Abstract
Insurers, banks, mutual funds, sovereign wealth funds and also individuals invest money in the financial markets in order to generate financial returns. Hereby the investor allocates capital to different investments, such as low-risk low-expected return investments (e.g. high quality government bonds, bank accounts) or higher-risk higher-expected return investments (e.g. stocks, real estate, commodities). It is one of the core problems in finance to provide decision making tools for the optimal (or: efficient) allocation of capital. Optimality in this context depends on the decision maker’s liquidity needs and risk aversion. This chapter will introduce the classical mean-variance optimization framework in a static one-period setup, and proceed to continuous-time portfolio optimization problems.
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Notes
- 1.
In this chapter we will use a fair amount of vector/matrix notation to keep things compact. Vectors are printed in bold (e.g. a). a ′ is the transposed vector of a (e.g. if a has dimensions n ×1 (a ‘column’ vector), then \({\mathbf{a}}^{\boldsymbol{{\prime}}}\) will be the corresponding 1 ×n matrix (a ‘row’) with a i, 1 = a′ 1, i (1 ≤ i ≤ n)). For n ×1 dimensional vectors a and b, we have \(\mathbf{a}\prime\mathbf{b} =\sum _{ i=1}^{n}a_{i}b_{i}\) (the so-called scalar product of a and b). We define 1 = (1, 1, . . . , 1)′ so that a ′ 1 = ∑ i a i is simply the sum of the elements of vector a. The inverse matrix of A (dimension n ×n) is denoted by A − 1 and we have \({A}^{-1}A = A{A}^{-1} = I_{n}\), with I n being the n ×n identity matrix (i.e. ‘1’s in the main diagonal, and ‘0’ entries otherwise).
- 2.
Harry Markowitz (1927–) was awarded the 1990 Nobel Memorial Prize in Economic Sciences, together with Merton H. Miller (1923–2000) and William F. Sharpe (1934–), ‘for his pioneering work in the theory of financial economics’.
- 3.
This formulation is equivalent to \(\max _{\mathbf{a}}\!\!\left (\mathbb{E}[W(T)] -\mathrm{Var}[W(T)]\right )\). In the sequel, transaction costs and taxes are neglected.
- 4.
The mean and the variance parameters capture all features of the return distribution in the multi-variate normal case, however, this will not necessarily be the case for other distribution classes (e.g. which are skewed or have relatively more tail mass).
- 5.
Whoever is not used to matrix algebra can also understand the below formulas componentwise, e.g. the first vector component of \(\partial L/\partial \boldsymbol{\theta }\) is simply ∂L ∕ ∂θ 1, etc. Obtaining the solutions to the mean-variance problem in matrix form will greatly facilitate the implementation in computer programs for larger n.
- 6.
The covariance matrix is calculated in this case as \(\Sigma =\boldsymbol{\sigma } \prime\mathbf{I}_{3}(\rho _{ij})\mathbf{I}_{3}\sigma\), where I 3 is the 3 ×3 identity matrix and (ρ ij ) is the correlation matrix.
- 7.
This extension of the initial Markowitz formulation of the problem for only risky assets was suggested by James Tobin (1918–2002), who was awarded the Nobel Memorial Prize in Economic Sciences in 1981 for his contributions in the field of portfolio theory.
- 8.
The Sharpe ratio is a popular measure to compare investments, as it divides the mean return in excess of the risk-free rate (μ − r f, also: risk premium) by a number related to the involved risk (σ).
- 9.
The CAPM was initially developed by William Sharpe (1964), see the footnote on page 156, and John V. Lintner (1965).
- 10.
In general, the efficient frontier with the long-only constraint will run below the unconstrained efficient frontier.
- 11.
Positive deviations (excess profits) from the mean return are equally penalized as negative deviations. In connection to this, the idea of using a risk measure that only punishes for negative deviations from the mean return already goes back to Markowitz.
- 12.
For example, both the Basel II and Solvency II accords define value-at-risk as a standard risk measure.
- 13.
John von Neumann (1903–1957) and Oskar Morgenstern (1902–1977) are also often credited for having built the foundation of game theory.
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Albrecher, H., Binder, A., Lautscham, V., Mayer, P. (2013). Portfolio Optimization. In: Introduction to Quantitative Methods for Financial Markets. Compact Textbooks in Mathematics. Birkhäuser, Basel. https://doi.org/10.1007/978-3-0348-0519-3_14
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