“Even the youngest of us may be wrong sometimes.” [George Bernard Shaw]

## Abstract

Value at Risk (VaR) is the most commonly used measure of the amount that could be lost from a position or a portfolio.

VaR is understood to be the maximum loss which could occur at a given confidence level over a specified time horizon. More precisely, it is the threshold value such that the probability that the mark-to-market loss on the portfolio over the given time horizon exceeds this value is the given probability level. We assume normal markets and no trading in the portfolio.

VaR calculations often assume that returns are normally distributed over the time horizon. Common parameters are 1% and 5% probabilities and a one day or 10 days (2 weeks) horizon. The inputs for VaR calculations include details of the portfolio, the time horizon and the parameters which drive the distribution of the underlyings (the average growth rate, volatilities and correlations). For short time horizons the growth rate is negligible.

## Confidence Level

If you are working with standard deviations only then you can go from one confidence level to another by adding the corresponding difference of their distances of standard deviations from the mean:

Degree of Confidence # of Standard Deviations from Mean
95% 1.64485362695147
99% 2.32634787404084
99.9% 3.09023230616781

You can move from one time horizon to another by multiplying or dividing with the square root of time. This is only true if the difference between time horizons is short enough to ignore the growth rate.

## Different Approaches

There are three different basic approaches to calculate VaR:

1. Variance / Covariance Method

2. Historical Simulation

3. Monte Carlo Simulation

The variance / covariance method requires only the two factors average return and standard deviation. Therefore it is also called the parametric method. It assumes that returns are normally distributed over the time horizon.

The historical simulation requires more computational data and produces a more accurate VaR. It also assumes that history repeats itself.

The Monte Carlo simulation is more complex to implement. Its advantage is that you can design future patterns to differ from historical patterns.