# Bond Futures why CTD driven by yields?

Can you please explain with a numerical example why long duration bonds (low coupon, long maturity) are CTD when yields are significantly greater than the contract standard coupon and when yields fall below the contract standard coupon short-duration bonds (high coupon, short maturity) become the CTD?

The price of a bond future is related to underlying bonds in the deliverable basket by means of the net basis equation. The net basis is a simple calculation, it is the forward price of the bond at delivery minus the invoice price of the deliverable bond determined from the futures price:

$$NB_i = P_{i,t} - I_i = \underbrace{P_{i,t}}_{\text{Forward price}} - (\underbrace{P_f}_{\text{Future's price}} \times \underbrace{cf_i}_{\text{Conversion factor}})$$

The CTD bond is that with the lowest net basis, i.e. relative to the current market future's price, the bond whose combination of forward price and conversion factor give the lowest net basis.

What happens when yields rise

Notice that the conversion factor for every bond is a static value. For each bond every tick in the future's price lower will increase the value of the part in brackets by a consistent amount. However, as yields rise the forward price of each bond also lowers. Bonds with higher durations experience a greater fall in price for the same change in yield so comparatively their net basis will typically fall faster than bonds with shorter durations.

What is the CTD cross over

When yields have risen sufficiently much that the net basis for a bond with higher duration has fallen enough to match the current CTD there will be a cross over point. At this point the future will have a new CTD and its behaviour (i.e. the duration on the future, i.e. its price relationship in relation to yields) will begin to more closely align with the new CTD.

The below chart shows a real image of the typical behaviour of the duration of a bond future and you can see that as yields rise the DV01 of the future tends to step up (as the CTD switches to a bond with higher duration)

From a quantitative perspective this demonstrates that bond futures with multiple CTD possibilities experience negative gamma - as yields rise their DV01s can rise, or at least not decline as they switch bonds. Whilst the corresponding DV01 on bonds will always decrease as yields rise. This means that a portfolio of "selling futures and buying cash bonds" is a favourable portfolio to own because you then own gamma (as opposed to the opposite). This type of portfolio is therefore in demand and the net basis for such a future will typically trade at premium (i.e. the net basis may be well above zero). How much of a premium depends on many factors such as the volatility of the market, the current market level and the correlation of the bonds which might become CTD.

You can see the code that generated this image in Python's rateslib cookbook documentation

• FWIW at the present time, in the US market, the CTD's tend to be bonds of short maturity and the Net Basis for these CTD tends to be negative. Commented Aug 2 at 10:24
• thanks very useful! Commented Aug 2 at 15:15
• Is this an arb if I can term fund to first delivery with negative net basis ? (Assuming away any complexity with delivery/boxing etc). @nbbo2 Commented Aug 2 at 17:28
• Yes it is an arb. In 2007 the 10y gilt basis was about -15cents. We arbed it to the tune of about 250mm notional or 400k GBP profit, and realised it. You need to be very careful to get the exact number of contracts and repo level to get it all to work. Wouldn't do it today though (in a bank anyway). It consumes too much balance sheet which may come with capital costs. Not sure about the rules for funds.
– Attack68
Commented Aug 2 at 17:57
• You also need to manage the EDSP since the the number of contracts you hold throughout the trade will not be the same as the number of contracts you want to deliver, so as the futures goes into the final EDSP settlement process you may need to buy/sell additional contracts. This is a transaction risk which may eat into the definition of "arb".
– Attack68
Commented Aug 2 at 18:00