Nonlinearity

What is Nonlinearity?

Nonlinearity is a statistical term that describes the relationship between dependent and independent variables. It describes a link that cannot be expressed with a straight line. If a system does not follow the linearity theorem, it is referred to as nonlinear.

A linear relationship is, therefore, one that can be expressed using a straight line. In a nonlinear relationship, a change in either of the inputs does not reflect a corresponding change in the output.

The Concept of Nonlinearity

Nonlinearity is a common phenomenon when assessing cause and effect associations. Such scenarios involve estimating models and testing hypotheses to conduct empirical inquiries. Unfortunately, a nonlinearity assumption can result in a wrong conclusion when linearity is the point of focus.

In investment, some managers may use the concept of nonlinearity to benchmark returns. For example, options are considered nonlinear derivations since the input variables do not guarantee a proportional change in the output variables. Using a high nonlinearity in the trade may generate concavity in the fund’s returns, making it unpredictable.

However, different pricing simulations apply when using nonlinear derivatives to estimate the value at risk (VaR) of investments. The estimations are market timers and can help managers curb the risk, as well as affect the fund’s Sharpe Ratio and other measures.

For example, while private investment companies are not subject to market constraints, mutual funds face constraints due to the use of nonlinear derivatives. The problem of nonlinear payoffs for performance assessment is recommended for portfolios that exhibit broad investment mandates because of the innumerable opportunity to trade dynamically.

Linearity vs. Nonlinearity

A linear relationship is a scenario where there is a correlation between an independent and a dependent variable, contrary to a nonlinear association. A linear relationship can also be expressed in a mathematical formula, just like a nonlinear relationship. It then follows that a linear relationship is a direct correlation between a variable and a constant.

Imposing change on an independent variable produces a proportional change in the dependent variable. Statistically, a linear equation is one that satisfies the equation:

y = ax + C

Where:

• a = slope
• C = y-intercept

Nonlinearity Models

Let y and x be the assay to the equation F(X) = Ğ (y/x), where x and y scalar random variables are defined by the function F(x). In investment, x can be the measure of risk for an asset and y the risk premium on the asset. On the other hand, the function F(x) defines a nonlinear return on trade-offs.

In another way, x can represent a traditional market index and y a manager’s portfolio. The function F(x) represents nonlinear exposure to the market. The model can also apply in microeconomics, where x is the unemployment rate and y is the inflation rate, while the function F(x) reflects a nonlinear Phillips Curve.

Nonlinearity in Investment and Options

An example of an investment with a high nonlinearity is the return market options impacted by multiple variables. Managers are given several choices to consider when trading options, including the time to maturity, asset price, current interest rate, and implied volatility. Most often, investors use the standard value at risk method to approximate the risk level.

Two scenarios abound when addressing the risk level of nonlinear payoffs in a managed portfolio. At one extreme, it is possible to trade other assets when replicating the nonlinearity in the fund’s returns. On the other end, it may be impossible to use other security returns when replicating the fund’s nonlinearities.

Generally, applying the standard value at risk approach to options is not always commendable given the higher level of nonlinearity. As a result, managers tend to use more advanced modeling techniques, such as Monte Carlo, to determine options for investors based on risks and returns.

Potential Cause of Nonlinearity in Trading and a Possible Solution

Interim trading is the likely cause of nonlinearity. The concept refers to a situation where managers trade with the fund’s returns. Managers can use a weight-based measure to address interim trading bias. The approach evaluates the covariance between the subsequent two-period return and the manager’s weights at the start of the first duration.

Managers in the financial industry use a nonlinear regression model to model nonlinear data against independent variables to show their association. Although the parameters in the nonlinear regression are nonlinear, the model can employ successive approximations to fit data.

It is worth noting that designing linear models is much easier than nonlinear models because of several attempts when defining outputs. Still, the models are valuable tools for investors focusing on evaluating risks and profits according to various variables.

Additional Resources

Thank you for reading CFI’s guide to Nonlinearity. To keep advancing your career, the additional resources below will be useful:

• Correlation Matrix
• Random Variable
• Elastic Net
• Implied Volatility (IV)
• See all data science resources