23
Sep 18

## Portfolio analysis: return on portfolio

Exercise 1. Suppose a portfolio contains $n_1$ shares of stock 1 whose price is $S_1$ and $n_2$ shares of stock 2 whose price is $S_2$. Stock prices fluctuate and are random variables. Numbers of shares are assumed fixed and are deterministic. What is the expected value of the portfolio?

Solution. The portfolio value is its market price $V=n_1S_1+n_2S_2$. Since this is a linear combination, the expected value is $EV=n_1ES_1+n_2ES_2$.

In fact, the portfolio analysis is a little bit different than suggested by Exercise 1. To explain the difference, we start with fixing two points of view.

View 1. I hold a portfolio of stocks. I may have inherited it, and it does not matter how much it cost at the moment it was formed. If I want to sell it, I am interested in knowing its market value. In this situation the numbers of shares in my portfolio, which are constant, and the market prices of stocks, which are random, determine the market value of the portfolio, defined in Exercise 1. The value of the portfolio is a linear combination of stock prices.

View 2. I have a certain amount of money $M^0$ to invest. Being a gambler, I am not interested in holding a portfolio forever. I am thinking about buying a portfolio of stocks now and selling it, say, in a year at price $M^1$. In this case I am interested in the rate of return defined by $r=\frac{M^1-M^0}{M^0}.$ $M^0$ is considered deterministic (current prices are certain) and $M^1$ is random (future prices are unpredictable). Thus the rate of return is random.

We pursue the second view (prevalent in finance). As it often happens in economics and finance, the result depends on how one understands the things. Suppose the initial amount $M^0$ is invested in $n$ assets. Denoting $M_i^0$ the amount invested in asset $i$, we have $M^0=\sum\limits_{i = 1}^nM_i^0$. Denoting $s_i=M_i^0/{M^0}$ the share (percentage) of $M_i^0$ in the total investment $M^0$, we have

(1) $M_i^0=s_iM^0,\ M^0=\sum\limits_{i = 1}^ns_iM^0.$

The initial shares $s_i$ are deterministic.

Let $M_i^1$ be what becomes of $M_i^0$ in one year and let $M^1=\sum\limits_{i = 1}^nM_i^1$ be the total value of the investment at the end of the year. Since different assets grow at different rates, generally it is not true that $M_i^1 =s_iM^1$. Denote $r_i=\frac{M_i^1-M_i^0}{M_i^0}$ the rate of return on asset $i$. Then

(2) $M_i^1=(1+r_i)M_i^0,$ $M^1=\sum\limits_{i = 1}^n(1+r_i)M_i^0.$

Exercise 2. The rate of return on the portfolio is a linear combination of the rates of return on separate assets, the coefficients being the initial shares of investment.

Solution. Using Equations (1) and (2) we get

(3) $r=\frac{M^1-M^0}{M^0}=\frac{\sum(1+r_i)M_i^0-\sum M_i^0}{M^0}=\frac{\sum r_iM_i^0}{M^0}=\frac{\sum r_is_iM^0}{M^0}=\sum s_ir_i .$

Once you know this equation you can find the mean and variance of the rate of return on the portfolio in terms of investment shares and rates of return on assets.