Problem $1$ Suppose that the stock market consists of two stocks $-$ Alfa Inc. $($referred

to as A$)$ and Beta Inc. $($referred to as B$).$ The end$-$of$-$month adjusted closing prices for the

last $4$ months are:

$(1)$ For each of the two stocks compute the expected return, variance $($of returns$),$ and

standard deviation $($of returns$).$ Also compute the covariance and correlation of

returns of the stocks.

$(2)$ Suppose that you are an asset manager helping rich private clients to invest their

money. Today you welcomed a new client who is very risk$-$averse. This client asked

you to form a portfolio of A and B that would have a standard deviation of returns

less than or equal to $7\%.$ Suppose that short sales are prohibited by your company's

policy. What is the closest portfolio $($in terms of risk$)$ that you can offer to your

client?

$(3)$ After a long discussion with the CEO, you managed to convince them to make

an exception and to allow short sales in order not to loose the client. The client

continues insisting on a portfolio with the risk as low as possible, disregarding the

expected return. What is the lowest standard deviation that you can offer to this

client?