Asset allocation: Is the parametric rule a breakthrough ?

Asset allocation: Is the parametric rule a

breakthrough ?

Xu Lujun

Student Number: s1940570 Email: l.xu.4@student.rug.nl

Master’s Thesis in Econometrics

Faculty of Economics and Business Rijksuniversiteit Groningen Groningen, the Netherlands

To my parents

Preface

This thesis is the final project for my Master of Science in Econometrics. I would like to take this chance to give my thanks. Glory to God, first. He is always there and always good.

I want to extremely appreciate my supervisor Prof. Bekker, without his help I cannot even start this thesis. For the past half year, he is not only my thesis supervisor, but also my course teacher. I have been learnt a lot from him both in academic knowledge and in academic attitude. I am a very careless and sloppy student. He teaches me with great rigour that writing without precise is wasting time, just as learning without thinking is wasting effort. This is the most valuable fortune for my further study. Additionally, I would like to thank him for sparing so much time and efforts on my thesis. He is always ready for help when I have problems even though he also has heavy teaching loads. A second word of thanks goes to my church and fellows in Groningen, they accompany me in the past four years. We have fellowships and worships every week. It it the place and the people that encourages me and supports me.

Contents

1 Introduction 7

2 Problem Formulation 13

2.1 Mean-Variance Maximization . . . 13 2.2 Direct Utility Maximization . . . 14 2.3 Parametric Portfolio Policies Maximization . . . 15

3 Consistency of MV maximization and Direct Utility maximization 16

3.1 Research agrees . . . 17 3.2 Research disagrees . . . 18

4 Consistency of MV Maximization and PPP Maximization 20

5 Methodology 22

5.1 Mean-Variance Analysis . . . 22 5.2 Naive Portfolio . . . 25 5.3 Parametric Portfolio Policies . . . 25

Asset allocation: Is the parametric rule a

breakthrough ?

Xu Lujun 2014-Jan-28

Abstract

Modern Portfolio Theory (MPT) plays an important role in portfolio diversifi-cation. It attempts to maximize the expected return for a target risk, or minimize the portfolio risk for a target expected return. In practice, MPT is widely used in financial institutions and investment departments. Compared to a direct utility max-imization, mean-variance maximization does not need an explicit form of investor’s utility function and requires fewer calculations. Despite this, there are also harsh criticisms leveled against it. A theoretical criticism is that mean-variance efficient portfolio is only optimal when the utility function is quadratic or when the returns are jointly normally distributed. A practical criticism is that mean-variance maxi-mization is difficult to implement for a large number of assets. Instead, Brandt et al. (2009) suggest to model directly the portfolio weights as a function of k asset’s attributes. They assert that their new approach is applicable to any size of asset groups and is superior to traditional mean-variance model. In this thesis, we want to show that the parametric rule of Brandt et al. is unnecessary. Because based on their rule we could also perform mean-variance analysis. We present an empirical analysis on five AEX stocks with two characteristics. The result shows that optimal portfolios found by Brandt et al.’s approach are all located on the mean-variance frontier of the three constructed portfolios.

1

Introduction

It has been sixty-one years since Markowitz (1952) first pioneered the mean variance max-imization(MV) criterion to approximate direct utility maximization. It is also sixty-one years full of controversy. The notion: ”mean-variance maximization” is quite obvious. An investor, who wants to invest in a number of assets, has to make a trade-off between the expected return and the risk. He or she either tries to minimize the portfolio risk at a given portfolio mean return, or maximize the portfolio return at a certain level of risk. There are a lot of suggestions for risk criteria, such as variance, mean absolute deviation, semivari-ance, value at risk or conditional value at risk. In the formulation of Markowitz (1952), variance is used as a measure of portfolio risk. Once Markowitz has proposed the mean-variance model, there are two aspects left to be discussed. First, whether mean-mean-variance maximization has practical advantages over the traditional direct utility maximization. Second, whether it gives precise approximation to direct utility maximization. These two aspects must be considered simultaneously. Only then one can claim that mean-variance maximization is to be preferred to direct utility maximization, or not. We will discuss these two aspects separately in the following part of this section.

read-(1984). The second computational advantage is that mean-variance maximization does not require a specific form of utility function. Actually, it is rather difficult to numericalize an individual’s preferences. For example, Von Neumann and Morgenstern (1947) attempt to numerically measure an individual’s preferences by asking the individual making pairwise comparisons between a certain return A and a combination return pB +(1−p)C at different probability level p. If A is preferred to both B and C, the individual will always choose the certain return A. On the other way round, he will always prefer the combination of B and C to a certain return A. If A is preferred to B, while C is preferred to A, we can find a probability p∗ such that he is indifferent between the certain return and a combination return. By doing this, we can rank the preferences for a particular individual and thereby his utilities become numerically measurable. However, this would be very troublesome when the individual has an inconsistent preference set or when his preferences are non transitive. In addition, Kroll et al. (1984) also indicate that mean-variance maximization is beneficial to investment consultants. A consultant will have lots of workload if he has to identify each of his clients’ utility function. It becomes even worse if he has many clients. While mean-variance maximization does not have such concerns, the consultant just needs to provide each of his clients a list of mean-variance efficient portfolios and let the client him/herself decides which portfolio he/her want to invest in. This makes life much more easy.

In spite of practical advantages, there are some criticisms on the performance of mean-variance approximation. The most famous opposition is the assertion: Mean-mean-variance approximation performs good only if the utility function is quadratic or the return is nor-mally distributed. 1 _{It is because a quadratic utility can be written as a function of mean}

return and variance of return by Taylor expansion. Similarly, the distribution of a nor-mally distributed random variable is also uniquely characterized by its mean and variance. Thus, the expected utility of a normally distributed return can also be represented by its

1_{Discussions of the relation between mean-variance accuracy and quadratic utility or return normality}

mean return and its variance. At this point, direct utility maximization is equivalent to mean-variance maximization. However, the assumptions of quadratic utility and normal returns are problematic. As Pratt (1964) points out in his paper, quadratic utility is not applicable when the portfolio return is higher than a certain level (a bliss point). Because after the bliss point, quadratic utility exhibits a negative marginal utility of return. Thus the economic relevance of quadratic utility is severely restricted in applications. Moreover, normal returns form rather an idealized hypothesis since it is likely to be rejected in real life. Thus, many scholars who are against mean-variance analysis believe mean-variance maximization can approximate direct utility maximization only under quadratic utility or normally distributed returns.

However, many researchers try to reconcile the controversy by showing portfolio selection based on mean-variance analysis is also applicable for non-quadratic utility functions and non-normal historical stock return distributions. Markowitz (1959) uses empirical results to show that for a logarithmic utility, mean-variance approximation error is as small as 3% when portfolio return falls into the interval [−0.3, 0.3]. Levy and Markowitz (1979) extend the analysis on logarithmic utility function to power and exponential utilities as well as to different return distributions. They try to fit the quadratic approximation to the utility function at three values of portfolio return.2 In general, they find an average correlation as high as 0.997 between mean-variance approximation and expected utility maximization. Based on this result, Levy and Markowitz conclude that mean-variance maximization and direct utility maximization are indistinguishable for various utilities and return distributions. Later on, Kroll et al. (1984) further challenge the approximate ability of mean-variance maximization by using annual data and by including short-selling. Because both Pulley (1981) and Levy and Markowitz (1979) mention that the higher the portfolio variance the higher likely that mean-variance maximization performs less well.

2_{The three values of return are (E − kpσ}2

P), (E), (E + kpσ 2

P), where E is the expected return, k is

a given number, andpσ2

Nevertheless, Kroll et al. (1984) obtain good results. They find that, at the same level of portfolio mean return, the optimal portfolios which are found by direct maximization have a portfolio risk only 3.8%-0.0% higher than the portfolio risk of mean-variance efficient portfolios. The differences are relative small and can be acceptable. In addition, Tsiang (1972) rewrites the Taylor expansion of negative exponential utility and constant relative risk aversion utility functions in terms of a polynomial in deviation from the mean. He finds that when risk is small relative to the total wealth of an individual, mean variance analysis performs good. Later on Kroll et al. (1984)

Theoretically, Cochrane (2007) provides a theoretical proof to show that for continuous time series, when the mean and variance of return are time invariant, any risk-averse in-vestor will hold an instantaneously mean-variance efficient portfolio. This result also holds true for a discrete time series. Thus, Cochrane (2007) concludes that ”Mean variance portfolios do not require quadratic utility” (p 47). However, since Cochrane (2007) uses diffusion processes that are locally normal, his proof can not show mean variance portfo-lios do not require normal distributions. In a recent paper, Markowitz (2014) explains why aforementioned researchers have such a myopic view on mean-variance approximation. He claims that it is because they mistake the sufficient condition for a necessary condition. Markowitz (2014) says ”Quadratical and normality are only sufficient but not necessary conditions for a mean-variance approximation. The necessary and sufficient condition is that a careful choice from the mean-variance frontier will almost maximize expected utility for a wide variety of concave (risk-averse) utility functions” (p 346).

of traditional direct utility maximization. First, they select several stock attributes that are correlated with stock returns. Such attributes, for example, can be size effect (Banz (1981), Van Rensburg and Robertson (2003)), price-to-earning ratios (P/E) (Van Rensburg and Robertson (2003)), or lagged returns ( Jegadeesh (1990), Lehmann (1990), De Bondt and Thaler (1985)). Subsequently, they parameterize the portfolio weight of each asset at each time as the sum of the specific asset’s weight at that time in a benchmark portfolio and a function of the asset’s own firm characteristics. Finally, they substitute the function of weights into the general direct utility model and maximize over it. For a given utility function, they can calculate the optimal asset allocation portfolio. Brandt et al. (2009) call this approach Parametric Portfolio Policies (PPP) because they parameterize the stock weights in terms of a list of stock characteristics. They claim that PPP method has two major advantages over mean-variances analysis. First, PPP dramatically reduce the di-mensions. For N stocks, traditional mean-variance maximization has to estimate N first moments and (N2_{+ N )/2 second moments of returns. In contrast, PPP just need to}

esti-mate several coefficients of characteristics. Second, PPP captures implicitly the relations between utilities and mean return, variance, skewness, kurtosis and even higher moments of returns. However, higher moments in a quadratic maximization model are neglected. In their empirical analysis, Brandt et al. (2009) use monthly firm-level data from CRSP for the time period Jan. 1964 - Dec.2002. They use size, book-to-market ratio and momen-tum as their stock characteristics. They find that the optimal parametric portfolio yields a slightly larger standard deviation (19%) than that of a value-weighted portfolio (16%) but a significantly larger mean return of 24% compared to 12% of a value-weighted portfolio. In terms of Sharpe ratio, they find that the Sharpe ratio of the optimal parametric portfolio based on monthly data is 0.964. It is at least two times the one found for a value-weighted portfolio (0.362).

claim that PPP method has several advantages over traditional mean-variance analysis, but they do not compare the performance of a mean-variance efficient portfolio with an optimal parametric portfolio. Instead, they compare the performance of a value-weighted portfolio with an optimal parametric portfolio and find that the later outperforms former. It might be that a mean-variance efficient portfolio also outperforms a value-weighted port-folio in terms of Sharpe ratio. Moreover, computationally speaking, mean-variance analysis might be more simple.

Given N assets and their corresponding k characteristics, Brandt et al.’s PPP method is essentially equivalent to first reducing these N assets to (k + 1) portfolios3 _{and then}

find-ing an optimal combination of these (k + 1) portfolios that maximize the expected utility. However, if N assets can be reduced to (k + 1) portfolios, we can also use mean-variance analysis. That is to say, we can apply mean-variance analysis exactly on these (k + 1) port-folios. When k is assumed to be sufficiently smaller than N , we get rid of the disadvantage claimed by Brandt et al. (2009) that mean-variance has to estimate N first moments and (N2 _{+ N )/2 second moments of returns. In this way, the amount of computation of a}

mean-variance analysis is the same as that of a PPP method.

The objective of this paper is to investigate whether mevariance maximization is an-alytically and computationally consistent with Brandt et al. (2009)’s PPP method. In order to make the analysis simple, we will use monthly returns of only 5 Dutch stocks that were traded on AEX from November 2006 to November 2013. We use two characteristics, viz. market capitalization and P/E ratio. That means, we can reduce the five stocks to three constructed portfolios. The same analysis can be applied to more stocks and more firm characteristics. First, we conduct JarqueCBera tests for each stock return and investigate whether the returns have the skewness and kurtosis matching a normal distri-bution. Second, based on the two characteristics, three reduced portfolios can be derived.

Third, we apply mean variance analysis exactly on these three reduced portfolios and draw an efficient frontier for them. Third, by varying the value of the parameter for a utility function we can find varies optimal parametric portfolios. Finally, we compare whether the optimal parametric portfolios are consistent with the mean-variance efficient portfolios.

The remainder of this paper is structured as follows. In Section 2, the traditional mean-variance maximization model, the traditional direct utility maximization model and PPP utility maximization model are formulated. In Section 3, I will discuss the consistency of mean-variance analysis and direct utility maximization. In Section 4, I will discuss the consistency of mean-variance analysis and PPP utility maximization. In Section 5, algorithms for all maximization models are given. The characteristics selection procedure based on zero-cost long-short portfolios is presented in Section 6. The properties of the utility we will use in utility maximization are discussed in Section 7. In Section 8, an empirical analysis is conducted. Section 9 concludes.

2

Problem Formulation

2.1

Mean-Variance Maximization

an efficient portfolio can be formulated as a solution to the following minimization problem4 Min 1 2w 0 Ωw s.t. N X i=1 wi = 1 and w0µ = µP (1) where

w = (w1, w2, ..., wN)0, are the weights for wealth allocation,

N is the total number of stocks,

Ω is the covariance matrix of stock returns,

µ = (µ1, µ2, ..., µN)0, is the vector of mean returns of N stocks,

µP is the expected portfolio return.

For different values of µP, we can calculate different mean-variance efficient portfolios.

2.2

Direct Utility Maximization

In the variance maximization model, we select the portfolio that is only mean-variance efficient. By contrast, direct utility maximization amounts to selecting the optimal portfolio among all possible allocation mixtures. It can be formulated as follows

Max EU ( N X i=1 wi,tri,t; γ) s.t. N X i=1 wi,t = 1. (2)

4_{This model allows short selling. If short selling is prohibited, linear constraints: w}

i ≥ 0, for i =

where

N is the total number of stocks, U (.) is the investor’s utility function,

ri,t is the rate of return of stock i from month t to t + 1 ,

wi,t is the portfolio weight of asset i at time t,

γ is the parameter in the utility function.

This is the traditional set up of a direct utility maximization model. For a risk-averse investor, a portfolio is optimal if it maximizes the expected utility.

2.3

Parametric Portfolio Policies Maximization

The approach of Brandt et al. (2009) to optimize assets weights is a variant of the tra-ditional direct utility maximization model. They parameterize portfolio weights wi,ts in

terms of a benchmark portfolio and firm-level characteristics. The benchmark portfolio return, is defined as rBP =

N

X

i=1

˜

wi,tri,t, where ˜wi,t can be equal weighted weights or value

weighted weights. Thus maximization problem (2) expands to

Max E[U (rp,t)] s.t. rp,t= N X i=1 wi,tri,t

and wi,t = ˜wi,t+

1 Nθ

>_x i,t

where

U (.) is investor’s utility function,

rp,t is the portfolio’s return from month t to month t + 1

ri,t is the rate of return of stock i from month t to t + 1 ,

wi,t is the portfolio weight of asset i at time t,

˜

wi,t is the portfolio weight of asset i at time t in a benchmark portfolio,

θ is a k by 1 vector of parameters to be estimated,

xi,t is a k by 1 vector of asset i’s own characteristics at time t.

The firm characteristics can be, for example, the market capitalization of the stock, the price-earning ratio, the firm’s lagged return and so on. A discussion of the choice of characteristics will be presented in the following section. Notice that, different from max-imization problems (1) and (2), (3) doest not impose the constraint that stock weights should sum up to one. However, when using the style characteristics xi,t, we standardize

them cross-sectionally to have sample mean zero and sample standard deviation one. Thus total stock weights sum up to one automatically at each time period. In the following content, without further statement, mean-variance maximization model is abbreviated to MV maximization, direct utility maximization is written as DU maximization and Brandt et al.’s variant to a direct utility maximization is abbreviated to PPP maximization.

3

Consistency of MV maximization and Direct Utility

maximization

distributed.

3.1

Research agrees

Most researches 5 agree on the efficiency of mean-variance maximization when the utility function is quadratic or when the returns have a normal distribution. If the first condition is absent, the second one must be present. The reason is quite obvious, the expected quadratic utility can be written as a function of mean return and variance of return. Similarly, a normally distributed random variable is uniquely characterized by its mean and variance. Thus, the expected utility of a normally distributed return can also be represented by the mean return and the variance of return. The elementary proof is as follows. Assume the utility function U (R) is quadratic, we can expand it at a by Taylor expansion. Levy and Markowitz (1979) used two values of a, namely zero and mean return E. They find that expanding at mean return E performs better than expanding at zero. Thus, we expand the Taylor series of U (R) around E,

U (R) = U (E) + U0(E)(R − E) + U

00_(E)

2 (R − E)

2_{, (4)}

taking the expectation yields

E(U (R)) = U (E) + U

00_(E)

2 E((R − E)

2_{). (5)}

Therefore, maximizing E(U (R)) is equivalent to max E(U (R)) = U (E) + max U

00_(E)

2 E((R − E)

2

), (6)

where E((R − E)2_{) is the variance of portfolio return. Because of the negativity of the}

second derivative for a quadratic utility, maximizing term U00₂(E)E((R − E)2_{) is equivalent}

to minimizing E((R − E)2_{). It is exactly the same as mean variance maximization. In}

validity of mean-variance analysis. For a normally distributed random variable R, the ith order moment

mi(R) =

E([R − E(R)]i)

σ(R)i (7)

is always a function of its mean and variance. Moreover, a linear combination of normally distributed random variables is also normally distributed. Thus under the above condi-tions, utility maximization model amounts to mean variance maximization.

3.2

Research disagrees

As I’ve already discussed in the Introduction that quadratic utility exhibits abnormal behaviour at a high level of return and in reality asset returns can be non-normal. Thus quadratic and normal assumptions are restrictive. However, disagreements arise when the aforementioned two conditions are relaxed. It should be explained that mean-variance maximization is reasonable as well for other utility functions and other distributions of the returns. Markowitz (1952) writes, ”The third moment M3 of the probability distribution

of returns from the portfolio may be connected with a propensity to gamble, ... for a great variety of investing institutions which consider yield to be a good thing; risk, a bad thing; gambling, to be avoided – E-V efficiency is reasonable as a working hypothesis and a working maxim” (p 90,91). Thus, Markowitz thinks skewness is not very relevant. For a non-quadratic utility function, the Taylor expansion around mean is no longer (4), but instead, more complicated

U (R) = U (E) + U0(E)(R − E) + U 00_(E) 2 (R − E) 2_{+ O(.), (8)} where O(.) = ∞ X n=3 Un_(E) n! (R − E) n

Many researches give empirical results to show how well a function of mean and variance can approximate various utility functions. Markowitz (1959) examines the goodness of fit of mean-variance analysis on logarithmic utility function U (R) = ln(1 + R). He figures out that for return R in the interval [−0.3, 0.3], the difference between the two maximization models is small. Later on, Levy and Markowitz (1979) extend the analysis on logarithmic utility function to power and exponential utilities and as well as to different return distri-butions. They found similar results. On the other hand, Tsiang (1972) proposes another approach to justify the availability of mean-variance approximation. He claims that when risk is small relative to the total wealth of an individual, mean variance analysis performs good.

In a recent paper, Cochrane (2007) provides a proof of the validity of mean-variance analysis for other utility functions. He first assumes mean of the returns and variance of the returns are time-variant, thus state-variables must be introduced so as to identify the state of the period. He finds that, for a continuous time series, the optimal portfolio weights amount to

w = Σ−1(λ1ι + λ2µ) − β0η, (9)

where the first part is the mean-variance efficient portfolio weights6 and the second part is the portfolio weights for state-variables.7 _{Therefore, when the mean and variance of returns}

are time invariant, any risk-averse investor will hold an instantaneously mean-variance ef-ficient portfolio. As a result, Cochrane concludes ”Mean variance portfolios do not require quadratic utility” (p.47).

4

Consistency of MV Maximization and PPP

Maxi-mization

In the Introduction, I mentioned that the PPP method of Brandt et al. (2009) is essentially equivalent to first reduce N assets to a number of portfolios and then apply direct utility maximization on these portfolios. In this section, we discuss this in more detail. We first discuss in a general case. Assume there are N assets in the market and we observe k characteristics for each asset. Define the following k + 1 portfolios returns

P0 = N X i=1 ˜ wi,tri,t P1 = N X i=1 ( ˜wi,t+ 1 Nτ > 1 xi,t)ri,t · ·· Pk= N X i=1 ( ˜wi,t+ 1 Nτ > k xi,t)ri,t (10)

where τj is a k by 1 vector, j = 1, .., k. Let λ = (λ0, ..., λk)0 define a linear combination

of portfolio returns P = (P0, P1, ..., Pk) such that k

X

j=0

λj = 1. In addition, let λ− =

(λ1, ..., λk)0. Then the utility of λ0P at time t becomes

We can observe that (12) equals U (rp,t) = U ( _N X i=1  ˜ wi,t+ 1 Nθ > xi,t  ri,t ) , (13) for k X j=1 λjτj = θ, (14)

which is the utility calculated from Brandt et al.’s parametric portfolio policy method. Therefore, to maximize U (λ0P ) over λ is equivalent to maximize U (rp,t) over θ.

A simple example will illustrate the essence of this transformation. Assume now we only have 5 stocks and we observe two characteristics for each stock. Then we can construct three portfolio returns as follows:

P0 = 5 X i=1 ˜ wi,tri,t P1 = 5 X i=1 ( ˜wi,t+ 1 5τ > 1 xi,t)ri,t. P2 = 5 X i=1 ( ˜wi,t+ 1 5τ > 2 xi,t)ri,t. (15)

Let λ = (λ0, λ1, λ2) such that λ0+ λ1+ λ2 = 1. Then we find

max EU (λ0P ) = max EU ( ₅ X i=1 ˜ wi,tri,t+ 1 5 5 X i=1 λ1τ1>xi,tri,t+ 1 5 5 X i=1 λ2τ2>xi,tri,t ) , (16) is equivalent to max EU (rp,t) = max EU ( ₅ X i=1  ˜ wi,t+ 1 5θ > xi,t  ri,t ) . (17)

Hence, λ1τ1+ λ2τ2 = θ, for given vectors τ1 and τ2. Moreover, λ0 = 1 − λ1− λ2.

portfolios. Moreover, in Section 3 we also showed that MV maximization is consistent with DU maximization, regardless of utility style and return distribution. Thus theoretically speaking, we can conclude that the PPP approach of Brandt et al. (2009) amounts to mean-variance analysis.

5

Methodology

5.1

Mean-Variance Analysis

In this section, I will provide the algorithm for mean-variance maximization. In order to make it brief without sacrificing readability, I will reproduce Model (1) but without variable definitions. For mean-variance analysis, an efficient portfolio is a solution to the following minimization problem

Min 1 2w 0 Σw, s.t. N X i=1 wi = 1, and w0µ = µP. (1)

This model allows short selling. If short selling is prohibited, linear constraints: wi ≥

0, for i = 1, ..., N , should be added to the quadratic model. However, we will not discuss this case here.

Let us first construct the following Lagrangian equation

L = 1

2w

0

Σw − k1(ι0w − 1) − k2(w0µ − µP), (18)

the first order conditions of the Lagrangian equation. They are Σw − k1ι − k2µ = 0,

ι0w = 1, µ0w = µP.

(19)

Since the covariance matrix Σ is assumed to be nonsingular, rearranging (19) yields w = Σ−1(k1ι + k2µ),

1 = ι0Σ−1Σw = k1ι0Σ−1ι + k2ι0Σ−1µ,

µP = µ0Σ−1Σw = k1µ0Σ−1ι + k2µ0Σ−1µ.

(20)

Now, define three more auxiliary symbols

A = ι0Σ−1ι, B = ι0Σ−1µ, C = µ0Σ−1µ, (21)

Thus, the Lagrangian multipliers k1 and k2 are the solution to the following systems of

linear equations

1 = k1A + k2B, and µP = k1B + k2C. (22)

Solving for k1 and k2, yields

k1 =

C − BµP

AC − B2 and k2 =

AµP − B

AC − B2. (23)

For a target expected portfolio return: µP, we can find its mean-variance efficient portfolio

as well as the minimized portfolio variance σ2_P ,

w = Σ−1(k1ι + k2µ), and (24)

σ_P2 = w0Σw. (25)

elements of µ and Σ can be estimated as follows ˆ µi = 1 T T X t=1 ri,t, i = 1, ..., N, (26) ˆ σij = 1 T T X t=1 (ri,t− ˆµi)(rj,t− ˆµj), i, j = 1, ..., N, (27)

where ˆµi is the sample mean of ri,t and ˆσij is the sample covariance of ri,t and rj,t. The

correlation between return i and return j, is denoted by ρij = σij/

√

σiiσjj, where σij and

σii can be estimated by ˆσij and ˆσii.

We consider the global minimum variance portfolio (GMV). It is found by setting the first derivative of the portfolio variance (σ2

P) with respect to portfolio return (µP) equal to zero,

i.e. dσ2P

dµP = 0. Combining Equations (24) and (25) yields:

σ_P2 = (k1ι + k2µ)0Σ−1(k1ι + k2µ) = (k1ι + k2µ)0w = k1 + k2µP. (28)

Setting the derivative of σ_P2 with respect to µP equal to zero yields k2 = 0, which is

equivalent to µGM V = B/A. Moreover, we can find σ2GM V = 1/A. Since the GMV portfolio

is mean-variance efficient and it has the global minimum variance, a formulation of the efficient frontier in terms of the GMV portfolio is given by

σ_P2 = k1+ k2µP (29) = C − BµP AC − B2 + AµP − B AC − B2µP (30) = Aµ 2 P − 2BµP + C AC − B2 (31) = σ2_{GM V} +(µP − µGM V) 2 C − µGM V · B , (32)

where A, B and C are defined above. If we plot σP against µP, for varying values of µP,

5.2

Naive Portfolio

The naive portfolio selection strategy is to allocate wealth on the N risky assets equally. It can be modeled as follows:

µP = 1 N N X i=1 µi, (33) with σ_P2 = 1 N2 N X i=1 σ_ii2 + 1 N2 N X i=1i6=j N X j=1 σij. (34)

Undoubtedly, it has no free parameters to be estimated. Even though a naive portfolio is simple, it plays an important role in portfolio diversification. It can be used as a bench-mark portfolio in PPP maximization problem.

5.3

Parametric Portfolio Policies

Brandt et al. (2009) propose a new way to model the portfolio weights other than tradi-tional direct expected utility maximization. They parameterize the portfolio weight of an asset by writing it as a function of its own firm characteristics. Then they substitute the function into the expected utility maximization model. The optimal weight of each asset is chosen so as to maximize Max E[U (rp,t)] s.t. rp,t= N X i=1 wi,tri,t

and wi,t = ˜wi,t+

1 Nθ

>

xi,t

(35)

sum of wi,ts equals to 1 at time t, we need to standardize xi,t cross-sectionally to have a

sample mean zero and sample standard deviation one across all stocks at time t. Notice that the coefficients θ in (35) are time invariant.

For some presumed utility function (e.g., CRRA utility), the corresponding sample ana-logue of (35) is: max {wi,t}Ni 1 T T X t=1 U N X i=1  ˜ wi,t + 1 Nθ > xi,t  ri,t ! . (36)

The estimator ˆθ should satisfy the first order conditions: 1 T T X t=1 U0 N X i=1  ˜ wi,t + 1 Nθ > xi,t  ri,t ! 1 N N X i=1 xi,tri,t ! = 0. (37)

Then ˆθ can be estimated by applying the Generalized Method of Moments (GMM) (Hansen, 1982). Since the GMM objective function is nonlinear, we can use the Gauss-Newton(GN) algorithm to find the minimum. First, define:

g(θ) = U0(rp,t) 1 N N X i=1 xi,tri,t ! , (38) m(θ) = 1 T T −1 X t=1 g(θ), and (39) G(θ) = ∂m(θ) ∂θ = 1 T T −1 X t=1 U00(rp,t) 1 N N X i=1 xi,tri,t ! 1 N N X i=1 xi,tri,t !> . (40)

Then the vector function m(θ) satisfies: √ T m(θ) ∼ N (0, Ω(θ)),a (41) where Ω(θ) is estimated by ˆ Ω(θ) = 1 T T −1 X t=1 g(θ)g(θ)>. (42)

Finally, with some starting point θ(0) we iterate, cycling through

1. b = G>( ˆθ(j)) ˆΩ−1( ˆθ(j))G( ˆθ(j))

−1

G>( ˆθ(j)) ˆΩ−1( ˆθ(j))m( ˆθ(j)) (43)

We stop when the norm of b is less than a small number. In addition, under regularity assumptions the GMM estimator is asymptotically normal with limiting distribution

√ T  ˆθGM M − θ  _a ∼ N  0, plim T →∞ n G0( ˆθGM M) ˆΩ−1( ˆθGM M)G( ˆθGM M) o−1 . (45)

Therefore, the GMM estimator’s asymptotic covariance matrix is estimated by

Ψ = 1 T n G0( ˆθGM M) ˆΩ−1( ˆθGM M)G( ˆθGM M) o−1 . (46)

6

Selection of Characteristics

Previous work shows that returns on stocks are correlated with firm specific characteris-tics. For instance, Banz (1981) claims size effect exists. He shows that the stock of small firms have higher risk-adjusted returns than that of large firms. This can be referred to as the small firm effect. A more recent paper by Van Rensburg and Robertson (2003) provides evidence of the forecasting ability of a range of ’style-based’ characteristics. They investigate a list of 23 attributes of stocks 8_{, and test the effect of each of the attributes,}

on an individual base, on stock returns. They find that price-to-earning ratios (P/E), size, dividend yield, and price-to-profit have significant effects on stock returns. In addition, researchers have had also identified the relation between past returns and average stock returns. For example, there may exists short-term return reversals. Evidence in support of this position, can be found in Jegadeesh (1990)’s and Lehmann (1990)’s paper. Jegadeesh (1990) finds a negative first-order serial correlation in monthly stock returns and a positive

8_{The 23 attributes of stocks are: Price-to-earnings ratio, Dividend yield, Price-to-profit, Price-to-NAV,}

serial correlation in twelve-month stock returns. These two findings will be referred to as the short-term reversal (STR) and medium-term momentum (MTM). Regarding a longer period (3 to 5 years), evidence can be found that loser stocks, over the past 3 to 5 years, perform better in the following 3 to 5 years than a winner stock (De Bondt and Thaler, 1985). This result will henceforth be referred to as the long-term reversal (LTR). Further-more, Van Rensburg and Robertson (2003) also shed light on how well those attributes jointly explain the stock returns. They find that using size and P/E as explanatory vari-ables gives the optimal model to explain stock returns. Since the firm characteristics that have been discussed above are not explained in the CAPM model, they are called anomalies.

In this paper, we will examine 5 most widely discussed firm characteristics. They are: price-to-earnings ratio (P/E), market capitalization (Mcap), short-term reversal (STR) 9_,

medium-term momentum (MTM) 10 and long-term reversal (LTR) 11. The first one, P/E, is a measurement of value. The second characteristic, Mcap, represents the size effect. The last three attributes are price momentum. We can select proper characteristics by con-structing zero-cost long-short portfolios (Asness, Moskowitz, and Pedersen, 2013). Take the characteristic short-term reversal (STR) as an example. For each period, we can rank our assets in ascending order according to STR. That is to say, the asset which has the low-est previous month return will be put in the first place and etc. Since it has been discussed that there exists short-term return reversals, we create a long-short portfolio by having one euro long on the first decile of the list and one euro short on the bottom decile of the list. Thus we can calculate the return on this zero-cost long short portfolio. It is the return on the long position minus the return on the short position. For the whole time period, we can then calculate the mean return as well as the standard error of the STR-based zero-cost long-short portfolio. Notice that when examining characteristic MTM, we should have one euro short on the first decile of the portfolio and one euro long on the bottom decile of the

portfolio. After constructing the long-short portfolio, we can test whether return on the portfolio is significantly different from zero and we choose proper characteristics based on this. However, we may encounter the problem that none of the mean returns is statistically significantly different from zero. This is not a big concern as the firm-level characteristics are just used to specify an investment strategy.

7

Utility Function

An isoelastic utility of form

U (Q) = Q

1−γ

1 − γ, γ > 0 and γ 6= 1 (47)

will be used since we want to compare our results with Brandt et al.’s. This utility is a special case of hyperbolic absolute risk aversion (HARA) utility functions. The nonnega-tivity of its first derivative U (Q)0 = Q−γ implies that an investor prefers more than less. The second derivative: U (Q)00 = −γQ−γ−1 implies that the investor is risk averse. It’s coefficients of absolute/relative risk-aversion are given by

A(Q) = −U (Q) 00 U (Q)0 = γ Q, and R(Q) = −Q U (Q)00 U (Q)0 = γ. (48)

8

Empirical Application

8.1

Data

We have discussed the relation between traditional mean-variance maximization and Brandt et al.’s improved direct utility maximization. In order to show these two maximization problems are consistent, a simple empirical analysis will be conducted. We use monthly

data from November 2006 to November 2013 of 5 AEX-listed companies 12 _{on NYSE}

Euronext Amsterdam. For each company at the end of every month, we construct the fol-lowing variables: price-to-earnings ratio (P/E), market capitalization (Mcap), short-term reversal (STR), medium-term momentum (MTM) and long-term reversal (LTR). Observa-tions from November 2003 to November 2006 are also used as lag 36/12/1 months returns. In the following application, investors are restricted to only invest in the risky stocks. Thus investing as well in the risk-free asset and other equities is beyond the content of this pa-per. We assume the investors have CRRA utility functions. There are no transaction costs.

Table 1 presents the means, standard deviations as well as Jarque-Bera tests of the monthly returns of these 5 stocks. Means are scaled in percentages. Stock 2 has the highest mean return, it is as high as 2.06%. The lowest mean return is 0.44% for stock 4. These 5 stocks, on average, have a mean return of 1.192 % and a standard deviation of 0.07. In addition, all of the Jarque-Bera tests reject the null hypothesis that the distribution of monthly return is normal at 5 % significance level. As it is known, decreasing the significance level makes it harder to reject the null, however, even if we test at significance level 0.1%, only stock 1 and 3 failed to reject the null. Obviously, not all stock returns are normally distributed. Thus if we can show that mean-variance maximization is consistent with Brandt et al.’s maximization, it is not due to the normality of the returns.

Table 1: Mean, Std. Deviation and Jarque-Bera Test of the 5 stocks during 2006-2013

This table shows the means, standard deviations as well as Jarque-Bera tests of the monthly returns of the selected 5 stocks. Means are scaled in percentages. The first two rows give the means and standard deviations of the stocks. The means and the standard deviations are estimated using Equations (26) and (27). The last two rows show the Jarque-Bera test-statistics along with the p-values. The test statistic is defined as: N₆ s2₊(k−3)2

4



, where N is the number of observations, s is the sample skewness, and k is the sample kurtosis.

stock 1 2 3 4 5

mean(%) 1.59 2.06 0.95 0.44 0.92

Std. Deviation 0.08 0.09 0.06 0.08 0.06

Jarque-Bera test∗∗ 34.17 145.90 39.66 398.44 79.53

p-value 0.001∗ 0.000∗ 0.000∗ 0.001∗ 0.000∗

* Rejects the null hypothesis that the distribution is normal at 5 % significance level.

Table 3 reports the correlations of the returns. As we can see, most of the stocks have typical correlations of 0.5, or around 0.5, with each other. The highest correlation is be-tween stock 1 and stock 2, it is 0.5593. On the other hand, stock 1 is poorly correlated with stock 4 with a correlation only equals to 0.1299. Moreover, no stocks are perfectly positive correlated with others (i.e. correlation equals to 1). This guarantees a gain from portfolio diversification.

Table 2: Correlations of the Returns

The results of zero-cost long-short portfolios for each of the five characteristics are sum-marized in Table 3. All t-statistics are smaller than critical value 1.96. That means we cannot reject the null that the mean returns generated by the five long-short portfolios are statistically zero at 95% confidence level. If we test at 90% confidence level, only Mcap can reject the null. However, this barely matters since the firm characteristics are only used to specify a portfolio. In the section of characteristics selection, I mentioned that Van Rens-burg and Robertson (2003) shed light on how well the stock characteristics jointly explain the stock returns. They find that using size and P/E ratio as explanatory variables gives the optimal model to explain stock returns. Therefore we use Mcap and P/E ratio as firm characteristics. Characteristics other than Mcap and P/E ratio can be also included in the analysis.

Table 3: Zero-cost long-short portfolio for each characteristic

This table shows the summary statistics for the returns of all zero-cost long-short portfolios. The second row of this table illustrates whether we have one euro short on the bottom decile of the portfolio and one euro long on the first decile of the portfolio (first-minus-bottom), or the other way around (bottom-minus-first). The third and fourth rows show the mean returns and the standard errors of all portfolios. The last row contains t-statistics.

STR MTM LTR P/E Mcap

first-bottom bottom-first first-bottom firs-bottom first-bottom

µP 0.0045 0.0055 0.0047 0.0059 0.0053

Std. Err. 0.0038 0.0035 0.0029 0.0041 0.0030

t-value 1.1842 1.5714 1.6207 1.4390 1.7667

8.2

Empirical Results

For varying values of µP, we can calculate a series of mean-variance efficient portfolios

are located under the efficient frontier and no portfolio point can be constructed above it. Thus, investors only consider the portfolios on the efficient frontier. As we can seen, higher returns are associated with higher risks, thus investors need to concentrate their efforts on achieving a better trade-off between risk and return. Figure 1 also exhibits an obvious advantage of mean-variance maximization. For a stock broker, instead of first investigating each of his clients’ utility function, it is very convenient for the broker to provide all of his clients a plot like Figure 1. The clients will decide by themselves which portfolio they want to invest. In fact, neither the broker nor the client need to know the explicit form of the client’s utility function.

Figure 1: Efficient Frontier of Five Stocks

0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.18 0.2 −0.01 −0.005 0 0.005 0.01 0.015 0.02 0.025 0.03 0.035 0.04

Risk (standard deviation)

Mean Return Efficient Frontier Individual Stock Individual Stock Individual Stocks GMV Portfolio Individual Stock

has two characteristics. Thus we can construct the following three portfolio returns: P0 = 5 X i=1 ˜ wi,tri,t, P1 = 5 X i=1 ( ˜wi,t+ 1 5τ > 1 xi,t)ri,t, P2 = 5 X i=1 ( ˜wi,t+ 1 5τ > 2 xi,t)ri,t, (49)

where τ₁> = (1, 0) and τ₁> = (0, 1). For an equal-weighted portfolio benchmark, these portfolio returns reduced to:

P0 = 1 5 5 X i=1 ri,t, P1 = 5 X i=1 (1 5+ 1 5τ > 1 xi,t)ri,t, P2 = 5 X i=1 (1 5+ 1 5τ > 2 xi,t)ri,t. (50)

Applying mean-variance analysis exactly on these three portfolio returns and varying the value of expected return µP, we obtain different mean-variance efficient portfolios. The

re-turns and risks for ten selected efficient portfolios are summarized in Table 4. Moreover, Ta-ble 5 presents the results of Brandt et al’s direct utility maximization with equal-weighted benchmark for four CRRA utility functions. We can observe that standard deviation de-creases as the value of γ inde-creases. This is obvious, since a more risk-averse investor (i.e. an investor with a high γ) will prefer a lower level of risk.

Table 4: Mean-variance analysis on three constructed portfolio returns

1 2 3 4 5 6 7 8 9 10

µP 0.009 0.010 0.011 0.012 0.013 0.014 0.015 0.016 0.017 0.018

Table 5: Summary of five CRRA utilities γ γ = 2 γ = 3 γ = 4 γ = 5 U (Q) = Q_1−γ1−γ U (Q) = −Q−1 U (Q) = −Q₂−2 U (Q) = −Q₃−3 U (Q) = −Q₄−4 ˆ µP 0.019 0.015 0.014 0.012 ˆ σP 0.078 0.062 0.057 0.053 number of iterations 503 80 214 99

conclude that a naive portfolio is not mean-variance efficient because 10% difference is rather large. In general, the CRRA-optimum portfolios coincides with the mean-variance efficient portfolios. If we calculate optimal portfolios for more CRRA utilities, we can obtain a CRRA-utility based efficient frontier which is the same as mean-variance efficient frontier. Notice that Brandt et al. (2009) use utility of the form U (Q) = −Q₄−4, which is the point γ = 5 in Figure 2.

Figure 2: Efficient Frontier with CRRA-Optimum Portfolios

This figure plots the efficient frontier for three constructed portfolio returns along with four CRRA-optimum portfolios. The CRRA-optimum portfolios calculated using Mcap and P/E ratio as characteristics are marked with ’◦’. The three constructed portfolios are marked with ’+’. The GMV portfolio is marked with ’•’. The naive portfolio is represented by ’∗’.

0.05 0.1 0.15 −0.01 −0.005 0 0.005 0.01 0.015 0.02 0.025 0.03 Risk Mean Return Constructed P1 Constructed P0 Naive Portfolio Constructed P2

Efficient Frontier for 3 Constructed Portfolios γ=5

γ=4

Table 6: A Comparison of the Risk between MV Efficient Portfolios and CRRA-optimum Portfolios

This table shows the comparison of the risk between mean-variance efficient portfolios and CRRA-optimum portfolios given the same mean return. The second and fourth column gives the mean returns and risk of CRRA-optimum portfolios. The third column shows the risk of mean-variance efficient portfolios that have the same mean returns of the CRRA-optimum portfolios. Column 5 calculates the ratio of the two risks. Column 6 shows the difference of the risks in percentage scale. Utility function µP σM V ∗ P σCRRA ∗∗ P σM V P σCRRA P σCRRA P −σ M V P σM V P γ = 2 U (Q) = −Q−1 0.019 0.078 0.077 0.99 1.30 % γ = 3 U (Q) = −Q₂−2 0.015 0.062 0.062 1.00 0.00 % γ = 4 U (Q) = −Q₃−3 0.014 0.057 0.057 1.00 0.00 % γ = 5 U (Q) = −Q₄−4 0.012 0.053 0.052 0.98 1.92 % Naive Portfolio 0.011 0.055 0.050 0.91 10.00 %

* the risk of the mean-variance efficient portfolio ** the risk of the CRRA-optimum Portfolio

9

Conclusion

optimize the portfolio weights on a small number of characteristics. Second, PPP captures implicitly the relations between utilities and high moments of returns while in a quadratic maximization model higher moments are neglected. It seems that PPP performs very well. Yet, if we look ”under the surface”, mean-variance analysis and the PPP method essen-tially require the same amount of calculations. This is because when modeling N assets with k characteristics, the PPP approach is equivalent to first transfer these N assets to k + 1 portfolios and then model upon these k + 1 portfolios. If we take one step further, one can realize that if N assets can be reduced to k + 1 portfolios, mean-variance analysis can be used as well based on k + 1 portfolios. In this regard, the problem that traditional mean-variance maximization has to model an enormous number of first two moments is avoided. Moreover, as mean-variance analysis does not really require quadratic utility, we can conclude that even though the two methods (mean-variance analysis and PPP) appear differently, they are the same in essence.

In order to support our argument, we conduct a simple empirical analysis. We select five Dutch stocks and use market capitalization and P/E ratio as the firm-level characteristics. For different CRRA investors, we can find different optimal portfolios using Brandt et al.’s methods. Since there are only two characteristics, we conduct three reduced portfolios and draw an efficient frontier for the three portfolios. Using Brandt et al.’s parametric rules, we obtain four optimal portfolios for four CRRA utility functions. Then we plot these four portfolios along with the efficient frontier. We observe that the optimal portfolios are on or almost on the mean-variance efficient frontier. The errors are within the allowable range.

broker, he just needs to provide his clients a table of mean-variance efficient portfolios. The clients can decide by themselves which portfolios they want to invest. It is unnecessary for the stock broker to investigate his clients’ preferences. In fact, none of them need to know the explicit form of the utility function. This feature makes the mean-variance analysis easier to implement than other utility maximization models.

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