used. Since this is a premium bond, the market rate should be lower than 8.8 percent. Try i = 7% or i/2 = 3.5%. 1 1 (1 i ) n FV PB C n i (1 i ) 1 1 (1.035)10 $1,000 $1,064.86 $44 10 0 . 035 (1.035) $365.93 $708.92 $1,074.85 Try a higher rate, i = 7.2%, i/2 = 3.6%.
1 1 (1 i ) n FV PB C n i (1 i ) 1 1 10 (1.036) $1,000 $1,068.86 $44 10 0 . 036 (1.036) $364.09 $702.11 $1,068.04 The YTM is approximately 7.2 percent. The effective annual yield can be computed as: EAY (1 Quoted rate m) m 1
1.036 1 0.0733 7.33% 2
10
Enter
$44
N
i%
Answer
-$1,064.86
PMT
PV
$1,000 FV
3.6156%
The effective annual yield can be computed as: EAY (1 Quotedrate m) m 1
1.036156 1 0.0736 7.36% 2
8.22
Yield to maturity: Adrienne Dawson is planning to buy 10-year zero coupon bonds
issued by the U.S. Treasury. If these bonds have a face value of $1,000 and are currently selling at $404.59, what is the expected ex pected return on these bonds? Assume that interest compounds semiannually on similar coupon-paying bonds. LO 3, LO 4
Solution:
Years to maturity = n = 10
Coupon rate = C = 0% Current market rate = i Assume annual coupon payments. Present value of bond = PB = $404.59 0
1
2
3
20
├───────┼────────┼────────┼── $0
$0
─────────┤
$0
$0 $1,000
To solve for the YTM, a trial-and-error approach has to be used. Try YTM = 10%. PB
Fmn
1 i m
mn
$1,000
1.0520
$404.59 $376.89 Try a lower rate, YTM = 9%. PB
Fmn
1 i m
mn
$1,000
1.04520
$404.59 $414.64 Try YTM=9.25%. PB
Fmn
1 i m
mn
$1,000
1.0462520
$404.59 $404.85 The YTM is approximately 9.25 percent . EAY (1 Quoted rate m) m 1
1.04625 1 0.09464 9.46% 2
The expected return from this investment is 9.46 percent. Enter
20 N
Answer
$0 i%
-$404.59
PMT
4.63%
The effective annual yield can be computed as:
PV
$1,000 FV
EAY (1 Quoted rate m) m 1 2
1.046283 1 0.0947 9.47% 8.23
Realized yield: Brown & Co. issued seven-year bonds two years ago that can be called
after two years. The bond makes semiannual coupon payments at a coupon rate of 7.875 percent. Each bond has a market value of $1,053.40, and the call price is $1,078.7 5. If an investor purchased the bonds at par value when they were originally issued and the bonds are called by the firm today, what is the investor’s realized yield? LO 3, LO 4
Solution:
Purchase price of bond = $1,000 Years investment held = n = 2 Coupon rate = C = 7.875% Frequency of payment = m = 2 Annual coupon = $1,000 × (0.07875/2) = $39.375 Realized yield = i Call price of bond = CP = $1,078.75 Current market value = $1,053.40
To compute the realized return, either the trial-and-error approach or the financial calculator can be used. Since the price has increased, market rates must have decreased.
So, the realized return is going to be greater than the bond’s coupon. Try rates higher than the coupon rate. Try i = 10%, or i/2 = 5%.
1 1 (1 i ) C CP 2 P i (1 i ) 2 2 2 1 1 (1.05)4 $1,078.75 $1, $1, 000 $39 $39.37 .375 5 0 . 0 5 (1.05)4 $139. 9.62 62 $88 $887.4 7.49 9 $1, $1, 006 006.2 .26 6 $13 m n
B
m n
Try a higher rate, i = 11.48% or i/2 = 5.74%.
P B
1 1 (1 i ) C 2 i 2 2
m n
CP (1 i ) 2
m n
1 1 (1.0574)4 $1,078.75 $1, $1, 000 000 $39. $39.375 375 4 0 . 0 5 7 4 (1.0574) $137.25 25 $862. $862.91 91 $1, $1, 000.1 000.16 6 $137.
EAY EAY = (1+ Quo Quoted ted rate rate = 1. 1.0574
2
m)
m
-1
-1
= 0.11809 0.11809 = 11.81% The realized rate of return is approximately 11.81 percent . Using a financial calculator provided an exact yield of 11.49 percent .
Enter
4 N
Answer
$39.375 i %
PMT
5.74%
The effective annual yield can be computed as:
-$1,000 $1,078.75 PV
FV
EAY = (1+ Quoted Quoted rate rate m)m -1 = 1. 1.0574
2
-1
= 0.118 0.1181 1 = 11.81% 11.81% 8.24
Trevor Price bought 10-year bonds issued by b y Harvest Foods five years Realized yield: Trevor Price ago for $936.05. The bonds make semiannual coupon payments at a rate of 8.4 percent. If the current price of the bonds is $1,048.77 each, what is the yield that Trevor would earn by selling the bonds today? LO 3, LO 4
Solution:
Purchase price of bond = $936.05 Years investment held = n = 5 Coupon rate = C = 8.4% Frequency of payment = m = 2 Annual coupon = $1,000 × (0.084/2) = $42 Realized yield = i Selling price of bond = PB = $1,048.77
To compute the realized return, either the trial-and-error approach or the financial calculator can be used. Since the price has increased, market rates must have decreased. So, the realized return is going to be greater than the bond’s coupon. Try rates higher than the coupon rate.
Try i = 11%, or i/2 = 5.5%.
1 1 (1 i ) mn C FV 2 PB (1 i ) mn i 2 2 2 1 1 10 (1.055) $1,048.77 $936.05 $42 14 0 . 055 (1.055) $316.58 $613.98 $930.56
Try a lower rate, i = 10.8% or i/2 = 5.4%. 1 1 (1 i ) mn C 2 PB i 2 2
FV (1 i ) mn 2
1 1 (1.054)10 $1,048.77 $936.05 $42 14 0.054 (1.054) $318.10 $619.83 $937.94
EAY (1 Quoted rate m) m 1 2
1.054 1 0.1109 11.09% The realized rate of return is approximately 11.1 percent . Using a financial calculator provided an exact yield of 11.14 percent.
Enter
10 N
Answer
$42 i%
-$936.05
PMT
5.425%
The effective annual yield can be computed as:
PV
$1,048.77 FV
EAY (1 Quoted rate m) m 1 2
1.05425 1 0.1114 11.14% 8.25
Realized yield: You bought a six-year bond issued by Runaway Corp. four years ago. At
that time, you paid $974.33 for the bond. The bond pays a coupon rate of 7.375 percent, and coupon payments are paid semiannually. Currently, the bond is priced at $1,023.56. What yield can you expect to earn on this bond if you sell it today? LO 3, LO 4
Solution:
Purchase price of bond = $974.33 Years investment held = n = 4 Coupon rate = C = 7.375% Frequency of payment = m = 2 Annual coupon = $1,000 × (0.07375/2) = $36.875 Realized yield = i Selling price of bond = PB = $1,023.56 To compute the realized return, either the trial-and-error approach or the financial calculator can be used. Since the price has increased, market rates must have decreased.
So, the realized return is going to be greater than the bond’s coupon. Try rates higher than the coupon rate. Try i = 9%, or i/2 = 4.5%. 1 1 (1 i ) mn C 2 PB i 2 2
FV (1 i ) mn 2
1 1 8 (1.045) $1,023.56 $974.33 $36.875 8 0 . 045 (1.045) $243.22 $719.75 $962.98
Try a lower rate, i = 8.6% or i/2 = 4.3%. 1 1 (1 i ) mn C FV 2 PB (1 i ) mn i 2 2 2 1 1 (1.043) 8 $1,023.56 $974.33 $36.875 8 0.043 (1.043) $245.22 $730.87 $976.09 EAY (1 Quoted rate m) m 1 2
1.043 1 0.08785 8.79% The realized rate of return is approximately 8.79 percent . Using a financial calculator provided an exact yield of 8.84 percent.
8
Enter
$36.875
N Answer
i%
PMT
-$974.33
$1,023.56
PV
FV
4.327%
The effective annual yield can be computed as: EAY (1 Quoted rate m) m 1 2
1.04327 1 0.0884 8.84%
ADVANCED 8.26
Lopez Information Systems is planning to issue 10-year bonds. The Th e going market yield for such bonds is 8.125 percent. Assume that coupon payments will be made semiannually. The firm is trying to decide between issuing an 8 percent coupon bond or a zero coupon bond. The company needs to raise $1 million. a.
What will be the price of an 8 percent coupon bond?
b.
How many 8 percent coupon bonds would have to be issued?
c.
What will be the price of a zero coupon bonds?
d.
How many zero coupon bonds will have to be issued?
LO 1, LO 2
Solution: a.
Years to maturity = n = 10 Coupon rate = C = 8.125% Semiannual coupon = $1,000 × (0.08/2) = $40 Current market rate = i = 8.125% Present value of bond = PB 0
1
2
3
14
├───────┼────────┼────────┼── $40
$40
─────────┤
$40
$40 $1,000
1 1 2n 1 i F 2 C PB 2 i i 1 2 2 $540.62 $450.94 $991.55
2n
1 1 20 (1.040625) $1,000 $40 20 0.040625 (1.040625)
The firm can sell these bonds at $991.55 .
Enter
20
4.0625%
N
i%
$40 PMT
Answer
b.
$1,000 PV -$991.55
Amount needed to be raised = $1,000,000 Number of bonds sold = $1,000,000 / $991.55 = 1,009
c.
Years to maturity = n = 10 Coupon rate = C = 0% Current market rate = i = 8.125% Assume semiannual coupon payments.
FV
0
1
2
3
4
5
6
20
├───┼────┼───┼───┼───┼────┼── $0
$0
$0
$0
$0
─────┤
$0
$0 $1,000
PB
Fmn
1 m i
mn
20
Enter
$1,000
1.04062520
$450.94
4.0625%
N
i%
$0
$1,000
PMT
PV
Answer
d.
FV
-$450.94
At the price of $450.94, the firm needs to raise $1 million. To do so, the firm will have to issue: Number of contracts
8.27
= $1,000,000 / $450.94 = 2,218 contracts
Showbiz, Inc., has issued eight-year bonds with a coupon of 6.375 percent and
semiannual coupon payments. The market’s required rate of o f return on such bonds is 7.65 7. 65 percent. a.
What is the market price of these bonds?
b.
If the above bond is callable after five years at an 8.5 percent p ercent premium on the face value, what is the expected return on this bond?
LO 2, LO 4
Solution: a.
Years to maturity = n = 8 Coupon rate = C = 6.375% Semiannual coupon = $1,000 × (0.06375/2) = $31.875 Current market rate = i = 7.65% Present value of bond = PB
0
1
2
3
16
├───────┼────────┼────────┼── $31.875
$31.875
$31.875
─────────┤ $31.875 $1,000
1 1 2 n 1 i F 2 PB C 2 i i 1 2 2 $376.26 $548.49 $924.75
2n
1 1 (1.03825)16 $1,000 $31.875 16 0.03825 (1.03825)
The firm can sell these bonds at $924.75 .
b.
Purchase price of bond = $924.75 Years investment held = n = 5 Coupon rate = C = 6.375% Semiannual coupon = $1,000 × (0.06375/2) = $31.875 Frequency of payment = m = 2 Realized yield = i Call price of bond = CP = $1,000 × (1.085) = $1,085.00 To compute the expected return, either the trial-and-error approach or the financial calculator can be used. Try rates higher than the coupon rate. Try i = 8%, or i/2 = 4%. 1 1 C CP (1 i 2) P 2 i 2 (1 i 2) 1 - 1 (1.04)10 $1,085 $924.7 $92 4.75 5 $31 $31.8 .875 75 10 0 . 0 4 (1.04) $258.5 8.54 4 $73 $732.9 2.99 9 $99 $991. 1.53 53 $25 m n
B
m n
Try a higher rate, i = 9.67% or i/2 = 4.835%.
P B
1 1 C (1 i 2) 2 i 2
m n
CP (1 i 2)
m n
1 1 (1.04835)10 $1,085 $924. $92 4.75 75 $31 $31.87 .875 5 10 0.04835 (1.04835) $248.1 8.11 1 $67 $676.6 6.65 5 $92 $924. 4.77 77 $24 The realized rate of return is approximately 9.67% percent . Using a financial calculator provided an exact yield of 9.6705 percent.
10
Enter
N Answer
$31.875 % i
PMT
-$924.75 $1,085 PV
FV
4.835%
The effective annual yield can be computed as: EAY
(1 Quoted rate
m)
m
1
2
1.04835 1 0.0990 90 9.90 9.90% 0.09
8.28
Peabody Corp. has seven-year bonds outstanding. The bonds pay a coupon of 8.375 percent semiannually and are currently worth $1,063.49. The bonds can be called in three years at price of $1,075. a.
What is the yield to maturity of these bonds?
b.
What is the effective annual yield?
c.
What is the realized yield on the bonds if they are called?
d.
If you plan to invest in one of these bonds today, what is the expected yield on the investment? Explain.
LO 3, LO 4
Solution:
a.
Years to maturity = n = 7 Coupon rate = C = 8.375% Current market rate = i Semiannual coupon payments = $1,000 × (0.08375/2) = $41.875 Present value of bond = PB = $1,063.49 0
1
2
3
├───────┼────────┼────────┼── $41.875
$41.875
$41.875
14
─────────┤ $41.875 $1,000
To solve for the YTM, a trial-and-error approach has to be used. Since this is a premium bond, the market rate should be lower than 8.375 percent. Try i = 8% or i/2 = 4%. 1 1 (1 i ) n FV 2 PB C (1 i ) n i 2 2 1 1 (1.04)14 $1,000 $1,063.49 $41.875 14 0 . 04 (1.04) $442.33 $577.48 $1,019.81 Try a lower rate, i = 7.2%, or i/2 = 3.6%. 1 1 (1 i 2) n FV PB C i (1 i ) n 2 2 1 1 (1.036)14 $1,000 $1,063.49 $41.875 14 0.036 (1.036) $454.24 $609.49 $1,063.73 The yield-to maturity is approximately 7.2 percent.
b.
The effective annual yield can be computed as: EAY (1 Quoted rate m) m 1 2
1.036 1 0.0941 7.3% 14
Enter
$41.875
N
i%
Answer
PMT
-$1,063.49 $1,000 PV
FV
3.5998%
The effective annual yield can be computed as: EAY (1 Quoted rate m) m 1 2
1.035998 1 0.073292 7.3% c.
Purchase price of bond = $1,063.49 Years investment held = n = 3 Coupon rate = C = 8.375% Semiannual coupon payments = $1,000 x (0.08375/2) = $41.875 Frequency of payment = m = 2 Realized yield = i Selling price of bond = PB = $1,075
To compute the realized return, either the trial-and-error approach or the financial calculator can be used. Since the price has increased, market rates must have decreased.
So, the realized return is going to be higher than the bond’s coupon. Try rates higher than the coupon rate. Try i = 9%, or i/2 = 4.5%.
1 1 m n C FV (1 i 2) PB m n 2 i 2 (1 i 2) 1 1 (1.045)6 $1,075 $1,063. $1,063.49 49 $41 $41.8 .875 75 6 0 . 0 4 5 (1.045) $215.9 5.99 9 $82 $825.4 5.49 9 $1,041. $1,041.48 48 $21
Try a lower rate, i = 8.2% or i/2 = 4.1%. 1 1 m n C FV (1 i 2) PB m n 2 i 2 (1 i 2) 1 1 6 (1.041) $1,075 $1,063.4 $1,063.49 9 $41 $41.87 .875 5 6 0 . 0 4 1 (1.041) $218.8 8.80 0 $84 $844.7 4.70 0 $1, $1, 063 063.50 .50 $21
EAY (1 Quoted rate m)m 1 2
1.041 1 0.08368 68 8.368 8.368% 0.083
The realized rate of return is approximately 8.37 percent . Using a financial calculator provided an exact yield of 8.2 percent.
Enter
6 N
Answer
% i 4.1%%
$41.875 -$1,063.49
$1075
PMT
FV
PV
The effective annual yield can be computed as: EAY (1 Quoted rate m)m 1 2
1.04100 1 0.083 0.08368 68 8.368 8.368%
d.
Purchase price of bond = PB = $1,063.49 Years to maturity = n =7 Coupon rate = C = 8.375% Semiannual coupon payments = $1,000 × (0.08375/2) = $41.875 Frequency of payment = m = 2 Maturity value = FV = $1,000 Use the trial-and-error approach to compute the yield to maturity. Since we have a premium bond, market rates are lower than the bond’s coupon. Try i = 7%, or i/2 = 3.5%. 1 1 (1 i )mn C FV 2 PB i (1 i )mn 2 2 2 1 1 (1.035)14 $1,000 $1, $1, 063 063.4 .49 9 $41 $41.8 .875 75 14 0 . 0 3 5 (1.035) $457.3 7.30 0 $61 $617.7 7.78 8 $1,075. $1,075.08 08 $45
Try a higher rate, i = 7.2%, or i/2 = 3.6%.
1 1 n (1 i 2) FV PB C i (1 i ) n 2 2 1 1 14 (1.036) $1,000 $1,063.49 $41.875 14 0.036 (1.036) $454.24 $609.49 $1,063.73 The expected yield is approximately 7.2 percent which is the same as the yield to maturity obtained in (a). If the bond is not called and is held to maturity, then the expected yield is the yield to maturity. EAY (1 Quoted rate m) 1 m
2
1.036 1 = 0.073296 = 7.33%
The expected yield is approximately 7.33 percent . Using a financial calculator provided an exact yield 7.334 percent.
14
Enter
$41.875 -$1,063.49 $1,000
N
i%
Answer
PMT
PV
FV
3.602%
The effective annual yield can be computed as: EAY (1 Quoted rate
m)
m
1
2
1.03602 1 0.0733 3337 37 7.33 7.33% 0.07
8.29
The Maryland Department of Transportation has issued 25-year 25- year bonds that make semiannual coupon payments at a rate of 9.875 percent. The current market rate for similar securities is 11 percent. a.
What is the current market value of one of these bonds?
b.
What will be the bond’s price if rates in the market (i) decrease to 9 percent; (ii) increase to 12 percent?
c.
Refer to your answers in part b. How do d o the interest rate changes affect premium bonds and discount bonds?
d.
Suppose the bond were to mature in 12 years. How do the interest rate changes in part b affect the bond prices?
LO 2, LO 3, LO 4
Solution: a.
Years to maturity = n = 25 Coupon rate = C = 9.875% Semiannual coupon = $1,000 × (0.09875/2) = $49.375 Current market rate = i = 11% Present value of bond = PB 0
1
2
3
50
├───────┼────────┼────────┼── $49.375
$49.375
─────────┤
$49.375
$49.375 $1,000
1 1 2n 1 i F 2 C PB 2 i i 1 2 2 $835.99 $68.77 $904.76
2n
1 1 50 (1.055) $1,000 $49.375 50 0.055 (1.055)
The Maryland bonds will sell at $904.76.
Enter
50
5.5% N
Answer
$49.375 i%
$1,000 PMT -$904.76
PV
FV
b.
(i)
Current market rate = i = 9% 1 1 1 2n 1 i 50 (1.045) 1 F $1,000 2 PB C $ 49 . 375 2 n 2 i 0.045 (1.045) 50 i 1 2 2 $975.75 $110.71 $1,086.46
The Maryland bonds will increase in price to sell at $1,086.46 . 50
Enter
4.5%
$49.375
N
i%
$1,000 PMT
Answer
(ii)
PV
FV
-$1,086.46
Current market rate = i = 12%
1 1 1 2n 1 i 50 (1.06) $1,000 1 F 2 PB C $ 49 . 375 2n 2 i 0 . 06 (1.06) 50 i 1 2 2 $778.24 $54.29 $832.53
The Maryland bonds will drop in price to $832.53 . 50
Enter
6% N
$49.375 i%
Answer
c.
PMT
PV
FV
-$832.53
Bonds, in general, decrease in price when wh en interest rates go up. When interest rates decrease, bond prices increase.
d.
$1,000
(i)
Current market rate = i = 9% Term to maturity = 12 years
1 1 1 2n 1 i 24 1 F $1,000 (1.045) 2 PB C $ 49 . 375 2n 2 i 0.045 (1.045) 24 i 1 2 2 $715.71 $347.70 $1,063.42
The Maryland bonds will increase in price to sell at $1,063.42 . 24
Enter
4.5%
N
$49.375
i%
PMT
PV
Answer
(ii)
$1,000 FV
-$1,063.42
Current market rate = i = 12%
1 1 2n i 1 F 2 PB C 2 i i 1 2 2 $619.67 $246.98 $866.65
2n
1 1 (1.06) 24 $1,000 $49.375 24 0.06 (1.06)
The Maryland bonds will drop in price to $866.65 . Enter
24 N
6% i%
$49.375 PMT
Answer
$1,000 PV
FV
-$866.65
With shorter maturity, bond prices react the same way as in part b, but to a lesser extent. When interest rates increase, the bond’s price declines; but
the decline in price is less
than that for a longer term bond. When interest rates decrease, bond prices increase with longer-term bonds, increasing more than shorter-term bonds.
8.30
Rachette Corp. has 18-year bonds outstanding. These bonds, which pay interest semiannually, have a coupon rate of 9.735 percent and a yield to maturity of 7.95 percent.
a. Compute the current price of these bonds. b. If the bonds can be called in five years at a premium of 13.5 percent over par value, what
is the investor’s realized yield?
c. If you bought one of these bonds today, what is your expected rate of return? Explain. LO 2, LO 3, LO 4
Solution: a.
Years to maturity = n = 18 Coupon rate = C = 9.735% Semiannual coupon = $1,000 × (0.09735/2) = $48.675 Current market rate = i = 7.95% Present value of bond = PB 0
1
2
3
36
├───────┼────────┼────────┼── $48.675
$48.675
─────────┤
$48.675
$48.675 $1,000
1 1 2n 1 i F 2 C PB 2 i 1 i 2 2 $923.5 3.56 6 $24 $245 5,79 $1,169.34 $92
2n
1 1 (1.03975)36 $1, 000 $48.675 36 0.03975 (1.03975)
The bond’s current price is at $1,169.34 . 36
Enter
3.975%
N
% i
$48.675 PMT
Answer
b.
$1,000 PV
FV
-$1,169.34
Purchase price of bond = $1,169.34 (value today); Call price = $1135
To compute the realized return, either the trial-and-error approach or the financial calculator can be used.
Try rates lower than the coupon rate.
Try i = 8%, or i/2 = 4%. 1 1 (1 i ) mn C FV 2 PB i (1 i ) mn 2 2 2 1 1 (1.04)10 $1,135 $1,169. $1,169.34 34 $48.675 $48.675 10 0 . 0 4 (1.04) $394.8 4.80 0 $76 $766.7 6.77 7 $1,16 $1,161. 1.56 56 $39
Try a lower rate, i = 7.8% or i/2 = 3.9%. 1 1 m n (1 i ) C FV 2 PB i (1 i ) mn 2 2 2 1 1 10 (1.039) $1,135 $1,169. $1,169.34 34 $48.675 $48.675 10 0 . 0 3 9 (1.039) $396. 6.77 77 $77 $774. 4.18 18 $1,17 $1,170. 0.95 95 $39 EAY (1 Quoted rate m)m 1 2
1.039 1 0.0795 952 2 7.95 7.95% 0.07 The realized rate of return is approximately 7.95 percent . Using a financial calculator provided an exact yield of 7.834 percent.
Enter
10 N
Answer
$48.675 -$1,169.34 $1,135.00 i%
PMT
3.917%
The effective annual yield can be computed as:
PV
FV
EAY (1 Quoted rate m)m 1 2
1.03917 1 0.07987 7.99%
c.
Purchase price of bond = PB = $1,169.34 Years to maturity = n = 18 Coupon rate = C = 9.735% Semi-annual coupon = $1,000 × (0.09735/2) = $48.675 Frequency of payment = m = 2 Maturity value = FV = $1,000
Use the trial-and-error approach to compute the yield to maturity. Since we have a
premium bond, market rates are lower than the bond’s coupon. Try i = 8%, or i/2 = 4.0%. 1 1 (1 i ) mn C FV 2 PB i (1 i ) mn 2 2 2 1 1 36 (1.04) $1,000 $1,169. $1,169.34 34 $48.675 $48.675 36 0 . 0 4 (1.04) $920.3 0.36 6 $24 $243.6 3.67 7 $1,16 $1,164. 4.03 03 $92
Try i = 7.9%, or i/2 = 3.95%.
1 1 (1 i ) mn C 2 PB i 2 2
FV (1 i ) mn 2
1 1 36 (1.0395) $1,000 $1,169. $1,169.34 34 $48.675 $48.675 36 0.0395 (1.0395) $926.7 6.77 7 $24 $247.9 7.92 2 $1,17 $1,174. 4.69 69 $92
Thus the expected yield is between 7.9 percent and 8 percent. Using a financial calculator provided an exact yield of 7.95 percent.
36
Enter
$48.675
N
i%
Answer
PMT
-$1,169.34 $1,000 PV
FV
3.975%
The effective annual yield can be computed as: EAY (1 Quoted rate m)m 1 2
1.03975 1 = 0.08108 = 8.11%
8.31
Zippy Corporation just sold $30 million million of convertible bonds with a conversion ratio of 40. Each $1,000 bond
a. b.
is convertible into 25 shares of Zippy’s stock.
What is the conversion price of Zippy’s stock? If the current price of Zippy’s stock is $15 and the Company’s annual stock return is normally distributed with a standard deviation of $5, what is the probability that investors will find it attractive to convert the bond into Zippy stock in the next year?
LO 1, LO 2
Solution:
a.
The conversion price is $1,000/40 = $25.
b.
The stock price would have to increase by approximately two standard deviations (2 $5 = $10) for the price to increase to $25 and for conversion to become attractive to the investors. From Chapter 7 we know kno w that 95% of possible outcomes fall within two standard deviations of the mean (avera ge) value in a normal distribution. This means that there is a 5 percent chance that the stock price will move up or down by $10 or more. Since the normal distribution is
symmetric, this means that there is only a 2.5 percent pe rcent chance that Zippy’s stock price will increase enough for it to become attractive for the investors to to exercise the conversion option in the next year.
Sample Test Problems 8.1
Torino Foods issued 10-year bonds three years ago with a coupon of 6 percent. If the current market rate is 8.5 percent and the bonds make annual coupon payments, what is the current market value of one of these bonds?
Solution:
Years to maturity = n = 7 Coupon rate = C = 6% Annual coupon = $1,000 × 0.06 = $60 Current market rate = i = 8.5% Present value of bond = PB 0
1
2
3
4
7
├───┼────┼───┼───┼─── $60
$60
$60
$60
─────┤ $60 $1,000
n = 7;
C = 6%;
i = YTM = 8.5%
1 1 1 1 n 7 (1 i ) (1.085) $1,000 F PB C $ 60 n 7 ( 1 ) 0 . 085 i i (1.085) $307.11 $564.93 $872.04
8.2
Kim Sundaram recently bought a 20-year zero coupon bonds that compounds interest semiannually. If the current market rate is 7.75 percent, what is the maximum price he should have paid for this bond?
Solution:
Years to maturity = n = 20 Coupon rate = C = 0% Current market rate = i = 7.75%
Frequency of payments = m = 2
0
1
2
3
4
5
6
├───┼────┼───┼───┼───┼────┼── $0
$0
$0
$0
$0
40
─────┤
$0
$0 $1,000
F
PB 1
8.3
$1, 000
mn mn
i
1.03875
40
$218.55
m
Five-year bonds of Infotech Corporation are currently priced at $1,065.23. They make semiannual coupon payments of 8.5 percent. If you bought these bonds today, what would be the yield to maturity and effective annual yield that you would earn?
Solution:
Years to maturity = n = 5 Coupon rate = C = 8. 5% Current market rate = i Semiannual coupon payments = $1,000 × (0.085/2) = $42.50 Present value of bond = PB = $1,065.23 0
1
2
3
├───────┼────────┼────────┼── $42.50
$42.50 $42.50
10
─────────┤ $42.50 $1,000
To solve for the YTM, a trial-and-error approach has to be used. Since this is a premium bond, the market rate should be lower than 8. 5 percent. Try i = 8% or i/2 = 4%.
1 1 (1 i ) n FV 2 PB C (1 i ) n i 2 2 1 1 10 (1.04) $1,000 $1,065.23 $42.50 10 0 . 04 (1.04) $344.71 $675.56 $1,020.28 Try a lower rate, i = 7 %, or i/2 = 3.5%. 1 1 (1 i ) n FV 2 PB C (1 i ) n i 2 2 1 1 (1.035)10 $1,000 $1,065.23 $42.50 10 0.035 (1.035) $349.92 $708.92 $1,062.37 The yield to maturity is approximately 7 percent . The effective annual yield can be computed as: EAY (1 Quoted rate m) m 1 2
1.035 1 0.0712 7.12% 10
Enter
$42.50 N
Answer
6.93% -$1,065.23 $1,000
i%
PMT
3.467%
The effective annual yield can be computed as: EAY (1 Quoted rate m) m 1 2
1.03467 1 0.07054 7.05%
PV
FV
8.4
The Gold Company is applying for a five-year term loan from its bank. The lender determines that the firm should pay a default risk premium of 1.75 1 .75 percent over the Treasury rate. The five-year Treasury rate is currently 5.65 percent. The firm also faces a marketability risk premium of 0.80 percent. What is the total borrowing cost to the firm?
Solution:
Risk-free real rate of interest =
5.65%
Market risk premium
=
0.80%
Default risk premium
=
1.75%
Using Equation 8.6: k corp corp
=
irf + risk premium adjustments
=
irf + MRP + DRP
6.5% = =
5.65%+0.80%+1.75 8.2%
The company’s borrowing cost is 8.2 percent. 8.5
Trojan Corp. has issued seven-year bonds with a 7 percent semiannual coupon payment. If the opportunity cost for an investor is 8.25 percent, what is the maximum price that this investor would pay?
Solution:
Years to maturity = n = 7 Coupon rate = C = 7% Semi-annual coupon payments = $1,000 × (0.07/2) = $35 Current market rate = i = 8.25% Present value of bond = PB 0
1
2
3
4
14
├───┼────┼───┼───┼─── $35
$35
$35
$35
─────┤ $35 $1,000
n = 7; m = 2; C = 7%;
i = YTM = 8.25%
1 1 1 1 (1 i ) n (1.04125)14 F $1,000 PB C $ 35 n 14 i (1 i ) 0.04125 (1.04125) $366.68 $567.84 $934.52 The investor should pay no more than $934.52.
