13.1.1 UK Long Gilt Futures Contract

Details of the UK Long Gilt Future (on NYSE-EURONEXT) are given in Table 13.1. The bond deliverable in the contract is a ‘notional bond’ with a 4% coupon (with a maturity between 8.75 and 13 years). Surprisingly, this notional 4% bond does not actually exist! However, as we shall see, it provides a benchmark from which to calculate the price of possible bonds for delivery (which do exist).

The seller of the futures contract can decide the exact delivery date (within the delivery month) and exactly which bond she will deliver (from a limited set, designated by the exchange). In fact, the seller will select a bond which is known as the ‘cheapest-to-deliver’ (CTD). The CTD bond can be shown to depend on the ‘conversion factor’ (CF) for each possible deliverable bond. (These concepts are explained below.)

The futures contract size is for delivery of (face value) T-bonds and futures price quotes are expressed per £100 nominal (of deliverable T-bonds). The tick size is £0.01 per £100 nominal (e.g. 1-tick is a move from £95 to £95.01) – hence the tick value is £10 per contract .

13.1.2 US ‘Classic’ and ‘Ultra’ T-bond Futures Contracts

The principles underlying these futures contracts (traded on the CBOT) are very similar to those for the UK gilt-futures contract, except price quotes are in 1/32nd of 1% (see Table 13.2). Expiration months are March, June, September, and December. The last trading day is the business day prior to the last 7 days of the expiry month. The first delivery day is the 1st business day of the delivery month but delivery can take place on any business day in the delivery month.

Note that the notional bond deliverable in the futures contract is assumed to be a 6%-coupon bond. However, in practice the person with a ‘short’ futures position can choose from around 30 different eligible bonds to deliver (which have coupons different from 6% and the CF adjusts the delivery price to reflect the type of bonds actually delivered).

For the ‘classic’ futures contract the T-bonds delivered must have at least 15 years to maturity² and less than 25 years to maturity (from the first day of the delivery month).³ There are also futures on 2, 5 and 10-year T-notes, which only differ from the T-bond futures contract in the maturity of the T-notes which can be delivered in the contract. As with most futures contracts, delivery rarely takes place but it is the possibility of delivery and hence arbitrage profits which keeps the T-bond futures price in line with the spot/cash market price of T-bonds.

13.2.1 Conversion Factor (CF)

This section assumes the reader is largely familiar with the pricing of bonds in the spot market and the conventions used in calculating accrued interest (see Cuthbertson and Nitzsche 2008). The CF adjusts the price of the actual bond to be delivered, by assuming it has a 6% yield, which makes it equivalent to the notional 6% bond in the futures contract.

When computing the CF the maturity of the underlying bond is defined as the maturity on the first day of the delivery month. For example, if we assume the actual delivery date is 11 September 2018 and the underlying bond matures on 15 February 2038, then the maturity period used in the calculation of CF is 1 September (i.e. not 11 September) to 15 February 2038.

The CF is best understood using a specific example. To simplify matters we assume the counterparty who is short the futures contract does not yet hold a bond for delivery, so we are going to calculate the CF and CTD bond at the maturity date of the futures contract (=T).

Consider an actual 20-year US T-bond, paying semi-annual coupons of 8% (i.e. $4 per 6 months), with (6-month) periods to maturity (see Example 13.3). If the yield to maturity on this 8%-coupon bond is assumed to be 6% p.a. then its ‘fair’ or ‘theoretical’ price would be $123.1 (per $100 nominal) which gives a conversion factor . In essence the deliverable bond (with an 8% coupon) is worth 1.231 times as much as the notional 6% coupon bond (trading at par and hence with ).

The conversion factor adjusts the price of the actual bond to be delivered, relative to the notional 6% coupon bond in the futures contract:

The conversion factor will differ for bonds with different coupon payments and different maturities. The CF for any specific deliverable bond will change over time simply because the maturity date of the deliverable bond gets closer (although it will always exceed 15 years, otherwise it will cease to be an eligible bond). In calculating the CF the CBOT assumes the yield curve is flat at 6%. But in practice this is rarely (if ever) the case. This means that the ‘true price’ of the deliverable bond, which should be priced using spot rates (see Cuthbertson and Nitzsche 2008) will not equal that used in the calculation of the CF. Hence, there will usually be eligible bonds for delivery that are actually cheaper than the ‘cheapest-to-deliver’ bond.

Let us now consider the cash amount received by a trader (Ms Short) who is short T-bond futures, when she delivers the underlying bond at maturity . Let be the futures settlement price (on the ‘position day’ – see below). When Ms Short delivers the 8%-coupon, 20 year bond she will receive:

Ms Short's receipts at settlement

Note that is the ‘settlement price’ at (and not the price initially agreed at the outset of the futures contract). is the accrued interest at and is a fraction of the next coupon payment on the bond delivered against the futures contract. For example, if the maturity date of the futures is 11 September 2017 and the deliverable bond matures on 15 February 2038, then the deliverable bond has semi-annual coupons each year on 15 February and 15 August. Assume there are 184 days between 15 August and 15 February. The short therefore delivers a bond at T = 11 September 2017, which already has 27 days of accrued interest. Hence the long must pay the short for this loss of accrued interest of .

13.2.2 Cheapest-to-Deliver

Suppose it is now (= 11 September 2017) and there are three bonds designated by CBOT for actual delivery in September whose CF are 1.044, 1.033, and 1.065. In practice, the calculation of the CTD bond is quite complex but a rough idea of the CTD bond can be obtained by choosing that bond with the smallest raw basis:

(13.1)

where is the spot (‘clean’) price an eligible bond for delivery, is the settlement futures price and is the conversion factor of a deliverable bond.

Table 13.3 shows the calculation of the raw basis for three deliverable bonds. The final column indicates that Bond-C is the CTD. Although it is often the case that the bond actually delivered by the short is the CTD, nevertheless this is not always so since the seller may wish to preserve the duration of her own bond portfolio or provide a bond, other than the CTD, in order to minimise her tax bill. We now examine how Equation (13.1) for the raw basis arises. At settlement the short receives:

Deliverable bonds (maturity)	Spot price ($)	Conversion factor	Raw basis
1. Bond-A (2039)	112-4 (112.125)	1.044	0.156
2. Bond-B (2041)	112-8 (112.25)	1.033	1.461
3. Bond-C (2044)	114-8 (114.25)	1.065	0.029

Notes: F(September-futures) = 107-8 (107.25)

(13.2)

where is the accrued interest. The conversion factor adjusts the T-bond futures price for the fact that the deliverable bond has a coupon different from the notional 6% bond (stated in the T-bond futures contract). The 8%-coupon bond actually delivered would have a higher price when its coupons are discounted at the notional 6%. To purchase the actual deliverable bond in the cash market, the short will pay:

(13.3)

Hence, the net cost to the short of providing the deliverable 8%-coupon bond is just the raw basis defined above:

(13.4)

13.3.1 Portfolio Duration

For illustrative purposes assume a fund manager has two T-bonds in her portfolio, with market values in Bond-1 and in Bond-2, so that . The portfolio duration of her cash-market position is:

(13.7)

where , and is the duration of bond-i, for . Using portfolio duration to calculate assumes a parallel shift in the yield curve and small changes in yields (e.g. 25 bps) – otherwise the hedging error could be large.

13.3.2 Hedging a T-bond Portfolio

Consider a US pension fund manager (Ms Bond) on 1 May who wishes to hedge her portfolio of Corporate bonds (or T-bonds) with market value ⁴ and portfolio duration . Ms Bond fears a rise in interest rates over the next 3 months (i.e. 1 May to 1 August) and if this occurs the value of her cash-market bond portfolio will fall. The of the CTD bond is 1.12 (Table 13.4A). For a portfolio of corporate bonds this would be a cross-hedge, as the corporate bonds are hedged using T-bond futures.

Cash market – 1 May	Futures (September delivery) – 1 May
Market value of bond portfolio,	CF of CTD bond = 1.12 Size of one contract, z = $100,000 Futures Price, F0 = 103-16 ($103.5) V F = z(F0/100) = $103,500
Duration,	Duration (futures), Df = 18 Tick value 1/32 equals $31.25

Ms Bond, on 1 May, sells September T-bond futures contracts and assuming a parallel shift in the yield curve and no complications of accrued interest:

(13.8)

Suppose interest rates rise between 1 May and 1 August (Table 13.4B). Her cash-market bond portfolio falls by 5% to $1,045,000 – a capital loss of $55,000. But the futures price falls by 4 points from to and the gain on the short futures position is . The hedge shows a small loss of $3,000 on an initial cash-market value of about $1m (i.e. about 0.3%). The unhedged portfolio would have lost $55,000 (i.e. about 5.5%).

Cash market – 1 August	Futures (September delivery) – 1 August
Loss on bond portfolio = $1,100,000 – $1,045,000 = $55,000	September futures: ($99.50 per $100) Gain on 13 short futures = NFz(F0 – F1)/100 = 13($100,000)(4)/100 = $52,000 = 13 × 128 ticks × $31.25 = $52,000

13.4.1 Cross Hedge: Corporate Bond Portfolio

Suppose company-XYZ is going to raise funds by issuing corporate bonds in 6 months' time, after the legal details of the bond issue are completed. The treasurer of company-XYZ may be worried by the possibility of a rise in long-term interest rates over the next 6 months, which will raise the cost of issuing corporate bonds. Company-XYZ therefore hedges by shorting T-bond futures.

When hedging corporate bonds, it is important to estimate the relationship between (the change in the corporate bond yield) and (the yield on the deliverable bond in the T-bond future contract), since these may change by different amounts over the hedge period.

Regression techniques can be used but results may be subject to error, as changes in corporate ‘specific risk’ (e.g. IT failures, results of patent applications, environmental issues, default) will affect changes in the corporate bond yield – whereas changes in (government T-bond yields) should be largely unaffected by these ‘corporate risks’.

13.4.2 PVBP, Convexity, and Perturbation Analysis

Instead of using the duration-based hedge ratio we could obtain the same result using the ‘price value of a basis point’ (PVBP). This requires the calculation of the PVBP for the cash-market bond portfolio to be hedged and the PVBP for the US T-bond futures contract. This method usually uses the duration approximation for the change in T-bond prices and therefore assumes a parallel shift in the yield curve and small changes in interest rates.

Alternatively, we can calculate the PVBP of the cash-market (spot) bond portfolio using the ‘duration + convexity’ approximation (for a parallel shift in the spot yield curve of 1 bp):

where is the cash-market convexity, proportion of the total bond portfolio held in bond-i and .

Alternatively, we can use the ‘full’ (present value) pricing formula when considering the effect of a large change in interest rates on the ‘total’ change in the cash-market value of a T-bond portfolio – this will implicitly incorporate the convexity of the bonds. The ‘full valuation’ method can be done for either parallel or non-parallel shifts in spot yields – as spot rates might change by different amounts.

13.4.2.1 Non-parallel Shifts

We can also calculate the change in dollar value of one futures contract, using where the change is calculated with respect to changes in the spot yields which determine (a) the price of the deliverable bond and (b) the (future) value of any coupon payments . The number of futures in the hedge is then:

(13.9)

This is a form of perturbation analysis because we choose the size of the change in each spot interest rate along the yield curve and then calculate the impact on the numerator and denominator in Equation (13.9).

13.4.3 Hedging the Market Risk of an Underpriced Corporate Bond

In Chapter 6 we discussed how a hedge fund that buys underpriced stocks can hedge the market risk of the stocks by shorting stock index futures. Consider a similar situation where Ms Bond today, buys what she believes are underpriced corporate bonds (of several companies). She might believe the corporate bonds are underpriced because the rating agencies (and the average bond trader) have given the bond an A-rating but her assessment of their default risk would suggest a higher AA-rating.

She therefore expects a rise in the market price of the bond (even if risk-free government bond yields stay constant), as ‘market participants’ come to realise that the corporate bond is actually less risky than they first thought. Then she can close out her long corporate bond position at a profit.

However, Ms Bond is exposed to changes in long-term (risk-free) T-bond yields. If long-term risk-free yields increase, this will cause a fall in the cash market price of her corporate bonds, which may more than offset the gains she makes when the ‘credit risk underpricing’ is corrected. To hedge against future changes in T-bond yields, Ms Bond should short, T-bond futures, today:

(13.10)

where is the total dollar amount held in her portfolio of underpriced cash-market corporate bonds which have a portfolio duration of . If the risk-free T-bond rate increases, then the T-bond futures price will fall and the profit after closing out the futures will just compensate for any loss in value of the cash-market bond position – solely due to changes in risk-free T-bond yields.

Ms Bond can then capture any future rise in corporate bond prices due to the correction of the ‘credit risk’ mispricing. However, her portfolio of underpriced corporate bonds is still subject to any (residual) ‘specific risk’ (e.g. bankruptcy, regulatory changes, IT failures, reputational damage, etc.) of the bond issuers. The hedge using T-bond futures does not protect the ‘mispriced’ corporate bond portfolio from specific risks – but overall, specific risk may be small in a well-diversified corporate bond portfolio.

13.4.4 Risks in the Hedge

A hedged bond portfolio will not provide a perfect hedge because:

• the hedge period (e.g. 3 months from May to August) may not match the maturity date of the futures contract (e.g. September contract) – this gives rise to basis risk.
• calculating price changes in futures markets are subject to error, in part because it is difficult to ascertain the CTD bond (and hence its duration).
• shifts in the yield curve may not be parallel (there may be twists in the yield curve), so we cannot always assume duration or duration plus convexity provides a good approximation to changes in T-bond cash-market prices_.
• If we are hedging a corporate bond portfolio then the change in the value of the corporate bonds due to changes in risk-free T-bond yields can be hedged using T-bond futures but changes in corporate bond prices due to any change in (residual) specific/credit risk will not be hedged.

However, academic studies find that hedging using T-bond futures can reduce price risk, with the duration based hedge ratio performing particularly well. Hedging cash-market positions in corporate bonds using T-bond futures is also found to be effective, even though this is a cross-hedge and involves possible changes in credit risk.

13.7.1 Futures Price on Deliverable Zero-coupon Bond

Suppose today is time and the futures contract matures at , so the time to maturity is . The underlying asset to be delivered at maturity is a single zero-coupon bond. Today, we can borrow at and purchase the underlying bond for a price in the cash market and later deliver it against the short futures position. The usual risk free arbitrage argument ensures that:

(13.11a)

(13.11b)

(13.11c)

The implied repo rate () is the return from selling the futures at for and simultaneously buying the underlying bond at , so that (compound rate). In this case, risk-free arbitrage is possible if does not equal , the cost of financing the arbitrage strategy using the actual repo market.

13.7.2 Futures Price on Deliverable Coupon Paying Bond

In practice there is not a futures contract on a zero-coupon bond. As we saw in Chapter 3 if we ignore some important practical issues, the fair price (at ) of a T-bond futures contract using cash-and-carry arbitrage is:

where is the invoice price of the cash-market bond (i.e. the ‘clean price’ plus accrued interest), is the present value of any coupon payments (on the deliverable bond) between today and the maturity date of the futures contract and is the (continuously compounded) spot yield (for maturity ). Note that the T-bond futures price can also be written , where the future value at of the coupon payments is .

Unfortunately, in practice it is difficult to accurately calculate the fair T-bond futures price. This is because of the flexibility of the short's decision over the delivery date and the precise choice of the CTD bond and hence its invoice price .

First, note that the cost of creating a synthetic futures contract is the cost of buying the underlying cash-market bond at using borrowed funds, which accrues to a debt of at maturity (of the futures contract). The cost-of-carry is offset by the coupons received from the cash-market bond at times from today. When these coupons are reinvested at the (expected) risk-free rates (between and ) they have a future value at of:

(13.12)

The no-arbitrage futures price is then:

(13.13)

where is the interest rate over the period (). Considering the additional complexities of the conversion factor, accrued interest etc., the futures price for a contract which matures at (and has time to maturity) is:

(13.14)

Let's examine this from a more practical point of view, which requires some additional points to be made. Suppose it is 1 July 2017. The yield curve is flat and (3% p.a., continuously compounded). You are deciding whether cash-and-carry arbitrage is possible with the September T-bond futures contract, which has a known maturity date T = 11 September (Figure 13.1).

By buying a cash-market T-bond on 1 July and carrying it forward to 11 September, you create a synthetic T- bond future, since the T-bond can be delivered against the futures contract. Assume the CTD bond is a 10%-coupon T-bond which pays semi-annual coupons of $5 each year, on 15 August and 15 February and matures on 15 February 2038. It has a CF of 1.22.

Today at = 1 July to the maturity of the futures, the deliverable bond pays one coupon of on 15 August, which when invested over 27 days is expected to accrue to a future value at time T of:

On 1 July, borrowing at the risk-free (repo rate) and purchasing the bond in the cash market for an invoice price, leads to an amount owing at = 11 September, of (i.e. purchase cost plus the repo interest cost). Therefore at , the net cost-of-carry in the cash market is:

(13.15)

But the above strategy creates a synthetic T-bond future since it ensures that the bond purchased in the cash market on 1 July is available for delivery against the futures contract at = 11 September. At maturity of the futures , the invoice price the long pays the short for delivery of the underlying T-bond is the quoted price of plus accrued interest⁷:

(13.16)

is the accrued interest (between 15 August and T = 11 September) on the CTD bond delivered at (see Figure 13.1). is a proportion of the next coupon which accrues on 15 February 2018.

Since the actual futures contract and the synthetic futures both deliver one bond at then the cost today must be equal, otherwise risk-free arbitrage profits would be possible. Hence equating (13.15) and (13.16):

(13.17)

The no-arbitrage (fair) futures price is:

(13.18)

where is the cash-market bond price (i.e. clean price plus accrued interest payable when initially purchased at time t). If there are no coupon payments over the arbitrage period (i.e. ), no accrued interest () and the deliverable bond is the one specified in the futures contract (), then not surprisingly the above formula for reduces to that for the futures price on a zero-coupon bond, namely .

13.7.3 Arbitrage

Profitable risk-free arbitrage would ensue if the invoice price of the futures received at , exceeds the cost of the synthetic futures () at .⁸ Today at t, the arbitrageur borrows an amount at a cost of (the repo rate), purchases the bond and simultaneously sells T-bond futures. The amount owed at , the expiry of the futures contract is:

Hence the net (cash-market) cost at is , but the arbitrageur receives a larger amount . This arbitrage creates additional demand for cash-market bonds so increases and sales of bond futures will reduce until the equality in (13.17) is restored and the equilibrium futures price is given by Equation (13.18).

However, the arbitrage strategy is not completely risk-free because the coupon payments received from the cash market T-bond have to be re-invested at interest rates which are unknown at (1 July). At , the best guess of the reinvestment rate for any coupon receipts would be the implied forward rate – and to lock in this rate would require a strip of FRAs. Hence cash-and-carry arbitrage only provides an approximate formula for the T-bond futures price.

13.8.1 Turtle Trade: Arbitrage Profits

13.8.1.1 Buying the Spread

First, let us examine how an arbitrage ‘June-September’ T-bond futures spread position can be used to exploit any mispricing along the yield curve (Figure 13.2). Suppose at (1 April 2017) we:

1. buy a T-bond futures which matures at (1 June 2017)
2. sell a T-bond futures which matures at (1 September 2017) and
3. you borrow money at the forward repo rate (applicable between June and September) by selling the June-Eurodollar futures on 1 April.

We are buying the spread (i.e. we buy the nearby T-bond futures contract). When the nearby T-bond futures contract matures we take delivery of the eligible bond (which matures on 1 August 2038, say) and pay (plus any accrued interest, ) with finance raised by borrowing at the (forward) repo rate.

We then ‘carry’ this cash market bond until 1 September when it is delivered against the short September-futures position and we receive (plus any accrued interest , arising from the underlying bond's next coupon payment on 1 February 2018). The (compound) return from this long-spread strategy⁹ is the implied repo rate :

(13.22)

where is the (future) value of any coupons paid on the ‘carried’ cash-market bond on 1 August 2017, compounded to T = 1 September 2017. Note that the implied repo rate is a forward rate. If the implied repo rate (say) exceeds the cost of borrowing at the actual forward repo rate (applicable between 1 June and 1 September), then a long arbitrage spread trade undertaken on 1 April is guaranteed to be profitable.

The actual repo borrowing rate is the forward rate , when viewed from (1 April 2017). How can we ‘lock-in’ this forward rate on 1 April? Today we sell June-Eurodollar futures contracts at . When we close out the Eurodollar futures in June, the effective cost (between June and September) is the implied forward rate ).¹⁰ You use the cash inflow on 1 June from closing out several short Eurodollar futures contracts, to pay (plus accrued interest) for delivery of the eligible T-bond in the maturing June T-bond futures. Hence if the implied (forward) repo rate from buying the spread is and the cost of borrowing is (= 4.16%, the actual forward repo rate), we undertake the following turtle trade on 1 April:

Turtle trade

Buy the T-bond futures spread and sell June-Eurodollar futures.

When the implied repo rate (i.e. the ‘percentage return’ on the T-bond spread trade) exceeds the actual (forward) repo cost of borrowing, an overall profit will be made on the turtle trade.

First (for illustrative purposes), consider a cash-market (spot) position in two bonds, where = (invoice) price of bond-i and = number of bonds-i held ( if held long or if short-sold). The dollar value and change in the cash-market T-bond portfolio is:

Cash Market

(13.A.1)

(13.A.2)

where we use the duration approximation, . For a parallel shift in spot yields , then:

(13.A.3)

where , for and is the portfolio duration of the cash-market bond portfolio.

T-bond futures

The change in value of a portfolio consisting of ‘cash-market T-bonds+T-bond futures’ (ignoring accrued interest) is:

(13.A.4)

Assume the relationship between the change in the YTM of the cash-market T-bonds and the change in the yield (on the deliverable bond) in the T-bond futures contract is , then:

(13.A.5)

Hedging

To fully hedge your cash-market T-bond portfolio we set , hence:

(13.A.6)

which is sometimes referred to as the minimum variance hedge ratio.

Effective Duration and Market Timing

Suppose a trader holds a cash-market (spot) T-bond portfolio and she wants to change the effective duration of the portfolio to some desired level , using T-bond futures contracts. The desired change in her cash-market T-bond portfolio is:

(13.A.7)

Equating (13.A.5) and (13.A.7):¹¹

(13.A.8)

The cash-market T-bond position could be either net long or short – depending on the sign of – and the cash-market portfolio duration could be positive or negative depending on the value of . For the moment assume and are both positive. Then (13.A.8) gives the required number of futures contracts to achieve a ‘desired duration’:

Hence, to increase (decrease) the effective duration of an underlying (long) cash-market bond portfolio (with ), we go long (short) T-bond futures. This makes sense because if interest rates subsequently fall, then you make profits both from the cash market bond portfolio and the long T-bond futures contracts – this is equivalent to increasing the duration of the cash-market bond portfolio and is a form of ‘market timing’. If your forecasts are correct then you will make more profit by taking the long futures position rather than just holding your cash-market bond portfolio.

Note that (13.A.6) the minimum variance hedge ratio is a special case of (13.A.8), where the desired duration, :

The implied repo rate is another way of determining whether risk-free arbitrage profits are possible. The implied repo rate is the return from buying the underlying cash-market T-bond at for the invoice price and simultaneously selling the futures for delivery at . The return or implied repo rate (using discrete compounding) is:

(13.B.1)

where = futures price set at for payment at (when the underlying bond is delivered) and the conversion factor is . The arbitrageur holds the underlying bond and receives coupon payments which can be reinvested over the period to to give a future value . The accrued interest on the deliverable bond at maturity of the futures contract is . The cash-market bond has an invoice price of and this is financed by borrowing at the (actual) repo rate , at .

It is easy to see using (13.B.1) that if then:

(13.B.2)

Part of the reason for using the implied repo rate to determine possible arbitrage profits is that it is straightforward to compare the implied repo rate with the quoted repo rate (cost of borrowing funds). Using (13.B.2), the elimination of any arbitrage profits implies the fair (no-arbitrage) T-bond futures price is:

(13.B.3)

If we replace the discrete compound rate by the continuously compounded rate then Equation (13.B.3) is equivalent to the formula for the ‘no-arbitrage’ or ‘fair price’ given by Equation (13.14) in the text.