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.
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.

Longterm 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
longterm 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 longterm 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 longterm trend is permanent and establishes a new level
and trend. In others, the deviation is shortterm and the value reverts
back to the longterm 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 causeeffect 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 causeeffect 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:
dV_{t} = f(V_{mean}  V_{t}) + g(V_{t},
W_{t}, X_{t}, Y_{t}) * d_{t} + h(V_{t})
* 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, dV_{t}, is the change in the value of the
variable, V, from time t to t+1, i.e., V_{t+1}  V_{t}. This,
of course, is what we are solving for.
 f(V_{mean}  V_{t}) is a mean reversion function. It
indicates the degree to which the variable has a tendency to return back to
its longterm fundamental level whenever it moves away from it. Whenever the
current value, V_{t}, is above its mean level, V_{mean}, 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(V_{t}, W_{t}, X_{t}, Y_{t}) 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, shortterm deviations, and seasonality.
Often this is accomplished by breaking this term into several functions, each
of which addresses one of these characteristics.
 h(V_{t}) * 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(V_{t}) 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 * (V_{mean}  V_{t}) * dt. The mean
reversion coefficient (in this case it is 1.1) describes the rate at which a
shortterm deviation is expected to disappear. The "halflife" 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 V_{mean}, 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 of macroeconomic variables.
Let's 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
SchwartzSmith 2factor model of commodity prices. The "2factor"
refers to the fact that it models both the longterm trend in price as well as
the mean reversion from shortterm shocks, an advancement over traditional
models, which only analyzed longterm trends. It captures all the
characteristics of macroeconomic variables—longterm trend, mean reversion,
correlation among the two commodities, seasonality, and randomness.
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.
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.