# Can you calculate modified duration for swaps?

I know how to calculate them for bonds. But it came to my mind this.

In bonds, Macaulay duration technically is a weighted average of coupon payments. But can it be somehow calculated for swaps? Or when dealing with swaps, you always need to proxy duration as the "contractual duration"?

If you know how to calculate them for bonds, you know how to calculate them for swaps.

Assuming you refer to fixed-income swaps where a party receives a fixed rate and pays a floating rate or vice versa, the duration of a swap is the duration of the long position and the duration of your short position, which in this case will be a negative duration. Let's say a swap is entered where party 'A' will receive a floating rate and will pay a fixed rate. This is the same as issuing a fixed-rate bond and using the proceeds of such issuance to buy a floating-rate bond. Thus, the duration of the swap can be summarized as:

$$\text{duration of swap} = \text{duration of long position} - \text{duration of short position}$$

In our example, as party 'A' is borrowing at a fixed-rate it would be benefited with rising rates and a lower market value. In the same way, he will see the benefit of being long the floating-rate because future payments will reflect the rise in rates.

To finish, let's express the idea with numbers. Let's say the duration of the floater for party 'A' is 0.125 and the duration on the short side is 0.75. In this case the duration of the swap would be

$$0.125 - 0.75 = -0.625$$,

a negative duration. Effectively, when rates rise, his short position would be worth less. As a note of reference $$\text{change in price} = -\,\text{duration} \cdot \text{change in yield}$$. So when rates rise, the market yield will rise and the market value of the short position decreases. Entering the same swap again, would require party 'A' to pay a higher fixed rate. The opposite logic will apply to his long position.

• Hi, thanks for the response. If I wanted to support this professionally, is there any academic literature to back this methodology up? – FridaTheDog Mar 26 at 22:41
• I’m sure there are as I’ve studied it. After running a quick search on Google you may find something here oreilly.com/library/view/bond-math-the/9781576603062/… I must say I don’t know what Oreilly is. – teoeme139 Mar 27 at 12:29
• The Oreilly web site is based on the book BOND MATH: The Theory Behind the Formulas by Donald J. Smith (2011,2014) , Bloomberg Press, ISBN: 9781576603062 so that is your academic reference for you. – noob2 Mar 27 at 13:43

You can always calculate a duration for any interest rate instrument as the duration describe price sensitivity to interest rate changes.

The duration is a first derivative of price, so you can calculate it numerically with this formula:

$$D = \frac{1}{MV}\frac{\Delta MV}{\Delta i},$$

where $$\Delta MV$$ is change in market value of the instrument when interest rate is changed for $$\Delta i$$ and $$MV$$ is current market value.

Note that in this calculation I assumed parallel shift of interest curve and $$\Delta i$$ is small.