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 $15\%$.