lication of an index, the calculation of each individual Ri may be useful inaddition to that of R to identify which, if any, stock position deviates from its perfectvalue.
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FINANCIAL VALUATION AND RISK ANALYSIS
This same framework can be used to handle interest rate riskbecause again it is dif®cult to separate one particular rate from another.However, we have here one additional dif®culty because to hedge theposition, we need to enter a money market operation of a given maturity; therefore, we have to know what maturity is most exposed in theposition. The solution is to complement R by each individual Ri . Rcaptures the exposure of the position to a general move in the levelof interest rates, while the series Ri shows very clearly the maturities atrisk in the yield curve. For example, consider a position holding different instruments of maturities below 5 years. V is dependent uponthe entire yield curve below 5 years:V  V r1 WK ; r1 MO ; r2 MO ; r3 MO ; . . . ; r2 Y ; r3 Y ; r4 Y ; r5 Y Ri and R are respectively de®ned by:V r1 WK  ; r1 MO ; r2 MO ; . . . ; r5 Y ...
V r1 WK ; r1 MO ; r2 MO ; . . . ; r5 Y 
V r1 WK ; r1 MO ; r2 MO ; . . . ; r5 Y  
V r1 WK ; r1 MO ; r2 MO ; . . . ; r5 Y 
R1 WK 
R5 Y 
V r1 WK  ; r1 MO  ; r2 MO    . . .  r5 Y  V r1 WK ; r1 MO ; r2 MO ; . . . ; r5 Y RIf each rate were independent, we would have R  R1 WK R1 MO      R5 Y   Ri , but in general this relation does not hold.6Looking at the Ri , it is now possible to determine what is the implicitexposure in the portfolio and hedge it. For example, if we determinethat R12 MO is the predominant element in the series, we know thatprobably the whole portfolio should be hedged with a maturity closeto 12 months. As for the nominal at risk, it depends on the expectedbehavior of the actual market curve.7 If it is probable that a punctualmove will affect r12 MO speci®cally while leaving the remaining part of6
To see why this is possibly so, consider a 31/2-year loan. The relevant rate to value thisloan is an interpolation of the 3- and 4-year rates. If this interpolation is linearÐr3 r3  r4 =2Ðand the risk analysis is also linear, then R  R3 Y  R4 Y . If, however, theinterpolation is not linearÐpolynomial, for exampleÐthen, R 6 R3 Y  R4 Y .7Although Ri and R can be homogeneous to a number of shares or nominal whencomputed in stocks, for example, on interest rates they are always homogeneous toa nominal, representing the nominal at risk in the position for a move in interest ratesequal to .
1/2
CHAPTER 7 RISK ANALYSIS
335
the curve relatively unchanged, we should use R12 MO as an exact indication of the nominal to hedge. If, on the other hand, we expect ageneral move, we should probably use R, with the underlying assumption that the entire position will be adequately hedged by taking careonly of its most sensitive maturity.ConclusionThe above discussion shows that a risk analysis process is not as clearcut as a valuation because of the subtleties of the mandate and thedif®culties to describe appropriately the intrinsic randomness of themarket. In particular, two of the fundamental principles we used toperform a valuation are not valid anymore. On one side, we cannotconsider each instrument independently but rather we need to treat theportfolio as a whole and identify classes of risk across all assets.Similarly, depending on the type of position held, the methodologywill be different. For example, for a large portfolio it probably makesmore sense to compute a global R, assuming that risk is accuratelydescribed from the perspective of the entire market shifting in thesame direction. For a small position, a stock-by-stock perspectivemay be warranted, in which case the portfolio risk is more appropriately described by the family of Ri .In fact, an identical underlying may be analyzed differently intwo distinct positions. If it is held as part of an arbitrage, the assumedconvergence will dictate the type of analysis performed, in contrast toanother type of strategy. Assume, for example, that we try to evaluatethe risk embedded in a risk arbitrage between two stocks A and B.We can compute three risk factors RA , RB , and R on this portfolio,each capturing a different side of the risk exposure. However, none ofthese does a good job at looking at the most important risk, the factthat the spread between A and B might evolve unfavorably. Themetrics necessary to account for that particular risk is the arithmeticquantity S  r  PA PB where r is the merger ratio. If at the sametime somebody else on the trading ¯oor is looking at A and B from ageneral perspectiveÐthat is, without any emphasis on the plannedmergerÐthe corresponding risk analysis would not use S but simplyPA and PB .A risk analysis is therefore inherently highly context sensitive, insharp contrast to a valuation that we presented as an ``absolute'' process. If the purposeÐallowing a trader to make a relevant decisionÐisuniversal, and the toolsÐpartial derivativesÐcommonly used, theimplementation should be tailored to the situation. Back to the risk
336
FINANCIAL VALUATION AND RISK ANALYSIS
arbitrage example: RA , RB , and R do not serve the purpose adequately,but RS does if RS is de®ned by:RS 
V S  
V S
In essence, the risk analysis should extract the relevant informationfrom the position, and it belongs to the trader to make sure that thisis truly the case. Thus, our enumeration of risks cannot possibly beexhaustive but will present the most important categories and thosedirectly relevant to arbitrage opportunities.
Risk Management and ReportingInternal Risk ManagementLike a valuation, a risk analysis is reported every day from each individual trading desk to the management of the ®rm, with a degree ofaggregation depending upon the target audience. For example, thehead of an equity trading department will be interested in how eachindividual activity contributes to the risk of the entire ¯oor, which deskis most heavily exposed, and which risk factor is most critical. As aresult of this information, he or she acquires a vision broader than thatof individual traders and may be able to optimize the overall exposure.8 Higher in the hierarchy, the information needs to be compactedeven moreÐfor example, to include ®xed income and foreign exchangeas well as equity. Finally, at the higher executive level, the CEO, CFO,or COO are most probably interested in a maximum loss, in addition tothe recurrent level of risk carried by the ®rm as part of its day-to-daybusiness.This situation is very similar to what happens with a valuationstatement, with two major differences: The ®rst one is that a risk analysis is essentially a forward-looking prescriptive document because itsentire purpose is to enable hedging. A valuation statement, on theother hand, is usually backward looking and descriptive, and it doesnot call for action. Furthermore the ability to execute on the risk analysis disappears rapidly when going up the hierarchy. For example, the8
Consider, for example, that two traders have an opposite exposure on a given riskfactor. Independently each one would be tempted to hedge and possibly pay a fee to athird party for that. If they traded internally, on the other hand, they could price thetransaction at fair value so that each is appropriately hedged at zero cost.Needless to say, this broader vision is also absolutely necessary in evaluating individual performances.
CHAPTER 7 RISK ANALYSIS
337
head of a trading ¯oor may be in a position to recommend speci®ctrades, but the CEO will simply comment on the level of risk. Therefore,even though valuation and risk statements tend to travel togetherthroughout a ®rm, they cannot be aggregated in a similar way.9The second is related to the fact that a risk analysis is highlydependent on a strategy and its particular management, and thereforeit carries much more potential for misunderstanding. In other words, itis possible to reconcile and explain a valuation, good or bad, based onfacts, but much more dif®cult to justify a level of risk that mightdepend on the speci®c nature of the position as well as a marketopportunity. Index arbitrage is an excellent example of this potentialfor con¯ict. In general, positions of that type reach several billion dollars, and such an amount tends to make risk managers nervous. At thesame time, when properly hedged and managed, the intrinsic risk hereis much smaller than that, for example, of an option position on a muchsmaller nominal.To understand a valuation, a risk manager needs to be familiaronly with instruments and products, whereas when dealing withreported risk analyses, he or she has to understand the underlyingtrading strategy and its implications on the management of theposition. This speci®c knowledge is hard to acquire without practicaltrading experience, which makes the task of nontrader risk managersall the more demanding.
Time Horizons
Another important aspect of a risk analysis is its time horizon. A valuation is a snapshot as of today, and it does not have any time componentbecause the position is liquidated immediately. A risk analysis, in contrast, should have a time component because risk is likely to evolve inthe future. Typically the expiry of an instrument, derivative or not, at agiven time signi®cantly alters the risk pro®le of the entire portfolio.This problem can be addressed in two ways. On one hand,although the purpose of a risk report is to enable trading immediately,it should at the very least present a detailed schedule of expectedexpirations and other relevant events. Indeed, it should ideally providethe trader with an exhaustive calendar of events likely to affect theposition one way or another. Such a calendar would, for example,include corporate eventsÐdividends, splits, and so onÐindex inclu9
Typically a dollar earned is a dollar earned anywhere in a ®rm. In contrast, a dollar atrisk is not the same depending on which activity carries it.
338
FINANCIAL VALUATION AND RISK ANALYSIS
sions and/or deletions, and expirations of, say, money markets, stockloan agreements, and futures. Naturally monitoring such a diverseuniverse is a dif®cult task, and in practice, many traders manage thisprocess themselves, in general manually.On the other hand, and this is the most important aspect, some riskfactors are likely to depend on time and should be organized in a termstructure, similar to that of interest rates. The best example of this isvolatility, used to value an options position, but the same reasoningapplies also to the repo rate for stock loans and other parameters.Volatility is highly dependent on maturity, the same way interestrates are, and an accurate description of the associated risk requiresseveral estimators based on different expiries. Therefore, it is not onevolatility risk factor but several that must be included and analyzedtogether if the position is to be hedged correctly.
Magnitudes
A risk factor is characterized by two elements: its magnitude and itsprobability. In a risk analysis performed with partial derivatives, thenotion of probability is relatively silent because we ®x the elementaryvariation we want to use, and we do not really need to know theassociated probability. On the other hand, the probability becomesextremely important in a V@R, as we will see.Therefore, we will focus primarily on the order of magnitude of agiven class of risk. To illustrate this notion, consider a $100 millionposition on stocks. If the market loses 10 percent tomorrow, the portfolio is likely to lose close to $10 million. If the portfolio is instead aloan for 1 year on $100 million with a market rate of 5 percent, a changeof 10 percent in the rate leads to a loss of $500,000 (= $100,000,000  5percent  10 percent). Finally, if we hold again a stock position but thistime the dividend yieldÐassumed equal to 2 percentÐvaries unfavorably by 10 percent, the total loss is $200,000 (= $100,000,000  2 percent 10 percent).This short example shows that the absolute magnitude of the riskdepends on the nominal of the position but more importantly on thenature of the risk examined. It is customary to refer to the magnitude ofa given exposure as its order. In the above example the market riskrelated to asset prices is clearly of the ®rst order, interest rates of thesecond order, and dividends, the third order. This nomenclature ishighly subjective, however. For a treasury manager, interest ratesclearly represent a ®rst-order concern, while perfectly hedged indexarbitrageurs, on the other hand, would consider the change in asset
CHAPTER 7 RISK ANALYSIS
339
prices as of the second order because their exposure is usually differential, not outright.The purpose of this labeling is essentially practical. When dealingwith a particular position, it is essential, especially for junior traders, tomake very quickly a distinction between the different layers of risk,which is exactly what the notion of ``order'' tries to describe. In thefollowing examination of market risks, and even more in the last partpresenting speci®c arbitrage situations, we will try to qualify risks bytheir order of magnitude, to make their relative importance and impactclear.10
Interest Rates
In contrast to a valuation, the yield curve in a risk analysis does notneed to be zero coupon. All valuations are usually performed with zerocoupons, but the parameters used to represent interest rate riskÐthatis, Pi Ðdo not have to be. To understand what this means, we denotethe zero-coupon yield curve and ri the market curve, made ofrZCiswaps, eurofutures, or any other relevant product. We saw in a prefrom ri , a process we sumvious section how to derive the series rZCimarize by a function g such that:ZCrZC1 ; . . . ; rn   g r1 ; . . . ; rn 
We know that V uses rZCi , so we can de®ne the ``natural'' ith risk factorRi as:ZCZCZCZCV rZCV rZC1 ; . . . ; ri  ; . . . ; rn 1 ; . . . ; ri ; . . . ; rn Ri However, if we want to use the market yield curve, we can use:R0i 
Vg r1 ; . . . ; ri  ; . . . ; rn 
Vg r1 ; . . . ; ri ; . . . ; rn 
R0i is different from Ri because we apply the same change in rate, , butnot on the same curve; a shift of, for example, 0.10 percent of themarket curve is not equivalent to a shift of 0.10 percent on the zerocoupons.The reason behind this transformation is primarily practical. Eventhough zero coupons are essential in a valuation, the most liquid ratesactually traded in the market are swaps and eurofutures. Therefore,10
Alternatively, a ®rst-order risk is a risk to which the entire nominal of the position isexposed. A second-order risk is a risk to which only a fraction of the position isexposed, and so on for subsequent orders. This is consistent with the original discussion introduced in the ®rst chapter about nominal versus differential exposure.
340
FINANCIAL VALUATION AND RISK ANALYSIS
using a market rate as Pi gives a result that is directly relevant tomarket instruments. Consider, for example, that we determine a possible loss of $500,000 to an increase of 0.20 percent in the two-yearzero-coupon rate. To hedge this risk, we would need to enter a twoyear zero coupon such that an increase of 0.20 percent in the rategenerates $500,000 in additional pro®t. However, it may be dif®cult,or expensive, to trade this type of zero coupon in the market, whereastwo-year swaps are readily available. Assuming that a variation of 0.20percent in the two-year zero coupon is equivalent to the same variationin the two-year swap is a mistake because the relationship between thetwo is not linear. Therefore, we have no easy way to perform a hedge,and we need to translate the incremental 0.20 percent in swap prices. If,in contrast, the risk analysis had been carried out originally on swapprices, we would have the information necessary to hedge directlyavailable.In the remainder of this chapter, we do not specify if interest ratesPi are zero coupon or market. Except for the intricacies of translatingone into the other, the fundamental results and analysis are independent from the reference yield curve.
MARKET RISKSMarket risks incorporate all risks that are dependent upon one or aset of market parameters. For our purpose, a market parameter is anumerical quantity varying randomly continuously.11 For example,the price of a stock is known today, but it is unknown tomorrowor at any later date for that matter. More generally, the price discovery process occurring in the market is such that asset prices arealways instantaneously available, but they are systematicallyunknown for any future period. In contrast, other risks do notdepend on the market and may not even be parametric. For example,the risk of a system failure is uncorrelated to the market level anddif®cult to describe with a single ®gure.Market risks are the primary focus of risk managers' attention, ifonly because these risks need to be monitored at least daily. In contrast,nonmarket risks often tend to be overlooked, despite the fact that in11
Note that the variable may be discrete or continuous, and in fact, it does not need tovary very often, although the higher the variability, the more important the associated risk. Dividends, for example, vary only episodically and generally on theupside and still they generate a risk that must be accounted for.
CHAPTER 7 RISK ANALYSIS
341
some instances their magnitude is equivalent. Back to the example of asystem failure, a power outage, for example, on a futures expiry canvery well lead to signi®cant losses if a large number of orders cannot besent to the market at the right time or price.12In the following discussion, we separate market and nonmarketrisks, and we tend to discuss the former in more detail. The reasonfor this choice is the perspective we have chosen from the start. Atrader is directly accountable for the management of the market risksin his or her position, but his or her sphere of in¯uence and responsibility is rapidly constrained for nonmarket risks. Although they certainly should be kept in mind, they are usually better handled by morequali®ed parties. The legal and contractual aspect of OTC trades, forexample, should be proofed by experienced lawyers.
PricesDelta and GammaThe most important market risk is a ®rst-order risk related to the priceof an asset. This asset may be a stock, bond, or foreign currency, and inall cases its price varies continuously and has a direct impact on theexposure of the position. By analogy to the Greek used in an optionsposition, this risk is often referred to as the delta of the position:VV S   V SSAs we mentioned earlier, the delta may be computed asset by asset, oron the entire position at once. Similarly, it can be expressed as a number of units or nominal, depending on the environment. For example, ifthe position examined is a foreign exchange or a treasury portfolio, therelevant unit of measure is a nominal ®gure. This risk is very simple totrack and understand. In an arbitrage situation it corresponds to aposition for which the legs are not properly matched, in number ofunits or nominal.If today S  S0 and we do not hedge a move from S0 to S1 , thevaluation V of the position becomes:V S1   V S0   S1
12
S0    0
An interesting example is the Y2K phenomenon. Financial institutions worldwidespent hundreds of millions to alleviate their fears of an unrecoverable failure in thesystem.
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FINANCIAL VALUATION AND RISK ANALYSIS
As long as delta is constant and small, the hedge can be executed at anytime, and the incremental valuation will probably remain relativelysmall. A problem occurs, however, if delta is large or not constant.The case of delta's being too large indicates a position that was nothedged in the ®rst place, which may be the result of a mistake or anintentional choice to ``play'' the market in a given direction. Moreinteresting is a situation in which delta is not constant and varieswith the market. This phenomenon is typical of an options position,but it may be found in ``standard'' positions as well.13 The variation ofdelta is described with another Greek letter borrowed from optionstheory, gamma:!2 V S    S2SGamma makes a position much more dif®cult to manage, and for allpurposes much more risky, because the P&L generated by an error ingamma grows with the square of market movements:V S1   V S0   S1
S0   0 1/2 S1
S0 2 
0
In practice, the existence of a gamma exposure means that the positionshould be hedged very often in periods of volatile markets because theterm (S1 S0 )2 may become signi®cant. Ideally the hedge should becontinuous, but due to transaction costs, this is impossible, and the bestcompromise may be to hedge several times a day.Correlation and ConvergenceCorrelationsThe correlation risk is a second-order risk coming from the mismatch oftwo assets that should ¯uctuate very close to each other. If we use theframework we introduced earlier for a generic spread position in twoassets A and B, with a nominal such that NA  NB  N, the valuationis given by:V  pA  NA
pB  NB  pA
pB   NB  pA  N
where pA  PA =PA and pB  PB =PB denote the percentage variations of A and B. Assuming the position is originally hedged in nominal, NA  NB  N; V is then:13
For example, we showed how the arbitrage between the Nikkei 225 in Osaka andChicago required the management of a foreign exchange position with a deltadepending on the current spot rate and the level of the Nikkei 225.
CHAPTER 7 RISK ANALYSIS
V  pA
343
pB   N
The dependence of V on a differential performance creates a secondorder exposure, which is exactly the overall correlation of A and B.Considering what an arbitrage position is built upon, this risk is particularly relevant to our enterprise and, as we will see, it takes a slightlydifferent form depending on the situation.ConvergencesA convergence risk is nothing more than an extreme form of correlationexposure. When considering the dynamic relationship of two assets,we can make a distinction between a one-time equalityÐconvergenceÐand the existence of an ongoing pattern of similar returnsÐcorrelation. We know already that if a convergence exists, the correlation should be very high as is the case in index or risk arbitrage. Thereciprocal relation is not true, however, because two assets may exhibita strong historical relationship without converging with any certainty.This is typically what pair or technical trading is about.Convergence and correlation tend to evolve in the same directionbut do not capture the same information. Except for situations of absolute convergence in which both are relatively benign in magnitude, atrader would care more about one dimension than about the other. Inrisk arbitrage, for example, convergence is dominant because if itbreaks, correlation will automatically collapse. In contrast, pair or technical trading would focus more on correlation because convergence ishighly uncertain. As with all other risks, the focus is dictated by theunderlying strategy.Corporate EventsCorporate events constitute a risk only rarely, and usually only forholders of short positions.14 When terms are available in a timely manner, which is almost always the case, arbitrageurs have plenty of timeto analyze the proposed event and make the necessary adjustments totheir position. Short sellers, however, may be squeezed in speci®csituations, unable to close the risk at a reasonable price.In many cases an additional layer of risk comes from the uncertainty attached to the general acceptance of the event. If, for example,distribution of new shares in a rights offering is prorated based on the14
Naturally risk arbitrage is based on a corporate event, but we consider here only``incidental'' corporate eventsÐthat is, events affecting a position built for the purpose of pursuing a speci®c and distinct strategy.
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FINANCIAL VALUATION AND RISK ANALYSIS
overall subscription ratio, an arbitrageur may have to hedge a positionunknown at the time of the hedge.15Clearly the best way to protect a position against unforeseen dif®culties, especially on the short side, is to monitor announcements andevents very closely. First, a deep understanding of the details will leadto an appropriate decision and a clear assessment of the risks involved.Then it may be more economically ef®cient to close the relevant position early than to try to carry it through.
Interest Rates
In general, interest rate risk is relatively straightforward to handle,given adequate information. Depending on the maturity, swaps, eurofutures, or money market deposits can provide the necessary hedge,with or without actually changing the treasury balance. However, getting the adequate information is not easy considering the number ofdimensions involved and the possible movements within the yieldcurve. For a money market trader, hedging interest rate exposuremay be a complex process because of the multitude of instrumentsand maturities involved. For an equity trader, on the other hand, themanagement of this risk is simpler because it always appears as abyproduct of the main trading activity. Typically the interest raterisk created by trading a futures contract or forward, for example, iseasy to identify and neutralize because these are instruments with ade®ned expiry.In fact, in many instances the interest rate risk is simply ignoredbecause it is not relevant considering the speci®cs of the position. Forexample, a trader using an index futures contract only infrequently tohedge a minor part of his or her portfolio would probably not feel theneed to enter a short-term hedge on interest rates for the inducedexposure. Even if futures or other instruments such as foreign exchangederivatives are used often and for a large volume, if the time horizon ofa strategy is short enough, the real risk becomes very small.16We do not mean to imply that risks associated with interest ratesshould be neglected but rather that they can probably be considered15
A proration indicates an allocation of shares on a certain criterion. A company maydecide, for example, that a rights subscription is canceled if a certain threshold hasnot been met. In that case, the arbitrageur may have hedged the entire number ofshares he or she exercised and will eventually receive nothing.16For a position of typical life span equal to three months, for example, even a 0.50percent move in interest rates, fairly large, would create a risk of only 0.125 percent( 0:50 percent/4) on the total nominal.
CHAPTER 7 RISK ANALYSIS
345
from a distance in the particular context of an equity trading desk.Therefore, judgment and experience appear once more to be very valuable in making the most appropriate and ``cheapest'' decision byweighting the different factors.
Financing Risk
Financing risk is a subset of interest rate exposure, and it refers to theP&L generated in the very short term, typically because of the monetary balance carried overnight. This risk can be of two types: outrightand spread. An outright exposure indicates an additional ®nancingcost that needs to be accounted for and monitored. An example ofthis is typically the margin calls of futures. A spread exposure ariseswhen the overnight ®nancing is carried at a rate possibly different fromwhat has been implicitly priced in a traded instrument. The overnightsqueeze at the end of a ®scal year will serve as an example here.
Outright Risk
Suppose we buy 10 S&P 500 futures at $1,400, a position for which wehave to pay initial and maintenance margin calls every day. The initialmargin is 10  $20,438  $204,380, payable in cash or acceptable securities, typically treasury bills. If this position is kept for a week andexited at $1,400, the realized P&L is zero because the market did notmove. However, we had to pay ®nancing on $204,438 for a week. Thecost associated may be a cash market rate if we deposited cash or aborrowing fee if we borrowed securities. Therefore, the magnitudevaries, but the cost exists in both cases and applies to the entire initialmargin.If the market moves over the week, we pay or receive in additionmaintenance margins, for which again the deposit needs to be borrowed or lent. Here the funding cost may actually be a pro®t if wereceive margin payments. In any case, these daily payments vary withthe market level and as such can and should be hedged. In contrast,initial margin payments depend on the size of the position, not thelevel of the market and therefore cannot be hedged.To establish the algebraic expression of the hedge ratio, consider along position of n futures. The market prices are respectively FY and FTfor yesterday and today. Let r be the zero-coupon rate from today untilthe expiry of the futures. If we held n futures yesterday and the marketmoved from FY to FT , the maintenance margin call would be:MCT  PV  FT
FY   n
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FINANCIAL VALUATION AND RISK ANALYSIS
where PV is the futures point value. If the market remains unchangeduntil the expiry, the ®nancing generated by this margin call is MCT  r.In order for us to be hedged, we need to take an additional futuresposition  such that a move from FY to FT generates a loss equal to the®nancing of the margin call.  is then given by:  PV  FT
FY  
PV  FT
FY   n    r
The left term is the additional P&L generated by  futures, and the rightterm is the ®nancing on the margin call on (n  ) futures. This leads to:rn1rwhich in turn gives:

1nn1r

This result indicates that hedging the margin call ®nancing of a futuresposition requires changing the number of futures from n to n/(1 + r).17 If wetake r  5 percent over two months, 1= 1  r  1= 1  5 percent 2/12) = 0.99173. For a $1 billion positionÐthat is, 2,857 futuresÐtheadjustment is 23.6 futures, big enough to warrant attention.This example of ``parasitic'' ®nancing and the necessary hedge isvery common and widely used by futures traders.18 Note that thisadjustment is not necessary with forwards because of the absence ofmargin calls.Spread RiskConsider a portfolio in which we entered a swap to cover long-term®nancing needs. Having bought the swap, we pay the ®xed rate,receive the variable rate