Sailing through Rough Winds of Macroeconomic Risks
June 2000
As opposed to operational risk, a company
does not have control over such variables as currency fluctuation, increasing
fuel costs, and interest rate changes. This article suggests methods for quantifying
and managing these macroeconomic risks.
by Samir
Shah
Tillinghast–Towers Perrin
Who would have believed only 3 years ago that Asian currencies would lose
40-80 percent of their value? Or that stable currencies like the German mark
and the Japanese yen would experience a spread of 40 percent between high and
low against the U.S. dollar? Who would have expected this time last year that
oil prices would double? Or that interest rates would increase steadily over
the last year? Most large companies are financially exposed to the effects of
one or more of these macroeconomic variables. For them, heading into an environment
of volatile interest rates, currency exchange rates, and commodity prices is
risky business indeed. Unlike in the case of operational risk, a company does
not have control over the movements of these variables. But while you do not
control the wind, you can adjust the sails. The subject of this article is methods
for quantifying and managing these macroeconomic risks.
Modeling Risks using Scenario Generation
The technique most commonly used to model macroeconomic risks is scenario
generation. As the name suggests, it involves developing a scenario for the
possible path of a variable, e.g., oil price, over the period of study, e.g.,
5 years. Just as for forecasting the path of a hurricane, hundreds or thousands
of scenarios are generated using computer simulation. Each one represents a
unique path that the variable can take. The scenarios are then summarized into
probability distributions for the variable. It is important to keep in mind
that this is not a prediction of the movement of the price or rate. Rather,
scenario generation is a method for measuring the size and frequency of the
possible deviation of the movement from its expected path.
To model a macroeconomic variable, it is first necessary to understand its
fundamental characteristics. The following are some typical characteristics
of macroeconomic variables.
Long-term trend. For most variables, looking
at historical time series data over a long time period reveals secular trends
reflecting the fundamentals underlying the variable. For example, the long-term
trend of oil prices is based on fundamental factors like the current worldwide
supply, exploration activities, and worldwide demand for oil.
Jumps versus mean reversion. Although
variables may have varying long-term trends, in all cases there are deviations
from that trend caused by market events, like the OPEC production cut, or
the Federal Reserve Board announcing a change in its lending rate. In some
cases the deviation from the long-term trend is permanent and establishes
a new level and trend. In others, the deviation is short-term and the value
reverts back to the long-term trend. The former type is called a jump process
whereas the latter is called a mean reverting process. For example, individual
stocks sometimes exhibit jumps, whereas interest rates and many commodity
prices tend to exhibit mean reversion.
Relationship to other variables. There
are generally two kinds of relationships to other variables: causal and
correlation. A causal relationship is one where the cause-effect relationship
between the variables can be explicitly stated. For example, currency exchange
rates are directly affected by interest rates within the respective countries.
As the interest rates in the United States move up, so must the value of
the U.S. dollar, otherwise it would create arbitrage opportunities. There
is a formal mathematical relationship between currencies and interest rates
called the interest rate parity. In other cases, two variables may exhibit
some correlation, although it is difficult to explicitly connect the two
through cause-effect relationships. For example, oil and natural gas prices
move independently, although they exhibit some correlation. Reflecting these
relationships is vital to the generation of scenarios for all the variables
that are internally consistent.
Seasonality. Many commodities exhibit
seasonal cycles in their price movement. For example, natural gas cycles
between summer and winter prices as the demand shifts each season.
Randomness. Even after accounting for
the above characteristics, most variables still exhibit a significant degree
of uncertain behavior because of “noise.” The noise is the aggregate effect
of a multitude of small causal factors. It results in many temporary and
independent movements that are uncertain in both direction and size.
Mathematics of Scenario Generation
The mathematics most often used to capture these characteristics in developing
scenarios is called stochastic differential equations (SDEs). Differential equations
are so named because they express the difference between the value of a variable
at time, t, and the value 1 time period later, t+1. Thus, the process of generating
scenarios involves starting with the current value and using the differential
equation to generate by iteration the values for each subsequent time period.
It is a stochastic differential equation because the difference has a random
element representing our uncertainty as to its movement at each time step.
A generic form of the SDE is:
dVt = f(Vmean – Vt) + g(Vt, Wt,
Xt, Yt) * dt + h(Vt) * dZ
Let’s briefly go over each term of this equation to relate it to the characteristics
we are trying to model.
- The first term, dVt, is the change in the value of the variable,
V, from time t to t+1, i.e., Vt+1 - Vt. This, of course,
is what we are solving for.
- f(Vmean – Vt) is a mean reversion function. It
indicates the degree to which the variable has a tendency to return back
to its long-term fundamental level whenever it moves away from it. Whenever
the current value, Vt, is above its mean level, Vmean,
this function produces negative values, and vice versa. For variables exhibiting
“jumps,” this term is replaced by a function that models infrequent large
jumps to a new level.
- g(Vt, Wt, Xt, Yt) is a function
that indicates how the movements of the variable V are related to other
potential variables W, X, and Y. This function is used to capture causal
relationships, secular trend, short-term deviations, and seasonality. Often
this is accomplished by breaking this term into several functions, each
of which addresses one of these characteristics.
- h(Vt) * dZ is the term that captures the random movement
or noise. By contrast, the other terms are all deterministic. dZ is the
value of a standard normal random variable. Its probability distribution
has the familiar bell curve shape centered at zero with a standard deviation
of 1. When the SDE is used in an iterative fashion, at each time step, a
value for dZ is determined by randomly sampling this probability distribution.
h(Vt) is a scaling factor for the random movement. This is the
term in the SDE that has different values every time we generate a new scenario.
This term is also used to capture the correlation with other variables.
Each term on the right side of the SDE is a mathematical function whose structure
and parameters must be determined specifically for the variable that is being
studied. For example, the mean reversion function for crude oil may look like
this: -1.10 * (Vmean - Vt) * dt. The mean reversion coefficient
(in this case it is 1.1) describes the rate at which a short-term deviation
is expected to disappear. The “half-life” of the deviation—the time in which
a deviation away from the fundamental level is expected to halve—is equal to
ln(2)/mean reversion coefficient. In our example, this translates to 0.63 years
or about 7.5 months. Once the deviation is halved, it takes another 7.5 months
for it to halve again (¼ of the original deviation). This means that the rate
at which the value changes decreases as it gets closer to the fundamental level.
Note also that the rate of mean reversion is not dependent on the value of Vmean,
the fundamental level.
Defining the functions and the parameter values of the SDE involves rigorous
statistical analysis of historical data. SDEs are validated by mathematically
analyzing how well they capture the fundamental characteristics discussed above.
Lets look at some outputs from using an SDE to generate scenarios for the
behavior of crude oil and natural gas (Figures 1 and 2). These are based on
a set of SDEs developed by Dr. Eduardo Schwartz of UCLA and Dr. Jim Smith of
the Duke University Fuqua School of Business, called appropriately the Schwartz-Smith
2-factor model of commodity prices. The “2-factor” refers to the fact that it
models both the long term trend in price as well as the mean reversion from
short-term shocks, an advancement over traditional models, which only analyzed
long-term trends. It captures all the characteristics we have discussed above,
mean reversion, correlation among the two commodities, seasonality, and randomness.
Figure 1
Distribution of WTI Crude Oil Prices
Figure 2
Distribution of Natural Gas Prices
Figures 1 and 2 illustrate summary statistics from generating 1,000 scenarios
for the movement of crude oil and natural gas, respectively, over the next 24
months (this was done as of late November 1999). The black curve indicates,
for each month, the mean value of the price over all 1,000 scenarios. The other
colored lines indicate various percentiles for the price at each month. Although
only certain percentiles are plotted, since we have all the data from the 1,000
scenario simulations, we can plot a full probability distribution of the price
for each month.
Relating Risks to Financial Measures
The modeling of macroeconomic variables in itself is not worth much unless
we can translate it into the impact on corporate financial metrics, such as
cash flow or net income. It amounts to measuring wind direction and speed but
not how much it will take you off your set course. It is vital to analyze the
combined impact of all the risk variables on financial measures to capture the
portfolio risk diversification benefits. A simple method for relating financial
measures to macroeconomic variables is using statistical techniques like multiple
regression. The independent variables are the risk variables, such as commodity
prices, and the dependent variable is the financial measure, such as net income.
More advanced methods such as generalized linear models or neural networks can
also be applied to explore the relationships between the sources of risk and
financial measures. Explicitly linking a financial measure to the sources of
risk allows us to develop scenarios and probability distributions for the financial
measure—just as we had done for the risk variables.
Hedging Against Risks
Now that we have quantified the risks and their impact on the bottom line,
the final step is to determine how to mitigate the risk. The most common method
for mitigating the impact of macroeconomic risks is to hedge them in the financial
markets. The financial markets for hedging interest rates, exchange rates, and
commodity prices are well developed and offer considerable liquidity, thereby
minimizing transactional costs. Having developed scenarios for each risk variable,
analyzing the impact of alternative hedging strategies is straightforward.
For example, let’s assume that we are considering a zero cost option collar
on WTI crude for the next 12 months. A zero cost option collar involves purchasing
a put option and selling a call option of equal premium so that the net transactional
cost is zero. The following two charts show the uncertainty in net income before
and after implementing this hedging strategy.
Figure 3
Net Income Distribution before Hedging
Figure 4
Net Income Distribution after Hedging
Note that the hedge effectively places a floor on net income but takes away
some upside potential. Is giving up some potential income worth protecting the
downside? Of course, modeling cannot answer this question. The answer is affected
by the risk/reward preferences of the management team. However, modeling macroeconomic
risks provides invaluable information to help companies navigate through these
issues to make decisions aligned with their objectives.
Scenario generation using stochastic differential equations is a rigorous
modeling technique that requires skill and experience. Managers must take responsibility
for:
- Ensuring that the SDE properly represents the fundamental characteristics
of the sources of risk and validating the scenarios generated
- Employing analysts that are skilled in statistics, stochastic calculus,
and computer simulation
- Developing and analyzing financial hedging strategies in accordance
with their risk/reward preferences
Without proper modeling and managing of macroeconomic risks, a company may
as well be blowing in the wind.
Editor’s Note: Charts of scenarios were developed using Tillinghast
proprietary computer simulation models.
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.