ñòð. 92 
Yield to Maturity. The Rite Companyâ€™s bonds have 3 years remaining until maturity. Interest is
7.24
paid annually, the bonds have a $1,000 par value, and the coupon interest rate is 10percent. What
is the yield to maturity at a current market price of: (a) $1,052; (b) $1,000;and (c) $935?
SOLUTION
Annual interest = $100 (10% X $1,00)
M
V=C
â€œ I
+
(1 + r)â€
(1 + r)â€˜
r1
+ M(PV1Fr.n)
= Z(PVIFA,,,)
$100
+ $100 +
$1,052 =   *  * +  $l,OOo $100
+
(I + r)* (1 + r)* (1 + r)3 (1 + r)3
$1,052 = $100(PVIFAr,3)+ $1,000(PVIFr,3)
Since the bond is selling above par value, the bondâ€™s yield is below the coupon rate of 10 percent. At
r = 8%:
V = $100(2.5771) + $1,000(0.7938) = $257.71 + $793.80 = $1,051.51
Therefore, the annual yield to maturity of the bond is 8 percent.
Alternatively, the shortcut formula yields:
I+(MV)ln
Yield to maturity of a bond =
(M+ V)/2
196 RISK, RETURN, AND VALUATION [CHAP. 7
+ $l9000(PVIFr3)
$1,000 = $lOo(PVIFA,,)
(b)
Since the bond is selling at par value, the bond's yield should be the same as the coupon rate. At
r = 10%:
V = $100(2.4869) + $1,000(0.7513) = $248.69 + $751.30 = $999.99
Or, using the shortcut formula:
$100 + ($1,000  $1,000)/3 =$100

Yield = 10%
($l,OOo + $1,000)/2 $MJo
$935 = $lOO(PVIFArJ) + $l,OOo(PVIFAr3)
(4
Since the bond is selling at a discount under the par value, the bond's yield is above the going coupon
rate of 10 percent. At r = 12%:
V = $100(2.4018) + $1,000(0.7118) = $240.18 + $711.80 = $951.98
At 10 percent, the bond's value is above the actual market value of $935, so we must raise the rate.
At r = 13%:
V = $100(2.361) + $1,000(0.693) = $236.10 + $693 = $929.10
Since the bond value of $935 falls between 12 percent and 13 percent, find the yield by
interpolation, as follows:
Bond Value
$951.18 $951.18
12%
935.00
True yield
929.10
13%
$ 16.18 $ 22.08
Difference
16 18
12% + 0.73(1%) = 12.73%
Yield = 12% +(lYo) =
22.08
Alternatively, using the shortcut method:
$100 + ($l,OOO  $935)/3 =
 $121.67
12.58%
Yield =
($1,000 + $935)/2 $967.5
Yield to Maturity. Assume a bond has 4 years remaining until maturity and that it pays interest
7.25
semiannually (the most recent payment was yesterday). ( a ) What is the yield to maturity of the
bond if its maturity value is $1,000, its coupon yield is 8 percent, and it currently sells for $821?
(6) What if it currently sells for $1,070?
SOLUTION
2n
M
I12
V=C
+
(I + rD)' (1 + r/2)&
=I/2(PVIFArn.b) + M(PVIF,,Z,b)
= $40
"('OO
o$*O)
Semiannual interest =
2
By trial and error, we find that when r/2 = 7%:
V = $40(5.9713) + $1,000(0.5820) = $238.85 + $582 = $820.85
Therefore, the annual yield is 7% X 2 = 14%.
197
RISK, RETURN, AND VALUATION
CHAP. 71
Using the shortcut formula:
1 + (M V ) / n  $80 + ($l,OOO  $821)/4  $124.75
 = 13.7%
Yield =
($1,000 + $821)/2
(M+ V)/2 $910.5
40
+ (I$1,OOo
$40 ++***+
$1,070 =  40
+ rD)8
(I + r/2)8
(I + r/2)' (1 + r/2)2
$1,070 = $4O(PVIFA,n,e) + $l,OOO(PVIF,,˜˜)
By trial and error, we get the semiannual interest of 3 percent:
V = $40(7.0197) + $1,000(0.7894) = $280.79 + $789.40 = $1,070.19
Therefore, the annual yield is 3% X 2 = 6%.
Alternatively, using the shortcut method:
$80 + ($1,000  $1,070)/4  $62.5
= 6%
Yield =
($l,OOO + $1,070)/2 $1,035
Yield of a Note. You can buy a note at a price of $13,500. If you buy the note, you will receive
7.26
10 annual payments of $2,000,the first payment to be made immediately. What rate of return,
or yield, does the note offer?
SOLUTION
V = C  c,
r1 (1 + r)l
$2000 $2,000
+ $2,000

$13,500 = $2,000 ++ +
ñòð. 92 