Treasury Futures DV01, Conversion Factors, and Hedge Exposure

# Why is the economic exposure in a hedge using govi futures so different between cash and derivative?


# Why is the economic exposure in a hedge using govi futures so different between cash and derivative?












I have a question regarding swap spread trade. Let's assume I would like to bet on a swap spread widening. For this I could put a payer swap and going long the same maturity cash bond $DV01$ matched. Now assume that I would like to express the $DV01$ of the cash bond with Treasury futures. Now assume my target is to have a swap $DV01$, denoted by $d_s$ of $10'000$.

Using futures I would then check the DV01 of the CTD bond and its $DV01$ denoted by $d_c$. The $d_c$ per $100'000$ is $358$ and the conversion factor $cf$ is $0.76$. That implies a futures $DV01$, $d_f$ of

$$d_f = \frac{358}{0.76}=471$$

as this is already per $100'000$ we get the total amount of contracts required for matching $d_s$ to be $\frac{d_s}{d_f}=\frac{10000}{471}\approx 21$.

Now what puzzles me is the following. I would have intuitively assumed that the economic exposure of this future position should be very close to the economic exposure of a similar cash bond position used instead.

For the future, the total economic exposure is

$$P_f\cdot cf\cdot 1000\cdot n$$ with $P_f = 223, n=21$ implying an economic exposure of approximately $3'560'000$. However, using the data on Bloomberg the equivalent amount of face value of par of the Bond to get an $DV01$ of $d_s$ is $2'8000'000$ which is approximately $4'730'000$ of principal.

What is my initial gut feeling so wrong about this different economic exposure?

## Answer by kurtosis (score 2, accepted)

https://quant.stackexchange.com/a/57536

I think you are getting confused because the conversion factor is used to account for bond futures being standardized to a 6% coupon. Since the futures expect a 6% coupon, you have to adjust the invoice price lower if delivery will yield something less valuable.

For a futures price of 223, it looks like you are trading the Ultras (the true 30Y futures) or the 20Y (ZB or US, what used to be the "30Y"). None of the bonds in the 30Y delivery basket has a conversion factor near 0.76, so I will assume you are trading the 20Y.

Based on your conversion factor, that suggests we are considering a bond like the 3-7/8% maturing on 15 August 2040 (CUSIP=912810QK7). That has a conversion factor of 0.756 which means the bond price is 75.6% of what a 6% coupon bond with the same maturity would be worth. Treasury Direct puts the price of that bond today at about 150.23.

Based on some interpolation using a 4.75% bond, a 6% bond would be worth about 191.75. Lo and behold, 150.23/191.75 = 0.78. So maybe the conversion factor used yesterday's price, but the conversion factor seems correct.

The cash price for buying 21 bonds is 150.23$\times$\$1000$\times$21=\$3,154,830. Compare that to the invoice price of $P_f\cdot cf\cdot 1000\cdot n=$ $223\cdot 0.756\cdot 1000 \cdot 21$ = \$3,540,348. Therefore, delivering these bonds will net you a profit of \$385,518.

Your Bloomberg numbers seem to be off. Bloomberg says you would need 28 bonds to hedge the DV01 (hence \$2,800,000 face). Note that 21 bonds would be \$2,100,000 face or \$3,154,830 of principal. Bloomberg seems to think you need more bonds to hedge and that those bonds would be priced at 168.92857. I would look into why Bloomberg is giving you those numbers. My guess is it is choosing a different bond in the CTD basket.