‘Value at Risk’ useful, but beware the pitfalls 
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Tony Morley, winner of the treasury FTI Award for Excellence for 1999, writes that Value at Risk is a first order approximation for possible losses from adverse financial risk, but the numbers cannot be taken directly at face value without understanding the assumptions and flaws of the model. 
An inevitable component of every business activity is risk, and never has its importance been so prominent. Its measurement and management have become as integral a part of the business as the generation of profits. It has however been easier to measure the results of an action than its associated risks.
Researchers in the field of finance have long recognised the importance of measuring the risk of a portfolio of financial assets. Investors are generally assumed to be risk adverse, that is, are only willing to hold minimum risk for a given level of return.
While it is important to have information about the average return of the investment portfolio it is also necessary to know how much the portfolio's performance deviates from the average, i.e. the standard deviation of returns.
The assumption is that investors wish to achieve a portfolio with the highest rate of return for a given level of risk.
For example, suppose one has a simple portfolio containing only two risk assets. The actual return from holding these assets over a defined period is R1 and R2 with expected returns E[R1] and E[R2]. If the investor invests x1 of their funds in one of these assets and the balance x2 = (1x1) in another asset, the expected return and variance on the portfolio can be given by:
E[Rp] = x1E[R1] + x2E[R2]
And the variance of the portfolio is:
Var[Rp] ? sp2 = x12ss12+x22s222+2x1x2s12
Where si2 = variance of asset i.
The relationship between the returns on both assets as characterised by the correlation coefficient, s, is embedded in the covariance quantity
cov[R1,R2] ? s12 = rs1s2
Therefore the portfolio variance sp, which is a measure of the deviation, is dependent on the relationship between the two assets in the portfolio. The variance of the portfolio return, that is risk, as quantified by the standard deviation of return, depends not only on the individual assets but also on the extent to which their returns are correlated. The more negative the degree of correlation, (given by the correlation coefficient,r), the greater the benefits of diversification and the lower the overall level of risk incurred.
In recent years the growth of trading activity, the explosion of volatility and instances of severe financial market instability have prompted new studies on the need for market participants to develop reliable risk measurement techniques. One need only think of the losses made from recent currency and stock market crashes, as well as those resulting from the perilous positions taken, for example by Barings, Orange County, LTCM etc. One such technique advanced in recent years is the Value at Risk model (VaR).
Value at Risk is a commonly accepted methodology for measuring the loss magnitudes associated with rare tail events in financial markets. It aims to summarise risk by estimating the worst expected loss over a chosen time horizon with a given confidence interval. Suppose a firms’ daily VaR is ?100,000 with 95% probability; this means that, over the coming day there is only a 5% chance that a loss will be greater than ?100,000. The popularity of VaR is based on aggregation of several components of firmwide market risk into a single number.
While the concept of VaR is straightforward, its implementation is not. There are a variety of models and model implementation that produce very different estimates of risk for the same portfolio. The different approaches to VaR can essentially be classified into three categories.
The variance covariance approach
In short, creates a covariance matrix of returns S, between the variables. VaR is calculated as:
VaR = aspW = a[wSwT]1/2,
where W = initial investment and a is a scaling factor derived from the standard normal distribution.
Simulations
Historical VaR is calculated by mapping a portfolio into a historical price distribution. i.e. map the price movements of financial instruments over a set number of past days and apply these price movements to the current portfolio.
Perhaps the most complicated and computationally intensive. A large number of random draws from an estimated statistical model are simulated and the portfolio is revalued at each simulation and the change calculated.
Each has its own merits and demerits and the assumptions that underlie the model are critical to its robustness. For example (a) VaR generally assumes that market returns are normally distributed; and (b) with variancecovariance VaR and historical simulation VaR to interpret it as a measure of the risk due to adverse movements in market conditions requires the estimation of adverse future asset price movements using historical information on previous price movements and their correlation with other assets. For example, is the Japanese yen correlated to movements in the British pound or Mexican peso? VaR also requires that the user determine correlations across markets as well as within markets, e.g. how does the US bond market correlate with the US stock market?
LTCM assumptions
As with any model, it is assumptions such as the above that lead to the problems with Value at Risk models. VaR models aim to predict downside risk for a firm, i.e. the maximum it would lose given a certain market move. The problem with VaR however is that extreme market moves generally do not occur in normal markets, hence VaR could significantly underestimate the potential risk. E.g. LTCM VaR models estimated that its daily risk at one point should have been $45million, yet in one month, their portfolio lost $1.71 trillion. Using a normal distribution, such an event should have occured once every 800 trillion years or 40,000 times the age of the universe.
No model can be perfect; nevertheless what is trying to be achieved is a best estimate of potential losses from financial positions.
Every treasurer from the largest company to the smallest company is becoming more aware of the risk inherent in their financial dealings. Markets are becoming increasingly volatile, and shareholders will look to treasurers not only to maximise their revenues but to do this in the most risk efficient manner.
The expectation will be that the treasurer should be able to estimate with a certain degree of confidence the maximum loss given a change in market conditions. VaR is one methodology that could be used for this.
Notwithstanding this however, it is important that companies recognise the importance of some form of risk measurement/scenario management of their positions.
The last few years have seen extreme moves in financial markets and provide good historical information for a hypothetical stress test. Information such as this can be obtained easily via Bank of Ireland's “Trendspotter” data. For example, the Mexican peso weakened versus the dollar by well over 200% between 1995 and 1999, this type of movement could have a significant impact on exporters to Mexico.
The point is that treasurers should have some idea in their mind of previous extremes of the currencies that they deal in and use this as a ballpark for maximum potential losses, while noting that history does not always repeat itself in the same guise.
VaR, while an important advance in risk management, is only one aspect of an overall riskmanagement program. It is a first order approximation for possible losses from adverse financial risk, and although it is a vast improvement than no risk measurement, the numbers cannot be taken directly at face value without understanding the assumptions and flaws of the model. It is an aid rather than a substitute to the treasurer’s experience. 
Tony Morley is a Senior Dealer at Bank of Ireland Treasury. The views of the article are his own.

Article appeared in the March 2000 issue.


