Modern Portfolio Theory: A Dialectic Approach

Stephen Slade

February 2008

Introduction

Modern portfolio theory is based on several key concepts, some of which have been recognized with the Nobel Prize in economics. If modern portfolio theory is the thesis, then behavioral finance is the Hegelian antithesis, which has also garnered a couple of Nobel Prizes. The synthesis is post modern portfolio theory.

The key players are:

• Harry Markowitz. Born 1927. Developed portfolio theory for his Ph.D. thesis at University of Chicago in 1955. Won Nobel Prize for economics in 1990. Key concepts: mean-variance analysis and efficient frontier.

• James Tobin. (1918 – 2002). Ph.D. in Economics from Harvard in 1947. Economic advisor to President Kennedy. Professor at Yale. Won Nobel Prize in Economics in 1981. Key concepts: portfolio selection theory, super-efficient portfolio, and the capital market line.

• William Sharpe. Born 1934. Student of Markowitz at UCLA. Earned Ph.D. in 1961. Won Nobel Prize in 1990. Key concepts: Capital Asset Pricing Model (CAPM) and Sharpe Ratio. Professor Emeritus from Stanford. Current interest: retirement economics (Financial Engines: https://www.financialengines.com/ ).

• Eugene Fama. Born 1939. Ph.D. from University of Chicago in 1965. Won Nobel Prize in 2013. Key concepts: efficient market hypothesis, random walk, and questions regarding CAPM (with Kenneth French, born 1954.)

• Stephen Ross. Born 1944. Ph.D. in economics from Harvard. Professor at MIT. Key concepts: Arbitrage Pricing Theory and Binomial options pricing model (1979; also known as the Cox-Ross-Rubinstein model).

• Herbert Simon. (1916 – 2001) Ph.D. in Political Science from University of Chicago in 1943. Won Nobel Prize in Economics in 1978. Many years at Carnegie-Mellon University. Fundamental contributions in cognitive psychology, computer science (artificial intelligence), and economics. Key concepts: bounded rationality and satisficing.

• Amos Tversky. (1937 - 1996) Ph.D. in psychology from University of Michigan in 1964. Fundamental contributions to cognitive science, psychology, and economics. Collaborator of Daniel Kahneman (see below). Professor at Stanford. Key concepts: loss aversion, base-rate fallacy, conjunction fallacy, prospect theory and behavioral finance.

• Daniel Kahneman. (Born 1934). Ph.D. from University of California, Berkeley in 1961. Currently at Princeton. Along with Tversky, developed behavioral economics. Won Nobel Prize in Economics in 2002 (after Tversky’s death).

• Robert Shiller. (Born 1946) Ph.D. from MIT in Economics in 1972. Professor at Yale. Won Nobel Prize in Economics in 2013. Key concepts: Case-Shiller index, behavioral finance, questioned efficient market hypothesis.

Mean-Variance Analysis

Put all your eggs in one basket -- and watch that basket!

Mark Twain, The Tragedy of Pudd'nhead Wilson

Before modern portfolio theory, investors had a common sense notion of risk and return. The basic idea was to select individual stocks that had the lowest expected risk and the highest expected return. In order to mitigate the risk associated with any single stock, the idea was to diversify the portfolio by selecting a number of these low-risk, high-return stocks.

In practice, this approach might mean identifying railroad stocks as low risk and high return, and therefore constructing a portfolio consisting only of railroad stocks. This notion of diversification left something to be desired.

In 1952, Markowitz suggested a different, more principled solution. Markowitz proposed a mathematical theory of diversification whereby an investor did not simply pick individual stocks, but instead fashioned an entire portfolio, according to quantitative principles of reward and risk.

Markowitz’ theory rests on the following assumptions.

• Investors want to maximize returns for a given level of risk. If an investor is given a choice of two assets with equal expected levels of risk, she will choose the asset with the higher expected rate of return.

• Investors are generally risk averse. If an investor is given the choice of two assets with equal expected rates of return, then risk aversion results in the investor’s selecting the investment with the lower perceived level of risk. This creates a positive relationship between expected return and expected risk.

• Each investment alternative has a probability distribution of expected returns over a given holding period.

• Investors maximize their expected utility for any one period and experience diminishing marginal utility of wealth.

• Risk is measured by the volatility of expected returns.

• Investors base decisions only on expected return and risk.

• If risk is constant, then higher returns are preferred to lower returns. If the returns are constant, then lower risk is preferred to higher risk.

We view the returns for a security as a random variable, which can have expected values, variances, and correlations. A portfolio is a collection of these random variables, for which we may calculate the cumulative expected return, variance and volatility.

• Expected Portfolio Return (where R is Return, w is the respective weight))

• Portfolio variance (ρ is the correlation. _i_jρ_ij is the covariance.)

• Portfolio volatility

• Portfolio return: (two-asset portfolio)

• Portfolio variance: (two-asset portfolio)

• Portfolio variance: (three-asset portfolio)

Inputs to Portfolio Optimization

If the world was perfect, it wouldn't be.
Yogi Berra

• The expected return of each asset(R_A)

• The standard deviation of the returns of each asset (_A).

• The covariance (or correlation) in the movements of returns for every pair of assets (COV_A,i or r_A,i) in the portfolio.

• The percentage of the total portfolio’s value invested in each asset.

The Markowitz Efficient Frontier

The illustration below plots the expected returns and standard deviations of a large number of portfolios. The set of portfolios representing the maximum expected return for each level of risk defines the Markowitz Efficient Frontier in the mean-variance plane.

Optimal Portfolios

All points lying on the efficient frontier offer the highest expected return relative to all other portfolios of comparable risk. Portfolios that lie on the efficient frontier are superior to portfolios that are located inside the frontier because they have higher return-to-risk ratios.

In the illustration, single-asset portfolios will be located well within the efficient frontier because these portfolios have high levels of market and specific risk. Multi-asset portfolios lie closer to the efficient frontier because diversification causes their specific risk to be reduced. Ultimately, portfolios lying on the efficient frontier will be those whose specific risks have been eliminated by diversification; they are efficiently diversified portfolios.

The principle of risk reduction through diversification is similar to the principle used to create noise cancelling headphones.

Risk-Return Relationship

The expected return-risk (mean-variance) relationship has a concave shape because the correlation of returns between different securities tends to reduce portfolio risk. This pushes the risk of a portfolio towards the left, producing the concave effect.

The objective of portfolio management is to find the optimal portfolio for an investor. Such a portfolio should have two characteristics:

1. It should lie on the efficient frontier.

2. It should have no more risk than the investor is willing to take.

The illustration depicts the efficient frontier, using the standard deviation of portfolio returns as the measure of risk. The slope of the efficient frontier at any point depicts how much additional expected return is obtained by taking a bit more risk. This is called the return/risk tradeoff.

Portfolio selection theory. (Tobin's theory of the speculative demand for money)

James Tobin elaborated on Keynes' theory of the speculative demand for money and developed into a form of the portfolio selection theory. In his portfolio theory Tobin attempted to avoid some of the weak (and criticized) points of Keynes' theory - mainly the too high level of the model’s abstraction.

A necessary shortcoming of the original Keynes' analysis of speculative demand for money was the claim that subjects hold all their wealth either in the form of money or in the form of bonds (in the case of high interest rates). It is thus clear that he did not consider the possibility of diversification. Keynes' theory of speculative money holding assumed that subjects hold only bonds, if their expected yield is greater than the expected yield of money (which is, according to Keynes, nil), or only money, if the expected yield of bonds is less than the expected yield of money. Only in an exceptional case, where the expected yield of bonds as well as of money are identical, would people hold bonds as well as money. From Keynes' theory it thus results that practically no one would hold concurrently money and bonds (i.e. would not hold a diversified portfolio), which however does not correspond to reality.

Tobin attempted to counter these weak points. He worked from Keynes' assumptions that subjects hold wealth in money or in bonds, where money bears a zero yield. At the same time he elaborated on the Markowitz portfolio theory (1952), which shows that the variability of yields (rate of risk) may be reduced by investing in assets whose prices do not move in conjunction.

Tobin constructed a model of the speculative demand for money for the situation when an individual considers not simply the yield of assets, but also the level of their risk. Money is an asset with in usual circumstances a zero yield, but also with zero risk (ceteris paribus). On the other hand bonds may show a positive yield, but also represent a certain level of risk (potential loss). People have different levels of aversion to risk, and so it is also probable that they decide to hold a certain diversified portfolio of money and bonds and not solely money or solely bonds. On the basis of a comparison of yields and risks connected with various alternatives of securities holdings and - speculative money balances Tobin created an optimum portfolio structure model.

An important rule is the diversification of a portfolio into various assets types, or put differently, “don't put all your eggs in one basket”. The optimum portfolio is as a rule a combination of low risk and high risk assets. Rational investors therefore diversify their wealth across various assets with varying degrees of risk. It is in this that Tobin's theory differs significantly from Keynes’ approach. The main conclusion of both theories is however the same: the speculative demand for money remains a declining function of the interest rate.

Tobin's theory of risk aversion is a new interpretation of why in the asset portfolios of rationally behaving subjects there can be found money which does not directly bear any yield.

Mean-Variance Exercise

Below are the monthly returns for some major indexes for the past year. For each index, calculate the annualized return and standard deviation, and plot them on a mean-variance graph, with each index labeled. (Note: these data are included in the case study spreadsheet.)