1
$\begingroup$

I'm trying to solve for the following problem and I cannot get the right #.

You are given the spot rates at time $t=1,\ 2 \ \text{and} \ 3$ as $s_0(1)=.15,\ s_0(2)=.10,\ \text{and} \ s_0(3)=.05$ These are effective annual rates of interest for zero coupon bonds of 1, 2 and 3 years maturity, respectively. A newly issued 3-year bond with face amount $F=100$ has annual coupon rate $r=10\%$, with coupons paid once per year starting one year from now.

Find the price and effective annual yield to maturity of the bond.

I understand that the price of the bond is simply the present value of the cash flows, so

$$P=10(1.15^{-1}+1.10^{-2}+1.05^{-3})+100(1.05^{-3}) \approx 111.98$$

but I cannot get the yield rate correctly.

I want to say that the yield rate $j$ can be found from

$$F(1+(r-j)a_{\overline{3}\rceil j})=P$$

using the formula for bonds with the redemption value being the same as the face value.

I get $j \approx 2.94\%$ but the answer is supposedly $5.56\%$.

I would really appreciate any help.

$\endgroup$
1
$\begingroup$

The yield to maturity $y$ satisfies

$$111.98 = 100(1+y)^{-3} + 10\sum_{k=1}^{3} (1+y)^{-k} = 10\frac{(1+y)^{-1}[1-(1+y)^{-3}]}{1 - (1+y)^{-1}}+100(1+y)^{-3}\\=10\frac{1-(1+y)^{-3}}{y}+100(1+y)^{-3}.$$

Solve numerically using, for example, bisection to find $y = 0.0556$.

$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.