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There is a puzzling thing that is bothering me regarding bonds and I would like to have some help. The following is the situation I am dealing with.

A 20-yr 8% bond has semi=annual coupons and a face amount of 100. It is quoted at a purchase price of 70.400. Smith purchases this bond and sells it to Jones in 5 yrs right after Smith receives his 10th coupon. Jones pays 112.225. Smith is able to reinvest his coupons at a bank that offers 6% nominal interest convertible semiannually.

I understand that the internal rate of return earned by Smith is equal to 19.0% because

$$70.4=112.225v^{10}_j+4a_{\overline{10}\rceil j}$$

has a solution $j \approx 9.5\%$

I also understand that Smith can reinvest his coupons which amounts to

$$4s_{\overline{10}\rceil j} \approx 45.86$$

at the time he sells his bond.

However, the average annual rate of return can be calculated fro

$$112.225+45.86 = 70.4(1+i)^5$$

where $i \approx 17\%$

This makes no sense to me. Why is it that Smith reinvested his money and the rate of return becomes smaller from 19% to 17%?

I would appreciate any help.

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I think you should use:

$112.225+45.86=70.4(1+i)^9$

which gives approx. $9.4\%$ for half-yearly rate of return.

To be more accurate, the $6\%$ interest applies only on $9$ coupon deposits, since the bond is sold immediately after receiving the last ($10th$) coupon of $\$4$.

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