Chapter 4 Risk and Return

Let \(A(0) = 100\) and \(A(1) = 110\) dollars, as before, but \(S(0) = 80\) dollars and \[ S(1) = \begin{cases} 100, &\text{with probability }0.8\\ 60, &\text{with probability }0.2 \end{cases} \] Suppose that you have $10, 000 to invest in a portfolio. You decide to buy \(x = 50\) shares, which fixes the risk-free investment at \(y = 60\). Then

\[ V(1) = \begin{cases} 11600, &\text{if stock goes up,}\\ 9600, &\text{if stock goes down,} \end{cases} \] \[ K_V = \begin{cases} 0.16, &\text{if stock goes up,}\\ -0.04, &\text{if stock goes down.} \end{cases} \]

The expected return, that is, the mathematical expectation of the return on the portfolio is

\[ E(K_V)=0.16\times 0.8-0.04\times 0.2 = 0.12, \]

that is, 12%. The risk of this investment is defined to be the standard deviation of the random variable \(K_V\) :

\[\sigma_V = \sqrt{(0.16-0.12)^2\times 0.8+(-0.04-0.12)^2\times 0.2} = 0.08\]

that is 8%. Let us compare this with investments in just one type of security. If \(x = 0\), then \(y = 100\), that is, the whole amount is invested risk-free. In this case the return is known with certainty to be \(K_A = 0.1\), that is, 10% and the risk as measured by the standard deviation is zero, \(\sigma_A = 0\).

On the other hand, if \(x = 125\) and \(y = 0\), the entire amount being invested in stock, then

\[ V(1) = \begin{cases} 12500, &\text{if stock goes up,}\\ 7500, &\text{if stock goes down.} \end{cases} \] and \(E(K_S) = 0.15\) with \(\sigma_S = 0.20\), that is, 15% and 20%, respectively.

Given the choice between two portfolios with the same expected return, any investor would obviously prefer that involving lower risk. Similarly, if the risk levels were the same, any investor would opt for higher return. However, in the case in hand higher return is associated with higher risk. In such circumstances the choice depends on individual preferences.