Approximating Interest Rate Swap Duration from Discounting

Summary

The document examines a simplified duration expression for a fixed-for-floating interest rate swap and relates it to the sensitivity of swap present value to a parallel rate shift. In a single-curve framework, the response writes the swap value as discounted floating-leg cash flows less discounted fixed-leg cash flows, then differentiates that value with respect to the rate shift. This produces a duration-like sensitivity involving payment times and discount factors.

A further simplification makes the sensitivity approximately equal to the swap maturity. A Taylor expansion of the proposed exponential expression shows why it also approaches maturity when the rate-times-maturity term is small. These are approximations grounded in the stated single-curve setup; the document does not establish accuracy across market conditions, explain multi-curve valuation, or fully address other swap structures. The original question asks about applicability, but the response only derives the fixed-for-floating case.

Key ideas

• In a single-curve framework, swap value can be expressed using discounted floating and fixed cash flows.
• The rate sensitivity depends on the timing and discounting of swap payments.
• A simplified derivative approximation makes the sensitivity close to the maturity.
• The exponential duration expression also approaches maturity under a small rate-times-maturity expansion.
• The derivation addresses a simple fixed-for-floating swap and does not establish broader applicability.

Tags

# Simplified formula for duration of interest rate swap


# Simplified formula for duration of interest rate swap












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 by Kermittfrog (score 3)

https://quant.stackexchange.com/a/73969

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.

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This summary was written by Stratmill's research agent from the original; it is not a copy of the source.