Interest-Rate Swaps 91 Inserting y=-O.87 change in contracts per bp gives the small result
$dS, ,,,,,,,n = -$l.lK
.
The sum gives the “Math Calc” change $dSMo,,,~c ,The result for $dSM(,f,,,,c is equal to the scenario revaluation difference to this accuracy, $dSs,e,,,,i, = $427.5K. Actually, this result is “too accurate” because a real-world hedge would just involve an even multiple of 100, i.e. 1700 contracts.
Now we need to check the change in delta dA,,,,,, due to gamma. We should have to a good approximation
- df (‘ObP)
‘contracts - Ydcontrucis I bp ’ Scenurio
Plugging in gamma as y = -0.87 gives the change in the number of contracts as dAco,,fr~CfS = -8.7, in agreement with the reval result.
Hedges
The following instruments are available as possible hedges:
0 Short-term money-market cash instruments
0 Eurodollar (ED) futures
m s
Otherswaps
0 Treasuries
0 Treasury futures
The swap can be divided up into constituent swaplets, as we have seen. These swaplets have individually different cash flow specifications with definite rules for determining the reset values of the rate on which the swap is based, the cash flow datesg, etc. according to the contract”’ between BD and ABC”. The following complications theoretically enter into the hedging of the swap.
Hedging with Cash Instruments
The swap may have an uncertain cash flow before the first Eurodollar future IMM date”. This leads to “spot risk”, and can be hedged with the short-term
’
Cross Currency Swaps: If payments are made in more than one currency, we have a“cross currency swap”. Extra complications involving FX hedging then occur. One distinction with single-currency swaps is that the notionals for a cross-currency swap (in the different currencies) are exchanged at the beginning and then reversed at the end of the cross-currency swap. If there is only one currency, notionals are not exchanged, only interest payments. Cross-currency swaps can get very complicated as cash flows are converted into several currencies successively in the same deal. See Beidleman (Ref.)
Payments in Arrears: The various swap cash-flow dates are generally “in arrears”, which means that a cash payment is made some time after the rate determining that cash payment is made (or “reset” or “fixed”). An extra discount factor for the extra time period from the reset time to the cash-flow time is included. If a payment is not “in arrears”, the payment is said to be “up front”.
l o The All-Important “Term Sheet“ and Why You Need It: The legal contract for the
deal will be preceded (before the deal is transacted) by a “term sheet” containing the details of the proposed transaction. These term sheets may or may not be generic, and they can change up to the last minute depending on the transaction dynamics between the desk BD and the customer ABC. In early discussions, the term sheet may only be schematic. The quant should always get a term sheet - hopefully the latest one, even if indicative or provisional, and even if the salesman argues that he doesn’t have it yet - before wasting time modeling, pricing, and hedging using wrong assumptions regarding the nature of the deal. Having said this, a fair amount of back-and-forth quantitative analysis often occurs for a given deal if there are non-generic aspects.
“Initial Cash flow in the Swap: New swaps can also have an initial cash flow that is set in the contract. A swap already in the books may have a cash flow that has been reset but not yet paid. There is a risk due to the uncertainty of the discounting from the analysis
Interesr-Rate Swaps 93 money market cash instruments. S o m e t i m e s this is not done, and the spot risk will be approximately hedged with t h e nearest ED future”.
Hedging with ED Futures
The cash flows originating in t h e forward time period where t h e ED futures are relatively liquid can be hedged with ED futuresI3. In general, t h e cash flows for t h e swap will occur a t various times that will n o t coincide with t h e ED future IMM datesi4. The following picture should clarify things.
r S w a p l e t reset times located between IMM dates
Reset dates for a typical swap
I I I
IMM dates for ED futures
I
Ip T l
date until the payment date for the entire cash flow. However, if the swap is sold, the cash to be paid from the reset to the transaction date is sometimes set on an “accrual” basis not including discounting.
I ’ The End-Year Effect: There is an “end-year” effect that happens at the end of the year
which leads to anomalies in the money market and which needs to be hedged separately.
For the year 2000, this effect was very large.
l 3 Hedging with ED Futures: The liquidity of the E D futures is greatest for times within
the front few years, and the implementation of ED future transaction is more difficult further out in time. In addition, futures are generally transacted in groups of 100 (anything smaller is called an “odd lot” and is inconvenient to transact). Often an ED hedge will be transacted first quickly using the front few contracts for the whole swap risk and then some time later the hedge will be distributed among hedge instruments with other transactions. Various residual risks corresponding to the inexactness of the hedging for each of the points raised above will occur.
l 4 IMM Swaps: Some swaps, generally short term swaps, do have fixing dates at the ED
future IMM dates (these are imaginatively called IMM swaps). IMM swaps are very quickly transacted and are popular with short-term swap desks.
An interpolation of the risks in time then needs t o be performed. In general, this is just done proportionately to the time intervals from a given swaplet cash flow to the IMM dates before and after that cash flow.
For a mneumonic, see the footnoteI5.
Hedging with FRAs
An FI2A’” (forward rate agreement) is the same as a one-period swap, i.e. a swapletI6. FRAs have the advantage that the periods over which they exist are measured in months from today, not at IMM dates. For this reason the dates can be tailored to match those of a new swap.
Hedging with Other Swaps
If the ABC swap happens t o be a swap depending on other rates” (e.g. CP, Prime, Muni, C M T etc.) then a Libor swap can be transacted in the opposite direction. The residual risks here involve “basis risk” due t o the difference in the behavior of the rate from Libor. Date or time risk also occurs since the cash flow dates of the two swaps will generally not be identical. There is also normalization risk since the ABC swap notional or principal” may not match the hedge swap
l 5 ED Futures and Swaps Hedging Mnemonic: Put your hands together with
interlocking fingers. Your right hand intersects at the swap-reset times and your left hand intersects at the IMM dates. You need to interpolate the risks of each of the swap-reset dates of each right-hand finger with the neighboring two futures of the neighboring fingers on your left hand.
l 6 FRAs: FRA = Forward Rate Agreement. Either say the letters “F.R.A.” or say the
acronym “fra” pronounced to rhyme with “bra”. FRAs are short-term money market instruments (see Stigum, Ref.). The floating side of an FRA is a forward rate starting at z = p months and ending at T = q months. FRAs are represented as pxq (e.g. 1x4, 3x6, 6x12). For 3x6, say “Threes Sixes”. FRAs are cash settled proportional to the difference of the floating rate and the contract rate. There is a consistency between the FRA market and the futures market.
I7 Other Rates: Examples include CP = Commercial Paper, Prime = Prime rate set by banks, CMT = Constant Maturity Treasury, Muni = A municipal index rate. These
“basis” rates have their own swap rates often expressed as a “spread” or difference with respect to Libor swap rates (except for Muni rates which use ratios with respect to Libor rates depending on tax considerations). Some of these swaps (e.g. CP) involve complicated rules regarding averaging the rates over various date periods.
l 8 Notional and Principal: “Notional” and “principal” mean the same thing. The notional
is a normalization factor that multiplies the overall expression for the swaplets comprising the swap. Sometimes the swap market is characterized by the total notional in all transactions.
Interest-Rare Swaps 95 principal. See the ISDAI9 reference” for data on
nationals''.
If the ABC swap is an amortizing swap, (the principal is different for the different swaplet cash flows), further complications exist21.Hedging with Treasuries
U.S. treasuries (which are very liquid) can also be used as hedges. Again there is basis risk since treasury rates are generally not the same as the rate on which the swap is based, date risk corresponding to mismatched cash flows, normalization risk, etc. An exception is for CMT swaps, which have a natural hedge in treasuries. The market for “repo” becomes involved 22,23.
l 9 ISDA: This is the International Swaps and Derivatives Association, Inc. ISDA
describes itself as the global trade association representing leading participants in the privately negotiated [fixed income] derivatives industry. ISDA was chartered in 1985 and today has over 550 members around the world. ISDA Master Agreements are always used for contracts for plain-vanilla deals (see the Documentation Euromoney Book Ref).
You can find out more at www.isda.org.
”The Total Swap Notional and Why You Don’t Care: A recent figure for the total outstanding notional for interest rate swaps, interest rate options and currency swaps is over $50T (Trillion) according to ISDA News (Ref.). “Outstanding” means all deals already done that have not expired or closed. This initially somewhat scary number has very little to do with the real risk in swaps which is many orders of magnitude less. For example we saw above that a $100MM notional swap has a risk of around $40K per bp change of rates to the B D if the swap is unhedged. The B D will generally hedge this risk down to a very small fraction of $40K. Finally, swaps on the other side (receive fixed vs.
pay fixed) will behave in the opposite fashion. Forget about the $50T.
” Notional or Principal Schedules for Amortizing Swaps: Amortizing swaps are often specified by the customer ABC according to a “schedule” of the different notionals for the different cash flows, corresponding to specific ABC needs. For example, the first cash flow could have $100MM notional and the second cash flow $90MM notional, etc.
Naturally, ABC will have to pay an extra amount to BD for such custom treatment, but part of this will be eaten up by the residual risk forced on the BD.
22 Repo: The rep0 market is an art unto itself. The simplest version is overnight repo, where a dealer sells securities to an investor who has a temporary surplus of cash, agreeing to buy them back the next day. This amounts to the BD paying an interest rate (repo) to finance the securities. The arrangement can be made such that the BD keeps the coupon accrual. The complexities of rep0 (including “specials”) enter the hedging considerations. We give an illustrative example of a repo deal in Ch. 16.
l 3 Acknowledgement: I thank Ed Watson for discussions on repo and many other topics.
Hedging with Treasury Futures
Treasury futures involve complications, including the “delivery option” for the
“cheapest to deliver”24. The inclusion of these complications into risk systems is standard.
Example of Swap Hedging
We already considered the basic idea in a previous section. Again consider the above 5-year swap, non-amortizing with $lOOMM notional, based on 3-month Libor with payments in arrears, where the BD pays a fixed rate to the customer ABC. The risk to BD from the ABC swap is that rates decrease. This is because then BD would then receive less money from ABC determined by the decreased floating Libor rate that ABC pays to BD. Therefore the pay-fixed swap of the BD must be hedged with instruments that increase in value as rates decrease. These include for example buying bonds or buying ED contracts or
As we saw, the total delta or DVO127 from a swap model for this swap is short 1714 equivalent ED contracts2*. This means that if ED futures were to constitute
’‘Bond Futures and the CTD: A bond future (as opposed to ED futures) requires delivery of a bond to the holder of the bond future from a party that is short the bond future. However, this does not mean delivery of a definite bond, but rather of any bond chosen at liberty from a set of bonds specified for that particular bond future. One of these possible deliverable bonds (the “cheapest to deliver or CTD’) is economically the best choice for the holder of the short position, who gets to choose. The CTD bond (with today’s rates) thus determines the characteristics of the bond future today. However, since the future behavior of rates is uncertain, another different bond may wind up being the cheapest to deliver when it actually becomes time to deliver a bond. This uncertainty shows up as a correction to the bond future’s price and its sensitivity to interest rates.
Other complexities exist for bond futures.
25 “Short and Long the Market” for Swaps: The pay-fixed swap makes the BD “short the market” and the pay-fixed swap loses money as rates decrease. In order to hedge the pay-fixed swap, the BD must go “long the market”, buying instruments that make money as rates decrease. Generally, going long the market is in the same direction as going long (buying) bonds, whose prices increase as rates drop.
More Swap Jargon: Say that the bid swap spread is 40 bp/yr and the offer 44 bp/yr.
This means that a potential fixed rate payer is ready to pay 40 bp/yr and a potential fixed rate receiver wants to receive 44 bp/yr (above the corresponding treasury rate). A “bid side swap” for a BD means that the BD pays the fixed rate. Since the BD in that case receives floating rate payments, which increase when the swap spread increases for fixed treasury rate, the BD who pays fixed is said to be “long the spread” and “long the swap”.
’’
DVOl Warning: Careful. Some desks use the convention that DVOl or delta corresponds to a rate move up, and some desks use the convention that DVOl corresponds to a rate move down. This complicates aggregation of risk between desks.The story gets worse. See the following footnote.
28 More on Different DVOl Conventions and Risk Aggregation Problems: Other conventions for expressing delta or DVOl exist. One convention uses the notional for a
both25,26.
Interest -Rate Swaps 97
the total hedge for the swap, the BD would have to buy (“go long”) 1714 ED futures. For a one basis point ( I / ] 00 of I %) decrease in Libor, one ED future increases in value by $25, so the risk to BD for the ABC swap is a loss of 1714*$25 or around $43,000 for each bp of overall (parallel) rate increase29.
There is also a small correction from second-order derivative “gamma” terms3’.
The “Delta Ladder”, or Bucketing: Breaking Up the Hedge
The “ladder” or set of “buckets” is a breakup of the total hedge in maturity steps.
For example, we can use 3-month “buckets”. The units are equivalent ED future contracts. An approximate hedge for this front part of the swap could consist of buying 100 of the first three “ ~ t r i p s ” ~ ’ . That is, the approximate hedge for the first given bond or swap ( e g 10 year). That is, the risk is expressed in terms of the number of 10-year treasuries it would take to hedge the overall risk. Sometimes the risk is expressed in terms of zero-coupon rate movements. Sometimes the risk is expressed in terms of a mathematical rate present in a model which is used to generate the dynamics, but which actually has no physical meaning. The difference between the reported DVOI using these different conventions can easily be on the order of a few percent. Finally, sometimes the magnitude of the rate move will be different between desks (e.g. 1 bp, 10 bp, 50 bp) and thus may include some convexity. This is often done as a compromise involving numerical stability issues of the models dealing with options that are also in the portfolio.
These conventions will sometimes but not always be marked on the desk risk reports. All this can generate confusion when aggregating risk. Similar anarchy unfortunately exists for other risks as well. Sometimes this whole problem is ignored, or goes unrecognized.
29 The Movements of Rates is Roughly Parallel: This risk is expressed in the simplest case for all forward rates determining the swap increasing together by one bp. While different forward rates by no means move by the same amount in the real world, nonetheless in practice, the approximation of this “parallel shift” covers much of the risk and is the measure commonly used for a first-order approximation. The extent to which the forward rates do not move in parallel will present risks insofar as the individual swaplet deltas are not individually hedged. One way to look at rate movements is using principal components, which we treat in Ch. 45 and in Ch 48 (Appendix B).
30 Gamma for a Swap: A Libor swap has small gamma (second derivative) risk, so most of the risk is just due to the first derivative or delta (or DVOI). The gamma risk is expressed by specifying the change in the number of equivalent contracts per bp increase in rates. For this swap, this number is about -0.9 for the entire swap. As we shall see below, the details of bucketing gamma in forward time is complicated since gamma is really a matrix of mixed second partial derivatives corresponding to the different forward rates comprising the swap. Thus, there is really no exact gamma ladder. However, an effective approximate gamma ladder will be constructed.
3 ’ More Handy ED Futures Jargon: A “strip” is a set of four neighboring ED futures
contracts corresponding to one year. These strips are given names. The “Front Four” are first contracts (ED1 - ED4). The “Reds” are the next four (ED5 - ED@, then the
“Greens”, the “Blues”, the “Golds”, etc. Moreover, the months corresponding to expiration are abbreviated (e.g. December is called DEC pronounced “Deece”). The names are convenient shorthand for the traders.
3 years could consist of buying 100 contracts of each of the first 12 contracts EDl-ED12. The "spot" risk up to the first contract would need to be hedged separately, as would the tail risk from 3-5 years of 561 equivalent contracts (see discussion below). Residual risks would be put into the portfolio.
A graph of the first 3 years of the delta ladder hedge needed for the swap is shown below.
Hedge for 3 yrs of 5-yr swap (Equivalent ED contracts)
120 I
Future Label
Hedging Long-Dated Swaps
The hedging of a long-dated swap is achieved by first choosing hedging instruments and then minimizing the risk. The risk can be broken up into buckets, but in general, it will not be possible to make the risk zero in every bucket unless the buckets are suitably chosen large enough.
Back Chaining Hedging Procedure: Example
Consider, for example, an amortizing 15-year Libor swap. Aside from back-to- back offsetting of this swap with a similar 15-year swap, there is no single natural hedge. A general strategy is to employ a mixture of the hedging instruments above. For illustrative purposes, we use five, ten, and thirty-year treasuries along with the first 2 years of ED futures. The buckets are thus 0-2 years, 2-5 years, 5- 10 years, and 10-30 years.
The logic is shown in the next figure.