CHAPTER 2 PRICING OF BONDS

CHAPTER 2 PRICING OF BONDS CHAPTER SUARY This chapter will focus o the time value of moey ad how to calculate the price of a bod. Whe pricig a bod it is ecessary to estimate the expected cash flows ad determie the appropriate yield at which to discout the expected cash flows. Amog other aspects of a bod, we will look at the reasos why the price of a bod chages REVIEW OF TIE VALUE OF ONEY oey has time value because of the opportuity to ivest it at some iterest rate. Future Value The future value of ay sum of moey ivested today is: P P0(1 + r) where umber of periods, P future value periods from ow, P0 origial pricipal, r iterest rate per period, ad the expressio (1 + r) represets the future value of $1 ivested today for periods at a compoudig rate of r. Whe iterest is paid more tha oe time per year, both the iterest rate ad the umber of periods used to compute the future value must be adjusted as follows: r aual iterest rate / umber of times iterest paid per year, ad umber of times iterest paid per year times umber of years. The higher future value whe iterest is paid semiaually, as opposed to aually, reflects the greater opportuity for reivestig the iterest paid. Future Value of a Ordiary Auity A auity ivolves equal cash flows beig ivested (or paid) over a fiite period of time with equal legths of time betwee each equal cash flow. Whe the first auity paymet occurs oe period from ow, it is referred to as a ordiary auity. The equatio for the future value of a ordiary auity is: where A is the amout of the auity. P (1 r) 1 A r Copyright 2016 Pearso Educatio, Ic. 14

Example of Future Value of a Ordiary Auity Usig Aual Iterest: (1 r) 1 If A $2,000,000, r 0.08, ad 15, the P A r P15 15 (1 0.08) 1 3.17217 1 $2,000,000 0.08 $2,000,000 0.08 $2,000,000[27.152125] $54,304.250. Because 15($2,000,000) $30,000,000 of this future value represets the total dollar amout of aual iterest paymets made by the issuer ad ivested by the portfolio maager, the balace of $54,304,250 $30,000,000 $24,304,250 is the iterest eared by reivestig these aual iterest paymets. Example of Future Value of a Ordiary Auity Usig Semiaual Iterest: Cosider the same example, but ow we assume semiaual iterest paymets. If A $2,000,000 / 2 $1,000,000, r 0.08 / 2 0.04, 2(15) 30, the P (1 r) 1 A r P30 30 (1 0.04) 1 $1,000,000 0.04 3.24341 $1,000,000 0.04 $1,000,000[56.085] $56,085,000. The opportuity for more frequet reivestmet of iterest paymets received makes the iterest eared of $26,085,000 from reivestig the iterest paymets greater tha the $24,304,250 iterest eared whe iterest is paid oly oe time per year. Preset Value The preset value is the future value process i reverse. We have: PV 1 1 r. For a give future value at a specified time i the future, the higher the iterest rate (or discout rate), the lower the preset value. For a give iterest rate (discout rate), the further ito the future that the future value will be received, the the lower its preset value. Preset Value of a Series of Future Values To determie the preset value of a series of future values, the preset value of each future value must first be computed. The these preset values are added together to obtai the preset value of the etire series of future values. Copyright 2016 Pearso Educatio, Ic. 15

Preset Value of a Ordiary Auity Whe the same dollar amout of moey is received each period or paid each year, the series is referred to as a auity. Whe the first paymet is received oe period from ow, the auity is called a ordiary auity. Whe the first paymet is immediate, the auity is called a auity due. The preset value of a ordiary auity is: 1 1 1 r PV A r where A is the amout of the auity. The term i brackets is the preset value of a ordiary auity of $1 for periods. Example of Preset Value of a Ordiary Auity Usig Aual Iterest: 1 1 If A $100, r 0.09, ad 8, the: PV A 1 r r 1 1 $100 1.99256 0.09 1 0.501867 $100 0.09 $100[5.534811] $553.48. 1 1 1.09 8 $100 0.09 Preset Value Whe Paymets Occur ore Tha Oce Per Year If the future value to be received occurs more tha oce per year, the the preset value formula is modified so that (i) the aual iterest rate is divided by the frequecy per year, ad (ii) the umber of periods whe the future value will be received is adjusted by multiplyig the umber of years by the frequecy per year. PRICING A BOND Determiig the price of ay fiacial istrumet requires a estimate of (i) the expected cash flows, ad (ii) the appropriate required yield. The required yield reflects the yield for fiacial istrumets with comparable risk. I geeral, the curret or market price of a bod ca be calculated by usig the followig formula: t P C + t. t 1 1 r 1+ r Copyright 2016 Pearso Educatio, Ic. 16

where P price, umber of periods (umber of years times 2), C semiaual coupo paymet, r periodic iterest rate (required aual yield divided by 2), maturity value, ad t time period whe the paymet is to be received. Computig the Value of a Bod: A Example: Cosider a 20-year 10% coupo bod with a par value of $1,000 ad a required yield of 11%. Give C 0.1($1,000) / 2 $50, 2(20) 40 ad r 0.11 / 2 0.055, the preset value of the coupo paymets is: 1 1 1 1 r P C 1 40 $50 1.055 r 0.055 $50 16.046131 $802.31. 1 1 $50 8.51332 0.055 1 0.117463 $50 0.055 The preset value of the par or maturity value of $1,000 is: 1 r $117.46. $1,000 (1.055) 40 $1,000 8.51331 The price of the bod (P) preset value coupo paymets + preset value maturity value $802.31 + $117.46 $919.77. Pricig Zero-Coupo Bods For zero-coupo bods, the ivestor realizes iterest as the differece betwee the maturity value ad the purchase price. The equatio is: P 1 r where is the maturity value. Thus, the price of a zero-coupo bod is simply the preset value of the maturity value. Zero-Coupo Bod Example Cosider the price of a zero-coupo bod that matures 15 years from ow, if the maturity value is $1,000 ad the required yield is 9.4%. Give $1,000, r 0.094 / 2 0.047, $1,000 ad 2(15) 30, we have: P 1 r 30 (1.047) $1,000 3.99644 $252.12. Price-Yield Relatioship A fudametal property of a bod is that its price chages i the opposite directio from the Copyright 2016 Pearso Educatio, Ic. 17

chage i the required yield. The reaso is that the price of the bod is the preset value of the cash flows. Relatioship Betwee Coupo Rate, Required Yield, ad Price Whe yields i the marketplace rise above the coupo rate at a give poit i time, the price of the bod falls so that a ivestor buyig the bod ca realize capital appreciatio. The appreciatio represets a form of iterest to a ew ivestor to compesate for a coupo rate that is lower tha the required yield. Whe a bod sells below its par value, it is said to be sellig at a discout. A bod whose price is above its par value is said to be sellig at a premium. Relatioship Betwee Bod Price ad Time if Iterest Rates Are Uchaged For a bod sellig at par value, the coupo rate is equal to the required yield. As this bod moves closer to maturity, the bod will cotiue to sell at par value as log as the coupo rate remais equal to the required yield with its price thus remaiig costat as the maturity date is reached. The price of a bod will ot remai costat for a bod sellig at a premium or a discout as it reaches its maturity. For a discout bod, its price icreases as it approaches maturity assumig that the required yield does ot chage. For a premium bod, the opposite occurs. For both bods, the price will equal par value at the maturity date. Reasos for the Chage i the Price of a Bod The price of a bod ca chage for three reasos: (i) there is a chage i the required yield owig to chages i the credit quality of the issuer; (ii) there is a chage i the price of the bod sellig at a premium or a discout, without ay chage i the required yield, simply because the bod is movig toward maturity; or, (iii) there is a chage i the required yield owig to a chage i the yield o comparable bods (i.e., a chage i the yield required by the market). COPLICATIONS The framework for pricig a bod assumes the followig: (i) the ext coupo paymet is exactly six moths away; (ii) the cash flows are kow; (iii) the appropriate required yield ca be determied; ad, (iv) oe rate is used to discout all cash flows. Next Coupo Paymet Due i Less tha Six oths Whe a ivestor purchases a bod whose ext coupo paymet is due i less tha six moths, the accepted method for computig the price of the bod is as follows: C P + v t1 v 1 (1 + r ) (1 + r ) (1 + r )(1 + r ) t1 where v (days betwee settlemet ad ext coupo) / (days i six-moth period). Cash Flows ay Not Be Kow Copyright 2016 Pearso Educatio, Ic. 18

For most bods, the cash flows are ot kow with certaity. This is because a issuer may call a bod before the stated maturity date. Determiig the Appropriate Required Yield All required yields are bechmarked off yields offered by Treasury securities. From there, we must still decompose the required yield for a bod ito its compoet parts. Oe Discout Rate Applicable to All Cash Flows A bod ca be viewed as a package of zero-coupo bods, i which case a uique discout rate should be used to determie the preset value of each cash flow. PRICING FLOATING-RATE AND INVERSE-FLOATING-RATE SECURITIES The cash flow is ot kow for either a floatig-rate or a iverse-floatig-rate security; it will deped o the referece rate i the future. Price of a Floater The coupo rate of a floatig-rate security (or floater) is equal to a referece rate plus some spread or margi. The price of a floater depeds o (i) the spread over the referece rate ad (ii) ay restrictios that may be imposed o the resettig of the coupo rate. Price of a Iverse Floater I geeral, a iverse floater is created from a fixed-rate security. The security from which the iverse floater is created is called the collateral. From the collateral two bods are created: a floater ad a iverse floater. A floater may have a maximum coupo rate called a cap or a miimum coupo rate called a floor. The price of a floater will trade close to its par value as log as the spread above the referece rate that the market requires is uchaged, ad either the cap or the floor is reached. The price of a iverse floater equals the collateral s price mius the floater s price. PRICE QUOTES AND ACCRUED INTEREST Price Quotes A bod sellig at par is quoted as 100, meaig 100% of its par value. A bod sellig at a discout will be sellig for less tha 100; a bod sellig at a premium will be sellig for more tha 100. Accrued Iterest Whe a ivestor purchases a bod betwee coupo paymets, the ivestor must compesate the Copyright 2016 Pearso Educatio, Ic. 19

seller of the bod for the coupo iterest eared from the time of the last coupo paymet to the settlemet date of the bod. This amout is called accrued iterest. For corporate ad muicipal bods, accrued iterest is based o a 360-day year, with each moth havig 30 days. The amout that the buyer pays the seller is the agreed-upo price plus accrued iterest. This is ofte referred to as the full price or dirty price. The price of a bod without accrued iterest is called the clea price. The exceptios are bods that are i default. Such bods are said to be quoted flat, that is, without accrued iterest. KEY POINTS The price of a bod is the preset value of the bod s expected cash flows, the discout rate beig equal to the yield offered o comparable bods. For a optio-free bod, the cash flows are the coupo paymets ad the par value or maturity value. The higher (lower) the required yield, the lower (higher) the price of a bod. For a zero-coupo bod, there are o coupo paymets. The price of a zero-coupo bod is equal to the preset value of the maturity value, where the umber of periods used to compute the preset value is double the umber of years ad the discout rate is a semiaual yield. A bod s price chages i the opposite directio from the chage i the required yield. The reaso is that as the required yield icreases (decreases), the preset value of the cash flow decreases (icreases). A bod will be priced below, at par, or above par depedig the bod s coupo rate ad the required yield required by ivestors. Whe the coupo rate is equal to the required yield, the bod will sell at its par value. Whe the coupo rate is less (greater) tha the required yield, the bod will sell for less (more) tha its par value. Over time, the price of a premium or discout bod will chage eve if the required yield does ot chage. Assumig that the credit quality of the issuer is uchaged, the price chage o ay bod ca be decomposed ito a portio attributable to a chage i the required yield ad a portio attributable to the time path of the bod. The price of a floatig-rate bod will trade close to par value if the spread required by the market does ot chage ad there are o restrictios o the coupo rate. The price of a iverse floater depeds o the price of the collateral from which it is created ad the price of the floater. Accrued iterest is the amout that a bod buyer who purchases a bod betwee coupo paymets must pay the bod seller. The amout represets the coupo iterest eared from the time of the last coupo paymet to the settlemet date of the bod. Copyright 2016 Pearso Educatio, Ic. 20

ANSWERS TO QUESTIONS FOR CHAPTER 2 (Questios are i bold prit followed by aswers.) 1. A pesio fud maager ivests $10 millio i a debt obligatio that promises to pay 7.3% per year for four years. What is the future value of the $10 millio? To determie the future value of ay sum of moey ivested today, we ca use the future value equatio, which is: P P0 (1 + r) where umber of periods, P future value periods from ow, P0 origial pricipal ad r iterest rate per period. Isertig i our values, we have: P4 $10,000,000(1.073) 4 $10,000,000(1.325558466) $13,255,584.66. 2. Suppose that a life isurace compay has guarateed a paymet of $14 millio to a pesio fud 4.5 years from ow. If the life isurace compay receives a premium of $10.4 millio from the pesio fud ad ca ivest the etire premium for 4.5 years at a aual iterest rate of 6.25%, will it have sufficiet fuds from this ivestmet to meet the $14 millio obligatio? To determie the future value of ay sum of moey ivested today, we ca use the future value equatio, which is: P P0 (1 + r) where umber of periods, P future value periods from ow, P0 origial pricipal ad r iterest rate per period. Isertig i our values, wehave:p4.5 $10,400,000(1.0625) 4.5 $10,400,000(1.313651676) $13,661,977.43. Thus, it will be short by: $13,661,977.43 $14,000,000 $338,022.57. 3. Aswer the below questios. (a) The portfolio maager of a tax-exempt fud is cosiderig ivestig $500,000 i a debt istrumet that pays a aual iterest rate of 5.7% for four years. At the ed of four years, the portfolio maager plas to reivest the proceeds for three more years ad expects that for the three-year period, a aual iterest rate of 7.2% ca be eared. What is the future value of this ivestmet? At the ed of year four, the portfolio maager s amout is give by: P P0 (1 + r). Isertig i our values, we have P4 $500,000(1.057) 4 $500,000(1.248245382) $624,122.66. I three more years at the ed of year seve, the maager amout is give by: P7 P4(1 + r) 3. Isertig i our values, we have: P7 $624,122.66(1.072) 3 $624,122.66(1.231925248) $768,872.47. (b) Suppose that the portfolio maager i Questio 3, part a, has the opportuity to ivest the $500,000 for seve years i a debt obligatio that promises to pay a aual iterest rate of 6.1% compouded semiaually. Is this ivestmet alterative more attractive tha the oe i Questio 3, part a? At the ed of year seve, the portfolio maager s amout is give by the followig equatio, which adjusts for semiaual compoudig. We have: P P0(1 + r/2) 2(). Isertig i our values, we have P7 $500,000(1 + 0.061/2) 2(7) $500,000(1.0305) 14 $500,000(1.522901960) $761,450.98. Thus, this ivestmet alterative is ot more attractive. It is less by the amout of Copyright 2016 Pearso Educatio, Ic. 21

$761,450.98 $768,872.47 $7,421.49. 4. Suppose that a portfolio maager purchases $10 millio of par value of a eight-year bod that has a coupo rate of 7% ad pays iterest oce per year. The first aual coupo paymet will be made oe year from ow. How much will the portfolio maager have if she (1) holds the bod util it matures eight years from ow, ad (2) ca reivest all the aual iterest paymets at a aual iterest rate of 6.2%? At the ed of year eight, the portfolio maager s amout is give by the followig equatio, which adjusts for aual compoudig. We have: P (1 + r) - 1 A Par Value r where A coupo rate times par value. Isertig i our values, we have: 0.062 $6,978,160.38 + $10,000,000 $16,978,160.38. 8 (1 + 0.062 ) - 1 P 8 0.07($10,000,000) $1,000 $700,000[9.9688005] + $10,000,000 5. Aswer the below questios. (a) If the discout rate that is used to calculate the preset value of a debt obligatio s cash flow is icreased, what happes to the price of that debt obligatio? The price will fall. A fudametal property of a bod is that its price chages i the opposite directio from the chage i the required yield. The reaso is that the price of the bod is the preset value of the cash flows. As the required yield icreases, the preset value of the cash flow decreases; thus the price decreases. The opposite is true whe the required yield decreases: The preset value of the cash flows icreases, ad therefore the price of the bod icreases. (b) Suppose that the discout rate used to calculate the preset value of a debt obligatio s cash flow is x%. Suppose also that the oly cash flows for this debt obligatio are $200,000 four years from ow ad $200,000 five years from ow. For which of these cash flows will the preset value be greater? Cash flows that come earlier will have a greater value. As log as x% is positive ad the amout is the same, the preset value will be greater for the $200,000 four years from ow compared to five years from ow. 1 1 This ca also be see by otig that if x > 0 the. The latter iequality 1x 4 1x 5 1 1 implies $2,000 $2,000 4 5 will hold. 1x 1x Copyright 2016 Pearso Educatio, Ic. 22

6. The pesio fud obligatio of a corporatio is calculated as the preset value of the actuarially projected beefits that will have to be paid to beeficiaries. Why is the iterest rate used to discout the projected beefits importat? It is importat because the preset value icreases as the discout rate (or iterest rate) decreases ad it decreases as the discout rate icreases. Thus, i order to project the beefits accurately, we eed a accurate estimate of the discout rate. If we uderestimate the discout rate the we will be projectig more available pesio fuds tha we will actually have. 7. A pesio fud maager kows that the followig liabilities must be satisfied: Years from Now Liability (i millios) 1 $2.0 2 $3.0 3 $5.4 4 $5.8 Suppose that the pesio fud maager wats to ivest a sum of moey that will satisfy this liability stream. Assumig that ay amout that ca be ivested today ca ear a aual iterest rate of 7.6%, how much must be ivested today to satisfy this liability stream? To satisfy year oe s liability ( 1), the pesio fud maager must ivest a amout today that is equal to the future value of $2.0 millio at 7.6%. We have: 1 PV P 1 + r 1 $ 2,000,000 (1.076 1 ) $ 2,000,000 0.929368030 $1,858,736.06. To satisfy year two s liability ( 2), the pesio fud maager must ivest a amout today that is equal to the future value of $3.0 millio at 7.6%. We have: 1 PV P 1 + r 1 (1.076) $3,000,000 2 $3,000,000 0.863724935 $2,591,174.80. To satisfy year three s liability ( 3), the pesio fud maager must ivest a amout today that is equal to the future value of $5.4 millio at 7.6%. We have: 1 PV P 1 + r 1 $5,400,000 3 (1.076) $5,400,000 0.802718341 $4,334,679.04. To satisfy year four s liability ( 4), the pesio fud maager must ivest a amout today that is equal to the future value of $5.8 millio at 7.6%. We have: Copyright 2016 Pearso Educatio, Ic. 23

1 PV P 1 + r 1 $ 5,800,000 4 (1.076) $ 5,800,000 0.746020763 $4,326,920.42. If we add the four preset values, we get $1,858,736.06 + $2,591,174.80 + $4,334,679.04 + $4,326,920.42 $13,111,510.32, which is the amout the pesio fud maager eeds to ivest today to cover the liability stream for the ext four years. 8. Calculate for each of the followig bods the price per $1,000 of par value assumig semiaual coupo paymets. Bod Coupo Rate (%) Years to aturity Required Yield (%) A 8 9 7 B 9 20 9 C 6 15 10 D 0 14 8 Cosider a 9-year 8% coupo bod with a par value of $1,000 ad a required yield of 7%. Give C 0.08($1,000) / 2 $40, 2(9) 18 ad r 0.07 / 2 0.035, the preset value of the coupo paymets is: 1 1 1 1 r 1 1 P C r 1.035 18 1 $40 0.035 $40 1.857489196 0.035 1 0.538361140 $40 $40 13.189681727 $527.587. 0.035 The preset value of the par or maturity value of $1,000 is: 1+ r $1,000 18 (1.035) 1.8574892 $538.361. Thus, the price of the bod (P) preset value of coupo paymets + preset value of par value $527.587 + $538.361 $1,065.95. Cosider a 20-year 9% coupo bod with a par value of $1,000 ad a required yield of 9%. Give C 0.09($1,000) / 2 $45, 2(20) 40 ad r 0.09 / 2 0.045, the preset value of the coupo paymets is: 1 1 1 1 r P C r 1 1.045 40 $45 0.045 1 1 $45 5.81863645 0.045 1 0.1719287 $45 0.045 $45[18.401584] $828.071. Copyright 2016 Pearso Educatio, Ic. 24

The preset value of the par or maturity value of $1,000 is: 1+ r 40 1.045 5.81863645 $171.929. Thus, the price of the bod (P) $828.071 + $171.929 $1,000.00. [NOTE. We already kew the aswer would be $1,000 because the coupo rate equals the yield to maturity.] Cosider a 15-year 6% coupo bod with a par value of $1,000 ad a required yield of 10%. Give C 0.06($1,000) / 2 $30, 2(15) 30 ad r 0.10 / 2 0.05, the preset value of the coupo paymets is: P C 1 1 1 1 r r 1 1.05 30 $30 0.05 1 1 $30 4.3219424 0.05 1 0.2313774 $30 0.05 $30[15.372451] $461.174. The preset value of the par or maturity value of $1,000 is: 1+ r 30 1.05 $1,000 4.3219424 $231.377. Thus, the price of the bod (P) $461.174 + $231.377 $692.55. Cosider a 14-year 0% coupo bod with a par value of $1,000 ad a required yield of 8%. Give C 0($1,000) / 2 $0, 2(14) 28 ad r 0.08 / 2 0.04, the preset value of the coupo paymets is: P C 1 1 1 r r $0 1 1 (1.04) 0.04 28 $0 1 1 2.998703319 1 0.33477471 $0 0.055 0. 055 $0[16.66306322] $0. [NOTE. We already kew the aswer because the coupo rate is zero.] The preset value of the par or maturity value of $1,000 is: 1 + r $333.48. Thus, the price of the bod (P) $0 + $333.48 $333.48. 28 1.04 2.99870332 9. Cosider a bod sellig at par ($100) with a coupo rate of 6% ad 10 years to maturity. (a) What is the price of this bod if the required yield is 15%? We have a 10-year 6% coupo bod with a par value of $1,000 ad a required yield of 15%. Give C 0.06($1,000) / 2 $30, 2(10) 20 ad r 0.15 / 2 0.075, the preset value of the coupo paymets is: Copyright 2016 Pearso Educatio, Ic. 25

1 1 1 1 r 1 20 P C r (1.075) $30 0.075 $30[10.1944913] $305.835. $30 1 1 4.2478511 0.075 1 0.2354131 $30 0.075 The preset value of the par or maturity value of $1,000 is: 1 + r 20 1.075 $235.413. Thus, the price of the bod (P) $305.835 + $235.413 $541.25. 4.2478511 (b) What is the price of this bod if the required yield icreases from 15% to 16%, ad by what percetage did the price of this bod chage? If the required yield icreases from 15% to 16%, the we have: P C 1 1 1 r r 1 1 (1.08) $30 0.08 20 $ 309.8181474 $294.544. The preset value of the par or maturity value of $1,000 is: 1+ r Thus, the price of the bod (P) $294.544 + $214.548 $509.09. 20 1.08 $214.548. The bod price falls with percetage fall is equivalet to about 5.94%. $509.09 $541.25 $541.25 0.059409 or (c) What is the price of this bod if the required yield is 5%? If the required yield is 5%, the we have: 1 1 1 r P C r 1 1 (1.025) $30 0.025 The preset value of the par or maturity value of $1,000 is: 1+ r Thus, the price of the bod (P) $467.675 + $610.271 $1,077.95. 20 $30 15.5891623 $467.675. 20 1.025 $610.271. (d) What is the price of this bod if the required yield icreases from 5% to 6%, ad by what percetage did the price of this bod chage? If the required yield icreases from 5% to 6%, the we have: Copyright 2016 Pearso Educatio, Ic. 26

1 1 1 r P C r 1 1 (1.03) $30 0.03 The preset value of the par or maturity value of $1,000 is: 1+ r 20 $30 14.87747486 $446.324. 20 (1.03) $553.676. The price of the bod (P) $446.324 + $553.676 $1,000.00. [NOTE. We already kew the aswer would be $1,000 because the coupo rate equals the yield to maturity.] The bod price falls with the percetage fall equal to ($1,000.00 $1,077.95) / $1,077.95 0.072310 or about 7.23%. (e) From your aswers to Questio 9, parts b ad d, what ca you say about the relative price volatility of a bod i a high-iterest-rate eviromet compared to a low-iterestrate eviromet? We ca say that there is more volatility i a low-iterest-rate eviromet because there was a greater fall of 7.23% for the 5% to 6% rise compared to the fall of oly 5.94% for the 15% to 16% rise. 10. Suppose that you purchased a debt obligatio three years ago at its par value of $100,000 ad ie years remaiig to maturity. The market price of this debt obligatio today is $90,000. What are some reasos why the price of this debt obligatio could have declied from time you purchased it three years ago? The price of a bod will chage for oe or more of the followig three reasos: (i) There is a chage i the required yield owig to chages i the credit quality of the issuer. (ii) There is a chage i the price of the bod sellig at a premium or a discout, without ay chage i the required yield, simply because the bod is movig toward maturity. I case, the bod is sellig at a discout. (iii) There is a chage i the required yield owig to a chage i the yield o comparable bods (i.e., a chage i the yield required by the market). The first ad third reasos are the most likely reasos for the drop i value sice the bod is still ie years away from maturity Thus, the bod has plummeted from $100,000 to $90,000 maily because the credit quality of the issuer has falle ad/or the bod has plummeted because the yield o comparable bods has icreased. 11. Suppose that you are reviewig a price sheet for bods ad see the followig prices (per $100 par value) reported. You observe what seem to be several errors. Without calculatig the price of each bod, idicate which bods seem to be reported icorrectly, ad explai why. Copyright 2016 Pearso Educatio, Ic. 27

Bod Price Coupo Rate (%) Required Yield (%) U 90 6 9 V 96 9 8 W 110 8 6 X 105 0 5 Y 107 7 9 Z 100 6 6 If the required yield is the same as the coupo rate the the price of the bod should sell at its par value. This is the case of bod Z sice par values are typical at or ear a $100 quote. If the required yield decreases below the coupo rate the the price of a bod should icrease. This is the case for bod W. This is ot the case for bod V so this bod is ot reported correctly. If the required yield icreases above the coupo rate the the price of a bod should decrease. This is the case for bod U. This is ot the case for bods X ad Y so these bods are ot reported correctly. Thus, bods V, X, ad Y are icorrectly reported because the chage i the bod price is ot cosistet with the differece betwee the coupo rate ad the required yield. 12. What is the maximum price of a bod? Cosider a extreme case of a 100-year 20% coupo bod with a par value of $1,000 that after oe year falls so that the required yield is 1%. Give C 0.2($1,000) / 2 $100, 2(99) 198 ad r 0.01 / 2 0.005, the preset value of the coupo paymets is: P C 1 1 1 r r 1 1 (1.005) $100 0.005 198 1 1 $100 2.684604 0.005 1 0.3724944 $100 0.005 $1,000[1,125.51012] $12,550.112. The preset value of the par value of $1,000 is: 1+ r (1.005) 198 $372.494. 2.684604 Thus, the price of the bod (P) $12,550.112 + $372.494 $12,922.61. This is a percet icrease of ($12,922.6 $1,000) / $1,000 11.92606 or about 1,192.61%. If the required yield falls to 0.001%, the the bod price would icrease to $20,778.33, which would be a percet icrease of about 1,977.83%. If the required yield falls to 0.00001%, the the bod price would icrease to $20,778.33, which would be a percet icrease of about 1,977.83%. If the required yield falls to 0.0000000001%, the the bod price would icrease to $20,801.76, which would be a percet icrease of about 1,980.18%. Copyright 2016 Pearso Educatio, Ic. 28

Thus, we see that eve for these extreme umbers (that are highly ulikely ad probably ot eve possible), we fid there appears to be a limit o how high a bod price might rise assumig that rates do ot reach egative umbers. If the required yield is a egative umber the there would be o limit to how high a bod price might rise. For example, if the required yield becomes a egative 1%, the the bod price would icrease to $70,468.18. If it becomes a egative 10%, the the bod price becomes $2,296,218,049,925.23 or about $2.3 trillio. 13. What is the dirty price of a bod? The dirty (or full ) price is the amout that the buyer agrees to pay the seller, which is the agreed-upo price plus accrued iterest. The price of a bod without accrued iterest is called the clea price. The exceptios are bods that are i default. Such bods are said to be quoted flat, that is, without accrued iterest. 14. Explai why you agree or disagree with the followig statemet: The price of a floater will always trade at its par value. Oe would disagree with the statemet: The price of a floater will always trade at its par value This ca be see from the equatio where compoets of the equatio ca chage as see from the fact that the coupo rate of a floatig-rate security (or floater) is equal to a referece rate plus some spread or margi. For example, the coupo rate of a floater ca reset at the rate o a three-moth Treasury bill (the referece rate) plus 50 basis poits (the spread). Next, the price of a floater depeds o two factors: (1) the spread over the referece rate ad (2) ay restrictios that may be imposed o the resettig of the coupo rate. For example, a floater may have a maximum coupo rate called a cap or a miimum coupo rate called a floor. The price of a floater will trade close to its par value as log as (1) the spread above the referece rate that the market requires is uchaged ad (2) either the cap or the floor is reached. However, if the market requires a larger (smaller) spread, the price of a floater will trade below (above) par. If the coupo rate is restricted from chagig to the referece rate plus the spread because of the cap, the the price of a floater will trade below par. 15. Explai why you agree or disagree with the followig statemet: The price of a iverse floater will icrease whe the referece rate decreases. As explaied below, oe would disagree with the statemet: The price of a iverse floater will icrease whe the referece rate decreases. The factors that affect the price of a iverse floater are affected by the referece rate oly to the extet that it affects the restrictios o the floater s rate. This is quite a importat result. Some ivestors mistakely believe that because the coupo rate rises, the price of a iverse floater should icrease if the referece rate decreases. This is ot true. The key i pricig a iverse floater is how chages i iterest rates affect the price of the collateral. The referece rate is Copyright 2016 Pearso Educatio, Ic. 29

importat oly to the extet that it restricts the coupo rate of the floater. ore details are give below. I geeral, a iverse floater is created from a fixed-rate security. The security from which the iverse floater is created is called the collateral. From the collateral two bods are created: a floater ad a iverse floater. The two bods are created such that (1) the total coupo iterest paid to the two bods i each period is less tha or equal to the collateral s coupo iterest i each period, ad (2) the total par value of the two bods is less tha or equal to the collateral s total par value. Suppose the total par value of the floater ad iverse floater equals the par value of the collateral. Regardless of the level of the referece rate, the combied coupo rate for the two bods is equal to the coupo rate of the collateral. However, if the referece rate exceeds a certai percetage, the the formula for the coupo rate for the iverse floater will be egative. To prevet this from happeig, a floor is placed o the coupo rate for the iverse floater. Typically, the floor is set at zero. Because of the floor, the coupo rate o the floater must be restricted so that the coupo iterest paid to the two bods does ot exceed the collateral s coupo iterest. Thus, whe a floater ad a iverse floater are created from the collateral, a floor is imposed o the iverse ad a cap is imposed o the floater. The price of a iverse floater is foud by determiig the price of the collateral ad the price of the floater. This ca be see as follows: collateral s price floater s price + iverse s price. Therefore, iverse s price collateral s price floater s price. Copyright 2016 Pearso Educatio, Ic. 30