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# Portfolio: expectation, standard deviation and covariance

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An investor wants to invest 1000 euro. He can choose between the shares of the electronic company Solips and the food supply chain Bhold. The value of the first share is 1 euro at the moment the investment is made, the price of the second share is 2 euro. How should the investor divide the 1000 euro between the two types of shares? That is, how should he choose his portfolio? Information about the joint probability distribution of the (random)prices S and B of the two shares after one year, is depicted in Table 1:

b
1.9 2 2.1
0.9 0.1 0.1 0
s 1 0 0.2 0.1
1.1 0 0.2 0.3

Table 1: Joint probability distribution of S and B.

So, after one year from now, the price S of one share Solips can take three possible values: 0.9 euro, 1 euro, 1.1 euro. The price B of one share Bhold can also take three values: 1.9 euro, 2 euro, 2.1 euro. For example, the joint probability that S and B take the values 1.1 and 2.1 respectively, is equal to 0.3.

a) Calculate the expectation and the variance of both S and B. Also, calculate the covariance and the coefficient of correlation of S and B.
b) Suppose that the investor decides to spend the whole amount of money on Solips
shares. Determine the expectation, the variance, and the standard deviation of the
value of the portfolio after one year.
c) Suppose that the investor invests the whole amount in Bhold shares. Determine the
expectation, the variance, and the standard deviation of the value of this portfolio
after one year
d) Suppose the investors spends one half of the amount an Solips shares and the other
half on Bhold shares. Determine the expectation, the variance, and the standard
deviation of the random variable X that represents the value of that portfolio
after one year
e) Suppose the investor is a risk lover He decides not to border about the level of
risk incorporated in the standard deviation of next year 's value of the portfolio,
but to base the choice of the 'best 'portfolio purely on its next year 's expected
value. How will his portfolio be constituted?
f) Suppose the investor is risk averse. Since all possible Solips-Bhold portfolios will
have a positive expected return, He decides to choose that portfolio that minimizes
the standard deviation of next year 's value. What portfolio will he choose?
g) For the portfolios in b, c, and d, determine the expectations and the variances of the rates of returns.

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