# Value at Risk

Risk metrics traditionally included valuation, sensitivity analysis, scenario analysis, and maybe even Monte Carlo simulations. VaR goes further: it blends the price- yield relationship with the likelihood of a market movement that is unfavorable. Correlation and leverage are taken into consideration, and a summary measure of portfolio risk is expressed in a single probabilistic statement (Jorion 2001, p. 27).

VaR was initially developed to measure market risk and has many applications, including risk management and measurement, financial control and reporting, and the computation of regulatory capital requirements. Investors can use VaR to analyze, with a given level of confidence, what is the worst-case scenario or how much they may lose in a given time period. Formally, "VaR described the quantile of the projected distribution of gains and losses over the target horizon. If *c* is the selected confidence level, VaR corresponds to the 1 – c lower-tail level" (Jorion, p. 22).^{[1]} Intuitively, "VaR summarizes the worst loss over a target horizon with a given level of confidence" (p. 22).

# History, Characteristics, and Assumptions of VaR

Value at risk is a risk management tool developed by Till Guldimann at J.P. Morgan in the 1980s. It was developed as a result of discussions surrounding the importance of "value risks" or "earnings risks." The parties determined that value risks were of greater consequence, and VaR was born.

VaR is applicable to many different assets, including stocks, bonds, and derivatives as well as single assets or portfolios of assets. There are several methods that can be used to calculate VaR, including (1) the historical method (nonparametric delta normal), which uses past data; (2) the parametric method, which only requires the mean and standard deviation to be used;^{[2]} or (3) the Monte Carlo method, which uses future or forecasted data. Additionally, either percentage VaR or dollar VaR can be obtained, depending on the preferred output result. It is important to note that back-testing is extremely important with this technique. VaR is only an estimated worst-case scenario, and actual losses may surpass VaR. A loss that exceeds VaR is termed a "VaR break." VaR is rooted in the statistical and probabilistic foundations of portfolio theory. There is no one VaR value. In fact, there are multiple VaRs, depending on the circumstances and inputs.

There are five primary underlying assumptions for VaR. They are as follows:

1. *Stationarity.* A 1 percent fluctuation in returns is equally likely to occur at any point in time.

2. *Random walk of intertemporal unpredictability.* Day-to-day fluctuations in returns are independent.

3. *Nonnegativity.*^{[3]} Financial assets with limited liability cannot attain negative values.

4. *Time consistency.* All single-period assumptions hold over the multiperiod time horizon.

5. *Distributional.* Daily return fluctuations follow a normal distribution with a mean of zero and a standard deviation of 100 bp^{[4]} (Allen, Boudoukh, and Saunders 2004, 8-9).

With respect to the assumptions, the most obvious flaw is with the distributional assumption. Stock returns, in particular, have repeatedly been shown to not follow a normal distribution historically. However, using log returns can compensate for this issue.

- [1] Alternatively, some sources will use z as the confidence level instead of c. The approaches are identical; it is simply a notational difference.
- [2] The parametric method for calculating VaR is commonly used, and the only variables needed to do the calculation are the estimated mean and standard deviation of the portfolio. This method assumes that returns from portfolios are normally distributed.
- [3] Derivatives can violate this assumption.
- [4] Here, bp represents the common abbreviation of basis points. A basis point is 1/100th of a percent.