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Monte Carlo – Page 2 – Strategy @ Risk

Strategy @ Risk

Tag: Monte Carlo

Glossary: Statistics

Certain or Uncertain

You may know the values your variables will take in the time frame of your model — they are certain, or what statisticians call “deterministic”. Conversely, you may not know the values they will take — they are uncertain, or “stochastic”. If your variables are uncertain you will need to describe the nature of their uncertainty. This is done with probability distributions, which give both the range of values that the variable could take (minimum to maximum), and the likelihood of occurrence of each value within the range.

Correlation

Correlation is a quantitative measurement of the strength of a relationship between two variables. The most common type of correlation is linear correlation, which measures the linear relationship between two variables. The rank order correlation value can vary between -1 and 1. A value of 0 indicates there is no correlation between variables; they are independent. A value of 1 indicates a complete positive correlation between the two variables; when the input value samples “high,” the output value will sample “high.” A value of -1 indicates a complete inverse correlation between the two variables; when the input value samples “high,” the output value will sample “low.” Other correlation values indicate a partial correlation; the output is affected by changes in the selected input, but may be affected by other variables as well.

Deterministic

The term deterministic indicates that there is no uncertainty associated with a given value or variable.

Independent or Dependent

In addition to being certain or uncertain, variables in a Risk Analysis model can be either “independent” or “dependent”. An independent variable is totally unaffected by any other variable within your model. For example, if you had a financial model evaluating the profitability of an agricultural crop, you might include an uncertain variable called Amount of Rainfall. It is reasonable to assume that other variables in your model such as Crop Price and Fertilizer Cost would have no effect on the amount of rain — Amount of Rainfall is an independent variable. A dependent variable, in contrast, is determined in full or in part by one or more other variables in your model. For example, a variable called Crop Yield in the above model should be expected to depend on the independent variable Amount of Rainfall. If there’s too little or too much rain, then the crop yield is low. If there’s an amount of rain that is about normal, then the crop yield would be anywhere from below average to well above average. Maybe there are other variables that affect Crop Yield such as Temperature, Loss to Insects, etc.

Mean/average

The mean or average of a set of values is the sum of all the values in the set divided by the total number of values in the set; or the average value of the set.

Monte Carlo sampling

Monte Carlo sampling refers to the traditional technique for using random or pseudo-random numbers to sample from a probability distribution. The term Monte Carlo was introduced during World War II as a code name for simulation of problems associated with development of the atomic bomb. Today, Monte Carlo techniques are applied to a wide variety of complex problems involving random behaviour. A wide variety of algorithms are available for generating random samples from different types of probability distributions.

Monte Carlo sampling techniques are entirely random — that is, any given sample may fall anywhere within the range of the input distribution. Samples, of course, are more likely to be drawn in areas of the distribution, which have higher probabilities of occurrence. In the cumulative distribution shown earlier, each Monte Carlo sample uses a new random number between 0 and 1. With enough iterations, Monte Carlo sampling “recreates” the input distributions through sampling. A problem of clustering, however, arises when a small number of iterations are performed.

In the illustration shown here, each of the 5 samples drawn falls in the middle of the distribution. The values in the outer ranges of the distribution are not represented in the samples and thus their impact on your results is not included in your simulation output.

Clustering becomes especially pronounced when a distribution includes low probability outcomes, which could have a major impact on your results. It is important to include the effects of these low probability outcomes. To do this, these outcomes must be sampled. But, if their probability is low enough, a small number of Monte Carlo iterations may not sample sufficient quantities of these outcomes to accurately represent their probability.

Skewed distribution

Skewness is a measure of the shape of a distribution. Skewness indicates the degree of asymmetry in a distribution. Skewed distributions have more values to one side of the peak or most likely value — one tail is much longer than the other is. A skewness of 0 indicates a symmetric distribution, while a negative skewness means the distribution is skewed to the left. Positive skewness indicates a skew to the right.

This distribution is skewed to the right, indicating upside potential rather that downside risk.

Standard deviation

The standard deviation is a measure of how widely dispersed the values are in a distribution or how much they deviate – on average – from the mean or average value. Equals the square root of the variance.

Stochastic

Stochastic is a synonym for uncertain, risky.

Value @ Risk

Anybody who owns a portfolio of investments knows there is a great deal of uncertainty about the future worth of the portfolio. Recently the concept of value at risk (VaR) has been used to help describe a portfolio’s uncertainty. Simply stated, value at risk of a portfolio at a future point in time is usually considered to be the fifth percentile of the loss in the portfolio’s value at that point in time. In short, there is considered to be only one chance in 20 that the portfolio’s loss will exceed the VAR. To illustrate the idea, suppose a portfolio today is worth $100. We simulate the portfolio’s value one year from now and find there is a 5% chance that the portfolio’s value will be $80 or less. Then the portfolio’s VaR is $20 or 20%.

Conficence levels:

68,0 % Std.dev.*1
90,0 % Std.dev.*1,65
95,0 % Std.dev.*2
99,7 % Std.dev.*3

Variance

The variance is a measure of how widely dispersed the values are in a distribution, and thus is an indication of the “risk” of the distribution. It is calculated as the average of the squared deviations about the mean. The variance gives disproportionate weight to “outliers”, values that are far away from the mean. The variance is the square of the standard deviation.

Volatility

Volatility can be measured as the Standard deviation * square root of time, or

24/02/2009
Corporate Risk Analysis

This entry is part 2 of 6 in the series Balance simulation

Strategy @Risk has developed a radical and new approach to the way risk is assessed and measured when considering current and future investment. A key part of our activity in this sensitive arena has been the development of a series of financial models that facilitate understanding and measurement of risk set against a variety of operating scenarios.

We have written a paper which outlines our approach to Corporate Risk Analysis to outline our approach. Read it here.

Our purpose in this paper is to show that every item written into a firm’s profit and loss account and its balance sheet is a stochastic variable with a probability distribution derived from probability distributions for each factor of production. Using this approach we are able to derive a probability distribution for any measure used in valuing companies and in evaluating strategic investment decisions. Indeed, using this evaluation approach we are able to calculate expected gain, loss and probability when investing in a company where the capitalized value (price) is known.

03/02/2009
The weighted average cost of capital
This entry is part 1 of 2 in the series The Weighted Average Cost of Capital

A more extensive version of this article can be read here in .pdf format.

The weighted cost of capital (WACC) and the return on invested capital (ROIC) are the most important elements in company valuation, and the basis for most strategy and performance evaluation methods.

WACC is the discount rate (time value of money) used to convert expected future cash flow into present value for all investors. Usually it is calculated both assuming a constant cost of capital and a fixed set of target market value weights ((Valuation, Measuring and Managing the Value of Companies. Tom Copeland et al.)) , throughout the time frame of the analysis. As this simplifies the calculations, it also imposes severe restrictions on how a company’s financial strategy can be simulated.

Now, to be able to calculate WACC we need to know the value of the company, but to calculate that value we need to know WACC. So we have a circularity problem involving the simultaneous solution of WACC and company value.

In addition all the variables and parameters determining the company value will be stochastic, either by themselves or by being functions of other stochastic variables. As such WACC is a stochastic variable– determined by the probability distributions for yield curves, exchange rates, sale, prices, costs and investments. But this also enables us – by Monte Carlo simulation –to estimate a confidence interval for WACC.

Some researchers have claimed that the free cash flow value only in special cases will be equal to the economic profit value. By solving the simultaneous equations, giving a different WACC for every period, we will always satisfy the identity between free cash flow and economic profit value. In fact we will use this to check that the calculations are consistent.

We will use the most probable value for variables/parameters in the calculations. Since most of the probability distributions involved are non-symmetric (sale, prices etc), the expected values will in general not be equal to the most probable values. And as we shall see, this is also the case for the individual values of WACC.

WACC

To be consistent with the free cash flow or economic profit approach, the estimated cost of capital must comprise a weighted average of the marginal cost of all sources of capital that involves cash payment – excluding non-interest bearing liabilities (in simple form):

WACC = ${C_d}(1-t)*{D/V} + {C_e}*{E/V}$

${C_d}$ = Pre-tax debt nominal interest rate
${C_e}$ = Opportunity cost of equity,
t = Corporate marginal tax rate
D = Market value debt
E = Market value of equity
V = Market value of entity (V=D+E).

The weights used in the calculation are the ratio between the market value of each type of debt and equity in the capital structure, and the market value of the company. To estimate WACC we then first need to establish the opportunity cost of equity and non-equity financing and then the market value weights for the capital structure.

THE OPPORTUNITY COST OF EQUITY AND NON-EQUITY FINANCING

To have a consistent WACC, the estimated cost of capital must:
1. Use interest rates and cost of equity of new financing at current market rates,
2. Be computed after corporate taxes,
3. Be adjusted for systematic risk born by each provider of capital,
4. Use nominal rates built from real rates and expected inflation.
However we need to forecast the future risk free rates. They can usually be found from the yield curve for treasury notes, by calculating the implicit forward rates.

THE OPPORTUNITY COST OF EQUITY

The equation for the cost of equity (pre investor tax), using the capital asset pricing model (CAPM) is:

C =

R = risk-free rate,
= the levered systematic risk of equity,
M = market risk premium,
L = liquidity premium.

If tax on dividend and interest income differs, the risk-free rate and the market premium has to be adjusted, assuming tax rate -ti, for interest income:

R = (1-)*R and M = M+*R.

= Investor tax rate,
R = tax adjusted risk-free rate,
M = tax adjusted market premium

The pre-tax cost of equity can then be computed as:

$R/(1-t_d)+{beta}*{M/(1-t_d)}+{LP/(1-t_d)}$

(pre-tax) = = $R/(1-t_d)+{beta}*{M/(1-t_d)}+{LP/(1-t_d)}$

Where the first line applies for an investor with a tax rate of -td, on capital income, the second line for an investor when tax on dividend and interest differs ((See also: Wacc and a Generalized Tax Code, Sven Husmann et al., Diskussionspapier 243 (2001), Universität Hannover)) .

The long-term strategy is a debt-equity ratio of one, the un-levered beta is assumed to be 1.1 and the market risk premium 5.5%. The corporate tax rate is 28%, and the company pays all taxes on dividend. The company’s stock has low liquidity, and a liquidity premium of 2% has been added.

In the Monte Carlo simulation all data in the tables will be recalculated for every trial (simulation), and in the end produce the basis for estimating the probability distributions for the variables. This approach will in fact create a probability distribution for every variable in the profit and loss account as well as in the balance sheet.

THE OPPORTUNITY COST OF DEBT

It is assumed that the pre-tax debt interest rate can be calculated using risk adjusted return on capital (RAROC) as follows:

Lenders Cost =

= Lenders Funding Cost (0.5%),
= Lenders Average Expected Loss (1.5%),
= Lenders Administration Cost (0.8%),
= Lenders Risk Premium (0.5%).

The parameters (and volatility) have to be estimated for the different types of debt involved. In this case there are two types; short -term with a maturity of four years and long-term with a maturity of 10 years. The risk free rates are taken from the implicit forward rates in the yield curve and lenders cost are set to 3.3%.

In every period the cost and value of debt are recalculated using the current rates for that maturity, ensuring use of the current (future) opportunity cost of debt.

THE MARKET VALUE WEIGHTS

By solving the simultaneous equations, we find the market value for each type of debt and equity:

And the value weights:

Multiplying the value weights by the respective rate and adding, give us the periodic most probable WACC rate:

As can be seen from the table above, the rate varies slightly from year to year. The relative small differences are mainly due to the low gearing in the forecast period.

MONTE CARLO SIMULATION

In the figure below we have shown the result from simulation of the company’s operations, and the resulting WACC for year 2002. This shows that the expected value of WACC in is 17.4 %, compared with the most probable value of 18.9 %. This indicates that the company will need more capital in the future, and that an increasing part will be financed by debt. A graph of the probability distributions for the yearly capital transactions (debt and equity) in the forecast period would have confirmed this.

In the figure the red curve indicates the cumulative probability distribution for the value of WACC in this period and the blue columns the frequencies. By drawing horizontal lines on the probability axis (left), we can find confidence intervals for WACC. In this case there is only a 5% probability that WACC will be less than 15%, and a 95% probability that it will be less than 20%. So we can expect WACC for 2002 with 90% probability to fall between 15% and 20%. The variation is quite high – with a coefficient of variation of 6.8 ((Coefficient of variation = 100*st.dev/mean)).

VALUATION

The value of the company and the resulting value of equity can be calculated using either the free cash flow or the economic profit approach. Correctly done, both give the same value. This is the final test for consistency in the business model. The calculations are given in the tables below, and calculated as the value at end of every year in the forecast period.

As usual, the market value of free cash flow is the discounted value of the yearly free cash flow in the forecast period, while the continuing value is the value of continued operation after the forecast period. All surplus cash are paid, as dividend so there is no excess marketable securities.

The company started operations in 2002 after having made the initial investments. The charge on capital is the WACC rate multiplied by the value of invested capital. In this case capital at beginning of each period is used, but average capital or capital at end could have been used with a suitable definition of capital charge.
Economic profit has been calculated by multiplying RIOC – WACC with invested capital, and the market value at any period is the net present value of future economic profit. The value of debt as the net present value of future debt payments – is equal for both methods.

For both methods using the same series of WACC when discounting cash the flows, we find the same value for the both company and equity. This ensures that the calculations are both correct and consistent.

Tore Olafsen and John Martin Dervå
08/09/2008
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
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