Futures
18.3 Attribution on bond futures
18.4 Futures contracts on other fixed income securities
18.5 Heuristics for dealing with futures
18.1 INTRODUCTION
Futures contracts on bonds, or bond futures, are not complex, but tend to cause more difficulties in attribution analysis than any other asset class. The reasons include:
- their often substantial effective weights in managed funds, meaning that their contributions to return and residuals are typically much greater than individual stocks;
- the cheapest-to-deliver feature of bond futures contracts;
- ways in which optionality and noise in the price of the futures contract can distort attribution results.
For these reasons, any attribution scheme must treat futures on fixed income securities with especial care – and even then futures often generate substantial residuals. At the end of this chapter, I suggest some pragmatic ways to help treat this asset class.
18.2 HOW FUTURES WORK
A bond futures contract replicates the behaviour of a physical bond, including contribution to profit and loss, without the requirement to trade a physical security. Bond futures are designed to form a highly liquid market, allowing positions to be set up and unwound quickly and cheaply. The future acts as a proxy for the underlying bond by delivering the same exposure with very small transaction costs.
In addition, a short position can be taken in a bond future relatively easily. This is much harder to transact with a physical bond, which typically requires trading in the repo market or buying a put option.
Like futures on currencies or commodities, a bond futures contract promises delivery of a bond at a certain date in the future.1 Bond futures trade in the form of standardised contracts, with delivery in March, June, September and December.
Unlike the commodities markets, however, the owner of a bond futures contract does not know in advance precisely what bond they will receive at maturity. Rather than a single known bond, the seller of the contract can deliver one of a number of bonds from a predefined pool. For instance, at the time of writing the UK Long Gilt contract is based on a bond with maturity between 8.75 and 13 years, and a nominal coupon of 6% (the nominal bond). This means that if you own a long gilt contract and hold it to maturity, you will receive a bond with a maturity between these limits. The coupon will probably not be 6%, but the yield will be adjusted to the yield of the nominal bond, using a published quantity called the conversion factor. The price of the bonds in the pool will vary, so the seller of the contract will usually deliver the cheapest security, known as the cheapest to deliver (CTD).
There are several good reasons why it makes sense to make delivery from a pool of bonds, rather than a single instrument.
One is that the availability of any individual bond issue is limited. If a futures contract was based on one single bond, and liquidity became poor, then liquidity in the futures contract would become poor too.
Another is to avoid the possibility of a squeeze, where a speculator buys a large quantity of both the futures contract and the underlying bond. As the delivery date for the futures contract approaches, other market participants who are short the contract would need to buy them back at higher prices. The threat of such a situation would discourage traders from taking short positions and drive down liquidity.
However, the use of the CTD bond does imply some extra complexity for the attribution analyst. One of the most important is that the CTD can change during the lifetime of the futures contract. I cover this case in the following section.
18.2.1 Contracts
Many countries have active bond futures exchanges on which standardised bond and bill contracts are traded.
Bond futures contracts typically trade using standardised expiration or maturity dates. The maturity month is denoted using the codes H, M, U, Z, with the year following the month. For instance, the code for the US 30-year T-Bond contract maturing in December 2013 is USZ3.
If a fund wants to keep a futures position beyond the maturity date of the contract, the manager will typically sell the existing contracts a few weeks before they mature and buy contracts for the following contract. This is called rolling over.
Do not confuse the maturity date of the contract with the maturity date of the underlying bond. These dates will be completely different. In particular, the maturity date of the contract is independent of the interest-rate sensitivity of the future.
Exchanges often introduce new contracts if market research shows sufficient market appetite. The coupon of contracts can also change according to market conditions. For instance, during October 2012 the notional coupon on the UK long gilt was revised downwards from 6% to 4% for the March 2012 and subsequent contracts, reflecting the lower average coupons of bonds on issue.
18.2.2 The theoretical price of a future
At any given time, a bond future is associated with an underlying CTD bond, which is the bond that will be delivered at the maturity of the contract.
For attribution purposes it is necessary to know the details of this bond, which has an associated conversion factor by which the price of the contract is multiplied to ensure that its yield to maturity on the delivery day of the contract equals to the notional coupon of the contract. However, for reasons shown subsequently, the conversion factor is not required when running attribution. The CTD is available from sources such as Bloomberg (page DLV) or futures exchanges.
The theoretical price of a bond future is closely related to that of its CTD. Broadly speaking, this is the price of the physical bond (including accrued interest), plus financing costs, minus any coupon and reinvestment income accrued received during the life of the contract. More precisely,
where
- Pfuture is the futures price;
- Pbond is the (dirty) bond price, including accrued interest;
- r is the repo rate applicable to the bond future;
- t is the fraction of a year to the maturity of the contract;
- Ci is the ith coupon;
- N is the number of coupons paid between the current date and the maturity date of the contract;
- t(i,del) is the fraction of a year between coupon i and the maturity date of the contract;
- AIdel is the accrued interest payable on the bond on the maturity date of the contract;
- CF is the bond conversion factor.
This is a rather formidable formula, so here is a worked example.
Example
On 18 January 2013, the CTD bond for the 30-year Government of Canada bond future (LGB) was the Canadian 5% 1 June 2037. The clean price of this CTD is $109.31 per $100 face value, and the conversion factor is 1.1691. What was the theoretical price of the June future?2
On January 18, the previous coupon date was 1 December. There are therefore 48 days of accrued interest due on a semi-annual coupon payment of 2.5%, so on a nominal face value of $100,000, the CTD bond costs
$100, 000 × (109.31/100) + 48/183 × 0.025) = $109, 965
This is known as the dirty price of the bond.
There are 164 days from settlement to the futures maturity date. The financing costs of holding this position to the futures maturity date using the repo rate are
$109, 965 × 0.0495 × 164/365 = $2446
The income received during the holdings period includes a coupon paid 29 days before the future matures, together with interest paid on that coupon at the repo rate:
$2500 + ($2500 × 0.0495 × 29/365) = $2510
The total costs of the bond position are therefore
$109, 965 + $2446 − $2510 = $109, 901
Subtract the accrued interest generated on the bond between next coupon (1 June) and futures delivery gives
$109, 901 − ($100, 000 × 0.05 × 29/365) = $109, 591
Divide the result by the conversion factor of 1.1691 (which is constant over the future’s lifetime, and is typically published on the exchange website) to get the final theoretical price of $93,739, or $93.739 per $100 face value.
The market price of the futures contract on this date was $93.74 per $100 face value, so there is a very small arbitrage opportunity.
More generally, to find the theoretical price of a bond future:
- start with the dirty (including accrued interest) price p of the underlying CTD bond;
- calculate the financing costs of the bond
f = p × r × δT
where r is the repo rate, and δT is the fraction of a year to run before the expiration of the futures contract; - calculate the income i of the bond during the contract’s lifetime. i is the accrued interest owing on the bond at expiration of the contract, times the repo rate;
- if a coupon is to be paid between now and the expiration of the futures contract, add this to the financing costs.
Since there is an explicit relationship between the theoretical and the market price of the future, it is, in principle, possible to arbitrage this relationship. Traders who buy a bond and simultaneously sell the futures contract, and close out positions at the expiry of the contract, are engaging in cash and carry trading. The difference between the cash price and the futures price is called the basis, and traders who arbitrage price differences between the price of the CTD and the future are basis trading.
18.2.3 Data required to price a bond futures contract
In addition to the usual quantities required to price a vanilla CTD bond, a futures contract also requires levels of the repo rate and the bond conversion factor:
- The repo rate is relatively stable during the contract’s lifetime and may be approximated as a constant, or as the risk-free rate.
- The bond conversion factor is known at the time the contract is issued and remains constant throughout the lifetime of the contract.
The conversion factor simply rescales all futures prices by a constant amount. If we wish to measure the return of the future rather than its absolute price, it is not necessary to source this quantity, as it will not affect the attribution returns.
More explicitly, if the price of the future at time t is CF × Pt, then the return of the future over the interval [t, t + 1] will be
In other words, the conversion factor drops out and is not needed.
18.2.4 Changes in the CTD
If there are large changes in the level of the yield curve during the lifetime of a contract, the CTD bond can change. In very active markets, the CTD can change several times in a day.
While this can cause difficulties for arbitrage traders, changes in the CTD seldom present much of a problem for attribution analysts. As long as the supporting software allows effective dating capabilities for security modelling, a change in the CTD simply involves redefining the structural parameters of the underlying bond (coupon, maturity).
Some commercial attribution systems require the user to maintain all bonds in the delivery bucket for each futures contract held, together with the data required to price these securities, in order to ensure that every bond future is linked to its correct CTD.
In my experience, this is a great deal of work. Although there may be cases where the CTD changes during the lifetime of a contract, this only happens relatively rarely, and the user may prefer to monitor such occurrences manually.
18.3 ATTRIBUTION ON BOND FUTURES
There are several ways in which a bond future can be treated in an attribution scheme.
A first-principles approach will use formula (18.2) for a futures price. In general terms, one might reprice the future under a range of yield curve scenarios, and calculate the resulting returns.
An alternative route is to consider the theoretical price F of a bond futures contract
where
- S is the spot price of the bond;
- PV is the present value of the coupons payable by the bond during the life of the futures contract;
- r is the risk-free rate at the futures maturity;
- t is the life, in years, of the futures contract.
Assuming that there are no coupons payable between the present time and the maturity of the contract, we can calculate the return of the futures contract as follows.
Denote the price of the futures contract Ft at time t as
and
Then the return of the futures contract RF is given by
Assuming S0 ≈ S1, rt << 1 gives
Ignoring convexity, the perturbational equation for the return R of a bond is
Using this expression in (18.8) gives
where
- RF is the bond future’s return;
- y is the yield to maturity of the CTD;
- r is the risk-free rate;
- δt is the elapsed time;
- MD is the modified duration of the CTD;
- δy is the change in the yield to maturity of the CTD.
This is the same expression for the return of a vanilla bond, with the single difference that the carry return term has been modified by subtracting the risk-free rate. The holder of a bond future should expect their carry return to be lessened because they are in effect borrowing the cost of the underlying security in order to have it delivered at the contract expiration date. The reduction in carry return reflects this borrowing cost. Therefore, (18.10) is the perturbational equation for a bond future, and should be used in place of (18.9) for all assets of this type.3
18.3.1 Sensitivity to risk numbers
One difficulty with the perturbational approach to attribution on futures is that the results rely heavily on the accuracy of the supplied risk numbers.
Consider a bond future with a modified duration of six years, a yield to maturity of 1% at the start and end of a 1-day period, and a zero repo rate. Over this day the return of the future will be
There is no return due to duration as yields are unchanged.
Now suppose that the yield to maturity at the end of the interval is incorrectly quoted as 1.01%, or 1 basis point above the true value. In this case the future return will be
or nearly 6 basis points away from the true value. Given that long-dated futures of this type often have a large effective exposure in many portfolios, one should take especial care to ensure yields for this asset class are as accurate as possible. For instance, if the future has 20% effective exposure in a portfolio and a yield 10 bp away from the true value, it will contribute 5.7 × 10 × 0.2 = 11.5 basis points to the portfolio’s residual, which is substantial and can in some cases exceed the returns from all other effects combined.
18.3.2 Noise in the attribution calculation
Given that the relationship between the futures price and the CTD is so close, it is worth asking how any noise (or basis) can appear at all. Causes of residual in a futures attribution calculation include:
- noise in the attribution calculation for the CTD bond;
- optionality implied by the choice of which contract to deliver;
- US traders can choose when in the month they want to deliver the underlying bond to the contract owner.
These factors can (and do) affect the pricing levels of a bond future. However, the details are highly technical and make little difference to the overall report. For more information, see Burghardt et al., 2005.
The net result is that attribution residuals frequently occur on bond futures, even with completely accurate data on the current CTD bond. They are largely unavoidable, and the attribution analyst should be able to explain why they arise.
18.3.3 Other pitfalls
Some institutions hedge their portfolios using a probability-weighted price for bond futures. Since the CTD is not known in advance, the probability of the various deliverable bonds becoming the CTD at the contract’s maturity is calculated and a weighted price calculated, along with probability weighted analytics (yield, modified duration, convexity).
This yield is not suitable for attribution, since it does not measure the yield of the cheapest to deliver but is the yield of a basket of securities that might become the CTD. The CTD’s yield and maturity should always be used for attribution.
18.4 FUTURES CONTRACTS ON OTHER FIXED INCOME SECURITIES
While bonds are the most common underlying instrument for fixed income futures, futures contracts are also available on a range of other security types. These include Eurodollars, bank bills and interest rate swaps.
Interest rate futures are a particularly simple asset class. The price P of a short sterling contract is given by
where y is the interest rate at expiry at the expiration date of the contract. For instance, if a short sterling future is trading at a price of 94.880, this implies the 3-month LIBOR rate will be 5.12% at the maturity date of the future.
In an attribution framework, these securities may be modelled as follows.
Since δP = −δy, we have
where r is the return of the future and δy is the future’s change in yield.
Note that there is no carry term, and that no use is made of modified duration. Interest rate futures should therefore be treated differently to conventional securities in an attribution framework, as the usual perturbational equation does not apply.
Since the calculation is exact, residuals should never be generated for this asset class.
The future’s yield should be read from the appropriate part of the yield curve. For instance, since the above security is priced off a 3-month LIBOR rate, yields should be read at the 3-month point (note that this will require conversion from zero rate to forward rate, since Eurodollar yields are actually forward yields).
18.5 HEURISTICS FOR DEALING WITH FUTURES
Given that a future is designed to behave like an idealised bond, it should be possible to treat it in the same way.
If bond futures are generating large attribution residuals, one possible solution is to assign this residual to ‘duration’, because curve movements are what drive most of the future’s return. The client is happy because futures residuals automatically drop to zero, and the attribution report now reflects the underlying investment strategy without being polluted by bad analytics data.
1 The exception are the Australian and New Zealand markets, where the cash equivalent of the bond is delivered; see Appendix C.
2 This example is from the brochure ‘30-year Government of Canada Bond Futures’, published by the Bourse de Montréal, www.m-x.ca.
3 Note that Australian and New Zealand bond futures do not generate any carry return. This case is described in more detail in Appendix C.