Here we explain how to convert the value at risk (VAR) of one time period into the equivalent VAR for a different time period and show you how to use VAR to estimate the downside risk of a single equity investment.

## Converting from one time period to another

In Part 1we calculate VAR for the Nasdaq 100 Index (teleprinter: QQQ) and establish that VAR answers a three-part question: “What is the worst loss I can expect in a given period with a certain level of confidence?”

Since the time period is a variable, different calculations can specify different time periods – there is no “correct” time period. commercial banksfor example, usually calculate a daily VAR, wondering how much they can lose in a day; pension fundson the other hand, often calculate a monthly VAR.

To briefly recap, let’s repeat our calculations of three VARs in Part 1 using three different methods for the same “QQQ” investment:

Because of the time variable, VAR users need to know how to convert one time period to another, and they can do this based on a classic idea in finance: the standard deviation stock returns tend to increase with the square root of time. If the standard deviation of daily returns is 2.64% and there are 20 trading days in a month (T = 20), then the monthly standard deviation is represented by the following:

$$p

Monthly

≅

p

Daily

×

J

≅

2.64

%

×

20

\sigma_{\text{Monthly}}\ \cong\ \sigma_{\text{Daily}}\ \times\ \sqrt{T}\ \cong\ 2.64\%\ \times\ \sqrt{20}

pMonthly ≅ pDaily × J ≅ 2.64% × 20

To “scale” the daily standard deviation to a monthly standard deviation, we multiply it not by 20 but by the square root of 20. Similarly, if we want to scale the deviation daily standard deviation to an annual standard deviation, we multiply the daily standard deviation by the square root of 250 (assuming 250 trading days in a year). If we had calculated a monthly standard deviation (which would be done using monthly returns), we could convert to an annual standard deviation by multiplying the monthly standard deviation by the square root of 12.

## Applying a VAR Method to a Single Stock

Both historical and Monte Carlo simulation The methods have their proponents, but the historical method requires dealing with historical data, and the Monte Carlo simulation method is complex. The simplest method is variance–covariance.

Below, we incorporate the time conversion element into the variance-covariance method for a single stock (or single investment):

Now let’s apply these formulas to QQQ. Recall that the daily standard deviation of the QQQ since its creation is 2.64%. But we want to calculate a monthly VAR, and assuming 20 trading days in a month, we multiply by the square root of 20:

***Important note:** These worst losses (-19.5% and -27.5%) are losses below the expected or average return. In this case, we keep things simple by assuming that the expected daily return is zero. We’ve rounded down, so the worst loss is also the net loss.

So, with the variance-covariance method, we can say with 95% confidence that we will not lose more than 19.5% in any given month. QQQ is clearly not the most conservative investment! You may notice, however, that the above result is different from the one we got in the Monte Carlo simulation, which indicated that our maximum monthly loss would be 15% (under the same 95% confidence level).

## Conclusion

VaR is a special type of downside risk measure. Rather than producing a single statistic or expressing absolute certainty, it makes a probabilistic estimate. With a given level of confidence, it asks: “What is our maximum expected loss over a specified period of time?” There are three methods for calculating VAR: historical simulation, variance-covariance method and Monte Carlo simulation.

The variance-covariance method is the simplest because you only need to estimate two factors: the average return and the standard deviation. However, it assumes that returns behave well according to the symmetric normal curve and that historical patterns will repeat in the future.

Historical simulation improves the accuracy of the VAR calculation, but requires more computer data; it also assumes that “the past is a prologue”. Monte Carlo simulation is complex but has the advantage of allowing users to adapt their ideas about future models that deviate from historical models.