Stochastic analysis generates a large number of scenarios to give a better understanding of risk interactions, business linkages, optionality, and contracts designed to mitigate risk.
Even the most sophisticated models are still subject to model risk; and they do not likely capture all the risks affecting an enterprise.
Example of a Stochastic Pro Forma
Consider the case of a company that has experienced six months of cash flows this year and wants to forecast the next six months. The usual way to do this is to predict a cash flow growth rate—expected, high, and low—and to base the analysis on these choices. A sample cash flow projection might be illustrated graphically in Figure 1.
In reality, of course, several different cash flow patterns might emerge for the last six months of the year. Using the same risk model, we could run a large number of simulations and see what the outcomes might be. Eight possible outcomes are plotted in Figure 2.
Clearly the stochastic analysis, albeit more realistic, is not as simple and not as attractive at first blush as deterministic analysis. And there are many situations where stochastic analysis is not needed. Yet there are certain results that one can get from stochastic analysis that cannot be gained from deterministic analysis. Table 1 gives some examples.
|Probabilities of outcomes||What is the likelihood we will need to borrow?|
|Risk of outcomes||What is the most likely range for annual cash flows at year-end?|
|Interactions||If we invest more in capital expenditures only when cash flows are up, how do we reflect that in the analysis, and what impact does it have?|
|Options||Our loan contracts have floating rates, but the rates are capped. How does this affect the probabilities of different cash flow levels?|
|Worst case||We probably won’t have the worst case revenues and the worst case costs in the same year; how does that reflect on our expectation of the worst case?|
|Events||There is a 10% chance we get a major contract that will increase our cash flows significantly. How do we incorporate this in the model?|
|Risk mitigation||The treasurer wants to lock in foreign exchange rates for our foreign buyers. How will this affect cash flow volatility?|
|Capital structure||What is our capacity to make interest payments on debt with 99% certainty?|
Stochastic analysis is needed in situations where risk assessment is required, where the future company decisions depend on an unknown variable, where options are present, and when the company wants to study risk mitigation strategies.
Stochastic modeling of the income statement can be done at the aggregate level as it has been demonstrated here, or the components can be broken down into smaller components, such as the prices of products, inputs, interest rates, foreign exchange rates, and the like. The benefit of breaking down the income statement into its market-driven components is that we can find much more information on market-quoted prices and rates. This historical information is usually used as a starting point in determining how best to model these prices and rates.
Modeling Market Risk
Risk analysts need to spend significant time and effort to model the risk of the inputs to their stochastic models correctly. Incorrect specifications for market prices will lead to incorrect results. There are several models available to model market price risk. The choice of the best model generally is made by looking at the market’s historical performance and making judgments about market price behavior.1
For example, if our risk model depends on fluctuations in the stock market index, a popular approach is to represent the index as following a random walk in percentage terms. Thus, any given day’s return is normally distributed with a constant mean and standard deviation, and statistically independent from the previous day’s return. This approach was popularized in the Black–Scholes (1973) and Merton (1973) papers on option pricing. The random walk works reasonably well, except that with specialized knowledge one could argue that the average return should not be constant, the volatility should not be constant, and there are sometimes events which cause stock prices not to be normally distributed. For this reason, the S&P 500 index may reasonably follow a random walk, but the stock of a small pharmaceutical company will not, since it is prone to occasional major events such as FDA drug approval or discovery of legal liability.
Other market prices, such as interest rates, do not follow random walks. Overly high and overly low interest rates tend to correct over time to equilibrium levels. Although that equilibrium level may change over time, the general character of interest rates is that they are mean-reverting—i.e., they revert to a long-run mean over time. The same is true of commodity prices. High commodity prices stimulate production, which causes future prices to fall. Low prices discourage production, causing future prices to rise. Therefore, interest rates and commodities need to be modeled in a similar way. Some currencies exhibit mean-reverting behavior and some do not.
Finally, every market price may have unique characteristics. The volatility of natural gas and heating oil changes by season. Power prices spike rapidly when generation fails and bounce back immediately as generation comes back on line. Careful modeling of critical market price inputs will lead to the best models of stochastic results.
Modeling Risk Interactions
It is not enough to have good models of security prices, interest rates, foreign exchange rates, and commodity prices. We must also understand how those prices and rates interact. For example, higher security prices are generally correlated with low interest rates. The Australian dollar exchange rate is correlated to gold prices, due to the importance of gold mining in its economy. In many cases, simplistic correlation is fine to establish a linear relationship between changes in the risk variables. However, in other cases, the correlations may not be linear, requiring a more subtle approach. For some firms, that subtlety will be important enough to build a precise model of the interaction between two risks of importance to the company.
Modeling Event Risk
Every corporation is subject to risks from significant events, such as losing a major lawsuit, or obtaining a patent on its proprietary technology. Also, the company can be affected by market-related events, such as the bankruptcy of a key supplier. In many modeling situations, these events play an important role in determining the probability distributions of future cash flows.
It is tempting to think of event risks as being random outcomes, independent of everything else in the model. This is the biggest mistake a modeler can make. The credit crisis of 2008, for example, showed vividly how default risks across investment banks were correlated, owing to the similarity of their risk-taking activities.
Aggregating Cash Flow Risks to the Income Statement
Once all the drivers of the income statement have been modeled, they are compiled to the income statement in the same way that a pro-forma income statement would normally be generated. For example, suppose a refinery in Brazil buys crude oil in dollars, sells products in reais, shuts down production when it is not profitable to produce, and runs the risk of operational failures according to some statistical model. In this case, the modeler could build stochastic formulas for the price of crude, the price of products, the dollar foreign exchange rate, the shutdown policy of the firm, and the unplanned outage rates due to operational risk. The result is a determination of net income for each particular simulated environment. These net income numbers can be simulated as many times as required to determine the volatility of cash flows, the value of the shutdown option, and the answer to any of the questions posed above.
Modeling Risks Other Than Cash Flow
Some risks may not affect cash flows but could affect earnings, such as a mark-to-market liability. In these cases, similar risk models can be built to model earnings risk, or to model the likelihood of a credit downgrade. Stochastic models can be simple or extremely complex, but they all are built fundamentally to make deterministic models more realistic and able to answer questions related to risk, risk management, optionality, capital structure, and much, much more.
An ethanol-producing company may be reluctant to issue more debt because of the high volatility of its cash flows and the increased risk of being put into bankruptcy.
A bank has proposed a transaction where the company would reduce its risk by selling its ethanol to customers at a price agreed today—i.e., entering forward contracts. If it did so, the bank would lend additional funds at the same rate. The company is reluctant to accept the bank’s proposal because the sales price falls below the level at which the company thinks it can sell ethanol, costing the company $2 million per year. How can the company compare the benefit of higher debt with the cost of selling at a distressed price? And how can the company and the bank determine an appropriate level of additional debt?
A stochastic pro forma analysis could be done for the company before and after the proposed transaction. Before the transaction, the average earnings before interest and tax (EBIT) is estimated at $100 million with a standard deviation of $50 million. Shown in Figure 3 are five outcomes simulated over an eight-year period. The current annual debt service is $49 million.
By selling its ethanol forward, the company expects to lose $2 million per year, but to reduce the standard deviation to $25 million. The resulting stochastics demonstrate that the company can now prudently afford to make higher interest payments without having much risk of failure to pay (Figure 4).
The company can afford to pay $65 million in interest safely, after hedging its results. Should the company accept the hedging program? The answer depends on taxes. If the ethanol company is not in a tax-paying situation, it has lost an expected $2 million per year in value, so it should not hedge unless there are other reasons to do so. A taxpaying firm in the 40% bracket, however, will be able to deduct the interest expense from taxable income, saving $6.4 million per year (40% of 65 minus 49). The taxpaying firm should hedge, barring other considerations that might cause the firm not to want to hedge.
Making It Happen
Begin with a project or corporate pro forma.
Consider every assumption and ask if it is vulnerable to risk.
Produce a risk model to simulate all the assumptions consistently and simultaneously.
Use this model (stochastic pro forma) to design best and worst cases.
Simulate outcomes of all key financial variables and communicate the risks.
Stochastic pro-forma analysis answers many financial questions that cannot be addressed with usual deterministic pro-forma analysis. The case study demonstrates how hedging and capital structure may be evaluated using stochastic pro formas. Other applications include evaluation of real options,2 the study of credit ratings, and the development of probability statements around cash flow or earnings.
Like any other type of analysis, poor assumptions lead to poor conclusions. Good simulation models take a great deal of care in specifying the correct models for all the risk drivers and the interactions between them. Finally, more realistic risk-based models lead to better corporate financial decisions.
In the final analysis, however, even a very sophisticated model is still a model, and is therefore subject to model risk. Thus, the model may not fully identify or quantify all risks that affect an enterprise, and can thereby lead to a false sense of security. Accordingly, decision-makers should consider model risk as one of the components of any financial decision based on stochastic pro-forma analysis.
1 Analysts should never expect that historical price behavior will represent future price behavior—only to realize that there is usually no better source of information for modeling purposes.
2 See the article on “Real Options: Opportunity from Risk” in this volume.