2.4 Application to portfolio management

  • The portfolio theory is a risk management framework through the concept of diversification

  • When investing, the theory attempts to maximize portfolio expected return or minimize portfolio risk for a given level of expected return by choosing the proportions of various assets

  • The risk of any asset is commonly measured as the variance of returns, also known as the volatility

  • For simplicity we will consider only the case of two risky assets or two stocks \(X\) and \(Y\)

  • The return is usually understood as the price change of an asset in the current period against previous period, and calculated as the first difference of the natural logs

\[\begin{align} r_t&=\bigg(log(p_t)-log(p_{t-1})\bigg)100\%=log\bigg(\frac{p_t}{p_{t-1}}\bigg)100\% \\ r_t&= \text{percenatge change of the price of an asset at time }t \\ \\ p_t&= \text{the price of an asset at time }t \\ \\ p_{t_1}&= \text{the price of an asset at time }t-1 \\ \tag{2.8} \end{align}\]

  • We introduce the following notations, assuming that portfolio \(Z\) is a nonempty set, and consists of \(w\) portion of stock \(X\) with returns \(x_1\), \(x_2\),…, and (\(1-w\)) portion of the stock \(Y\) with returns \(y_1\), \(y_2\),…

\[\begin{align} Z&=wX+(1-w)Y \\ \\ E(Z)&=wE(X)+(1-w)E(Y) \\ \\ Var(Z)&=w^2 Var(X)+(1-w)^2 Var(Y)+2w(1-w)Cov(X,Y) \\ \\ Cov(X,Y)&=E(XY)-E(X)E(Y) \\ \\ \rho_{X,Y}&=\frac{Cov(X,Y)}{\sqrt{Var(X)Var(Y)}} \\ \tag{2.9} \end{align}\]

Example 2.5 The joint probability distribution of returns for two stocks \(X\) and \(Y\) is presented by contingency table 2.3.

TABLE 2.3: Portfolio consisting of two stocks
Returns on \(X/Y\) 10% \((y_1)\) 20% \((y_2)\) \(p(x_i)\)
2% \((x_1)\) 0.1 0.2 0.3
5% \((x_2)\) 0.2 0.3 0.5
8% \((x_3)\) 0.1 0.1 0.2
\(p(y_j)\) 0.4 0.6 1
  1. Calculate expected return and risk for both stocks \(X\) and \(Y\), i.e. \(E(X)\), \(Var(X)\), \(E(Y)\) and \(Var(Y)\).

  2. Which stock is more risky?

  3. Calculate the covariance of returns between two stocks \(Cov(X,Y)\) and indicate if returns of two stocks are independently distributed.

  4. Compute and explain the value of correlation coefficient \(\rho_{X,Y}\).

  5. What is the expected return of portfolio which consists of \(20\%\) of \(X\) stocks and \(80\%\) of \(Y\) stocks? Compute the risk of a such portfolio.

  6. Determine the portions of two stocks if we would like expected return of portfolio to be \(15\%\).