Bond Price

Study guide: Exam FM-2

The bond price $P$ is the present value of all future cash flows from a bond, discounted at the yield rate $j$ per period.

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

Here $F$ is the face value, $r$ is the coupon rate per period (so $Fr$ is the periodic coupon amount), $C$ is the redemption value (often equal to $F$ ), $n$ is the total number of coupon periods, and $v = 1/(1+j)$ .
An equivalent form is the premium/discount formula: $P = C + (Fr - Cj) \cdot a_{\overline{n}|j}$ , which makes the premium or discount explicit.
When $Fr > Cj$ the bond sells at a premium ( $P > C$ ); when $Fr < Cj$ it sells at a discount ( $P < C$ ); when $Fr = Cj$ it sells at par ( $P = C$ ).

Bond Price Calculation A $ $1{,}000$ face value 10-year bond pays semi-annual coupons at a coupon rate of $8\%$ per year. The bond is priced to yield $6\%$ per year convertible semi-annually. Find the bond price.
Answer Per-period values: $F = C = 1000$ , $r = 4\% = 0.04$ , $j = 3\% = 0.03$ , $n = 20$ .

$Fr = 1000(0.04) = 40$

$v^{20} = (1.03)^{-20} = 0.5536758$

$a_{\overline{20}|3\%} = \frac{1 - v^{20}}{0.03} = 14.877475$
 $\begin{align*} P &= 40(14.877475) + 1000(0.5536758) \\ &= 595.0990 + 553.6758 \\ &= 1148.7748 \\ &\approx \$1{,}148.77 \end{align*}$ 
Since $Fr = 40$ exceeds $Cj = 30$ (with $C = F$ , the coupon rate per period, $4\%$ , exceeds the yield rate per period, $3\%$ ), the bond sells at a premium above its $ $1{,}000$ redemption value.