Python for Finance

Portfolio Theory
Introduction to portfolio theory
The keyword for the portfolio theory is diversification, while the keyword for
diversification is correlation. In other words, correlation is used to measure how
closely two stocks or portfolios are moving together. The objective of portfolio
theory is to allocate our assets optimally with respect to risk and return. Markowitz
(1952) argues that we should consider only the first two moments of a security's
return distribution: mean and variance. For financial markets, several important
assumptions are made, such as stock markets are inefficient, a typical investor is
rational, and an arbitrage opportunity would not last long. For the preferences
between two stocks, for a given risk, a rational investor would prefer stock with
a higher expected return; for a given return, a rational investor prefers stock with
a lower risk level. Sometimes, a single period portfolio optimization is called
Markowitz Portfolio Optimization. The input includes a return matrix, and a variance
and covariance matrix, while the output is an efficient portfolio. By connecting
numerous efficient portfolios, an efficient frontier is formed. Here, we start with the
simplest scenario: a two-stock portfolio.
A 2-stock portfolio
Clearly, a 2-stock portfolio is the simplest one. Let's assume that the weights of those
two stocks are w1 and w2. The portfolio returns are given here:
Here, Rp,t, is the portfolio return at time t, w1 (w2) is the weight for stock 1 (2), and
R1,t (R2,t) is return at time t for stock 1 (2). When talking about expected return or
mean, we have a quite similar formula:
Here, is the mean or expected portfolio returns and is the mean or
expected returns for stock 1 (2). The variance of such a 2-stock portfolio is given here:
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