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Published in:
Preliminaries on Financial Markets Read chapter Read first chapter

2016 | OriginalPaper | Chapter

3. Markowitz Portfolio Theory

Authors : Arlie O. Petters, Xiaoying Dong

Published in: An Introduction to Mathematical Finance with Applications

Publisher: Springer New York

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Abstract

We introduce Harry Markowitz’s mathematical model for how to distribute an initial capital across a collection of risky securities to create an efficient portfolio, namely, one with the least risk given an expected return and largest expected return given a level of portfolio risk. This chapter covers: the set up of the Markowitz portfolio model, which includes modeling security returns, the issue of multivariate normality, weights, short selling, portfolio return, portfolio risk, and portfolio log returns; two-security portfolio theory; the efficient frontier for N securities with and without short selling; the global minimum-variance portfolio, diversified portfolio, and Mutual Fund Theorem; utility functions and utility maximization; and diversification.

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previous chapter The Time Value of Money
next chapter Capital Market Theory and Portfolio Risk Measures
Footnotes
1
Harry Markowitz, Merton Miller, and William F. Sharpe shared the 1990 Nobel Prize in Economic Sciences. Markowitz won for his work on portfolio selection (see Press Release at Novelprize.org).
 
2
No lending or borrowing of money will be done in the current chapter, but it will be part of the modeling in Chapter 4, which generalizes the Markowitz model.
 
3
The theoretical importance of the positive definite property will be seen in Section 3.3.
 
4
Note that the returns are notationally the reverse of the case for future times: for past times t′ j  < t′ j−1, we use \(\widehat{R}(t'_{j},t'_{j-1})\) instead of R(t j−1, t j ), which is for future times t j−1 < t j .
 
5
For example, see the Correlation Tracker at: http://​www.​sectorspdr.​com/​correlation.
 
6
Recall from Chapter 2 that interest rate is per annum, unless otherwise stated.
 
7
This issue pertains to the Weak Efficient Market Hypothesis.
 
8
If \(B\,\boldsymbol{\mu } - C\,\boldsymbol{e} =\boldsymbol{ 0}\), then the linear independence of \(\boldsymbol{\mu }\) and \(\boldsymbol{e}\) implies \(B = C = 0\). This contradicts C > 0. Hence, \(B\,\boldsymbol{\mu } - C\,\boldsymbol{e}\neq \boldsymbol{0}\).
 
9
See (3.5) on page 89.
 
10
We used the free online Correlation Tracker tool at (www.​sectorspdr.​com) and Stock Correlation Calculator at Buyupside (www.​buyupside.​com/​calculators).
 
11
Note that the ticker symbol of American Airlines at the time was AAMRQ, which changed after the merger with US Airways.
 
12
Do not attempt to show the equivalence via the Lagrange conditions (i.e., (3.64) to (3.66)). Simply use the statement of the problems and logically imply one from the other.
 
13
For optimization problems with inequality constraints, the Karush-Kuhn-Tucker Theorem is employed.
 
14
A strictly concave function f on an interval I satisfies \(f(x\,a + (1 - x)\,b)> x\,f(a) + (1 - x)\,f(b)\) for all 0 < x < 1, and a, b in I. In the example, the strict inequality follows using f = u with \(x\,a + (1 - x)\,b = \mathbb{E}(R_{B})\), where \(x = \mathbb{P}\big(R_{B} = 0.05\big)\), \(1 - x = \mathbb{P}\big(R_{B} = 0.15\big)\), a = 0. 05, and b = 0. 15.
 
15
See (2.​28) on page 31.
 
16
A strictly convex function f on an interval I satisfies \(f(x\,a + (1 - x)\,b) <x\,f(a) + (1 - x)\,f(b)\) for all 0 < x < 1 and a, b in I. In the example, the strict inequality follows using f = u with \(x\,a + (1 - x)\,b = \mathbb{E}(R_{B})\), where \(x = \mathbb{P}\big(R_{B} = 0.05\big)\), \(1 - x = \mathbb{P}\big(R_{B} = 0.15\big)\), a = 0. 05, and b = 0. 15. Also, see Jensen’s inequality.
 
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Metadata
Title
Markowitz Portfolio Theory
Authors
Arlie O. Petters
Xiaoying Dong
Copyright Year
2016
Publisher
Springer New York
Book
An Introduction to Mathematical Finance with Applications

Print ISBN: 978-1-4939-3781-3

Electronic ISBN: 978-1-4939-3783-7

Copyright Year: 2016

DOI
https://doi.org/10.1007/978-1-4939-3783-7_3

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