The Markowitz Portfolio Theory

Published Date: 02 Nov 2017

Abstract

There are two parts in this paper, the first one is about the mean variance model and the semi-variance model. Afterwards, ten stocks will be chosen for the portfolio optimization. The second part focuses on two Bayesian methods for estimation risk, i.e. diffuse prior and conjugate prior. The purpose of this paper is to discuss and compare the portfolio compositions and performance of the mean variance model and semi-variance model in portfolio optimization. Moreover, the impact of estimation risk is also talked about when optimize portfolio.

1 Introduction

Portfolio selection is one of the most significant problems in investment management. The first paper in the field go back to the "Portfolio Selection" of Markowitz (1952), which became the pioneer work of modern portfolio theory. The main idea of the theoretical framework is to maximize the expected return for a given amount of risk or to minimise the risk when giving a specified level of return by choosing the weight of various assets.

Undoubtedly, Markowitz mean variance model plays a significant role in investment management. However, empirical studies have shown that the assumptions of the mean variance model may be wrongly founded as the distributions of the common stock returns may not be normal. To overcome this limitations, different measures of risk is required. Semi-variance model is suggested by Markowitz in 1959 to be an alternative risk measure.

In addition, the mean variance model often performs poorly out-of-sample due to estimation risk in the mean return and covariance matrix. As a consequence, minimum-variance portfolios may produce unstable weights that affect stability over time. It may also lead to extreme portfolio weights and dramatic fluctuation in weights with only unimportant changes in expected returns or the covariance matrix because of the loss of stability. Therefore, for the sake of achieving better stability properties, two Bayesian methods(diffuse prior and conjugate prior) has been introduced to reduce the estimation risk.

There are two sections in the paper. The first is comparisons between the mean variance model and semi-variance model under different criterions, such as the minimum risk rule. Then, ten stocks are selected for portfolio. Secondly, the two popular Bayesian approach: diffuse prior and conjugate prior are discussed.

2 Background

In this section, some main concepts of portfolio theory used in this paper will be introduced, including assumptions, risk aversion etc.

2.1 Some background assumptions

Before presenting the portfolio theory, we introduce some general assumptions for the theory. One basic assumptions in the portfolio theory is that investors maximize the returns of the total sets of investment for a specified risk level. It also assumes that investors are risk averse, which denotes that the investor will choose the asset with the lower level of risk when two assets have the same rate of return.

2.2 Markowitz portfolio theory

In the 1960s, the investment community discussed the risk, however, there is no specific measure for it. Investors need to quantify the risk in order to build a portfolio model. The basic portfolio model was proposed by Harry Markowitz(1952,1959), who derived the expected rate of return for a portfolio of assets and an expected risk measure. Markowitz obtained the variance of the expected rate of retrun of assets was a good measure for quantifying the risk under some assumptions. Moreover, he derived the formula that was used to calculate the variance. The formula not only showed the significance of diversifying investment to reduce the total risk of the portfolio but also denoted how to diversify effectively. The Markowitz theory is based on the following assumptions on the investor behaviors:

1 Investors choose each portfolio by the probability distribution of expected returns over some holding period

2 Investors aim to maximize the expected utility for one period, and the utility curve shows the diminishing marginal utility of wealth

3 Investors use the variability of expected return to estimate the risk of the portfolio

4 Investors select a portfolio by the expected return and risk, so the utility are the function of expected return and variance(or standard deviration) of returns solely.

5 For a given expected return level, investors prefer less risk to more risk. Likewise, for a specified level of risk, investors prefer higher expected returns than lower returns.

Under these assumptions, a portfolio assets or a single asset is considered to be optimal if no other portfolio assets or other asset can offer higher expected returns with the same or less risk or less risk with the same or higher returns.

2.2.1 Expected rates of return

The most basic concept in any investment is the return or the rate of return over a given investment period. Thus, in a single period, the return or the rate of return can be calculated by following formula:

=

where is the amount invested at the beginning of the period and is the amount received by the investor at the end of the period.

In fact, the rate of return in a specified investment priod is always regarded as a random variable r. The value of expected return (E(r)) is significant in portfolio optimisation. Expected rates of return for a portfolio of investments is the weighted average of the expected rates of return for the individual investments in the portfolio. The wights are the percentage of total value for the individual investment.[]

Therefore, the formula for the expected rates of return E(r) of m assets is:

E(r) =

Where indicates the weight of an individual asset in the portfolio

denotes the expected rate of return of asset i

2.2.2 Measures of risk

For most investors, risk means the uncertainty of future outcomes. An alternative definition might be the probability of an adverse outcome.[]

In the classical mean variance model, variance or the standard deviation of returns is used to quantify risk.

The variance or the standard deviation of an individual investment E(r) is

Variance==

Where is the probability of rate of return

Standard deviation is the square root of the variance.

Standard deviation= =

Futhermore, the standard deviation of a portfolio is defined by

where , indicates the weight of an individual asset i,j in the portfolio respectively, is the standard deviation of and is the covariance between asset i and asset j.

This formula denotes the standard deviation for a portfolio is a function of the weighted average of the individual variances and the weighted covariances between all the assets in the portfolio.

2.23 The efficient frontier

The envelop curve that contains all possible combinations that have the best expected return for any risk level is reffered to as the efficient frontier[]. The efficient frontier represents the set of portfolios that has the lowest level of risk for every given level of retrun or the highest return for every given level of risk and is located in the upward-sloped part of the left boundary. Below is an example of an efficient frontier.

Investors select a point on the efficient frontier based on their utility function. The optimal portfolio is the efficient portfolio that has the highest utility.

According to the descriptions of Luenberger [1998], the market portfolio is the

summation of all assets if everyone invests the same amount of risky asset and assigns to thesame means, variances as well as covariances for the returns, namely, it must incorporate every asset in proportion to that asset's representation in the total market. The straight line passing through the risk free rate point and the market portfolio point is called captital market line, which can be seen in the figure. Obviously, this line shows the relationship between the expected return and the risk measured by variance for efficient assets or portfolios. In addition, the curve above the minimum-variance point is the efficient frontier.

Furthermore, under certain assumptions, Fama [13] demonstrates that the tangency portfolio which is the feasible point that maximizes the angle between a line drawn from the risk-free asset to a point in the feasible region and the horizontal axis must contain all assets available to investors, and each asset must be included in proportion to its market value associating with the entire market. Thus, the tangency portfolio is usually considered as the market portfolio.

Meanwhile, if the angle is indicated by , then, of tangency portfolio can be derived by

=

where is the expected return of tangency portfolio, is the risk-free rate and is the standard deviation of tangency portfolio. Actually, this is also equal to the Sharpe ratio of the tangency portfolio. Paticularly, all portfolios have a Sharp ratio no more than the market portfolio. Therefore, the tangency portfolio is often regarded as the portfolio of risky assets with the highest Sharpe ratio.

2.3 Portfolio Performance Measures

2.3.1 Sharpe Ratio

The sharpe ratio is suggested by William Forsyth Sharpe [1996] as a composite measure to evaluate the performance of mutual funds. The Shape ratio is defined as follows:

S=

Where E(r) is the expected return of portfolio, is the risk free return of portfolio and is the standard deviation of rate of return for portfolio

This performance measure concentrates on measuring the additional return (or risk premium) earned per unit of dispersion in an investment asset or a trading strategy which is considered as risk, that is, a variance risk measure. Finally, the standard deviation of rate of retun can be calculated by using either portfolio returns in excess of the risk free rate or total portfolio returns.

2.3.2 Sortnio Ratio

The Sortnio ratio measures the actual return above investors’ target return per unit of downside return. In fact, it is a modification of the Sharpe ratio , which measures the risk-adjusted return of an investment asset, portfolio or strategy. Unlike Sharpe ratio that penalizes both upside and downside volatility equally, the Sortnio ratio only penalize those returns below the investor’s target or given rate of return.The ratio is calculated as follows:

S=

Where E(r) is the return of an asset or portfolio, T is the target or required rate or return and was initially known as the minimum acceptable return(MAR) and the denominator is the square root where a return is less than the required rate of return, which should be positive otherwise 0 if the return is greater than the required rate of return.

2.3.3 Omega Ratio

The Omega ratio is also a measure of risk of an investment asset, portfolio or strategy based on the level of returns in return for the risk of investing in assets.It is a ratio of weighted gains to weighted losses — a ratio that includes information about the probability of each level of returns. Copmared to its predecessors, the Omega ratio differs according to the shape of the distribution of an asset’s returns. This allows investors to distinguish between assets with different risk profiles. The omega ratio can be calculate as follows:

where F is the cumulative distribution function, r denotes the threshold defining the gain versus the loss. The higher the ratio the better.

3 Markowitz portfolio selection models

3.1 Mean-Variance Model

The mean variance model is regarded as one of the basic theories investment management. It is a single period model that provides investors the optimal tradeoff between expected return and risk. It solves the portfolio problems by using two parameters: expected return(mean return) and risk(variance of return).

3.1.2 Assumptions

Markowitz made the following assumptions when developing the model:

All investors aim to maximize utility.

Only one period investment is considered.

Investors are risk averse.

Investors select a portfolio by the mean and the variance.

There are no taxes and transaction costs. And all securities can be divided into any size.

Actions of investors do not affect prices.

Correlations between assets are fixed and constant forever

3.1.3 Classical framework for portfolio optimization

·Minimum Variance Portfolio

For a standard portfolio problem of m assets with expected returns(i=1,2,…,m). And they are independently across time as normal, with variance . Let be the weight that the portfolio value invested in the asset i, which sums to 1. And let = (,…,), w= and 1=.Negative weights as well as weights that are greater than 1 are allowed, corresponding to short selling. The expected return and variance of the portfolio are = and =w respectively. Therefore the portfolio selection problem of minimum variance criterion can be written as:

Min =w

Subject to =

1=1 (1)

for all levels of expected portfolio return above that of the minimum variance portfolio. Investors select a particular portfolio investment on Markowitz’s efficient frontier in maximizing the expected utility. This portfolio is defined as the optimal portfolio with the optimal weights , expected return and variance .

The first constrain denote the expected return is specified. There are two constrains and we introduce two Lagrangian multipliers and to get the Lagrangian function:

L(w, ,)=w- 2(-)-2（ 1-1）

The optimal condition is w-2-21=0 （2）

-2=0 (3)

1-2=0 (4)

Assume is reversible, from equation 2) , we can obtain:

W=+1) (5)

Put (5) in (3), we can obtain:

+1=B+A= （6）

Put (5) in (4), we can obtain:

1+11=A+C=1 (7)

And A=1,B=,C=1,D=BC-

Since is reversible, B,C>0 and =B(BC-)>0

So D=BC->0 ,from (6), (7), we can obtain

=(C-A)

=(B-A)

In equation (5), we can get

W=(C-A)+ (B-A)1)

And the corresponding variance is

=w=(+1))= +1=+

=(C-2A+B)= + (8)

Equation (8) is equal to (9)

is the expected return and is the corresponding variance.

In ( dimensions, equation (9) is the standard form of hyperbola whose center point is (0, and axis of symmetry are =0 and =. We only consider the part in the first quadrant as the standard deviation() is greater than 0.

a

-

From the picture above. the point a is the vertex of hyperbola in the first quadrant. Also as can be seen, this point has the lowest variance. So the optimal solution in this case is:

·Maximum Expected Return Portfolio

Alternatively, if the investors aim to maximize their expected return, the problem can be formulated as follows:

Max =

Subject to w =

1=1 (1)

3.2 Semi-Variance Model

The variance has become one of the most popular measure of risk since the introduction of mean-variance model. However, empirical studies shown that the assumption of the mean variance approach may be wrongly founded. The studies have also demonstrated that the distribution of common stocks may not always be normal. Therefore, the classical mean variance model may not be a satisfactory portofolio selection model. As a consequence, probably, a different measure of risk are needed: semi-variance.

The semi-variance measure of risk is also proposed by Markowitz in 1959. Semi-variance aims to measure the movements of below average returns. The definition is given as follows:

=E(()

The semi-variance approach for portfolio optimization attempts to minimize under-performance and does not penalize over-performance of expected return of the portfolio. In order to find the optimal portfolio using semi-variance, it is unnecessary to calculate the covariance matrix, instead, the joint distribution of assets is required. This approach attempts to minimize the dispersion of return from the expected return only when the dispersion return is less than the expected return. If all distributions of returns are normal, or have the same degree of asymmetry, the optimal portfolios choose by the variance and semi-variance approaches are same.

Let be observed return on asset i. i=1.2….,m,at time t, t=1.2….T, and the data

Then the the portfolio selcetion problem can be formulated as follows (Markowitz et al. (1993)):

Min =

Subject to -), j=1,2,…T

0, j=1,2,…,T

=

1=1 (1)

The first two constrains denote that when an observed return is less than the expected return, then the corresponding variable y will be strictly positive. As the object function is to minimize the sum of , the correspongding variable y will take the difference between the expected return and the observed return. On the other hand, when an observed return is greater than the expected return, the variable y will be negative but because every variables need to be non-negative, the variable will take the value greater than or equal to zero. Meanwhile, for the sake of minimizing , the optimal value of y will be zero. Therefore, the object function and the constrain above will solve the portfolio optimization problems by semi-variance approach. The given expected return of the portfolio will lie between minmum possible portfolio return and maximum possible portfolio return . The minimum possible portfolio return can be obtained by solving the following similar problem.

Min =

Subject to -), j=1,2,…T

0, j=1,2,…,T

=

1=1 (1)

Likewise, the maximum possible return can be derived by solving the closely related problem that replace the of the problem above to . The main problem is solved for different values between and . The corresponding semi-variance is obtained for any given return . Thus the efficient frontier are plotted by the expected returns and semi-variances. The corresponding w represents the weight that invest in each asset to obtain the given return and the particular optimal portfolio. Any point on the efficient frontier is optimal. And investors can choose one point on the efficient frontier that satisfy their needs.

4 Estimation Risk

The Markowitz model typically assumes investors know the true value of parameters(expected return and variance), however, in practice, these parameters are usually not known and need to be estimated.

4.1 Classical Method

The classical method, ignoring the estimation risk, simply regard the estimates as the true values and plug them into the optimal portfolio under the mean variance model.

Under the assumption that is identical independent normal distributed, the sample mean and covariance matrix and by using equations below:

(4)

And (5)

Putting (4) and (5) into the equations (2) and (3) to obtain the final results.

The calculation methods above are known as the plug in methods.

The plug in method regard as the estimated values and as the real values, and thus ignore the errors, which may lead to better portfolio choice existing. Best and Grauer also said the optimal portfolio choice obtained by the plug in method is sensitive to the values of parameter especially the mean value. Increasing or decreasing several samples may have a small influence on values of parameter estimators, but the small change may lead to the optimal portfolio solution changing. In addition, investors invest most of their assest in the portfolio with high expected return and the low variance when the optimal portfolio is obtained by this method. However, this type of assets tend to have higher estimation error compared to other assets. Therefore, the optimal portfolio choosen by this method usually performs bad and is not stable.

4.2 Bayesian approach

There is much information available to predict future rate of return on individual assets. The information not only includes the past observations for rate of return but also can contain accounting information, economic news and so on.

Another approach, Bayesian approach is also used to estimate the parameters. In this model, the parameters are not regarded as the fixed values, instead, they are treated as random variables. The probability of all values of parameters is considered when predicting the distribution of returns.

The Bayesian approach is based on the minimizing the variance from knowledge of joint predictive distribution of security returns. There are a large number of Bayesian posterior efficient frontier, since the posterior estimated efficient frontier are the function of the prior distribution, which assumes the multivariate normal distribution.

Optimal portfolio choice: Noninformative or invariant priors

In this section, we consider the case of noninformative or invariant priors and derive the optimal portfolio choice. The joint probablility distribution of asset returns R is assumed to be multivariate normal distributed with mean and variance matrix . Assuming investors know the data of return until time t, then we should estimate the return at time t+1. Let be observed return on asset i. i=1.2….,m,at time t, t=1.2….T, and the data

, consist of observations from a multivariate normal distribution with mean vector and variance matrix (A TXT identity matrix) and And therefore the portfolio return P=R is normally distributed with mean = and variance =w. Finally, in the noninformative or invariant prior condition, the prior distribution of unknown parameter =(is

Let =(,…,),=, and so is

=

Let r be the T X m matrix with T observations on the m assets , that is to say

r= , define = =

The and are regarded as the estimators of and , respectively. And therefore. The predictive distribution is a student t distribution with T-m degrees of freedom. The parameters and are

=

Note that the predictive distribution of portfolio return is an induced distribution. We cannot obtain this distribution by assigning a prior to the parameters of the univariate portfolio return distribution. Instead, the induced predictive distribution of portfolio return is derived by the multivariate distribution for individual assets and then assign a prior to the parameters of this distribution.

In addition. and are independent in the prior. And small changes in the data for return of assets will alter largely the posterior probability distribution function for ( since the prior for ( is noninformative.Thus, the data play a import role in determining the posterior distribution for paremeters.

The mean and covariance matrix of predictive of returns will be given by (Brown 1976)

So =var（=（1+

Estimation risk does not affect the optimal portfolio choice as and are proportional to and , and the optimal portfolio sets are still derived with comparisons of and for student t distributions.

Conjugate prior distributions for covariance matrices

The conjugate prior retains the same class of distributions, and is a natural and common

informative prior on any problem in decision making. In this case, the conjugate

specification considers a normal prior for (conditional on ) and inverted Wishart prior

for . The conjugate prior is

where is the prior mean, is a coefficient that reflects the prior precision of , and similarly is also the prior mean and is a coeffficient that reflects the prior precision of .

Under this prior, the posterior distribution of and obeys the same form as that based on the conjugate prior, except that now the posterior mean of is given by a weighted average of the prior and sample means.

=

Likewise, is given by

=+

which is a weighted average of the prior variance, sample variance, and deviations of

from .

Frost & Savarino (1986) provide an interesting application of the conjugate prior,

assuming a priori that all assets exhibit identical means, variances, and patterned covariances.

They find that such a prior improves ex post performance. This prior is related the

well-known 1/N rule that invests equally across the N assets.

www.

The Bayesian approach use the predictive distribution to deal with estimation risk. Firstly, the joint predictive distribution of security rates of return should be determined which in turn determins the predictive distribution of return for all feasible portfolios. Therefore, we discuss which type of information determines the predictive distribution.

There is much information investors can assess the future security return.

While the plug in methods ignore the estimation risk, the Bayesian approach based on the predictive distribution, proposed by Zellner and Chetty(1965) provides a general framework which considers estimation risk.

Under the plug in methods, the utility is defined with parameters and . In the Bayesian approach, the investors cares about the expected utility with the predictive distribution, which is determined by the historical data and prior. With a good choice of prior, it is undoubltable that the Bayesian approach preforms better than the plug in methods.

Empirical analysis

In this section, ten stock is used to compare mean variance model and semi-variance model under different criterions. The data used in this study are form January 1998 to February 2013. The portfolio are constructed based on the past returns over the 62 months. The portfolio performance is calculated using the reward per

risk equation:

Portfolio Performance = mean return / risk

Standard Deviation

Semi-Variance

Maximum Sharpe

Maximum Omega

Maximum Return

Minimum Risk

Maximum Sharpe

Maximum Omege

Maximum Return

Minimum Risk

NSANY

0

0

0

0.00001

0

0

0

0.00015

TM

0

0

0

0.44043

0

0

0

0.50336

EBAY

0

0

0

0.00048

0

0

0

0.00129

DD

0

0

0

0.00059

0

0

0

0.00039

ADB

0

0

0

0,00068

0

0

0

0.0009

ORCL

0

0

0

0.09842

0

0

0

0.08712

BA

0

0

0

0.13063

0

0

0

0.00219

GE

0

0

0

0.00024

0

0

0

0.00047

EMC

0

0

0

0.00037

0

0

0

0.00115

EW

1

1

1

0.32814

1

1

1

0.40298

Table Weight of stocks in the optimal porfolio

Results denote that the optimal portfolios generated by the two risk measures do not differ very much. The difference is the weight of stocks. According to Byrne and Lee (2004), the difference in weight is probably due to the non-normality displayed by data.

Criterion

Return

Variance

Sharpe

Omega

Performance

Variance

Maximum Sharpe

0.024903

0.007086

0.253065

1.623806

3.514216

Maximum Omega

0.024903

0.007086

0.253065

1.623806

3.514216

Maximum Return

0.024903

0.007086

0.253065

1.623806

3.514216

Minimum Risk

0.010786

0.003039

0.130347

1.037855

3.548724

Semi-variance

Maximum Sortino

0.024903

0.005816

0.279332

1.623806

4.281586

Maximum Omega

0.024903

0.005816

0.279332

1.623806

4.281586

Maximum Return

0.024903

0.005816

0.279332

1.623806

4.281586

Minimum Risk

0.012159

0.002742

0.163443

1.105955

4.434061

Table Summary statistics of the optimal portfolio

It can be seen above value of performance(1.89512) of mean variance model under minimum risk condition is larger than semi-variane model(1.87062), so the semi-variance model outperforms the mean variance model. Simliarly, the same conclusions can also be obtained under other criterions(maximum omega and maximum return) since the values of performance of the mean variance model is always less than the semi-variance model. The different result of the two models is due to the fact that the distributions of asset return are not normal.

The data consisting of 62 historical monthly returns (January,1998 - February

0.2013) of ten stocks chosen by section 3 is used to compare the classical method and the Bayesian method(diffuse prior).

The efficient frontiers of the classical method and diffuse prior method are shown in the picture below. As what discussed before, investors encounter higher level of risk under the diffuse prior method, which is derived by right shift of the efficient frontier of the classical method. If we increase the number of data, the two efficient frontiers will coincide with each other.