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Erschienen in:
The Real Estate Investment Market: The Current State and Why Advances Are Needed Kapitel lesen Erstes Kapitel lesen

2022 | OriginalPaper | Buchkapitel

3. Modern Portfolio Theory

verfasst von : W. Brent Lindquist, Svetlozar T. Rachev, Yuan Hu, Abootaleb Shirvani

Erschienen in: Advanced REIT Portfolio Optimization

Verlag: Springer International Publishing

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Abstract

The basic elements of modern portfolio theory are covered in this Chapter. Starting from the basics of price return time series, the authors introduce Markowitz’s mean variance optimization and the central concept of the efficient frontier. Extensions to other risk measure optimization methods within the portfolio theory framework are covered, including: tangent portfolio optimization which exploits the relationship between the efficient frontier and the capital market line; minimization of the conditional value-at-risk, a tail-risk measure replacing the variance; and the Black–Litterman model, designed to address issues appearing in mean variance optimization. The classical implementation of these optimization techniques using moving windows of historical asset return data is developed.

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Vorheriges Kapitel The Data
Nächstes Kapitel Historical Portfolio Optimization: Domestic REITs
Fußnoten
2
See, for example, Chap. 2 of Tsay (2010).
 
3
By default, a vector v is assumed to be a column vector, whereas its transpose, written vT, is a row vector. Thus, the column vector, v, having elements v1, …, vn can be written v = (v1, …, vn)T, and the row vector vT can be written vT = (v1, …, vn).
 
4
“Daily” refers only to trading days.
 
5
These are assumed to be simple return values, ri(t) = (Pi(t) − Pi(t − 1))/Pi(t − 1), where Pi(t) is the daily closing prices of asset i.
 
6
The weights are applied at the beginning of trading day t and are assumed not to change during the trading day.
 
7
en = (1, …, 1)T.
 
8
Because L(w, q, θ0) is quadratic in w with a positive coefficient on the quadratic term, the optimizing condition is equivalent to a minimizing one.
 
9
Specifically, wlb,i ≤ wi ≤ wub,i,  i = 1, …, n.
 
10
They are the vectors \( {\boldsymbol{\theta}}_1^T=\left({\theta}_{1,1},\dots, {\theta}_{1,n}\ \right) \), \( {\boldsymbol{\theta}}_2^T=\left({\theta}_{2,1},\dots, {\theta}_{2,n}\ \right) \).
 
11
Some constraints may not, if fact, affect the minimizing solution. Standard techniques recognize when this occurs and eliminate such constraints from consideration.
 
12
For a long-only strategy, it is sufficient to require 0 ≤ wi,  i = 1, …, , n and \( {\boldsymbol{e}}_n^T\boldsymbol{w}=1 \).
 
13
It takes some computation to show that (3.20) satisfies (3.12a)—that is, that (σm, \( {\overline{r}}_m \)) is on the efficient frontier and that the slope of the efficient frontier determined by Eq. (3.12a) at (σm, \( {\overline{r}}_m \)) is indeed the slope of the CAPM line.
 
14
The Basel III regulatory framework for banks requires CVaR as the risk measure.
 
15
Recall that VaR is defined as positive for losses.
 
16
In addition, CVaR satisfies the desirable attributes of a coherent risk measure and is consistent with performance relations of risk-averse investors (see Pflug, 2000).
 
17
It is common in the literature to define f(w, r) alternatively as the loss probability function, whose samples provide negative return values.
 
18
The returns in the set S are negative.
 
Literatur
Best, M., & Grauer, R. (1991). On the sensitivity of mean-variance-efficient portfolios to changes in asset means: Some analytical and computational results. Review of Financial Studies, 4(2), 315–342.CrossRef
Black, F., & Litterman, R. (1991). Asset allocation combining investor views with market equilibrium. Journal of Fixed Income, 1(2), 7–18.CrossRef
Bloch, M., Guerard, J. B., Markowitz, H., Todd, P., & Xu, G. (1993). A comparison of some aspects of the U.S. and Japanese equity markets. Japan and the World Economy, 5, 3–26.CrossRef
He, G., & Litterman, R. (1999). The intuition behind Black-Litterman model portfolios. Investment Management Research.
Kolm, P. N., & Ritter, G. (2017). On the Bayesian interpretation of Black-Litterman. European Journal of Operational Research, 258(2), 564–572.CrossRef
Krokhmal, P., Palmquist, J., & Uryasev, S. (2002). Portfolio optimization with conditional value-at-risk objective and constraints. Journal of Risk, 4(2), 11–27.
Lee, W. (2000). Advanced theory and methodology of tactical asset allocation. Wiley.
Markowitz, H. (1952). Portfolio selection*. Journal of Finance, 7(1), 77–91.
Michaud, R. (1989). The Markowitz optimization enigma: Is optimized optimal? Financial Analysts Journal, 45(1), 31–42.CrossRef
Pflug, G. C. (2000). Some remarks on the value-at-risk and the conditional value-at-risk. In S. P. Uryasev (Ed.), Probabilistic constrained optimization: Methodology and applications (pp. 272–281). Springer.CrossRef
Rockafellar, R. T., & Uryasev, S. (2000). Optimization of conditional value-at-risk. Journal of Risk, 2(3), 21–41.CrossRef
Rockafellar, R. T., & Uryasev, S. (2002). Conditional value-at-risk for general loss distributions. Journal of Banking & Finance, 26, 1443–1471.CrossRef
Sharpe, W. F. (1994). The Sharpe ratio. Journal of Portfolio Management, 21, 49–58.CrossRef
Tsay, R. S. (2010). Analysis of financial time series. John Wiley & Sons.CrossRef
Tütüncü, R. H., Toh, K. C., & Todd, M. J. (2003). Solving semidefinite-quadratic-linear programs using SDPT3. Mathematical Programming, 95(2), 189–217.CrossRef
Metadaten
Titel
Modern Portfolio Theory
verfasst von
W. Brent Lindquist
Svetlozar T. Rachev
Yuan Hu
Abootaleb Shirvani
Copyright-Jahr
2022
Buch
Advanced REIT Portfolio Optimization

Print ISBN: 978-3-031-15285-6

Electronic ISBN: 978-3-031-15286-3

Copyright-Jahr: 2022

DOI
https://doi.org/10.1007/978-3-031-15286-3_3