4) Suppose that you have $50,000 in wealth to invest for one year. You are considering buying stocks. There are many companies whose stock you could potentially buy. Suppose that each company you are considering is very risky: in one year, the company’s stock will either be worth nothing, or worth $125,000, with each outcome equally likely (that is, each outcome has a probability of 0.5). Assume for simplicity that there are no dividends and inflation. Assume that each company’s fate is independent of each other company’s fate. Finally, assume for now that there are no brokerage or other transactions costs to buying stocks.
1. (a) Suppose you invest your entire $50,000 in one company’s stock. What is probability distribution of your wealth after one year? (In other words, what are the possible outcomes, and what are their probabilities)? What is the expected (or mean) value of your wealth after one year?
2. (b) Now suppose that split your savings between 2 company’s stocks, buying $25,000 worth of each. (in one year, each company’s stock will either be worth nothing, or worth $62,500, with each outcome equally likely (that is, each outcome has a probability of 0.5). What is the probability distribution of your wealth after one year? (What are the possible outcomes for wealth after one year, and what are the probabilities of each outcome?) What is the expected value of your wealth after one year?
[HINT: recall that the companies’ outcomes are independent–whether one company succeeds or fails does not depend on whether the other company fails. Thus the probability that both company A and company B both succeed (for instance) is similar to the probability that two coin flips will both come up heads].
3. (c) Explain why in this example it might be a better idea to be diversified–that is, to own two companies’ stocks rather than just one company’s stock. Does having two companies’ stock increase the expected value of your portfolio? If not, then why is diversification a good thing in this example?
4. (d) Calculate the probability that you will end up with nothing, and the probability that you will end up with $125,000, for each of the following cases:
splitting your money evenly between 3 stocks, between 5 stocks, and between 10 stocks. What is the expected value in each case?
[HINT: if you flip a coin 3 times, the probability that it will come up “heads” all 3 times is one-half to the 3rd power, or (1/2)*(1/2)*(1/2) = 1/8. The probability that it will come up tails all three times is the same: (1/2)*(1/2)*(1/2). Similarly, if you flip a coin 5 times, the probability that it will come up heads all 5 times is one half to the fifth power, or (1/2)*(1/2)*(1/2)*(1/2)*(1/2), and so on].
(e) In this example, more diversification is always better–if there are a million stocks available then your best strategy would be to buy a tiny amount of each. But now suppose there is a fixed brokerage fee of, say, $10 for each company’s stock that you purchased, independent of how many shares you purchased, so that if you bought shares in a million companies you’d have to pay the $10 fee a million times. How would that affect your
optimal degree of diversification? Can this provide an explanation of why many people own stock mutual funds instead of buying individual stocks?
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