Chapter 5 Bonds, Bond Prces and the Determnaton of Interest Rates Problems and Solutons 1. Consder a U.S. Treasury Bll wth 270 days to maturty. If the annual yeld s 3.8 percent, what s the prce? $100 P = = $97. 24 9 /12 (1 + 0.038) 2. Whch of these $100 face value bonds wll have a hgher yeld to maturty? Why? a. 6 percent coupon bond sellng for $85 b. 7 percent coupon bond sellng for $100 c. 8 percent coupon bond sellng for $115 a. $ 85 $6 $100 + 1+ 1+ = 24.71% $7 $100 b. $ 100 = + = 7% 1+ 1+ $8 $100 c. $ 115 = + = - 6.1% 1+ 1+ Opton (a.) has the hghest yeld to maturty. 3. You are consderng purchasng a consol that promses annual payments of $4. a. If the current nterest rate s 5 percent, what s the prce of the consol? b. You are concerned that the nterest rate may rse to 6 percent. Compute the percentage change n the prce of the consol and the percentage change n the nterest rate. Compare them. c. Your nvestment horzon s one year. You purchase the consol when the nterest rate s 5 percent and sell t a year later, followng a rse n the nterest rate to 6 percent. What s your holdng perod return?
$4 a. P = = $ 80 0.05 $4 b. newp = = $66. 67 0.06 P falls by 16.7%; rses by 20% $4 $66.67 - $80 c. + = - 11.7% $80 $80 4. In a recent ssue of The Wall Street Journal, locate the prces and yelds on U.S. Treasury ssues. For one bond sellng above par and one sellng below par (assumng they both exst), compute the current yeld and compare t to the coupon rate and the ask yeld prnted n the paper. (From the Aprl 5, 2004 Wall Street Journal) a. $100, 4% note maturng Feb 2014: ask prce=$98.26, ask yeld=4.15% Current yeld = $4/$98.26 = 4.1% Prce<Par value: coupon rate<current yeld<ask yeld b. $100, 7.5% bond maturng Nov 2016: ask prce=$128.13, ask yeld=4.52% Current yeld = $7.50/$128.13 = 5.9% Prce>Par value: coupon rate>current yeld>ask yeld 5. There s consderable concern that the government of Transaxa, an emergng economy n Eastern Europe, wll default on ts debt. Some years ago t ssued 8 percent coupon bonds that now have one year to maturty. Lookng at the market, you fnd that the yeld on these bonds s 24 percent. If the current rsk-free nterest rate s 5 percent, and assumng that nvestors do not demand a rsk premum for holdng Transaxan bonds, what s the probablty that the Transaxans wll default? Yeld = Pr ce = $108 0.24 = - 1 Pr ce = $87.10 Pr ce ExpectedValue ExpectedPV = $87.10 = 1.05 ExpectedValue = $91.46 Expected Value = $91.46 = Probablty of Payoff * $108 Prob of Payoff=84.7%
Probablty of Default = 100% - 84.7% = 15.3% 6. The Transxans do, n fact, default. As a result, nvestors become more attuned to the chances that other small emergng-market economes may default, and mmedately move to reduce ther exposure to them. Descrbe the consequences of ths ncrease n rsk averson for the prces of both low-rsk bonds and rsker emergng-market bonds. Investors wll want to shft ther money nto relatvely low-rsk bonds. The demand for emergng-market bonds shfts left, whch causes prces to fall and yelds to rse. The demand for low-rsk bonds shfts rght, ncreasng prces and reducng yelds. 7. A 10-year zero-coupon bond has a yeld of 6 percent. Through a seres of unfortunate crcumstances, expected nflaton rses from 2 percent to 3 percent. a. Compute the change n the prce of the bond. b. Suppose that expected nflaton s stll 2 percent, but the probablty that t wll move to 3 percent has rsen. Descrbe the consequences for the prce of the bond. a. Prce (wth 2% expected nflaton) = 100/(1.06) 10 = $55.84 Prce (wth 3% expected nflaton) = 100/(1.07) 10 = $50.83 The prce has fallen by $5.01 b. There s ncreased nflaton rsk. Investors wll requre compensaton for takng on addtonal rsk, so the prce wll fall and the yeld wll rse. 8. Assume that forecasts for the U.S. economy have taken a sudden turn for the worse. Everyone had expected healthy growth of 3-4 percent n the comng year, but now a recesson s predcted, wth output contractng by as much as 2 percent. Offcals expect unemployment to rse and corporate profts to plummet. Descrbe the consequences for prces and nterest rates for both government and prvate-sector bonds. Consder both the supply and demand effects. Use graphs to support your answer. Corporatons wll be more lkely to default durng the mpendng recesson, so the perceved rskness of corporate bonds relatve to government bonds has ncreased and demand for corporate bonds wll shft left. It wll also become more dffcult for corporatons to borrow, shftng the supply of corporate bonds to the left. Ths combnaton could cause prces of corporate bonds to ncrease or decrease. (See fgure 6.6)
Because the perceved rskness of corporate bonds has ncreased, demand for low-rsk government bonds wll ncrease, whch wll cause prces of government bonds to rse and prces to fall. (See fgure 6.4) 9. What mpact would a stock-market collapse have on bonds? Why? The perceved rskness of bonds relatve to stocks would fall and nvestors would shft ther money nto bonds. Demand for bonds would shft rght, causng prces to rse and yelds to fall. 10. The government proposes cuttng taxes on nvestment by mplementng a credt for nvestment n nformaton technology equpment. The proposal would reduce government tax revenues. Descrbe the lkely mpact on the bond market. Frms would have an ncentve to ncrease ther nvestments; they would borrow more, shftng the supply of corporate bonds to the rght, whch would decrease prces and rase yelds. Because government tax revenues have fallen, the government wll need to borrow more. Ths wll shft the supply of government bonds to the rght, decreasng prces and rasng yelds. 11. Other muscans, among them Rod Stewart, James Brown, and Dusty Sprngfeld, have ssued bonds smlar to those created by Davd Bowe (as descrbed n the artcle n the chapter). What s lkely to make such bond ssues successful? Because the success of these artsts s uncorrelated wth any fnancal markets, nvestors see these bonds as an opportunty to reduce ther overall rsk through dversfcaton. 12. Use supply and demand n the bond market to explan each of the followng news headlnes. Show n a dagram what shfts and explan why. a. Treasury prces fell for the sxth tme n seven sessons as nvestors made room for new debt ssues. b. Government bonds contnued ther month-long rse as data ndcated that the U.S. economy s stll progressng, but at a moderate pace. c. U.S. Treasurys rse sharply as stocks take dve. d. Treasury prces are rsng on expected future rate cut. e. Illqudty s crpplng bond world. Answer. a. The supply of bonds has ncreased, causng prces to fall.
Prce of Bonds S 0 S 1 D Quantty of Bonds b. People s wealth has ncreased, shftng demand to the rght. Frms have more nvestment opportuntes, whch shfts supply to the rght. If prces are rsng, t must mean that the shft of demand s stronger than the shft of supply. Prce of Bonds S 0 S 1 D 1 D 0 Quantty of Bonds c. The perceved rskness of stocks has rsen and nvestors have shfted ther resources nto low-rsk Treasurys. Demand has shfted rght, ncreasng prces and decreasng yelds. Prce of Bonds S Increased Demand for Bonds Quantty of Bonds D 0 D 1 d. Investors expect nterest rates to fall (and prces to rse). Ths shfts demand to the rght (see dagram for part c). e. If bonds are vewed as llqud, demand shfts left, causng prces to fall and nterest rates to rse.
Prce of Bonds S Decreased Demand for Bonds Quantty of Bonds D 0 D 1 13. As you read the mornng paper, you come across an ad for a bond mutual fund, a fund that pools the nvestments from a large number of people and then purchases bonds, gvng the ndvduals shares n the fund. The company clams the fund has had a return of 13½ percent over the last year. But you remember that nterest rates have been pretty low, 5 percent at most. A quck check of the numbers n the busness secton you re holdng tells you that your recollecton s correct. Explan the logc behnd the mutual fund s clam n the ad. There are two possble explanatons for the hgh return. The frst s that the mutual fund s so much hgher than the rsk-free rate t must mean that they are nvestng n relatvely rsky bonds (and are beng compensated for takng on ths rsk wth hgher returns). The second s that the fund was holdng bonds durng a perod when nterest rates were fallng, so the holdng perod return far exceeded the nterest rate. Remember that when nterest rates fall, the prces of bonds rse gvng the owner a captal gan. If nterest rates are now low, then the lkelhood s that they wll rse, causng a captal loss to the owners. Chances are that the hgh return s a consequence of the nterest rate declne, so not only wll t not be repeated, t s lkely to be followed by a low or even negatve return when nterest rates rse. 14. Your Fnancal World: Interest-Only Mortgages descrbes the monthly payments a person would have to make under varous crcumstances. Show how to derve the results n parts a-c f you borrow $400,000 at an annual nterest rate of 6 percent: a. An nterest-only mortgage requres a payment of $1,947 a month. b. A 30-year, fxed-rate mortgage requres 360 equal payments of $2,357. c. A 25-year fxed-rate mortgage requres 300 equal payments $2,538. Note: To answer these questons you wll need to compute the monthly compound nterest rate and use the formulas n the Appendx 4. 1 m 12 a. = ( 1.06-1) = 0. 0048676
$400,000*0.0048676=$1947 $400,000* 0.0048676 b. C = = $ 2357 1 (1 - ) 360 1.0048676 $400,000* 0.0048676 c. C = = $ 2538 1 (1 - ) 300 1.0048676 15. At the dnner table, your father s extollng the benefts of nvestng n bonds. He nssts that as a conservatve nvestor he wll make only nvestments that are safe, and what could be safer than a bond, especally a U.S. Treasury bond? What accounts for hs vew of bonds? Explan why you thnk t s rght or wrong. Lke most people, your father beleves that the government guarantee means that he wll get hs nvestment back. He s rght that the U.S. Treasury s extremely unlkely to default, so there s vrtually no default rsk. But he s wrong about nterest-rate and nflaton rsk. The value of the bond wll fluctuate when the nterest rate changes (movng nversely) and the purchasng power of the coupon and prncpal repayment wll fluctuate wth nflaton. So, the bond s not rsk free.