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Not authorized for duplication or distribution.Reprinted fromAFP Exchange, September/October 2002.

Copyright2002 by Association for Financial Professionals (AFP). All rights reserved.

risk management

W

hen companies usederivative instru-

ments to hedge eco-nomic exposures, they gener-ally want to apply special hedgeaccounting. Without this spe-cial treatment, derivative gainsor losses and the gains or loss-es associated with the riskbeing hedged would hit earn-ings in different time periods.The resulting income volatili-ty masks the objectives of thehedging strategy; and with

hedge accounting, this volatil-ity may be largely avoidable.Hedge accounting is elective,but the problem is that com-panies must qualify for thistreatment.

Perhaps the most onerousrequirement to qualify forhedge accounting is that thederivatives results must beexpected to be highly effec-

tive in offsetting the changesin fair value or cash flows asso-

ciated with the risk beinghedged. Besides requiring acompany to make this assess-ment prospectively, Statementof Financial Accounting Stan-dards No. 133,Accounting forDerivative Instruments andHedging Activities (SFAS 133)also requires a retrospectiveassessment to be made on anongoing basis at eachreporting period, and no less

frequently than quarterly.FAS 133 does not prescribe

a single method for assessinghedge effectiveness; and in fact,different methods for prospec-tive and retrospective testingare permitted. What isrequired, however, is that theselected approach must be rea-sonable; it must also be con-sistent with the companys risk

Ira G. Kawaller, Ph.D.

Kawaller & Company, LLC

HedgeEffectiveness

TestingUsing Regression Analysis

Reva B. Steinberg

BDO Seidman LLP

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management strategy; it mustbe documented at the incep-tion of the hedging relation-ship; and it must be consis-tently applied throughout the

term of the hedge. Further,although different methodsmay be used for differenthedges, a company must beable to justify the use of differ-ent methodologies for similarhedges.

The simplest approach toassessing hedge effectiveness isthe dollar offset method.Under this method, the changein the value of the derivative iscompared to the change in thevalue of the hedged item. Inpractice, the term highly effec-tive is interpreted to mean thatthis ratio takes on a value with-in a range of 80-120 percent.

The drawback to thismethod, however, is that it isoften difficult to achieve higheffectiveness consistently, fromperiod to period1; and failingthe hedge effectiveness test pre-cludes the use of hedge

accounting. However, Deriva-tive Implementation Group(DIG) Issue E7, Methodology toAssess Effectiveness of Fair Valueand Cash Flow Hedges, permitscompanies to use regression orother statistical methods to testhedge effectiveness, therebypossibly permitting hedgeaccounting when the dollar off-set method would otherwisedisallow it.

Terminology

Regression analysis is a statis-tical technique that providesquantitative information aboutthe relationship between twoor more variables. In the con-text of FAS 133, the need toshow that a derivative will behighly effective translates to

showing that the price (orinterest rate or currencyexchange rate) associated withthe hedged item bears a closerelationship to the price asso-

ciated with the hedging deriv-ative.2 Simple regression (i.e.,a regression designed to explainthe relationship between twovariables) provides a summarystatistic, the correlation coeffi-cient, which quantifies thecloseness of the relationship.

Correlation coefficients mayrange in value from -1.0 to+1.0, where 1.0 is indicative ofa case where the two respectivevariables, X and Y, are perfect-ly correlated. That is, X and Yperform perfectly in a linearrelationship such that the databehave consistently and per-fectly, according to the follow-ing equation:

EQUATION (1): Y = aX + b

In this equation a is theslope coefficient and b isthe intercept; and the regres-

sion program provides numer-ical values for these two param-eters.

If the correlation coefficientis -1.0, X and Y are said to beperfectly inversely correlated.The regression equation wouldshow that the value of the slopecoefficient, a, in such a casewould be negative, such thatlarger values for X would con-sistently be associated with

smaller values for Y, and viceversa.

A correlation coefficientequal to zero suggests that noreliable linear relationshipbetween X and Y exists. Rather,the relationship between thesetwo variables is perfectly ran-dom. In this case a would havea value not statistically differ-ent from zero.

A related concept to the cor-relation coefficient is the coef-ficient of determination, or theR-Squared. The R-Squared isfound simply by squaring the

correlation coefficient, so thepossible range of the R-Squaredstatistic is from zero to one.While FAS 133 does not spec-ify how to distinguish highlyeffective hedges from those thatare less effective, in practice,highly effective is interpretedto describe a hedging relation-ship where the R-Square is ofat least 0.80.

Using a RegressionProgram

Most regression programs arefairly easy to use if the data aregiven. Collecting and prepar-ing the data, however, requiressome time and thought.

To perform the analysis,data for the X and Y variablesmust be collected data relat-ing to the price (or interest rateor currency exchange rate)

associated with the hedgeditem and the hedging deriva-tive, respectively. In assemblingthese data, the analyst mustmake judgments about thetime span of the data (i.e., thetime between the first and thelast observations) and the fre-quency (i.e., daily, weekly,monthly, etc.).

Often, however, the deci-sion may default to whatever

data are available. In any case,for statistically reliable results,25-30 observations should beconsidered to be the bare min-imum requirement; and if thisthreshold cannot be met, analternative to regression analy-sis should be explored.

Clearly, if daily data wereavailable, the company onlywould have to collect data for

a little more than a month tomeet the minimum require-ments, but a larger data set,starting at an earlier point intime, would definitely be pre-

ferred. Ideally, the companyshould try to incorporate asmuch relevant information aspossible. Determining what isrelevant requires considerable judgment. This judgmentshould incorporate a perspec-tive as to what has happenedand what could happen, to theextent possible and practical.

For instance, in choosing atime span, if earlier data werethought to be unrepresentativeof the current price relationship say, because of some struc-tural change in the market (e.g.,a change in tax treatment or anew regulatory regime) itwould be appropriate to use ashorter, more recent data set.Put another way, a companyshould not use historical datathat may not represent currentor future market conditions.

On the other hand, too

short a time period might beequally problematic. For exam-ple, suppose over the last sev-eral months interest rates wererelatively stable. A regressionanalysis based on this shorttime span would offer a poorindication of how a hedgemight perform when interestrates become more volatile.

An Example

Consider the case where a com-pany wants to assess whethera pay fixed /receive 3-monthLIBOR (i.e., the London Inter-bank Offer Rate on Eurodollardeposits) interest rate swapcould be used to hedge thevariable interest expense asso-ciated with a prime-based loan.Under the terms of the loan, the

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interest rate is reset wheneverthe prime rate changes.

Interest on the swap is reseton the first business day of eachcalendar quarter. Ideally, theclient should find these twodata series, but sometimes theright data (i.e., the data thatprecisely correspond to theprice, interest rate, or curren-

cy that is under consideration)simply do not exist. In that casea close substitute will have tosuffice.

Other times, seemingly sim-ilar data may be readily avail-able often for free whilebetter data may be availablefor a price. Here, the trade-offneeds to be evaluated on a case-by-case basis.

Returning to the example at

hand, LIBOR is the interest rateunderlying the variable inter-est exposure, and therefore thecompany would want to col-lect an historical time series forLIBOR. The company evaluat-ing this hedge might refer tothe Federal Reserve SystemsH15 report for historical inter-est rates.3 Unfortunately, despitebeing an especially popular

benchmark interest rate, thesedata seem to be available onlyon a paid subscription basis.

It is important to under-stand exactly what the dataused in a regression represent,and the explanatory footnotesin the H15 report provide thisinformation. While the H15report does not provide a price

history of LIBOR, it does havedata on bid rates for 3-monthEurodollar deposits.

LIBOR, however, as earliernoted, stands for the LondonInterbank Offer Rate of theEurodollar deposit market.What is at issue here is thespread between the bid and theoffer. This spread tends to vary but only slightly typical-ly ranging between 1/16 and

1/8 of a percent.Those with greater market

experience are obviously in abetter position to gaugewhether this compromise isacceptable or not; and some-times, consultation might beneeded to make the requireddetermination. In the currentexample, because the spreadbetween the bids and offers for

Eurodollar deposits tend torange between 1/16 and 1/8 ofa percent, and this spread isonly a tiny fraction of the levelof interest rates, we have con-structed a proxy series forLIBOR by adding a spread of0.0009375 (= 3/32nds of a per-cent) to the bid rate.4

Both series from the H15

report are available by businessday, week, month, and annu-ally. A closer look at the foot-notes, however, shows that theweekly data are not directlycomparable, as the Eurodollardeposit rates reflect averagesfor weeks ending on Fridays,while the prime rates reflectaverages for weeks ending on Wednesdays. Monthly andannual data are comparable,

reflecting averages rates overthe same time periods.

At this point it is important tofully appreciate the way the inter-est expense under the prime-rate-based loan and the swaps cashflow settlements are calculated.For the swap, LIBOR on any ratesetting date determines the cashflow requirement relevant to thecoming three months.

For example, assumingquarterly cash flow settlements,if LIBOR is set on January 15,the associated cash flow for thisinterest rate will be made three

months later on April 15

5

(assuming this date is a work-ing business day). Alternative-ly, the interest on a prime-rate-based loan generally is adjustedimmediately with a change inthe prime rate, such that theeffective interest rate is the aver-age prime rate throughout theterm.

To construct appropriatedata for the regression analy-sis, we start by importing thebusiness day data for the twoseries into an Excel spread-sheet. On some business days,one or the other of the seriesmay be missing data, general-ly because of banking holidays.Our adjusted data set spans January 2, 1997, throughDecember 31, 2001, includingonly those dates for which bothprice series are reported.

Exhibit 1 shows an abbre-

viated Excel spreadsheet.Columns A, B, and C show thedates, the Eurodollar bid ratesand the prime rate, respective-ly the unaltered original datafrom the H15 release. ColumnD is the estimated LIBOR,found by adding 3/32nds of abasis point to the rates shownin Column B. Column E showsthe average prime rates over thecoming three months. Reflect-

ing the fact that rates are notquoted for weekends and hol-idays, the average prime ratecalculation assumes that eachquarter has 63 business days.6

With the data arrayed in thisway, the regression tool may beemployed. All that is requiredis to specify the data in columnsD and E for the X and Y vari-ables, respectively. Note that

A B C D E1 Date ED-bid Prime LIBOR Avg Prime

2 1.2.97 5.44 8.25 5.468 8.2773 1.3.97 5.44 8.25 5.468 8.2814 1.6.97 5.44 8.25 5.468 8.2855 1.7.97 5.44 8.25 5.468 8.2896 1.8.97 5.44 8.25 5.468 8.3937 1.9.97 5.44 8.25 5.468 8.2978 1.10.97 5.44 8.25 5.468 8.3019 1.13.97 5.44 8.25 5.468 8.305

10 1.14.97 5.44 8.25 5.468 8.30911 1.14.97 5.44 8.25 5.468 8.31312 1.16.97 5.44 8.25 5.468 8.316

1189 9.26.01 2.51 6.00 2.538 5.219

Exhibit 1. Setting up a Spreadsheet

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because of the averagingrequirement, the data used inthe regression run only throughthe September 26, 2001 obser-vation.

As far as the R-Square Sta-tistic is concerned, the order isnot important (i.e., LIBOR mayserve as the X variable and AvgPrime may be Y, or vice versa).The order does matter in termsof the resulting parameter esti-mates for a and b. Exhibit 2shows the sample results froman Excel regression package,where LIBOR served as the Yvariable, and Avg Prime servedas the X variable.

Multiple R is the same as thecorrelation coefficient. Squar-ing this value then gives the R-Square, which is the key sta-tistic for the purposes of hedgeeffectiveness testing.

Another statistic related tothe R-Square is the adjustedR-Square. In cases where onlytwo variables are compared, theadjusted R-Square and the R-Square will be identical.7

At the bottom of thistable, the intercept and slope

coefficients are given (-2.208and 0.932 respectively). Thus,according to the regressionresults, the relationshipbetween LIBOR and the aver-age prime rate can be expressedby the following equation:

EQUATION (2): LIBOR =0.932 Avg Prime 2.208

One might expect that ifequation (2) were the trueequation relating the averageprime rate to LIBOR (remem-ber, the regression equation onlyestimates the true relationship),then we should be able torearrange the terms in Equation

(2) and solve for LIBOR. Theresult would be the following:

EQUATION (3): Avg Prime =1.073 LIBOR + 2.370

However, if we re-run theregression, this time specifyingAvg Prime as the Y variable, weget something else. (See Exhib-it 3.) Specifically:

EQUATION (4): Avg Prime =1.019 LIBOR + 2.666

The discrepancy arisesbecause the regression analysishas two distinct objectives inmind for the two respective tri-als. That is, in the first case, wherethe average prime rate is the Yvariable, the regression providesestimates for parameters a andb

that minimize the differencesbetween the observed LIBORsand those predicted by theregression equation. In the sec-ond case, the regression line min-imizes the difference between theobserved Avg Primes and the AvgPrimes predicted by the regres-sion equation.

To illustrate, Exhibit 4shows a plot of the pairs of data

SUMMARY OUTPUT

Regression Statistics

Multiple R 0.975

R Square 0.950

Adj. R Square 0.950

Standard Error 0.176

Observations 1188

ANOVA

df SS MS F Signit F

Regression 1 692.620 692.620 22373.488 0.000

Residual 1185 36.550 0.031

Total 1167 729.171

Lower Upper Lower Upper

Coeffs Std Error t Stat P-Value 95% 95% 95% 95%Intercept -2.208 0.052 -42.676 0.000 -2.310 -2.107 -2.310 -2.107

X Variable 1 0.932 0.006 149.915 0.000 0.920 0.944 0.920 0.944

Exhibit 2. Sample Regression Output (LIBOR = Y Variable)

SUMMARY OUTPUT

Regression Statistics

Multiple R 0.975

R Square 0.950

Adj. R Square 0.950

Standard Error 0.184

Observations 1188

ANOVA

df SS MS F Signit F

Regression 1 757.597 757.597 22474.488 0.000

Residual 1186 39.979 0.034

Total 1187 797.576

Lower Upper Lower Upper

Coeffs Std Error t Stat P-Value 95% 95% 95% 95%

Intercept -2.666 0.036 70.440 0.000 2.592 2.740 2.592 2.740

X Variable 1 1.019 0.007 149.915 0.000 1.006 1.033 1.006 1.033

Exhibit 2. Sample Regression Output (Avg Prime = Y Variable)

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Not authorized for duplication or distribution.Reprinted fromAFP Exchange, September/October 2002.

Copyright2002 by Association for Financial Professionals (AFP). All rights reserved.

risk management

LIBOR and the averageprime rate for all datesbetween January 2, 1997 andSeptember 26, 2001. In thiscase, the average prime rate

is shown as the Y variable. Alternatively, the secondregression result minimizesthe horizontal differencesbetween the dots and thesolid line. These are two dif-ferent objectives, and there-fore they result in two dif-ferent regression equations.

In the Long Run

SFAS 133 requires testingfor hedge effectiveness bothprospectively and retro-spectively. Testing is notrequired on an ongoing basis,but at least quarterly. We rec-ommend that regression beused for the prospective testand dollar offset for retrospec-tive tests. We make these rec-ommendations even though weexpect the dollar offset retro-spective test may likely fail with

high frequency.In the event that the retro-

spective dollar offset ratio fallsbetween 0.80 and 1.20, this

result validates the expectationof high effectiveness, and hedgeaccounting may be applied. Inthis case, no additional prospec-tive regressions analysis wouldbe needed. However, if the dol-lar offset retrospective test fails,DIG Issue E7 permits a com-pany to continue hedge

accounting without interrup-tion, so long as the prospectiveregression test satisfied anew andthus the hedging relationship may

still be expected to be highly effec-tive on a go-forward basis. Thus,assuming this updated regres-sion again passes, the failureof the dollar offset ratio retro-spective test would not prohibithedge accounting.

The revised regressionanalysis would necessarily have

to incorporate the recent obser-vations from the period justpassed; and the documentationwould have to be specific about

how the data set will be altered.For instance, the analyst mayeither extend the original dataset or re-run the regressionusing a constant number of

observations (i.e., dropping offthe older data points andadding the newer). Eitherapproach is acceptable, as longas the procedure is fully docu-mented and followed consis-tently.

In the long run, we expectthat this approach to allowfor hedge accounting tobe applied with the fewestinterruptions.

Ira G. Kawaller, Ph.D., is the

founder of Kawaller & Com-

pany, LLC, a Brooklyn, NY-

based financial consulting

company that specializes in

assisting commercial enter-

prises in their use of deriv-

ative instruments.

Reva B. Steinberg is a direc-

tor in the national SECdepartment of BDO Seid-

man LLP, the U.S. member

of BDO International.

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Avg.

PrimeRates

Exhibit 4. Scatter Diagram

Endnotes1 See Kawaller, I.G., The 80/125 Problem, Derivatives Strategy, March 2001 for a more detailed discussion of some of the shortcomings of applying dollar offset method for testing hedgeeffectiveness.2 Paragraph 75 of FAS 133 asserts that if the price of the hedged item and the price of the hedging derivative are highly correlated, it is reasonable to expect the hedge to be highly effec-tive in offsetting price changes. For an expanded discussion of the validity of this point, see Kawaller, I.G. and Paul Koch, Meeting the Highly Effective Expectation Criterion for Hedge

Accounting, The Journal of Derivatives, Summer 2000.3 See http://www.federalreserve.gov/releases/H15/data.htm#top.4 Using the unadjusted bid rates (as apposed to this adjusted series) would yield the same R-Square statistic. On the other hand, the adjustment would affect the intercept parameter, b,in the regression equation.5 Sometimes, the swap documentation will distinguish between a trade date and a value date. The trade date is the date at which an interest rate is determined, but the value date isthe start of the period for which the interest rate applies. In Eurodollar deposit market, the normal convention calls for the value date to follow the trade date by two business days, andthe swap documentation may adopt this convention. If that were true in the case of the current example, cash settlements would occur on three-month anniversaries of the value date,rather than on three-month anniversaries of the trade date.6 Because the columns are truncated, the average prime rates cannot be validated. Column E reflects the fact that the prime rate rose to 8.50 percent on March 26, 1997 and remained atthat higher rate until September 30, 1998, as they reflect the fact that the prime rate increased.7 Formally, the R-Square is said to measure the proportion of the total variance that is explained by the linear equation. If other right-hand side variables are introduced in the equation,the R-Square would necessarily increase. The adjusted R-Square, makes a correction to the R-Square statistic, such that adjusted R-Squares are comparable, irrespective of the number ofright-hand side variables. In other words, the adjusted R-Square is a statistic that reveals whether the inclusion of additional right hand side variable adds to the explanatory capability ofthe equation. As the sample size expands, however, adjusted R-Squares will tend to converge to the unadjusted R-Square. In the example shown, the difference is lost in the rounding.

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