What is value at risk?

Value at risk, or VaR, states a loss level and a confidence: for example, there is a 95% chance that losses over one day will not exceed 1 million. It condenses a portfolio's downside into one number that can be compared and limited. It matters because it gives managers a common yardstick for how much risk they are taking.

How it works

VaR is a percentile of the distribution of possible losses. To get it, you describe how the portfolio value could change over the period, using history, a formula, or many randomly generated outcomes, then sort the losses from worst to best and read off the point that cuts off the worst 5% (for a 95% level). If outcomes were roughly bell-shaped, a 95% one-day VaR is about 1.65 times the daily standard deviation times the portfolio value.

The key limit is that VaR says nothing about how bad things get beyond that point. A 95% VaR means that in 1 day out of 20 the loss is at least that large, but it could be only slightly more or a great deal more. That is why it is usually paired with a look at tail risk and stress tests.

Example

Say a portfolio is worth 10,000,000 and its daily return has a standard deviation of 1%, so the daily standard deviation in money is 100,000. A 95% one-day VaR is about 1.65 × 100,000 = 165,000. This means that on about one day in 20 the portfolio loses at least 165,000. It does not say whether that bad day loses 170,000 or 2,000,000. These are illustrative numbers.

How to model it in Machinations

Create a Pool for the portfolio value and a Register that draws a random daily return each step. A state connection applies it to the Pool. Play thousands of times at once and read the distribution of the loss after one step; the value at the 5th percentile of the outcomes is the 95% VaR. Change the volatility Lever to see how the VaR scales with it.