the traditional approach to asset allocation

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Chapter 2 The Traditional Approach to Asset Allocation Abstract After presenting the preliminary denitions and statistics that are necessary to correctly formulate portfolio risk and return, the chapter illustrates the appropriate way of constructing portfolios according to the Markowitz model, also named Mean-Variance Optimization. Its application requires optimization inputs (expected returns, risks, correlations) that are not observable ex-ante, thus they have to be estimated usually from past data. Then the standard implementation of Markowitzs model follows the plug-inrule. In practice, the estimated parameters are processed into the mean-variance optimizer that treats them as if they were the true parameters. This renders the portfolio construction process deterministic as the parameters uncertainty is completely neglected. This chapter gives evidence and clarication of the different undesirable features and deciencies of the optimized portfolios (counter-intuitive nature, instability, erroneously supposed uniqueness and poor out-of-sample performance) triggered by implementing the Mean- Variance Optimization without recognizing the existence of estimation risk inherent in the input parameters. Amongst other, estimation errors in expected returns are the most crucial and costly ones. Keywords Asset allocation Bayesian approaches Ef cient frontier Estimation risk Heuristic approaches Mean-Variance Optimization Markowitz model Parameters uncertainty Plug-in rule Portfolio instability 2.1 A Short Review of the Traditional Markowitz Model for Asset Allocation Markowitzs seminal works (1952, 1959) provide the standard model to solve asset allocation problems. It is called Mean-Variance Optimization or Mean- Variance Analysis and is generally regarded as the cornerstone of Modern Portfolio Theory. Its foundations consist of the two following main ideas that have become central in nance literature and practice: © The Author(s) 2016 M.D. Braga, Risk-Based Approaches to Asset Allocation, SpringerBriefs in Finance, DOI 10.1007/978-3-319-24382-5_2 3

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Chapter 2The Traditional Approach to AssetAllocation

Abstract After presenting the preliminary definitions and statistics that arenecessary to correctly formulate portfolio risk and return, the chapter illustrates theappropriate way of constructing portfolios according to the Markowitz model, alsonamed Mean-Variance Optimization. Its application requires optimization inputs(expected returns, risks, correlations) that are not observable ex-ante, thus they haveto be estimated usually from past data. Then the standard implementation ofMarkowitz’s model follows the “plug-in” rule. In practice, the estimated parametersare processed into the mean-variance optimizer that treats them as if they were thetrue parameters. This renders the portfolio construction process deterministic as theparameters uncertainty is completely neglected. This chapter gives evidence andclarification of the different undesirable features and deficiencies of the optimizedportfolios (counter-intuitive nature, instability, erroneously supposed uniquenessand poor out-of-sample performance) triggered by implementing the Mean-Variance Optimization without recognizing the existence of estimation risk inherentin the input parameters. Amongst other, estimation errors in expected returns are themost crucial and costly ones.

Keywords Asset allocation � Bayesian approaches � Efficient frontier � Estimationrisk � Heuristic approaches � Mean-Variance Optimization � Markowitz model �Parameters uncertainty � Plug-in rule � Portfolio instability

2.1 A Short Review of the Traditional Markowitz Modelfor Asset Allocation

Markowitz’s seminal works (1952, 1959) provide the standard model to solveasset allocation problems. It is called Mean-Variance Optimization or Mean-Variance Analysis and is generally regarded as the cornerstone of Modern PortfolioTheory. Its foundations consist of the two following main ideas that have becomecentral in finance literature and practice:

© The Author(s) 2016M.D. Braga, Risk-Based Approaches to Asset Allocation,SpringerBriefs in Finance, DOI 10.1007/978-3-319-24382-5_2

3

• diversification is the ideal way of managing risk. Markowitz proves formallythat the benefits of diversification depend on correlations/covariances amongasset classes and not solely on their different stand alone risk. This, of course,also means that diversification is not only affected by the number of the selectedasset classes in a portfolio;

• investors act (or should act), when taking decisions to allocate wealth, in a twodimensional space. Markowitz wrote (1952): “the investor does (or should)consider expected return a desirable thing and variance of return an undesirablething”. This is a novel idea and obviously not consistent with the choice ofselecting the portfolio with the maximum expected return that was previouslyrecommended to his contributions.

Based on the second idea, the so called mean-variance criteria or rule is for-mulated: let A and B be two different portfolios, denote with μA and μB theirexpected return and with σA and σB their expected risk, then A is dominating B ormust be preferred to B if μA ≥ μB and σA ≤ σB with at least one strong inequalitysatisfied.

Before illustrating the appropriate way of constructing portfolios according tothe Markowitz model, we present the preliminary definitions and statistics that arenecessary to correctly formulate the return and risk of a portfolio.

The portfolio weights are given in an N × 1 vector (w) where wi is the percentageholding of asset class i. Note that the following condition must be satisfied:PNi¼1

wi ¼ 1.

w ¼

w1

w2

. . .

. . .

wi

. . .

. . .

wN

2666666666666664

3777777777777775

We then denote with μ and σ, respectively, the vectors N × 1 of the expectedreturns and standard deviations for the selected asset classes in the investmentuniverse, where μi is the return of asset class i and σi is its risk.

4 2 The Traditional Approach to Asset Allocation

l ¼

l1l2. . .

. . .

li. . .

. . .

lN

2666666666666664

3777777777777775

r ¼

r1r2. . .

. . .

ri. . .

. . .

rN

2666666666666664

3777777777777775

Finally, we summarize the relevant correlation coefficients in the N × N matrixC. We denote with Σ the corresponding covariance matrix. Obviously ρij and σij arethe generic correlation and covariance parameters between asset class i and j suchthat σij = ρijσiσj and σii = σi

2.

C ¼

q11 q12 . . . . . . q1i . . . . . . q1Nq21 q22 . . . . . . q2i . . . . . . q2N. . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . .qi1 qi2 . . . . . . qii . . . . . . qiN. . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . .qN1 qN2 . . . . . . qNi . . . . . . qNN

266666666664

377777777775

R ¼

r11 r12 . . . . . . r1i . . . . . . r1Nr21 r22 . . . . . . r2i . . . . . . r2N. . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . .ri1 ri2 . . . . . . rii . . . . . . riN. . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . .rN1 rN2 . . . . . . rNi . . . . . . rNN

266666666664

377777777775

Using these notations and statistics, μP, the expected return of a portfolio, isgiven by:

lP ¼XNi¼1

liwi ð2:1Þ

that is equivalent, in matrix form, to the following expression:

2.1 A Short Review of the Traditional Markowitz … 5

lP ¼ w0l ð2:2Þ

The variance of the portfolio, r2P, is computed, alternatively, by expressions (2.3)or by (2.4):

r2P ¼XNi¼1

XNj¼1

wiwjrij ;XNi¼1

XNj¼1

wiwjrirjqij ;XNi¼1

wirið Þ2 þXNi¼1

XNj ¼ 1j 6¼ i

wiwjrij ð2:3Þ

r2P ¼ w0Rw ð2:4Þ

According to Markowitz’s works, the investor’s asset allocation problem entailschoosing the portfolio weights such that, for a given expected return target, theportfolio variance or standard deviation is kept to a minimum. The portfolioscorresponding to this description are called mean-variance efficient portfoliosbecause they present the best return and risk pairs. In practice these portfolios mustbe sought by running an optimization algorithm usually referred to as Mean-Variance Optimization. Mathematically, it consists of three fundamental compo-nents. The first one is the objective function given by the generic formulation of theportfolio variance, thus it is a quadratic objective function to be minimized. Thesecond element is the set of the unknown variables representing the optimal port-folio weights to identify and upon which the objective function is dependent. Thelast component is the set of constraints. Naturally, this includes the return con-straint. Beyond this, also equality and inequality constraints are included respec-tively to impose that portfolio weights add up to one (referred to as budgetconstraint) and to restrict short selling (referred to as non-negativity constraint orlong-only constraint).

In formal terms, the Mean-Variance Optimization proposed by Markowitz iswritten as follows:

Minw�

XNi¼1

wiri� �2 þ

XNi¼1

XNj ¼ 1

j 6¼ i

wiwjrirjqij

0BB@

1CCA

s:t:

XNi¼1

liwi ¼ l�P

XNi¼1

wi ¼ 1

wi � 0

ð2:5Þ

6 2 The Traditional Approach to Asset Allocation

The same problem in matrix form is stated as follows:

Minw� w0Rw

s:t:

w0l ¼ l�Pw0e ¼ 1

w½ � � 0

ð2:6Þ

where e is a vector N × 1 of ones.Given the type of restrictions imposed on portfolio weights, the optimization

problem above cannot be solved analytically. Therefore solutions for w* are foundusing numerical techniques that are of an iterative nature.

Now that the classical Mean-Variance Optimization has been underlined, it isworthwhile looking at an example. For this purpose, we consider an investmentuniverse of four asset classes. Their expected returns, standard deviations andcorrelations are summarized in Table 2.1. The position that asset classes have in arisk-return space is marked with a circle in Fig. 2.1. Our first action is to randomlysimulate 1000 portfolios and plot them in Fig. 2.1 as well. In this way we are able toprovide an approximate illustration of the feasible portfolios. Next, we runMarkowitz’s algorithm (for different choices of l�P) and obtain the set of theoptimized portfolios. As previously stated, the optimized portfolios are calledmean-variance efficient portfolios given that we have found the combination ofassets that, for a particular level of risk, is expected to give the highest return or,equivalently, the combination of assets that, for a particular level of expected return,bears the lowest risk.

The set of all the optimized portfolios is shown by the concave function inFig. 2.1 that is called Mean-Variance Efficient Frontier or simply EfficientFrontier.1 We observe that our asset classes are (with one exception) below the

Table 2.1 Inputs for Mean-Variance Optimization

Expectedreturns (%)

Standarddeviations (%)

Assetclass 1

Assetclass 2

Assetclass 3

Assetclass 4

2.50 5.00 Assetclass 1

1 0.40 0.30 0.10

3.00 5.50 Assetclass 2

0.40 1 0.50 0.40

6.00 8.00 Assetclass 3

0.30 0.50 1 0.80

7.50 10.00 Assetclass 4

0.10 0.40 0.80 1

1For further information, we observe that the same Efficient Frontier can be derived using differentbut equivalent formulations of the Mean-Variance Optimization problem. We refer to the expected

2.1 A Short Review of the Traditional Markowitz … 7

Fig.2.1

Feasible

portfolio

sandMean-VarianceEfficientFron

tier

8 2 The Traditional Approach to Asset Allocation

Efficient Frontier and that the dominating portfolios computed by the Mean-Variance Optimization allow a significant improvement of the risk-return profilewith respect to the individual constituents.

2.2 An Analysis of Markowitz’s Portfolios: Drawbacksand Motivations

The Mean-Variance Optimization illustrated in Sect. 2.1 provides a clear, rationalframework to face the portfolio construction problem of investors. Its applicationrequires knowledge of both expected returns and covariance matrix of asset classesreturns or, equivalently, of expected returns, risks and correlations.

In reality, these model parameters are not observable ex-ante, thus they must beestimated. This task is commonly accomplished with the download of time series ofhistorical returns for the selected asset classes and then consequently with thestatistical computation of the relevant parameters using past data. This means thatthe following sample estimates for expected returns (l̂i), risks (r̂i) and covariancesor correlations (respectively, R̂ij or q̂ij) are considered:2

l̂i ¼1T

XTt¼1

Rt;i ð2:7Þ

r̂i ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1

T � 1

XTt¼1

Rt;i � l̂i� �2

vuut ð2:8Þ

R̂ij ¼ 1T � 1

XTt¼1

Rt;i � l̂i� �� Rt;j � l̂j

� �

or q̂ij ¼

1T � 1

XTt¼1

Rt;i � l̂i� �� Rt;j � l̂j

� �

r̂ir̂j

ð2:9Þ

(Footnote 1 continued)

return maximization formulation and to the risk aversion formulation. In the first case, theformulation of the portfolio return becomes the objective function to maximize and the level of riskbecomes a constraint. In the second case, the trade-off between risk and return is explicitlymodelled using a utility function to maximize for different risk aversion coefficients.2The exclusive consideration for these statistical moments shows that the multivariate normality isassumed by the Markowitz model. Alternatively, it can be assumed that investors have a quadraticutility function. The result is the same because in this case the utility is a function of the portfoliomean and standard deviation.

2.1 A Short Review of the Traditional Markowitz … 9

The standard implementation of the Markowitz model then usually follows anapproach named by Kan and Zhou (2007) the “plug-in” rule. In practice, theestimated parameters are processed into the mean-variance optimizer that treatsthem as if they were the true parameters. Handling the asset allocation problem inthis way has an important and serious implication: the procedure becomes totallydeterministic as the parameters uncertainty is completely neglected. However, inthe portfolio optimization context (as in many other financial situations) the risingof estimation risk is almost unavoidable. From a logical point of view, it can bedefined as the (very real) possibility to make estimation errors that is to record adifference between any parameter’s estimated value and that parameter’s true value.Of course, we are not able to know a priori how far we are from the true parameters.As indicated by a wide number of authors, Fernandes et al. (2012), Drobetz (2001),Eichorn et al. (1998), Grauer and Shen (2000), Herold and Maurer (2006), Jobsonand Korkie (1981), Jorion (1985, 1992), Michaud (1989), implementing the Mean-Variance Analysis without recognizing the existence of estimation risk inherent inthe input parameters has a huge impact on the optimized portfolios and leads toseveral undesirable features and deficiencies. They can be summarized in the theircounter-intuitive nature, instability, erroneously supposed uniqueness and poorout-of-sample performance.

With reference to the above mentioned, the issue deserving attention is the lackof diversification of optimal portfolios which paradoxically contrasts with one ofthe main ideas of Markowitz. Efficient allocations frequently completely excludesome asset classes of the chosen investment universe and give extremely largeweights to some others. They also show sudden shifts along the Efficient Frontier.In addition, not so rarely the most northeast point of the Mean-Variance EfficientFrontier lacks completely diversification and is equivalent to a mono-asset port-folio.3 These are consequences of the high discriminatory and impassive manneradopted by the portfolio construction method in dealing with data inputs which areconsidered extremely accurate. The mean-variance optimizer tends to heavily weighthose asset classes that show high estimated returns compared to low variances (orstandard deviations) and negative correlations. These are also the asset classes thatare most likely to suffer from large estimation errors. For this reason, Michaud(1989) wrote: “The unintuitive character of many optimized portfolios can be tracedto the fact that MV optimizers are, in a fundamental sense, estimation-errormaximizers”.

The second major drawback of mean-variance efficient portfolios is theirinstability. This attribute defines the high sensitivity of portfolio allocations to smallchanges in the estimated parameters. Especially, in the case we have in theinvestment universe couples of asset classes with similar risk-return profile, smallperturbations may completely alter the distribution of the portfolio weights because

3This situation is verified in the simple asset allocation example presented in Sect. 2.1. Theextreme portfolio on the right side of the Efficient Frontier in Fig. 2.1 is 100 % composed of thefourth asset class.

10 2 The Traditional Approach to Asset Allocation

they may cause an alternation between the dominant and the dominated asset class.This circumstance proves that optimized portfolios are not reasonable under dif-ferent set of plausible input parameters. Since they are estimated from historicalobservation of asset classes’ returns, the occurrence of perturbations must be takenfor granted and therefore the problem of instability has a practical impact.4 Theliterature (Best and Grauer 1991) has suggested that especially changes in expectedreturns tend to impact dramatically on optimal portfolio weights.

The third limitation consists of failing to recognize the non-uniqueness ofoptimal portfolios. As sharply noted by Michaud (1989) “Optimizers, in general,produce, a unique optimal portfolio for a given level of risk. This appearance ofexactness is highly misleading, however. The uniqueness of the solution dependson the erroneous assumption that the inputs are without statistical estimation error”.Hence, in practice, it would be helpful, in order to reach the goal of identifyingrecommended portfolio structures, to consider many statistically equivalent port-folios and take the average weights resulting from their different compositions.5

The last but potentially the most serious drawback affecting mean-varianceefficient portfolios is the poor out-of-sample performance: outside the sample periodused to estimate input parameters, classical Markowitz’s portfolios show a con-siderable deterioration of performance with respect to the expected one and thesame occurs in terms of risk-adjusted performance. As observed by DeMiguel et al.(2009) and Jobson and Korkie (1981), the extent to which they fall short of theoriginal targeted performance is such that very simple investment criteria candominate Mean-Variance Optimization. Thus, this means that the “plug-in rule”cannot be validated.

Ignoring the possible misspecification of the input parameters has major con-sequences especially if we refer to expected returns. Sample means are subject tolarge fluctuations and their predictive power seems poor. The errors in variancesand covariances/correlations are considered less costly than estimation errors in theexpected returns. Chopra and Ziemba (1993) found that errors in mean are at least10 times as important as errors in variances and errors in variances are about twiceas important as errors in covariances.

To prove the impact of fluctuations of sample estimates on the identification ofthe optimal portfolios, we consider an investment universe including 9 asset classes6

4Perturbations can simply result from the use of a rolling-window approach to get estimates ofinput parameters from historical returns.5Michaud and Michaud (2008) propose the Resampled Efficiency technique. It allows, by meansof Monte Carlo methods, to implement a simulation approach that draws numerous randomsamples of returns from a multivariate normal distribution described by the vector of means andcovariance matrix derived from the actual sample of historical returns. Different sets of new inputparameters can then be calculated from the simulated sample of returns and based on them manystatistically equivalent portfolios can be obtained.6They are represented by the following benchmarks: Jpm Euro Cash 12 M, Jpm Emu GovernmentAll Maturities, CGBI WGBI World non Euro All Maturities, Bofa ML Global Broad Corporate,MSCI Emu, MSCI North America, MSCI Pacific ex Japan Free, MSCI Emerging Markets.

2.2 An Analysis of Markowitz’s Portfolios … 11

for which 216 euro-denominated monthly returns are available from January 1997 toDecember 2014. We calculate sample estimates using a rolling estimation windowof 120 months (10 years) and then we run the Mean-Variance Optimizationexploiting the “plug-in” rule. We obtain the 97 historical or classical EfficientFrontiers shown in Fig. 2.2; each frontier consists of 100 optimized portfolios. Wenote that the range of return displayed in the historical Efficient Frontiers goes from2.45 to 33.24 % while the range displayed by risk is from 0.30 to 26.01 %. Recallingour preceding consideration that uncertainty in the sample means is more critical

Fig. 2.2 Analysis of the fluctuating behaviour of historical Mean-Variance Efficient Frontiers with10 years estimation window

Fig. 2.3 Analysis of the fluctuating behaviour of historical Mean-Variance Efficient Frontiers with12 years estimation window

12 2 The Traditional Approach to Asset Allocation

than uncertainty in variances (or standard deviations) and covariances (or correla-tions), we repeat the exercise using a rolling estimation window of 144 months(12 years). In this case, we obtain the 73 historical Efficient Frontiers represented inFig. 2.3 that shows, relatively to Fig. 2.2, a strong alteration especially in the rangeof returns.

It must be emphasized that instability in the optimal risk-return pairs has atendency to decrease with an increase in the number of the sample observations inthe rolling window and has a tendency to increase with size of the investmentuniverse. This means the historical Efficient Frontiers would appear more/lesscompacted should we use a small/large set of asset classes and a larger/smallerestimation window.

Another way of demonstrating the impact of fluctuations in estimated inputparameters is to by illustrating the composition of portfolios that are associated tothe same rank relative to the 100 portfolios computed in each of the 97 historicalEfficient Frontiers of our first application of the “plug-in” rule. We propose thisexercise with reference to portfolios ranked as 25th and 50th. Figure 2.4 providestheir 97 asset allocations. This confirms the important differences in portfolioweights using rolling samples of historical data.

The analysis and the discussion developed in this section demonstrate the severedrawbacks suffered by the Markowitz portfolios that prevent them from having aninvestment value and from being externally marketable by the asset managers. Themotivation for these limitations is not a fault in the analytical formulation of theoptimization algorithm or any other intrinsic or theoretical error. The cause is of anoperative nature and refers to the way the optimizer uses investment informationdelivered by means of the optimization inputs. It works as if it had exactly thecorrect inputs available without accounting for any estimation error. Therefore, thefundamental problem is that the level of power of the optimization procedure inmanaging investment information is far greater than the quality of the inputs. Inother words, we observe an over-fitting of inaccurate information represented bysample estimates. Michaud and Michaud (2008) contributed to this argumentwriting: “The procedure overuses statistically estimated information and magnifiesthe impact of estimation errors. It is not simply a matter of garbage in, garbage out,but rather a molehill of garbage in, a mountain of garbage out”.

Given the implications of the presence of estimation risk for investment portfo-lios, several attempts have been made to overcome them. From the mid 1980sonwards, alternative methodologies have been developed to deal with the problem ofparameters uncertainty. They can be distinguished between heuristic and Bayesianapproaches. The first group includes using the Resampling technique patented byMichaud and Michaud (2008) and the application of additional weight constraints(besides the classical long-only and budget constraints) suggested by Frost andSavarino (1988) and Jagannathan and Ma (2003). Bayesian approaches can beimplemented in different ways. One application consists in the use of shrinkageestimators that propose estimation of expected returns by shrinking the sample meantoward a common mean (Jorion 1985). A second application is represented by the

2.2 An Analysis of Markowitz’s Portfolios … 13

0%10%

20%

30%

40%

50%

60%

70%

80%

90%

100%

15

913

1721

2529

3337

4145

4953

5761

6569

7377

8185

8993

97

Weights for portfolio n. 25

His

toric

al E

ffici

ent F

ront

ier

Num

ber

0%10%

20%

30%

40%

50%

60%

70%

80%

90%

100%

15

913

1721

2529

3337

4145

4953

5761

6569

7377

8185

8993

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Weights for portfolio n. 50

His

toric

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ent F

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Num

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.4Com

positio

nforpo

rtfolio

snu

mber25

and50

ofthehistorical

EfficientFron

tiers

14 2 The Traditional Approach to Asset Allocation

Black and Litterman model (1992) which combines two sources of information(equilibrium returns and investor’s views) to obtain predictive returns.

In practice, heuristic approaches and Bayesian approaches take different actionsin response to the impact of the unavoidable imprecision of estimated inputs,especially of expected returns, on optimal portfolios construction. The former actson the model side to make the optimization process less deterministic or to force itto produce more diversified portfolios. The latter acts on the estimation side toimprove expected returns estimates with respect to their sample estimates.Obviously, it also makes sense to use methodologies that combine features ofheuristic and Bayesian approaches as proposed by Fernandes et al. (2012).

In conclusion, it is worthwhile emphasizing that neither heuristic approaches norBayesian approaches modify the classical mean-variance portfolio optimizationframework. They try to make its response more reliable and robust by making someimprovements concerning how inputs are used or produced.

References

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optimal portfolio choice. J Portfolio Manage. 19:6–11DeMiguel V, Garlappi L, Uppal R (2009) Optimal versus naive diversification: how inefficient is

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http://www.springer.com/978-3-319-24380-1