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 stock investment.

## Converting One Time Period to Another

In Part 1, we calculate VAR for the Nasdaq 100 index (ticker: QQQ) and establish that VAR answers a three-part question: "What is the worst loss that I can expect during a specified time period with a certain confidence level?"

Since the time period is a variable, different calculations may specify different time periods - there is no "correct" time period. Commercial banks, for example, typically calculate a daily VAR, asking themselves how much they can lose in a day; pension funds, on the other hand, often calculate a monthly VAR.

To recap briefly, let's look again at our calculations of three VARs in part 1 using three different methods for the same "QQQ" investment: * We do not need a standard deviation for neither the historical method (because it just re-orders returns lowest-to-highest) or the Monte Carlo simulation (because it produces the final results for us). Image by Sabrina Jiang © Investopedia 2021

Because of the time variable, users of VAR need to know how to convert one time period to another, and they can do so by relying on a classic idea in finance: the standard deviation of stock returns tends 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:

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

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 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). Had we calculated a monthly standard deviation (which would be done by using month-to-month 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 the historical and Monte Carlo simulation methods have their advocates, but the historical method requires crunching historical data and the Monte Carlo simulation method is complex. The easiest 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 the QQQ. Recall that the daily standard deviation for the QQQ since inception 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: