Value at Risk (“VaR”) is a common risk management metric which quantifies the risk of a portfolio over a period of time. It is easy to explain to top management while answering the question of how much could a portfolio lose within a given probability. However, VaR can’t describe how bad could the portfolio’s losses be when things really go bad. In other words, VaR could underestimate the portfolio loss during stressed scenarios.
After considering its strength and weakness, the company still view that VaR is a good metric to be included into it periodic risk management report. The next natural question is how to compute VaR?
In order to compute VaR, the following are needed:
- Time Frame
- Portfolio Constituents
- Confident Percentage
Some market participants invest by the minute while some invest over a decade or longer. Therefore, logically, the VaR value for short term investors should be smaller than long term investors.
The reason is to study the relationship between assets in the portfolio and compute the correlation matrix among assets. One of the questions to ask during this study is would asset A goes up when asset B goes up?
Confident percentage could be view as how likely that the VaR value be realized. However, the percentage holds true under the assumption of the model used to compute the VaR value and not the real world.
The commonly accepted models to compute VaR are: (i) historical, (ii) Variance Covariance and (iii) Monte Carlo simulation. The order is arrange in increasing difficulty but with increasing comprehensiveness.
Model – Historical
For example, suppose that the confident percentage is 90%, the 10% worst historical portfolio loss would be VaR value with losses computed over the intended time frame (point 1). This method doesn’t require the correlation matrix (point 2).
Model – Variance Covariance
First, either based on historical data or forward looking data, the mean of the portfolio’s return, and the standard deviation of the portfolio’s return, is computed. Note that the standard deviation is computed with correlation being considered. Second, denoting the portfolio’s value as , the VaR using the following formula:
where is the normal distribution’s z-value correspond to the percentile.
Model – Monte Carlo Simulation
It is similar to the normal distribution, however, the mean and the standard deviation could take a range of number. The above formula above is used to compute the VaR value for each simulation run. The final VaR value would be the average of all simulation runs.