Zero-coupon vs. $10\%$ coupon problem

Question

I am working on Bonds and I am having trouble solving this problem.

A zero-coupon bond pays no coupons and only pays a redemption amount at the time the bond matures. Greta can buy a zero-coupon bond that will pay $10,000$ at the end of $10$ years and is currently selling for $5,083.49$. Instead she purchases a $10\%$ bond with coupons payable semi-annually that will pay $10,000$ at the end of 10 years. If she pays $X$ she will earn the same annual effective interest rate as the zero coupon bond. Calculate $X$.

This is what I know so far.

1), Greta can pay $5,083.49$ in order to receive $10,000$ in 10 years. Since she receives no coupon, the redemption value is simply $10,000$. Therefore, the yield rate $j$ per conversion period can be calculated by

$$5,083.49 = 10,000v^{20}_j$$

and I got $$j\approx 3.44\%$$

2), The final value that Greta receives is $10,000$. This time she receives $X \times 10\%$ each conversion period $20$ times, and at the end she will receive a certain redemption fee, say, $C$. so

$$(0.1)Xa_{\overline{20}\rceil i}+Cv^{20}_i=10,000v^{20}_i$$

for some yield rate $i$ (per conversion).

Here is the part that I am confused.

I am not sure how the problem can be solved from here, because there is simply too many variables left. I tried assuming that $i=j$ which might be attainable from "she will earn the same annual effective interest rate" but that still leaves me wondering what $C$ is. I also tried assuming that $X=C$, but I do not see where in the problem I would be able to assume that (the answer was not right, anyways so it must be wrong.)

Can I have some help?

The answer is supposedly $X=12,229$

Answer 1

Your calculation for 1 looks good. If you use the rule of $72$, the bond should double in $20$ periods if the interest is $3.6\%$ per period. It doesn't quite double in $20$ periods and you have an interest rate a little less. Bingo. For 2, we need to know what she does with the cash that comes in during the period. Naively, you would be expected to assume no interest on it-she stuffs it in the mattress. In that case she has $20,000$ at the end of $10$ years, half from the interest and half from the principal. Then she should pay twice the $5083.49$, because she winds up with twice the money. Since the expected answer is higher, they assume some interest on the cash that comes from the interest, maybe (I think without justification, because the terms are different) the rate you have calculated. Now I would make a spreadsheet and root find for the value.

Answer 2

Looks right what you have above, What I did I looked at the future.

X*(1.0344)^20 = 10,000(0.01/2)S(20,0.0344) + 10,000

Left hand side is simply what would happen if he puts his money in a bank for 20 periods (10 years), right hand side is - coupons 10% semi-annually means that r = 0.05 coupon rate , so F*r=10,000*0.05=500 , and this money in his savings account that earns same effective interest rate which is 1,07^(1/2) = 0.0344 for 20 periods (semi-annual) (10 years) , basically he gets coupons and put them into savings account and at the end of year 10 (period 20) he gets redemption = 10,000 . solving for X = 12,229.578

Answer 3

$$ 5,083.49=10,000\times v_j^{20}\qquad\Longrightarrow\quad j=3.44\% \;\text{(six-month yield rate)} $$ Then $$ X=10,000\times v_j^{20}+500\times a_{\overline{20}|j}=12,229 $$