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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.

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1 Answer 1

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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 $y = 0.0556$.

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