Expert Commentary

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.

Quantitative Methods

June 2000

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.

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.

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 long-term 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, 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(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 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 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 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.

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.

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.

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.

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