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 $$15\%$$.