The Garraty Company Company has two bond issues outstanding. outstanding. Both bonds pay $100 annual interest interest plus $1,000 at maturity. maturity. Bond L has a maturity maturity of 15 years, years, and Bond a maturity of 1 year. year. !"#5 a.
%hat %hat will will be the the &alue &alue of ea'h ea'h of these these bond bondss when when the goin going g rate rate of inte interes restt is (1) (1) 5*, 5*, (+) *, and (-) 1+* /ssume that there is only one more interest interest payment to be made on Bond .
b.
%hy does the longer#term (15#year) bond flu'tuate more when interest rates 'hange than does the shorter#term bond (1#year)
ome of the maor 'onsiderations with bond uestions are2 (i)3ow many periods per year, that is, how many times per year is interest paid(ii) 3ow many periods, 4n, are left until maturity This is what matters, not how long was the bond6s life when it was first issued.(iii) 7emember that 4i is the periodic interest rate and refers to the le&el of interest rates in the e'onomy. The general euation for finding the &alue of a bond 'an be e8pressed as follows2 9sing tables2# :B ; /i,n) ? @(=:<>i,n) 9sing the euation2# :B ;
1
?
@AA
i( 1 ? i)n (1 ? i)n
!i
%e will use ea'h method to wor an e8ample, using 5* 1+*. in'e they ha&en6t indi'ated the number of periods per year, we will assume it is 1, ie, 'oupon 'oup on payments are made on'e per year. 9sing tables, the &alue of the long bond D 5* isE :L ; 100(=:<>/5*,15) ? 1000(=:<>5*,15) ; (100 810.-"F") ? (1,000 8 .10) ;$10-".F" ? $1.00 ; $1,51.F" /t 1+*, :L ; (100 8 H.10F) ? (1,000 8 .1+") ; $H1.0F ? $1+."0 ; $H-."F The &alue of the short bond at 5*, : ; 100(=:<>/5*,1) ? 1000(=:<>5*,1) ; (100 8 .F5+) ? (1,000 8 .F5+) ; $F5.+ ? $F5+.0 ; $1,0".H /t 1+*, : ; (100 8 .F+F) ? (1,000 8 .F+F) ; $F.+F ? $+F.F0 ; $F1F.1F %e are not surprised, are we, that for both the long and short bond, at 5* we ha&e a premium bond and at 1+* we ha&e a dis'ount bond The siIe of the dis'ount and a nd premium differs with the siIe of 4n, and again we are not surprised, be'ause the number of period remaining has a large effe't on the amount of &alue left on the bond. 9sing the euation D 5*, :L ; 100! 1 #
1
? 1000 ; (100 8 10.-"5) ? 1.0+
!.05 .05(1.05)15
(1.05)15
:L ; $1,0-".5 ? $1.0+ ; $1,51.5H /t 1+*, :L ; 100! 1 #
1
? 1000 ; (100 8 .F5+) ? F5+.- ; $1,0".H+
!.05 .05(1.05)
(1.05)
=ra'ti'e using both methods to find the remaining &alues.
(b) Joti'e the 'hanges in bond &aluation between the long and short bonds as interest rates 'hange. Between 5* and 1+*, the long bond &alues mo&e from $151F." to $H-."F, a differen'e of $H5H.0 (-*), while those for the short bond mo&e from $10".H to $F1F.1F, a differen'e of $1+.5 (1+.-*). %hy %hen interest rates rise, they ha&e an ad&erse effe't on all bonds, but ha&e a greater effe't on on long bonds. This is be'ause long bonds still ha&e mu'h (probably most) of their 'oupon payments as well as the maturity &alues stret'hing far into the future and dis'ount fa'tors get more se&ere as 4n gets larger. The long bond in this 'ase has n ; 15 whereas the short bond has n ; 1. This is 'alled interest rate risk , and is more se&ere on longer bonds. %hen interest rates fall, the opposite o''urs. Both bonds are affe'ted in that they both benefit, but the long bond benefits more. The short bond will soon mature and the in&estor will be for'ed to re#in&est his 'oupon payments and maturity &alue at the new, lower rate now a&ailable. The long bond will not be maturing for another 15 years, and so only the 'oupon payments ha&e to rein&ested at the lower rate. The maturity &alue (whi'h is most of the money) is lo'ed away (at the old, higher rate) for another 15 years and will not be a 'andidate for rein&estment. This is reinvestment rate risk, and brings a greater benefit to the long bond.
