0
$\begingroup$

Please illustrate that a bond with maturity N years that has coupon equal to its yield is associated with the conversion factor of 1.

I do this by writing out $$\frac1{100} \left( \sum_{t=1}^N \left[ \frac{100 (0.06)}{1.06^t} \right]+\frac{100}{1.06^N} \right)$$ but I do not get that this = 1.

I use the formula:

$$\sum_{k=m}^n a^k = \begin{cases}\frac{a^{n+1} - a^m}{a-1}, \quad &a \neq 1\\n-m+1, \quad &a=1\end{cases}$$

$\endgroup$
  • $\begingroup$ I solved it now. $\endgroup$ – jacob Sep 14 '15 at 16:54
2
$\begingroup$

We are given a bond with Coupon = Yield = $6 \%$ and Maturity $N$. We want to check that the conversion factor = 1, in other words that $$\frac1{100} \left( \frac{100}{1.06^N} + \sum_{t=1}^N \frac{100 \cdot 0.06}{1.06^t} \right) = 1 $$ or equivalently $$ \frac{1}{1.06^N} + \sum_{t=1}^N \frac{0.06}{1.06^t} = 1. $$

The first term is $\frac{1}{1.06^N} = \left( \frac{1}{1.06} \right)^N.$ The second term can be simplified using the formula for geometric sums $$\sum_{t=1}^N \frac{0.06}{1.06^t} = 1 - \left( \frac{1}{1.06} \right)^N.$$ Now we add the first term and the second term and see that $$\left( \frac{1}{1.06} \right)^N \quad + \quad 1 - \left( \frac{1}{1.06} \right)^N = 1. $$

$\endgroup$
  • $\begingroup$ Well done! Also note that in practice, conversion factor for a 6% coupon bond maybe not be exactly 1.0 (could be 0.9999) due to the rounding of the maturity date and the rounding convention of the conversion factor. $\endgroup$ – Helin Sep 14 '15 at 22:34

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.