Evaluation of Value-at-Risk Models Using Historical Data

2020-10-02 来源：爱问旅游网

Evaluation of Value-at-Risk ModelsUsing Historical DataDarryll HendricksThe views expressed in this article are those of the authors and do not necessarily reﬂect the position of the FederalReserve Bank of New York or the Federal Reserve System.

The Federal Reserve Bank of New York provides no warranty, express or implied, as to the accuracy, timeliness, com-pleteness, merchantability, or ﬁtness for any particular purpose of any information contained in documents producedand provided by the Federal Reserve Bank of New York in any form or manner whatsoever.

esearchers in the ﬁeld of ﬁnancial economicshave long recognized the importance of mea-suring the risk of a portfolio of ﬁnancialassets or securities. Indeed, concerns go back

at least four decades, when Markowitz’s pioneering workon portfolio selection (1959) explored the appropriate deﬁ-nition and measurement of risk. In recent years, thegrowth of trading activity and instances of ﬁnancial marketinstability have prompted new studies underscoring theneed for market participants to develop reliable risk mea-surement techniques.1

One technique advanced in the literature involves

the use of“value-at-risk” models. These models measure themarket, or price, risk of a portfolio of ﬁnancial assets—thatis, the risk that the market value of the portfolio willdecline as a result of changes in interest rates, foreignexchange rates, equity prices, or commodity prices. Value-at-risk models aggregate the several components of pricerisk into a single quantitative measure of the potential forlosses over a speciﬁed time horizon. These models are clearlyappealing because they convey the market risk of the entireportfolio in one number. Moreover, value-at-risk measuresfocus directly, and in dollar terms, on a major reason forassessing risk in the ﬁrst place—a loss of portfolio value.

Recognition of these models by the ﬁnancial and

regulatory communities is evidence of their growing use.For example, in its recent risk-based capital proposal(1996a), the Basle Committee on Banking Supervisionendorsed the use of such models, contingent on importantqualitative and quantitative standards. In addition, theBank for International Settlements Fisher report (1994)urged ﬁnancial intermediaries to disclose measures ofvalue-at-risk publicly. The Derivatives Policy Group, afﬁli-ated with six large U.S. securities ﬁrms, has also advocatedthe use of value-at-risk models as an important way tomeasure market risk. The introduction of the RiskMetricsdatabase compiled by J.P. Morgan for use with third-partyvalue-at-risk software also highlights the growing use ofthese models by ﬁnancial as well as nonﬁnancial ﬁrms.

Clearly, the use of value-at-risk models is increas-

FRBNYECONOMICPOLICYREVIEW/APRIL199639

ing, but how well do they perform in practice? This articleexplores this question by applying value-at-risk models to1,000 randomly chosen foreign exchange portfolios overthe period 1983-94. We then use nine criteria to evaluatemodel performance. We consider, for example, how closelyrisk measures produced by the models correspond to actualportfolio outcomes.We begin by explaining the three most commoncategories of value-at-risk models—equally weighted mov-ing average approaches, exponentially weighted movingaverage approaches, and historical simulation approaches.Although within these three categories many differentapproaches exist, for the purposes of this article we select ﬁveapproaches from the ﬁrst category, three from the second,and four from the third.By employing a simulation technique using thesetwelve value-at-risk approaches, we arrived at measures ofprice risk for the portfolios at both 95 percent and 99 per-cent conﬁdence levels over one-day holding periods. The con-ﬁdence levels specify the probability that losses of aportfolio will be smaller than estimated by the risk mea-sure. Although this article considers value-at-risk modelstwelve approaches we examine is superior on every count.In addition, as the results make clear, the choice of conﬁ-dence level—95 percent or 99 percent—can have a sub-stantial effect on the performance of value-at-riskapproaches.INTRODUCTIONTOVALUE-AT-RISKMODELSA value-at-risk model measures market risk by determin-ing how much the value of a portfolio could decline over agiven period of time with a given probability as a result ofchanges in market prices or rates. For example, if thegiven period of time is one day and the given probabilityis 1 percent, the value-at-risk measure would be an estimateofthe decline in the portfolio value that could occur with a1 percent probability over the next trading day. In otherwords, if the value-at-risk measure is accurate, lossesgreater than the value-at-risk measure should occur lessthan 1 percent of the time.The two most important components of value-at-risk models are the length of time over which market risk isto be measured and the conﬁdence level at which market riskis measured. The choice of these components by risk manag-ers greatly affects the nature of the value-at-risk model.The time period used in the deﬁnition of value-at-risk, often referred to as the “holding period,” is discretion-ary. Value-at-risk models assume that the portfolio’s com-position does not change over the holding period. Thisassumption argues for the use of short holding periodsbecause the composition of active trading portfolios is aptto change frequently. Thus, this article focuses on thewidely used one-day holding period.3Value-at-risk measures are most often expressed aspercentiles corresponding to the desired conﬁdence level.For example, an estimate of risk at the 99 percent conﬁ-dence level is the amount of loss that a portfolio isexpected to exceed only 1 percent of the time. It is alsoknown as a 99th percentile value-at-risk measure becausethe amount is the 99th percentile of the distribution ofpotential losses on the portfolio.4 In practice, value-at-riskestimates are calculated from the 90th to 99.9th percen-tiles, but the most commonly used range is the 95th to99th percentile range. Accordingly, the text charts and theClearly, the use of value-at-risk models isincreasing, but how well do theyperform in practice?only in the context of market risk, the methodology isfairly general and could in theory address any source of riskthat leads to a decline in market values. An important lim-itation of the analysis, however, is that it doesnot considerportfolios containing options or other positions with non-linear price behavior.2We choose several performance criteria to reﬂectthe practices of risk managers who rely on value-at-riskmeasures for many purposes. Although important differ-ences emerge across value-at-risk approaches with respectto each criterion, the results indicate that none of the40FRBNYECONOMICPOLICYREVIEW/APRIL1996

tables in the appendixreport simulation results for each ofthese percentiles.

In the sections below, we describe the individualfeatures of the two variance-covariance approaches to value-at-risk measurement.THREECATEGORIESOFVALUE-AT-RISKAPPROACHES

Although risk managers apply many approaches when cal-culating portfolio value-at-risk models, almost all use pastdata to estimate potential changes in the value of the port-folio in the future. Such approaches assume that the futurewill be like the past, but they often deﬁne the past quitedifferently and make different assumptions about howmarkets will behave in the future.

The ﬁrst two categories we examine, “variance-covariance” value-at-risk approaches,5 assume normalityand serial independence and an absence of nonlinear posi-tions such as options.6 The dual assumption of normalityand serial independence creates ease of use for two reasons.First, normality simpliﬁes value-at-risk calculationsbecause all percentiles are assumed to be known multiplesof the standard deviation. Thus, the value-at-risk calcula-tion requires only an estimate of the standard deviation ofthe portfolio’s change in value over the holding period.Second, serial independence means that the size of a pricemove on one day will not affect estimates of price moves onany other day. Consequently, longer horizon standard devi-ations can be obtained by multiplying daily horizon stan-dard deviations by the square root of the number of days inthe longer horizon. When the assumptions of normalityand serial independence are made together, a risk managercan use a single calculation of the portfolio’s daily horizonstandard deviation to develop value-at-risk measures forany given holding period and any given percentile.

The advantages of these assumptions, however,

must be weighed against a large body of evidence suggest-ing that the tails of the distributions of daily percentagechanges in ﬁnancial market prices, particularly foreignexchange rates, will be fatter than predicted by the normaldistribution.7 This evidence calls into question the appeal-ing features of the normality assumption, especially forvalue-at-risk measurement, which focuses on the tails ofthe distribution. Questions raised by the commonly usednormality assumption are highlighted throughout the article.

EQUALLYWEIGHTEDMOVINGAVERAGEAPPROACHESThe equally weighted moving average approach, the morestraightforward of the two, calculates a given portfolio’svariance (and thus, standard deviation) using a ﬁxedamount of historical data.8 The major difference amongequally weighted moving average approaches is the timeframe of the ﬁxed amount of data.9 Some approachesemploy just the most recent ﬁfty days of historical data onthe assumption that only very recent data are relevant toestimating potential movements in portfolio value. Otherapproaches assume that large amounts of data are necessaryto estimate potential movements accurately and thus relyon a much longer time span—for example, ﬁve years.The calculation of portfolio standard deviationsusing an equally weighted moving average approach is(1)σt=1---------------(k–1)t–1s=t–k∑(xs–µ)2,whereσt denotes the estimated standard deviation of theportfolio at the beginning of dayt. The parameterk speci-ﬁes the number of days included in the moving average(the “observation period”),xs, the change in portfolio valueon days, andµ, the mean change in portfolio value. Fol-lowing the recommendation of Figlewski (1994),µ isalways assumed to be zero.10Consider ﬁve sets of value-at-risk measures withperiods of 50, 125, 250, 500, and 1,250 days, or about twomonths, six months, one year, two years, and ﬁve years ofhistorical data. Using three of these ﬁve periods of time,Chart 1 plots the time series of value-at-risk measures atbiweekly intervals for a single ﬁxed portfolio of spot for-eign exchange positions from 1983 to 1994.11As shown,the ﬁfty-day risk measures are prone to rapid swings. Con-versely, the 1,250-day risk measures are more stable overlong periods of time, and the behavior of the 250-day riskmeasures lies somewhere in the middle.FRBNYECONOMICPOLICYREVIEW/APRIL199641

EXPONENTIALLYWEIGHTEDMOVINGAVERAGEAPPROACHESExponentially weighted moving average approachesemphasize recent observations by using exponentiallyweighted moving averages of squared deviations. In con-trast to equally weighted approaches, these approachesattach different weights to the past observations containedin the observation period. Because the weights declineexponentially, the most recent observations receive muchmore weight than earlier observations. The formula for theportfolio standard deviation under an exponentiallyweighted moving average approach ist–1the equally weighted moving averages, the parameterµ isassumed to equal zero.Exponentially weighted moving average approachesclearly aim to capture short-term movements in volatility,the same motivation that has generated the large body of lit-erature on conditional volatility forecasting models.12 Infact, exponentially weighted moving average approaches areequivalent to the IGARCH(1,1) family of popular condi-tional volatility models.13Equation 3 gives an equivalentformulation of the model and may also suggest a more intu-itive understanding of the role of the decay factor:(3)σt=λσt2–1+(1–λ)(xt–1–µ)2.(2)σt=(1–λ)s=t–k∑λt–s–1(xs–µ)2.The parameterλ, referred to as the “decay factor,”determines the rate at which the weights on past observa-tions decay as they become more distant. In theory, for theweights to sum to one, these approaches should use an inﬁ-nitely large number of observationsk. In practice, for thevalues of the decay factorλ considered here, the sum of theweights will converge to one, with many fewer observa-tions than the 1,250 days used in the simulations. As withAs shown, an exponentially weighted average onany given day is a simple combination of two components:(1) the weighted average on the previous day, whichreceives a weight ofλ, and (2) yesterday’s squared devia-tion, which receives a weight of (1 -λ). This interactionmeans that thelower the decay factorλ, thefaster the decayin the inﬂuence of a given observation. This concept isillustrated in Chart 2, which plots time series of value-at-risk measures using exponentially weighted moving aver-Chart 1Value-at-Risk Measures for a Single Portfolio over Time󰀀Equally Weighted Moving Average Approaches Millions of Dollars1050 days8250 days61,250 days4201983848586878889909192939495Source: Author’s󰀀 calculations.42FRBNYECONOMICPOLICYREVIEW/APRIL1996

ages with decay factors of 0.94 and 0.99. A decay factor of0.94 implies a value-at-risk measure that is derived almostentirely from very recent observations, resulting in thehigh level of variability apparent for that particular series.

On the one hand, relying heavily on the recent

past seems crucial when trying to capture short-termmovements in actual volatility, the focus of conditionalvolatility forecasting. On the other hand, the reliance onrecent data effectively reduces the overall sample size,increasing the possibility of measurement error. In the lim-iting case, relying only on yesterday’s observation wouldproduce highly variable and error-prone risk measures.

HISTORICALSIMULATIONAPPROACHES

The third category of value-at-risk approaches is similar tothe equally weighted moving average category in that itrelies on a speciﬁc quantity of past historical observations(the observation period). Rather than using these observa-tions to calculate the portfolio’s standard deviation, how-ever, historical simulation approaches use the actualpercentiles of the observation period as value-at-risk mea-sures. For example, for an observation period of 500 days,the 99th percentile historical simulation value-at-risk mea-

sure is the sixth largest loss observed in the sample of 500outcomes (because the 1 percent of the sample that shouldexceed the risk measure equates to ﬁve losses).

In other words, for these approaches, the 95th and

99th percentile value-at-risk measures will not be constantmultiples of each other. Moreover, value-at-risk measuresfor holding periods other than one day will not be ﬁxedmultiples of the one-day value-at-risk measures. Historicalsimulation approaches do not make the assumptions ofnormality or serial independence. However, relaxing theseassumptions also implies that historical simulationapproaches do not easily accommodate translationsbetween multiple percentiles and holding periods.

Chart 3 depicts the time series of one-day 99th

percentile value-at-risk measures calculated through his-torical simulation. The observation periods shown are 125days and 1,250 days.14 Interestingly, the use of actual per-centiles produces time series with a somewhat differentappearance than is observed in either Chart 1 orChart 2. Inparticular, very abrupt shifts occur in the 99th percentilemeasures for the 125-day historical simulation approach.

Trade-offs regarding the length of the observation

period for historical simulation approaches are similar to

Chart 2Value-at-Risk Measures for a Single Portfolio over Time󰀀Exponentially Weighted Moving Average Approaches Millions of Dollars10Lambda = 0.948Lambda = 0.9964201983848586878889909192939495Source: Author’s󰀀 calculations.FRBNYECONOMICPOLICYREVIEW/APRIL199643

those for variance-covariance approaches. Clearly, thechoice of 125 days is motivated by the desire to captureshort-term movements in the underlying risk of the port-folio. In contrast, the choice of 1,250 days may be drivenby the desire to estimate the historical percentiles as accu-rately as possible. Extreme percentiles such as the 95th andparticularly the 99th are very difﬁcult to estimate accu-rately with small samples. Thus, the fact that historicalsimulation approaches abandon the assumption of normal-ity and attempt to estimate these percentiles directly is onerationale for using long observation periods.

portfolios over this time span. Second, the results giveinsight into the extent to which portfolio composition orchoice of sample period can affect results.

It is important to emphasize, however, that nei-ther the reported variability across portfolios nor variabil-ity over time can be used to calculate suitable standarderrors. The appropriate standard errors for these simulation

The simulation results provide a relativelycomplete picture of the performance of selectedvalue-at-risk approaches in estimating themarket risk of a large number of portfolios.SIMULATIONSOFVALUE-AT-RISKMODELS

This section provides an introduction to the simulationresults derived by applying twelve value-at-risk approachesto 1,000 randomly selected foreign exchange portfolios andassessing their behavior along nine performance criteria(see box). This simulation design has several advantages.First, by simulating the performance of each value-at-riskapproach for a long period of time (approximately twelveyears of daily data) and across a large number of portfolios,we arrive at a clear picture of how value-at-risk modelswould actually have performed for linear foreign exchange

results raise difﬁcult questions. The results aggregateinformation across multiple samples, that is, across the1,000 portfolios. Because the results for one portfolio arenot independent of the results for other portfolios, we can-not easily determine the total amount of information pro-

Chart 3Value-at-Risk Measures for a Single Portfolio over Time󰀀Historical Simulation Approaches Millions of Dollars108125 days1,250 days64201983848586878889909192939495Source: Author’s󰀀 calculations.44FRBNYECONOMICPOLICYREVIEW/APRIL1996

DATAANDSIMULATIONMETHODOLOGY

This article analyzes twelve value-at-risk approaches. Theseinclude ﬁve equally weighted moving average approaches (50days, 125 days, 250 days, 500 days, 1,250 days); three expo-nentially weighted moving average approaches (λ=0.94,λ=0.97,λ=0.99); and four historical simulation approaches(125 days, 250 days, 500 days, 1,250 days).

The data consist of daily exchange rates (bid prices

collected at 4:00 p.m. New York time by the Federal ReserveBank of New York) against the U.S. dollar for the followingeight currencies: British pound, Canadian dollar, Dutch guil-der, French franc, German mark, Italian lira, Japanese yen,and Swiss franc. The historical sample covers the periodJanuary 1, 1978, to January 18, 1995 (4,255 days).

Through a simulation methodology, we attempt to

determine how each value-at-risk approach would have per-formed over a realistic range of portfolios containing the eightcurrencies over the sample period. The simulation methodol-ogy consists of ﬁve steps:1.

Select a random portfolio of positions in the eight curren-cies. This step is accomplished by drawing the position ineach currency from a uniform distribution centered onzero. In other words, the portfolio space is a uniformlydistributed eight dimensional cube centered on zero.1

Calculate the value-at-risk estimates for the random port-folio chosen in step one using the twelve value-at-riskapproaches for each day in the sample—day 1,251 to day4,255. In each case, we draw the historical data from the1,250 days of historical data preceding the date for whichthe calculation is made. For example, the ﬁfty-dayequally weighted moving average estimate for a givendate would be based on the ﬁfty days of historical datapreceding the given date.

Calculate the change in the portfolio’s value for each dayin the sample—again, day 1,251 to day 4,255. Withinthe article, these values are referred to as the ex post port-folio results or outcomes.

Assess the performance of each value-at-risk approach forthe random portfolio selected in step one by comparingthe value-at-risk estimates generated by step two withthe actual outcomes calculated in step three.

Repeat steps one through four 1,000 times and tabulatethe results.

The upper and lower bounds on the positions in each currency are +100 million U.S. dollars and -100 million U.S. dollars, respectively.In fact,however, all of the results in the article are completely invariant to the scale of the random portfolios.

vided by the simulations. Furthermore, many of theperformance criteria we consider do not have straightfor-ward standard error formulas even for single samples.15

These stipulations imply that it is not possible

to use the simulation results to accept or reject speciﬁcstatistical hypotheses about these twelve value-at-riskapproaches. Moreover, the results should not in any way betaken as indicative of the results that would be obtained forportfolios including other ﬁnancial market assets, spanningother time periods, or looking forward. Finally, this articledoes not contribute substantially to the ongoing debateabout the appropriate approach to or interpretation of“backtesting” in conjunction with value-at-risk model-ing.16 Despite these limitations, the simulation results doprovide a relatively complete picture of the performance ofselected value-at-risk approaches in estimating the marketrisk of a large number of linear foreign exchange portfoliosover the period 1983-94.

For each of the nine performance criteria, Charts 4-12

provide a visual sense of the simulation results for 95thand 99th percentile risk measures. In each chart, the verti-cal axis depicts a relevant range of the performance crite-rion under consideration (value-at-risk approaches arearrayed horizontally across the chart). Filled circles depictthe average results across the 1,000 portfolios, and theboxes drawn for each value-at-risk approach depict the5th, 25th, 50th, 75th, and 95th percentiles of the distri-bution of the results across the 1,000 portfolios.17 In somecharts, a horizontal line is drawn to highlight how theresults compare with an important point of reference.Simulation results are also presented in tabular form inthe appendix.

FRBNYECONOMICPOLICYREVIEW/APRIL199645

MEANRELATIVEBIAS

The ﬁrst performance criterion we examine is whether thedifferent value-at-risk approaches produce risk measures ofsimilar average size. To ensure that the comparison is notinﬂuenced by the scale of each simulated portfolio, we use afour-step procedure to generate scale-free measures of therelative sizes for each simulated portfolio.

First, we calculate value-at-risk measures for each

of the twelve approaches for the portfolio on each sampledate. Second, we average the twelve risk measures for eachdate to obtain the average risk measure for that date for theportfolio. Third, we calculate the percentage differencebetween each approach’s risk measure and the average riskmeasure for each date. We refer to these ﬁgures as daily rel-ative bias ﬁgures because they are relative only to theaverage risk measure across the twelve approaches ratherthan to any external standard. Fourth, we average the dailyrelative biases for a given value-at-risk approach across allsample dates to obtain the approach’s mean relative bias forthe portfolio.

Intuitively, this procedure results in a measure of

size for each value-at-risk approach that is relative to theaverage of all twelve approaches. The mean relative bias for

a portfolio is independent of the scale of the simulatedportfolio because each of the daily relative bias calculationson which it is based is also scale-independent. This inde-pendence is achieved because all of the value-at-riskapproaches we examine here are proportional to the scale ofthe portfolio’s positions. For example, a doubling of the

Actual 99th percentiles for the foreign exchangeportfolios considered in this article tend to belarger than the normal distribution wouldpredict.scale of the portfolio would result in a doubling of thevalue-at-risk measures for each of the twelve approaches.

Mean relative bias is measured in percentage

terms, so that a value of 0.10 implies that a given value-at-risk approach is 10 percent larger, on average, than theaverage of all twelve approaches. The simulation resultssuggest that differences in the average size of 95th percen-

Chart 4aChart 4bMean Relative Bias󰀀95th Percentile Value-at-Risk MeasuresPercent0.2Mean Relative Bias󰀀99th Percentile Value-at-Risk MeasuresPercent0.30.20.10.100-0.1-0.150d250d125dhs250hs1250λ󰀀=0.971250dλ󰀀=0.94hs125hs500λ=0.99500d-0.250d250d125dhs250hs1250λ󰀀=0.971250dλ=0.94hs125hs500λ󰀀=0.99500dSource: Author’󰀀s calculations.󰀀Notes: d=days; hs=historical simulation; λ=exponentially weighted.Source: Author’󰀀s calculations.Notes: d=days; hs=historical simulation; λ=exponentially weighted.46FRBNYECONOMICPOLICYREVIEW/APRIL1996

Chart 5aChart 5bRoot Mean Squared Relative Bias󰀀95th Percentile Value-at-Risk MeasuresPercent0.25Root Mean Squared Relative Bias󰀀99th Percentile Value-at-Risk MeasuresPercent0.350.300.200.250.150.200.150.100.050.0500250d125dhs250hs1250λ󰀀=0.971250dλ󰀀=0.94hs125hs500λ󰀀=0.99500d0.1050d250d125dhs250hs1250λ󰀀=0.971250dλ=0.94hs125hs500λ=0.99500d50dSource: Author’󰀀s calculations.Notes: d=days; hs=historical simulation; λ=exponentially weighted.Source: Author’󰀀s calculations.Notes: d=days; hs=historical simulation; λ=exponentially weighted.tile value-at-risk measures are small. For the vast majorityof the 1,000 portfolios, the mean relative biases for the95th percentile risk measures are between -0.10 and 0.10(Chart 4a). The averages of the mean relative biases acrossthe 1,000 portfolios are even smaller, indicating that acrossapproaches little systematic difference in size exists for95th percentile value-at-risk measures.

For the 99th percentile value-at-risk measures,

however, the results suggest that historical simulationapproaches tend to produce systematically larger risk mea-sures. In particular, Chart 4b shows that the 1,250-day his-torical simulation approach is, on average, approximately13 percent larger than the average of all twelve approaches;for almost all of the portfolios, this approach is more than5 percent larger than the average risk measure.

Together, the results for the 95th and 99th percen-tiles suggest that the normality assumption made by all ofthe approaches, except the historical simulations, is morereasonable for the 95th percentile than for the 99th percen-tile. In other words, actual 99th percentiles for the foreignexchange portfolios considered in this article tend to belarger than the normal distribution would predict.

Interestingly, the results in Charts 4a and 4b also

suggest that the use of longer time periods may producelarger value-at-risk measures. For historical simulationapproaches, this result may occur because longer horizonsprovide better estimates of the tail of the distribution. Theequally weighted approaches, however, may require a dif-ferent explanation. Nevertheless, in our simulations thetime period effect is small, suggesting that its economicsigniﬁcance is probably low.18

ROOTMEANSQUAREDRELATIVEBIAS

The second performance criterion we examine is the degreeto which the risk measures tend to vary around the averagerisk measure for a given date. This criterion can be com-pared to a standard deviation calculation; here the devia-tions are the risk measure’s percentage of deviation fromthe average across all twelve approaches. The root meansquared relative bias for each value-at-risk approach is cal-culated by taking the square root of the mean (over allsample dates) of the squares of the daily relative biases.

The results indicate that for any given date, a dis-persion in the risk measures produced by the differentvalue-at-risk approaches is likely to occur. The average rootmean squared relative biases, across portfolios, tend to fall

FRBNYECONOMICPOLICYREVIEW/APRIL199647

largely in the 10 to 15 percent range, with the 99th per-centile risk measures tending toward the higher end(Charts 5a and 5b). This level of variability suggests that,in spite of similar average sizes across the different value-at-risk approaches, differences in the range of 30 to 50 per-cent between the risk measures produced by speciﬁcapproaches on a given day are not uncommon.

Surprisingly, the exponentially weighted average

approach with a decay factor of 0.99 exhibits very low rootmean squared bias, suggesting that this particularapproach is very close to the average of all twelveapproaches. Of course, this phenomenon is speciﬁc to thetwelve approaches considered here and would not necessar-ily be true of exponentially weighted average approachesapplied to other cases.

ANNUALIZEDPERCENTAGEVOLATILITY

The third performance criterion we review is the tendencyof the risk measures to ﬂuctuate over time for the sameportfolio. For each portfolio and each value-at-riskapproach, we calculate the annualized percentage volatilityby ﬁrst taking the standard deviation of the day-to-daypercentage changes in the risk measures over the sample

period. Second, we put the result on an annualized basis bymultiplying this standard deviation by the square root of250, the number of trading days in a typical calendar year.We complete the second step simply to make the resultscomparable with volatilities as they are often expressed inthe marketplace. For example, individual foreign exchangerates tend to have annualized percentage volatilities in therange of 5 to 20 percent, although higher ﬁgures some-times occur. This result implies that the value-at-riskapproaches with annualized percentage volatilities inexcess of 20 percent (Charts 6a and 6b) will ﬂuctuate moreover time (for the same portfolio) than will most exchangerates themselves.

Our major observation for this performance cri-terion is that the volatility of risk measures increases asreliance on recent data increases. As shown in Charts 6aand 6b, this increase is true for both the 95th and 99thpercentile risk measures and for all three categories ofvalue-at-risk approaches. This result is not surprising, andindeed it is clearly apparent in Charts 1-3, which depicttime series of different value-at-risk approaches over thesample period. Also worth noting in Charts 6a and 6b isthat for a ﬁxed length of observation period, historical sim-

Chart 6aChart 6bAnnualized Percentage Volatility󰀀95th Percentile Value-at-Risk MeasuresPercent1.25Annualized Percentage Volatility󰀀99th Percentile Value-at-Risk MeasuresPercent1.251.001.000.750.750.500.500.250.25050d250d125d500d1250dhs125hs250hs1250λ󰀀=0.97λ=0.94hs500λ󰀀=0.990.0050d250d125dhs250hs1250λ󰀀=0.971250dλ=0.94hs125hs500λ󰀀=0.99500dSource: Author’󰀀s calculations.Notes: d=days; hs=historical simulation; λ=exponentially weighted.Source: Author’󰀀s calculations.Notes: d=days; hs=historical simulation; λ=exponentially weighted.48FRBNYECONOMICPOLICYREVIEW/APRIL1996

ulation approaches appear to be more variable than the cor-responding equally weighted moving average approaches.

FRACTIONOFOUTCOMESCOVERED

Our fourth performance criterion addresses the fundamentalgoal of the value-at-risk measures—whether they cover theportfolio outcomes they are intended to capture. We calculatethe fraction of outcomes covered as the percentage of resultswhere the loss in portfolio value is less than the risk measure.

For the 95th percentile risk measures, the simula-tion results indicate that nearly all twelve value-at-riskapproaches meet this performance criterion (Chart 7a).For many portfolios, coverage exceeds 95 percent, and onlythe 125-day historical simulation approach captures lessthan 94.5 percent of the outcomes on average across all1,000 portfolios. In a very small fraction of therandomportfolios, the risk measures cover less than 94 percentof the outcomes.

Interestingly, the 95th percentile results suggest

that the equally weighted moving average approaches actu-ally tend to produce excess coverage (greater than 95 per-cent) for all observation periods except ﬁfty days. Bycontrast, the historical simulation approaches tend to pro-

vide either too little coverage or, in the case of the 1,250-day historical simulation approach, a little more than thedesired amount. The exponentially weighted movingaverage approach with a decay factor of 0.97 producesexact 95percent coverage, but for this approach the results

All twelve value-at-risk approaches eitherachieve the desired level of coverage or come veryclose to it on the basis of the percentageof outcomes misclassiﬁed.are more variable across portfolios than for the 1,250-dayhistorical simulation approach.

Compared with the 95th percentile results, the

99th percentile risk measures exhibit a more widespreadtendency to fall short of the desired level of risk coverage.Only the 1,250-day historical simulation approach attains99 percent coverage across all 1,000 portfolios, as shown inChart 7b. The other approaches cover between 98.2 and

Chart 7aChart 7bFraction of Outcomes Covered󰀀95th Percentile Value-at-Risk MeasuresPercent0.97Fraction of Outcomes Covered󰀀99th Percentile Value-at-Risk MeasuresPercent0.9950.960.9900.950.9850.940.9800.930.97550d250d125d500d1250dhs125hs250hs1250λ󰀀=0.97λ=0.94hs500λ󰀀=0.9950d125d250dhs250hs1250λ=0.971250dλ=0.94hs125hs500λ󰀀=0.99500dSource: Author’󰀀s calculations.Notes: d=days; hs=historical simulation; λ=exponentially weighted.Source: Author’󰀀s calculations.Notes: d=days; hs=historical simulation; λ=exponentially weighted.FRBNYECONOMICPOLICYREVIEW/APRIL199649

98.8 percent of the outcomes on average across portfolios.Of course, the consequences of such a shortfall in perfor-mance depend on the particular circumstances in whichthe value-at-risk model is being used. A coverage level of98.2percent when a risk manager desires 99 percentimplies that the value-at-risk model misclassiﬁes approxi-mately two outcomes every year (assuming that there are250 trading days per calendar year).

Overall, the results in Charts 7a and 7b support

the conclusion that all twelve value-at-risk approacheseither achieve the desired level of coverage or come veryclose to it on the basis of the percentage of outcomes mis-classiﬁed. Clearly, the best performer is the 1,250-day his-torical simulation approach, which attains almost exactcoverage for both the 95th and 99th percentiles, while theworst performer is the 125-day historical simulationapproach, partly because of its short-term construction.19One explanation for the superior performance of the 1,250-day historical simulation is that the unconditional distri-bution of changes in portfolio value is relatively stable andthat accurate estimates of extreme percentiles require theuse of long periods. These results underscore the problemsassociated with the assumption of normality for 99th per-centiles and are consistent with ﬁndings in other recentstudies of value-at-risk models.20

from the desired outcome, requires a multiple of only 1.04.On the whole, none of the approaches considered hereappears to understate 95th percentile risk measures on asystematic basis by more than 4 percent, and several appearto overstate them by small amounts.

For the 99th percentile risk measures, most value-at-risk approaches require multiples between 1.10 and1.15 to attain 99 percent coverage (Chart 8b). The 1,250-day historical simulation approach, however, is markedlysuperior to all other approaches. On average across all port-

Shortcomings in value-at-risk measures thatseem small in probability terms may be muchmore signiﬁcant when considered in terms of thechanges required to remedy them.MULTIPLENEEDEDTOATTAINDESIREDCOVERAGE

The ﬁfth performance criterion we examine focuses on thesize of the adjustments in the risk measures that would beneeded to achieve perfect coverage. We therefore calculateon an ex post basis the multiple that would have beenrequired for each value-at-risk measure to attain thedesired level of coverage (either 95 percent or 99 percent).This performance criterion complements the fraction ofoutcomes covered because it focuses on the size of thepotential errors in risk measurement rather than on thepercentage of results captured.

For 95th percentile risk measures, the simulation

results indicate that multiples very close to one are sufﬁ-cient (Chart 8a). Even the 125-day historical simulationapproach, which on average across portfolios is furthest

folios, no multiple other than one is needed for thisapproach to achieve 99 percent coverage. Moreover, com-pared with the other approaches, the historical simulationsin general exhibit less variability across portfolios withrespect to this criterion.

The fact that most multiples are larger than one is

not surprising. More signiﬁcant is the fact that the size ofthe multiples needed to achieve 99 percent coverage exceedsthe levels indicated by the normal distribution. For example,when normality is assumed, the 99th percentile would beabout 1.08 times as large as the 98.4th percentile, a level ofcoverage comparable to that attained by many of theapproaches (Chart 7b). The multiples for these approaches,shown in Chart 8b, are larger than 1.08, providing furtherevidence that the normal distribution does not accuratelyapproximate actual distributions at points near the 99thpercentile. More generally, the results also suggest that sub-stantial increases in value-at-risk measures may be neededto capture outcomes in the tail of the distribution. Hence,shortcomings in value-at-risk measures that seem small inprobability terms may be much more signiﬁcant when con-sidered in terms of the changes required to remedy them.

50FRBNYECONOMICPOLICYREVIEW/APRIL1996

These results lead to an important question: what

distributional assumptions other than normality can beused when constructing value-at-risk measures using avariance-covariance approach? The t-distribution is oftencited as a good candidate, because extreme outcomes occurmore often under t-distributions than under the normaldistribution.21 A brief analysis shows that the use of at-distribution for the 99th percentile has some merit.

To calculate a value-at-risk measure for a single

percentile assuming the t-distribution, the value-at-riskmeasure calculated with the assumption of normality ismultiplied by a ﬁxed multiple. As the results in Chart 8bsuggest, ﬁxed multiples between 1.10 and 1.15 are appro-priate for the variance-covariance approaches. It followsthat t-distributions with between four and six degrees offreedom are appropriate for the 99th percentile risk mea-sures.22 The use of these particular t-distributions, how-ever, would lead to substantial overestimation of 95thpercentile risk measures because the actual distributionsnear the 95th percentile are much closer to normality.Since the use of t-distributions for risk measurementinvolves a scaling up of the risk measures that are calcu-lated assuming normality, the distributions are likely to be

useful, although they may be more helpful for some per-centiles than for others.

AVERAGEMULTIPLEOFTAILEVENTTORISKMEASURE

The sixth performance criterion that we review relates tothe size of outcomes not covered by the risk measures.23Toaddress these outcomes, we measure the degree to whichevents in the tail of the distribution typically exceed thevalue-at-risk measure by calculating the average multipleof these outcomes (“tail events”) to their correspondingvalue-at-risk measures.

Tail events are deﬁned as the largest percentage

of losses measured relative to the respective value-at-riskestimate—the largest 5 percent in the case of 95th per-centile risk measures and the largest 1 percent in the caseof 99th percentile risk measures. For example, if thevalue-at-risk measure is $1.5 million and the actual port-folio outcome is a loss of $3 million, the size of the lossrelative to the risk measure would be two. Note that thisdeﬁnition implies that the tail events for one value-at-risk approach may not be the same as those for anotherapproach, even for the same portfolio, because the risk

Chart 8aChart 8b

Multiple Needed to Attain 95 Percent Coverage󰀀

95th Percentile Value-at-Risk Measures

Multiple1.2Multiple Needed to Attain 99 Percent Coverage󰀀

99th Percentile Value-at-Risk MeasuresMultiple1.31.11.21.01.10.91.00.80.950d250d125d

500d

1250dhs125

hs250hs1250λ=0.97λ󰀀=0.94hs500λ󰀀=0.99

50d250d125d

hs250hs1250λ󰀀=0.971250dλ=0.94hs125hs500λ󰀀=0.99500d

Source: Author’󰀀s calculations.

Notes: d=days; hs=historical simulation; λ=exponentially weighted.Source: Author’󰀀s calculations.

Notes: d=days; hs=historical simulation; λ=exponentially weighted.

FRBNYECONOMICPOLICYREVIEW/APRIL199651

measures for the two approaches are not the same. Hori-zontal reference lines in Charts 9a and 9b show where theaverage multiples of the tail event outcomes to the riskmeasures would fall if outcomes were normally distrib-uted and the value-at-risk approach produced a true 99thpercentile level of coverage.

In fact, however, the average tail event is almost

always a larger multiple of the risk measure than is pre-dicted by the normal distribution. For most of the value-at-risk approaches, the average tail event is 30 to 40 percentlarger than the respective risk measures for both the 95thpercentile risk measures and the 99th percentile risk mea-sures. This result means that approximately 1 percent ofoutcomes (the largest two or three losses per year) willexceed the size of the 99th percentile risk measure by anaverage of 30 to 40 percent. In addition, note that the 99thpercentile results in Chart 9b are more variable across port-folios than the 95th percentile results in Chart 9a; the aver-age multiple is also above 1.50 for a greater percentage ofthe portfolios for the 99th percentile risk measures.

The performance of the different approaches

according to this criterion largely mirrors their perfor-mance in capturing portfolio outcomes. For example, the1,250-day historical simulation approach is clearly supe-Chart 9arior for the 99th percentile risk measures. The equallyweighted moving average approaches also do very well forthe 95th percentile risk measures (Chart 7a).

MAXIMUMMULTIPLEOFTAILEVENTTORISKMEASURE

Our seventh performance criterion concerns the size of themaximum portfolio loss. We use the following two-stepprocedure to arrive at these measures. First, we calculatethe multiples of all portfolio outcomes to their respectiverisk measures for each value-at-risk approach for a particu-lar portfolio. Recall that the tail events deﬁned above arethose outcomes with the largest such multiples. Ratherthan average these multiples, however, we simply select thesingle largest multiple for each approach. This procedureimplies that the maximum multiple will be highly depen-dent on the length of the sample period—in this case,approximately twelve years. For shorter periods, the maxi-mum multiple would likely be lower.

Not surprisingly, the typical maximum tail event

is substantially larger than the corresponding risk measure(Charts 10a and 10b). For 95th percentile risk measures,the maximum multiple is three to four times as large as therisk measure, and for the 99th percentile risk measure, it is

Chart 9bAverage Multiple of Tail Event to Risk Measure󰀀95th Percentile Value-at-Risk MeasuresMultiple1.75Average Multiple of Tail Event to Risk Measure󰀀99th Percentile Value-at-Risk MeasuresMultiple1.751.501.501.251.251.001.0050d250d125d500d1250dhs125hs250hs1250λ=0.97λ󰀀=0.94hs500λ󰀀=0.9950d250d125dhs250hs1250λ󰀀=0.971250dλ=0.94hs125hs500λ󰀀=0.99500dSource: Author’󰀀s calculations.Notes: d=days; hs=historical simulation; λ=exponentially weighted.Source: Author’󰀀s calculations.Notes: d=days; hs=historical simulation; λ=exponentially weighted.52FRBNYECONOMICPOLICYREVIEW/APRIL1996

approximately 2.5 times as large. In addition, the resultsare variable across portfolios—for some portfolios, themaximum multiples are more than ﬁve times the 95th per-centile risk measure. The differences among results for thisperformance criterion, however, are less pronounced than1percent of events can in extreme cases entail losses substan-tially in excess of the risk measures generated on a daily basis.CORRELATIONBETWEENRISKMEASUREANDABSOLUTEVALUEOFOUTCOMEThe eighth performance criterion assesses how well the riskmeasures adjust over time to underlying changes in risk. Inother words, how closely do changes in the value-at-riskmeasures correspond to actual changes in the risk of theportfolio? We answer this question by determining the cor-relation between the value-at-risk measures for eachapproach and the absolute values of the outcomes. This cor-relation statistic has two advantages. First, it is not affectedby the scale of the portfolio. Second, the correlations are rel-atively easy to interpret, although even a perfect value-at-risk measure cannot guarantee a correlation of one betweenthe risk measure and the absolute value of the outcome.For this criterion, the results for the 95th percen-tile risk measures and 99th percentile risk measures arealmost identical (Charts 11a and 11b). Most striking is thesuperior performance of the exponentially weighted mov-ing average measures. This ﬁnding implies that theseapproaches tend to track changes in risk over time moreaccurately than the other approaches.Chart 10b

It is important not to view value-at-riskmeasures as a strict upper bound on the portfoliolosses that can occur.for some other criteria. For example, the 1,250-day histori-cal simulation approach is not clearly superior for the 99thpercentile risk measure—as it had been for many of theother performance criteria—although it does exhibit loweraverage multiples (Chart 9b).These results suggest that it is important not toview value-at-risk measures as a strict upper bound on theportfolio losses that can occur. Although a 99th percentilerisk measure may sound as if it is capturing essentially all ofthe relevant events, our results make it clear that the otherChart 10a

Maximum Multiple of Tail Event to Risk Measure󰀀

95th Percentile Value-at-Risk Measures

Multiple6Maximum Multiple of Tail Event to Risk Measure󰀀

99th Percentile Value-at-Risk MeasuresMultiple4.54.053.543.02.532.021.550d250d125d

500d

1250dhs125

hs250hs1250λ=0.97λ󰀀=0.94hs500λ󰀀=0.99

50d250d125d

500d

1250dhs125

hs250hs1250λ󰀀=0.97λ=0.94hs500λ󰀀=0.99

Source: Author’󰀀s calculations.

Notes: d=days; hs=historical simulation; λ=exponentially weighted.Source: Author’󰀀s calculations.

Notes: d=days; hs=historical simulation; λ=exponentially weighted.

FRBNYECONOMICPOLICYREVIEW/APRIL199653

Chart 11aChart 11bCorrelation between Risk Measure and Absolute󰀀Value of Outcome󰀀95th Percentile Value-at-Risk MeasuresPercent0.4Correlation between Risk Measure and Absolute󰀀Value of Outcome󰀀99th Percentile Value-at-Risk MeasuresPercent0.40.30.30.20.20.10.100-0.150d250d125d500d1250dhs125hs250hs1250λ=0.97λ󰀀=0.94hs500λ󰀀=0.99-0.150d250d125d500d1250dhs125hs250hs1250λ󰀀=0.97λ=0.94hs500λ󰀀=0.99Source: Author’󰀀s calculations.Notes: d=days; hs=historical simulation; λ=exponentially weighted.Source: Author’󰀀s calculations.Notes: d=days; hs=historical simulation; λ=exponentially weighted.In contrast to the results for mean relative bias

(Charts 4a and 4b) and the fraction of outcomes covered(Charts 7a and 7b), the results for this performance crite-rion show that the length of the observation period isinversely related to performance. Thus, shorter observationperiods tend to lead to higher measures of correlationbetween the absolute values of the outcomes and the value-at-risk measures. This inverse relationship supports theview that, because market behavior changes over time,emphasis on recent information can be helpful in trackingchanges in risk.

At the other extreme, the risk measures for the

1,250-day historical simulation approach are essentiallyuncorrelated with the absolute values of the outcomes.Although superior according to other performance criteria,the 1,250-day results here indicate that this approach revealslittle about actual changes in portfolio risk over time.

accomplished on an ex post basis by multiplying the riskmeasures for each approach by the multiples needed toattain either exactly 95 percent or exactly 99 percent cover-age (Charts 8a and 8b). These scaled risk measures provide

Because market behavior changes over time,emphasis on recent information can be helpful intracking changes in risk.MEANRELATIVEBIASFORRISKMEASURESSCALEDTODESIREDLEVELOFCOVERAGE

The last performance criterion we examine is the mean rel-ative bias that results when risk measures are scaled toeither 95 percent or 99 percent coverage. Such scaling is

the precise amount of coverage desired for each portfolio.Of course, the scaling for each value-at-risk approachwould not be the same for different portfolios.

Once we have arrived at the scaled value-at-risk

measures, we compare their relative average sizes by usingthe mean relative bias calculation, which compares theaverage size of the risk measures for each approach to theaverage size across all twelve approaches (Charts 4a and4b). In this case, however, the value-at-risk measures havebeen scaled to the desired levels of coverage. The purposeof this criterion is to determine which approach, once suit-

54FRBNYECONOMICPOLICYREVIEW/APRIL1996

ably scaled, could provide the desired level of coveragewith the smallest average risk measures. This performancecriterion also addresses the issue of tracking changes inportfolio risk—the most efﬁcient approach will be the onethat tracks changes in risk best. In contrast to the correla-tion statistic discussed in the previous section, however,this criterion focuses speciﬁcally on the 95th and 99thpercentiles.

Once again, the exponentially weighted moving

average approaches appear superior (Charts 12a and 12b).In particular, the exponentially weighted average approachwith a decay factor of 0.97 appears to perform extremelywell for both 95th and 99th percentile risk measures.Indeed, for the 99th percentile, it achieves exact 99 percentcoverage with an average size that is 4 percent smaller thanthe average of all twelve scaled value-at-risk approaches.

The performance of the other approaches is similar

to that observed for the correlation statistic (Charts 11aand 11b), but in this case the relationship between efﬁ-ciency and the length of the observation period is not aspronounced. In particular, the 50-day equally weightedapproach is somewhat inferior to the 250-day equallyweighted approach—a ﬁnding contrary to what is observed

in Charts 11a and 11b—and may reﬂect the greater inﬂu-ence of measurement error on short observation periodsalong this performance criterion.

At least two caveats apply to these results. First,

they would be difﬁcult to duplicate in practice because thescaling must be done in advance of the outcomes ratherthan ex post. Second, the differences in the average sizes ofthe scaled risk measures are simply not very large. Never-theless, the results suggest that exponentially weightedaverage approaches might be capable of providing desiredlevels of coverage in an efﬁcient fashion, although theywould need to be scaled up.

CONCLUSIONS

A historical examination of twelve approaches to value-at-risk modeling shows that in almost all cases the approachescover the risk that they are intended to cover. In addition,the twelve approaches tend to produce risk estimates thatdo not differ greatly in average size, although historicalsimulation approaches yield somewhat larger 99th percen-tile risk measures than the variance-covariance approaches.

Despite the similarity in the average size of the

risk estimates, our investigation reveals differences, some-

Chart 12aChart 12bMean Relative Bias for Risk Measures Scaled to󰀀Cover Exactly 95 Percent󰀀95th Percentile Value-at-Risk MeasuresPercent0.10Mean Relative Bias for Risk Measures Scaled to󰀀Cover Exactly 99 Percent󰀀99th Percentile Value-at-Risk MeasuresPercent0.100.050.0500-0.05-0.05-0.1050d250d125d500d1250dhs125hs250hs1250λ=0.97λ󰀀=0.94hs500λ󰀀=0.99-0.1050d250d125dhs250hs1250λ󰀀=0.971250dλ=0.94hs125hs500λ󰀀=0.99500dSource: Author’󰀀s calculations.Notes: d=days; hs=historical simulation; λ=exponentially weighted.Source: Author’󰀀s calculations.Notes: d=days; hs=historical simulation; λ=exponentially weighted.FRBNYECONOMICPOLICYREVIEW/APRIL199655

times substantial, among the various value-at-riskapproaches for the same portfolio on the same date. Interms of variability over time, the value-at-risk approachesusing longer observation periods tend to produce less vari-able results than those using short observation periods orweighting recent observations more heavily.

Virtually all of the approaches produce accurate

95th percentile risk measures. The 99th percentile riskmeasures, however, are somewhat less reliable and gener-ally cover only between 98.2 percent and 98.5 percent ofthe outcomes. On the one hand, these deﬁciencies are smallwhen considered on the basis of the percentage of outcomesmisclassiﬁed. On the other hand, the risk measures wouldgenerally need to be increased across the board by 10 per-cent or more to cover precisely 99 percent of the outcomes.Interestingly, one exception is the 1,250-day historicalsimulation approach, which provides very accurate cover-age for both 95th and 99th percentile risk measures.

The outcomes that arenot covered are typically 30

to 40 percent larger than the risk measures and are alsolarger than predicted by the normal distribution. In somecases, daily losses over the twelve-year sample period areseveral times larger than the corresponding value-at-riskmeasures. These examples make it clear that value-at-riskmeasures—even at the 99th percentile—do not “bound”possible losses.

Also clear is the difﬁculty of anticipating or tracking

changes in risk over time. For this performance criterion, theexponentially weighted moving average approaches appear tobe superior. If it were possible to scale all approaches ex post toachieve the desired level of coverage over the sample period,these approaches would produce the smallest scaled riskmeasures.

What more general conclusions can be drawn

from these results? In many respects, the simulation esti-mates clearly reﬂect two well-known characteristics ofdaily ﬁnancial market data. First, extreme outcomes occurmore often and are larger than predicted by the normaldistribution (fat tails). Second, the size of market move-ments is not constant over time (conditional volatility).Clearly, constructing value-at-risk models that performwell by every measure is a difﬁcult task. Thus, althoughwe cannot recommend any single value-at-risk approach,our results suggest that further research aimed at combin-ing the best features of the approaches examined here maybe worthwhile.

56FRBNYECONOMICPOLICYREVIEW/APRIL1996

APPENDIX:VALUE-AT-RISKSIMULATIONRESULTSFOREACHPERFORMANCECRITERIONThe nine tables below summarize for each performance cri-terion the simulation results for the 95th and 99th percen-tile risk measures. The value-at-risk approaches appear atthe extreme left of each table. The ﬁrst column reports theaverage simulation result of each approach across the 1,000portfolios for the particular performance criterion. Thenext column reports the standard deviation of the resultsacross the 1,000 portfolios, a calculation that providesinformation on the variability of the results across portfo-lios. To indicate the variability of results over time, theremaining four columns report results averaged over the1,000 portfolios for four subsets of the sample period.Table A1MEANRELATIVEBIASEntire Sample PeriodStandardMean acrossDeviation acrossPortfoliosPortfolios1983-85Mean acrossPortfolios-0.00-0.00-0.010.010.08-0.04-0.03-0.020.05-0.02-0.01-0.001986-88Mean acrossPortfolios-0.05-0.020.030.080.06-0.060.000.050.03-0.07-0.050.001989-91Mean acrossPortfolios0.010.010.00-0.010.05-0.03-0.02-0.050.02-0.010.000.011992-94Mean acrossPortfolios-0.03-0.000.030.070.01-0.040.000.03-0.02-0.04-0.020.01PANELA:95THPERCENTILEVALUE-AT-RISKMEASURES50-day equally weighted-0.02125-day equally weighted-0.00250-day equally weighted0.01500-day equally weighted0.041,250-day equally weighted0.05125-day historical simulation250-day historical simulation500-day historical simulation1,250-day historical simulationExponentially weighted (λ=0.94)Exponentially weighted (λ=0.97)Exponentially weighted (λ=0.99)-0.04-0.010.000.02-0.03-0.020.000.010.010.010.020.030.030.030.030.030.010.010.01PANELB:99THPERCENTILEVALUE-AT-RISKMEASURES50-day equally weighted-0.05125-day equally weighted-0.04250-day equally weighted-0.03500-day equally weighted-0.001,250-day equally weighted0.01125-day historical simulation250-day historical simulation500-day historical simulation1,250-day historical simulationExponentially weighted (λ=0.94)Exponentially weighted (λ=0.97)Exponentially weighted (λ=0.99)-0.010.060.080.13-0.07-0.06-0.030.020.020.020.020.030.030.040.040.050.020.020.02-0.03-0.03-0.04-0.020.04-0.030.020.040.18-0.05-0.04-0.03-0.09-0.06-0.010.040.02-0.000.080.110.13-0.10-0.08-0.04-0.03-0.03-0.04-0.050.010.010.070.050.13-0.05-0.04-0.04-0.06-0.04-0.010.03-0.02-0.000.080.110.09-0.08-0.06-0.03APPENDIXFRBNYECONOMICPOLICYREVIEW/APRIL199657

APPENDIX:VALUE-AT-RISKSIMULATIONRESULTSFOREACHPERFORMANCECRITERION(Continued)Table A2ROOTMEANSQUAREDRELATIVEBIASEntire Sample PeriodStandardMean acrossDeviation acrossPortfoliosPortfolios1983-85Mean acrossPortfolios0.170.100.080.130.180.150.120.140.170.200.130.051986-88Mean acrossPortfolios0.150.100.090.130.140.130.110.130.130.170.110.041989-91Mean acrossPortfolios0.140.080.080.080.140.130.100.100.130.170.100.041992-94Mean acrossPortfolios0.160.110.090.130.140.140.110.140.150.190.130.05PANELA:95THPERCENTILEVALUE-AT-RISKMEASURES50-day equally weighted0.16125-day equally weighted0.10250-day equally weighted0.09500-day equally weighted0.131,250-day equally weighted0.16125-day historical simulation250-day historical simulation500-day historical simulation1,250-day historical simulationExponentially weighted (λ=0.94)Exponentially weighted (λ=0.97)Exponentially weighted (λ=0.99)0.140.110.130.150.180.120.050.010.010.010.020.040.020.010.020.030.010.010.01PANELB:99THPERCENTILEVALUE-AT-RISKMEASURES50-day equally weighted0.16125-day equally weighted0.10250-day equally weighted0.09500-day equally weighted0.121,250-day equally weighted0.14125-day historical simulation250-day historical simulation500-day historical simulation1,250-day historical simulationExponentially weighted (λ=0.94)Exponentially weighted (λ=0.97)Exponentially weighted (λ=0.99)0.180.160.160.220.190.130.060.010.010.010.020.030.030.030.040.060.010.010.010.170.110.090.130.160.150.140.150.240.200.140.060.160.110.090.120.130.190.150.180.200.190.130.060.140.080.090.100.130.170.160.120.190.170.110.050.160.110.090.120.140.170.160.170.190.190.130.0658FRBNYECONOMICPOLICYREVIEW/APRIL1996APPENDIX

APPENDIX:VALUE-AT-RISKSIMULATIONRESULTSFOREACHPERFORMANCECRITERION(Continued)Table A3ANNUALIZEDPERCENTAGEVOLATILITYEntire Sample PeriodStandardMean acrossDeviation acrossPortfoliosPortfolios1983-85Mean acrossPortfolios0.490.180.100.060.030.380.200.110.040.940.490.181986-88Mean acrossPortfolios0.420.190.090.050.020.390.190.090.040.880.430.141989-91Mean acrossPortfolios0.440.170.090.050.020.400.190.100.040.890.440.151992-94Mean acrossPortfolios0.450.200.110.050.020.410.210.100.040.940.490.17PANELA:95THPERCENTILEVALUE-AT-RISKMEASURES50-day equally weighted0.45125-day equally weighted0.19250-day equally weighted0.10500-day equally weighted0.051,250-day equally weighted0.02125-day historical simulation250-day historical simulation500-day historical simulation1,250-day historical simulationExponentially weighted (λ=0.94)Exponentially weighted (λ=0.97)Exponentially weighted (λ=0.99)0.400.200.100.040.910.470.160.050.030.020.010.000.040.020.010.010.090.060.03PANELB:99THPERCENTILEVALUE-AT-RISKMEASURES50-day equally weighted0.45125-day equally weighted0.19250-day equally weighted0.10500-day equally weighted0.051,250-day equally weighted0.02125-day historical simulation250-day historical simulation500-day historical simulation1,250-day historical simulationExponentially weighted (λ=0.94)Exponentially weighted (λ=0.97)Exponentially weighted (λ=0.99)0.550.300.150.060.910.470.160.050.030.020.010.010.070.050.030.020.100.060.030.490.180.100.060.030.490.270.160.060.940.490.180.420.190.090.050.020.550.280.130.050.880.430.140.440.170.090.050.020.510.270.140.060.880.440.150.450.200.110.050.020.570.310.150.060.940.490.17APPENDIXFRBNYECONOMICPOLICYREVIEW/APRIL199659

APPENDIX:VALUE-AT-RISKSIMULATIONRESULTSFOREACHPERFORMANCECRITERION(Continued)Table A4FRACTIONOFOUTCOMESCOVEREDEntire Sample PeriodStandardMean acrossDeviation acrossPortfoliosPortfolios1983-85Mean acrossPortfolios0.9480.9500.9460.9460.9540.9430.9430.9420.9510.9480.9500.9501986-88Mean acrossPortfolios0.9470.9530.9600.9630.9590.9460.9550.9590.9560.9460.9500.9571989-91Mean acrossPortfolios0.9490.9510.9500.9470.9540.9430.9450.9410.9510.9470.9500.9511992-94Mean acrossPortfolios0.9480.9530.9560.9580.9500.9460.9520.9520.9450.9460.9500.956PANELA:95THPERCENTILEVALUE-AT-RISKMEASURES50-day equally weighted0.948125-day equally weighted0.951250-day equally weighted0.953500-day equally weighted0.9541,250-day equally weighted0.954125-day historical simulation250-day historical simulation500-day historical simulation1,250-day historical simulationExponentially weighted (λ=0.94)Exponentially weighted (λ=0.97)Exponentially weighted (λ=0.99)0.9440.9490.9480.9510.9470.9500.9540.0060.0060.0050.0060.0060.0020.0030.0030.0040.0060.0060.006PANELB:99THPERCENTILEVALUE-AT-RISKMEASURES50-day equally weighted0.983125-day equally weighted0.984250-day equally weighted0.984500-day equally weighted0.9841,250-day equally weighted0.985125-day historical simulation250-day historical simulation500-day historical simulation1,250-day historical simulationExponentially weighted (λ=0.94)Exponentially weighted (λ=0.97)Exponentially weighted (λ=0.99)0.9830.9870.9880.9900.9820.9840.9850.0030.0030.0030.0030.0030.0010.0010.0010.0010.0030.0030.0030.9850.9840.9820.9810.9840.9830.9840.9850.9900.9840.9860.9850.9820.9840.9870.9890.9880.9850.9910.9910.9920.9810.9830.9860.9820.9820.9820.9810.9840.9820.9860.9860.9890.9820.9830.9830.9830.9840.9860.9870.9830.9840.9890.9900.9890.9830.9840.98660FRBNYECONOMICPOLICYREVIEW/APRIL1996APPENDIX

APPENDIX:VALUE-AT-RISKSIMULATIONRESULTSFOREACHPERFORMANCECRITERION(Continued)Table A5MULTIPLENEEDEDTOATTAINDESIREDCOVERAGELEVELEntire Sample PeriodStandardMean acrossDeviation acrossPortfoliosPortfolios1983-85Mean acrossPortfolios1.011.001.021.020.951.051.051.060.981.010.990.991986-88Mean acrossPortfolios1.020.980.930.900.931.030.960.940.951.031.000.951989-91Mean acrossPortfolios1.010.991.001.020.971.051.031.060.991.021.000.991992-94Mean acrossPortfolios1.020.980.950.931.001.030.980.991.041.031.000.96PANELA:95THPERCENTILEVALUE-AT-RISKMEASURES50-day equally weighted1.01125-day equally weighted0.99250-day equally weighted0.98500-day equally weighted0.971,250-day equally weighted0.97125-day historical simulation250-day historical simulation500-day historical simulation1,250-day historical simulationExponentially weighted (λ=0.94)Exponentially weighted (λ=0.97)Exponentially weighted (λ=0.99)1.041.011.011.001.021.000.970.050.040.040.040.050.010.020.020.030.050.040.04PANELB:99THPERCENTILEVALUE-AT-RISKMEASURES50-day equally weighted1.15125-day equally weighted1.13250-day equally weighted1.13500-day equally weighted1.131,250-day equally weighted1.11125-day historical simulation250-day historical simulation500-day historical simulation1,250-day historical simulationExponentially weighted (λ=0.94)Exponentially weighted (λ=0.97)Exponentially weighted (λ=0.99)1.141.061.051.001.141.121.100.060.070.070.080.080.030.030.030.040.060.060.061.111.121.171.221.121.151.111.131.001.121.091.111.191.111.061.031.041.130.990.980.941.191.151.081.191.171.201.201.131.181.121.101.011.141.151.171.141.131.111.101.171.161.041.021.041.161.121.09APPENDIXFRBNYECONOMICPOLICYREVIEW/APRIL199661

APPENDIX:VALUE-AT-RISKSIMULATIONRESULTSFOREACHPERFORMANCECRITERION(Continued)Table A6AVERAGEMULTIPLEOFTAILEVENTTORISKMEASUREEntire Sample PeriodStandardMean acrossDeviation acrossPortfoliosPortfolios1983-85Mean acrossPortfolios1.401.391.431.461.351.471.491.531.391.391.371.381986-88Mean acrossPortfolios1.411.351.281.241.271.451.341.291.311.421.381.301989-91Mean acrossPortfolios1.411.391.411.431.351.491.461.481.391.411.381.381992-94Mean acrossPortfolios1.411.391.361.341.431.501.441.431.501.421.381.34PANELA:95THPERCENTILEVALUE-AT-RISKMEASURES50-day equally weighted1.41125-day equally weighted1.38250-day equally weighted1.37500-day equally weighted1.381,250-day equally weighted1.36125-day historical simulation250-day historical simulation500-day historical simulation1,250-day historical simulationExponentially weighted (λ=0.94)Exponentially weighted (λ=0.97)Exponentially weighted (λ=0.99)1.481.431.441.411.411.381.350.070.070.070.080.080.040.050.060.070.070.070.07PANELB:99THPERCENTILEVALUE-AT-RISKMEASURES50-day equally weighted1.46125-day equally weighted1.44250-day equally weighted1.44500-day equally weighted1.461,250-day equally weighted1.44125-day historical simulation250-day historical simulation500-day historical simulation1,250-day historical simulationExponentially weighted (λ=0.94)Exponentially weighted (λ=0.97)Exponentially weighted (λ=0.99)1.481.371.371.301.441.421.400.120.110.130.140.140.070.070.090.100.110.110.111.481.451.491.561.431.511.441.461.281.451.431.441.451.411.341.291.311.471.281.251.201.441.401.351.481.421.441.461.391.461.371.341.251.441.411.421.471.501.501.471.551.551.411.401.401.481.451.4462FRBNYECONOMICPOLICYREVIEW/APRIL1996APPENDIX

APPENDIX:VALUE-AT-RISKSIMULATIONRESULTSFOREACHPERFORMANCECRITERION(Continued)Table A7MAXIMUMMULTIPLEOFTAILEVENTTORISKMEASUREEntire Sample PeriodStandardMean acrossDeviation acrossPortfoliosPortfolios1983-85Mean acrossPortfolios3.253.013.033.253.053.133.033.353.123.163.133.031986-88Mean acrossPortfolios2.562.542.452.332.352.842.612.442.442.552.462.401989-91Mean acrossPortfolios2.732.562.592.662.602.782.622.732.672.752.572.551992-94Mean acrossPortfolios2.983.093.073.043.213.493.313.303.373.032.992.96PANELA:95THPERCENTILEVALUE-AT-RISKMEASURES50-day equally weighted3.59125-day equally weighted3.59250-day equally weighted3.67500-day equally weighted3.861,250-day equally weighted3.97125-day historical simulation250-day historical simulation500-day historical simulation1,250-day historical simulationExponentially weighted (λ=0.94)Exponentially weighted (λ=0.97)Exponentially weighted (λ=0.99)3.913.854.094.143.583.533.550.930.981.011.081.101.021.101.161.120.990.990.96PANELB:99THPERCENTILEVALUE-AT-RISKMEASURES50-day equally weighted2.50125-day equally weighted2.50250-day equally weighted2.56500-day equally weighted2.701,250-day equally weighted2.77125-day historical simulation250-day historical simulation500-day historical simulation1,250-day historical simulationExponentially weighted (λ=0.94)Exponentially weighted (λ=0.97)Exponentially weighted (λ=0.99)2.582.342.482.492.482.462.470.610.700.730.780.770.520.570.630.650.640.660.682.262.092.112.272.142.182.002.081.892.202.182.111.831.821.751.661.671.971.661.601.541.831.761.721.911.791.811.851.811.861.721.701.631.921.791.782.082.152.142.132.242.252.022.052.022.102.082.06APPENDIXFRBNYECONOMICPOLICYREVIEW/APRIL199663

APPENDIX:VALUE-AT-RISKSIMULATIONRESULTSFOREACHPERFORMANCECRITERION(Continued)Table A8CORRELATIONBETWEENRISKMEASURESANDABSOLUTEVALUEOFOUTCOMEEntire Sample PeriodStandardMean acrossDeviation acrossPortfoliosPortfolios1983-85Mean acrossPortfolios0.210.170.120.010.050.160.10-0.000.060.260.230.171986-88Mean acrossPortfolios0.150.130.150.070.050.110.120.060.050.180.170.151989-91Mean acrossPortfolios0.120.070.020.05-0.040.040.020.03-0.030.150.140.091992-94Mean acrossPortfolios0.190.140.130.05-0.020.120.100.01-0.050.240.210.17PANELA:95THPERCENTILEVALUE-AT-RISKMEASURES50-day equally weighted0.19125-day equally weighted0.16250-day equally weighted0.13500-day equally weighted0.061,250-day equally weighted0.01125-day historical simulation250-day historical simulation500-day historical simulation1,250-day historical simulationExponentially weighted (λ=0.94)Exponentially weighted (λ=0.97)Exponentially weighted (λ=0.99)0.140.110.030.000.230.220.170.050.050.050.040.030.050.050.040.040.050.050.04PANELB:99THPERCENTILEVALUE-AT-RISKMEASURES50-day equally weighted0.19125-day equally weighted0.16250-day equally weighted0.13500-day equally weighted0.061,250-day equally weighted0.01125-day historical simulation250-day historical simulation500-day historical simulation1,250-day historical simulationExponentially weighted (λ=0.94)Exponentially weighted (λ=0.97)Exponentially weighted (λ=0.99)0.120.100.050.010.230.220.170.040.050.050.040.040.060.070.050.040.050.050.040.210.170.120.020.060.160.100.030.050.260.230.170.150.120.150.070.040.070.090.040.040.180.170.150.120.070.020.05-0.040.060.010.06-0.020.150.140.090.190.150.130.06-0.020.130.120.060.000.240.220.1764FRBNYECONOMICPOLICYREVIEW/APRIL1996APPENDIX

APPENDIX:VALUE-AT-RISKSIMULATIONRESULTSFOREACHPERFORMANCECRITERION(Continued)Table A9MEANRELATIVEBIASFORRISKMEASURESSCALEDTODESIREDCOVERAGELEVELSEntire Sample PeriodStandardMean acrossDeviation acrossPortfoliosPortfolios1983-85Mean acrossPortfolios-0.00-0.01-0.000.020.010.000.010.030.01-0.02-0.02-0.021986-88Mean acrossPortfolios-0.00-0.00-0.010.010.010.01-0.000.010.01-0.01-0.02-0.021989-91Mean acrossPortfolios0.00-0.00-0.00-0.010.010.010.00-0.000.010.01-0.01-0.021992-94Mean acrossPortfolios-0.00-0.01-0.010.010.020.01-0.010.020.03-0.01-0.02-0.02PANELA:95THPERCENTILEVALUE-AT-RISKMEASURES50-day equally weighted-0.00125-day equally weighted-0.01250-day equally weighted-0.01500-day equally weighted0.011,250-day equally weighted0.02125-day historical simulation250-day historical simulation500-day historical simulation1,250-day historical simulationExponentially weighted (λ=0.94)Exponentially weighted (λ=0.97)Exponentially weighted (λ=0.99)0.00-0.000.020.02-0.01-0.02-0.020.020.010.010.020.020.010.020.020.020.020.010.01PANELB:99THPERCENTILEVALUE-AT-RISKMEASURES50-day equally weighted-0.02125-day equally weighted-0.02250-day equally weighted-0.01500-day equally weighted0.021,250-day equally weighted0.02125-day historical simulation250-day historical simulation500-day historical simulation1,250-day historical simulationExponentially weighted (λ=0.94)Exponentially weighted (λ=0.97)Exponentially weighted (λ=0.99)0.030.020.030.03-0.04-0.04-0.030.030.020.020.030.030.030.030.030.040.030.020.02-0.03-0.030.000.060.04-0.000.020.050.04-0.05-0.06-0.040.02-0.02-0.020.00-0.010.06-0.000.01-0.010.01-0.01-0.030.00-0.000.01-0.00-0.010.050.050.000.00-0.05-0.03-0.01-0.03-0.02-0.010.020.030.050.020.030.03-0.04-0.05-0.04APPENDIXFRBNYECONOMICPOLICYREVIEW/APRIL199665

ENDNOTES

1. See, for example, the so-called G-30 report (1993), the U.S. GeneralAccounting Office study (1994), and papers outlining sound riskmanagement practices published by the Board of Governors of theFederal Reserve System (1993), the Basle Committee on BankingSupervision (1994), and the International Organization of SecuritiesCommissions Technical Committee (1994).

2. Work along these lines is contained in Jordan and Mackay (1995) andPritsker (1995).

3. Results for ten-day holding periods are contained in Hendricks (1995).This paper is available from the author on request.

4. The 99th percentileloss is the same as the 1st percentilegain on theportfolio. Convention suggests using the former terminology.5. Variance-covariance approaches are so named because they can bederived from the variance-covariance matrix of the relevant underlyingmarket prices or rates. The variance-covariance matrix containsinformation on the volatility and correlation of all market prices or ratesrelevant to the portfolio. Knowledge of the variance-covariance matrix ofthese variables for a given period of time implies knowledge of thevariance or standard deviation of the portfolio over this same period.6. The assumption of linear positions is made throughout the paper.Nonlinear positions require simulation methods, often referred to asMonte Carlo methods, when used in conjunction with variance-covariance matrices of the underlying market prices or rates.

7. See Fama (1965), a seminal paper on this topic. A more recentsummary of the evidence regarding foreign exchange data and “fat tails”is provided by Hsieh (1988). See also Taylor (1986) and Mills (1993) forgeneral discussions of the issues involved in modeling financial timeseries.

8. The portfolio variance is an equally weighted moving average ofsquared deviations from the mean.

9. In addition, equally weighted moving average approaches may differin the frequency with which estimates are updated. This article assumesthat all value-at-risk measures are updated on a daily basis. For acomparison of different updating frequencies (daily, monthly, orquarterly), see Hendricks (1995). This paper is available from the authoron request.

10. The intuition behind this assumption is that for most financial timeseries, the true mean is both close to zero and prone to estimation error.

Thus, estimates of volatility are often made worse (relative to assuming azero mean) by including noisy estimates of the mean.

11. Charts 1-3 depict 99th percentile risk measures and are derived fromthe same data used elsewhere in the article (see box). For Charts 1 and 2,the assumption of normality is made, so that these risk measures arecalculated by multiplying the portfolio standard deviation estimate by2.33. The units on the y-axes are millions of dollars, but they could beany amount depending on the definition of the units of the portfolio’spositions.

12. Engle’s (1982) paper introduced the autoregressive conditionalheteroskedastic (ARCH) family of models. Recent surveys of theliterature on conditional volatility modeling include Bollerslev, Chou,and Kroner (1992), Bollerslev, Engle, and Nelson (1994), and Dieboldand Lopez (1995). Recent papers comparing specific conditionalvolatility forecasting models include West and Cho (1994) and Heynenand Kat (1993).

13. See Engle and Bollerslev (1986).

14. For obvious reasons, a fifty-day observation period is not well suitedto historical simulations requiring a 99th percentile estimate.15. Bootstrapping techniques offer perhaps the best hope for standarderror calculations in this context, a focus of the author’s ongoing research.16. For a discussion of the statistical issues involved, see Kupiec (1995).The Basle Committee’s recent paper on backtesting (1996b) outlines aproposed supervisory backtesting framework designed to ensure thatbanks using value-at-risk models for regulatory capital purposes faceappropriate incentives.

17. The upper and lower edges of the boxes proper represent the 75th and25th percentiles, respectively. The horizontal line running across theinterior of each box represents the 50th percentile, and the upper andlower “antennae” represent the 95th and 5th percentiles, respectively.18. One plausible explanation relies solely on Jensen’s inequality. If thetrue conditional variance is changing frequently, then the average of aconcave function (that is, the value-at-risk measure) of this variance willtend to be less than the same concave function of the average variance.This gap would imply that short horizon value-at-risk measures shouldon average be slightly smaller than long horizon value-at-risk measures.This logic may also explain the generally smaller average size of theexponentially weighted approaches.

66FRBNYECONOMICPOLICYREVIEW/APRIL1996NOTES

ENDNOTES(Continued)

19. With as few as 125 observations, the use of actual observationsinevitably produces either upward- or downward-biased estimates ofmost specific percentiles. For example, the 95th percentile estimate istaken to be the seventh largest loss out of 125, slightly lower than the95th percentile. However, taking the sixth largest loss would yield a biasupward. This point should be considered when using historicalsimulation approaches together with short observation periods, althoughbiases can be addressed through kernel estimation, a method that isconsidered in Reiss (1989).

20. In particular, see Mahoney (1995) and Jackson, Maude, andPerraudin (1995).

21. See, for example, Bollerslev (1987) and Baillie and Bollerslev (1989).

22. The degrees of freedom,d, are chosen to solve the following equation,da*z(0.99)=t(0.99,d) /-------------, wherea is the ratio of the observed 99thd–2percentile to the 99th percentile calculated assuming normality,z(0.99)is the normal 99th percentile value, andt(0.99,d) is the t-distribution99th percentile value ford degrees of freedom. The term under the squareroot is the variance of the t-distribution withd degrees of freedom.23. This section and the next were inspired by Boudoukh, Richardson,and Whitelaw (1995).The author thanks Christine Cumming, Arturo Estrella, Beverly Hirtle,JohnKambhu, James Mahoney, Christopher McCurdy, Matthew Pritsker,Philip Strahan, and Paul Kupiec for helpful comments and discussions.NOTESFRBNYECONOMICPOLICYREVIEW/APRIL199667

REFERENCES

Baillie, Richard T., and Tim Bollerslev. 1989. “The Message in DailyExchange Rates: A Conditional-Variance Tale.” JOURNALOFBUSINESSANDECONOMICSTATISTICS 7: 297-305.Bank for International Settlements. 1994. “Public Disclosure of Market andCredit Risks by Financial Intermediaries.” Euro-currency StandingCommittee of the Central Banks of the Group of Ten Countries[Fisher report].Basle Committee on Banking Supervision. 1994.RISKMANAGEMENTGUIDELINESFORDERIVATIVES._____.1996a.SUPPLEMENTTOTHECAPITALACCORDTOINCORPORATEMARKETRISKS._____. 1996b.SUPERVISORYFRAMEWORKFORTHEUSEOF“BACKTESTING”INCONJUNCTIONWITHTHEINTERNALMODELSAPPROACHTOMARKETRISKCAPITALREQUIREMENTS.Board of Governors of the Federal Reserve System. 1993.EXAMININGRISKMANAGEMENTANDINTERNALCONTROLSFORTRADINGACTIVITIESOFBANKINGORGANIZATIONS.Bollerslev, Tim. 1987. “A Conditionally Heteroskedastic Time SeriesModel for Speculative Prices and Rates of Return.”REVIEWOFECONOMICSANDSTATISTICS69: 542-7.Bollerslev, Tim, Ray Y. Chou, and Kenneth F. Kroner. 1992. “ARCHModeling in Finance: A Review of the Theory and EmpiricalEvidence.”JOURNALOFECONOMETRICS 52: 5-59.Bollerslev, Tim, Robert F. Engle, and D.B. Nelson. 1994. “ARCH Models.”In Robert F. Engle and D. McFadden, eds.,HANDBOOKOFECONOMETRICS.Vol.4.Amsterdam: North-Holland.Boudoukh, Jacob, Matthew Richardson, and Robert Whitelaw. 1995. “Expectthe Worst.”RISK 8, no. 9 (September): 100-1.Derivatives Policy Group. 1995.FRAMEWORKFORVOLUNTARYOVERSIGHT.Diebold, Francis X., and Jose A. Lopez. 1995. “Modeling VolatilityDynamics.” National Bureau of Economic Research TechnicalWorking Paper no. 173.Engle, Robert F. 1982. “Autoregressive Conditional Heteroskedasticitywith Estimates of the Variance of U.K. Inflation.”ECONOMETRICA50: 987-1008.

Engle, Robert F., and Tim Bollerslev. 1986. “Modeling the Persistence ofConditional Variance.”ECONOMETRICREVIEWS 5: 1-50.Fama, Eugene F. 1965. “The Behavior of Stock Market Prices.”JOURNALOFBUSINESS38: 34-105.Figlewski, Stephen. 1994. “Forecasting Volatility Using Historical Data.”New York University Working Paper no. 13.Group of Thirty Global Derivatives Study Group. 1993.DERIVATIVES:PRACTICESANDPRINCIPLES. Washington, D.C. [G-30 report].Hendricks, Darryll. 1995. “Evaluation of Value-at-Risk Models UsingHistorical Data.” Federal Reserve Bank of New York. Mimeographed.Heynen, Ronald C., and Harry M. Kat. 1993. “Volatility Prediction: AComparison of GARCH(1,1), EGARCH(1,1) and Stochastic VolatilityModels.” Erasmus University, Rotterdam. Mimeographed.Hsieh, David A. 1988. “The Statistical Properties of Daily ExchangeRates: 1974-1983.”JOURNALOFINTERNATIONALECONOMICS13:171-86.International Organization of Securities Commissions Technical Committee.1994.OPERATIONALANDFINANCIALRISKMANAGEMENTCONTROLMECHANISMSFOROVER-THE-COUNTERDERIVATIVESACTIVITIESOFREGULATEDSECURITIESFIRMS.Jackson, Patricia, David J. Maude, and William Perraudin. 1995. “CapitalRequirements and Value-at-Risk Analysis.” Bank of England.Mimeographed.Jordan, James V., and Robert J. Mackay. 1995. “Assessing Value-at-Riskfor Equity Portfolios: Implementing Alternative Techniques.”Virginia Polytechnic Institute, Pamplin College of Business, Centerfor Study of Futures and Options Markets. Mimeographed.J.P. Morgan. 1995.RISKMETRICSTECHNICALDOCUMENT. 3d ed. NewYork.Kupiec, Paul H. 1995. “Techniques for Verifying the Accuracy of RiskMeasurement Models.” Board of Governors of the Federal ReserveSystem. Mimeographed.Mahoney, James M. 1995. “Empirical-based versus Model-basedApproaches to Value-at-Risk.” Federal Reserve Bank of New York.Mimeographed.

68FRBNYECONOMICPOLICYREVIEW/APRIL1996NOTES

ENDNOTES(Continued)

Markowitz, Harry M. 1959.PORTFOLIOSELECTION:EFFICIENTDIVERSIFICATIONOFINVESTMENTS. New York: John Wiley & Sons.Mills, Terence C. 1993.THEECONOMETRICMODELINGOFFINANCIALTIMESERIES. Cambridge: Cambridge University Press.Pritsker, Matthew. 1995. “Evaluating Value at Risk Methodologies:Accuracy versus Computational Time.” Board of Governors of theFederal Reserve System. Mimeographed.Reiss, Rolf-Dieter. 1989.APPROXIMATEDISTRIBUTIONSOFORDERSTATISTICS. New York: Springer-Verlag.

Taylor, Stephen. 1986.MODELINGFINANCIALTIMESERIES. New York:John Wiley & Sons.U.S. General Accounting Office. 1994.FINANCIALDERIVATIVES:ACTIONSNEEDEDTOPROTECTTHEFINANCIALSYSTEM. GAO/GGD-94-133.West, Kenneth D., and Dongchul Cho. 1994. “The Predictive Ability ofSeveral Models of Exchange Rate Volatility.” National Bureau ofEconomic Research Technical Working Paper no. 152.

NOTESFRBNYECONOMICPOLICYREVIEW/APRIL199669

70FRBNYECONOMICPOLICYREVIEW/APRIL1996

因篇幅问题不能全部显示，请点此查看更多更全内容

查看全文

全部栏目

Evaluation of Value-at-Risk Models Using Historical Data