Back to Basics: Value at Risk (VaR)

Teri Geske
Senior Vice President, Product Development


Over the past few years a tremendous amount of work has been done in the area of "Value at Risk" (referred to as "VaR" or "V-A-R"). There have been countless seminars and conferences on VaR, many books and articles have been written on the subject and there is no shortage of vendors touting their VaR systems as the sin qua non of risk management. VaR was originally designed for banks with significant trading operations covering several markets (fixed income, foreign exchange, derivatives, etc.) to quantify the institution’s risk in a systematic way. VaR is now used not only as an internal management tool, it has been adopted by international bank regulators in determining whether or not an institution is adequately capitalized. Although VaR has been embraced by most large banks, other members of the financial community (insurance companies, investment managers and plan sponsors) are still determining how, if at all, VaR fits into their business. Nonetheless, even though your firm may not yet be using VaR, it is a concept that is most likely here to stay. Therefore, we thought it might be useful to review the basics of VaR , including some of the strengths and weaknesses of this approach to risk management.

VaR is defined as the expected loss in value, given a statistical level of confidence, due to adverse movements in underlying risk factors. VaR allows us to state that "over the next x days, the portfolio is expected to lose no more than $y (or y%) in value with z% confidence" where z% is typically 95% or 99% (the Bank for International Settlements (BIS) standards use VaR in terms of a 3-day horizon with a 99% confidence interval.). Now, the statement that "99% of the time losses will not exceed $y" may sound rather comforting, but this also means that 1% of the time (one out of a hundred observations) we expect that losses will exceed the dollar value resulting from the VaR analysis. If we are using a 1-day VaR and a 99% confidence level, given that there are about 200+ trading days in a year we are saying that losses will be more severe than the VaR amount approximately twice a year. Furthermore, VaR says nothing about how bad the loss might be that 1% of the time – this is why risk managers realize that VaR should be combined with stress testing to determine what might happen under extreme conditions.

There are three approaches used to compute VaR, referred to as the "variance/covariance", "historical simulation" and "Monte Carlo simulation" methods. We will briefly summarize them here, mentioning some strengths and weaknesses of each. The variance/covariance approach assigns (or "maps") each asset to one or more equivalent risk position based on the factor(s) that affect the asset’s value. For example, a portfolio consisting of a 5 year bond and futures contracts on the S&P500 and would be represented as exposures to movements in the five year U.S. interest rate and the S&P500. The VaR of the portfolio is computed based on the variance and covariance of the individual risk factors over the VaR time horizon. So, if the daily change in the 5 year U.S. Treasury rate has a standard deviation of 4bps, the interest rate risk component of the portfolio’s one-day VaR with a 99% confidence interval would be based on a 2.33x4bp approximately 9bp move. To compute VaR (in dollars), the change in each risk factor associated with the chosen confidence level is multiplied by the "delta equivalent" value of the position – for fixed income securities, this is the dollar-duration (i.e., the change in dollar value given a small change in interest rates). If there is some negative correlation among the risk factors (e.g. if the S&P500 tends to move up when the Treasury prices go down and vice versa), the covariance between these risk factors would make the VaR of the portfolio something less than the sum of the VaRs of two separate portfolios, one holding the S&P500 futures and the other a 5 year Treasury.

The primary advantage of the variance/covariance approach is that it is fairly easy to compute. There are a number of sources of variances and covariance "matrices" for key risk factors (exchange rates, interest rates, commodity prices, etc.) that can be downloaded into spreadsheet programs designed to compute VaR using this method. However, there are a number of drawbacks to this approach; the most important for fixed income portfolios is that the price sensitivity of options, or of bonds with embedded options (callable bonds, mortgage-backed securities, etc.) cannot be adequately described by the variance/covariance method. This method implicitly assumes that prices change at a constant rate with respect to a change in a market risk factor, but this assumption is not valid for options. The price behavior of options is "non-linear" – in other words, not constant. For example, as interest rates rise a mortgage-backed security’s duration (its sensitivity to interest rate risk) can increase considerably due to a change in the value of the embedded prepayment option. The variance/covariance approach does not capture this and can significantly underestimate the true VaR for a portfolio containing options.

The Historical Simulation VaR method observes the actual level of market risk factors (such as yield curves, exchange rates, commodity prices, etc.) over a period of time and revalues each asset in the portfolio given each observed risk factor. For example, if a portfolio consisted solely of 30 year zero coupon bonds, we would observe the 30 year Treasury (spot) rate over each of the past 100 days and would revalue the portfolio 100 times, given these different interest rate levels. A one-day VaR with 95% confidence would be computed as the 5th largest decline in the portfolio’s value of the 100 daily observations. One advantage to this approach over the variance/covariance method is that it does reflect the non-linear price behavior of options. A disadvantage of the historical simulation method is that the VaR number is highly sensitive to the time period used to observe the market risk factors. For example, the VaR of a corporate bond portfolio calculated as the 5th worst loss using six months of credit spread changes observed over two years from June 1996 – June 1998 would be markedly different than the 5th worse loss over the two years from January 1997 to January 1999.

The Monte Carlo VaR approach overcomes the historical method’s dependence on a particular time period by generating a (random) distribution of changes in each key market risk factor based on parameters specified for each factor. The portfolio is revalued under each set of market conditions generated by the Monte Carlo simulation and, as with the Historical method, the changes in portfolio value are ordered so that the VaR is observed as the loss in value corresponding to the desired confidence level, e.g., the 5th worst loss out of one hundred observations for a 95% confidence level. The Monte Carlo approach is quite robust but requires the most sophisticated analytical systems and the greatest data collection effort.

Some problems with VaR: In addition to the drawbacks of each method cited above, VaR suffers from a number of shortcomings. First, the three calculation methods can produce radically different results – this makes it difficult (if not impossible) to compare VaR numbers reported by different institutions. There is the issue raised earlier that even a 99% confidence level says nothing about how severe the loss might be at the "tail" of the distribution. Two firms (or two portfolios) might have the same VaR at a 95% confidence interval, but at the 96th percentile one’s loss might be twice as large as the other’s. If a firm relies exclusively on VaR for risk management, the potential for catastrophic loss due to extreme changes in risk factors could grow unchecked over time. That possibility leads us back to the importance of stress testing as mentioned earlier. There are many other practical as well as theoretical issues to address in deciding which (if any) VaR analysis to use, and there is no shortage of financial literature devoted to the topic.

Finally, we should ask ourselves, is VaR appropriate for investment management? Traditionally, the time horizon for VaR analyses has been measured in terms of days (one day, three days, a week) and in terms of dollar value. Since investment managers typically measure performance monthly, and usually relative to a benchmark, a one-month "relative" VaR might be more appropriate. Using this approach, a VaR analysis using a 99% confidence level would state that in one month out of 100 the portfolio is expected to underperform by more than y% relative to a benchmark. Frankly, it becomes increasingly difficult to compute, back-test and interpret VaR numbers for these longer time horizons because the necessary data is difficult to collect. In this example, back testing would require us to collect 8+ years of monthly observations before we could determine whether or not our actual loss exceeded our computed VaR more than 1% of the time. Over that lengthy period, the parameters used to compute the original VaR number may no longer reflect actual market conditions and the portfolio’s exposure to different risk factors would most likely change. So, perhaps it is better to stay with short VaR time horizons despite the longer-term perspective of most investment managers.

Despite these drawbacks, VaR can be a useful tool. It promotes risk awareness, can be used to evaluate a firm’s risk profile over time or to compare asset managers across different sectors and so on.