internas de calidad
6. DIAGRAMA DE FLUJO
Even though simulation requires more work than merely using historical data, I find that because of the high importance of the volatility parameter in real option valuation and the fact that historical data in many cases are inadequate, it is here that the PE firms should invest some time and effort to get as accurate an estimation as possible.
Copeland and Antikarov (2001) present a method for estimating volatility in real option analysis using Monte Carlo simulation. Their simulation is based on the assumption of Market Asset Disclaimer (chapter 7.3 above) and on the management’s predictions/expectations of the future, in the form of a static valuation method such as the DCF analysis. The method is consolidated which means the output is a single estimate of volatility, built up from the different variables (e.g. price, quantity, costs etc.) that contribute to it. The idea behind the method is illustrated in figure 11
Figure 11 – Monte Carlo simulation of real option volatility
Source: Copeland % Antikarov (2001) p.245
Their method is based on Monte Carlo simulations using Monte Carlo simulation programs such as Crystal Ball and At Risk. Doing a manual Monte Carlo simulation with multiple, and sometimes correlated, variables is a very cumbersome and extremely difficult mathematical process. The purpose of the thesis is not to present how Monte Carlo simulation works, but simply to use it as a tool for applying Real Options to PE firms. I therefore find it appropriate to use a specialized program for simulating volatility when historical data are insufficient, and this will save time and space for the analysis of Real Options and the result will be no different from a manual simulation. I therefore compromise by using Crystal Ball for the Monte Carlo simulation.
The basic idea behind the method is to simulate the evolution of a few key variables which have a great influence on value of the company and thereby retrieving at large number of different company values using NPV valuation. The volatility based on these values is then used as an estimate of the volatility of the underlying asset.
The method includes:
Making a NPV estimation of the company value
Determine the variables witch influence the company value
For each variable determine: Mean, Standard deviation, Distribution, and auto-correlation as well as cross-sectional correlation
Use Monte Carlo simulation to generate volatility based on the NPV and variables.
Present Value Model Uncertainty 3 Uncertainty N Uncertainty 2 Uncertainty 1 0 PV Probability of PV
Inputs Monte Carlo
Simulation Output
NPV estimation of firm value
The first step is to set up a NPV valuation of the underlying asset using the appropriate valuation method. The Holding option and the Post Exit option have different characteristics and therefore different volatility, which is the main reason for separating the option value into two different options (chapter 6.1).
When calculating the value of the underlying asset for the PE Exit options, the PE firm is interested in the value it can get from exiting i.e. what a potential buyer can pay. Hence, it should be the valuation method that gives the best indication of the PE target’s market value which is used to value the underlying asset in the PE Exit options. But what model does that? The LBO model gives the maximum value that a buyer can pay, given the same conditions for leverage, active ownership etc. Using the LBO model is therefore likely to give a gap between the value that the target has for the PE firm, and could thereby give an unrealistic high value of the option. The DCF model, on the other hand, is maybe the most used and recognised valuation method. It gives a more general picture of the market value of the company by using the assumption from Capital Asset Pricing Model (CAPM). This entails using the WACC instead of a individual IRR as discount rate, and is therefore more consistent with capturing the true enterprise value (market value) of the company. Furthermore, contrary to using multiples, the DCF is not dependent on finding comparable companies with identical characteristics on business areas, capital structure, and risk. Based on the above, I find using a traditional DCF to estimate the value of the underlying asset to be the most appropriate.
Furthermore capital structure must be assumed to have reached a normal level, which can be maintained by the new owners making a DCF valuation possible. It is also the model recommended by Copeland & Antikarov (2002) when simulating volatility this way, and is the method I recommend using.
A DCF model is therefore the best way to calculate the value of the underlying asset in both PE options, and to distinguish them from each other I will call them DCFHolding and DCFExit for the PE holding- and Post Exit options respectively.
Basically it is a traditional DCF that is needed for both options. But depending on the variables that are simulated, the DCF must be build to include these variables, which is not necessarily the variables included in the traditional DCF.
Using the DCF as valuation tool raises the question of the capital structure, as this is assumed to be constant in a DCF (chapter 3.2) But since the underlying asset in the Holding Option is not considered a LBO but simply a normal market valuation, this is not an implicating factor here. The only question is what capital structure should be used, as this will affect the WACC and thereby the value. Since it is hard to precisely estimate the optimal capital structure of a company, I recommend using the capital structure reached at exit in the LBO valuation as the capital structure can be assumed to have reached a normal level by then, which can be maintained by the new owners. Hence, it would be the best estimate of a general capital for the company.
Determining relevant variables
Determining the relevant variables for the valuation is the critical point in this method. The PE firm will need to come up with variables which have a significant influence on the value of the portfolio company for both options. This of cause is again very case specific. Copeland & Antikarov 2001 mentions sales price, sales amount, and variable costs as the most common variables. Because the PE firm will make firm specific changes in the holding period, and the target is assumed to go into a steady stage after the planned exit, it influences which variables are most relevant in the PE Holding option compared to post exit period. Consequently the two options will have different volatilities, hence the division into two options. In order to determine which variables have the greatest influence on firm value a sensitive analysis is needed. One way to do this is to construct a Tornado diagram.
Tornado diagram
As mentioned above a tornado diagram help identify which variables in the DCF model has the greatest influence of firm value. A tornado diagram can be made in different ways, but the main purpose is to analyse the sensitivity of the variables in the DCF model one at a time. One way of doing this is to increase and decrease each variable with +/- 25 percent respectively holding the other variables constant, and thereby resulting in a diagram over the absolute change in firm value from a +/- 25 % change in the variables. A graphic illustration will show which variables the DCF is most sensitive to. Often it will be clear that the vast amount of sensitivity come from 2-3 variables (Copeland & Antikarov 2003, p 236). Most attention should of cause be on the variables with the greatest influence on firm value, as they contribute the most to volatility of firm value, ignoring any offsetting correlations. It is recommended though, that the analysis is not over complicated by using too many variables.
Characteristics of variables
After the variables are determined, their probability distribution, mean and standard deviation must be determined in order for the simulation tool to simulate their values. This again raises the question of using historical data or not. This will be very dependent on each variable and the situation. Is it for example a variable which the PE firm can affect, it is best to use the management’s subjective estimates. On the other hand if it is variables which are out of the control of the PE firm, and where they have little knowledge of the future development, estimation using historical data is preferred.
Volatility
Copeland & Antikarov’s (2001) method for estimating the volatility of each parameter using managers’ predictions assumes that the variable follows a geometric Brownian motion. The procedure is to find the expected values of the variable at different points in time (Vt) and
calculate the continuous growth factor (ri) for each period. This is usually done when calculating
the DCF model. In some cases, the development of the variable is determined using the growth factor ri which will be constant. The next step is to determine a confidence band where the value
is certain to be within with 95% confidence, meaning the highest (upper VT) and lowest (lower VT) value the variable is expected to have.
The volatility can then be estimated using:
= =
∑ D 89 E FGHIJKL FM N O DPQ 7√, = =
89EFGHIJKLFM N ∑ODPQD 7√Source: Copeland and Antikarov, 2002, p. 262
It can be discussed whether these ‘qualified guesses’ from management are to arbitrary to give a correct estimation of the variables volatility. But it has to be kept in mind that this method should only be used when it is assessed that the management has better knowledge of the future development of the variable than historical data can give. Furthermore it is a fast and simple way for the PE firm to get an estimate of the variables characteristics.
Distribution
The distribution of the variables can also be estimated, but I believe that the effort it takes to do this is not worth it. A simple assessment of the comparison of each variable to the general
distributions which I based on a standard deviation and mean value would be adequate. This is supported by Copeland and Antikarov (2001) which gives the example prices. Prices can be assumed to follow a log-normal distribution, as it is unlikely that prices will ever be negative.
Correlation and Autocorrelation
After each variable is determined and their general characteristics are estimated, the correlation between variables and their autocorrelation with themselves need to be determined. These correlations are also entered into Crystal Ball, and will increase accuracy of the estimated volatility. The correlation in Crystal Ball is measured using r-squared values, where a positive r- squared means that there is a positive correlation and a negative r-squared value means that there is a negative correlation3.
The correlation and autocorrelation can again either be estimated using historical data, or be set by management using their assumption of the correlation. For example, it is reasonable to assume that an increase in prices is followed by another increase or vice versa a decrease is followed by a decrease, which entails a positive autocorrelation. Management can then set the autocorrelation percentage after their assumption of the degree of correlation.
Simulating the volatility
After all the variables as set up, the volatility can be simulated using Crystal Ball.
Notice that the volatility is estimated on the return by using the change from one year to the next, by holding the first year constant.
R = STUVUV
4
Source: Copeland and Antikarov, 2001, p. 246
Simulation using Crystal Ball will automatically generate some very useful data in the form of both statistical data and graphic illustrations. Example hereof can been seen in figure 12 below.
3
Figure 12 - Examples of output from Crystal Ball
Source: Own construction in Crystal Ball