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VALUE AT RISK (VAR)
What is the most I can lose on this investment? This is a question that almost
every investor who has invested or is considering investing in a risky asset asks at some
point in time. Value at Risk tries to provide an answer, at least within a reasonable bound.
In fact, it is misleading to consider Value at Risk, or VaR as it is widely known, to be an
alternative to risk adjusted value and probabilistic approaches. After all, it borrows
liberally from both. However, the wide use of VaR as a tool for risk assessment, especially in financial service firms, and the extensive literature that has developed
around it, push us to dedicate this chapter to its examination.
We begin the chapter with a general description of VaR and the view of risk that
underlies its measurement, and examine the history of its development and applications. We then consider the various estimation issues and questions that have come up in the
context of measuring VAR and how analysts and researchers have tried to deal with
them. Next, we evaluate variations that have been developed on the common measure, in
some cases to deal with different types of risk and in other cases, as a response to the
limitations of VaR. In the final section, we evaluate how VaR fits into and contrasts with the other risk assessment measures we developed in the last two chapters.
What is Value at Risk? In its most general form, the Value at Risk measures the potential loss in value of
a risky asset or portfolio over a defined period for a given confidence interval. Thus, if
the VaR on an asset is $ 100 million at a one-week, 95% confidence level, there is a only a 5% chance that the value of the asset will drop more than $ 100 million over any given
week. In its adapted form, the measure is sometimes defined more narrowly as the
possible loss in value from “normal market risk” as opposed to all risk, requiring that we
draw distinctions between normal and abnormal risk as well as between market and non- market risk.
While Value at Risk can be used by any entity to measure its risk exposure, it is
used most often by commercial and investment banks to capture the potential loss in
value of their traded portfolios from adverse market movements over a specified period;
this can then be compared to their available capital and cash reserves to ensure that the
losses can be covered without putting the firms at risk. Taking a closer look at Value at Risk, there are clearly key aspects that mirror our
discussion of simulations in the last chapter:
- To estimate the probability of the loss, with a confidence interval, we need to define
the probability distributions of individual risks, the correlation across these risks and the effect of such risks on value. In fact, simulations are widely used to measure the VaR for asset portfolio.
- The focus in VaR is clearly on downside risk and potential losses. Its use in banks
reflects their fear of a liquidity crisis, where a low-probability catastrophic occurrence creates a loss that wipes out the capital and creates a client exodus. The demise of Long Term Capital Management, the investment fund with top pedigree Wall Street traders and Nobel Prize winners, was a trigger in the widespread acceptance of VaR.
- There are three key elements of VaR – a specified level of loss in value, a fixed time
period over which risk is assessed and a confidence interval. The VaR can be specified for an individual asset, a portfolio of assets or for an entire firm.
- While the VaR at investment banks is specified in terms of market risks – interest rate
changes, equity market volatility and economic growth – there is no reason why the risks cannot be defined more broadly or narrowly in specific contexts. Thus, we could compute the VaR for a large investment project for a firm in terms of competitive and firm-specific risks and the VaR for a gold mining company in terms of gold price risk.
In the sections that follow, we will begin by looking at the history of the development of
this measure, ways in which the VaR can be computed, limitations of and variations on
the basic measures and how VaR fits into the broader spectrum of risk assessment approaches.
A Short History of VaR While the term “Value at Risk” was not widely used prior to the mid 1990s, the
origins of the measure lie further back in time. The mathematics that underlie VaR were
largely developed in the context of portfolio theory by Harry Markowitz and others,
at Risk, with wide variations on how it was measured. In the aftermath of numerous
disastrous losses associated with the use of derivatives and leverage between 1993 and 1995, culminating with the failure of Barings, the British investment bank, as a result of
unauthorized trading in Nikkei futures and options by Nick Leeson, a young trader in
Singapore, firms were ready for more comprehensive risk measures. In 1995, J.P.
Morgan provided public access to data on the variances of and covariances across various
security and asset classes, that it had used internally for almost a decade to manage risk, and allowed software makers to develop software to measure risk. It titled the service
“RiskMetrics” and used the term Value at Risk to describe the risk measure that emerged
from the data. The measure found a ready audience with commercial and investment
banks, and the regulatory authorities overseeing them, who warmed to its intuitive
appeal. In the last decade, VaR has becomes the established measure of risk exposure in financial service firms and has even begun to find acceptance in non-financial service
firms.
Measuring Value at Risk There are three basic approaches that are used to compute Value at Risk, though
there are numerous variations within each approach. The measure can be computed analytically by making assumptions about return distributions for market risks, and by
using the variances in and covariances across these risks. It can also be estimated by
running hypothetical portfolios through historical data or from Monte Carlo simulations.
In this section, we describe and compare the approaches.^1
Variance-Covariance Method Since Value at Risk measures the probability that the value of an asset or portfolio
will drop below a specified value in a particular time period, it should be relatively
simple to compute if we can derive a probability distribution of potential values. That is
basically what we do in the variance-covariance method, an approach that has the benefit
(^1) For a comprehensive overview of Value at Risk and its measures, look at the Jorion, P., 2001, Value at Risk: The New Benchmark for Managing Financial Risk, McGraw Hill. For a listing of every possible reference to the measure, try www.GloriaMundi.org.
of simplicity but is limited by the difficulties associated with deriving probability
distributions.
General Description
Consider a very simple example. Assume that you are assessing the VaR for a
single asset, where the potential values are normally distributed with a mean of $ 120
million and an annual standard deviation of $ 10 million. With 95% confidence, you can
assess that the value of this asset will not drop below $ 80 million (two standard deviations below from the mean) or rise about $120 million (two standard deviations
above the mean) over the next year.^2 When working with portfolios of assets, the same
reasoning will apply but the process of estimating the parameters is complicated by the
fact that the assets in the portfolio often move together. As we noted in our discussion of
portfolio theory in chapter 4, the central inputs to estimating the variance of a portfolio are the covariances of the pairs of assets in the portfolio; in a portfolio of 100 assets, there
will be 49,500 covariances that need to be estimated, in addition to the 100 individual
asset variances. Clearly, this is not practical for large portfolios with shifting asset
positions. It is to simplify this process that we map the risk in the individual investments in
the portfolio to more general market risks, when we compute Value at Risk, and then
estimate the measure based on these market risk exposures. There are generally four steps
involved in this process:
- The first step requires us to take each of the assets in a portfolio and map that asset on
to simpler, standardized instruments. For instance, a ten-year coupon bond with annual coupons C, for instance, can be broken down into ten zero coupon bonds, with matching cash flows: C C C C C C C C C FV+C
The first coupon matches up to a one-year zero coupon bond with a face value of C, the second coupon with a two-year zero coupon bond with a face value of C and so
(^2) The 95% confidence intervals translate into 1.96 standard deviations on either side of the mean. With a 90% confidence interval, we would use 1.65 standard deviations and a 99% confidence interval would require 2.33 standard deviations.
Implicit in the computation of the VaR in step 4 are assumptions about how
returns on the standardized risk measures are distributed. The most convenient assumption both from a computational standpoint and in terms of estimating probabilities
is normality and it should come as no surprise that many VaR measures are based upon
some variant of that assumption. If, for instance, we assume that each market risk factor
has normally distributed returns, we ensure that that the returns on any portfolio that is
exposed to multiple market risk factors will also have a normal distribution. Even those VaR approaches that allow for non-normal return distributions for individual risk factors
find ways of ending up with normal distributions for final portfolio values.
The RiskMetrics Contribution
As we noted in an earlier section, the term Value at Risk and the usage of the
measure can be traced back to the RiskMetrics service offered by J.P. Morgan in 1995. The key contribution of the service was that it made the variances in and covariances
across asset classes freely available to anyone who wanted to access them, thus easing the
task for anyone who wanted to compute the Value at Risk analytically for a portfolio.
Publications by J.P. Morgan in 1996 describe the assumptions underlying their computation of VaR:^3
- Returns on individual risk factors are assumed to follow conditional normal
distributions. While returns themselves may not be normally distributed and large outliers are far too common (i.e., the distributions have fat tails), the assumption is that the standardized return (computed as the return divided by the forecasted standard deviation) is normally distributed.
- The focus on standardized returns implies that it is not the size of the return per se
that we should focus on but its size relative to the standard deviation. In other words, a large return (positive or negative) in a period of high volatility may result in a low standardized return, whereas the same return following a period of low volatility will yield an abnormally high standardized return.
(^3) RiskMetrics – Technical Document, J.P. Morgan, December 17, 1996; Zangari, P., 1996, An Improved Methodology for Computing VaR, J.P. Morgan RiskMetrics Monitor, Second Quarter 1996.
The focus on normalized standardized returns exposed the VaR computation to the risk of
more frequent large outliers than would be expected with a normal distribution. In a subsequent variation, the RiskMetrics approach was extended to cover normal mixture
distributions, which allow for the assignment of higher probabilities for outliers. Figure
7.1 contrasts the two distributions:
Figure 7.
In effect, these distributions require estimates of the probabilities of outsized returns
occurring and the expected size and standard deviations of such returns, in addition to the
standard normal distribution parameters. Even proponents of these models concede that
estimating the parameters for jump processes, given how infrequently jumps occur, is difficult to do.
still fall a multivariate normal distribution.^5 These and other papers like it develop interesting variations but have to overcome two practical problems. Estimating inputs for non-normal models can be very difficult to do, especially when working with historical data, and the probabilities of losses and Value at Risk are simplest to compute with the normal distribution and get progressively more difficult with asymmetric and fat-tailed distributions. Second, other research has been directed at bettering the estimation techniques to yield more reliable variance and covariance values to use in the VaR calculations. Some suggest refinements on sampling methods and data innovations that allow for better estimates of variances and covariances looking forward. Others posit that statistical innovations can yield better estimates from existing data. For instance, conventional estimates of VaR are based upon the assumption that the standard deviation in returns does not change over time (homoskedasticity), Engle argues that we get much better estimates by using models that explicitly allow the standard deviation to change of time (heteroskedasticity).^6 In fact, he suggests two variants – Autoregressive Conditional Heteroskedasticity (ARCH) and Generalized Autoregressive Conditional Heteroskedasticity (GARCH) – that provide better forecasts of variance and, by extension, better measures of Value at Risk.^7 One final critique that can be leveled against the variance-covariance estimate of VaR
is that it is designed for portfolios where there is a linear relationship between risk and portfolio positions. Consequently, it can break down when the portfolio includes options,
since the payoffs on an option are not linear. In an attempt to deal with options and other
non-linear instruments in portfolios, researchers have developed Quadratic Value at Risk
measures.^8 These quadratic measures, sometimes categorized as delta-gamma models (to
(^5) Hull, J. and A. White, 1998, Value at Risk when daily changes are not normally distributed, Journal of Derivatives, v5, 9 6 - 19. Engle, R., 2001, Garch 101: The Use of ARCH and GARCH models in Applied Econometrics, Journal of Economic Perspectives, v15, 157 7 - 168. He uses the example of a $1,000,0000 portfolio composed of 50% NASDAQ stocks, 30% Dow Jones stocks and 20% long bonds, with statistics computed from March 23, 1990 to March 23, 2000. Using the conventional measure of daily standard deviation of 0.83% computed over a 10-year period, he estimates the value at risk in a day to be $22,477. Using an ARCH model, the forecast standard deviation is 1.46%, leading to VaR of 8 $33,977. Allowing for the fat tails in the distribution increases the VaR to $39,996. Britten-Jones, M. and Schaefer, S.M., 1999, Non-linear value-at-risk, European Finance Review, v2, 161-
contrast with the more conventional linear models which are called delta-normal), allow
researchers to estimate the Value at Risk for complicated portfolios that include options and option-like securities such as convertible bonds. The cost, though, is that the
mathematics associated with deriving the VaR becomes much complicated and that some
of the intuition will be lost along the way.
Historical Simulation Historical simulations represent the simplest way of estimating the Value at Risk
for many portfolios. In this approach, the VaR for a portfolio is estimated by creating a hypothetical time series of returns on that portfolio, obtained by running the portfolio
through actual historical data and computing the changes that would have occurred in
each period.
General Approach
To run a historical simulation, we begin with time series data on each market risk factor, just as we would for the variance-covariance approach. However, we do not use
the data to estimate variances and covariances looking forward, since the changes in the
portfolio over time yield all the information you need to compute the Value at Risk.
Cabedo and Moya provide a simple example of the application of historical simulation to measure the Value at Risk in oil prices.^9 Using historical data from 1992 to
1998, they obtained the daily prices in Brent Crude Oil and graphed out the prices in
Figure 7.2:
187;2 p p 1 6 1 - 1 8 7 Rouvinez, C. , 1997, Going Greek with VAR, Risk, v10, 57 - 65.
(^9) J.D. Cabedo and I. Moya, 2003, Estimating oil price Value at Risk using the historical simulation approach, Energy Economics, v25, 239-253.
Assessment
While historical simulations are popular and relatively easy to run, they do come with baggage. In particular, the underlying assumptions of the model generate give rise to
its weaknesses.
a. Past is not prologue: While all three approaches to estimating VaR use historical data,
historical simulations are much more reliant on them than the other two approaches for the simple reason that the Value at Risk is computed entirely from historical price changes. There is little room to overlay distributional assumptions (as we do with the Variance-covariance approach) or to bring in subjective information (as we can with Monte Carlo simulations). The example provided in the last section with oil prices provides a classic example. A portfolio manager or corporation that determined its oil price VaR, based upon 1992 to 1998 data, would have been exposed to much larger losses than expected over the 1999 to 2004 period as a long period of oil price stability came to an end and price volatility increased.
b. Trends in the data: A related argument can be made about the way in which we
compute Value at Risk, using historical data, where all data points are weighted equally. In other words, the price changes from trading days in 1992 affect the VaR in exactly the same proportion as price changes from trading days in 1998. To the extent that there is a trend of increasing volatility even within the historical time period, we will understate the Value at Risk. c. New assets or market risks: While this could be a critique of any of the three
approaches for estimating VaR, the historical simulation approach has the most difficulty dealing with new risks and assets for an obvious reason: there is no historic data available to compute the Value at Risk. Assessing the Value at Risk to a firm from developments in online commerce in the late 1990s would have been difficult to do, since the online business was in its nascent stage.
The trade off that we mentioned earlier is therefore at the heart of the historic simulation
debate. The approach saves us the trouble and related problems of having to make
specific assumptions about distributions of returns but it implicitly assumes that the
distribution of past returns is a good and complete representation of expected future
returns. In a market where risks are volatile and structural shifts occur at regular intervals,
this assumption is difficult to sustain.
Modifications
As with the other approaches to computing VaR, there have been modifications
suggested to the approach, largely directed at taking into account some of the criticisms
mentioned in the last section.
a. Weighting the recent past more: A reasonable argument can be made that returns in the recent past are better predictors of the immediate future than are returns from the distant past. Boudoukh, Richardson and Whitelaw present a variant on historical simulations, where recent data is weighted more, using a decay factor as their time weighting mechanism.^11 In simple terms, each return, rather than being weighted equally, is assigned a probability weight based on its recency. In other words, if the decay factor is .90, the most recent observation has the probability weight p, the observation prior to it will be weighted 0.9p, the one before that is weighted 0.81p and so on. In fact, the conventional historical simulation approach is a special case of this approach, where the decay factor is set to 1. Boudoukh et al. illustrate the use of this technique by computing the VaR for a stock portfolio, using 250 days of returns, immediately before and after the market crash on October 19, 1987.^12 With historical simulation, the Value at Risk for this portfolio is for all practical purposes unchanged the day after the crash because it weights each day (including October 19) equally. With decay factors, the Value at Risk very quickly adjusts to reflect the size of the crash.^13
b. Combining historical simulation with time series models: Earlier in this section, we
referred to a Value at Risk computation by Cabado and Moya for oil prices using a historical simulation. In the same paper, they suggested that better estimates of VaR could be obtained by fitting at time series model through the historical data and using the parameters of that model to forecast the Value at Risk. In particular, they fit an
(^11) Boudoukh, J., M. Richardson and R. Whitelaw, 1998. "The Best of Both Worlds," Risk, v 11, 64-67. (^12) The Dow dropped 508 points on October 19, 1987, approximately 22%. (^13) With a decay factor of 0.99, the most recent day will be weighted about 1% (instead of 1/250). With a decay factor of 0.97, the most recent day will be weighted about 3%.
!
(^) 0.6 * 1%). Their approach requires day-specific estimates of variance that change over the historical time period, which they obtain by using GARCH models.^14 Note that all of these variations are designed to capture shifts that have occurred in the recent past but are underweighted by the conventional approach. None of them are designed to bring in the risks that are out of the sampled historical period (but are still relevant risks) or to capture structural shifts in the market and the economy. In a paper comparing the different historical simulation approaches, Pritsker notes the limitations of the variants.^15
Monte Carlo Simulation In the last chapter, we examined the use of Monte Carlo simulations as a risk assessment tool. These simulations also happen to be useful in assessing Value at Risk, with the focus on the probabilities of losses exceeding a specified value rather than on the entire distribution.
General Description The first two steps in a Monte Carlo simulation mirror the first two steps in the Variance-covariance method where we identify the markets risks that affect the asset or assets in a portfolio and convert individual assets into positions in standardized instruments. It is in the third step that the differences emerge. Rather than compute the variances and covariances across the market risk factors, we take the simulation route, where we specify probability distributions for each of the market risk factors and specify how these market risk factors move together. Thus, in the example of the six-month Dollar/Euro forward contract that we used earlier, the probability distributions for the 6- month zero coupon $ bond, the 6-month zero coupon euro bond and the dollar/euro spot rate will have to be specified, as will the correlation across these instruments. While the estimation of parameters is easier if you assume normal distributions for all variables, the power of Monte Carlo simulations comes from the freedom you have
(^14) Hull, J. and A. White, 1998, Incorporating Volatility Updating into the Historical Simulation Method for Value at Risk, Journal of Risk, 15 v1, 5-19. Pritsker, M., 2001, The Hidden Dangers of Historical Simulation, Working paper, SSRN.
to pick alternate distributions for the variables. In addition, you can bring in subjective
judgments to modify these distributions. Once the distributions are specified, the simulation process starts. In each run, the
market risk variables take on different outcomes and the value of the portfolio reflects the
outcomes. After a repeated series of runs, numbering usually in the thousands, you will
have a distribution of portfolio values that can be used to assess Value at Risk. For
instance, assume that you run a series of 10,000 simulations and derive corresponding values for the portfolio. These values can be ranked from highest to lowest, and the 95%
percentile Value at Risk will correspond to the 500th^ lowest value and the 99th^ percentile
to the 100th^ lowest value.
Assessment
Much of what was said about the strengths and weaknesses of the simulation approach in the last chapter apply to its use in computing Value at Risk. Quickly
reviewing the criticism, a simulation is only as good as the probability distribution for the
inputs that are fed into it. While Monte Carlo simulations are often touted as more
sophisticated than historical simulations, many users directly draw on historical data to make their distributional assumptions.
In addition, as the number of market risk factors increases and their co-
movements become more complex, Monte Carlo simulations become more difficult to
run for two reasons. First, you now have to estimate the probability distributions for hundreds of market risk variables rather than just the handful that we talked about in the
context of analyzing a single project or asset. Second, the number of simulations that you
need to run to obtain reasonable estimate of Value at Risk will have to increase
substantially (to the tens of thousands from the thousands).
The strengths of Monte Carlo simulations can be seen when compared to the other two approaches for computing Value at Risk. Unlike the variance-covariance approach,
we do not have to make unrealistic assumptions about normality in returns. In contrast to
the historical simulation approach, we begin with historical data but are free to bring in
both subjective judgments and other information to improve forecasted probability
the sampling process in Monte Carlo simulations and report a substantial savings in time and resources, without any appreciable loss of precision.^18 The trade off in each of these modifications is simple. You give some of the power and
precision of the Monte Carlo approach but gain in terms of estimation requirements and
computational time.
Comparing Approaches Each of the three approaches to estimating Value at Risk has advantages and
comes with baggage. The variance-covariance approach, with its delta normal and delta gamma variations, requires us to make strong assumptions about the return distributions
of standardized assets, but is simple to compute, once those assumptions have been made.
The historical simulation approach requires no assumptions about the nature of return
distributions but implicitly assumes that the data used in the simulation is a representative
sample of the risks looking forward. The Monte Carlo simulation approach allows for the most flexibility in terms of choosing distributions for returns and bringing in subjective
judgments and external data, but is the most demanding from a computational standpoint.
Since the end product of all three approaches is the Value at Risk, it is worth
asking two questions.
- How different are the estimates of Value at Risk that emerge from the three
approaches?
- If they are different, which approach yields the most reliable estimate of VaR?
To answer the first question, we have to recognize that the answers we obtain with all three approaches are a function of the inputs. For instance, the historical simulation and
variance-covariance methods will yield the same Value at Risk if the historical returns
data is normally distributed and is used to estimate the variance-covariance matrix.
Similarly, the variance-covariance approach and Monte Carlo simulations will yield
roughly the same values if all of the inputs in the latter are assumed to be normally distributed with consistent means and variances. As the assumptions diverge, so will the
(^18) Glasserman, P., P. Heidelberger and P. Shahabuddin, 2000, Efficient Monte Carlo Methods for Value at Risk, Working Paper, Columbia University.
answers. Finally, the historical and Monte Carlo simulation approaches will converge if
the distributions we use in the latter are entirely based upon historical data. As for the second, the answer seems to depend both upon what risks are being
assessed and how the competing approaches are used. As we noted at the end of each
approach, there are variants that have developed within each approach, aimed at
improving performance. Many of the comparisons across approaches are skewed by the
fact that the researchers doing the comparison are testing variants of an approach that they have developed against alternatives. Not surprisingly, they find that their approaches
work better than the alternatives. Looking at the unbiased (relatively) studies of the
alternative approaches, the evidence is mixed. Hendricks compared the VaR estimates
obtained using the variance-covariance and historical simulation approaches on 1000
randomly selected foreign exchange portfolios.^19 He used nine measurement criteria, including the mean squared error (of the actual loss against the forecasted loss) and the
percentage of the outcomes covered and concluded that the different approaches yield
risk measures that are roughly comparable and that they all cover the risk that they are
intended to cover, at least up to the 95 percent confidence interval. He did conclude that all of the measures have trouble capturing extreme outcomes and shifts in underlying
risk. Lambadrais, Papadopoulou, Skiadopoulus and Zoulis computed the Value at Risk
in the Greek stock and bond market with historical with Monte Carlo simulations, and
found that while historical simulation overstated the VaR for linear stock portfolios, the results were less clear cut with non-linear bond portfolios.^20
In short, the question of which VaR approach is best is best answered by looking
at the task at hand? If you are assessing the Value at Risk for portfolios, that do not
include options, over very short time periods (a day or a week), the variance-covariance
approach does a reasonably good job, notwithstanding its heroic assumptions of normality. If the Value at Risk is being computed for a risk source that is stable and
where there is substantial historical data (commodity prices, for instance), historical
(^19) Hendricks, D., 1996, Evaluation of value-at-risk models using historical data, Federal Reserve Bank of New York 20 , Economic Policy Review, v2,, 39– 70. Lambadiaris, G. , L. Papadopoulou, G. Skiadopoulos and Y. Zoulis, 2000, VAR: Hisory or Simulation?, www.risk.net.
