Raising the bar

Chapter 5. Discrete Random Variables and Probability Distributions

Unit 5.17. Portfolio analysis

In Linearity we already defined the portfolio value, and that definition corresponds to what NCT has. In fact, the portfolio analysis is a little bit different. To explain the difference, we start with fixing two points of 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 Linearity. The value of the portfolio is a linear combination of stock prices.
2. I have a certain amount of money $latex {M^0}$ to invest. Being a gambler, I am not interested in holding a portfolio. I am thinking about buying a portfolio of stocks now and selling it, say, in a year at price $latex {M^1}$. In this case I am interested in the rate of return defined by

$latex r = \frac{M^1 - M^0}{M^0} = \frac{M^1}{M^0} - 1.$

$latex {M^0}$ is considered deterministic (current prices are certain) and $latex {M^1}$ is random (future prices are unpredictable). The rate of return thus is random.

Main result. The rate of return on the portfolio is a linear combination of the rates of return on separate assets.

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

$latex {M^0} = \sum\limits_{i = 1}^n {{s_i}{M^0}} ,\,\,M_i^0 = {s_i}{M^0}. \qquad (\textrm{Eq. 5.20})$

The shares $latex {s_i}$ are deterministic.

Denote $latex M_i^1$ what becomes of $latex M_i^0$ in one year and by $latex {M^1} = \sum\limits_{i = 1}^n {M_i^1}$ 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 $latex M_i^1 = {s_i}{M^1}$. Denote

$latex {r_i} = \frac{{M_i^1 - M_i^0}}{{M_i^0}} \qquad (\textrm{Eq. 5.21})$

the rate of return on asset $latex i$. Then using Eq. (5.20) and Eq. (5.21) we get

$latex r = \frac{{{M^1} - {M^0}}}{{{M^0}}} = \frac{{\sum {(1 + {r_i})M_i^0 - \sum {M_i^0} } }}{{{M^0}}} = \frac{{\sum {{r_i}M_i^0} }}{{{M^0}}} = \frac{{\sum {{r_i}{s_i}{M^0}} }}{{{M^0}}} = \sum {{s_i}{r_i}}.$

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

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