If you have a bond with the following properties:
- Issued at price $X$;
- Redeemed at price $Y$;
- Has a coupon rate of $p$ ;
- Has a term of $n$ years;
then I am told, in my accounting studies, that the effective interest rate, $r$, can be approximated as
$r\approx(\frac{Y}{X})^\frac{1}{n}-1+p$.
It seems to do a decent job, but I cannot figure out for the life of me why this should work as an approximation.
In attempting to figure this out, I have got to the point of seeing that
$X((1+r)^n+p(\frac{1-(1+r)^n}{r}))=Y$,
but I am unable to use this to get to the above approximation for $r$.
I would really appreciate if someone could help out, as, although the derivation is not directly relevant to my accounting studies, it has been bugging me for some time!
$\require{enclose}$ I'll use $i$ for the interest rate, and $v=\frac1{1+i}$.
We have $$ X=pXa_\enclose{actuarial}{n}+Yv^n\\ X(1-pXa_\enclose{actuarial}{n})=Yv^n\\ X((1+i)^n-ps_\enclose{actuarial}{n})=Y\\ \frac YX=(1+i)^n-ps_\enclose{actuarial}{n}\\ \left(\frac YX\right)^{1/n}=\left((1+i)^n-ps_\enclose{actuarial}{n}\right)^{1/n} $$ First the intuitive explanation. The left hand side show that $i$ is the rate of interest we'd have to earn if we put the bond proceeds into a trust fund, out which we paid the coupons, and eventually redeemed the bonds. Each year, we pay out $pX$ and if $x$ and $Y$ are not very different, as is usual, we earn about $iX$. That explains the result on an intuitive level.
For a more analytic approach, note that $$\left((1+i)^n-ps_\enclose{actuarial}{n}\right)^{1/n}=(1+i)\left(1-pa_\enclose{actuarial}{n}\right)^{1/n}$$ and $$pa_\enclose{actuarial}{n}=p\frac{1-(1+i)^{-n}}i=p\frac{1-(1-ni+o(i))}i\approx np$$ so that $$\left(1-pa_\enclose{actuarial}{n}\right)^{1/n}\approx(1-np)^{1/n}\approx1-p$$
This gives $$\left(\frac YX\right)^{1/n}\approx(1+i)(1-p)=1+i+p-ip\approx1+i-p$$
Basically, this computation just assumes that all second-degree and higher terms, like $i^2$, $p^2$ and $ip$ are negligible, which is reasonable for typical interest rates.