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ANSWERS TO QUESTIONS
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1. Investment projects with different risks can affect the valuation of the firm by suppliers of capital. A project that provides a 20 percent expected return may add so much risk as to more than offset the expected return. In other words, the company's overall capitalization rate rises sufficiently to more than offset the incremental expected return. Either the cash flows or the required rate must be evaluated to bring the risk level of different projects into the analysis. The focus of the chapter is on how to develop information on project risk.
2. The standard deviation is a measure of the absolute dispersion of the probability distribution. It is an appropriate measure of risk for a relatively symmetrical distribution, provided the person using the measure associates risk with dispersion.
(To the extent that a distribution is skewed and a person is concerned with skewness, a between measure might be the semi- variance. The semivariance is the variance of the distribution to the left of the expected value and may be thought of as representing a measure of downside risk. If a person is concerned with risk in a particular state of the world, a state preference approach may be best. However, the standard deviation continues as the most widely used measure of risk. One reason is that it can be
calculated mathematically in a relatively easy manner. Higher moments of a probability distribution cannot be determined so easily.)
One alternative measure that is easy to use is the coefficient of variation (CV). Mathematically, it is defined as the ratio of the standard deviation of a distribution to the expected value of the distribution. This measure of relative dispersion is an index of risk per unit of expected value.
3. To standardize the dispersion of a probability distribution, one takes differences from the expected value (mean) of the distribution and divides them by the standard deviation. The difference could be associated with the NPV of zero or less or any other stated NPV or IRR. The standardized value obtained is then used to determine the probability of greater (lesser) differences occurring or not occurring. These probabilities are found in Table V at the end of the textbook. They are based on the normal, bell- shaped distribution. As long as the distribution is unimodal, the probabilities are reasonably accurate even though the distribution may not be normal. By determining the probabilities that various NPVs or IRRs will occur, one obtains a better understanding of the risk of the project.
4. For the riskless project the probability distribution would have no dispersion. It would be a straight line that touched the horizontal axis at the expected value of return for the project.
The extremely risky project would be characterized by a probability distribution that was quite wide.
5. The coefficients of variation for the two projects are:
CVA = $400/$200 = 2.00; and CVB = $300/$140 = 2.14.
On the basis of relative risk alone, we would say that project B was the more risky.
6. The initial probabilities are those for outcomes in the first period. Conditional probabilities are those for outcomes in subsequent periods conditional on the outcome(s) in the previous period(s). For a particular branch, the conditional probabilities for the next period associated with the various sub branches must total 1.00. A joint probability is the joint product of multiplying the initial probability and all subsequent conditional probabilities for a particular branch times each other. This gives the probability of the overall (complete) branch occurring.
7. The risk-free rate is used to discount future cash flows so as not to double count for risk. If a premium for risk, particularly a large premium, is included in the discount rate, a risk adjustment occurs in the discounting process. The larger the premium, the narrower the distribution of NPVs -- that is, the lower the
standard deviation of the probability distribution of NPVs. Risk would then be evaluated in a comparison of the standard deviation with the expected value. To avoid the double evaluation of risk, a risk-free rate should be used in discounting.
8. Simulation gives the analyst an idea of the dispersion of likely returns from a project as well as the shape of the distribution.
However, the results are only as good as the assumptions used in the model. In other words, the results follow from assumptions regarding cash flows, probabilities, and interrelationships between cash flows.
9. The greater the correlation of net present values among projects, the greater the standard deviation of the portfolio of projects, all other things the same. By acquiring assets with low degrees of correlation with each other, the standard deviation of risk of a portfolio can be reduced relative to its expected value. (Whether this company-provided diversification is a thing of value to investors in the company's stock is questionable, as we take up in Chapter 15.) The effect of the correlation coefficient on the standard deviation of a portfolio of projects is shown in Equations (14-6) and (14-7) in the chapter.
10. A portfolio of assets dominates another if it has a higher expected return and the same or lower level of risk (e.g., standard deviation), or a lower level of risk and the same or a higher
expected value. Using the concept of dominance, some combinations of assets can be dismissed because they are dominated by one or more others.
11. When a decision maker decides on a portfolio of assets, that determines the acceptance or rejection of investment projects under consideration. New projects included in the portfolio are accepted; those excluded are rejected.
12. A managerial option has to do with management's flexibility to make a decision after a project is accepted that will alter the project's subsequent expected cash flows and/or its life. It also includes the option to postpone. With uncertainty about the future, the presence of managerial options (flexibility) enhances the worth of an investment project. This worth is equal to the net present value of the project, determined in the traditional way, plus the value of any option(s).
13. The present value of a managerial option is determined by the likelihood that it will be exercised and the magnitude of the resulting cash-flow benefit. The greater the uncertainty or volatility of possible outcomes, the greater the value of the option. It is the same as with a financial option -- the driving force to option valuation is the volatility of the associated asset's price.