Quantitative Methods

Samir Shah | January 1, 2001

Rapid changes in business have made planning more difficult and risk management more important. Whereas business planning is based on forecasts of expected outcomes, risk management is focused on deviations from expected outcomes. Learn how GARCH models provide a significant advancement in the ability to forecast potential deviations by characterizing volatility as an evolving process.

In today's growing Internet-based economy, managers are keenly aware that it's no longer business as usual. Rapid technological change is reshaping the business landscape and emerging new businesses and industries are redefining business models and the fundamentals of competition. The rapid pace of change in the business environment has increased the uncertainty in the outcomes of management decisions and placed a greater emphasis on risk management. Assessing that uncertainty using both qualitative and quantitative techniques is the fundamental objective of risk analysis. This article discusses techniques for quantifying uncertainty and presents a recent advancement called GARCH modeling.

Uncertainty is measured in terms of the volatility of financial and operational metrics, such as share price, interest rates, foreign exchange rates, sales volume, commodity prices, labor cost, employee turnover, etc. Within the scope of this discussion, volatility is simply annualized variance, which is a measure of the deviation of performance from expected values. Rapid changes in business activities manifest themselves in deviations of financial and operational performance from expectations, thus increasing volatility. The task of any volatility model is to describe the typical historical pattern of volatility and use this to forecast future episodes. The more reliable the forecast of volatility, the easier it is to protect against surprises by explicitly reflecting volatility forecasts in management decision making.

Implicit in the definition of "volatility" is an expectation of future outcome. Expectations of future outcomes are based on forecasts of financial and operational metrics, which in turn are derived from extrapolation of historical trends. This typically involves developing a mathematical regression or time series model that best represents historical movements. Historical volatility is then determined by measuring the deviations around the fit, i.e., the residuals represented by the error term in the regression model.

In order to quantify risk, forecasting models must explicitly reflect the uncertainty in the forecast. Typically, a stochastic differential equation (SDE) is used to generate hundreds or thousands of alternative scenarios that are then summarized into probability distributions. An SDE has two parts: one part to forecast the expected outcome and another to forecast the variance around the expected outcome. The forecast of expected outcome is developed by fitting a regression or time series model to historical data. The variance forecast is typically modeled as a constant value equal to the historical variance around the fitted regression model.

For example, a forecast for weekly sales volume could be derived from historical data shown in Figure 1. A naïve forecast of sales would be to use the average of historical sales volume. The historical volatility would then be measured in terms of the historical deviations around the average. A more enlightened forecast would be to develop a regression model that reflects both the seasonal fluctuations and the long-term increasing trend (Figure 2). It's clear from a comparison of the forecasts that because the regression model better represents historical movements in weekly sales volume, the historical volatility is much less than under the naïve model.

**The historical average is used as the forecast resulting in significant uncertainty in the forecast.**

**A regression model reflecting how the trend and seasonality are used to develop the forecast, resulting in a decrease in the uncertainty of the forecast.**

It should be apparent from this example that a forecast of the volatility depends on the reliability of the model used to develop expected outcomes. Naturally, the better our understanding of the drivers of historical fluctuations of sales volume, the less uncertainty (risk) there will be in our forecast. Investment in time and effort to develop a good model of expected performance pays off in reducing risk and the cost of risk management, and increasing the reliability of the risk management plan.

A fundamental drawback of this approach however, is the assumption that volatility will be constant in the future. Recall that volatility is variance over 1 year. If volatility is constant, then for example, the variance of a 2-year forecast is twice the variance of a 1-year forecast. Generalizing this means that the variance in the forecast increases at a constant rate as the length of the forecast time period increases. The standard deviation, which is the square root of the variance, therefore increases as a square root of the length of the forecast time period. This explains the shape of the uncertainty in the forecast in Figure 1, which resembles a square root function. The same is true in Figure 2 after adjusting for the seasonality.

Often, however, constant volatility is not a valid assumption. Financial markets in particular have exhibited changes in volatility over time. A rapid succession of changes in business environment—as evident in the Internet age—result in rapid changes in financial and operational performance. These bursts of activity result in volatility clusters, i.e., successive periods of high volatility and low volatility. In other words, volatility is not constant! In these instances, approaches based on constant volatility can result in poor performance of the risk management plan.

As mentioned previously, the part of the SDE used to forecast variance is typically expressed as a constant. Of course, it doesn't have to be expressed this way. Like the forecast of expected outcome, the variance forecast can also be expressed more fully as a regression or time series model. This is in fact what Robert Engle, professor of Economics at University of California, San Diego, did when he developed an ARCH model in 1982.

ARCH stands for Autoregressive Conditional Heteroscedasticity. "Autoregressive" means that the volatility at each point in time is expressed in terms of volatility in prior periods, i.e., the independent variable in the regression model is volatility in prior periods. "Conditional Heteroscedasticity" means changing variance, or "volatility clustering." Tim Bollerslev, who was Engle's graduate student, later developed a generalized version of ARCH called, appropriately, Generalized Autoregressive Conditional Heteroscedasticity (GARCH), in 1986. Although the approach was developed more than 15 years ago, it has become widely used in the financial markets only recently as practitioners have become more adept in the underlying mathematics.

First, the forecast of expected outcome is determined as usual by fitting a regression model to historical data. Let's call this the Model of Expected Outcome. This model will explain much—but not all—of the historical fluctuations depending on the "goodness of fit."

Next, another regression is performed to model historical variance. Let's call this the Model of Expected Variance. It has the following form:

** σ**^{2}_{t} = c + a**ε**^{2}_{t-1} + b**σ**^{2}_{t-1}

where:

**σ**^{2}_{t} is the variance at time t,

**ε**^{2}_{t-1} is the square of the difference between the actual historical value at time t-1 and the value based on the Model of Expected Outcome,

a, b, and c are parameters that will provide the best fit of the regression model to historical data. The parameters are constrained such that a, b, and c > 0 and a + b < 1.

Once the model parameters (a, b, and c) are determined, then, after much math, the following equation can be derived for forecasting volatility:

** σ**^{2}_{t+s} = c + (a + b)^{s} (**σ**^{2}_{t} - c)

where **σ**^{2}_{t+s} is the forecast of the expected variance at some future time t+s. In this forecast, the volatility changes over time and the variance of the forecast does not increase linearly with the forecast time period.

Notice that GARCH produces two forecasts: one for the expected outcome and one for the variance around the expected outcome. Let's examine the fundamental characteristics of the variance forecast.

The variance forecast has the property of mean reversion when c > 0 (as we have constrained it to be). This means that although volatility is not constant, there is a long-term constant level of volatility around which there are short-term fluctuations. The value of c is the mean long-term volatility level. At the time of the forecast, the volatility may be either greater or less than this long-term level.

The forecast will start at the current level and revert to its long-term level over time. The rate at which it will revert is based on the values of the parameters a and b. The sum of a and b will be between 0 and 1 due to the constraints imposed in the regression. The closer the sum is to 0, the faster the volatility will shrug off a short-term period of extraordinarily high or low volatility to return to its long-term level. If the sum is closer to 1, it will revert more slowly. Thus volatility forecasts using GARCH have a term structure as shown in Figure 3.

Note that this mean reversion model captures the volatility clustering behavior of some markets discussed above. When volatility is higher than its long-term level, the forecast also generates higher variance forecasts until the effect gradually dissipates. Of course, the same characteristic exists also when volatility is lower than its long-term level.

GARCH models are commonly used in the financial markets to value derivatives whose prices are sensitive to volatility forecasts. Improvements in volatility forecasting have allowed many in fact to trade volatility itself rather than the underlying asset. The success of GARCH modeling in the financial markets suggests opportunities to apply the approach to other areas of corporate financial and operational risk management.

**The forecast of volatility either decreases or increases over the forecast period, depending on whether the current period is experiencing higher or lower volatility, respectively, than the long-term level.**

Volatility is a fundamental concept in managing risk. Whereas business planning is based on forecasts of expected outcomes, risk management is focused on deviations from expected outcomes. GARCH provides a significant advancement in our ability to forecast the range of potential deviations from business plans by characterizing volatility as a process that evolves over time rather than as a constant parameter.

The rapid changes in the business environment have made business planning more difficult and risk management more important. Indeed, some companies have reorganized themselves to elevate risk management to the same level as business planning—creation of a new Chief Risk Officer (CRO) position, for example. GARCH models provide a valuable tool for CROs and other senior managers to reliably forecast risk in order to develop effective risk management plans.

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