Risk management and Monte Carlo simulation: an illustration and cautionary tale

Establishing risk thresholds for any business endeavour is essential. Organizations that can quantify risk have better insights on risk management

By Anthony A. Atkinson, CMA, FCMA

Governance authorities argue that one of management’s major responsibilities is risk management. Risk management includes identifying the sources of risk, choosing the appropriate level of risk, and monitoring actual risk levels to identify situations in which the chosen risk levels might, or have been, breached. Evidence has shown that the straightforward failure to identify, manage and monitor risks can reduce shareholder value, hinder organizations from meeting business objectives, and lead to significant financial losses or business failure. Risk management plays an integral part in making certain such problems don’t occur.

This article focuses on identifying, describing, and quantifying the risk associated with capital investments. As the risk profile becomes more complicated, it is necessary to use modelling tools to assist in decision making. However, having the proper inputs for such models are a critical part of making the process a useful and effective one.

Risk and uncertainty

In 1921 Frank Knight, in a seminal book Risk, Uncertainty and Profit, defined risk as a situation where decision makers can assign statistical probabilities to the randomness that they face, and defined uncertainty as situations in which they cannot. More recently there has been a trend to define uncertainty as randomness in the events or circumstances that can affect an organization’s ability to achieve its objectives and risk as the potential that the organization or an important organization system will not achieve its objectives. For example, for a couple planning an outdoor summer wedding, one uncertainty is whether it will rain — risk is the potential for the weather uncertainty to disrupt the ceremony, dining, and entertainment activities. This article will use the terms uncertainty and risk in their latter meanings since they are more consistent with their implied meanings in the extant corporate governance literature.

The capital budgeting decision — a basic model case study

The management of Whistle Mines is considering developing a mine. The local government authorities have indicated that the mine could be operated for four years with no extensions. Whistle Mine’s required pre-tax return on investment is 15% for investments with this risk level¹. Management and staff have developed the following estimates relating to this project:

• Initial development costs — $10,000,000.
• Fixed annual licensing fees and operating costs — $2,000,000.
• Mine yield — 150,000 tons per year.
• The selling price, net of royalties, selling, and logistical costs — $100 per ton.
• The variable mining costs — $55 per ton.
• Shut down costs relating to land reclamation — $5,000,000.

The net present value is $702,381 and conventional financial analysis would suggest that this project be accepted since the expected net present value exceeds zero. Alternatively we can say that, the rate of return on this investment exceeds the organization’s required rate of return on a project of this type.

We can compute the break-even contribution margin (BE - earning the target 15% on the investment) such that the project’s net present value is zero for this project as follows:

0 = {[(BE x 150,000)-2,000,000] x present value annuity factor @15% for 4 years} - 10,000,000 - (5,000,000 x present value factor @ 15% in 4 years)

Solving this equation within the indicated factor implies that BE = $43.36. This value is of no immediate interest or significance; it is used as a benchmark in what follows below.

Simple uncertainty

If management can specify its beliefs about a project’s objects of uncertainty, it can undertake various forms of analyses to consider the implications of uncertainty in decision making and begin to quantify the risk associated with the investment. In particular, when management can express project uncertainty using a probability distribution, such as a normal distribution, it can identify directly the probability of achieving specific levels of return — an important element in characterizing, assessing, and managing project risk.

To provide an illustration of dealing with uncertainty in a simple setting, suppose that management believes that the uncertainty in the Whistle Mines project relates to product price and variable cost. Specifically, this uncertainty will be captured in assumptions about the project’s contribution margin. (This illustration explores the effects of a single element of uncertainty on the decision, of course. Another simple risk analysis in this case would be to use the number of tons of product mined and sold each year. More complex models follow below.)

Distribution of returns

Suppose management believes that the contribution margin per ton of product will lie between $40 and $50 per ton, with all values on this interval and its end points equally likely. The mean of this distribution is a contribution margin of $45 per ton and the expected value of the project is the same as in the basic scenario above. Note that this is simple uncertainty because we are assuming that the quantity sold is known.

However, if management can express its uncertainty in a standard probability density function, it can easily quantify the effect of uncertainty on project return. Given the contribution margin uncertainty assumption specified in the previous paragraph, the probability of breaking even is 66.4%² or about two chances in three. Capturing the effect of uncertainty on overall project return and therefore risk is useful since many decision makers seem to prefer to consider risk from the perspective of the overall project return rather than in terms of the uncertainty of individual project parameters.

This approach works for any standard distributional form. For example, if management believed that the contribution margin was normally distributed with a mean of $45 and a standard deviation of 1.67, the probability of breaking even would be assessed as 83.7%.³

Mitigating risk

One approach to mitigating project risk is to develop additional information about the elements of project uncertainty. Of course, additional studies are costly and often convey only partial information about the project. Therefore, acquiring additional information for the purpose of risk mitigation should involve comparing the expected value of additional information with its cost.

Returning to the Whistle Mines example, suppose the company can undertake marketing and cost information studies that will identify situations in which the project contribution margin will be less than $42 but uncertainty will remain about the actual value of the contribution margin — that is, once the information study is concluded, Whistle Mines will know either that the contribution margin will be less than or that it will be over the $42 to $50 interval.

Prior to undertaking these information studies, management, based on its prior beliefs, will assess a probability of 20% that these studies will predict that the contribution margin will be less than $42. This is the range of outcomes that the study will eliminate.

Since the breakeven point is $43.63, if the contribution is reported as less than $42, the project, would be abandoned. The expected value of the project, when the contribution margin range is reported to lie on the interval of $42 to $50, is $1,130,628 and the probability of at least breaking even, if the project is undertaken, is 83%. Since management would expect to undertake the project 80% of the time after receiving the results of the studies, the expected value of decision making with these studies is ,502 ($1,130,628 x 80%). On the previous page is a decision tree for the information investment decision.

The result is that the information studies have increased the expected return of the project by $202,121 ($904,502 - $702,381). This is the most that Whistle Mines should be willing to pay for this information study. The information has reduced the probability of not breaking even by 17% (83 — 66). Not only does this information provide management with a powerful form of risk mitigation but it also provides a basis for cost/benefit evaluations of the information studies. In more complex and therefore realistic environments, management can quantify and therefore assess the cost, through information studies, of risk mitigation.

The current decision-making literature often refers to this type of decision making as a “wait and see” strategy; the organization can defer making an investment and wait to see the results of some test. The best known examples of wait and see strategies are test marketing and drilling test holes in oil field development.

More complex uncertainty

As management adds more elements of uncertainty to the model, making the example more realistic, it loses the ability, except in very exceptional cases, to express the distribution of project returns in the specific distributional forms suggested above. When the expected distribution of results cannot be expressed in a statistical form, Monte Carlo simulation models allow management to use its probabilistic estimates to assess the expected value of projects.

To illustrate, return to the Whistle Mines example with the following assumptions:

• Initial development costs — 25% $9,000,000, 50% $10,000,000, 25% $11,000,000.
• Fixed annual licensing fees and operating costs — between $1,900,000 and main,100,000, with all values on the interval equally likely.
• Mine yield — normally distributed with a mean of 150,000 tons per year  and a standard deviation of 5,000 tons.
• Contribution Margin — $40 to $50.
• Shut down costs relating to land  reclamation — 33% ,500,000,  34% $5,000,000, 33% $5,500,000.

Using these data in a simulation model (1,000,000 trials of a Monte Carlo simulation) generated the following distribution of outcomes (demonstrated in the chart above).

The mean of this distribution is $702,771 and the probability of at least breaking even is 76.55%. Curve fitting suggested that the best fit was provided by a normal distribution with a standard deviation of $361,966. With this information the decision maker can provide management with probabilistic estimates that effectively characterize the project’s risk.

Once again, management can use its collective information to provide a quantification of project risk and evaluate information study proposals that alter the project’s expected results.

Caveats

Like all modelling, the quality and insight provided by the results depend critically on the adequacy of the model inputs. The decision maker’s ability to reasonably describe the uncertainty inherent in the decision will determine the quality of the results. This includes identifying not only the parameters of the uncertainty in the model but also their shape. This is not a trivial task and is fraught with subjectivity that can profoundly affect the results.

In this regard, the Monte Carlo method is only a tool that allows informed decision maker’s to apply their insight and knowledge to estimate project risk. The Monte Carlo method should only be used when the decision is too complex to model directly. Although extant software provides a smorgasbord of options for data input and specification, those options are relevant and valuable only in the hands of a thoughtful and knowledgeable decision maker. Absent reasonable input, the Monte Carlo results will be meaningless at best and misleading at worst.

It should also be noted that there are many individuals who question the validity of the Monte Carlo method in general and the meaning and interpretation provided by the results of large scale simulations of a one-shot project. Many of these critics point out that the Monte Carlo method is often used in situations where it is not warranted — for example in a situation when there is a reasonably simple approximation of the underlying uncertainty. In summary, when there is uncertainty in several decision-making parameters, when the uncertainty in more than one parameter is potentially material, when there is no practical way to model the resulting uncertainty in a convenient parametric form, and when the decision makers are comfortable with their estimates of underlying uncertainty, then the Monte Carlo model can provide useful insights into the risk inherent in a decision.

Anthony (Tony) Atkinson, Ph.D, CMA, FCMA, is a professor and Management

Accounting Area Head in the School of Accountancy at the University of Waterloo.

¹ An alternative approach to deal with uncertainty in capital budgeting is to identify certainty equivalents for the uncertainty parameters in a project (such as selling price, demand, capital investment) and then discount the certainty equivalent net cash flows using a risk free rate. (An uncertainty equivalent is an amount for which a decision maker would exchange an uncertain prospect.) My experience in practice is that decision makers do not find the certainty equivalent approach either intuitive or meaningful.

 ² The calculation is:

³ The assumption in the original model was that the contribution margin lay on the interval $40 to $50. The alternative uses a normal distribution with a mean of $45 and a standard deviation of 1.67 for comparability purposes. This normal distribution has the same mean as the previous example and there is only a 0.27% chance that the contribution margin will lie outside the range of $40 to $50.

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