Suppose that you have $50,000 in wealth to invest for one year.

4) Suppose that you have $50,000 in wealth to invest for one year. You are consideringbuying 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’sstock 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 nodividends and inflation. Assume that each company’s fate is independent of each othercompany’s fate. Finally, assume for now that there are no brokerage or other transactionscosts to buying stocks.
1. (a) Suppose you invest your entire $50,000 in one company’s stock. What isprobability distribution of your wealth after one year? (In other words, what arethe possible outcomes, and what are their probabilities)? What is the expected (ormean) 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 worthnothing, or worth $62,500, with each outcome equally likely (that is, eachoutcome has a probability of 0.5). What is the probability distribution of yourwealth 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 ofyour wealth after one year?
[HINT: recall that the companies’ outcomes are independent–whether onecompany succeeds or fails does not depend on whether the other company fails.Thus the probability that both company A and company B both succeed (forinstance) 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 havingtwo companies’ stock increase the expected value of your portfolio? If not, thenwhy is diversification a good thing in this example?
4. (d) Calculate the probability that you will end up with nothing, and theprobability 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 10stocks. 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 timesis one-half to the 3rd power, or (1/2)*(1/2)*(1/2) = 1/8. The probability that it will comeup 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 stocksavailable then your best strategy would be to buy a tiny amount of each. But now supposethere 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 millioncompanies 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 peopleown stock mutual funds instead of buying individual stocks?

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