Elon, Gruber, Brown, and Goezmann oluions o Tex Problems: Chaper Chaper : Problem We can use he cash lows bonds A and B o replicae he cash lows o bond C. Le YA be he racion o bond A purchased and YB be he racion o bond B purchased. (Noe ha hese are no invesmen weighs ha sum o.) Then we have: : $00 YA + $80 YB $90 : $,00 YA + $,080 YB $,090 olving he above wo equaions simulaneously gives YA YB /. o buying / o bond A and / o bond B gives he same cash lows as buying bond C (or, equivalenly, buying bond A and bond B gives he same cash lows as buying o bond C). Thereore, i he Law o One Price held, he bonds curren prices would be relaed as ollows: / PA + / PB PC Bu, since we are given ha PA $970, PB $96 and PC $980, we have insead: / $970 + / $96 $95 < $980 The Law o One Price does no hold. Given ha he uure cash lows o he porolio o bonds A and B are idenical in iming and amoun o hose o bond C, and assuming ha all hree bonds are in he same risk class, an invesor should purchase bond A and bond B raher han o bond C. Chaper : Problem A. A bond s curren yield is simply is annual ineres paymen divided by is curren price, so we have: Curren Yield $00 $960 0.0 (0.%) Elon, Gruber, Brown, and Goezmann - oluions To Tex Problems: Chaper
B. A bond s yield o mauriy is he discoun rae ha makes he sum o he presen values o he bond s uure cash lows equal o he bond s curren price. ince his bond has annual cash lows, we need o ind he rae, y, ha solves he ollowing equaion: 5 $00 $000 $960 + 5 y + + y We can ind y ieraively by rial and error, bu he easies way is o use a inancial calculaor and inpu he ollowing: PV 960 PMT 00 FV 000 N 5 Aer enering he above daa, compue I o ge I y.08%. Chaper : Problem In general, he nominally annualized spo rae or period (0) is he yield o mauriy or a -period zero-coupon (pure discoun) insrumen: P 0 F + 0 where P0 is he zero s curren marke price, F is he zero s ace (par) value, and is he number o semi-annual periods le unil he zero maures. The zero-coupon bonds in his problem all have ace values equal o $,000. Elon, Gruber, Brown, and Goezmann - oluions To Tex Problems: Chaper
I semi-annual periods are assumed, hen bond A is a one-period zero, bond B is a wo-period (one-year) zero, bond C is a hree-period zero, and bond D is a our-period (wo-year) zero. o we have: 000 960 0 + 000 90 0 + 000 885 0 + 000 855 0 + 0 0 0 0 0.08 (8.%) 0.085 (8.5%) 0.08 (8.%) 0.0799 (7.99%) The nominally annualized implied orward raes (,+j) can be obained rom he above spo raes. A general expression or he relaionship beween curren spo raes and implied orward raes is: + j j 0, + j +, + j 0, + where is he semi-annual period a he end o which he orward rae begins, j is he number o semi-annual periods spanned by he orward rae, and boh and j are inegers greaer han 0. Elon, Gruber, Brown, and Goezmann - oluions To Tex Problems: Chaper
Elon, Gruber, Brown, and Goezmann - oluions To Tex Problems: Chaper We can an obain a se o one-period orward raes by seing j equal o and varying rom o in he preceding equaion: (7.0%) 0.070.06.000 (7.9%) 0.079.06.06 (8.70%) 0.0870.07.06 0 0 0 0 0 0 I insead we waned he expeced spo yield curve one period rom now under he pure expecaions heory, we can se equal o and vary j rom o in he preceding equaion: (7.89%) 0.0789.07.000 0.08 (8.%).07.06 (8.70%) 0.0870.07.06 0 0 0 0 0 0
Chaper : Problem We can use he cash lows bonds A and B o replicae he cash lows o bond C. Le YA be he racion o bond A purchased and YB be he racion o bond B purchased. Then we have: : $80 YA + $,00 YB $0 : $,080 YA + $0 YB $,0 olving he above wo equaions simulaneously gives: Y A, 0,080 8 7 08 97 Y B 8 0 80 7, 00,0,0 7 7 9,700 7,000 7 7 9,700,000 9,700 0 97 o buying 08/97 o bond A and 0/97 o bond B gives he same cash lows as buying bond C (or, equivalenly, buying 08 o bond A and 0 o bond B gives he same cash lows as buying 97 o bond C). Thereore, i he Law o One Price held, he bonds curren prices would be relaed as ollows: 08/97 PA + 0/97 PB PC Bu, since we are given ha PA $98, PB $880 and PC $,00, we have insead: 08/97 $98 + 0/97 $880 $,08 > $,00 The Law o One Price does no hold. For he Law o One Price o hold, bond C would have o sell or $,08. Elon, Gruber, Brown, and Goezmann -5 oluions To Tex Problems: Chaper
Chaper : Problem 5 I he Law o One Price holds, hen he same discoun rae (which is a spo rae) applies or he cash lows in a paricular period or all hree bonds. Also, in he presence o axes, he price o each bond mus equal he sum o he presen values o is uure aer-ax cash lows, where he presen values are calculaed using he spo raes. Each bond s capial gain or loss is simply is principal (par) value minus is price. Given ha each bond has a par value o $,000, bond A has a capial gain o $,000 $985 $5, bond B has a capial gain o $,000 $900 $00, and bond C has a capial loss o $,000 $,00 $0. Given ha he periods shown are annual, ha axes mus be paid on capial gains and can be deduced on capial losses, and ha he capial gain or loss ax rae is one-hal o he ordinary income ax rae, we need o ind he discoun acors and ordinary income ax rae ha makes he ollowing se o equaions hold simulaneously: where T 80 d T 00 ( T ) d 00 d + $,000 d $ 900 T 0 T d + $0 T d + $0 d + $,000 d ( T ) d + $80 ( T) d $5 d + $,000 $ 985 $ $ $, 00 $ T he ordinary income ax rae; d he wo-semi-annual-period (one-year) discoun acor; 0 + d he our-semi-annual-period (wo-year) discoun acor. 0 + The soluion o he above se o simulaneous equaions is: T 0.0; d 0.8568; d 0.89 Elon, Gruber, Brown, and Goezmann -6 oluions To Tex Problems: Chaper