The value at risk (VaR) is a statistical risk management technique that determines the amount of financial risk associated with a portfolio. There are generally two types of risk exposures in a portfolio: linear or nonlinear. A portfolio that contains a significant amount of nonlinear derivatives is exposed to nonlinear risk exposures.

The VaR of a portfolio measures the amount of potential loss within a specified time period with a degree of confidence. For example, consider a portfolio that has a 1% one-day value at risk of $5 million. With 99% confidence, the expected worst daily loss will not exceed $5 million. There is a 1% chance that the portfolio could lose more than $5 million on any given day.

### Nonlinear Considerations

Nonlinear risk exposure arises in the VaR calculation of a portfolio of derivatives. Nonlinear derivatives, such as options, depend on a variety of characteristics, including implied volatility, time to maturity, underlying asset price, and the current interest rate. It is difficult to collect the historical data on the returns because the option returns would need to be conditioned on all of the characteristics to use the standard VaR approach. Inputting all of the characteristics associated with options into the Black-Scholes model or another option pricing model causes the models to be nonlinear.

Therefore, the payoff curves, or the option premium as a function of the underlying asset prices, are nonlinear. For example, suppose there is a change in the stock price, and it is input into the Black-Scholes model. The corresponding value is not proportional to the input due to the time and volatility portion of the model since options are wasting assets.

The nonlinearity of derivatives leads to nonlinear risk exposures in the VaR of a portfolio with nonlinear derivatives. Nonlinearity is easy to see in the payoff diagram of the plain vanilla call option. The payoff diagram has a strong positive convex payoff profile before the option's expiration date, with respect to the stock price. When the call option reaches a point where the option is in the money, it reaches a point where the payoff becomes linear. Conversely, as a call option becomes increasingly out of the money, the rate at which the option loses money decreases until the option premium is zero.

### The Bottom Line

If a portfolio includes nonlinear derivatives, such as options, the portfolio returns distribution will have positive or negative skew or high or low kurtosis. The skewness measures the asymmetry of a probability distribution around its mean. Kurtosis measures the distribution around the mean; a high kurtosis has fatter tail ends of the distribution, and a low kurtosis has skinny tail ends of the distribution. Therefore, it is difficult to use the VaR method that assumes the returns are normally distributed. Instead, the VaR calculation of a portfolio containing nonlinear exposures is usually calculated using Monte Carlo simulations of options pricing models to estimate the VaR of the portfolio.