Project Economic Analysis
2.6 Risk Adjusted Value (RAV)
Then, run the random numbers as per the “Mid square Method” (Von Neumann and Metropolis, 1940s), which produces pseudo-random four digit numbers:
1. Start with a four digit seed number.
2. Square the seed and extract the center most four digits. (This is your sampling parameter.)
3. Use the sampling parameter as the seed for the next trial. Go to step two.
4. Random number generators usually return a value between zero and one.
For any software you use to perform the simulation for the cost, time, or other risk criteria, the process can be summarized as follows.
Any random variable less than one, in this case this number is assumed to be the cumulative curve value, and by knowing the probability distribution curve, then, by the software definition, the value of that variable corresponds to these random numbers, which correspond to the cumulative value and do that for all variables and put these values in the deterministic model.
From the deterministic model, obtain the value of the output and other parameters. Then, store the data for this trial and repeat these steps again for 8,000 and 10,000 times, so you have 10,000 output for the variable and can calculate the arithmetic mean and standard deviation. Then, draw the distribution or histogram curve and the cumulative curve.
willing to accept in lieu of the risky investment. This point defines the value at which the manager is indifferent between the two alternatives.
To illustrate, consider an investment with an EPV of ten million dollars.
EPV includes a return, chance of success, and a potential loss. The actual outcome could be higher or lower than this value. Should someone start offering the manager a guaranteed amount less than twelve million dollars, say nine million, then eight million, and so on until the manager said yes, the value eliciting the yes is the certainty equivalent or the indifference value.
Two companies have different, but identical prospects worth ten million dollars (EPV), each with a 100 percent WI, and an offer to farm-out to a third party. The first party accepts a guaranteed offer of six million dollars, while the second party accepts an offer for four million dollars. Why would the indifference points differ? Because most managers are risk adverse, contrary to the primary assumption of EPV and the degree of risk aversion depends on two basic components: the wealth of the firm (hence, freedom from bankruptcy) and the budget level. In this example, both firms are risk averse because they would accept a lower amount to reduce risk. Had either accepted the EPV they would be risk neutral, and occasionally we see investors who would want more (a true risk taker).
Cozzolino (1977) introduced the term risk adjusted value to integrate these concepts, as defined in Equation 2.40.
RAV r1 Ln P es r R C P es rC
( ) (1 ) (2.40)
where r equals risk aversion level of the firm, Ps equals probability of success, R equals NPV of success, C equals NPV of failure, E equals expo- nential function, and Ln equals natural logarithm.
In the Cozzolino (1977) format examples, like those above that are used to solve for r, assume that RAV is already established. If R, C, RAV, and P are known, the corporate risk aversion can be determined. Without per- forming an example, larger values for r imply more risk aversion, while smaller values reflect lower risk aversion. Evidence suggests that an inverse relationship exists between capital budget size and risk aversion level.
Smaller companies tend to be more risk averse and, thus, tend to spread their risk across as many projects as possible.
This basic format has been extended by Bourdaire et al. (1985) to elimi- nate the need to estimate the risk component. By employing the elements of subjectivity and assuming an exponential utility function, Equation 2.42 results.
RAV m s B
2
2 (2.42)
where m equals the mean NPV, s2 =equals the standard deviation of distri- bution, and B equals total monies budgeted for risky investments.
RAV, under this format, can be based on information typically gener- ated in the evaluation. RAV also depends on the estimated value relative to the dispersion of the NPV outcome. More importantly, high dispersion projects may be ranked above projects with lower standard deviation if the dispersion relative to the budget is low. RAV depends on two basic rela- tionships: m relative to s2 and s2 relative to B.
If the mean value of NPV that was calculated from the Monte-Carlo simulation is equal to ten million dollars, then a standard deviation illus- trates the dominance of the dispersion term, since $10 – (15)2 will be a very negative value. Now, suppose that the investor is a large oil company with a budget of $1,000 million. The RAV of the project is
RAV 10 225 million 2 1000 $ .9 9 .
For a smaller investor with a budget of only $200 million, RAV becomes RAV 10 225 million
2 200 $ .9 4 . The breakeven RAV value for B is found by solving
RAV breakeven s
( ) m . .
2
2
225
2 10 11 25
We are not aware of anyone presently ranking on RAV, although more people are discussing it. Like other ideas portrayed in this book, we believe it should be included as part of the evaluation for a period of time. If RAV aids in decision-making, then include it permanently.
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