You have two 4-year annual-coupon bonds, each one of them has a face value of 8000 and a redemption value of 8000. The coupon rate of first bond is 7% and its price is 7908.57, while the second has a coupon rate of 8% and a price of 8101.55. Find the annual yield on a 4-year zero-coupon bond.
The thing that trouble's me is this.
I get the following equations
$$7908.57=560a_{\overline{4}\rceil i}+8000v^{4}$$
$$8101.55=640a_{\overline{4}\rceil i}+8000v^{4}$$
I am thinking "hey, I can actually just calculate the yield rate $i$ and that should be the answer", and to my surprise both equations have different values that what the problem says.
The first yield I got was $7.3\%$ and the other was $7.6\%$.
The answer is supposedly $5.1\%$.
Maybe I am not quite understanding the meaning of yield rate, since I know that a 0-coupon bond pays no coupons until the maturity date and it simply pays back $F(1+r)$ where $F$ is the face value and $r$ is the coupon rate.
Can I get some help?
The price $P_i$ of bond $i$, $i=1,2$ with face value $F$ is
$$P_i=\frac{c_i}{1+r_1}+\frac{c_i}{(1+r_2)^2}+\frac{c_i}{(1+r_3)^3}+\frac{F+c_i}{(1+r_4)^4}$$
So, form the difference $c_1P_2-c_2P_1$ to find
$$c_1P_2-c_2P_1=\frac{(c_1-c_2)F}{(1+r_4)^4}$$
where upon solving for the $4$-year zero $r_4$ reveals that
$$r_4=\left(\frac{(c_1-c_2)F}{c_1P_2-c_2P_1}\right)^{1/4}-1$$
Now, just plug in the numbers and you'll find $r_4 \approx 5.0956$%!