End-Of-Chapter Questions

CHAPTER 7

Bonds Valuation

ANSWERS TO

END-OF-CHAPTER QUESTIONS

7-1.Book value is the asset's historical value and is represented on the balance sheet as cost minus depreciation. Liquidation value is the dollar sum that could be realized if the assets were sold individually and not as part of a going concern. Market value is the observed value for an asset in the marketplace where buyers and sellers negotiate a mutually acceptable price. Intrinsic value is the present value of the asset's expected future cash flows discounted at an appropriate discount rate.

7-2.The value of a security is equal to the present value of cash flows to be received by the investor. Hence, the terms value and present value are synonymous.

7-3.The first two factors affecting asset value (the asset characteristics) are the asset's expected returns and the riskiness of these returns. The third consideration is the investor's required rate of return. The required rate of return reflects the investor's risk-return preference.

7-4.The relationship is inverse. As the required rate of return increases, the value of the security decreases, and a decrease in the required rate of return results in a price increase.

7-5.(a)The par value is the amount stated on the face of the bond. This value does not change and, therefore, is completely independent of the market value. However, the market value may change with changing economic conditions and changes within the firm.

(b)The coupon interest rate is the rate of interest that is contractually specified in the bond indenture. As such, this rate is constant throughout the life of the bond. The coupon interest rate indicates to the investor the amount of interest to be received in each payment period. On the other hand, the investor's required rate of return is equivalent to the bond’s current yield to maturity, which changes with the changing bond's market price. This rate may be altered as economic conditions change and/or the investor's attitude toward the risk-return trade-off is altered.

7-6.In the case of insolvency, claims of debt in general, including bonds, are honored before those of both common stock and preferred stock. However, different types of debt may also have a hierarchy among themselves as to the order of their claim on assets.

Bonds also have a claim on income that comes ahead of common and preferred stock. If interest on bonds is not paid, the bond trustees can classify the firm insolvent and force it into bankruptcy. Thus, the bondholder's claim on income is more likely to be honored than that of common and preferred stockholders, whose dividends are paid at the discretion of the firm's management.

7-7.Ratings involve a judgment about the future risk potential of the bond. Although they deal with expectations, several historical factors seem to play a significant role in their determination. Bond ratings are favorably affected by (1) a greater reliance on equity, and not debt, in financing the firm, (2) profitable operations, (3) a low variability in past earnings, (4) large firm size, and (5) little use of subordinated debt. In turn, the rating a bond receives affects the rate of return demanded on the bond by the investors. The poorer the bond rating, the higher the rate of return demanded in the capital markets.

For the financial manager, bond ratings are extremely important. They provide an indicator of default risk that in turn affects the rate of return that must be paid on borrowed funds.

7-8.The term debentures applies to any unsecured long-term debt. Because these bonds are unsecured, the earning ability of the issuing corporation is of great concern to the bondholder. They are also viewed as being more risky than secured bonds and as a result must provide investors with a higher yield than secured bonds provide. Often the issuing firm attempts to provide some protection to the holder through the prohibition of any additional encumbrance of assets. This prohibits the future issuance of secured long-term debt that would further tie up the firm's assets and leave the bondholders less protected. To the issuing firm, the major advantage of debentures is that no property has to be secured by them. This allows the firm to issue debt and still preserve some future borrowing power.

A mortgage bond is a bond secured by a lien on real property. Typically, the value of the real property is greater than that of the mortgage bonds issued. This provides the mortgage bondholders with a margin of safety in the event the market value of the secured property declines. In the case of foreclosure, the trustees have the power to sell the secured property and use the proceeds to pay the bondholders. In the event that the proceeds from this sale do not cover the bonds, the bondholders become general creditors, similar to debenture bondholders, for the unpaid portion of the debt.

7-9.(a)Eurobonds are not so much a different type of security as they are securities, in this case bonds, issued in a country different from the one in whose currency the bond is denominated. For example, a bond that is issued in Europe or in Asia by an American company and that pays interest and principal to the lender in U.S. dollars would be considered a Eurobond. Thus, even if the bond is not issued in Europe, it merely needs to be sold in a country different from the one in whose currency it is denominated to be considered a Eurobond.

(b)Zero and very low coupon bonds allow the issuing firm to issue bonds at a substantial discount from their $1,000 face value with a zero or very low coupon. The investor receives a large part (or all on the zero coupon bond) of the return from the appreciation of the bond at maturity.

(c)Junk bonds refer to any bond with a rating of BB or below. The major participants in this market are new firms that do not have an established record of performance. Many junk bonds have been issued to finance corporate buyouts.

7-10.The expected rate of return is the rate of return that may be expected from purchasing a security at the prevailing market price. Thus, the expected rate of return is the rate that equates future cash flows with the actual selling price of the security in the market.

7-11.When the coupon interest rate does not equal the bondholder's required rate of return, the bond will be issued at either a premium or discount. If the investor's required rate of return is higher than the coupon interest rate, the bond will be issued at a discount to the investor. If the coupon rate is higher that the investor's required rate, the bond will be issued at a premium.

7-12.A premium bond is issued when the coupon rate is higher than the bondholder's required rate of return. The premium is the excess of the market value over the face value of the bond. A discount bond is issued when the bondholder's required rate of return is higher than the coupon rate. The discount is the excess of the face value of the bond over the market value.

Over time, the premium or discount on a bond is amortized. This amortization allows the bondholder to realize an effective yield equal to their required rate of return.

7-13.A change in current interest rates (required rate of return) causes a change in the market value of a bond. However, the impact on value is greater for long-term bonds than it is for short-term bonds. The reason long-term bond prices fluctuate more than short-term bond prices in response to interest rate changes is simple. Assume an investor bought a 10-year bond yielding a 12 percent interest rate. If the current interest rate for bonds of similar risk increased to 15 percent, the investor would be locked into the lower rate for 10 years. If, on the other hand, a shorter-term bond had been purchased, say one maturing in 2 years, the investor would have to accept the lower return for only 2 years and not the full 10 years. At the end of year 2, the investor would receive the maturity value of $1,000 and could buy a bond offering the higher 15 percent rate for the remaining 8 years. Thus, interest rate risk is determined, at least in part, by the length of time an investor is required to commit to an investment.

7-14.The duration of a bond is simply a measure of the responsiveness of its price to a change in interest rates. The greater the relative percentage change in a bond price in response to a given percentage change in the interest rate, the longer the duration.

SOLUTIONS TO

END-OF-CHAPTER PROBLEMS

7-1A.Value (Vb)=

= $80(6.194) + $1,000 (.257)

= $495.52 + $257.00

= $752.52

1000

ANSWER-752.23

7-2A.If the interest is paid semiannually:

Value (Vb)=

Vb= $45 (11.652) + $1,000 (0.534)

Vb= $1,058.34

1000

ANSWER-1058.26

If interest is paid annually:

Value (Vb)=

Vb= $90(5.747) + $1,000 (0.540)

Vb= $1,057.23

1000

ANSWER1057.47

7-3A.$900 =

At 5%: $40(12.462) + $1,000(0.377) = $875.48

At 4%: $40(13.590) + $1,000(0.456) ≈ $1,000.00

Interpolation:

Expected rate of return = 4% + = 4.8%

900

1000

ANSWER4.79semiannual rate

The rate is equivalent to 9.6 percent annual rate compounded semiannually, or 9.8 percent (1.0482 - 1) compounded annually.

7-4A.$945 =

At 10%: $90(8.514) + $1,000(0.149) = $915.26

At 9%: $90(9.129) + $1,000(0.178) ≈ $1,000.00

Interpolation:

9% $1,000.00 $1,000.00

945.00

10% 915.26

$55.00 $84.74

= 9% + = 9.65%

945

1000

ANSWER9.63

7-5A.$1,150 =

At 6%: $70(8.384) + $1,000(0.497) = $1,083.88

At 5%: $70(8.863) + $1,000(0.557) = $1,177.41

Interpolation:

5% $1,177.41 $1,177.41

1,150.00

6% 1,083.88

$27.41 $93.53

= 5% + = 5.29%

1150

1000

ANSWER5.28

7-6A.a.$1,085 =

Interpolation:

At 7% $80(9.108) + $1,000(0.362) = $1,090.64

At 8% $80(8.559) + $1,000(0.315) ≈ $1,000.00

7% $1,090.64 $1,090.64

1,085.00

8% 1,000.00

$5.64 $90.64

= 7% + = 7.06%

1085

1000

ANSWER7.06

b.Vb =

Vb = $80(7.606) + $1,000(0.239)

Vb = $847.48

1000

ANSWER-847.88

c.Since the expected rate of return, 7.06 percent, is less than your required rate of return of 10 percent, the bond is not an acceptable investment. This fact is also evident because the market price, $1,085, exceeds the value of the security to the investor of $847.48.

7-7A.a.Value

Par Value $1,000.00

Coupon $ 100.00

Required Rate of Return 0.12

Years to Maturity 15

Market Value $ 863.78

b.Value at Alternative Rates of Return

Required Rate of Return 0.15

Market Value $ 707.63

Required Rate of Return 0.08

Market Value $1,171.19

c.As required rates of return change, the price of the bond changes, which is the result of "interest-rate risk." Thus, the greater the investor's required rate of return, the greater will be his/her discount on the bond. Conversely, the less his/her required rate of return below that of the coupon rate, the greater the premium will be.

d.Value at Alternative Maturity Dates

Years to Maturity 5

Required Rate of Return 0.12

Market Value $ 927.90

Required Rate of Return 0.15

Market Value $ 832.39

Required Rate of Return 0.08

Market Value $1,079.85

e.The longer the maturity of the bond, the greater the interest rate risk the investor is exposed to, resulting in greater premiums and discounts.

7-8A.$1,250=

At 7%: $90(9.108) + $1,000(0.362) = $1,181.72

At 6%: $90(9.712) + $1,000(0.417) = $1,291.08

Interpolation:

6% $1,291.08 $1,291.08

b% 1,250.00

7% 1,181.72

$41.08 $109.36

b =6% + = 6.376%

1250

1000

ANSWER6.364

7-9A.(a)Vb =

Vb =$110(9.123) + $1,000(0.178)

Vb=$1,181.53

110

1000

ANSWER-1,182.57

(b)(i)Vb=

Vb=$110(7.469) + $1,000(0.104)

Vb=$925.59

110

1000

ANSWER-925.31

(b)(ii)Vb=

Vb=$110(11.470) + $1,000(0.312)

Vb=$1,573.70

110

1000

ANSWER-1,573.50

(c)We see that value is inversely related to the investor's required rate of return.

7-10A.

Bond P Q R S T

$1,080 $960 $1,268 $1,000 $872

Years Ct *tPV(Ct) Ct t*PV(Ct) Ct t*PV(Ct) Ct t*PV(Ct) Ct t*PV(Ct)

1 $100 $93 $70 $65 $120 $111 $80 $74 $65 $60

2 100 171 70 120 120 206 80 137 65 111

3 100 238 70 167 120 286 80 191 65 155

4 100 294 70 206 120 353 80 235 65 191

5 1,100 3,743 1,070 3,641 120 408 80 272 65 221

6 120 454 80 302 65 246

7 120 490 80 327 65 265

8 120 519 80 346 65 281

9 120 540 80 360 65 293

10 1,120 5,188 1,080 5,002 65 301

11 65 307

12 65 310

13 65 311

14 65 310

15 1,065 5,036

4,539 4,198 8,554 7,247 8,398

Duration 4.20 4.37 6.74 7.25 9.63

Solution to Integrative Problem

1.Bank of America Bond Value (Vb)=

= $78 (7.3601) + $1,000 (.5584)

= $574.09 + $558.39

= $1,132.48

1000

ANSWER-$1,132.48

Hilton Bond Value (Vb)=

= $75.00 (8.5436) + $1,000 (.2311)

= $640.77 + $231.10

= $871.87

75.00

1000

ANSWER-$871.85

Time Warner Bond Value (Vb)=

= $79.75 (3.3121) + $1,000 (.7350)

= $264.14 + $735.00

= $999.14

79.75

1000

ANSWER-$999.17

2.To compute the expected rate of return for each bond, use a financial calculator to solve the following equation:

Time Warner:

$1,035=

1,035

79.75

1000

ANSWER 6.94%

Bank of America:

$1,030=

1,030

1000

ANSWER7.37%

Hilton Hotels:

$973=

973

75.00

1000

ANSWER7.79%

3.i.Time Warner Bond Value (Vb)=

= $79.75 (3.1024) + $1,000 (.6587)

= $247.42 + $658.70

= $906.12

79.75

1000

ANSWER- $906.15

Bank of America Bond Value (Vb)=

= $78 (6.4177) + $1,000 (.4224)

= $500.58 + $422.40

= $922.98

1000

ANSWER- $922.99

(Vb)=

= $75.00 (7.2497) + $1,000 (.1456)

= $543.73 + $145.60

= $689.33

75.00

1000

ANSWER- $679.62

3.iiTime Warner Bond Value (Vb)=

= $79.75 (3.546) + $1,000 (.8227)

= $282.79 + $822.70

= $1,105.49

79.75

1000

ANSWER- $1,105.49

Bank of America Bond Value (Vb)=

= $78 (8.5302) + $1,000 (.7441)

= $665.36 + $744.10

= $1,409.46

1000

ANSWER- $1,409.45

(Vb)=

= $75.00 (10.4773) + $1,000 (.3714)

= $785.80 + $371.40

= $1,157.20

75.00

1000

ANSWER- $1,157.16

4.As the interest rates rise and fall, we see the different effects on the bond prices depending on the length of time to maturity and whether the investor's required rate of return is above or below the coupon interest rate. If the investor’s required rate of return is above the coupon interest rate, the bond will sell at a discount (below par value), but if the investor’s required rate of return is below the coupon interest rate, the bond will sell at a price above its par value (premium).

5.Duration of bonds

Time-Warner / Bank of America / Hilton Hotels
Bond Value / $1,035.00 / $1,030.00 / $ 973.00
Required rate of return / 8% / 6% / 9%
Year / Ct / t* PV(Ct) / Ct / t* PV(Ct) / Ct / t* PV(Ct)
1 / $ 79.75 / $ 73.84 / $ 78.00 / $ 73.58 / $ 75.00 / $ 68.81
2 / 79.75 / 136.75 / 78.00 / 138.84 / 75.00 / 126.25
3 / 79.75 / 189.92 / 78.00 / 196.47 / 75.00 / 173.74
4 / 1,079.75 / 3,174.59 / 78.00 / 247.13 / 75.00 / 212.53
5 / 78.00 / 291.43 / 75.00 / 243.72
6 / 78.00 / 329.92 / 75.00 / 268.32
7 / 78.00 / 363.12 / 75.00 / 287.19
8 / 78.00 / 391.51 / 75.00 / 301.12
9 / 78.00 / 415.51 / 75.00 / 310.79
10 / 1,078.00 / 6,019.50 / 75.00 / 316.81
11 / 75.00 / 319.71
12 / 75.00 / 319.98
13 / 75.00 / 318.02
14 / 75.00 / 314.21
15 / 75.00 / 308.86
16 / 75.00 / 302.24
17 / 1,075.00 / 4222.86
Sum of t*PV(Ct) / 3,575.11 / 8,467.02 / 8,415.17
Duration / 3.45 / 8.22 / 8.65

6.Although the Bank of America bonds and the Hilton Hotel bonds have different terms to maturity, the duration of the two bonds is very similar. These two bonds will likely have similar sensitivity to changes in interest rates as evidenced by their duration values.

7.The Time Warner and Hilton Hotels bonds have lower expected rates of return than your required rate of return. Bank of America’s expected rate of return is greater than your required rate of return. So we would buy Bank of America and not Time Warner or Hilton Hotels.

Solutions to Problem Set B

7-1B.Value (Vb)=

= $90(5.018) + $1,000 (.247)

= $451.62 + $247.00

= $698.62

1000

ANSWER-698.87

7-2B.If the interest is paid semiannually:

Value (Vb)=

Vb= $50 (13.784) + $1,000 (0.380)

Vb= $1,069.20

4.5

1000

ANSWER-1068.92

If interest is paid annually:

Value (Vb)=

Vb= $100(6.805) + $1,000 (0.388)

Vb= $1,068.50

100

1000

ANSWER-1068.05

7-3B.$950 =

At 5%: $45(10.838) + $1,000(0.458) = $945.71

At 4%: $45(11.652) + $1,000(0.534) = $1,058.34

Interpolation:

Expected rate of return = 4% + = 4.96%

950

1000

ANSWER4.96

The rate is equivalent to 9.92 percent annual rate, compounded semiannually or 10.17 percent (1.04962 - 1) compounded annually.

7-4B.$975 =

At 10%: $100(8.514) + $1,000(0.149) ≈ $1,000.00

At 11%: $100(7.963) + $1,000(0.124) = $920.30

Interpolation:

10% $1,000.00 $1,000.00

975.00

11% 920.30

$25.00 $79.70

= 10% + = 10.32%

975

100

1000

ANSWER10.30

7-5B.$1,175 =

At 6%: $80(9.712) + $1,000(0.417) = $1,193.96

At 7%: $80(9.108) + $1,000(0.362) = $1,090,64

Interpolation:

6% $1193.96 $1,193.96

1,175.00

7% 1,090.64

$18.96 $103.32

= 6% + = 6.18%

1175

1000

ANSWER6.18

7-6B.a.$1,100 =

Interpolation:

At 7% $90(8.746) + $1,000(0.388) = $1,175.14

At 8% $90(8.2445) + $1,000(0.341) = $1,082.96

7% $1,175.14 $1,175.14

1,100.00

8% 1,082.96

$75.14 $92.18

= 7% + = 7.82%

1100

1000

ANSWER7.80

b.Vb =

Vb = $90(7.367) + $1,000(0.263)

Vb = $926.03

1000

ANSWER-926.33

c.Since the expected rate of return, 7.82 percent, is less than your required rate of return of 10 percent, the bond is not an acceptable investment. This fact is also evident because the market price, $1,100, exceeds the value of the security to the investor of $926.03.

7-7B.

a.Value

Par Value $1,000.00

Coupon $ 80.00

Required Rate of Return 0.07

Years to Maturity 20

Market Value $ 1,105.94

b.Value at Alternative Rates of Return

Required Rate of Return 0.10

Market Value $ 829.73

Required Rate of Return 0.06

Market Value $1,229.40

c.As required rates of return change, the price of the bond changes, which is the result of "interest-rate risk." Thus, the greater the investor's required rate of return, the greater will be his/her discount on the bond. Conversely, the less his/her required rate of return is below that of the coupon rate, the greater the premium will be.

d.Value at Alternative Maturity Dates

Years to Maturity 10

Required Rate of Return 0.07

Market Value $ 1,070.24

Required Rate of Return 0.10

Market Value $ 877.11

Required Rate of Return 0.06

Market Value $1,147.20

e.The longer the maturity of the bond, the greater the interest-rate risk the investor is exposed to, resulting in greater premiums and discounts.

7-8B.$1,110 =

At 5%: $70(9.899) + $1,000(0.505) = $1,197.93

At 6%: $70(9.295) + $1,000(0.442) = $1,092.65

Interpolation:

5% $1,197.93 $1,197.93

b% 1,110.00

6% 1,092.65

$87.93 $105.28

b =5% + = 5.84%

1110

1000

ANSWER5.83

7-9B.(a)Value (Vb)=+

=$70(8.825) + $1,000 (0.250)

=$617.75 + $250.00

=$867.75

8.5

1000

ANSWER-867.62

(b)(i)Value (Vb)=+

=$70(7.549) + $1,000 (0.170)

=$528.43 + $170.00

=$698.43

1000

ANSWER-698.05

(b)(ii)Value (Vb)=+

=$70(10.477) + $1,000 (0.371)

=$733.39 + $371.00

=$1,104.39

1000

ANSWER-1,104.77

(c)We see that value is inversely related to the investor's required rate of return.

7-10B.

Value Bond A

Par Value $1,000.00

Coupon $ 90.00

Required Rate of Return 0.07

Years to Maturity 5

Market Value $ 1,082.00

Value Bond B

Par Value $1,000.00

Coupon $ 60.00

Required Rate of Return 0.07

Years to Maturity 5

Market Value $ 959.00

Value Bond C

Par Value $1,000.00

Coupon $ 120.00

Required Rate of Return 0.07

Years to Maturity 10

Market Value $ 1,351.18

Value Bond D

Par Value $1,000.00

Coupon $ 90.00

Required Rate of Return 0.07

Years to Maturity 15

Market Value $ 1,182.16

Value Bond E

Par Value $1,000.00

Coupon $ 75.00

Required Rate of Return 0.07

Years to Maturity 15

Market Value $ 1,045.54

BondABCDE

$1,082 $959 $1,351$1,182 $1,046

Years Ct t*PV(Ct) Ct t*PV(Ct) Ct t*PV(Ct) Ct t*PV(Ct) Ct t*PV(Ct)

1 $90 $84 $60 $56 $120 $112 $90 $84 $75 $70

2 90 157 60 105 120 210 90 157 75 131

3 90 220 60 147 120 294 90 220 75 184

4 90 275 60 183 120 366 90 275 75 229

5 1,090 3,886 1,060 3,779 120 428 90 321 75 267

6 120 480 90 360 75 300

7 120 523 90 392 75 327

8 120 559 90 419 75 349

9 120 587 90 441 75 367

10 1,120 5,694 90 458 75 381

11 90 470 75 392

12 90 480 75 400

13 90 486 75 405

14 90 489 75 407

15 1,090 5,926 1,075 5,844

4,622 4,270 9,252 10,977 10,053

Duration 4.27 4.45 6.85 9.29 9.6