Master Class: Calculating Value at Risk (VaR): VaR Methods
There are three primary methods used for calculating Value at Risk (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 (VaR). 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 for Value at Risk
- 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 Value at Risk (VaR)
Variance Covariance Approach for calculating Value at Risk
This method assumes that the daily returns follow a normal distribution. From the distribution of daily returns we estimate the standard deviation (?). The daily Value at Risk VaR is simply a function of the standard deviation and the desired confidence level. In the Variance-Covariance VaR method the underlying volatility may be calculated either using a simple moving average (SMA) or an exponentially weighted moving average (EWMA).
Mathematically, the difference lies in the method used to calculate the standard deviation (?).
This approach is generally utilized if it is believed that the daily returns during the look back period have followed normal market conditions. The SMA approach places equal importance to all returns in the series whereas the EWMA approach places greater emphasis on returns of more recent durations.
The Variance-Covariance VaR 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 Value at Risk (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.
Value at Risk (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:
- Separate the portfolio in a long side and a short side.
- Calculate the return series for the long side and the short side.
- Use the return series to calculate the correlation and variances for the long and short sides
- 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 for calculating Value at Risk (VaR)
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. Value at Risk (VaR) is the value that is equaled or exceeded the required percentage of times (1, 5, 10).
Historical simulation is a non-parametric approach of estimating VaR, i.e. the returns are not subjected to any functional distribution. VaR is estimated directly from the data without deriving parameters or making assumptions about the entire distribution of the data. This methodology is based on the premise that the pattern of historical returns is indicative of future returns.
Monte Carlo Simulation for calculating Value at Risk (VaR)
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.
A Monte Carlo simulator uses random numbers to simulate the real world. A Monte Carlo VaR model using the following sequence of steps
- Generate randomly simulated prices
- Calculate daily return series
- Repeat the steps in the historical simulation method described below
Quick Review of Value at Risk (VaR) methods
|Comparison of Value at Risk Methods|
Variance / Covariance VaR
Monte Carlo Simulation VaR
Risks of portfolios that contain options
VaR Computation performed quickly
VaR method easy to explain to senior management
Misleading VAR estimates when recent past is not representative
Unable to examine alternative assumptions about distribution of market factors
Implementing Value at Risk (VaR)
The objective of a Value at Risk (VaR) implementation is to perform daily VaR analysis of positions within a portfolio. Such a process would be the first step in shifting the current emphasis from calculating VaR to managing VaR. Within the process the focus should be on:
- Positions with low coverage levels.
- Positions with VaR beyond a set threshold.
- Positions with significant VaR changes.
- VaR analysis for the Desk. (All clients, All accounts, All positions)
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