Why a Five-Year Interest Rate Swap and Par Treasury Have Similar DV01

# "The five year swap has the same dv01 as a par five year treasury bond" Why?


# "The five year swap has the same dv01 as a par five year treasury bond" Why?












Am reading a book (The Complete Practitioner's Guide to the Bond Market by Steven Dym, 2009) where the author gives an example of someone buying a 5 year par 4.65% treasury and someone else entering an 5 year interest rate swap agreeing to receive 5.75%. The treasury yield moves down .15% to 4.5% The swap spread on the IRS stays the same so they enter an offsetting swap at 5.6%, and therefore locking in .15% for five years. The part I'm trying to understand is this:

> (referring to the IRS)..is it really the same as transacting with the actual bond? In terms of profit, essentially yes. Recall from Chapter 12 that the degree of price change in a bond brought about by a change in yield depends on its maturity, modified somewhat by the size of the coupon. This is quantified by the bond's dv01 (duration). Had the trader purchased the five-year Treasury when it yielded 4.65% and sold soon thereafter when it yielded 4.5% her profit would have reflected the effect of the 15 basis point drop -- 15 times the bond's dv01. You know what? The trader with the pair of swaps in our example is now entitled to 15 basis points, net, each year multiplied by the face amount. And the present value of that is the same as the profit on the Treasury note! The five-year swap has the same dv01 (duration) as a par five-year bond.

I get that the Treasury buyer's profit is 15 x the dv01. I don't get how the 15 bp locked in through the swap is supposed to equal that same number. Does someone have the calculation on how one equals the other?

I like to believe it because it basically says you should just enter a swap rather than buy a Treasury, it's so much better. But I don't understand it enough and almost sounds too good to be true. Like why would anyone buy Treasuries?

## Answer by dm63 (score 1)

https://quant.stackexchange.com/a/71293

The book appears to be rather basic and is making the point that the dv01 of the swap and bond are approximately the same rather than exactly the same. The difference mostly comes from the discount rate applicable in each case. Using the numbers given, the p/l on the swap is the value of a 15bp annuity, discounted on a SOFR curve (assuming the swap is a standard cleared swap). The SOFR curve is not given in the book, so we don’t know what that is. On the other hand, the Treasury p/l, assuming the bond starts at par, is the value of a 15bp annuity discounted at the treasury yield of 4.5%. So this is not exactly the same as the swap, but usually it is pretty close. Of course market participants know this so they just adjust the notional amount of Treasuries to get the dv01 to line up with the swap, if needed. Just a final note- the numbers in the book are way off the current market. Swap rates are not 110bp higher than Treasury yields, in fact they are slightly lower.

## Answer by Michael Williamson (score 0)

https://quant.stackexchange.com/a/69542

There are two legs to an interest rate swap:

- the fixed leg, where interest is paid at the rate on the swap (i.e. 5.75%)

- the floating leg, where interest is paid at the current floating rating (usually a 1m or 3m rate that changes every 1m/3m)

It is similar to buying a bond with a loan.

Entering the offsetting swap is just a mechanic of the market. You can't exit a swap, so instead you create a new swap. The floating legs will always offset each other perfect, and so the P&L on the fixed legs will be paid out over time.

If you were to sell the bond, you would instead earn difference in the present value of the 4.65% coupon.