Determining Value at Risk Using Monte Carlo Simulation

Value at risk (VAR) is a commonly used risk measure in the finance industry.  Monte Carlo simulation is one of the methods that can be used to determine VAR.

There are two things we need to specify when stating value at risk:

  1. The time horizon.  This may be daily for some portfolios or a longer period for less liquid assets.
  2. The confidence level, 95% is common.

If the time horizon is one day and the confidence level is 95%, VAR is the most we would lose 95% of the time.  One out of twenty days we would expect to lose more than VAR.

An important topic beyond the scope of this article is if VAR is exceeded in the 5% of cases, how much could we expect to lose.  This is expected shortfall or conditional VAR.  Refer to the article on expected shortfall for more information.

Before doing a VAR simulation, there needs to be a model of the portfolio returns.  This model should also include any correlations among returns of each portfolio member.  Developing a model with appropriate return probability distributions is the most important and time consuming part of determining VAR.

A simple model consisting of five investments is shown below.  It is assumed that the return of each investment is normally distributed.  A Spearman rank correlation matrix is included to account for correlation among the investments.  Four of the investments are positively correlated while investment C is negatively correlated with the other investments.

The output of the simulation is calculated in the Ending Portfolio Value cell, B5.

VAR Model

The output of the simulation is shown below.  To calculate value at risk for a 95% confidence level we subtract the 5th percentile ending portfolio value from the initial portfolio balance.

VAR = 10,000,000 - 9,958,839 = 41,161

VAR Simulation Output

Value at risk is the maximum loss 95% of the time.  Also, 5% of the time we can expect to lose more than the value at risk.

Simulation was performed using Simulation Master.

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