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Uncertainty – lack of information
This entry is part 3 of 6 in the series Monte Carlo Simulation

Every item in a budget or a profit and loss account represents in reality a probability distribution. In this framework all items whether from the profit and loss account or from the balance sheet will have individual probability distributions. These distributions are generated by the combination of distributions from factors of production that define the item.

Variance will increase as we move down the items in the profit and loss account. The message is that even if there is a low variance in the input variables (sales, prices, costs etc.) metrics like NOPLAT, Free Cash Flow and Economic Profit will have a much higher variance.

The key issue is to identify the various items and establish the individual probability distribution. This can take place by using historical data, interviewing experts or comparing data from other relevant sources. There are three questions we need to answer to define the proportions of the uncertainty:
- What is the expected value?
- What is the lowest likely value?
- What is the highest likely value?
When we have decided the limits where we with 95% probability estimate the result to be within we then decide what kind of probability distribution is relevant for the item. There are several to choose among, but we will emphasize three types here.
1. The Normal Distribution
2. The Skewed Distribution
3. The Triangular Distribution
The Normal Distribution is being used when we have situations where there is a likeliness for a symmetric result. It can be a good result but has the same probability of being bad.

The Skew Distribution is being used when it can occur situations where we are lucky and experience more sales than we expected and vice versa we can experience situations where expenditure is less than expected.

The Triangular Distribution is being used when we are planning investments. This is due to the fact that we tend to know fairly well what we expect to pay and we know we will not get merchandise for free and there is a limit for how much we are willing to pay.

When we have defined the limits for the uncertainty where we with 95% probability estimate the result to be within we can start to calculate the risk and prioritize the items that matters in terms of creating value or loss.
01/09/2008
Risk – Exposure to Gain and Loss
This entry is part 4 of 6 in the series Monte Carlo Simulation

It is first when the decision involves consequences for the decision maker he faces a situation of risk. A traditional way of understanding risk is to calculate how much a certain event varies over time. The less it varies the minor the risk. In every decision where historical data exists we can identify historical patterns, study them and calculate how much they varies. Such a study gives us a good impression of what kind of risk profile we face.
- Risk – randomness with knowable probabilites.
- Uncertainty – randomness with unknowable probabilities.
Another situation occurs when little or no historical data is available but we know fairly well all the options (e.g. tossing a dice). We have a given resource, certain alternatives and a limited number of trials. This is equal to the Manahattan project.

In both cases we are interested in the probability of success. We like to get a figure, a percentage of the probability for gain or loss. When we know that number we can decide whether we will accept the risk or not.

Just to illustrate risk, budgeting makes a good example. If we have five items in our budget where we have estimated the expected values (that is 50% probability) it is only three percent probability that all five will target their expectation at the same time.

0.5^5 = 3,12%

A common mistake is to summarize the items rather than multiplying them. The risk is expressed by the product of the opportunities.
01/09/2008
Decisions – Criteria for selection

This entry is part 5 of 6 in the series Monte Carlo Simulation

The risk is best expressed by using a graph illustrating the probability curve. The slope tells us about the uncertainty involved, the steeper the curve the less uncertainty involved.

Having alternatives a study of the probability curve will ease the decision process. Since we can calculate the probability curve for any relevant item or metric like NOPLAT, EBIT, profit etc. a comparison between the alternatives makes the priority process more objective. Any board member or decisions maker can by the look at the probability curve understand the risk involved.

The argumentation is also logical and follows the principle that it can be audited and tested. The discussion can rather debate the premises and their defined uncertainties since they give the consequences.

Ordinary budgets not taken uncertainty into account is based on a deterministic and unrealistic assumtion and tells nothing about the uncertainty and risk involved.

01/09/2008
The advantages of simulation modelling
This entry is part 6 of 6 in the series Monte Carlo Simulation

All businesses need the ability to, if not predict the future; assess what its future economic performance can be. In most organizations this is done using a deterministic model, which is a model which does not consider the uncertainty inherent in all the inputs to the model. The exercise can best described as pinning jelly to a wall; it is that easy to find the one number which correctly describes the future.

The apparent weakness of the one number which is to describe the future is usually paired with so called sensitivity analysis. Such analysis usually means changing the value of one variable, and observe what the result then is. Then another variable is changed, and again the result is observed. Usually it is the very extreme cases which are analyzed, and some times these sensitivities are even summed up to show extreme values and improbable downsides.

Such a sensitivity analysis is as much pinning jelly to the wall as is the deterministic model itself. The relationship between variables is not considered, and rarely is the probability of each scenario stated.

What the simulation model does is to model the relationship between variables, the probability of different scenarios, and to analyze the business as a complex whole. Each uncertain variable is assessed by key decision makers giving their estimates for
- The expected value of the variable
- The low value at a given probability
- The high value at a corresponding probability level
- The shape of the probability curve
The relationship between variables is either modeled by its correlation coefficient or a regression.

Then a simulation tool is needed to do the simulation itself. The tool uses the assigned probability curves to draw values from each of the curves. After a sufficient number of simulations, it will give a probability curve for the desired goal function(s) of the model, in addition to the variables themselves.

As decision support this is an approach which will give you answers to questions like:
- The probability of a project NPV being at a given, required level
- The probability of a project or a business generating enough cash to run a successful business
- The probability of default
- What the risk inherent in the business is, in monetary terms
- And a large number of other very useful questions
The simulation model gives you a total view of risk where the sensitivity analysis or the deterministic analysis gives you only one number, without any known probability. And it also reveals the potential upside which is in every project. It is this upside which must be weighted against the potential downside and the risk level which is appropriate for each entity.

The S@R-model

The S@R-model is simulation tool which is built on proven financial and statistical technology. It is written in a language especially made for modelling financial decision problems, called Pilot Lightship. The model output is in the form of both probabilities for different aspects of a financial or business decision, and in the form of a fully fledged balance sheet and P&L. Hence, it gives you what you normally expect as output from at deterministic model, and in addition it gives you simulated results given defines probability curves and relationships between variables.

The operational part of the business can be modeled either in a separate subroutine, or directly into the main part of the simulation tool, called the main model. For complex goal functions with numerous variables and relationships, it is recommended to use the subroutine, as it gives greater insight into the complexity of the business. Data from the operational subroutine is later transferred to the main model as a compiled file.
01/09/2008
Risk and Monte Carlo simulation

This entry is part 1 of 6 in the series Monte Carlo Simulation

Risk, when does it occur? Whenever the outcome of a situation is not perfectly certain you have uncertainty or risk. Investment decisions taken under these circumstances involve a probability for an outcome that will differ from your estimated target. Decisions taken under uncertainty are a reality and a constraint manager’s face. In order to reduce the risk (probability of gain/loss) you have basically two ways of doing it, reduce the exposure or try to reduce the uncertainty by gathering more information.

Risk – randomness with knowable probabilities.

Uncertainty – randomness with unknowable probabilities.

The problem with information is very often the lack of it due to cost and time factors. A major point in this context is that uncertainty can be reduced but risk can be calculated.

We will illustrate this by describing a typical investment decision and look into the decisions and how they can be enhanced by taking advantage of calculating the risk by using Monte Carlo Simulation. This is a method especially developed to handle situations with uncertainty and to calculate the risk involved. The logic is fairly simple and the applications are numerous.

Most business concepts involve various proportions of income, costs and investments. We will in the following use the philosophy that every decisions shall be taken in order to maximize shareholder value, corporate competitiveness and customer satisfaction.

We have here split the decision process into various steps in order illustrate actually how easy it is to do it. By clicking on each theme you will see how we have given a flavor on how the problem can be solved.

22/02/2008
Risk, price and value
This entry is part 3 of 4 in the series A short presentation of S@R

Having arrived at the probability distribution for the value of equity (see full story) we are able to calculate expected gain, loss and their probability when investing in a company where the capitalized value (price) is known. (see “The Probability of Gain and Loss”)

In the figure below we have illustrated the investment and speculative area. The investment area comprice the part of the cumulative probability distribution below 50%.

The speculative area is the area above 50%. The expected value is given at the 50% probability point (stapled line). The literature advices, and successful investors insists, on having a safety margin (discount) of at least 20% between expected value (intrinsic value) and the market price, as shown by the yellow area in the figure below. Graham and Dodd in Security Analysis introduced the concept of a margin of safety in 1934.

In a stochastic framework as ours it is better to set the safety margin at one of the percentiles or quartiles giving directly the value of the safety margin. A fixed percentage safety margin will always give a different probability for gain (loss), depending on the shape of the cumulative probability distribution.

An investor having a portfolio of stocks should thus use percentiles as a margin – having the same probability for gain (loss) throughout the portfolio. In the case below a 20% safety margin coincide with the first quartile, – giving a 25% probability for loss and 75% probability for gain. The expected value of the company is 1.452 the first quartile is 1.160 giving an exepcted gain of 292 or more with 75% probability (dotted lines).

We know that the total risk of any individual asset is the sum of the systematic and unsystematic risk. When computing the figure above we have used the company’s appropriate beta to account for the systematic risk (in calculating WACC). The unsystematic risk is given by the variance in the figure above.

In a well-diversified portfolio the expected value of the unsystematic return is assumed to be zero. When investing in a single asset we should be looking for assets with a high unsystematic return. In our context companies with a capitalized value below the percentile set as limit of the safety margin.

References
1. Security Analysis: The Classic 1934 Edition by Benjamin Graham, David L. Dodd. October 1, 1996, McGraw-Hill Professional Publishing; ISBN: 0070244960
2. and an interesting webiste The Graham-Buffett Teaching Endowment
16/02/2008