Value at Risk


One of the most pertinent questions in risk management has been: How much do you stand to lose, over a certain period and with a certain probability? The most common answer to this question today is Value at Risk, a risk measure that expresses itself as one number. What is that number and what does it stands for? In order to interpret this number we first have to assume that:

•	Everything assumed in the (VaR) calculations/process is true
•	All approximations made are accurate
•	The future follows the past and whatever risk you are analyzing only exists for the specified (certain) period

If you are reasonably comfortable with all of the above then the one number to your question is Value at Risk: A worst case loss with limits on time period and probability.

VaR has been around for quite some time now. However the effectiveness of the concept has been the subject of persistent debate. Like all tool, its efficacy depends on how the tool is put to use. Some of the key issues raised in current literature about VaR are:

•	Is the one number sufficient by itself to completely capture the risk in a position?
•	Whether the users of VaR understand the limitations (of the tool) and the implications of those limitations?
•	Risk management is concerned with extreme events or large deviations from what is expected. The most common tool used for measuring the above is variance, an average (of sorts) of all the deviations from the mean. Although it is the key tool used in calculating VaR, its not the most appropriate. Higher order factors that measure symmetry or length & thickness of tails would be more accurate.
•	VaR uses data from all events to evaluate the impact of extreme events. VaR is forced to do this because by definition, extreme events do not occur frequently enough to generate sufficient data. The downside is that extreme events have much higher means and variances. This means that if VaR (somehow) did use extreme events, it would lead to a much higher Value at Risk estimate.
•	Given that the object of risk management is to understand risk exposures and neutralize them, there is a strong emphasis on supplementing VAR with scenario analysis or sensitivity testing.

There are three current methods used for calculating VaR

a.	Variance /Covariance method
b.	Historical simulation method
c.	Monte Carlo simulation method

All methods have a common base but then diverge in how they actually calculate Value at Risk. They also have a common problem in assuming that the future will follow the past. This shortcoming is normally addressed by supplementing any VAR figures with appropriate sensitivity analysis and/or stress testing. In general the VAR calculation follows five steps:

• Identification of positions
• Identification of risk factors affecting valuation of positions.
• Assignment of probabilities (or statistical distribution) to possible risk factors values.
• Creation of pricing functions for positions as a function of values of risk factors.
• Calculation of VAR


Variance/Covariance method
The Variance-Covariance method makes a number of assumptions. The accuracy of the results depends on how valid these assumptions are. The method gets its name from the variance-covariance matrix of securities that is used to calculate VaR.

The method starts by calculating the standard deviation and correlation for the risk factor and then uses these values to calculate the standard deviations and correlation for the changes in the value of the individual securities that form the position. If price, variance and correlation data is available for individual securities then this information is used directly. The values are then used to calculate the standard deviation of the portfolio.

VAR for a specific confidence interval is then calculated by multiplying the standard deviation by the appropriate normal distribution factor.

In some cases a method equivalent to the variance covariance approach is used to calculate VAR. This method does not generate the variance covariance matrix and uses the following approach:

1. Separate the portfolio in a long side and a short side.
2. Calculate the return series for the long side and the short side.
3. Use the return series to calculate the correlation and variances for the long and short sides
4. Use the results in (3) to calculate the VaR.

The modified approach can be used where, due to the nature of the institutions strategies, a number of positions would net close to zero on a portfolio basis and also where the set of securities employed is so large that a variance - covariance approach would have significant resource/time requirements.


Historical Simulation Method
This approach requires fewer statistical assumptions for underlying market factors. It applies the historical (100 days) changes in price levels to current market prices in order to generate a hypothetical data set. The data set is then ordered by the size of gains/losses. VAR is the value that is equaled or exceeded the required percentage of times (1, 5, 10).


Monte Carlo Simulation
The approach is similar to the Historical simulation method described above except for one big difference. The hypothetical data set used is generated by a statistical distribution rather than historical price levels. The assumption is that the selected distribution captures or reasonably approximates price behavior of the modeled securities.

	Comparison of Value at Risk Methods Historical Simulation Variance / Covariance Monte Carlo Simulation Risks of portfolios that contain options Yes, regardless of the option content No, except when computed using a short holding period with limited or moderate option content Yes, regardless of the option content Computation performed quickly Yes Yes No, except for relatively small portfolios Easy to explain to senior management Yes No No Misleading VAR estimates when recent past is not representative Yes Yes, except that alternative correlation may be used Yes, except that alternative correlation may be used Scenario analysis No Easy to examine assumptions about variances and correlation Unable to examine alternative assumptions about distribution of market factors Yes