Simplified formula for duration of interest rate swap

Question

Lets consider the simple interest rate swap instrument as 5-year maturity interest rate swap. I found an interesting simplification to calculate the duration of such swap as,

$\frac{\left(1 - e^{-r_t * T}\right)}{r_t}$

In above expression the $r_t$ is current level of interest rate and $T$ is the swap maturity i.e. in this case 5.

Could you please help to obtain explanation how the duration is an interest rate swap looks like this? Also, is such approximation is applicable only naive fixed vs floating interest rate swap?

Answer 1

That's an interesting approximation, I have not yet seen this one.

The PV of a fix-for-float IRS, in a single-curve-world, is:

$$ PV=\sum_i\Delta^{float}_iF_iD(t_i)-q\sum_j\Delta^{fix}_jD(t_j)=1-D(t_n)-q\sum_j\Delta^{fix}_jD(t_j) $$

The first derivative w.r.t. a parallel shift is

$$ \frac{\partial PV}{\partial r}=t_nD(t_n)+q\sum_j\Delta_j^{fix} t_jD(t_j) $$

From here, you can derive all sorts of simplifications, one being

$$ \frac{\partial PV}{\partial r}\approx t_n $$

If we now take your formula and Taylor expand, we arrive at:

$$ \frac{1-e^{-r_nt_n}}{r_n}=\frac{1-\left(1-r_nt_n+O(r_n^2t_n^2)\right)}{r_n}\approx t_n $$

Which yields a similar approximation result.