Abstract
Given the importance of strategic asset allocation in explaining the ex post performance of any type of investment portfolio, this chapter provides an in-depth analysis of asset allocation methods, illustrating the different theoretical and operational solutions available to institutional investors. This chapter focuses on the concepts and applications of traditional approaches to asset allocation, based on Mean-Variance Optimisation, with a focus on estimation risk and practical problems with Markowitz approach. This chapter also investigates the possible solutions to estimation errors: the additional weight constraints method and the resampling method, illustrating the application of resampling, the properties of resampled portfolios, and the discretionary choices in resampling. The chapter goes on with an in-depth analysis of a Bayesian strategic asset allocation method based on the Black-Litterman model, introducing the first information set of the model (equilibrium or implicit returns), the second information set (the views), the blending of information sets, and the application of the Black-Litterman model in practice.
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Notes
- 1.
A more extensive and detailed discussion of Markowitz’s contribution to the subject of portfolio construction can be found in the book he published some years later. See: Markowitz (1959).
- 2.
Markowitz (1952), p. 89.
- 3.
In his key study of 1952 Markowitz used variance as a measure of risk; in practice and in the subsequent literature, however, standard deviation was generally preferred. Since standard deviation is the square root of variance, it makes no difference which measure is used. Markowitz himself, in a later work, writes: “…although the article noted that the same portfolios that minimize standard deviation for given E [expected return] also minimize variance for given E [expected return], it failed to point out that standard deviation (rather than variance) is the intuitively meaningful measure of dispersion”, Markowitz (1999), p. 6.
- 4.
In other words, the numerical optimisation techniques work by generating a sequence of approximate solutions which move progressively closer to each other, converging on the true solution. As the latter is not known, and the sequence cannot continue indefinitely, these numerical techniques employ a termination criterion or convergence criterion which, when satisfied, interrupts the iterations.
- 5.
The calculation of the global minimum variance portfolio is the only one for which returns (μ i ) do not need to be taken into consideration, as in this case, the vector of the weights is totally dependent on the covariance matrix. If there is a budget constraint and a long-only constraint, the allocation is obtained by resolving the following optimisation problem:
$$ \begin{array}{l}\underset{\mathbf{w}}{ \min }\ \mathbf{w}\mathbf{\hbox{'}}\boldsymbol{\Sigma} \mathbf{w}\\ {}\kern1.75em \mathrm{subject}\ \mathrm{t}\mathrm{o}\\ {}\kern1.75em \mathbf{w}\mathbf{\hbox{'}}\mathbf{e}=1\\ {}\kern1.75em \left[w\right]\ge 0\end{array} $$For the sake of completeness (even if this is not particularly relevant to usual asset allocation activities), it should also be said that if the long-only constraint is removed, the problem could also be resolved directly as:
$$ {\mathbf{w}}_{GMVP}=\frac{{\boldsymbol{\Sigma}}^{-1}\mathbf{e}}{\mathbf{e}\mathbf{\hbox{'}}{\boldsymbol{\Sigma}}^{-1}\mathbf{e}} $$ - 6.
Note that, in this case, aggregation of risks over time uses the so-called square-root-of-time rule.
- 7.
- 8.
In its standardised version, the examined parameter is defined as follows: \( S=\operatorname{E}\left[{\left(\frac{x-\mu }{\sigma}\right)}^3\right] \). Its sample estimate can instead be calculated by the expression: \( \widehat{S}=\frac{1}{T}{\displaystyle \sum_{t=1}^T{\left(\frac{x_t-\widehat{\mu}}{\widehat{\sigma}}\right)}^3} \).
- 9.
- 10.
In its standardised version, kurtosis is defined as follows: \( K=\operatorname{E}\left[{\left(\frac{x-\mu }{\sigma}\right)}^4\right] \). Its sample estimate, on the other hand, can be calculated by the expression: \( \widehat{K}=\frac{1}{T}{\displaystyle \sum_{t=1}^T{\left(\frac{x_t-\widehat{\mu}}{\widehat{\sigma}}\right)}^4} \).
- 11.
For the sake of completeness, despite limited empirical evidence, we should also remember the case where the frequency of asset returns considerably above or below the mean is lower than that indicated by the “bell curve” and that distribution is consequently “thin-tailed”. In this case, such distribution would be called platykurtic and have a kurtosis value lower than 3.
- 12.
In empirical studies of the behaviour of asset class returns, as an alternative to kurtosis, the value for excess kurtosis can also be considered, i.e. kurtosis minus 3, which would be the normal value of kurtosis.
- 13.
Implicitly, an initial wealth equal to 1 was arbitrarily chosen.
- 14.
It is useful to recall that \( E\left[{\left(W-\overline{W}\right)}^2\right] \) is the variance and that the expected value of differences from the mean, i.e. \( E\left[\left(W-\overline{W}\right)\right] \), is zero.
- 15.
Sarnat (1974).
- 16.
Absolute risk aversion is measured by the ratio \( \raisebox{1ex}{$-{U}^2(W)$}\!\left/ \!\raisebox{-1ex}{${U}^1(W)$}\right. \). With decreasing absolute risk aversion, the first derivative of this ratio is lower than zero, while increasing absolute risk aversion (a condition considered to be contradictory and problematic) is confirmed by a first derivative greater than zero. This latter situation leads, in turn, to increasing relative risk aversion, i.e. a situation in which the proportion of wealth invested in risky assets falls as its absolute value rises.
- 17.
See Pratt (1964) for an analysis of risk-averse measures.
- 18.
Scott and Horwath (1980).
- 19.
- 20.
In support of this claim, note that if, for example, in an optimisation exercise with ten asset classes, the required number of unique (non-redundant) correlation/covariance parameters is 45, the required number of unique co-skewness and co-kurtosis parameters will be 220 and 715, respectively. These figures are obtained from the calculation of \( N\left(N+1\right)\left(N+2\right)/3! \) and of \( N\left(N+1\right)\left(N+2\right)\left(N+3\right)/4! \), respectively. For a more detailed examination of the estimation problems in strategic asset allocation when using moments higher than the second, see Jondeau and Rockinger (2006) and Martellini and Ziemann (2010).
- 21.
- 22.
Mossin (1968), p. 216.
- 23.
Mossin (1968), p. 220.
- 24.
Mossin (1968), p. 221.
- 25.
Even though, as we have said, the present study does not aim to offer a review of the attempts made to develop multiperiod asset allocation models, it is useful to point out that the latter have exploited stochastic programming, dynamic programming and robust optimisation techniques. A brief discussion of the methodology can be found in Chap. 10 of Fabozzi et al. (2007).
- 26.
Michaud (1989), p. 40.
- 27.
As an argument in the square root in (4.12), Σ indicates the determinant of the covariance matrix.
- 28.
- 29.
Michaud (1989), p. 33.
- 30.
In his well-known study “The Markowitz Optimisation Enigma: is “Optimized” Optimal?”,Michaud (1989) wrote (p. 31): “Given the success of the efficient frontier as a conceptual framework, and the availability for nearly 30 years of a procedure for computing efficient portfolios, it remains one of the outstanding puzzles of modern finance that MV optimisation has yet to meet with widespread acceptance by the investment community, particularly as a practical tool for active equity investment management. Does this Markowitz optimisation enigma reflect a considerable judgment (by the investment community) that such methods are not worthwhile, or is it merely another case of deep-seated resistance to change?”.
- 31.
Michaud (1989), p. 33.
- 32.
Implicitly, the claim implies the belief that asset diversification should be considered positively, not only as a way to manage financial risk, but also as a form of estimation risk management.
- 33.
Michaud (1989), p. 35.
- 34.
For a full discussion of statistically indifferent or equivalent portfolios, see Sect. 4.7.1.
- 35.
- 36.
Ceria and Stubbs (2006).
- 37.
To use a common expression in the financial community, it is a matter of “garbage in, garbage out”.
- 38.
On this point, Jobson and Korkie (1981, p. 72) write: “The number of historical observations of monthly returns required to give reasonably unbiased estimates of the optimal risk and return is at least two hundred” and (p. 72) “The traditional Markowitz procedure for predicting the optimal risk-return is extremely poor with conventional sample sizes of four to seven years”. Note that the frequent reference in the literature on asset allocation to monthly rather than daily or weekly sample observations is not accidental. The reason is that in the first case there is greater probability of finding a stationary (as well as normal) series than in the other two cases. On the topic, see also Jobson and Korkie (1980).
- 39.
Chopra and Ziemba (1993).
- 40.
Choosing a geographical criterion as a diversification driver implicitly sees factors such as monetary and fiscal policy, exchange-rate dynamics, the state of public finances, the evolution of GDP, and the institutional and political framework as features able to generate at one and the same time similarities between the securities belonging to a given geographic area/country and differences compared to those in other areas/countries. In contrast, distinction by sector sees a different set of factors, such as, for example, availability of certain raw materials, presence of technological innovation, demand for certain goods/products, as features able to create a certain level of similarity among the securities used in the same sector and, at the same time, heterogeneity with respect to those found in other sectors.
- 41.
From a logical and practical point of view, the alternative between hedging and non-hedging means, in simple terms, accepting an a priori forward premium or discount on the one hand, and an unknown profit or loss as a result of the exchange rate risk on the other. On this topic, see Chap. 16 and Schmittmann (2010).
- 42.
The Jarque-Bera test is based on the (JB) statistic, defined as follows: \( JB=T\left[\frac{\widehat{S}}{6}+\frac{{\left(\widehat{K}-3\right)}^2}{24}\right] \), which has an asymptotic chi-squared distribution with 2 degrees of freedom. As is well-known, the terms Ŝ and \( \widehat{K} \) are the sample values of skewness and kurtosis, which, in a normal distribution, are 0 and 3, respectively.
- 43.
Essentially, the Lilliefors test involves:
-
Estimating the mean and variance (or standard deviation) based on the data available;
-
Searching for the maximum discrepancy between the empirical distribution function and the cumulative distribution function (cdf) of the normal distribution, the mean and variance (or standard deviation) having been previously identified from the data;
-
Verifying whether or not this maximum discrepancy is big enough to be statistically significant and so reject the null hypothesis of a normal distribution.
-
- 44.
- 45.
Michaud (1989).
- 46.
Actually, the MSCI EMU is only found in one efficient portfolio (precisely, the one with minimum variance) with a very marginal exposure of 0.074 %.
- 47.
Amenc et al. (2011).
- 48.
Frost and Savarino (1988).
- 49.
For example, it might have tested various solutions aimed at balancing the wish for portfolio diversification with limited disadvantage and reduction in the proposed risk-return combinations.
- 50.
Jagannathan and Ma (2003).
- 51.
Eichhorn et al. (1998).
- 52.
In the example, this would be the case for MSCI Emu.
- 53.
Grauer and Shen (2000).
- 54.
The procedure described below forms the object of a patent from December 1999 (US patent 6,003,018) entitled “Portfolio Optimisation by Means of Resampled Efficient Frontier” granted to Richard and Robert Michaud. The licensing rights belong exclusively to New Frontier Advisors LLC, Boston, USA.
- 55.
Michaud and Michaud (2008).
- 56.
Michaud and Michaud (2008), p. 44.
- 57.
- 58.
- 59.
Michaud and Michaud (2008), p. 16.
- 60.
Markowitz (1987), p. 57. A close reading of Markowitz’s 1952 contribution reveals that, while he used sample estimates as inputs, he also suggested that different methods should be used to form the inputs: “My feeling is that the statistical computations should be used to arrive at a tentative set of μ i and σ ij . Judgment should then be used in increasing or decreasing some of these μ i and σ ij on the basis of factors or nuances not taken into account by the formal computations”. He added: “One suggestion as to tentative μ i , σ ij is to use observed μ i , σ ij for some period of the past. I believe that better methods, which take into account more information, can be found. I believe that what is needed is essentially a “probabilistic” reformulation of security analysis. I will not pursue this subject here, for this is another story”. Markowitz (1952), p. 91.
- 61.
- 62.
- 63.
- 64.
Indeed, implied equilibrium risk premium (and return) are the returns that, in a given instance, provide a balance between the supply and demand of the asset.
- 65.
In algebraic terms, according to this model, for a generic asset class i among the N considered can be written as:
$$ {R}_i={R}_f+{\beta}_i\left({\overline{R}}_{P\kern-0.15em \_mkt}-{R}_f\right) $$which, considering the definition of β i , can be re-written as:
$$ {R}_i={R}_f+\frac{\operatorname{cov}\left({R}_i;{R}_{P\_mkt}\right)}{\sigma_{P\kern-0.15em \_mkt}^2}\left({\overline{R}}_{P\kern-0.15em \_mkt}-{R}_f\right) $$Algebraically, the latter is identical to:
$$ {R}_i={R}_f+\frac{\left({\overline{R}}_{P\kern-0.15em \_mkt}-{R}_f\right)}{\sigma_{P\kern-0.15em \_mkt}^2}\operatorname{cov}\left({R}_i;{R}_{P\kern-0.15em \_mkt}\right) $$The re-writing extended to N asset classes in matrix form coincides with (4.34):
$$ {\boldsymbol{\Pi}}_{Eq.}={R}_f\mathbf{e}+\frac{\left({\overline{R}}_{P\kern-0.15em \_mkt}-{R}_f\right)}{\sigma_{P\kern-0.15em \_mkt}^2}\boldsymbol{\Sigma} {\mathbf{w}}_{Eq.} $$ - 66.
- 67.
- 68.
Adoption of a departure point for reverse optimisation other than the market portfolio is usual in cases where the Black-Litterman model is used to support tactical rather than strategic asset allocation. On tactical asset allocation, which is not the object of this work, see: Lee (2000) and Braga and Natale (2012).
- 69.
Bevan and Winkelmann (1998).
- 70.
See Sect. 4.8.3 below.
- 71.
- 72.
- 73.
He and Litterman (1999), p. 6.
- 74.
Idzorek (2002).
- 75.
For the sake of completeness only, we note that in the definition of matrix Ω, the author of this approach follows the logic in (4.4.7) and suggests a way to identify τ. This should result from the ratio between the variance of the view portfolio and the quantity \( \frac{{\displaystyle \sum_{k=1}^K\left(\frac{1}{\alpha_k}CF\right)}}{K} \) which is equal to the average value of the diagonal elements in Ω.
- 76.
Meucci (2005).
- 77.
In support of this, note that the objective function in (4.49) envisages the minimisation of the distance (the so-called Mahalanobis distance) between two information sets.
- 78.
Idzorek (2002, p. 10) underlines this aspect of the Black-Litterman model, observing that: “…it is not uncommon for a single view to cause the return of every asset in the portfolio to change from its Implied Equilibrium return, since each individual return is linked to the other returns via the covariance matrix of returns (Σ)”.
- 79.
Lee (2000), p. 130.
- 80.
He and Litterman (1999).
- 81.
Idzorek (2002).
- 82.
- 83.
Meucci (2005).
- 84.
On the issue of the combination between the heuristic and the Bayesian methods, see Barros Fernandes et al. (2012).
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Braga, M.D. (2016). Strategic Asset Allocation with Mean-Variance Optimisation. In: Basile, I., Ferrari, P. (eds) Asset Management and Institutional Investors. Springer, Cham. https://doi.org/10.1007/978-3-319-32796-9_4
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