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Measuring Risk: Value-at-Risk

In a previous article, we talked about the derivatives “greeks“, tools derivatives end-users employ to describe and to characterize the various exposures to fluctuations in financial prices inherent in a particular position or portfolio of instruments. Such a portfolio of instruments may include cash instruments, derivatives instruments, borrowing and lending. In this article, we will introduce two additional techniques for measuring and reporting risk: Value-at-Risk assessment and scenario analysis.


Financial institutions and corporate Treasuries require a method for reporting their risk that is readily understandable by non-financial executives, regulators and the investment public and they also require that this mechanism be scientifically rigorous. The answer to this problem is Value-at-Risk (VaR) analysis. VaR is a number that expresses the maximum expected loss for a given time horizon and for a given confidence interval and for a given position or portfolio of instruments, under normal market conditions, attributable to changes in the market price of financial instruments.

What does this mean in English? Suppose that we are investment managers with positions in foreign exchange, fixed income and equities. We need an assessment of what we can expect the worst case to be for the position overnight with a 95% degree of confidence. The VaR number gives us this measurement. For example, the portfolio manager might have 100 million dollars under management and an overnight-95% confidence interval VaR of 4 million dollars. This means that 19 times out of 20 his biggest loss should be less than 4 million dollars. Hopefully, he is making money instead of losing money. You can also express VaR as a percentage of assets, in this case 4%.

VaR is also useful when we want to compare the riskiness of different portfolios. Let us now consider two portfolio managers. Each of them starts the year with 100 million dollars under management. Bob makes a return of 30%, handily beating his target of 20%. Jerry makes a return of 20%, coming in on target. Who is the better fund manager? The answer is, as economists always say, it depends. To make an accurate judgment, many people believe that we need to compare the risk involved.

Let’s say that Bob’s average overnight-95% VaR was 7 million dollars and Jerry’s average overnight-95% VaR was 2 million dollars. One way of calculating Bob’s return on risk capital is as follows: 30 million dollars/7 million dollars=428.6% Using the same method, Jerry’s return on risk capital is: 20 million dollars/2 million dollars=1000.0% It could be reasonably argued that Jerry is a better fund manager in that he used his risk capital more efficiently. How many people when they invest in mutual funds know anything about the risk that their portfolio managers take in generating a return? Most mutual funds do not report this kind of risk-adjusted number, although some of them could use it to justify or explain their actions.

This is especially important when evaluating how closely a portfolio manager conformed to the stated risk tolerance of his fund. If the fund is advertising itself as a very low-risk investment vehicle suitable for widows and orphans, the average daily VaR as a percentage of assets is an interesting number, especially when compared to the same number for more openly risky investments. Corporate Treasuries and Banks use VaR for the same purpose. They need to have an idea of how their market exposures behave under normal market conditions. It is a risk management cliché but you know that you have a bad risk management regime in place if you are surprised by the extent of any gains or losses that you sustain.

Calculating Value-At-Risk

Value-at-Risk is scientifically rigorous in that it utilizes statistical techniques that have evolved in physics and engineering. VaR is questionable in that it makes assumptions in order to use these statistical techniques. Chief among these assumptions is that the return of financial prices is normally distributed with a mean of zero. The return of a financial price may be thought of as the capital gain/loss that one might expect to accrue from holding the financial asset for one day.

For example, in the case of foreign exchange, if I own one Canadian dollar against being short 1 US dollar, I will earn a return overnight if the Canadian dollar appreciates against the US dollar. One way of expressing the return is the difference between the current price and the previous period’s price, divided by the previous period’s price. JP Morgan has developed a methodology for calculating VaR for simple portfolios (i.e. portfolios that do not include any significant options components) called RiskMetrics. The success of RiskMetrics has been so great that Morgan has spun off the RiskMetrics group as a separate company.

Risk Metrics forecasts the volatility of financial instruments and their various correlations. It is this calculation that enables us to calculate the VaR in a simple fashion. Volatility comes into play because if the underlying markets are volatile, investments of a given size are more likely to lose money than they would if markets were less volatile. Volatility here refers to the distribution of the return around the mean. A volatile market is one in which the returns can vary greatly around the mean while a calm market is one in which the returns vary little around the mean.

Correlation is important, too. Modern portfolio theory is familiar to many people who intentionally diversify their investments. If we invest all of our money in a set of financial instruments that move in the same direction and with the same relative speed, that is a riskier portfolio than if we invest in a portfolio of financial instruments that move in different directions at different speeds. If the instruments in the former portfolio all move down, we will lose money on each of these instruments whereas we would expect to make money on some instruments and lose money on the remaining instruments in the latter portfolio. Hopefully, in the case of the latter portfolio, we make more money than we lose, on average.

Earlier, we stated that volatility was both dynamic and persistent. That is to say, volatility changes over time but it moves in a trending fashion. Correlation is dynamic, too. Correlations move with less persistence than volatilities. It is easy to see how complex the management of financial price risk can be with a portfolio containing more than two or three instruments. For more information, visit the RiskMetrics web site at http://www.riskmetrics.com. Once optionality is involved, it becomes computationally difficult to calculate the VaR, in some cases requiring statistical simulation of the portfolio. The reason for this is because of the convexity of option products. Straightforward VaR calculation underestimates or overestimates the VaR, depending on whether or not one is long or short convexity (i.e. whether one owns or has sold options).

Scenario Analysis

In describing VaR, I have emphasized the fact that VaR is only good for calculating an expected maximum loss under normal market conditions. Because of the generally idiosyncratic nature of financial prices, we must have a way of understanding the implications for our portfolio of abnormal market conditions. Scenario analysis is the tool we use for this purpose. Consider the portfolio manager from our original example in this article who has an overnight-95% VaR of 4 million dollars on underlying assets of 100 million dollars. The VaR number that our calculator generates describes his expected loss under normal market conditions. An important, critical adjunct procedure to VaR measurement is scenario analysis. In scenario analysis, the portfolio manager will simulate various hypothetical evolutions of events in order to determine their effect on the value of the portfolio.

For example, a Canadian bond portfolio manager in September 1995 would have been engaging in some heavy-duty scenario analysis to determine at all times what the effects of a Yes vote in the October 1995 Quebec separation referendum would have been. There are an infinite number of possible scenarios that the portfolio manager or investor could consider. However, it is possible to reduce this universe of possibilities to a set of important tests of the stressors of a portfolio of financial instruments.

Any portfolio manager must understand what the weak spot is in his portfolio. Naturally, this is the first set of scenarios to simulate. By determining the change in value of his portfolio under stressful conditions (called “stress-testing”), the portfolio manager has a better perception of where the risks in his portfolio lie. At that point, he can make trades that reduce this risk to levels with which he is comfortable. At the very least, he has an appreciation of what will happen so that if the worst-case does take place unexpectedly, he can act more decisively and more quickly to manage his portfolio. Without this kind of stress-testing, he will be forced to react in a moving market, a situation that can exacerbate his market losses. In a complex derivatives portfolio, stress-testing that reveals excessively risky exposures either to movements in the underlying cash rate or shifts in implied volatility or interest rates (or combinations of these factors) is said to identify “risk holes.”

For example, an options portfolio that is short a great deal of short-dated options around a particular strike is said to have a “gamma hole” around that strike and date (analogous to space and time, in a physical sense). If the underlying rate were to move to the same level as the strike price around the same time as the options were maturing or just before, the portfolio would become very difficult to manage and the profitability of the portfolio could become intolerably volatile. The bottom line here is that all of these ways of measuring risk must be interpreted in terms of the preferences of the investor or the institution managing the risk.