This article summarizes several methods of calculating Value at Risk, and provides pricing spreadsheets. Value at Risk is an important tool for estimating capital requirements, and is now a standard risk-management tool.
This approach for calculating the value at risk is also known as the delta-normal method. It needs the average returns, variances and correlation coefficients (derived from historical data). The variance-covariance method assumes that historical returns are normally distributed, and that the future will mirror the past.
The calculation is straightforward, and for a one-asset portfolio is given by this equation.
xα is the αth percentile of a normal distribution, and P is the portfolio value. xα is calculated with Excel’s NORM.S.INV() function. The equations for a two-asset portfolio are given here
This method employs historical returns data to assemble the cumulative distribution function, and does not place any assumptions on the shape of the distribution.
A historical simulation simply sorts the returns by size. If the sample include 100 returns, the value at risk at a confidence of 95% is the fifth largest loss.
Several criticisms are often made of this approach.
- Historical simulation assumes that returns are independent and identically distributed. This not necessarily the case; real-world data often displays volatility clustering.
- Returns in the recent-past and far-past are given equal weighting. However, recent returns have greater bearing on future behavior than older returns.
- This method requires a large set of historical data for accuracy; this, however, is not always available.
- Because the method is entirely reliant on historical data, the result cannot be influenced by subjective information (as with Monte-Carlo simulation). This may be significant if a fund manager predicts large changes in the business environment.
Quadratic or Delta-Gamma Method
The variance-covariance approach can only be used for portfolios with a linear relationship between investment weights and risk. However, options have nonlinear payoffs, which becomes significant close to expiry. Standard variance-covariance methods cannot necessarily be applied.
Quadratic methods (also known as delta-gamma methods) were developed to estimate the Value at Risk for portfolios with options.
The Monte-Carlo Method involves running multiple trials to calculate the portfolio returns. Generally, this method involves the following steps.
- Generate simulated returns by sampling a probability distribution.
- Order the returns by size
- Compute the VaR at the required confidence level. For a simulation of 1000 returns, the 95% percentile would correspond to the 50th lowest value.
Monte-Carlo simulation is an extremely flexible method for calculating Value at Risk. This is because any probability distribution can be selected for all the significant risk factors. However, for a large investment universe, Monte-Carlo simulation can be computationally intensive.
This spreadsheet uses VBA for the Monte-Carlo simulation of VaR. It assumes a normal distribution, but can be adapted to any other distribution.