# Portfolio: expectation, standard deviation and covariance

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.