FM for Actuaries

Bond Yields and the Term Structure 215
7.2 Yield to Maturity
Given the yield rate applicable under the prevailing market conditions, we can compute
the bond price using one of the pricing formulas in Chapter 6. The computed
price reflects the fair value of the bond based on the prevailing rate of interest. This
is especially important if an investor has to price a new issue of bonds. The pricing
formula can be extended to the case of a general term structure with given spot rates
i S t , forward rates i F t or accumulation function a(t). The bond price is then computed
using (6.10) (for an annual coupon bond) or (6.11) (for a semiannual coupon
bond).
In practice, however, the yield rate or the term structure are not observable,
while the transaction price of the bond can be observed from the market. 1 Given
the transaction price, we can solve for the rate of interest that equates the discounted
future cash flows (coupon payments and redemption value) to the transaction price.
This rate of interest is called the yield to maturity (or the yield to redemption),
which is the return on the bond investment if the investor holds the bond until it
matures, assuming all the entitled payments are realized.
It can be seen that the yield to maturity is indeed the internal rate of return of
the bond investment. Specifically, for a n-year annual coupon bond with transaction
price P , the yield to maturity, denote by i Y , is the solution of the following
equation:
n∑ 1
P = Fr
(1 + i Y ) j + C
(1 + i Y ) n . (7.1)
j=1
In the case of a n-year semiannual coupon bond, we solve i Y from the equation
(the coupon rate r is now per half-year)
P = Fr
2n∑
j=1
1
[
1+ i ] j
+
Y
2
C
[
1+ i Y
2
] 2n
. (7.2)
Thus, for an annual coupon bond i Y is an annual effective rate, while for a semiannual
coupon bond i Y is a nominal rate (per annum) convertible semiannually.
Note that equations (6.10) and (6.11) are applicable from the pricing perspective,
while equations (7.1) and (7.2) evaluate the return of the bond investment when an
investor purchases it at the price P and holds it until it matures.
To calculate the solutions of equations (7.1) and (7.2) numerical methods are
required. The Excel Solver may be used for the computation.
1 We shall discuss the estimation of the term structure in Sections 7.5 and 7.6.