Estimation and Test Theory of Optimal Portfolios: Evidence
from the International Portfolio
Leila Sati
II Semester 2018 Second cycle, 15 credits Master thesis
¨
Orebro University School of Business
Supervisor: Dr. Stepan Mazur, ¨Orebro University Examiner: Dr. Nicklas Pettersson, ¨Orebro University
This thesis analyses the weights of the expected utility (EU) portfolio and the global min-imum variance (GMV) portfolio, the former obtained by maximizing the expected quadratic utility function and the latter derived by minimizing the variance for a given level of the ex-pected return. A Monte Carlo simulation study is performed to obtain the point estimates of the EU portfolio weights. The density function of the estimated GMV portfolio weights is presented. The mean vector and the covariance matrix are estimated from the historical data. We assume returns to be independent and multivariate normally distributed. An empirical study considers international portfolios with different risk-aversion levels. As a result, the point estimates of the EU portfolio weights approach the estimated GMV portfolio weights by increasing the risk-aversion coefficient. Furthermore, a general linear hypothesis test of the EU and GMV portfolio weights is obtained.
Keywords: Optimal portfolio · Expected utility portfolio · Global minimum variance port-folio · General linear hypothesis test.
Acknowledgements
I would like to express my sincere gratitude towards my supervisor, Assistant Professor Stepan Mazur at the ¨Orebro University. This thesis would not have been possible without his continuous encouragement, guidance, and support.
I am grateful to my teachers at the Department of Statistics ( ¨Orebro University) for their aca-demic support and guidance. I also would like to thank my university mates for their friendship and encouragement.
A special word of gratitude is due for my family for their love which kept me motivated and inspired in carrying out this thesis.
1 Introduction 1
2 Optimal Portfolio Selection 3
2.1 What is a portfolio? . . . 3
2.2 Portfolio weights . . . 3
2.3 Returns of an asset . . . 4
2.4 Risk and expected return . . . 5
2.5 Efficient frontier . . . 6
2.6 Reducing risk by diversification . . . 7
3 EU and GMV Portfolios 9 3.1 Expected utility hypothesis . . . 9
3.2 EU portfolio - optimization problem . . . 9
3.3 Expected return and variance of the EU portfolio . . . 11
3.4 GMV portfolio - optimization problem . . . 12
3.5 Expected return and variance of the GMV portfolio . . . 13
4 Statistical Inference of Optimal Portfolios 14 4.1 Estimation . . . 14 4.2 Distributional properties . . . 15 4.3 Test theory . . . 16 5 Empirical Results 18 5.1 Estimation . . . 18 5.2 Test theory . . . 21
6 Conclusion and Discussion 24
References 25
1 1. Introduction
1
Introduction
In 1952 Sir Harry Markowitz introduced the concept of the mean-variance efficient portfolio that minimizes the portfolio variance for a given level of the expected return [Markowitz,1952]. The measurement of risk is determined by a concept of variance that complements the expected return as the fundamental criteria of the portfolio construction. The measure of return is the mean of the portfolio returns. The main approach is to select an optimal portfolio on the efficient frontier with the lowest risk for a given expected return. We focus on the portfolio weights derived by maximizing the expected quadratic utility (EU) function. This portfolio has been analyzed by several authors (e.g. Ingersoll [1987], Okhrin and Schmid[2006], Bodnar and Schmid [2011] ). According toMerton[1980] it is essential to consider the effect of changes in the level of risk. In this thesis we consider the risk-aversion measures for utility functions. According toRoss[1981] the risk-aversion coefficient is a comparative measure of the risk tolerance level and it is used to compare the behaviour of the investor in risky choice situations. Given the same expected return, a risk-averse investor prefers the portfolio with the lower level of risk [Fabozzi et al.,
2012]. We consider investor’s level of tolerance for risk in terms of the risk aversion coefficients α ∈ {5, 10, 50, ∞}. Changing the coefficient of risk aversion allows us to analyze the weights of any portfolio on the efficient frontier curve. We also consider a limiting case of the expected utility (EU) portfolio defined as the global minimum variance (GMV) portfolio with risk aversion coefficient set to infinity. It is a special case of the fully risk-averse investor. Markowitz[1952] introduced the GMV portfolio which obtains the smallest variance among the optimal mean-variance portfolios. The significance of the GMV portfolio in finance is discussed by Merton
[1980],Best and Grauer[1991] and others.
The major concern of the optimal portfolio is that the parameters of the asset return pro-cess, namely mean vector and covariance matrix, are commonly unknown [Bodnar and Schmid,
2011]. Therefore, in this thesis, we examine the estimated parameters from historical data. The distribution of the estimated optimal portfolio performs an essential role in constructing finan-cial investment strategies [Okhrin and Schmid, 2006]. The relevance of the distribution of the weights is discussed byBarberis[2000] andFleming et al.[2001]. They showed the importance of the distribution in developing statistical tests for various mean-variance portfolios and
evalu-ation of the efficiency of the portfolio. According toBodnar et al.[2017], the distribution can be applied for the assessment of uncertainty and for re-balancing the entire portfolio. For example, the distributional characteristics for the weights of the EU portfolio and the weights based on the Sharpe ratio were proved byOkhrin and Schmid[2006]. They derived an exact multivariate den-sity function of the GMV portfolio and a conditional denden-sity for the Sharpe ratio weights. They consider two different cases of finite and infinite sample size. Moreover,Jobson and Korkie[1980] analyzed the weights derived from the Sharpe ratio approach under the assumption of normally distributed returns. They examined the sampling properties at various sample sizes and obtain sampling distributions for estimators of the weights, mean and variance of the Markowitz port-folio. Another analysis under the assumption of normality is performed byBritten-Jones[1999]. As a result, the introduced an exact distribution of the normalized weights.
This thesis aims to consider the statements about the finite sample distribution of the EU and GMV portfolio weights. We consider a statement of the density function of the estimated weights for the EU and GMV portfolios. Moreover, we apply a stochastic representation introduced by
Bodnar and Schmid [2011] which we use for the Monte Carlo approach. Additionally, to make inference about the efficiency of the portfolio or the changes that affect the efficiency of the port-folio, we apply a test for the general hypothesis of the EU and GMV portfolio weights. Our test statistics are obtained similarly as it was suggested byRao and Toutenburg[1995] for linear mod-els. We follow the approach presented byBodnar and Schmid[2008] that the distribution of the returns is under a weak assumption of n > k, where n stands for the number of observations and k is a total number of assets in a portfolio. Okhrin and Schmid[2006] proved that under this assumption all marginal distributions of the estimated weights follow a multivariate t - distribu-tion. In an empirical study, we apply in practice the results obtained theoretically for analyzing the properties of the estimated EU and GMV portfolio weights.
3 2. Optimal Portfolio Selection
2
Optimal Portfolio Selection
2.1
What is a portfolio?
A portfolio is a combination of financial assets owned by the same investor or organization. A portfolio of k-assets at time t is a vector (x1(t), x2(t), ..., xk(t)). At time t = 0 the investor
constructs the portfolio, i.e. he/she buys xi units of asset i, i = 1, ..., k. Corresponding initial
value (initial investment wealth) V0 of such portfolio with the price Pi(t)at time t is given by
V0 = k X i=1 xi(t)Pi(t), (2.1)
2.2
Portfolio weights
The portfolio weight wi of the asset i is obtained by measuring the ith value of the asset relative
to the total investment value V0 at time t = 0, i.e.
wi =
xi(0)Pi(0)
V0
for i = 1, 2, ..., k.
where xi is the number of shares of the ith asset, and Pi(0)is the price of the ith asset at time t
= 0. The sum of the weights should always be equal to 1 and shown by
k X i=1 wi = k X i=1 xi(0)Pi(0) V0 = V0 V0 = 1.
The necessary condition is defined by
wT1 = 1,
where w = (w1, ..., wk)T and 1 stands for a suitable vector of ones.
Portfolio weights may be positive or negative. The case when wi < 0is referred to the short
selling. This means that the investor borrows a risky asset and sells it. However, he needs to repurchase it and return it to the initial owner. Hence, the investor assumes a decrease in the value of the borrowed asset to make a profit when he/she closes the short position. In our analysis, short selling is allowed.
2.3
Returns of an asset
A return is the measure of profit or loss on an investment over a specified period of time. It is also referred to the rate of return. A commonly used way for accomplishing the return of an asset in a particular period is to compute the difference of the asset price at the end of the period and the asset price at the beginning of the period. Returns can be defined as simple (arithmetic) and logarithmic returns. Simple return is known to be more numerous compared to the logarithmic returns.
One-period returns
Let us assume there are no dividends paid and an investor holds an asset from time t − 1 to time t, i.e. one time unit with the asset prices Pt−1 and Ptrespectively. The net one-period simple
returnis the profit rate of holding the asset for a given period of time and defined by
Rt= Pt− Pt−1 Pt−1 = Pt Pt−1 − 1.
The gross one-period simple return is a ratio given by
1 + Rt=
Pt
Pt−1
. (2.2)
Multi-period returns
The investor may hold the asset for more than one time unit i.e. k ≥ 1, gross k-period simple returnfor time period from t − k to k is given by
1 + Rt(k) =
Pt
Pt−k
. (2.3)
Multi-period returns can be expressed as a product of one-period returns since: Pt Pt−k = Pt Pt−1 Pt−1 Pt−2 ...Pt−k+1 Pt−k .
Then equation (2.3) can be written as
Rt(k) = Pt Pt−k − 1 = Pt Pt−1 − 1 + 1 Pt−1 Pt−2 − 1 + 1 ... Pt−k+1 Pt−k − 1 + 1 − 1 = (Rt+ 1) (Rt−1+ 1) ... (Rt−k+1+ 1) − 1. (2.4)
The equation (2.4) leads to an approximation
5 2. Optimal Portfolio Selection
This approxiamtion is valid for a small unit of time. The logarithmic return is a continuously compounded return. More precisely it is a natural logarithm of the gross one-period simple return (i.e of (2.2)) of the asset [Campbell and Shiller,1988] and defined as
rt = log(1 + Rt) = log Pt Pt−1 = log(Pt) − log(Pt−1).
Hudson and Gregoriou[2015] describe advantages of continuously compounded returns as ”for non-stochastic processes, such as the returns on risk-free fixed interest securities held to maturity, when logarithmic returns are used, the frequency of compounding does not matter and returns across assets can more easily be compared”. This means that the continuously compounded k-period return is the sum of the continuously compounded one-k-period returns
rt(k) =log(1 + Rt(k)) = log Pt Pt−k =log Pt Pt−1 Pt−1 Pt−2 ...Pt−k+1 Pt−k = k X i=1 log Pt−i+1 Pt−i =rt+ rt−1+ rt−2+ ... + rt−k+1. (2.6)
Let us point out that the equations (2.4) and (2.5) are approximately the same for small values of returns:
rt= log(1 + Rt) ≈ Rt. (2.7)
Jacquier et al. [2003] and McLean [2011] discussed the appropriate assessment of logarith-mic or simple investment returns. There are several advantages and disadvantages in applying logarithmic and simple returns. In this thesis, we consider logarithmic returns only.
2.4
Risk and expected return
Suppose a portfolio consisting of k assets. The simple returns at time t are R1(t), R2(t), ..., Rk(t).
The return of this portfolio is
Rp = w1R1(t) + w2R2(t) + ... + wkRk(t) = k X i=1 wiRi(t), wherePk
i=1wi = 1and wi represents the proportion of wealth invested in asset i relative to the
total investment value (see also equation (2.4)). The expected return of the given portfolio is
µp = E[Rp] = E " k X i=1 wiRi(t) # = k X i=1 wiE[Ri(t)] = k X i=1 wiµi = wTµ,
where w = (w1, w2, ..., wk)T, µi = E[Ri(t)] is the expected return of an asset i at time t and
µ = [µ1, µ2, ..., µk]T is a k-dimensional vector of all expected returns.
The risk σ2
p of the portfolio p shows how far from the mean µp our true value of return Rp
could be. The risk can be obtained by calculation of the variance of the portfolio return defined as σp2 = V ar(Rp) = V ar k X i=1 wiRi(t) ! = Cov k X i=1 wiRi(t), k X j=1 wjRj(t) ! = k X i=1 k X j=1 wiwjCov(Ri(t), Rj(t)) = wTΣw,
where Σ denotes the k × k covariance matrix with σij = Cov(Ri, Rj)and given by
Σ = σ11 σ12 ... σ1k σ21 σ22 ... σ2k : : : : : : : : σk1 σk2 ... σkk ,
where the diagonal elements of the matrix Σ are the variances of returns, i.e. σii= Cov(Ri, Ri) =
V ar(Ri)for i = 1, ...k.
2.5
Efficient frontier
An efficient frontier represents a relation between standard deviation and expected return rates. It shows an optimal distribution of the weights such that for a given return rate a minimum level of risk is obtained. According toMarkowitz[1959] it is a curve on a graph with expected risk on x-axis and expected return on y-axis. The assets that are not on the efficient frontier curve are considered to be non-optimal. The Figure 2.1 below illustrates the efficient frontier curve, where Rf stand for a risk free asset [Tobin, 1958]. We observe that by increasing the risk we
get higher return. A tangent line to the efficient frontier curve represents Capital Market Line (CML)[Hogan and Warren, 1974]. The intersection of the CML and the efficient frontier curve represents a tangency portfolio. The tangency portfolio is considered to be the most efficient portfolio. It consists only of one portfolio and it is found by maximizing the Sharpe ratio. The
7 2. Optimal Portfolio Selection
Sharpe ratio is a performance measure given as the ratio of the portfolio expected return in excess of the risk free rate over the portfolio’s standard deviation [Jorion, 1985]. The main concept it to choose the portfolio with the highest ratio of expected return to standard deviation, hence by maximizing the Sharpe ratio (Sharpe[1966],Sharpe[1994]).
Figure 2.1: Efficient frontier curve.
2.6
Reducing risk by diversification
One possible way to reduce the risk is investing in multiple assets that are uncorrelated. This phenomena is known as portfolio diversification [Lessard,1976]. Let us consider an example of a portfolio with two assets, i.e. k = 2. Then the weights w1 and w2 can be written in the form
of one vector, i.e. w2 = 1 − w1. The performance of such portfolio at time t defined by rate of
return, expected return and variance of return by using weights w1and 1 − w2 given as
Rp = w1R1(t) + (1 − w1)R2(t), µp = E[Rp] = wTµ = (w1 w2) µ1 µ2 = w1µ1+ (1 − w1)µ2, σp2 = V ar(Rp) = wTΣw = (w1 w2) σ11 σ12 σ21 σ22 w1 w2
= (w1 1 − w1) σ11 σ12 σ21 σ22 w1 1 − w1 = w12σ21+ 2w1(1 − w1)σ12+ (1 − w1)2σ22 = w12σ21+ 2w1(1 − w1)ρσ1σ2+ (1 − w1)2σ22, where ρ = σ12
σ1σ2 is a correlation coefficient between R1and R2 such that −1 ≤ ρ ≤ 1, and hence
σ12 = ρσ1σ2. We observe that one possible approach of risk reduction is under the assumption
of ρ = 0, i.e. when R1 and R2 are uncorrelated. Moreover, the assumption of ρ < 0 represents
negative correlation and is more welcomed by the investor since it reduces the riskiness of the portfolio [Arditti,1967].
The main concern of constructing an optimal portfolio is to find such combination of wi that
9 3. EU and GMV Portfolios
3
EU and GMV Portfolios
3.1
Expected utility hypothesis
The problem of decision making under uncertainty often refers to the Expected Utility Hypothesis. Let us consider a k asset portfolio case with two periods of time, t0 and t1. Assume an investor is
interested to allocate the initial wealth in the portfolio construction which will be held until time t1. We denote initial wealth to be invested as V0(see (2.1)). Then the wealth at time t1is defined
as V1 = 1 + k X i=1 wiRi ! V0.
At time t1the investor derives utility from consuming goods or services obtained by that wealth.
This connection between utility and wealth is defined by a Von Neumann-Morgenstern utility function, or just expected utility function, U (·) [Von Neumann and Morgenstern,2007]. Accord-ing to Expected Utility Hypothesis the investor will choose the composition of portfolio weights so that the expected value of the utility function is maximized [Chopra and Ziemba,2011], i.e.
E [U (V1)] = E ( U " 1 + k X i=1 wiRi ! V0 #)
−→ max such that
k
X
i=1
wi = 1.
Note that each individual or organization could have a unique expected utility function.
3.2
EU portfolio - optimization problem
The Expected Utility Portfolio maximizes the expected utility function. Assume a portfolio con-sisting of k assets and Xn represents a k-dimensional vector of asset returns at time n. The wi
stands for the i-th asset’s weight in the portfolio. Let w = (w1, ..., wk) be the k-dimensional
vector of portfolio weights such that wT1 = 1, where 1 is a k-dimensional vector of ones. The mean vector is µ and the covariance matrix Σ of the asset returns exist and Σ is positive definite. Then the expected return of the portfolio weights w equals to wTµ and the variance is wTΣw. The portfolio weights obtained by solving the maximization problem E[Rp]−α2V ar(Rp)subject
to w1+ w2+ ... + wk= 1, i.e.
wTµ− α
2w
where α represents the level of investor’s risk aversion [Bodnar and Schmid,2011]. We apply the method of Lagrange multipliers to derive a solution of (3.1).
Let α ≥ 0 then the Lagrange function under the matrix notation with a Lagrange multiplier λis given bySteinbach[2001] as follows
L(w, λ) = wTµ−α 2w
TΣw − λ(wT1 − 1).
The first order conditions for a maximum are ∂L(w, λ) ∂wT = µ − αΣw − λ1 = 0, ∂L(w, λ) ∂λ = w T1 − 1 = 0. (3.2)
Then solving equation (3.2) for w we obtain
w∗ = 1
αΣ
−1(µ − λ1). (3.3)
Next, we need to solve (3.2) and (3.3) with respect to λ
1Tw∗ = 1 ⇒ 1TΣ−1 µ− λ1TΣ−1 1 = α ⇒ λ = 1 TΣ−1µ− α 1TΣ−11 ⇒ λ = γ − α β , (3.4) where γ := 1TΣ−1µ and β := 1TΣ−11.
Finally, by substituting the value of λ from (3.4) to the equation of the optimal weights (3.3) we obtain the desired portfolio weights
w∗ = 1 α Σ−1µ− γ − α β Σ −1 1 = Σ −11 β + 1 α βΣ−1µ− γΣ−11 β = Σ −11 1TΣ−11 + 1 α (1TΣ−11)Σ−1µ− (1TΣ−1µ)Σ−11 1TΣ−11 . (3.5)
Therefore, the solution of the maximization problem is given as
wEU = Σ−11 1TΣ−11 + α −1 Rµ with R = Σ−1−Σ −1 11TΣ−1 1TΣ−11 . (3.6)
11 3. EU and GMV Portfolios
3.3
Expected return and variance of the EU portfolio
The expected return of the EU portfolio weights are derived by calculating
µEU = wEUT µ = Σ−11 1TΣ−11 + 1 α (1TΣ−11)Σ−1µ− (1TΣ−1µ)Σ−11 1TΣ−11 T µ = 1 TΣ−1µ 1TΣ−11 + 1 α (1TΣ−11)(µTΣ−1µ) − (1TΣ−1µ)2 1TΣ−11 = 1 TΣ−1µ 1TΣ−11 + 1 αµ TRµ. (3.7)
Note that, since the matrix Σ is symmetric, therefore its inverse is too. The transpose of the inverse covariance matrix is equal to itself, i.e. (Σ−1)T = Σ−1and RT = R.
The variance of the EU portfolio obtained by σEU2 = wTEUΣwEU. For simplicity purposes we
apply the formula (3.5) below with γ := 1TΣ−1µ, β := 1TΣ−11and ν := µTΣ−1
µ σEU2 = wTEUΣwEU = 1 β2 1TΣ−1+ 1 α(βµ TΣ−1− γ1TΣ−1 ) Σ Σ−11 + 1 α(βΣ −1 µ− γΣ−11) . (3.8)
Above computation of (3.8) requires the following information
1TΣ−1ΣΣ−11 = 1TΣ−11 = β,
1TΣ−1Σ(βΣ−1µ− γΣ−11) = βγ − γβ = 0,
(βµTΣ−1− γ1TΣ−1)ΣΣ−11 = βγ − γβ = 0,
(βµTΣ−1− γ1TΣ−1)Σ(βΣ−1µ− γΣ−11) = β2ν − βγ2 − γ2β + γ2β = β2ν − βγ2.
(3.9)
Then by substituting (3.9) in our calculation of (3.8) leads to
σEU2 = 1 β2 β + 1 α2(β 2ν − βγ2) = 1 β + 1 α2 βν − γ2 β = 1 1TΣ−11+ 1 α2 (1TΣ−11)(µTΣ−1µ) − (1TΣ−1µ)2 1TΣ−11 = 1 1TΣ−11+ 1 α2µ TRµ. (3.10)
3.4
GMV portfolio - optimization problem
The Global Minimum Variance Portfolio is the limit case of the EU portfolio that is when α → ∞, i.e. when an investor is fully risk-averse the weights of the EU portfolio and the weights of the Sharpe ratio convert to the GMV weights [Okhrin and Schmid,2006]. Assume the mean vector is µ and the covariance matrix Σ of the asset returns exist and Σ is positive definite. Then the expected return of the portfolio weights w equals to wTµ and the variance is wTΣw. The
portfolio weights obtained by solving the minimization problem V ar(Rp)subject to w1 + w2+
... + wk= 1, i.e. in matrix algebra notation
wTΣw −→min such that wT1 = 1.
Let α ≥ 0 then the Lagrange function under the matrix notation with a Lagrange multiplier τ is given bySteinbach[2001] as
L(w, τ ) = wTΣw − τ (wT1 − 1).
The first order conditions for a minimum are ∂L(w, τ ) ∂wT = 2Σw − τ 1 = 0, ∂L(w, τ ) ∂τ = w T1 − 1 = 0. (3.11)
Then solving equation (3.11) for w we obtain
w∗ = 1 2τ Σ
−1
1. (3.12)
Next, we need to solve (3.11) and (3.12) with respect to τ
1Tw∗ = 1 ⇒ τ 1TΣ−1 1 = 2 ⇒ τ = 2 1TΣ−11. (3.13)
Finally, by substituting τ in (3.12) we obtain the weights of the global minimum variance portfolio:
wGMV =
Σ−11 1TΣ−1
13 3. EU and GMV Portfolios
3.5
Expected return and variance of the GMV portfolio
The expected return of the GMV portfolio derived as follows:
µGM V = wTµ = Σ−11 1TΣ−11 T µ = 1 T Σ−1T µ 1TΣ−11 = 1TΣ−1 µ 1TΣ−11. (3.15)
The variance of the GMV portfolio is calculated by
σGM V2 = wTΣw = Σ−11 1TΣ−11 T ΣΣ−11 1TΣ−11 = 1T Σ−1T µ 1TΣ−11 1 1TΣ−11 = 1 TΣ−1 1 1TΣ−1 11TΣ−1 1 = 1 1TΣ−11. (3.16)
4
Statistical Inference of Optimal Portfolios
4.1
Estimation
In order to determine wEUand wGMVwe need to know the values of µ and Σ. In this thesis, the
as-sumption of independently and normally distributed asset returns is applied, i.e. Xi ∼ Nk(µ, Σ),
i = 1, ..., n. This assumption is considered to be appropriate since it follows the consistency with the mean-variance rule and agrees with the assumptions of the Capital Asset Pricing Model [Stiglitz,1989]. According toFama[1976] monthly stock returns can be well approximated un-der the assumption of Normal distribution. As shown inTu and Zhou[2004] the loss caused by ignoring the fat tails of the asset returns distribution is relatively small. We estimate unknown parameters of the asset returns by collecting the historical data and applying the maximum like-lihood estimators (MLE). The expected returns and the covariance matrix are obtained by
ˆ µ= 1 n n X j=1 Xj = ¯X and Σ =ˆ 1 n n X j=1 (Xj − ¯X)(Xj− ¯X)T, (4.1)
where under the assumption of normality ˆµ∼ Nk(µ,n1Σ)and n ˆΣ ∼ Wk(n − 1, Σ)and they are
independently distributed. We use symbol Wk(n − 1, Σ)to denote the k dimensional Wishart
distribution with n-1 degrees of freedom and the the parameter matrix Σ as it is shown in Chapter 3 ofMuirhead[1982].
Consequently, by substituting derived estimators from (4.1) into the equation (3.6) and (3.13) we obtain estimators of the optimal portfolio weights given by
ˆ wEU = ˆ Σ−11 1TΣˆ−11 + α−1Rˆˆµ with R = ˆˆ Σ−1− Σˆ −1 11TΣˆ−1 1TΣˆ−11 , ˆ wGM V = ˆ Σ−11 1TΣˆ−11.
In order to monitor the optimal portfolio we consider p linear combinations of the estimated portfolio weights that are given by
ˆ θEU = M ˆwEU = L ˆΣ−11 1TΣˆ−11 + α−1M ˆRˆµ, (4.2) ˆ θGM V = M ˆwGMV = M ˆΣ−11 1TΣˆ−11 , (4.3)
15 4. Statistical Inference of Optimal Portfolios
where MT = (m
1, ..., mp)with mi ∈ Rk,i = 1, ..., p. Let us point out that for the EU portfolio
weights p ≤ k − 2 and for GMV portfolio weights p ≤ k − 1Bodnar and Schmid[2011].
4.2
Distributional properties
The distributional properties of the weights were discussed widely by Gibbons et al. [1989],
Okhrin and Schmid [2006] and others. Later on,Bodnar and Schmid [2008] obtained the joint finite distribution of linear combinations of the estimated GMV portfolio weights. In this section, we apply the results derived byBodnar and Schmid[2008] for the GMV portfolio weights. For the EU portfolio weights we implement stochastic representation introduced byBodnar and Schmid
[2011]. Afterwards, we calculate the mean of the estimator and the standard error for the EU portfolio weights and compare them with estimators derived by the Monte Carlo simulations.
From Bodnar and Schmid[2011] we have the stochastic representation of ˆθEU = mTwˆEU
with an arbitrary vector of constants m given by
ˆ θEU d = u1 + ˜ α−1 u2 mTRµ + q (1 + n−k+2k−2 u4)mTRm √ n u3 , (4.4) where u1 ∼ t(n − k + 1, θGM V,n−k+11 mRmT/1TΣ−11), u2 ∼ χ2n−k+1, u3 ∼ N (0, 1), and
u4 ∼ F (k−22 ,n−k+22 , nΛ) with Λ = µTRµ − (mTRµ)2/mRmT and ˜α = α/n. The random
variables u1, u2, u3, u4 are mutually independently distributed. For the Monte Carlo method the
simulated data consists of N = 105independent variables which applied to fit the corresponding
kernel density estimators with Gaussian density. To calculate corresponding exact estimators of the EU weights, we apply the following formulas
E(ˆθEU) = mΣ−11 1TΣ−11 + n n − k − 1α −1 mRµ, V ar(ˆθEU) = 1 n − k − 1 mRlT 1TΣ−11 + α −2 (c1mRµµTRmT+ c2µTRµmRmT) +α −2 n c1+ c2(k − 1) + n2 (n − k + 1)2 mRmT, (4.5) where c1 = n 2(n−k +1) (n−k )(n−k −1)2(n−k −3) and c2 = n 2 (n−k )(n−k −1)(n−k −3).
FromBodnar and Schmid[2008] the distribution of the estimated GMV portfolio weights is given by ˆ θGM V ∼ t n − k + 1 , θGM V, 1 n − k + 1 mRmT 1TΣ−11 . (4.6)
Using the characteristics of the multivariate t - distribution we obtain estimators of the estimated GMV portfolio weights by E(ˆθGM V) = θGM V, V ar(ˆθGM V) = 1 n − k − 1 mRmT 1TΣ−11. (4.7)
Formula (4.6) shows that the linear combinations of the estimated GMV portfolio weights are t-distributed. Since the distribution of ˆθGM V still involves matrix Σ, we can not implement a test
for θGM V if we are unable to estimate the matrix Σ from the historical data.
4.3
Test theory
In this section we consider a test of the general linear hypothesis for the EU and GMV portfolio weights. The test theory of the efficiency of a portfolio has been a subject of discussion in a number of literature studies (e.g. Gibbons et al. [1989], Britten-Jones [1999] and others). For example,Shanken[1985] discussed the multivariate test for the efficiency of the mean-variance portfolio. Gibbons et al.[1989] obtained exact F-tests for given portfolios. Britten-Jones [1999] derived testing procedures for the efficiency of a portfolio based on a single linear regression. For the investor it is important to determine the efficiency of the holding portfolio and examine the market changes that affect the efficiency of the previous portfolio. We aim to test the efficiency of the selected portfolios. The general linear hypothesis given by
H0,EU : mTwEU = r against H1,EU : mTwEU 6= r;
H0,GM V : mTwGM V = r against H1,GM V : mTwGM V 6= r,
where r is a p - dimensional vector. m and r are assumed to be known.
We are interested to know whether the EU and GMV portfolio weights satisfy p linear restric-tions [Bodnar and Schmid,2008]. This testing includes special cases of general testing problems
17 4. Statistical Inference of Optimal Portfolios
discussed byGreene[2003]. We test whether our weights equal to zero or not. This type of test-ing may assist in removtest-ing correspondtest-ing weights and reduce possible transaction costs [Walker and Weber,1984].
According to Bodnar and Schmid[2011] the test statistics for the EU portfolio weights with r = 0 and p = 1 is given by TEU = (n − k + 1) 1TΣˆ−11 mTwˆ EU 2 m ˆRmT 1 + 1TΣˆ−11ˆµTRˆˆµ− 1TΣˆ−11m ˆRˆµµTRmˆ T . (4.8)
The statisic TEU has asymptotically a non-central χ2-distribution with p degrees of freedom and
non-centrality parameter λEU, where
λEU = n
1TΣ−11 mTwEU
2
mRmT 1 + 1TΣˆ−11µTRµ− 1TΣ−11mRµµTRmT
.
Under the null hypothesis TEU is asymptotically χ2-distributed with p degrees of freedom.
Based on the distributional properties of ˆθGM V Bodnar and Schmid[2011] obtained the following
test statistic for r = 0 and p = 1
TGMV = (n − k)
1TΣˆ−11 mTwˆ GM V
2
m ˆRmT . (4.9)
The density of TGM V is given by
fTGM V(x ) = fp,n−k(x )(1 + λGM V) −(n−k+p)/2 ×2F1 n − k + p 2 , n − k + p 2 , p 2; px n − k + px λGM V 1 + λGM V , where λGM V = 1TΣ−11(r − mTwGM V)2 mRmT −1
and 2F1(a, b, c; x)represents a
hypergeo-metric function defined by2F1(a, b, c; x) = Γ(a)Γ(b)Γ(c)
P∞
i=0
Γ(a+i)Γ(b+i) Γ(c+i)
zi
i! [Abramowitz and Stegun,
1984].
Under the null hypothesis it holds that T ∼ Fp,n−k. Corresponding test statistics with different
5
Empirical Results
In this section, we apply the methods discussed above in practice. We derive our estimators from the historical data applying sample statistics. We consider monthly data of the Morgan Stanley Capital International country stock index since monthly stock returns can be well approximated by Normal distribution [Fama,1976]. Data is converted to the US Dollars for the equity markets of the Group of Seven countries with the seven largest economies in the world (Canada, France, Germany, Italy, Japan, UK, and the USA) and three developed countries (Australia, Singapore, and Sweden). The period is from March 2008 to April 2018. The data was imported to the statistical software R on which most of the empirical study was carried out.
5.1
Estimation
We aim to compare the results of an optimal international portfolio obtained by maximizing the expected quadratic utility function and the global minimum variance portfolio achieved by setting the risk aversion coefficient to infinity. The investor interested in international trading can consider it as a benchmark portfolio for his/her investment.
Table 1 presents the estimators for the weights of the mentioned international EU portfolio for the 10-year period with different risk coefficients, α ∈ {5, 10, 50, ∞}. Portfolio exhibits positive and negative weights, negative weights stand for borrowing or short selling. According to Alexander [1993] short selling can be easily incorporated into the mean-variance analysis. For α = 5 we have short positions of approximately 47 %, 76 %, 71 %, 71 %, 5 %, 141 % of the total portfolio securities into the Australia, Canada, Germany, Italy, Japan and the UK markets respectively. Let us point out that the US market obtains the most significant weight in all cases. For α = 5 it is equal to 3.88 and for α → ∞ it is equal to 1.12. The figures of the Japan and UK markets show a considerable reduction of the short positions. For α = 5 Japan and UK markets obtain short positions of 5 % and 141 % respectively. For α → ∞ corresponding figures become 45 % for Japan market and 27 % for UK market. We can also observe that the short positions of the Australia, Canada, Germany and Italy markets reduce to 18 %, 11 %, 47 % and 11 % respectively for α → ∞. Additionally the France market decreases from 81 % for α = 5 to 20 % for α → ∞. Moreover, the figures for the Sweden market change from the long position of 17 % for α = 5
19 5. Empirical Results
to the short position of 17 % for α → ∞. We observe that estimators vary each from another for those that we obtain for different risk aversion coefficients. However, we do not obtain any robust weights.
Applying statistical methods to the results derived by Monte Carlo simulation of N = 105
repetitions, we obtain 95 % and 99 % confidence intervals for each market index separately. We observe that exact estimators derived by the Formula (4.5) and the ones obtained by Monte Carlo approach (4.4) are close enough. Whereas, standard errors have a considerable difference. The second moment of the estimator for the weights of the EU portfolio under the condition of n > k + 1, exists. We apply the Formula (4.5) to calculate it for each estimator separately. For the standard error of the GMV weights we apply the Formula (4.7) and take a square root of the variance.
Table 1 shows that the standard error is quite large when the coefficient of risk aversion is equal to 5 and 10. By increasing the α, we can observe that estimated standard error and exact standard error differ slightly compared to those calculated with the small value of α. Setting α → ∞for the weights of the GMV portfolio, we obtain the lowest standard error values for all ten stocks. This means that the investor is fully risk-averse and the uncertainty of the mean vector of the asset returns are of the less importance. The standard errors for France and the USA weights are the highest at about 150 % and 115 % respectively in the case when α = 5. These figures go down to approximately 22 % for France and 16 % for the USA weights. The smallest standard error for α = 5 is obtained for the Australia and Japan markets at the level of 67 % and 78 % respectively. In the case when α → ∞ the standard error decreases approximately 7 times to the level of 10 % and 11 % for Australia and Japan figures respectively. We find that the standard error is dependent on the coefficient of risk aversion.
Australia Canada France Germany Italy Japan Singapore Sweden UK USA α = 5 Exact Estimator -0.475362 -0.765131 0.816555 -0.714123 -0.708075 -0.052533 0.258696 0.176225 -1.412998 3.876746 MC Estimator -0.478415 -0.766263 0.814720 -0.716874 -0.705238 -0.052363 0.257826 0.173915 -1.417461 3.879168 Exact SE 0.666137 0.829646 1.497997 1.030907 0.791905 0.718291 0.718026 0.815030 1.255060 1.135395 MC SE 0.706918 0.876684 1.592696 1.093325 0.839281 0.761282 0.759136 0.8674682 1.318568 1.169504 Low Limit (95%) -1.885981 -2.534399 -2.302499 -2.888491 -2.387433 -1.587293 -1.229843 -1.515420 -4.108936 1.709777 Upper Limit (95%) 0.906046 0.933246 3.981126 1.433587 0.927442 1.428911 1.770331 1.907574 1.089676 6.314953 Low Limit (99%) -2.406801 -3.184436 -3.416064 -3.661945 -3.002783 -2.113659 -1.749363 -2.110703 -5.104793 1.035495 Upper Limit (99%) 1.379741 1.500158 5.120618 2.171542 1.486126 1.925895 2.299087 2.520611 1.909281 7.222651 α = 10 Exact Estimator -0.328762 -0.437677 0.509363 -0.595524 -0.407636 0.201490 0.126470 0.001973 -0.569765 2.500068 MC Estimator -0.326658 -0.436497 0.506342 -0.594851 -0.406830 0.202398 0.127918 0.000723 -0.567063 2.501053 Exact SE 0.343907 0.428268 0.773377 0.532249 0.408792 0.370799 0.370702 0.420777 0.647644 0.585193 MC SE 0.361993 0.451106 0.815521 0.563460 0.432238 0.391565 0.390439 0.444810 0.679152 0.603923 Low Limit (95%) -1.048989 -1.337485 -1.080953 -1.712313 -1.275613 -0.577033 -0.629650 -0.874328 -1.959869 1.373295 Upper Limit (95%) 0.377011 0.438587 2.136058 0.517543 0.431654 0.956943 0.899165 0.888510 0.726418 3.758344 Low Limit (99%) -1.301215 -1.654017 -1.638749 -2.091832 -1.586555 -0.858140 -0.896655 -1.169070 -2.456866 1.025912 Upper Limit (99%) 0.617536 0.734577 2.727325 0.891513 0.720631 1.215828 1.182513 1.202504 1.140827 4.233495 α = 50 Exact Estimator -0.211482 -0.175713 0.263609 -0.500645 -0.167285 0.404708 0.020689 -0.137429 0.104822 1.398726 MC Estimator -0.211758 -0.174851 0.264825 -0.500183 -0.168167 0.404731 0.020550 -0.137788 0.105486 1.397943 Exact SE 0.118838 0.147802 0.267264 0.183997 0.141101 0.128015 0.128120 0.145406 0.222730 0.198787 MC SE 0.121172 0.150694 0.272619 0.188439 0.143658 0.130858 0.130956 0.147555 0.225966 0.201426 Low Limit (95%) -0.449825 -0.472809 -0.271771 -0.871836 -0.451853 0.147628 -0.235285 -0.427042 -0.340838 1.006251 Upper Limit (95%) 0.027682 0.120900 0.801040 -0.131772 0.112959 0.659750 0.277785 0.152632 0.550311 1.796580 Low Limit (99%) -0.527282 -0.568528 -0.442581 -0.992554 -0.546264 0.065357 -0.321768 -0.515930 -0.489334 0.882360 Upper Limit (99%) 0.106031 0.212118 0.978405 -0.014778 0.200228 0.739705 0.360214 0.248942 0.690618 1.929904 α → ∞ Estimator -0.182163 -0.110223 0.202171 -0.476925 -0.107197 0.455513 -0.005756 -0.172279 0.273469 1.123390 SE 0.098909 0.122937 0.222452 0.153173 0.117371 0.106499 0.106643 0.121023 0.184929 0.163994 Low Limit (95%) -0.201199 -0.139631 0.105880 -0.522579 -0.134003 0.433443 -0.027886 -0.200779 0.206923 1.071058 Upper Limit (95%) -0.163126 -0.080814 0.298461 -0.431272 -0.080391 0.477583 0.016374 -0.143779 0.340015 1.175722 Low Limit (99%) -0.207340 -0.149119 0.074815 -0.537307 -0.142651 0.426323 -0.035025 -0.209974 0.185454 1.054175 Upper Limit (99%) -0.156985 -0.071326 0.329526 -0.416543 -0.071743 0.484703 0.023513 -0.134584 0.361483 1.192605
Table 1: The estimators for the weights of the EU and GMV portfolios. We use monthly data of the Morgan Stanley Capital International country stock index returns. Returns are converted into the US Dollar. The period is from March 2008 to April 2018. The exact weights sum to one. The standard error (SE) is the square root of the variance.
21 5. Empirical Results
5.2
Test theory
We aim to test the efficiency of the EU and GMV portfolios. In our case, we test whether the EU and GMV portfolio weights are equal to 0 or not. This test has application in decision making whether the portfolio equal to zero or not and hence reduce transaction costs [Walker and Weber,
1984].
In Table 1 the test statistics with different coefficients of risk aversion for each country are presented. The null hypothesis for the EU portfolio weights states that H0,EU : mTwEU = 0
and TEU follows the χ2 distribution with p degrees of freedom, in our case p = 1. We reject null
hypothesis if TEU < χ2q/2,p or TEU > χ21−q/2,p. For q = 5 % and p = 1 corresponding critical
values are 0.000982 and 5.023886. Hence the null hypothesis for α = 5 should be rejected for the Japan and USA markets. For α = 10 the null hypothesis is rejected for Sweden and USA markets. Finally, for α = 50 the null hypothesis should be rejected for the Singapore weights only. All these results coincide with the given p-values.
Under the null hypothesis a test statistic TGM V of the GMV portfolio weights follows an F
-distribution with p and n − k degrees of freedom, i.e. TGM V ∼ Fp,n−k. The null hypothesis
of the test statistic is rejected if TGM V < F1,n−k,q/2 or TGM V > F1,n−k,1−q/2 where q denotes
the significance level. For q = 5 % the corresponding quantiles of the F-distribution equal to 0.000987 and 5.164244. The null hypothesis of a weight equal to zero: H0,GM V : mTwGM V = 0
is not rejected for the Australia, Canada, France, Italy, Singapore, Sweden and UK returns. That is the null hypothesis for the Germany, Japan and USA returns should be rejected. For q = 1 % the critical region is given by 0.000039 and 8.206842. The null hypothesis should be rejected for Germany, Japan and USA weights. These results coincide with corresponding p − values of the test statistics. Moreover, from Table 1 we obtain the sum of the estimated GMV weights of the Germany, Japan and USA equal to 0.982 for the lower bound for a 95 % confidence interval. Corresponding sum for a 99 % confidence interval is equal to 0.943 for the Germany, Japan and USA markets. This provides the advantages of a portfolio allocation to the Germany, Japan and USA markets.
Australia Canada France Germany Italy Japan Singapore Sweden UK USA EU α = 5 0.249867 0.418285 0.141571 0.244787 0.391575 0.000039 0.058392 0.172531 0.676363 15.488540 (0.765662) (0.964115) (0.203194) (0.556614) (0.937055) (0.010008) (0.005516) (0.203738) (0.765662) (0.000166) α = 10 0.123517 0.139820 0.056720 0.174541 0.132673 0.055901 0.013890 0.000162 0.103585 6.644521 (0.549497) (0.583080) (0.376485) (0.647785) (0.568646) (0.373808) (0.187637) (0.020339) (0.549498) (0.019892) α = 50 0.054157 0.024232 0.016302 0.126876 0.024034 0.185414 0.000354 0.016390 0.006179 2.224322 (0.368036) (0.247408) (0.203194) (0.556614) (0.246403) (0.666478) (0.030039) (0.203738) (0.368036) (0.271705) GMV α → ∞ 3.423036 0.811225 0.833544 9.783714 0.841795 18.462070 0.002940 2.044993 2.206839 47.355750 (0.133953) (0.739452) (0.726493) (0.004507) (0.721782) (0.000075) (0.086285) (0.311084) (0.2805206) (0.000000)
Table 2: The test statistic TEU for the EU portfolio and TGM V for the GMV portfolio.
Correspond-ing p-values are provided in parentheses.
Table 3 and Figure 5.1 below complement the Table 2. Table 3 shows the expected return and variance of the EU and GMV portfolio weights. Corresponding figures derived by the formulas (3.7) and (3.10) for the EU portfolio. For the expected GMV portfolio returns and variance we applied formulas (3.15) and (3.16) respectively. Table 3 illustrates that by increasing the risk-aversion coefficient we obtain less variance. The variance of the portfolio decreases almost twice from setting α = 5 to α = 10. Moreover we observe that the variance of the EU portfolio approaches the variance of the GMV portfolio by increasing the risk-aversion coefficient. The expected returns for the EU portfolio for α = 5 is equal to 0.007084, for α = 10 is equal to 0.007203 and for α = 50 is equal to 0.00730. The expected return of the GMV portfolio is equal to 0.007321. These results coincide with the results of the test theory derived in section 5.2. and provides advantages of the portfolio allocation to the Germany, Japan, Singapore and USA markets. The Figure 5.1 below shows the allocation of portfolios in a two-way scatter plot. It illustrates that the highest return and minimum risk within the portfolio are obtained by the GMV portfolio. Furthermore, we can see that the closest to the GMV portfolio is the EU portfolio with risk-aversion coefficient α = 50. Next is the EU portfolio with α = 10 and finally the EU portfolio with α = 5 provides the least return with a higher level of standard deviation.
23 5. Empirical Results
Australia Canada France Germany Italy Japan Singapore Sweden UK USA EU α = 5 Expected return 0.000377 0.000278 0.000038 -0.000168 0.002016 0.000338 -0.000202 -0.000555 -0.000009 0.004972 Variance 0.001029 0.000528 0.000018 0.002794 0.000918 0.000075 0.000344 0.000948 0.000002 0.001438 α = 10 Expected return 0.000265 0.000180 -0.000040 -0.000138 0.001297 0.000576 -0.000103 -0.000386 -0.000192 0.005744 Variance 0.000509 0.000221 0.000021 0.001882 0.000380 0.000219 0.000089 0.000460 0.000057 0.001920 α = 50 Expected return 0.000175 0.000101 -0.000103 -0.000114 0.000722 0.000766 -0.000024 -0.000252 -0.000337 0.006362 Variance 0.000223 0.000007 0.000137 0.001282 0.000118 0.000387 0.000005 0.000195 0.000177 0.002355 GMV α → ∞ Expected return 0.000153 0.000008 -0.000118 -0.000108 0.000579 0.000814 -0.000004 -0.000218 -0.000374 0.006516 Variance 0.000170 0.000045 0.000182 0.001150 0.000076 0.000437 0.000002 0.000146 0.000217 0.002471
Table 3: The expected return and variance for the EU and the GMV portfolios.
Figure 5.1: A two-way scatter plot of the EU and GMV portfolios. The risk-aversion coefficient α is given in the parantheses for each EU portfolio.
6
Conclusion and Discussion
In this thesis we have compared the estimators of two optimal portfolios that lie on the efficient frontier. As optimal portfolios we considered the EU portfolio, derived by maximizing the ex-pected quadratic utility, and the GMV portfolio, obtained by minimizing the variance for a given level of return. We assume that returns are independent and follow multivariate normal dis-tribution and hence apply monthly stock returns in our empirical study. In order to derive the estimates of the EU portfolio weights we applied the Monte Carlo simulation of equation (4.4). The simulated data consists of N = 105 independent variables. Corresponding exact estima-tors derived by formula (4.5). The aim is to compare the mean point estimates of the weights of Monte Carlo approach and the exact point estimates. We can observe that there is not much difference between them. The estimates of the GMV portfolio weights are obtained as a solution of the optimization problem discussed in Section 4 and the density function of the equation (4.6). Corresponding standard errors for each portfolio point estimates were provided. Moreover, fol-lowing the theory, the empirical results showed that the estimates of the EU portfolio weights approach the estimates of the GMV portfolio weights by increasing the risk-aversion coefficient α. Moreover, the values of the standard error are dependent on the risk aversion coefficient. For example, by increasing the risk-aversion coefficient we obtain that the values of the standard error are decreasing.
Furthermore, we applied a generalised linear hypothesis test for the EU and the GMV portfolio weights given by equations (4.8) and (4.9) respectively. As a result we obtained that the null hypothesis of the zero return for the EU portfolio weights should be rejected for Japan and USA markets in case of α = 5, for α = 10 the null hypothesis is rejected for Sweden and USA markets, and for α = 50 only for the Singapore market. The null hypothesis for the GMV portfolio weights should be rejected for Germany, Japan and USA markets despite that the mean point estimate of the Germany weight is negative. In addition to the hypothesis test results we can observe that the returns of the Germany, Japan, Singapore and USA markets increase by increasing the risk-aversion coefficient as it can be seen in Table 3 and Figure 5.1. Moreover, the USA market obtains the most significant weight in all cases.
25 References
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29 Appendix
Appendix
Australia Canada France Germany Italy Japan Singapore Sweden UK USA Australia 1.000000 0.805726 0.816352 0.804392 0.743487 0.683877 0.823161 0.830837 0.834583 0.828343 Canada - 1.000000 0.772699 0.757554 0.695935 0.638276 0.826931 0.770754 0.851504 0.830851 France - - 1.000000 0.944784 0.934136 0.703811 0.759167 0.888598 0.901364 0.863494 Germany - - - 1.000000 0.883593 0.733741 0.783555 0.877898 0.868293 0.875710 Italy - - - - 1.000000 0.658526 0.678264 0.804790 0.845100 0.774627 Japan - - - 1.000000 0.703313 0.693232 0.736537 0.734162 Singapore - - - 1.000000 0.808109 0.805422 0.787246 Sweden - - - 1.000000 0.878013 0.850848 UK - - - 1.000000 0.876835 USA - - - 1.000000
