Portfolio Optimization
Modern or mean-variance portfolio theory (MPT) is a major cornerstone of financial
theory. Based on this theoretical breakthrough the Nobel Prize in Economics was awarded
to its inventor, Harry Markowitz, in 1990. Although formulated in the 1950s,
[ 41 ]
it is still a
theory taught to finance students and applied in practice today (often with some minor or
major modifications). This section illustrates the fundamental principles of the theory.
Chapter 5 in the book by Copeland, Weston, and Shastri (2005) provides a good
introduction to the formal topics associated with MPT. As pointed out previously, the
assumption of normally distributed returns is fundamental to the theory:
By looking only at mean and variance, we are necessarily assuming that no other statistics are necessary to
describe the distribution of end-of-period wealth. Unless investors have a special type of utility function
(quadratic utility function), it is necessary to assume that returns have a normal distribution, which can be
completely described by mean and variance.
The Data
Let us begin our Python session by importing a couple of by now well-known libraries:
In [ 33 ]: import numpy as np
import pandas as pd
import pandas.io.data as web
import matplotlib.pyplot as plt
%matplotlib inline
We pick five different assets for the analysis: American tech stocks Apple Inc., Yahoo!
Inc., and Microsoft Inc., as well as German Deutsche Bank AG and gold as a commodity
via an exchange-traded fund (ETF). The basic idea of MPT is diversification to achieve a
minimal portfolio risk or maximal portfolio returns given a certain level of risk. One
would expect such results for the right combination of a large enough number of assets
and a certain diversity in the assets. However, to convey the basic ideas and to show
typical effects, these five assets shall suffice:
In [ 34 ]: symbols = [‘AAPL’, ‘MSFT’, ‘YHOO’, ‘DB’, ‘GLD’]
noa = len(symbols)
Using the DataReader function of pandas (cf. Chapter 6) makes getting the time series
data rather efficient. We are only interested, as in the previous example, in the Close
prices of each stock:
In [ 35 ]: data = pd.DataFrame()
for sym in symbols:
data[sym] = web.DataReader(sym, data_source=‘yahoo’,
end=‘2014-09-12’)[‘Adj Close’]
data.columns = symbols
Figure 11-11 shows the time series data in normalized fashion graphically:
In [ 36 ]: (data / data.ix[ 0 ] * 100 ).plot(figsize=( 8 , 5 ))
