Risk Quantification in Rapidly Changing Business Environments
January 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.
by Samir
Shah
Tillinghast–Towers Perrin
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.
Current Methods of Modeling Volatility
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. (See the previous article in
this series, Sailing through Rough Winds of Macroeconomic
Risks, for a description of SDEs.) 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.
Figure
1
The historical average is used as the forecast resulting in significant uncertainty
in the forecast.
Figure
2
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.
Modeling Volatility Using GARCH Models
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.
How GARCH Works
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:
σ2t = c + aε2t-1 + bσ2t-1
where:
σ2t is the variance at time t,
ε2t-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:
σ2t+s = c + (a + b)s (σ2t - c)
where σ2t+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.
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.
Figure
3
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.
Conclusion
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.
Opinions expressed in Expert Commentary articles are those of the author and are
not necessarily held by the author’s employer or IRMI. This article does not purport
to provide legal, accounting, or other professional advice or opinion. If such advice
is needed, consult with your attorney, accountant, or other qualified adviser.