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Value-at-Risk (VaR): Three Calculation Approaches


There are three main approaches to calculating Value-at-Risk (VaR).  This article summarizes the three, and highlights their positives and negatives.

Variance-Covariance VaR

The Variance-Covariance approach to VaR assumes that portfolio returns are normally distributed around a mean return of zero.  Variance-Covariance VaR is often called "Delta Normal VaR" and is a parametric method of calculation.  Under this approach, the estimated return distribution is defined by its standard deviation (high to low volatility).  Variance-Covariance VaR is calculated using only by 2 factors: The estimated standard deviation of returns, and the percentage cutoff or scaling factor in the left loss tail of the distribution. 

For example, if we choose the 95% VaR cutoff or scaling factor, then 1.645 standard deviations of porftolio returns will give us the VaR number we are seeking.  At 99% cutoff, 2.33 standard deviations will give us the VaR.

General process for the Variance-Covariance approach:

- Gather the necessary historical input data sufficient for your chosen lookback period (1 year of daily trading days is the usual).  This includes units (ticks, cents, basis points, etc.), values (prices, rates), and quantities for each product and trade in the portfolio.

- Calculate the standard deviations (volatilities) of the individual instruments contained in the portfolio.  This is done by selecting a lookback period and data sample periodicity-- usually 1 year lookback and daily prices for market risk.

- Calculate a matrix of correlations between all risk factors and instruments.  This is most accurate if correlations are calculated using historical data.  For event risk stress testing, these matrix correlation values can be overridden.  Risk managers will typcially increase correlations signficantly to represent worst case scenarios.  Market panics cause correlations across normally non-correlated asset classes to converge (i.e. massive selloff and flight to cash and gold).  By stress testing with very high correlations, the Risk Manager can see the type of losses that may occur in rare but substantial market shocks.

- Calculate the expected portfolio standard deviation (volatility) using individual standard deviations and correlations.

- Multiply the daily portfolio mark to market by the calculated portfolio standard deviation to get the expected 1 standard deviation loss.  Then multiply this value by the chosen scaling factor (e.g. 95% = 1.645 standard deviations, 99% = 2.33 standard deviations) to get the expected VaR.

Benefits and Drawbacks of Variance-Covariance VaR:

The main benefits of Variance-Covariance VaR are speed, minimal system resources required for calculation, and broad support of data and software needs.  A wide range of software and data providers support Variance-Covariance VaR, including RiskMetrics (JPMorgan made VaR popular with RiskMetrics), Reuters, Bloomberg, Summit Systems, etc. 

The major drawback of Variance-Covariance VaR is the lack of accuracy in modeling "fat tails" associated with extreme market events.  By assuming a normal distribution of returns, Variance-Covariance VaR has the potential to miss substantial losses which occur in extreme market events when correlations increase substantially.  In addition, Variance-Covariance VaR does not capture return distribution skew or kurtosis associated with non-linear returns of options or products with embedded optionality.  Portfolios constructed solely of linear instruments such as cash equities and interest rate swaps have close to normally distributed returns under normal market conditions.  But the introduction of optionality and market shock "tail" events results in return distributions significantly different than the normal distribution assumption.  Failiure to capture "fat tails" leads to significantly underestimated VaR.  As a result, Variance-Covariance VaR must be used with caution, modified to incorporate nonlinearity and market shocks, or stress tested extensively.

Historical VaR (Full Revaluation)

The second method of calculating Value at Risk is Historical simulation.  Historical VaR is calculated using actual historical market data for each instrument in the current portfolio.  Every instrument in the portfolio is revalued under a variety of historical periods using the various risk factor inputs (prices, rates, volatilities, etc.), creating a distribution of returns.  VaR is calculated by multiplying the left loss tail by the cutoff scaling factor (e.g. 95% = 1.645 standard deviations, 99% = 2.33 standard deviations) to arrive at an estimated loss.

General process for the Historical approach

Let's assume we have a portfolio consisting of 10 stocks and 3 bonds.  Each of these products has 10 years of price history.  We can segment the historical data by year to create 10 data sets. Some of these years are quite volatile with market shocks while others are calm.  Using each annual data set, today's portfolio is revalued using daily prices to create a 1 year portfolio distribution of values.

This process is repeated for each year in the historical period, and the results are saved. 


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