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  • July 19, 2006
  • By Marcia Gulesian
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Appendix 1. The Assumption of Symmetry

Throughout much of the two previous articles in this series on capital budgeting (see Reference 1 and 2), the assumption of symmetry in probability distributions was made. However, as shown below, there can be weighty consequences to not looking beyond this assumption.

The three most commonly used measures of the location of central tendency are the mode, the median, and the mean. In brief, the mode is the point or region within a distribution where the largest number of individual measures congregate, the median is the midpoint of all the individual measures, and the mean is the arithmetic average of all the individual measures.

As a rule, the only time you will find the mode and median to be precisely coincident with the mean is when the distribution is unimodal and perfectly symmetrical. In skewed distributions, the mean, median, and mode will tend to be separated from one another, with the mean falling toward the tail of the skew, the mode falling away from the tail, at the peak, and the median falling somewhere in between. Thus, in a positively skewed distribution the mean will be to the right, the mode to the left, and the median in between, whereas in a negatively skewed distribution the mean will be to the left, the mode to the right, and the median in between. These relationships among the three measures of central tendency are shown below in Figure 6.

Figure 6. Relationships of the mean, median, and mode in symmetrical and skewed distributions.

If skew is not considered, then looking only at expected returns (for example, mean or median) and risk (standard deviation), a positively skewed project might be incorrectly chosen! Then, if the horizontal axis represents the net revenues of a project, clearly a left or negatively skewed distribution might be preferred because there is a higher probability of greater returns as compared to a higher probability for lower-level returns.

Failure to account for a project's distributional skewness may mean that the incorrect project may be chosen (for example, two projects may have identical mean and standard deviation—that is, they both have identical returns and risk profiles—but the distributional skews may be very different).

Finally, the introduction of probability distributions in the form of Monte Carlo simulation into decision tree analysis is beyond the scope of this article. But, at the very least, you should ask questions about the underlying statistical nature of the data presented to you in the decision tree.

Appendix 2. Worst-Case Scenario (Given Catastrophic Losses) vs. Regret

Figure 7. Probability distribution function with kurtosis (dashed line) higher than that of normal curve (solid line).

The area under the lower probability distribution function in Figure 7 is thicker at the tails with less area in the central body. This condition has major impacts on risk analysis. Although the returns and risks are identical with a normal distribution with the same mean and standard deviation, the probabilities of extreme and catastrophic events (potential large losses or large gains) occurring are higher for a high kurtosis distribution. (See Reference 12.)

Kurtosis is a measurement of the peakedness (broad or narrow) of a frequency distribution. Higher kurtosis means more of the variance is due to infrequent extreme deviations, as opposed to frequent modestly sized deviations.

When you calculate the value of the worst-case scenario given catastrophic losses, you also should think about regret. That is, if a decision is made to pursue a particular project, but if the project becomes unprofitable and suffers a loss, the level of regret is simply the difference between the actual losses compared to doing nothing at all.

Appendix 3. A Palimpsest on Real Options

Real options is now taught in most if not all MBA programs, and what's more, it's an interdisciplinary subject, found not just in finance but also in strategy and information-systems courses.

The classical application of real options, and the point of much research, is to show that a given investment with a negative NPV may in fact have substantial value, thanks to its embedded options. But, in today's capital-rationed environment, all IT investments are presumed to have a positive NPV, and a substantial one at that. Therefore, some calculate the real-options value of positive-NPV projects, to arrive at an "expanded" NPV for each—and an optimal ranking of IT investments.

The trouble is, although there is widespread interest in taking a portfolio approach to managing IT investments, few companies—24 percent—actually optimize such portfolios, according to a recent survey of 130 senior IT executives conducted by the Kellogg School, DiamondCluster International, and the Society for Information Management. None of the executives surveyed used real options.

Real options discounts management realities. Is the strength of real options also its Achilles' heel? Critics say that because real options don't expire according to contract as financial options do, managers can't be counted on to pull the plug on a project (exercise an "abandonment option") when they should. Also, projects assume lives of their own, and may not be easy to kill.

On the other hand, companies often yank NPV-sanctioned projects, while real options provides a detailed map for making such decisions, with far greater precision. Arguably, adopting a real-options approach would promote greater discipline in project management. But, the approach won't take if an organization doesn't embrace change, or if compensation systems aren't aligned accordingly. A manager can't be expected to exercise a growth option, for example, if she's being compensated for keeping costs down. Until these sorts of organizational process and governance issues are tackled, the lone analyst can't really do much more. I don't see companies at all interested in changing business processes right now. Everybody's glad they have jobs.


For readers not already familiar with the basics of Decision-Tree Analysis, an overview of this well-established decision-making methodology is available in References 3–8. For readers not already familiar with the basics of VoIP, an overview of this relatively new technology is available in References 9–11. The basics of other topics such a Net Present Value (NPV) and Monte Carlo Simulation mentioned in this article were covered in the previous two articles of this series. (See References 1 and 2.)


  1. http://www.developer.com/mgmt/article.php/3595036
  2. http://www.developer.com/mgmt/article.php/3601061
  3. Clemen, R., Reilly, T. Making Hard Decisions with DecisionTools, Duxbury Thomson (2004)
  4. Schuyler, J., Risk and Decision Analysis in Projects 2nd Ed., PMI (2001)
  5. Mun, J., Modeling Risk, Wiley (2006)
  6. Ross, S. et al, Corporate Finance, McGraw-Hill Irwin (2005)
  7. Kodukula, P., Papudesu, C., Project Valuation Using Real Options, J. Ross (2006)
  8. Brigham, E., Daves, P., Intermediate Financial Management, 8th Ed., Thomson (2004)
  9. Wallace, K., Voice over IP first-step, Cisco (2006)
  10. Porter, T. et al, Practical VoIP Security, Syngress (2006)
  11. Johnston,A., Piscitello, D., Understanding Voice over IP Security, Artech (2006)
  12. Rachev, S. et al, Fat-Tailed and Skewed Asset Return Distributions, Wiley (2005)
  13. Sanjay, A., Sarbanes-Oxley Guide for Finance and Information Technology Professionals, 2nd Ed., Wiley (2006)
  14. Taylor, H., The Joy of SOX, Wiley (2006)

About the Author

Marcia Gulesian has served as Software Developer, Project Manager, CTO, and CIO over an eighteen-year career. She is author of well more than 100 feature articles on Information Technology, its economics, and its management. You can e-mail Marcia at marcia.gulesian@verizon.net.

Copyright © 2006 Marcia Gulesian

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