# Diversification, Risk and Transactions Costs Term Paper

#### Excerpt from Term Paper :

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
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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.

Yes, mutual funds serve as the best evidence of reducing overhead costs for stocks purchase in case there exists brokerage fee.

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

Stock a Stock B

Probability

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

Expected return for stock a:

E (x)=.3*-1%+0.3*2%+.4*5%

E (x)=2.3% expected return for stock a Expected return for stock B:

E (x)=.3*-2%+.2* 2%+.5*6%

E (x)=2.8% expected return for stock B. b) What is the standard deviation of Stock a and B?

A standard deviation for stock a:

x|=?.3*(2.3+1)^2+.3*(2.3-2)^2+.4*(2.3-5)^2 x|=2.49% standard deviation for stock B:

x|=?.3*(2.8+2)^2+.2*(2.8-2)^2+.5*(2.8-6)^2 x|=3.48% c) if both Stock a and Stock B. have the same supply, which stock will sell at…

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