Yield to Maturity - New York University
嚜澳ebt Instruments and Markets
Professor Carpenter
Yield to Maturity
Outline and Suggested Reading
? Outline
每 Yield to maturity on
bonds
每 Coupon effects
每 Par rates
每 Yield vs. rate of return
? Buzzwords
每 Internal rate of return,
每 Yield curve
每 Term structure of
interest rates
Yield to Maturity
? Suggested reading
每 Tuckman, Chapter 3
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
K2
Kn
+
+ ... +
=P
2 t1
2t 2
(1 + y / 2)
(1 + y / 2)
(1 + y / 2) 2tn
or
n
Kj
﹉ (1 + y / 2)
j =1
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%:
0.0425
0.0425
1.0425
+
+
1
2
(1 + 0.0554 / 2) (1 + 0.0545 / 2) (1 + 0.0547 / 2)3
= 1.043066
0.0425
0.0425
1.0425
=
+
+
1
2
(1 + 0.054704 / 2) (1 + 0.054704 / 2) (1 + 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
c 2T
1
1
P= ﹉
+
2 s =1 (1 + y / 2) s (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
n
Math result:
1
﹉ (1 + r )
j =1
j
1
1
= (1 ?
)
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
[1 ? (
) 2T ] +
y
1+ y / 2
(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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