How can we compute the daily drop in gross basis?

Background

The implied repo rate (IRR) is essentially the carry for going long basis (buying the deliverable bond and selling the futures contract). For this reason, it rises in value day-by-day as we approach expiry, which can be seen in its formula:

$$IRR=\Big(\frac{P_{\text{invoice}}}{P_{\text{bond}}}-1\Big)\Big(\frac{365}{d}\Big),$$

where $$P_{\text{invoice}}$$ is the invoice price of the bond, $$P_{\text{bond}}$$ is the cash price of the bond, and $$d$$ is the number of days left to delivery.

Correspondingly, we see a daily drop in net basis, until it reaches roughly zero at delivery; which means there is also a daily drop in gross basis. These are calculated like so:

$$b_{\text{gross}}=P_{\text{bond}}-(CF\times P_{\text{futures}})$$

$$b_{\text{net}}=F_{\text{bond}}-(CF\times P_{\text{futures}})$$

where $$CF$$ is the conversion factor, and $$P_{\text{futures}}$$ is the market price of the futures contract, and $$F_{\text{bond}}$$ is the forward price of the bond.

In trading, it is important to be aware that tradable prices (such as the value of gross basis) will be lower at the open than they were at the close. This is particularly important on days like a Friday, when the deal date moves from $$T+1$$ to $$T+3$$, creating a more pronounced rise in IRR, and hence a more pronounced drop in gross basis. This is observed every day in the market.

Problem

How can we calculate the expected drop in gross basis?

For IRR, it is easy to compute the daily rise in gross basis. Just change the value of $$d$$ in the formula above to $$d-1$$, and this will give you the expected daily rise in IRR.

It is not so clear how to do this for gross and net basis. Obviously their drops are in line with the rise in IRR, but I'm unsure of how to explicitly calculate an expected drop.

Is there some other formula that relates basis to IRR?

Thanks.

The expected change of basis over time will be equal to the change of $$P_{bond}$$ over time; this is because he change of $$P_{futures}$$ (as market estimate of delivery price) is expected to be zero (it is not subject to drift, all else being equal).

For determining the change of $$P_{bond}$$, you would solve the first formula for $$P_{bond}$$, and besides changing $$d$$, be mindful to take the current value $$IRR$$, taking into account the term structure of repo rates, i.e. $$\mathit{IRR}=\mathit{IRR}(d)$$.

• Thanks. So you would be calculating the expected change in $P_{\text{bond}}$ by entering values for IRR (according to term structure) into the formula, and by reducing $d$ by one day, but then what about $P_{\text{invoice}}$? Would you just use the value of the previous day's close? The problem with that would be that surely an expected change in $P_{\text{bond}}$ would be accompanied by a change in the invoice price, so you couldn't use one to calculate the other... – quanty Feb 17 at 22:08
• I would assume that all else equal, the market will still have the same futures price at T+1. If you are saying the futures price would drift the same way as the bond price, then the basis would never converge to zero, which in fact it does. – ZRH Feb 17 at 22:59
• That's not what I meant: According to your method, we calculate the expected drop in the bond price by using a value for IRR (unchanged from the day before), but we know that IRR increases day-by-day. So how is this accounted for? – quanty Feb 19 at 6:36

The change in the gross basis would simply be due to carry. You know the spot price of the bond. If you lock in term repo to the forward date, you will know your forward price. The difference between forward and spot price is your carry. Convert the carry to a daily value and this reflects the daily drop in gross basis. Put differently, if we assume last delivery date then gross basis = net basis + carry. The gross basis should converge to net basis by the last delivery date because carry should be 0.

As commented by ZRH, the first step is to consider what the drop in $$P_{\text{bond}}$$ is, since this is the only changing value in the formula for gross basis:

$$b_{\text{gross}} = P_{\text{bond}}-(CF\times P_{\text{future}})$$

$$\implies \Delta b_{\text{gross}} = P_{\text{bond}}^{(1)}-(CF\times P_{\text{future}})-(P_{\text{bond}}^{(2)}-(CF\times P_{\text{future}}))=P_{\text{bond}}^{(1)}-P_{\text{bond}}^{(2)}=\Delta P_{\text{bond}}.$$

The reason the change in the futures price is not considered is this: The futures price is defined as the market expectation of the delivery price. Thus, the market does not expect any change in the futures price into delivery.

The appropriate way to compute the expected drop in the price of the bond is to consider the carry of holding the bond. This successfully captures the repo on the bond (which in turn captures the involvement of $$IRR$$, essential to considerations of basis). In order to do this, simply compute the forward price of the bond going into the next day, and take this away from the current price. This will give a drop in the bond price.