Solutions for Chapters 12: Questions and Problems
Answers to Questions
1. The present value equation is more useful for the bond investor largely because the bond investor has fewer uncertainties regarding future cash flows than does the common stock investor. By investing in bonds with relatively no default risk (i.e., government securities) the investor can value a bond based primarily on expected cash flows (coupon rate and par value), required return (market yield), and the number of periods in the investment horizon (maturity date). Even with corporate bonds, default (credit) risk is incorporated into bond ratings. Examining market data based on time to maturity and credit rating allows determination of the market interest rate on similar bonds. Each of these factors can be incorporated into the present value equation.
By contrast, common stocks have no stated maturity date and the valuation process is predominantly an estimate of future earnings. Although the present value method can be used for common stock analysis by estimating dividend payments and change in price over a given time frame, the uncertainties involved are much greater.
2. The most crucial assumption in both cases that the investor makes is that cash flows will be received in full and reinvested at the promised yield. This assumption is crucial because it is implicit in the mathematical equation that solves for promised yield. If the assumption is not valid, an alternative method must be used, or the calculations will yield invalid solutions.
3(a). RFR is the riskless rate of interest, I is the factor for expected inflation, and RP is the risk premium for the individual firm.
3(b). The model considers the firm’s business conditions. The risk of not breaking even would be reflected in the model through changes in the variable RP. This uncertainty would change the nature of the frequency distribution for earnings.
4 . The expectations hypothesis imagines a yield curve that reflects what bond investors expect to earn on successive investments in short-term bonds during the term to maturity of the long-term bond. The liquidity preference hypothesis envisions the generally upward-sloping yield curve owing to the fact that investors prefer the liquidity of short-term loans but will lend long if the yields are higher. The segmented market hypothesis contends that the yield curve mirrors the investment policies of institutional investors who have different maturity preferences. Student exercise as to which one best explains the alternative shapes of the yield curve; a viable response is to consider combinations of these three as helpful explanations of actual yield curve behavior.
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Copyright © 2010 by Nelson Education Ltd.

Solutions for Chapters 12: Questions and Problems
5(a). Given that you expect interest rates to decline during the next six months, you should choose bonds that will have the largest price increase, that is, bonds with long durations.
5(b). Case 1: Given a choice between bonds A and B, you should select bond B, since duration is inversely related to both coupon and yield to maturity.
Case 2: Given a choice between bonds C and D, you should select bond C, since duration is positively related to maturity and inversely related to coupon.
Case 3: Given a choice between bonds E and F, you should select bond F, since duration is positively related to maturity and inversely related to yield to maturity.
6. You should select portfolio A because it has a longer duration (5.7 versus 4.9 years) and greater convexity (125.18 versus 40.30), thereby offering greater price appreciation.
Portfolio A is also noncallable, therefore there is no danger of the bonds being called in by the issuer when interest rates decline (as you expect they will).
7(a). Call-adjusted duration takes into account the probability of a call and its impact on the actual duration of a bond. If a bond is noncallable, duration is based on all cash flows up to and including maturity. If interest rates drop and a call becomes likely, then duration should be calculated based on the time to call, which can be substantially sooner than maturity.
The range for duration now is 8.2 to 2.1 years. Because the bond is trading at par, its duration should be near 8.2 years.
7(b). If rates increase, then duration will drop (recall that duration is inversely related to yield to maturity.) However, call is now very unlikely, so call-adjusted duration will stay near the high end of the range.
7(c). If rates fall to 4%, call becomes highly probable, so call-adjusted duration will drop to the low end. (Remember here that duration to call is now greater than 2.1 years because of the inverse relationship between duration and yield to maturity).
7(d). Negative convexity refers to slow price increases of callable bonds as interest rates fall.
Indeed, at some point the price change will be zero. This is due to the call price placing a ceiling on the price of a callable bond.
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Copyright © 2010 by Nelson Education Ltd.

Solutions for Chapters 12: Questions and Problems
Answers to Problems
1(a). Assuming annual compounding
AYC

70

1,100 1,000
1,100
4

1,000

70

25
1050

95
1050

9 .
05 %
2
Of course, with financial calculators and spreadsheets such an approximation is not appropriate. It assume, for example, the average investment is $1,050 when the investor purchasing the bond four years ago invested $1,000
1(b). Using a financial calculator: PV= -1000, FV = 1100, PMT = 35, n= 8 as the bonds were purchased 4 years ago. The resulting periodic return is 4.564% or 9.13% annually.
1(c). Using a financial calculator: FV = 1000, PMT = 35, n= 42 as the bonds mature in 21 years. If the periodic interest rate is 5%/2 or 2.5%, the price of the bonds is $1,258.21.
2(a). Using a financial calculator: PV= -1012.50, FV = 1000, PMT = 40, n= 24 as the bonds have 12 years until maturity. The resulting periodic return is 3.919% or 7.84% annually.
2(b). Using a financial calculator: PV= -1012.50, FV = 1080, PMT = 40, n= 6 as the bond is callable in 3 years. The resulting periodic return is 4.932% or 9.86% annually.
3. (1) (2) (3)
Cash PV
(4) (5) (6)
PV of Flow
PV as % of Price (1) × (5) Period Flow
1
2
$ 40
40 at 5%
.9524
.9070
$ 38.10 .04014
36.28 .03822
.04014
.07644
3
4
5
6
40
40
40
1,040
.8638
.8227
.7835
.7462
34.55 .03640
32.91 .03467
.10920
.13868
31.34 .03302
776.05 .81756
$949.23 1.00000
.16510
4.90536
5.43492
The duration equals 5.43492 semiannual periods or 2.71746 years.
3(a).
Modified duration

1
2.71746

(.10/2)

2 .
71746
1 .
05

2.588
years
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Copyright © 2010 by Nelson Education Ltd.

Solutions for Chapters 12: Questions and Problems
3(b). Percentage change in price = -D mod
× Δi
= - (2.588) × (-50/100)
= 1.294%
The bond price should increase by 1.294% in response to a drop in the bonds YTM from
10% to 9.5%. If the price of the bond before the decline was $949.23, the price after the decline in the YTM should be approximately $949.23 × 1.01294 = $961.51.
4. Assuming semiannual compounding, 10 years, zero coupon, $1,000 par value, 12% YTM
Purchase Price = $1,000 (.31180) = $311.80 where .31180 is (1 / 1.06)
20
, namely the present value factor for 6% interest
(12% annually/2 interest payments per year) for 20 semi-annual periods (10 years × 2 interest payments per year)
After two years, assuming semiannual compounding, 8 years remaining, zero coupon,
$1,000 par value, 8% YTM
Selling Price = $1,000 (.53391) = $533.91 where .53391 is the present value factor for 4% interest
(8% annually/2 interest payments per year) for 16 semi-annual periods (8 years remaining
× 2 interest payments per year)
Assuming you bought the bond for $311.80 and sold it after two years for $533.91, the semi-annual return is (533.91 / 311.80) 1/4 minus 1 or 0.1439. The annualized return is
0.1439 x2 = 0.2878 namely 28.78%.
5(a). Assuming semiannual interest payments
Modified duration

1
5.7

(.095/2)

5 .
7
1 .
0475

5 .
442 years
Percentage change in price = -D mod
×

= -8.163% i
= - (5.442) × (+150/100)
This approximates the price change because the modified-duration line is a linear estimate of a curvilinear function. That is, convexity measures the rate of change in modified duration as yields change. The effect of convexity on price should be added to the effect of duration on price in order to obtain an improved approximation of the change in price given a change in yield.
5(b). Percentage change in price = -5.442 × -3% = + 16.33%
The 3% decline in rates may not elevate the bond price by 16.33% if the bond’s call price is violated and protection against a call has elapsed.
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Copyright © 2010 by Nelson Education Ltd.