Perspectives on Diversification
©Prof. P.V. Viswanath, 1997, 2000, 2001

Instructions: The following scenarios are designed to lead you to an understanding of what diversification is, and how it affects asset pricing.  The answer to the first part of each of the three questions is more or less subjective: there is no right or wrong answer.  However, once you have provided an answer to the first answer, that reduces the set of possible rational answers to the second answer.  In other words, there is a relationship between the first and second parts to each question.  Once you have answered both parts of questions 1 and 2, you should come to a realization about the specific aspect of the second part of each question that leads to the answer to that part being different from the answer to the first part.  You will then be ready to answer question 3.  

Answer question 3 in the same way.  Your answers to parts b) and c) relative to your answer to part a) will help you better understand the role of diversification in asset pricing.  The point in this exercise is not simply to get the solution to question 3, but to get the solution to question 3, and explain it in a way that parallels your answer to questions 1 and 2.

1. a. Suppose your boss tells you that he might ask you to put in an extra 2 hours of overtime next week. He is not sure that he will need your time because it depends on a request from another client. However, he thinks the chances of your client making the request is about 10%. The only problem is that if he needs the time he will have to have it during the beginning of next week. The problem for you is that there is a 40% chance that you are going to be running around a lot over the weekend taking care of family commitments (whether you'll have to do it or not depends on certain eventualities that you have no control over); if that happens, you expect to be really tired during the first part of next week. There is no company policy requiring you to put in overtime, but the boss doesn't want to take a chance that you won't be available if he needs you. In order to make sure that you'll be available if need, the boss agrees to compensate you for the additional demand on your time. How much would you require in compensation?

Note:  Try to ignore irrelevant "real-world" aspects of the problem.  For example, in the real world, you might not ask your boss for compensation for a small thing like this, especially, if that's going to get your boss unfavorably disposed towards you.  However, even in that case, you probably would think of how much the boss's request would inconvenience you, and then weigh that against how upset the boss would get if you asked for compensation.  Well, just get to the first part, i.e. figure out the compensation, and stop there!

b. The same scenario as before, except the boss is sure that he won't need you for the first part of the week. The chances that he will need you at all, is still 10%. How much would you require in compensation?

2. a. You have a chance to buy into a great investment in the defense industry. The only problem is that whether it is successful or not depends on whether the federal government okays a particular budget item. The chances that the decision will be positive are about 40%. If the decision is positive, you will get $10,000 in all (not net of your initial investment); if not, you'll lose your entire investment, that is, you will get $0. The word on this is going to come out of the Office of Management and Budgets next week, but you have to make the decision on whether to invest or not, right away. How much would you be willing to put into the deal?

b. The same scenario as above, with an added twist. You, yourself, work for the defense industry, and your job is on the line. If the federal government doesn't okay the budget item, it's quite likely that you'll be fired as well. Keep in mind, however, that your chances of losing your job overall are not any greater than in part a; it's just that whereas in part a., you might have been working for a company like MacDonald's, say, and losing your job would have had nothing to do with any investments that you might have made in the defense industry, in the current scenario, some of the situations where you would lose your job are also situations where the defense industry investment would not pan out.  Are you going to be willing to pay more to buy into the deal or less?

Note: The basic facts about the investment are given; no more information is available.  Even if you work for the defense industry, you cannot learn anything more about it!

3. a. You have a chance to invest $400 in a stock. Of course, the price of the stock could go up or down depending on a host of factors. You've computed the volatility (standard deviation) of returns at about 20% per year. What kind of average return would you require to be willing to make the investment?

b. The same scenario as above, with a new piece of information. This particular stock tends to really do well in bull markets and plunge in bear markets. On the other hand, most of the risk of price movements for the stock come from overall market movements; there are very few specifically firm-related uncertainties. The volatility is therefore no greater than in 2a; it's still about 20% per year. You have most of your money invested in a diversified portfolio that's about evenly spread out over the market. What kind of average return would you require to be willing to make the investment?

c. The same scenario as in 3a, but with a different kind of twist. The guy who gave you the information made a mistake; actually this stock does do better in bull markets, but only moderately better than it's normal performance; concomitantly, it does not plunge in bear markets, but is somewhat worse in bear markets, than its average performance. On the other hand, there are a lot of firm-specific uncertainties, so that the overall volatility of returns is still about 20% per year. What kind of average return would you require to be willing to make the investment?

Note: Keep in mind that more information doesn't always mean that you're better off!  For example, consider the following two scenarios, where you've bought a lottery ticket -- one, it's a day before the drawing, and you don't have any information about the winning number; two, it's a day after the draw, and you know that your ticket does not have the winning number.  You clearly have more information in the second case, but you're as clearly worse off in the second case.

State Space Approach to Valuation

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