Markowitz Portfolio Optimization Assumptions And Critical Thinking

The purpose of this article is to provide a brief explanation of Markowitz’s modern portfolio theory and how you can use it to more effectively allocate your investment portfolio.  Perhaps equally important to what will be covered is what is excluded: this is not a mathematical derivation of the model.  For a thorough explanation of the math behind the model, see this article in Wikipedia.  The objective of this article is to show how you can apply modern portfolio theory in real life to create an optimized portfolio.

Throughout chapters 1 through 4, we will refer to Excel files that will contain either the templates and the data.  We will also include instructional videos that will provide guidance on using the Excel files and applying the concepts behind modern portfolio theory.  In Chapter 5, we introduce R–a free open-source tool that is a more powerful and flexible alternative to Excel.

We have purposefully started this series by using Excel since most people are the most familiar with this tool.  However, we hope that you’ll read our chapter on R (and associated pages, posts, and video tutorials) and find that the process of downloading return data, plotting an efficient frontier, and finding the optimal portfolio much easier and faster.  In addition, R allows for flexibility that cannot be achieved in Excel.  We have provided all of the code necessary to get up and running with R in a matter of minutes.  We hope you find our series a compelling reason for using R as an alternative to Excel.

Introduction

In its simplest form, portfolio theory is about finding the balance between maximizing your return and minimizing your risk.  The objective is to select your investments in such as way as to diversify your risks while not reducing your expected return. It is actually simple to apply and effective.  While it does not replace the role of an informed investor, it can provide a powerful tool to complement an actively managed portfolio.

Models should never be blindly applied–see any number of articles on the role of models in the collapse of LTCM or other large funds.  But an understanding of how the portfolio theory works will enable you to make more informed decisions about which mutual funds to include in your 401k or which ETF’s to buy for your individual investor account.

Your portfolio (401k, IIA, etc.) probably consists of a number of stocks, bonds, ETF’s, and mutual funds.  The mix of these assets constitutes your portfolio allocation.  How your portfolio is allocated determines its performance.  During the first quarter of every year, investors typically spend a few hours reallocating their retirement accounts.  Most allocation decisions are based on past performance, gut feelings, or some arbitrary selection process. In this series, we’ll introduce you to the modern portfolio theory and demonstrate how you can use Microsoft Excel to construct an efficient and optimal portfolio.

Definitions and Assumptions

Before we begin to explain portfolio theory and its application, let’s begin by defining a number of key terms.  A common nomenclature is essential to correctly interpreting this series.

  • Return:  For many assets, this may include both capital appreciation (the price of the stock rises) and dividends.  For debt instruments, the return may include price appreciation (for example, when interest rates fall), the periodic interest payments, or the payment of the principal.  Expected returns may be based on historical performance; however, it is important to think critically about whether past performance is likely to continue in the future.  (For example, do you really expect to see a 50% rise in technology stocks year-over-year for the next 10 years?)
  • Risk:  This is perhaps the most contentious definition.  In the context of this series, risk is the measure of variability in the expected return.  We will use simple statistical tools to quantify risk.  Risk is typically based on past volatility; however, as with returns, investors should think critically about the assumptions underlying the estimates of risk.  If anything, the recent credit crisis has shown that two assets that appeared to be unrelated (uncorrelated, which we’ll cover later) may actually move together quite quickly under certain economic conditions.

Organization of this Series

We’ve organized this site to be accessible to both students and professionals.  For students, we’ve tailored the articles to answer common questions and provide context and real-world examples to the theory taught in the classroom.  For professionals, we aimed to make this a practical, hands-on, exercise with plenty of actual case studies.

  • Chapter 1 – Introduction to Portfolio Theory
    • A practical explanation of the ideas behind the theory
    • Definitions, assumptions, limitations, and other important points to keep in mind
  • Chapter 2 – Example Portfolio Data Set 
    • The mix of investments we’ll use throughout the articles to illustrate how to use the theory
    • Historical returns and volatility
  • Chapter 3 – Covariation Analysis–or simply, how related are the movements in individual investments
  • Chapter 4 – Efficient Frontier
    • What is the minimum risk for an expected return
    • Comparison with example portfolios
  • Chapter 5 – Using R as an Alternative to Excel
  • Chapter 6 – Tracking Portfolio Performance
    • How does our mean variance optimized portfolio perform against traditional allocations?

Why is this Important?

So why do you care about modern portfolio theory, a dead economist named Markowitz, or something called an efficient frontier? Simply because you can use this approach to lower your risk (portfolio variance) while maintaining (or increasing) your expected returns.  Which of the following portfolios would you prefer?

  • A mix of stocks and bonds that returned an average of 7% per year, but varied by as much as 10% per year (i.e., returns varied typically between -3% and +17%)
  • A mix of stocks and bonds that returned an average of 7% per year, but varied by only 2% per year (i.e., returns varied typically between 5% and 9%)

Although an annual return of 17% sounds good, keep in mind that it is also as likely that you’ll lose 3%!  A less volatile return of between 5% and 9% may be less exciting but will get you closer to your retirement goals faster.

>> Next Article in the Series:  Example Portfolio

"Portfolio analysis" redirects here. For theorems about the mean-variance efficient frontier, see Mutual fund separation theorem. For non-mean-variance portfolio analysis, see Marginal conditional stochastic dominance.

Modern portfolio theory (MPT), or mean-variance analysis, is a mathematical framework for assembling a portfolio of assets such that the expected return is maximized for a given level of risk, defined as variance. Its key insight is that an asset's risk and return should not be assessed by itself, but by how it contributes to a portfolio's overall risk and return.

Economist Harry Markowitz introduced MPT in a 1952 essay,[1] for which he was later awarded a Nobel Prize in economics.

Mathematical model[edit]

Risk and expected return[edit]

MPT assumes that investors are risk averse, meaning that given two portfolios that offer the same expected return, investors will prefer the less risky one. Thus, an investor will take on increased risk only if compensated by higher expected returns. Conversely, an investor who wants higher expected returns must accept more risk. The exact trade-off will be the same for all investors, but different investors will evaluate the trade-off differently based on individual risk aversion characteristics. The implication is that a rational investor will not invest in a portfolio if a second portfolio exists with a more favorable risk-expected return profile – i.e., if for that level of risk an alternative portfolio exists that has better expected returns.

Under the model:

In general:

where is the return on the portfolio, is the return on asset i and is the weighting of component asset (that is, the proportion of asset "i" in the portfolio).
  • Portfolio return variance:
,
where is the (sample) standard deviation of the periodic returns on an asset, and is the correlation coefficient between the returns on assets i and j. Alternatively the expression can be written as:
,
where for , or
,
where is the (sample) covariance of the periodic returns on the two assets, or alternatively denoted as , or .
  • Portfolio return volatility (standard deviation):

For a two asset portfolio:

  • Portfolio return:
  • Portfolio variance:

For a three asset portfolio:

  • Portfolio return:
  • Portfolio variance:

Diversification[edit]

An investor can reduce portfolio risk simply by holding combinations of instruments that are not perfectly positively correlated (correlation coefficient). In other words, investors can reduce their exposure to individual asset risk by holding a diversified portfolio of assets. Diversification may allow for the same portfolio expected return with reduced risk. These ideas have been started with Markowitz and then reinforced by other economists and mathematicians such as Andrew Brennan who have expressed ideas in the limitation of variance through portfolio theory.

If all the asset pairs have correlations of 0—they are perfectly uncorrelated—the portfolio's return variance is the sum over all assets of the square of the fraction held in the asset times the asset's return variance (and the portfolio standard deviation is the square root of this sum).

If all the asset pairs have correlations of 1—they are perfectly positively correlated—then the portfolio return’s standard deviation is the sum of the asset returns’ standard deviations weighted by the fractions held in the portfolio. For given portfolio weights and given standard deviations of asset returns, the case of all correlations being 1 gives the highest possible standard deviation of portfolio return.

Efficient frontier with no risk-free asset[edit]

Main article: Efficient frontier

See also: Portfolio optimization

This graph shows expected return (vertical) versus standard deviation. This is called the 'risk-expected return space.' Every possible combination of risky assets, can be plotted in this risk-expected return space, and the collection of all such possible portfolios defines a region in this space. The left boundary of this region is a hyperbola,[2] and the upper edge of this region is the efficient frontier in the absence of a risk-free asset (sometimes called "the Markowitz bullet"). Combinations along this upper edge represent portfolios (including no holdings of the risk-free asset) for which there is lowest risk for a given level of expected return. Equivalently, a portfolio lying on the efficient frontier represents the combination offering the best possible expected return for given risk level. The tangent to the hyperbola at the tangency point indicates the best possible capital allocation line (CAL).

Matrices are preferred for calculations of the efficient frontier.

In matrix form, for a given "risk tolerance" , the efficient frontier is found by minimizing the following expression:

where

The above optimization finds the point on the frontier at which the inverse of the slope of the frontier would be q if portfolio return variance instead of standard deviation were plotted horizontally. The frontier in its entirety is parametric on q.

Many software packages, including MATLAB, Microsoft Excel, Mathematica and R, provide optimization routines suitable for the above problem.

An alternative approach to specifying the efficient frontier is to do so parametrically on the expected portfolio return This version of the problem requires that we minimize

subject to

for parameter . This problem is easily solved using a Lagrange multiplier.

Two mutual fund theorem[edit]

One key result of the above analysis is the two mutual fund theorem.[2] This theorem states that any portfolio on the efficient frontier can be generated by holding a combination of any two given portfolios on the frontier; the latter two given portfolios are the "mutual funds" in the theorem's name. So in the absence of a risk-free asset, an investor can achieve any desired efficient portfolio even if all that is accessible is a pair of efficient mutual funds. If the location of the desired portfolio on the frontier is between the locations of the two mutual funds, both mutual funds will be held in positive quantities. If the desired portfolio is outside the range spanned by the two mutual funds, then one of the mutual funds must be sold short (held in negative quantity) while the size of the investment in the other mutual fund must be greater than the amount available for investment (the excess being funded by the borrowing from the other fund).

Risk-free asset and the capital allocation line[edit]

Main article: Capital allocation line

The risk-free asset is the (hypothetical) asset that pays a risk-free rate. In practice, short-term government securities (such as US treasury bills) are used as a risk-free asset, because they pay a fixed rate of interest and have exceptionally low default risk. The risk-free asset has zero variance in returns (hence is risk-free); it is also uncorrelated with any other asset (by definition, since its variance is zero). As a result, when it is combined with any other asset or portfolio of assets, the change in return is linearly related to the change in risk as the proportions in the combination vary.

When a risk-free asset is introduced, the half-line shown in the figure is the new efficient frontier. It is tangent to the hyperbola at the pure risky portfolio with the highest Sharpe ratio. Its vertical intercept represents a portfolio with 100% of holdings in the risk-free asset; the tangency with the hyperbola represents a portfolio with no risk-free holdings and 100% of assets held in the portfolio occurring at the tangency point; points between those points are portfolios containing positive amounts of both the risky tangency portfolio and the risk-free asset; and points on the half-line beyond the tangency point are leveraged portfolios involving negative holdings of the risk-free asset (the latter has been sold short—in other words, the investor has borrowed at the risk-free rate) and an amount invested in the tangency portfolio equal to more than 100% of the investor's initial capital. This efficient half-line is called the capital allocation line (CAL), and its formula can be shown to be

In this formula P is the sub-portfolio of risky assets at the tangency with the Markowitz bullet, F is the risk-free asset, and C is a combination of portfolios P and F.

By the diagram, the introduction of the risk-free asset as a possible component of the portfolio has improved the range of risk-expected return combinations available, because everywhere except at the tangency portfolio the half-line gives a higher expected return than the hyperbola does at every possible risk level. The fact that all points on the linear efficient locus can be achieved by a combination of holdings of the risk-free asset and the tangency portfolio is known as the one mutual fund theorem,[2] where the mutual fund referred to is the tangency portfolio.

Asset pricing[edit]

The above analysis describes optimal behavior of an individual investor. Asset pricing theory builds on this analysis in the following way. Since everyone holds the risky assets in identical proportions to each other—namely in the proportions given by the tangency portfolio—in market equilibrium the risky assets' prices, and therefore their expected returns, will adjust so that the ratios in the tangency portfolio are the same as the ratios in which the risky assets are supplied to the market. Thus relative supplies will equal relative demands. MPT derives the required expected return for a correctly priced asset in this context.

Systematic risk and specific risk[edit]

Specific risk is the risk associated with individual assets - within a portfolio these risks can be reduced through diversification (specific risks "cancel out"). Specific risk is also called diversifiable, unique, unsystematic, or idiosyncratic risk. Systematic risk (a.k.a. portfolio risk or market risk) refers to the risk common to all securities—except for selling short as noted below, systematic risk cannot be diversified away (within one market). Within the market portfolio, asset specific risk will be diversified away to the extent possible. Systematic risk is therefore equated with the risk (standard deviation) of the market portfolio.

Since a security will be purchased only if it improves the risk-expected return characteristics of the market portfolio, the relevant measure of the risk of a security is the risk it adds to the market portfolio, and not its risk in isolation. In this context, the volatility of the asset, and its correlation with the market portfolio, are historically observed and are therefore given. (There are several approaches to asset pricing that attempt to price assets by modelling the stochastic properties of the moments of assets' returns - these are broadly referred to as conditional asset pricing models.)

Systematic risks within one market can be managed through a strategy of using both long and short positions within one portfolio, creating a "market neutral" portfolio. Market neutral portfolios, therefore will have a correlations of zero.

Capital asset pricing model[edit]

Main article: Capital asset pricing model

The asset return depends on the amount paid for the asset today. The price paid must ensure that the market portfolio's risk / return characteristics improve when the asset is added to it. The CAPM is a model that derives the theoretical required expected return (i.e., discount rate) for an asset in a market, given the risk-free rate available to investors and the risk of the market as a whole. The CAPM is usually expressed:

  • , Beta, is the measure of asset sensitivity to a movement in the overall market; Beta is usually found via regression on historical data. Betas exceeding one signify more than average "riskiness" in the sense of the asset's contribution to overall portfolio risk; betas below one indicate a lower than average risk contribution.
  • is the market premium, the expected excess return of the market portfolio's expected return over the risk-free rate.

The derivation is as follows:

(1) The incremental impact on risk and expected return when an additional risky asset, a, is added to the market portfolio, m, follows from the formulae for a two-asset portfolio. These results are used to derive the asset-appropriate discount rate.

  • Updated market portfolio's risk =
Hence, risk added to portfolio =
but since the weight of the asset will be relatively low,
i.e. additional risk =
  • Market portfolio's expected return =
Hence additional expected return =

(2) If an asset, a, is correctly priced, the improvement in its risk-to-expected return ratio achieved by adding it to the market portfolio, m, will at least match the gains of spending that money on an increased stake in the market portfolio. The assumption is that the investor will purchase the asset with funds borrowed at the risk-free rate, ; this is rational if .

Thus:
i.e. :
i.e. :
Efficient Frontier. The hyperbola is sometimes referred to as the 'Markowitz Bullet', and is the efficient frontier if no risk-free asset is available. With a risk-free asset, the straight line is the efficient frontier.

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