Term Paper on "Diversification, Risk and Transactions Costs"

Quoted Instructions for "Diversification, Risk and Transactions Costs" Assignment:

answer the following questions:

1. Portfolio Diversification, Risk and Transactions Costs

Suppose that you have $100,000 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 2.1 times its current value, with each outcome equally likely (that is,

each outcome has a probability of one half). Assume for simplicity that there are no dividends

and no inflation. Assume that each company*****s fate is independent of each other company*****s

fate. Finally, assume for now that there is no brokerage or other transactions costs to buying

stocks.

(a) Suppose you invest your entire $100,000 in one company's stock. What is the expected (or

average, or mean) value of your stock after one year? (Use the formula for expected value in

Chapter 5 of Cecchetti). What is the probability that you will wind up with nothing? What is

the probability that you will wind up with $210,000?

(b) Now suppose that split your savings between 2 company*****s stocks, buying $50,000 worth of

each. What is the expected value of your portfolio after one year? What is the probability of

winding up with nothing? What is the probability of winding up with $210,000? What other

outcomes could occur (in terms of how much money you wind up with) and how likely are

they to occur?

[HINT: recall that the company's fates 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].

(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?

(d) Calculate the probability that you will end up with nothing, and the probability that you will

end up with $210,000, for each of the following cases: splitting your money evenly between 3

stocks, between 5 stocks, and between 10 stocks. What is happening to the probability of "inbetween"

outcomes as portfolio diversification increases in this example (no exact answer

required, just tell me whether the likelihood of in-between outcomes is going up or going

down)?

[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?

2) Given two stocks A and B, identical in characteristics, except the following:

Stock A Stock B

Probability Annual Return Probability Annual Return

.3 -1% .3 -2%

.3 2% .2 2%

.4 5% .5 6%

a) What is the expected return for Stock A and B?

b) What is the standard deviation of Stock A and B?

c) If both Stock A and Stock B have the same supply, which stock will sell at a higher

price? Why? [HINT: Think about investor tolerance (preference/attitude) toward risk]