Overview About Harry Markowitz Model

Published Date: 02 Nov 2017

Harry Markowitz (born in Chicago on August 24, 1927) is a talented American economist and an elite educator of finance. He was widely recognized as a pioneer in the financial field in the 1950s. After finishing his college career in the University of Chicago, Markowitz entered the RAND Corporation in 1952. Markowitz has worked as a professor of economics in the New York University since 1982.

With his outstanding contribution in fields of portfolio theory, SIMSCRIPT programming language and sparse matrix, Markowitz was granted the Von Neumann Prize in Operations Research Theory by the Operations Research Society of America and The Institute of Management Sciences in 1989. In 1990, Markowitz, together with William Sharp and Merton Miller, was awarded the Nobel Memorial Prize in Economic Sciences. The research which earned Markowitz that Nobel prize was first published in his article entitled "Portfolio Selection" (1952), and more comprehensively, in his book, "Portfolio Selection: Efficient Diversification" (1959). The Modern Portfolio Theory developed by Markowitz has been presented and analyzed in most college texts and courses on investments. It is currently applied by institutional investors worldwide to construct an optimal portfolio with risk control, asset management, return maximization and diversification.

Besides economics, Markowitz has a special passion for philosophy as well. In his autobiography (1990), he revealed that he had started reading original works of well-known philosophers in high school. One of the most intriguing books that inspired Markowitz’s interest in that field was "OII: Observation, Interpretation and Integration", written by Joseph J. Schwab (1923). Markowitz also strongly expressed his soul of arts. He enjoyed live performances of Shakespeare’s plays and poets composed by Robert Frost and Robert Browning.

At the moment, Markowitz is putting his great effort in writing a four-volume intensive book called "Risk-Return Analysis: Which, Why, When & How". The first volume of the book is planned to be published at the end of 2013.

1.2 Harry Markowitz Model of Portfolio Selection

1.2.1 Introduction

The Markowitz Portfolio model was first introduced in the Journal of Finance in 1952. Since its appearance, it has become a standard and reliable theoretical reference for most financial modern researchers and investors. The Markowitz model illustrates geometrically relations between beliefs and choice of portfolio relying on the "expected returns-variance of returns" rule (Markowitz. H, 1952). It is acknowledged as the scientific-based formalization of the investment diversification idea, which is "Don’t put all your eggs in one basket".

Before Markowitz developed his portfolio selection techniques, prior studies had mainly concentrated on assessing rewards generated by individual investment opportunity. Markowitz model retained the emphasis on return but it raised investors’ awareness about the equal importance of risk. At the same time, the concept of portfolio risk was first formally developed. "To reduce risk, it is necessary to avoid a portfolio whose securities are all highly correlated with each other." He affirmed in his book (1959). Markowitz was also the first to formally and clearly show how diversification could work to reduce the variance of a portfolio. Instead of constructing portfolios based on attractive risk-reward characteristics of individual securities, it was recommended for investors to rely on overall risk-reward characteristics of the whole portfolio. "Probably the most important aspect of Markowitz’s work was to show that it’s not security’s own risk that is important to investors but rather the contribution the security makes to the variance of his entire portfolio"(Rubinstein. M, 2002). To put it simply, it is rational to make risk-return evaluation on the whole portfolio, instead of on just single security. This perspective was absolutely missed from previous studies and findings.

Above all, Markowitz model targets to maximize the terminal wealth, while minimizing the risk using the variance as a criterion, which supports investors to seek highest return given their acceptable risk level.

1.2.2 Assumptions of Markowitz model

Regarding investors’ behaviors, there are five basic assumptions as follows:

Investors are rational and seek to maximise the expected return

Investors are risk averse so they require a higher expected return to compensate for higher risk accepted

Investors rely merely on expected returns and variance to make investment decisions.

Single period investment horizon is expected to be the same among investors

Investors are price-taker (cannot influence prices)

Risk is perceived by investors as the standard deviation of return

Five other following conditions of markets are assumed by Markowitz model:

Full and correct information on returns and risks are always available

New information is absorbed and reflected quickly and perfectly

There is no transaction cost or taxes

Short-selling is permitted without any restriction

Every asset is tradable at any point of time

1.2.3 Parameters of Markowitz model

Expected rate of Return

Return of a portfolio is the weighted average return of the securities consisted in the portfolio. The general formula of expected return for n assets is:

Where:

=

1.0;

N

=

Total number of securities;

=

Proportion or Weight of the total funds invested in security i;

=

Return on ith security and return on portfolio p;

Risk

Although there may be different ways to measure risk level, the most widely-used concept is variance. Variance measures the variability or the dispersion of realized returns around an expected value. The larger the variance indicates the higher the risk in the portfolio, all other things being equal. The rationale lies on the fact that the more disperse the expected returns the greater the uncertainty of those returns in the future. The square root of variance is called standard deviation.

Unlike portfolio expected return, portfolio variance is not simply the weighted average of all individual securities’ variances. Another crucial factor called covariance also has a strong impact on the result of portfolio variance. Covariance tells investors the degree to which two securities returns move together. Covariance with positive value implies that the rate of return for two securities tends to move in the same direction while a negative covariance implying opposite returns directions

Accordingly, portfolio variance is defined as the weighted average covariance of the returns on its individual securities:

Noted correlation is the covariance of two securities A and B divided by the product of the standard deviation of these two securities.

In which:

: Correlation coefficient between two rates of return

, : Standard deviations ofand

Correlation is bounded by -1 and +1.

A correlation of +1 is interpreted that the returns of the two securities always move in the same direction; they are perfectly positively correlated.

A correlation of zero refers that the two securities are uncorrelated and have no relationship to each other.

A correlation of –1 means the returns always move in the opposite direction and are negatively correlated

Therefore, another way to express the formula for computing portfolio variance is

All in all, in Markowitz model, every risky security in the market requires an estimate of:

expected returns

risk measured by standard deviation

correlation between each pair of risky securities.

1.2.4. Markowitz Diversification

Diversification is a risk management technique that combines a varied range of securities within a portfolio with a view to minimizing any negative influence of single security on the overall portfolio performance. Markowitz. H (1959) emphasized that diversification should not only aim at lowering the risk of a security by reducing its variability or standard deviation, but by lessening the covariance or interactive risk between securities in a portfolio. To put it in another way, portfolio risk could be effectively reduced by holding securities with returns that do not move in tandem with each other.

Being aware of two categories of risks would help to better understand the concept of Markowitz diversification

Systematic risk: the uncertainty inherent to the entire market that may negatively influence almost all market participants. It is virtually impossible to protect portfolio against systematic risk through diversification.

Unsystematic risk: the uncertainty that originates from the nature of a particular business or specific investment. Diversification can be utilized to effectively reduce this type of risk.

1.2.5 Capital allocation line

Taking consideration into portfolios including risky assets and risk-free security, with varying different weights for these types of assets, different portfolios could be created. All these possible combinations of the risk-free asset and the risky assets are illustrated by a straight line called Capital allocation line (CAL). Intercept of CAL is the Risk free rate and Slope of CAL is determined by the following formula:

Rf Return of risk-free asset

E(Ri) Expected return of risky assethttp://financetrain.com/assets/CAL.gif

Figure 2.1 Capital allocation line

1.2.5 Efficient frontier

In Markowitz portfolio, a portfolio of assets is defined as being efficient if there is no other portfolio providing higher expected return with the same level of risk or offering lower risk with the same expected return. An efficient portfolio requires an efficient asset combination.

The efficient frontier is the set of all these efficient portfolios. For any level of risk, the efficient frontier identifies the portfolio providing highest rewards. Similarly, the frontier points out the portfolio of lowest risk given any rate of return.

Figure 2.2 illustrates the set of all possible risk-return combinations offered by portfolios with 2 assets A and B, which is named the investment opportunity set. The curve passing through A and B represents that risk-return combinations. Rational investors would desire portfolios that are located on the northwest in Figure 2.2. These are portfolios providing high return rate (to the north of the figure) and low level of volatility (to the west of the figure)

A

Minimum Variance Portfolio (MVP)

0

B

B’

Investment

Opportunity Set

V

Standard Deviation

Expected Return

Z

Figure 2.2: Investment opportunity set for asset A and asset B

The area within the curve BVAZ is the attainable opportunity set available from all possible combinations of component securities. Portfolios lying below point V are regarded as inefficient because they do not satisfy the objective of maximizing rate of return for a given risk level or minimizing risk for a given return level. This can easily be seen by comparing the portfolio illustrated by two points B and B’. As a rule, investors are likely to prefer more expected return than less for a given risk level, B’ is perceived as being better than B. In Markowitz model, the portfolio at point V is called as the minimum-variance portfolio; since there are no other portfolios that can offer a lower standard deviation. The curve passing point V and point A shows all possible efficient portfolios and is identified as the efficient frontier.

0

Z’

Z

A

P

V

Expected Return

Standard Deviation

B

without short sales

with short sales

Figure 2.3: The efficient frontier of unrestricted/restricted portfolio

A rational investor is expected to select portfolios lying on the efficient frontier. Any portfolio that is positioned under the Efficient Frontier is considered to be Inefficient while portfolios with risk/return combinations above the Efficient Frontier are Impossible.

1.2.6 Optimal risky portfolio

After having the efficient frontier, the question is how investors identify the optimal portfolio allocation that best fit their personal risk preference. Hence, an optimal portfolio is the portfolio that carefully considers the investor's own risk aversion or investor’s willingness to trade off between risk and expected return. Risk aversion is quantified by the utility function as follows

U = E(r) - 0.005A

where: U is the utility value.

E(r)) is the expected return.

A is the index of investor's aversion.

is the standard deviation.

The constant number of 0.005 is a scaling convention that makes it possible to present the expected return and standard deviation in the utility equation as percentages instead of decimals. Following this function, the utility from a portfolio increases when the expected return rate rises, and it decreases when the variance declines. The relative magnitude of those changes is influenced by the coefficient of risk aversion, A. In case A = 0, investors are considered as risk-neutral, higher levels of risk aversion will be reflected in larger values for A.

The utility function could be demonstrated graphically by an indifference curve. The property of the indifferent curve is that the individual is indifferent among all portfolios lying on that curve. The indifferent curve illustrates the family of risk-return combinations identifying the trade-offs between risk and expected rate of return. It reflects the increment in return that an individual investor will expect so as to compensate for an increment in risk.

0

B

A

Standard Deviation

Expected Return

Minimum variance

portfolio

Efficient frontier of

risky assets

Figure 2.4 Indifference Curves and Efficient frontier

Portfolio selection is decided by graphing the efficient-frontier set of attainable investment opportunities and investors’ utility functions. In Figure X.6, there are two sets of indifference curves labeled andand one efficient frontier. The curve with higher slope, demonstrates a greater risk aversion level. Thecurve depicts a less risk-averse investor who has more willingness to accept relatively higher risk to achieve higher return. Optimal portfolio is the portfolio offering the highest utility—a point in the direction of northwest (lower risk and higher return). This optimal portfolio is located at the tangent of indifferent curve and the efficient frontier. A particular investor, therefore, would determine the optimal portfolio given his or her preference with respect to risk.

Capital allocation line (CAL), which reveals all available risk-return combinations from varied asset allocation possibility, is constructed to determine the optimal risky portfolio. To begin with, take consideration into portfolio construction consisting of a risk-free assets combined with two risky assets (asset A and asset B).

Final weights and are expected to produce the CAL with highest slope (or highest reward-to-variability ratio). Thus, the question is how to maximize the CAL’s slope (labeled as ) for any possible portfolio, P. The objective function, therefore is as follows

With regards to the portfolio formed by two risky assets, portfolio expected return and standard deviation are:

Maximize the objection function, is accompanied with satisfying the constraint that the sum of portfolio weights is equal to 1. Mathematically, it is expressed as:

Subject to

.As for portfolio of two risky assets, formula to solve weights of the optimal risky portfolio are

The next procedure is to establish an optimal complete portfolio from the optimal risky portfolio and the CAL constructed combining a risk-free asset at two- risky-asset portfolio. The level of risk aversion of a particular investor, A, is ready to compute the optimal proportion of portfolio investing in risky assets

Assuming that a risk-free rate is , and a risky portfolio with expected return and standard deviation . According to Markowitz model, the expected return of the complete portfolio is

The variance of the complete portfolio is

The investor targets to maximum utility, U, by selecting the best allocation to the risky asset, . This problem is mathematically written as follows:

The derivative of this above expression is equal to be zero, the optimal pproportion investing in the risky asset, , then is determined as follows:

The formula points out that the optimal position in the risky asset is negatively related to the degree of risk (measured by variance) and the degree of risk aversion whereas it positively proportional to the risk premium.

In short, the procedure to construct the complete optimal portfolio are summarized and generalized into following steps

Work out expected returns, standard deviations, covariances of component securities

Establish the optimal risky portfolio, find out optimal proportion investing in different assets included in the portfolio

Establish the optimal complete portfolio combining between risky portfolio and risk free asset. Allocate funds invested in each risky assets and risk-free asset

Indifference Curve

Opportunity Set of Risky Assets

Optimal Complete

Portfolio

CAL

0

Optimal Risky Portfolio

Standard Deviation

Expected Return

Figure 2.5: Determination of the optimal portfolio

In practice, when constructing optimal risky portfolios consisting of more than two risky assets, investors are recommended to resort to Microsoft EXCEL or other computer programs.

2. Advances on Modern Portfolio Theory

Markowitz’s pioneering work has inspired numerous innovations and serious studies to be developed in the portfolio construction area. Continuing efforts have been made to extend Markowitz’ mean-variance models.

James Tobin (1958) enabled leverage or deleverage portfolios on the efficient frontier by adding a risk-free asset to the portfolio. Concepts of super-efficient portfolio that outperforms the market and capital market line (CML) – a special case of CAL were accordingly born. According to James Tobin (1959), the super-efficient portfolio is the portfolio producing a rate of return even above the efficient frontier by taking into account the risk-free asset (the interest rate of government securities). The line that depicts the rates of return for efficient portfolios which are dependent on level of risk (measured by standard deviation) and the risk-free rate. Graphically, CML is a tangent line drawn from the intercept point of the efficient frontier and the point in which the expected rate of return equals the risk-free rate.

http://www.captaineconomics.fr/images/nov2012/capital-market-line-portolio.jpeg

Figure 2.6: Capital market line

Sharpe (1964) introduced the capital asset pricing model (CAPM) with new important parameter called beta . Beta is considered as a powerful measure of systematic risk. Calculated by CAMP, the expected return of a security equals a risk premium plus the rate of risk-free security.

E(Ri) = Rf + (E(Rm) - Rf)

where: E(Ri) is the expected rate of return on the risky asset

Rf is the risk-free rate of interest

is the beta determined by the expression:

E(Rm) - Rf is the risk premium

In case, the result of expected return does not exceed the required rate of return, the investment is not worth proceeding. By this way, CAMP makes great contribution to fairly price any securities or investments. CAPM is illustrated by Securities Market Line (SML) The x-axis of SML plots the risk (beta) while the y-axis shows the expected return and SML’s slope is specified by the market risk premium. When risk versus expected return of a security is located above the SML, it is considered as undervalued because investors are able to achieve a greater return for the given risk. When a security with risk-return graphed below the SML is called overvalued as investors would accept less return given the risk level assumed.

http://letslearnfinance.net/wp-content/uploads/2012/06/capm-security-market-line.gif

Figure 2.7: Security market line

In later years, there have been significant attempts to improve portfolio selection from single period model to dynamic multi-period or continuous-time models.

Multi-period utility models were put forward in papers of Mossin (1968), Samuelson (1969), Hakansson (1971), Elton and Gruber [1975], Francis (1976), Grauer and Hakansson (1993). As affirmed in those studies, the expected utility maximization model offered a natural framework for a rational investor to decide a portfolio allocation. Relied on standard assumptions on the utility function such as monotonicity and concavity, the optimal portfolio is well characterized in complete markets in both the single and multi-period settings .

As for continuous-time models, in 1973, the problem of utility maximization with market factors of diffusion process was first investigated by Merton. Subsequently, Duffie and Richardson (1991) and Schweizer (1996) found out the optimal dynamic strategy to deal with mean-variance hedging problem. A complete general solution was subsequently published by Rheinländer and Schweizer (1997) and Gourieroux et al. (1998). Papers on mean-variance hedging in discrete time, however, are relatively scarce. Schäl (1994) selected dynamic programming for constant investment opportunity set to investigate different intertemporal mean-variance criteria. Schweizer (1995) worked out the general problem with one asset and non-stochastic interest rate. After that, Laurent and Pham (1999) relied on the framework of duality theory and dynamic programming to reveal explicit characterization of the variance-optimal measure.

Zhou. X and Li. D (2000) developed a stochastic linear-quadratic model to continue assessing the continuous-time version of the Markowitz’s problem. One year later, the question of Stochastic linear quadratic control was further investigated through semidefinite programming by Yao. D and Zhou. X (2001). The purpose of this approach is to find out control function that maximizes the portfolio value subject to the condition that stock prices are modeled by stochastic differential equation. Yao, D and Zhou, X. (2006) published their studies titled "Tracking a financial benchmark using a few assets" which confirmed that the control of a portfolio affects not only the average return of the portfolio but also its volatility.

In addition, limitations of using variance as risk measurement in Markowitz model were identified (Roll, 1977, 1978, 1979), motivating number of studies to look for more suitable risk parameter. For example, Fishburn (1977), Bawa (1977) suggested mean-lower partial moment approach, Yitzhaki (1982); Shalit, Yitzhaki (1984) came up with mean-Gini portfolio selection model. Konno and Yamazaki (1991) advanced Mean-Absolute Deviation (MAD) approach.

The concept of Value at risk (VaR) was first mentioned in Linsmeier [Pearson 1996] and [Jorion 1997). Since its introduction, VaR has had tremendous applications financial areas. It is widely accepted as a risk measurement for any type of portfolios (including credit risk, insurance risk and market risk). Mathematically speaking, VaR estimates the expected loss in the worst condition over a specific time, given confidence level in normal market conditions. Another way of understanding is that VaR is the lowest quantile of the potential losses that can happen within a specified portfolio during a determined period of time. The basic time period T and the confidence level (or also called the quantile) q are two main parameters that are targeted to have appropriate selection with the overall goal of risk measurement. The time horizon can vary from a few hours for an active trading desk to a year for a pension fund. In case the primary aim is to satisfy external regulatory requirements, such as bank capital requirements, the quantile is quite small (for example, 1% of worst outcomes). However for an internal risk management model to control the risk exposure within the company, 5% of worst outcomes is the typical number. As regards to limitation of VaR, it was criticized for being unable to tackle the extent of the losses occurred beyond the threshold amount. It is impossible for VaR to differentiate between situations where losses that are worse; a little bit worse, and seriously worse. Actually, it only offers a lowest bound for losses in the tail of the loss distribution and has a bias toward optimism instead of the conservatism (Artzner et al. 1999)

Value at risk was extended to Conditional Value at risk (CVaR) by Rockafellar and Uryasev in 2000. Other names of this term are "Mean Excess Loss", "Mean Shortfall" and "Tail VaR".). CVaR was calculated by taking a weighted average between the VaR and losses exceeding the VaR. CvaR, therefore is capable of examining losses beyond VaR. It provides optimization short-cuts which simplify large-scale calculations via linear programming techniques. As a tool in optimization modeling, CVaR has superior usage in various aspects. It keep consistent with VaR by producing same results in the limited settings where VaR computations are tractable, i.e., for normal distributions (or perhaps ‘‘elliptical’’ distributions as in Embrechts et al. (2001)); for portfolios with simple distributions,working with CVaR, VaR, or minimum variance (Markowitz, 1952) are equivalent (Rockafellar and Uryasev, 2000). In addition, CVaR can be expressed by a remarkable minimization formula. This formula can readily be incorporated into problems of optimization that are designed to minimize risk or shape it within bounds. Significant shortcuts are thereby achieved while preserving crucial problem features like convexity. Such computational advantages of CVaR over VaR are turning into a major stimulus for the development of CVaR methodology, in view of the fact that efficient algorithms for optimization of VaR in high-dimensional settings are still not available, despite the substantial efforts that have gone into research in that direction (Andersen and Sornette, 1999; Basak and Shapiro, 1998; Gaivoronski and Pflug, 2000; Gourieroux et al., 2000; Grootweld and Hallerbach, 2000; Kast et al., 1998; Puelz, 1999; Tasche, 1999.

2.3. Review of studies in Vietnamese Stock Market

There have been a limited number of available articles and studies in Vietnam providing an insight into applications of Markowitz’s portfolio selection theory into domestic securities trading.

In August 2007, on the website of Saga which is one of the most comprehensive and reliable online site in Vietnam, the first study involved in assessing practical usage of Markowitz theory was published. Data for that paper was monthly prices of only five stock codes listed on Ho Chi Minh Stock Exchange, namely, AGF, GIL, GMD, NKD and REE during 3-year trading period from December 2004 to July 2007. A complete portfolio including those five stocks was finally constructed with the support of Crystal Ball add-in.

In December 2008, Mr. Do Hung Manh, a senior student of Hanoi University released his research paper with the topic "Do Markowitz and Sharp models improve investment performance of an investor in Vietnamese stock market". That research extended number of component stocks to 10, including. Mr Do figured out optimal portfolios by investigating the case of two stock portfolio

In 2010, Mr. Le Anh Tuan in his postgraduate thesis attempted to analyze current situations of portfolio investment activities in Vietnam. After running number of statistics models, his thesis proved that modern portfolio theories such as Markowitz’s and Sharp’s could be "applicable" and "feasible" in Vietnam stock market.

A detailed report of other eight students in 2011 figured out how to construct the efficient frontier based on weekly prices of 24 stocks 40 traded on both HOSE and HNX from September 2009 to June. In this report, both cases which are short-selling allowed and no short-selling are firstly raised and investigated