20-Term Structure Page 597 Wednesday, February 4, 2004 1:33 PM
Term Structure Modeling and Valuation of Bonds and Bond Options 597P C^1 – (^1 + y)- n
1
----- = ----- -------------------------------- + -------------------- (20.7)
M M y ( 1 + y)n
The yield to maturity is the solution to equation (20.7) for y, the
yield of an n-period bond. In equation (20.7) P/M is the so-called par
value relation, usually expressed as a percentage. If it is equal to one,
the bond sells “at par”; if it is larger than one, it sells at a “premium”;
and if it is less than one, it sells at a “discount.” C/M is the coupon rate
expressed as a ratio.
So far we have not specified the unit of time for measuring the fre-
quencies with which interest is computed and the coupons are paid.
Interest rates (and maturity) customarily are quoted per year (e.g., 7%
per year), and we shall follow this convention; this means that in equa-
tion (20.7) it is implicitly assumed that the coupon rate is C per year
and paid once a year. In fact, in the United States almost all bonds pay
interest twice a year. Each coupon payment therefore amounts to C/2,
which must be discounted twice a year at half the annual yield or y/2.
As a result, equation (20.7) is changed toP C 1 – ( 1 + y ⁄ 2 ) –^2 n 1
----- = --------- ------------------------------------------ + ------------------------------ (20.8)M 2 M y ⁄ (^2) ( 1 + y ⁄ 2 )^2 n
To illustrate calculation of the yield to maturity of a bond with semi-
annual coupon payments, consider a 7%, 20-year bond with a maturity
or par value of $100, and selling for 74.26%, or 74.26 cents per $1 of par
value. The cash flow for this bond per dollar of par value is: 40 six-month
payments of $0.035, and $1 received in 40 six-month periods from now.
The present value at various semiannual interest rates (y/2) is:
Interest rate (y/2): 3.5% 4.0% 4.5% 5.0% 5.5% 6.0% 6.5%
Present value (P/M): 1.0000 0.9010 0.8160 0.7426 0.6791 0.6238 0.5756
When a 5.0% semiannual interest rate is used, the present value of the
cash flows is equal to 0.7426 per $1 of par value, which is the price of
the bond. Hence, 5.0% is the semiannual yield to maturity.
The annual yield to maturity should, strictly speaking, be found by
compounding 5.0% for one year. That is, it should be 10.25. But the con-
vention adopted by the bond market is to double y/2, the semiannual
yield to maturity. Thus, the yield to maturity for the bond above is 10%
(two times 5.0%). The yield to maturity computed using this convention
of doubling the semiannual yield is called the bond equivalent yield.