Diversification, Risk and Transactions Costs

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

A 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?

Answer: there are two possible outcomes of the event with stock: either company's stock will equal to 0 after investing $100,000 or it will be worth 2.1 times of current value (of $100,000) or $210,000. Because these two events are independent probability of each event will be equal to 0.5:

P (0)=0.5

P (210,000)=0.5

Expected value can be found as folliwng:

E (x)= X1*p (x1)+X2*p (x2)

E (x)=0.5*0+210,000*0.5

E (x)=105,000

Expected value of stock will be equal to $105,000

Probability of winding up with $0 and probability of winding up with $210,000 will be equal to 0.5 or 50% 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?

We can make a table to see all possible events which can occur:

As we are supposed to by $50,000 of each company the maximum value of stock can be 2.1*$50,000=$105,000

First company

Second company

Total value

1st case

2nd case

3rd case

4th case

The probability of winding up with nothing is equal to the ratio of total number of cases (4 cases) to the number of favorable cases (1 case) and is equal to 0.25

The probability of winding up with 210,000 is equal to the ratio of total number of cases (4 cases) to the number of favorable cases (1 case) and is equal to 0.25

The probability of winding up with 105,000 is equal to the ratio of total number of cases (4 cases) to the number of favorable cases (2 cases) and is equal to 0.5

The expected value can be found as follows: E (x)=0*0.25+210,000*0.25+105,000*0.5

E (x)=$105,000

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?

Having the portfolio which consists of two equal stocks doesn't increase portfolio's value, which will remain the same ($105,000), yet it decreases the risk of winding with nothing, as the probability of winding with nothing in case portfolio consists of two companies' stocks is twice smaller (0.25 against 0.5) and the probability of favorable outcome is 0.75 compared to 0.5 for a one company stock portfolio.

A 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 "in-between" 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)?

Answer: probability for 3 stocks of ending with 0 and ending with 210,000 is equal to 0.5*(1/3)=1/6

For 5 stocks ending with 0 and ending with 210,000 is equal to 0.5*(1/5)=1/10

For 10 stocks ending with 0 and ending with 210,000 is equal to 0.5*(1/10)=1/20

The probability of in-between outcomes is growing as it can be found as total probability minus probability of ending with 0 and ending with 210,000:

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?

Answer: In case of diversification we would have to subtract the brokerage fee from the expected return if there exists brokerage fee for stocks.