Value-at-risk | RDP 9708: Measuring Traded Market Risk: Value-At-Risk and Backtesting Techniques (2024)

Internationally the use of VaR techniques has spread rapidly. This section definesthe VaR measure and discusses its use within the Australian banking industry.We then discuss the three methods most commonly used to calculate a VaR estimate– the variance-covariance approach, the historical-simulation approachand the Monte-Carlo simulation approach.

In addition to its inclusion in the Basle Committee on Banking Supervision'sguidelines for market risk capitaladequacy,^[1]a number of other official bodies and industry groups have recognised VaR asan important market-risk measurement tool. For example, the ‘Fisher Report’(Euro-currency Standing Committee 1994) issued by the Bank for InternationalSettlements in September 1994 made recommendations concerning the disclosureof market risk by financial intermediaries and advocated the disclosure ofVaR numbers in financial institutions' published annual reports. Moreover,private-sector organisations such as the Group of Thirty (1993) (an internationalgroup of bankers and other derivatives market participants) have recommendedthe use of VaR methodologies when setting out best-practice risk-managementstandards for financial institutions.

VaR is one of the most widely used market risk-measurement techniques by banks, otherfinancial institutions and, increasingly, corporates. Of the banks visitedby the Reserve Bank's Market-Risk On-Site review team as at end October1997, more than half had some form of VaR calculation in place.

The application of VaR techniques is usually limited to assessing the risks beingrun in banks' treasury or trading operations (such as securities, foreignexchange and equities trading). It is rarely only applied to the measurementof the exposure to the interest-rate and foreign-exchange risks that arisefrom more traditional non-traded banking business (for example, lending anddeposit taking). Use of VaR by Australian banks ranges from a fully integratedapproach (where VaR is central to the measurement and internal reporting oftraded market risk and VaR-based limits are set for each individual trader)to an approach where reliance is placed on other techniques and VaR is calculatedonly for the information of senior management or annual reporting requirements.At this stage, it is most common to see VaR used to aggregate exposures arisingfrom different areas of a bank's trading activities while individual tradersmanage risk based on simpler, market-specific risk measures.

2.1 Defining Value-at-risk

Value-at-risk aims to measure the potential loss on a portfolio that would resultif relatively large adverse price movements were tooccur.^[2]Hence, at its simplest, VaR requires the revaluation of a portfolio using aset of given price shifts. Statistical techniques are used to select the sizeof those price shifts.

To quantify potential loss (and the severity of the adverse price move to be used)two underlying parameters must be specified – the holding period underconsideration and the desired statistical confidence interval. The holdingperiod refers to the time frame over which changes in portfolio value are measured– is the bank concerned with the potential to lose money over, say, oneday, one week or one year. For example, a VaR measure based on a one-day holdingperiod reflects the impact of daily price movements on a portfolio. It is assumedthat the portfolio is held constant over the holding period. The Basle Committee'sstandards require that banks use a ten-day holding period – thus requiringbanks to apply ten-day price movements to their portfolios. The confidencelevel defines the proportion of trading losses that are covered by the VaRamount. For example, if a bank calculates its VaR assuming a one-day holdingperiod and a 99 per cent confidence interval then it is to be expected that,on average, trading losses will exceed the VaR figure on one occasion in onehundred trading days.

Thus, VaR is the dollar amount that portfolio losses are not expected to exceed,with a specified degree of statistical confidence, over a pre-specified periodof time.

2.2 An Example Portfolio

A simple portfolio of two spot foreign-exchange positions can be used to illustrateand compare three of the most common approaches to the calculation of VaR.In the following examples, a one-day holding period is assumed and VaR is definedin terms of both 95^th and 99^th percentile confidence levels.

The example portfolio consists of a spot long position in Japanese Yen and a spotshort position in US dollars (Table1). Thus the value of the portfoliowill be affected by movements in the JPY/AUD and USD/AUD exchange rates.

Estimation of a VaR figure is based on the historical behaviour of those market pricesthat affect the value of the portfolio. In line with the Basle Committee'srequirements we use 250 days of historical data, from 9 June 1995 to 5 June1996, to perform the VaR calculations below.

Figures 1 and 2 are histograms of the daily returns for the JPY/AUD and USD/AUD exchangerates. The smooth line in each chart represents a normal distribution withthe same mean and standard deviation as the data. In both the upper and lowertails of each series, the actual frequency of returns is greater than thatwhich would be expected if returns were normally distributed (that is, theobserved distributions of daily returns have ‘fatter tails’ thanimplied by the normal distribution). Thus both series of daily returns appearmore likely to be samples drawn from some distribution other than a normaldistribution (such as a t-distribution). The implications of this result forthe calculation of a VaR number will be considered later in this paper.

The starting point of all three VaR approaches is to revalue the portfolio at currentmarket prices. Table 2 shows the revalued portfolio given the foreign exchangerates on 5 June 1996.

2.3 The Variance-covariance Approach

In terms of the computation required, the variance-covariance method is the simplestof the VaR approaches. For this reason, it is often used by globally activebanks which need to aggregate data from a large number of trading sites. Variance-covarianceVaR was the first of the VaR approaches to be offered in off-the-shelf computerpackages and hence, is also widely used by banks with comparatively low levelsof trading activity.

The variance-covariance approach is based on the assumption that financial-assetreturns and hence, portfolio profits and losses are normally distributed. Theconsequence of these two assumptions is that VaR can be expressed as a functionof:

• the variance-covariance matrix for market-price returns; and
• the sensitivity of the portfolio to price shifts.

The first stage of the variance-covariance approach requires thecalculation of a variance-covariance matrix using the 250 days of historicaldata for the two series of daily exchange rate returns. The variance-covariancematrix for this example is expressed as:

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where σ_JPY² is the variance of the series ofdaily returns for JPY/AUD, σ_USD² is thevariance of the series of daily returns for USD/AUD and σ_JPY._USDis the covariance between the two series.

The second step in this approach is to calculate the market pricesensitivities or deltas of the portfolio; that is, the amounts bywhich the portfolio's value will change if each of the underlying marketprices change by some pre-specified amount. To do this, movements in each ofthe market prices which affect the value of the portfolio are examined separately.Table 3 shows the change in the portfolio given a 1 per cent move in each ofthe spot FX rates.

The third step in this approach is to calculate the standard deviationor volatility of total changes in portfolio value. Since total portfolio changesare assumed to be normally distributed, the volatility of portfolio changescan be expressed as a function of the deltas, the standard deviations of thetwo market-factor returns and the covariance between them. Let δbe the vector of market-price sensitivities or deltas. If the standard deviationof portfolio changes is ν and the variance-covariance matrix ofthe market prices is M then ν is expressed as:

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In this example ν is given by:

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The standard deviation of changes in the portfolio's total value is 46 AUD.

To establish the VaR number of the portfolio for a given level of confidence thestandard deviation must be multiplied by the relevant scaling factor, whichis derived from the standard normal distribution. For example, if a 99 percent level of confidence is desired the appropriate scaling factor is 2.33since the probability of occurrence of a number less than −2.33 is 1per cent. Scaling the standard deviation of the portfolio by this amount yieldsa VaR number which should only be exceeded 1 per cent of the time. Note thatthe choice of a 99 per cent confidence level and associated scaling factorof 2.33 assumes a one-tailed test in line with the Basle market risk requirements(that is, only large losses are considered, not large profits).

Table 4 shows the VaR amounts, given 95 and 99 per cent levels of confidence, forthe example portfolio. Clearly the higher the level of confidence, the largerthe VaR number will be: given the various assumptions there is a 5 per centprobability that the loss on the portfolio will exceed 76 AUD and only a 1per cent probability that the loss on the portfolio will be larger than 108AUD.

2.4 The Historical-simulation Approach

The historical-simulation method is more computationally intensive than the variance-covarianceapproach and its use emerged within the Australian banking industry a littlelater. While only three banks have been using historical simulation for sometime, the development of historical databases of market prices, together withmore powerful (and less expensive) computer technology, has led several otherbanks to move towards the use of this approach.

The historical-simulation approach also uses historical data on daily returns toestablish a VaR number, however, it makes no assumptions about the statisticaldistribution of these returns. The first step in this approach is to apply each of the past 250pairs of daily exchange rate movements to the portfolio to determine the seriesof daily changes in portfolio value that would have been realised had the currentportfolio been held unchanged throughout those 250 trading days.

To determine the revalued portfolio value two approaches can be used. The simplerapproach requires the previously calculated delta amount for each positionto be multiplied by each of the past changes in the relevant exchange rate.Recall that delta measures how much the position value will change if the exchangerate changes by 1 per cent. If the past actual change in the exchange rateis, say, 0.16 per cent then the portfolio value will change by 0.16 ×delta. The second, more arduous approach is to revalue each position in theportfolio at each of the past exchange rates. For linear positions (that is,positions the values of which change linearly with changes in the underlyingmarket prices) the two approaches will yield the same result. However, fornon-linear positions, such as positions in complex options, the first approachmay substantially under or overestimate the change in the value of the positionand thus may not generate an accurate measure of market risk exposure.

The second step is to sort the 250 changes in portfolio valuein ascending order to arrive at an observed distribution of changes in portfoliovalue. The histogram of these changes is shown in Figure 3. The VaR numberwill be equal to that percentile associated with the specified level of confidence.For a 95 per cent level of confidence, the VaR number is 68.17 AUD and equalsthe 5^th percentile of the distribution of changes in portfolio value.The k^th percentile means that the lowest k per cent of the sampleof changes in portfolio value will exceed the VaR measure. Since there are250 observations, essentially this means that 12.5 losses (or 5 per cent ofthe sample) will be larger than the VaR measure (the VaR measure is essentiallythe 13.5 lowest observation). Similarly, for a 99 per cent level of confidencethe VaR number is 102.11 and equals the first percentile. These results aresummarised in Table 5.

2.5 Monte-Carlo Simulation

This method is not widely used by Australian banks. Monte-Carlo techniques are extremelycomputer intensive and the additional information that these techniques provideis of most use for the analysis of complex options portfolios. To date, useof Monte-Carlo simulation has been limited principally to the most sophisticatedbanks and securities houses operating in the US. The Monte-Carlo method isbased on the generation, or simulation, of a large number of possible futureprice changes that could affect the value of the portfolio. The resulting changesin portfolio value are then analysed to arrive at a single VaR number.

Briefly, the method requires the following steps.

1. A statistical model of the market factor returns must be selected and the parametersof that model need to be estimated. For the purposes of our example, it isassumed that the two exchange-rate returns are drawn from a bivariate t-distributionwith 5 degrees of freedom and a correlation of0.63.^[3]A t-distribution was chosen as it is able to capture the fat-tails characteristicobserved in the data.
2. A large number of random draws from the estimated statistical model are simulated.This is done using a sampling methodology called Monte-Carlo simulation inwhich a mathematical formula is used to generate series of ‘pseudo-random’numbers to simulate the market factors. In this example, the two exchangerates are simulated 50,000 times.
3. The portfolio is revalued for each pair of simulated exchange rates and the changesin portfolio value between the current value and these revalued amounts arethen determined. Figure 4 shows the histogram of these changes.

In the same way as in the historical simulation approach, these changes in portfoliovalue are sorted in ascending order and the VaR number at a k per cent levelof confidence is determined as the (100-k) percentile of these sorted changes.The resulting VaR measures are shown in Table 6.

The Monte-Carlo process permits analysis of the impact of events that were not infact observed over the historical period but that are just as likely to occuras events that were observed. It is this capacity to evaluate likely eventsthat have not occurred that is one of the main attractions of this approach.

2.6 A Comparison of the Three Methods

The VaR numbers derived from the three approaches produce a wide range of results(Table7). In this example, the historical simulation method which takesinto account the actual shape of the observed distribution of profits and losses(shown in Figure3) yields the lowest risk estimates. The variance-covariancemethod's assumption of symmetry around a zero mean gives equal weight toboth profits and losses, resulting in VaR estimates which are slightly higherthan those of the historical simulation approach. The simulation of a bivariatet-distribution results in VaR estimates which are much larger than the estimatesgiven by the other two methods. A t-distribution with the same mean and varianceas a normal distribution will have a greater proportion of its probabilitymass in the tails of the distribution (in fact, in this case, the t-distributionalso has longer tails than the empirical distribution). The prime focus ofa VaR model is the probability of tail events, hence, the long tails of thet-distribution have a disproportionate effect on the VaR estimate. It can beseen that this effect becomes more marked the higher the confidence level.It should be noted that this ranking of results from the three methods is dependenton the data and also the statistical distribution used within the Monte-Carlosimulation technique. Other price series exhibiting different mean, skew andtail characteristics may result in the relative sizes of the three methods'VaR estimates being quite different.

2.7 Advantages and Shortcomings of VaR

While VaR is used by numerous financial institutions it is not without its shortcomings.First, the VaR estimate is based solely on historical data. To the extent thatthe past may not be a good predictor of the future, the VaR measure may underor overestimate risk. There is a continuing debate within the financial communityas to whether the correlations between different financial prices are sufficientlystable to be relied upon when quantifying risk. There is also debate as tohow best to model the behaviour of volatility in market prices. Nevertheless,if an institution wishes to avoid relying on subjective judgments regardinglikely future financial market volatility, reliance on history is necessary.

Second, a VaR figure provides no indication of the magnitude of losses that may resultif prices move by an amount which is more adverse than that amount dictatedby the chosen confidence level. For example, the dollar VaR provides no insightinto what would happen to a bank if a 1 in 10,000 chance event occurred. Toaddress the risks associated with such large price shifts, banks are developing,and bank supervisors are requiring, more subjective approaches such as stresstesting to be adopted in addition to the statistically based VaR approach.Stress testing involves the specification of stress scenarios (for example,the suspension of the European exchange rate mechanism) and analysis of howbanks' portfolios would behave under such scenarios.

Third, the comparative simplicity of a VaR calculation where exposures in a widearray of instruments and markets can be condensed into a single figure is botha strength and a weakness. This simplicity has been the key to the popularityof VaR, particularly as a means of providing summary information to a bank'ssenior management. The difficulty with this though, is that such a highly aggregatefigure may mask imbalances in risk exposure across markets or individual traders.

One of the chief advantages of the VaR approach is that it assesses exposure to differentmarkets (interest rates, foreign exchange, etc) in terms of a common base –losses relative to a standard unit of likelihood. Hence, risks across differentinstruments, traders and markets can be readily compared and aggregated. Inaddition, a dollar-value VaR can be directly compared to actual trading profitand loss results – both as a means of testing the adequacy of the VaRmodel and to assess risk-adjusted performance. Risk-adjusted returns can bequantified simply by looking at the ratio of realised profits/losses to VaRexposure. Such calculations provide a basis for a bank to develop sophisticatedcapital-allocation models and to renumerate individual traders not just forthe volume of trading done, but to reflect the riskiness of each trader'sactivities.