Yield Rate

Study guide: Exam FM-2

The yield rate $j$ (also called the yield to maturity) is the internal rate of return of a bond — the interest rate per period at which the present value of all future cash flows equals the current price $P$ .

$P = Fr \cdot a_{\overline{n}|j} + C \cdot v^n$

$F$ is the face amount, $r$ the coupon rate per period, $C$ the redemption value, $n$ the number of coupons and $v = 1/(1+j)$ ; the yield rate is the $j$ that solves this equation. Unlike the Coupon Rate $r$ , which is fixed by the bond's terms, the yield rate reflects the bond's market price.
A premium bond ( $P > C$ ) has $Fr > Cj$ — when $C = F$ , as it is unless stated otherwise, the coupon rate exceeds the yield rate; a discount bond ( $P < C$ ) has $Fr < Cj$ .
Bond prices and yields move in opposite directions: if the yield rises, the price falls, and vice versa.
The yield rate is the rate actually earned by an investor who buys the bond at $P$ and holds it until redemption.

Finding the Yield Rate A $ $1{,}000$ face value 3-year annual-coupon bond with coupon rate $5\%$ is currently priced at $ $950.26$ . Find its yield rate.
Answer Coupons are $Fr = 1{,}000(0.05) = 50$ per year and the redemption value is $C = 1{,}000$ , so $j$ solves $50\,a_{\overline{3}|j} + 1{,}000\,v^3 = 950.26$ Bracket the root by pricing at two trial rates:
 $\begin{align*} P(6.5\%) &= 50(2.64848) + 1{,}000(1.065)^{-3} \\ &= 960.27 \\ P(7\%) &= 50(2.62432) + 1{,}000(1.07)^{-3} \\ &= 947.51 \end{align*}$ 
The price $950.26$ lies between them, so interpolate:
 $\begin{align*} j &\approx 0.065 + 0.005 \cdot \frac{960.27 - 950.26}{960.27 - 947.51} \\ &= 0.065 + 0.005(0.78448) \\ &= 0.06892 \end{align*}$ 
Solving exactly (BA II Plus: $N = 3$ , $PV = 950.26$ , $PMT = -50$ , $FV = -1{,}000$ , CPT $I/Y$ ) gives $j = 6.89\%$ . The bond trades at a discount to its face value because its coupon rate ( $5\%$ ) is below the yield.