# Value at Risk

Value at risk is a financial risk measure which calculates the value of loss for a given significance level and time horizon.

Value at risk of $5 million for 1 week for 5% probability means that there is a 5% probability that the value of the portfolio will fall by more than $5 million in 1 week. An alternative interpretation is that there is 95% probability that 1 week loss will be no more than $5 million.

Value at risk can be calculated for the range of risks such as: market risk, cash flow risk, credit risk, etc. However, it is most appropriate for variables that can be approximated by normal distribution.

There are two methods for calculating value at risk: the analytical VaR method and the historical VaR.

The analytical α% VaR assumes that the variable can be approximated by normal distribution and calculates the value in the left-tail of distribution using the given z-statistic for the significance level α. The historical α% VaR involves extracting historical data on the variable, say daily rates of returns, and finding the highest value falling in the lowest α% of the values.

## Example

Megan McGraw and Jerry Chi are working as financial analysts with an investment research firm. They are tasked with finding the daily market risk VaR for trading activities for Bank of America (NYSE: BAC).

Megan pulls out the Annual Report of the Bank for 2012 and finds a table outlining estimates of daily market risk VaR for trading activities on page 112 of the annual report.

The model uses 99% confidence level and calculates daily values. The average daily VaR for foreign exchange is given as $21.4 million. Megan says that it means that in 1 out of 100 days, trading losses on foreign exchange transactions could have exceeded $21.4 million. Jerry says that it means there is 99% probability that daily losses on foreign exchange will not exceed $21.4 million on average.

Who is correct?

__Answer__

Both Megan and Jerry are correct. They are both giving two alternate interpretations of the VaR.

Written by Obaidullah Jan, ACA, CFA