# Can the carry of bond future be approximated by conventional yield and implied repo?

I have a question regarding bond futures, carry and convenience yield $$y$$. Suppose we look at the cheapest to deliver bond for a bond future. Suppose the CTD has a conventional yield of $$-0.72$$ and the implied repo rate $$r$$ is stated as $$-0.57$$.

As the conventional yield of the bond is related to the return of holding it until maturity and the implied repo is the return of being long the cash bond and short the future, can we say that the carry of being only short the future is

$$r-y=-0.57+0.72=0.15$$

or is my reasoning wrong?

I don't think you can make a such a statement.

As a simplification let's assume that there is only one bond eligible for delivery into the futures (hence the CTD does not change, and the short's switch option value is 0).

1. Carry of a long basis trade (i.e., long cash bond and short futures now, deliver bond into the futures at delivery) is decomposed into $$\text{Carry} = \text{Interest income} - \text{Financing/Repo cost}$$ which, in an absence of arbitrage, is equal to the basis of the bond (since there is no delivery option value): $$\text{Carry} = \text{Bond basis} = \text{Bond clean price} - \text{Futures price} \times \text{conversion factor}.$$
2. Implied repo rate (IRR) is the return of this long basis trade, excluding financing cost, i.e., $$\text{IRR} = \frac{\text{Invoice price} - \text{Purchase price}}{\text{Purchase price}} \times \frac{360}{n},$$ where $$n$$ is the length of the trade in days, and the invoice price is the price you would receive when delivering the cash bond into the futures: $$\text{Invoice price} = \text{Futures price}\times \text{conversion factor} + \text{Bond interest accrual at delivery}.$$ and Purchase price is the price you pay for the bond now: $$\text{Purchase price} = \text{Bond clean price} + \text{Interest accrual now}.$$

As you can see carry is a dollar amount of the trade, while IRR is a rate, therefore you cannot get one from another by simple addition/subtraction.

In fact, if you play with the two relations above, you should arrive at $$\text{IRR} = \frac{\text{Interest income}-\text{Carry}}{\text{Purchase price}} \times \frac{360}{n},$$ which is equal to the financing/repo rate absent of arbitrage (as the name IRR implies).

Both Carry and IRR are affected by the bond price, and in turn, its yield. But there is no simple relationship of the kind you suggested.

Edit:

If carry is defined as a rate: $$\frac{\text{Interest Income} - \text{Financing/Repo cost}}{\text{Purchase price}} \times \frac{360}{n},$$ and "yield" refers to the rate at which interest is earned, as a percentage of dirty price (note that this is different from YTM or current yield as conventionally defined): $$\text{yield''} = \frac{\text{Interest Income}}{\text{Purchase price}} \times \frac{360}{n},$$ then indeed we have $$\text{carry} = \text{yield"} - \text{financing/repo rate}.$$ But I don't think I have seem "yield" defined in this way.

Like many others I would recommend The Treasury Bond Basis for understanding the treasury futures products.

• Thanks for your answer. I'm not entirely sure if the statement "carry is a dollar amount" is correct. For example in papers.ssrn.com/sol3/papers.cfm?abstract_id=2298565 they define carry for bonds as a rate of return and this is also how market participants usually quote carry, e.g. 3 month carry of being long 7y BTP's is 6bps. Or am I missing here a point? Jul 4 at 9:54
• also in your definition of carry, interest income - financing cost. Both quantities could be specified as a rate Jul 4 at 9:56
• Hi @swissy, this is a good point. I went with The Treasury Bond Basis which refers to carry as a dollar amount. I've edited the answer with carry expressed as a rate, which hopefully answers the original question.
– GZ-
Jul 6 at 16:59