Yield to Maturity
Debt Instruments and Markets
Professor Carpenter
Yield to Maturity
Outline and Suggested Reading
? Outline ? Yield to maturity on bonds ? Coupon effects ? Par rates ? Yield vs. rate of return
? Buzzwords ? Internal rate of return, ? Yield curve ? Term structure of interest rates
? Suggested reading ? Tuckman, Chapter 3
Yield to Maturity
1
Debt Instruments and Markets
Professor Carpenter
General Definition
Suppose a bond (or portfolio of bonds) has price P and positive fixed cash flows K1, K2,..., Kn at times t1, t2,..., tn. Its yield to maturity is the single rate y that solves:
K1 (1+ y / 2)2t1
+
(1 +
K2 y / 2)2t2
+
... +
(1 +
Kn y / 2)2tn
=P
or
n
Kj
j=1 (1+ y / 2)2t j
=P
Note that the higher the price, the lower the yield.
Example
?Recall the 1.5-year, 8.5%-coupon bond. ?Using the zero rates 5.54%, 5.45%, and 5.47%, the bond price is 1.043066 per dollar par value. ?That implies a yield of 5.4704%:
(1 +
0.0425 0.0554 /
2)1
+
(1 +
0.0425 0.0545 /
2)2
+
(1 +
1.0425 0.0547 /
2)3
= 1.043066
=
(1 +
0.0425 0.054704 /
2)1
+
(1 +
0.0425 0.054704 /
2)2
+
(1 +
1.0425 0.054704 /
2)3
Yield to Maturity
2
Debt Instruments and Markets
Professor Carpenter
Yield of a Bond on a Coupon Date
For an ordinary semi-annual coupon bond on a coupon date, the yield formula is
P
=
c 2
2T s =1
(1 +
1 y / 2)s
+
(1 +
1 y / 2)2T
where c is the coupon rate and T is the maturity of the bond in years.
Formula for the Present Value of an Annuity
Math result:
n 1 = 1 (1- 1 ) j=1 (1+ r) j r (1+ r)n
Finance application: This formula gives the present value of an annuity of $1 to be received every period for n periods at a simply compounded rate of r per period.
Yield to Maturity
3
Debt Instruments and Markets
Professor Carpenter
Price-Yield Formula for a Bond
on a Coupon Date
Applying the annuity formula to the value of the coupon stream, with r=y/2 and n=2T:
P
=
c [1- ( 1 )2T y 1+ y/2
]+
(1 +
1 y / 2)2T
? The closed-form expression simplifies computation. ? Note that if c=y, P=1 (the bond is priced at par). ? If c>y, P>1 (the bond is priced at a premium to par). ? If c ................
................
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