# Determine the carry of a treasury bond futures contract?

Hi fellow financial market enthusiasts. I'm trying to understand my options as a retail investor. I want to leverage a cash bond portfolio but my broker does not allow that, so I want to use futures instead. How can I determine the carry of a treasury bond futures?

I can read from other posts that you need to do [CTD Yield] - [Implied Repo Rate].

So in our case right now on the ZF contract, we do 2.60 - 0.65 in order to get around 1.95% yearly carry if I buy a contract?

See attached snapshot of the CME treasury analytics page.

Can you help me understand this please. It doesn't need to be exact but at least a ballpark yield percent with .05 margin of error is fine.

Update: Thanks for the answers, really appreciate the help. Theoretically, as pointed out by Daniel below: because taking delivery only takes place in 3 months time the position would not accrue any income till futures delivery date.

Still, it doesn't mean the futures doesn't have a positive carry. This paper from CME illustrate my point for S&P500 futures: http://www.cmegroup.com/education/files/S-and-P-500-Implied-Financing.pdf - refer to "Positive and Negative Carry" section.

I assume same is valid for treasury futures: short-term interest rates being cheaper than the yield on the underlying bonds, resulting in a positive carry.

Update2: To further illustrate my point, I went through a rudimentary estimation exercise. According to my logic, measuring the difference in price of the ZF contract (mar18) when the yield is the same should give us a good approximation of the carry.

I'm looking for a more convenient way of doing this calculation.

Based on the your comments, I believe the issue lies with what you consider to be "carry." The reality is that there's no consensus. So let's take mini steps.

We'll start with what rates guys consider as "pure carry." In this most classical and fairly strict definition, carry is the deterministic component of expected returns – you know exactly what it is before you enter the trade. It is also very tangible, involving clear cash inflows/outflows. In this sense, @Daniel's answer is 100% correct as is: forwards, be it a forward bond position or even a forward starting swap, have no carry (no "pure carry" anyways).

There are several ways to think through this. First of all, recall that the futures price (excluding the embedded switch options, forward/futures difference, and other technicalities) is \begin{align*} \text{Forward Price} &= \text{Spot Price} - (\text{Coupon Income} - \text{Financing Cost}) \\ &= \text{Spot Price} - \text{Pure Carry}.\end{align*} This simple formula (and its equivalents) applies to all forward contracts. The futures price literally is the net result after the carry of the underlying has been removed.

Secondly, there is no tangible cashflows of any kind. By using a futures contract, you forgo coupon income from the underlying; nor do you pay a financing cost. Those have already been factored into the futures pricing.

Thirdly, as @Daniel has pointed out, carry basically provides you with a cushion – if carry is positive for a bond, yields can rise a little bit (by an amount equal to the difference between forward yield and spot yield) before you start losing money. For a futures contract, there's no such cushion at all. Yields start rising, you start losing money, because there's no coupon income to mitigate the capital losses.

This is not to say that there's no expected returns when you hold a futures contract and the world is static – now we're expanding the scope of the word carry. For clarify, I'll refer to this definition as "broad carry" – expected returns of an instrument when the world remains unchanged. When you allow for this broader definition, many things start to count.

For example, as you have pointed out, futures converge toward spot (by an amount equal to the underlying's pure carry). You can consider this to be a form of carry (I do!). Why isn't this "pure carry" though? Because it's neither tangible nor deterministic. It's not tangible because there are no real cashflows. It's not deterministic because bond futures allow for delivery one week after trading stops, so the classical futures/cash convergence may never happen.

Going further, bonds have expected rolldown returns that will flow through to futures – that could also count as a form of carry (some people do, others treat it as a separate concept). Bond futures also have an embedded delivery option, which can have time decay just like any other option. Bond futures may be mispriced relative to cash bonds, creating another source of convergence (toward fair value).

Anyways, I agree with @Daniel that strictly speaking, the carry of futures of zero. But if you're trying to think through what your expected returns might be if the world is unchanged, it's a much broader/messy definition. Depending how you trade and how you hedge, you are free to make discretionary decisions on what you want to count toward this "broad carry." It is probably best to list them out separately, so that you have a better idea about how reliable each component may be.

• Thank you Helin. I'm accepting your answer for lack of a better way to calculate futures convergence to spot price. I will continue to research the subject if anyone has additional information please share. – Yannick Feb 11 '18 at 14:19
• Hi @Yannick. Could you clarify? I actually thought you understood convergence very well – it's pretty much just the underlying's carry (i.e., coupon income minus cost of financing). – Helin Feb 11 '18 at 23:32
• So, what should I account for as the cost of financing? Just today's USD repo rate? We have simply: (5y yield rate) - (repo rate) ~ (implied carry)? The margin required to buy the futures contract is negligible? Thanks again. – Yannick Feb 12 '18 at 1:42
• 1/ This is what I was attempting to convey – in the case of bond futures, the convergence is not deterministic and it depends on how you think about things/how you trade. It's common to use the term repo rate corresponding to the cheapest to deliver (usually termed to last delivery date if carry is positive, or first delivery date if carry is negative), so you get back the pure carry equivalent; this ignores the switch option and assumes the future's rich/cheapness relative to cash will be unchanged. – Helin Feb 12 '18 at 2:03
• 2/ Another popular way is to use the CTD's implied repo rate. If you do this, you're counting the richness/cheapness of the futures toward convergence. Both methods are popular and neither is ideal. I personally prefer to separate things out so that it's easier to see all the sources of expected returns. Beware that if the cheapest to deliver changes, all of these calculations become completely meaningless. – Helin Feb 12 '18 at 2:06

If you think about carry as a cushion against a change in the forward yield then carry (in basis points) for the underlying bond equals with (coupon income of the bond - repo rate) / forward DV01 of the bond. (Carry could be also calculated as the forward yield - spot yield) That is how much the forward yield can rise before you start loosing money on the financed bond position.

We can apply the same logic for a bond futures contracts as well. Being long in a treasury futures means you are buying a treasury bond in 3 months time which is priced by the forward yield curve to get the present value (~or futures price) of the bond at the futures delivery date. Financing a futures contract is virtually zero and because taking delivery only takes place in 3 months time your position would not accrue any income till futures delivery date. Therefore your net carry is zero and your cushion against any changes in the forward YTM of the bond is zero.

• So according to your answer there is zero net carry? Hard to believe... this would mean going long 1 ZF contract is taking the same risks as going long 1 cash bond without any reward whatsoever? – Yannick Feb 3 '18 at 23:01
• But you are also not putting up any money (except for a small margin deposit on the future, USD 1080 per contract), unlike when you buy the cash CTD bond. A bond future is just a bet on the price of bonds, it is not a funded investment. – Alex C Feb 4 '18 at 0:14
• Just think about it, why would anybody hold the underlying if someone can get the same reward by buying a futures contract? – Daniel Feb 4 '18 at 8:40
• Of course I would not expect to have the same reward but the underlying "yield - financing cost" seemed logical to me. As a matter of fact, net carry on the ES future is positive and is equal ball park to dividend distribution - cost of financing the underlying. – Yannick Feb 4 '18 at 12:42
• You mix up two different things, the carry of the underlying vs carry of the futures. What you talking about is the carry of the underlying, not the carry of the futures contract. By buying the futures you forgo the income return of the underlying (that is net carry), and you only get the price return of the underlying. – Daniel Feb 4 '18 at 14:45