## Modern Portfolio Theory

Meanwhile, Modern Portfolio Theory (MPT) was also being developed. Markowitz (1932) made the distribution of possible returns, as measured by its variance, the measure of riskiness fiC tfcyportfolio. Formally, the population variance is defined by the following formula:

where a2 - variance rM - mean return r* ■» return observation

At the limit, the variance would measure the dispersion of possible returns around the average return. The square root of the variance, or standard deviation, measures the probability that the return deviates from the mean. If we use Osborne's concept of expected return, we can estimate the probability that actual return will deviate from average return. The wider the dispersion, the higher the standard deviation will be, and the riskier the stock would be. Using the variance requires that the returns be normally distributed. However, if stock returns follow a random walk and are I ID random variables, then the Central Limit Theorem of calculus (or the Law of Large Numbers) states that the distribution would be normal, and variance would be finite. Investors would thus desire the portfolio with the highest expected return for a level of risk. Investors were expected to be risk-averse. This approach became known as "mean/variance efficiency." The curve ahown in Figure 2.1 was called the "efficient frontier" because the dark curve contained the portfolios with the highest level of expected return for a given level of risk, or standard deviation. Investors would prefer these optimal portfolios, based on the rational investor model.

These concepts were extended by Sharpe (1964), Litner (196S), and Mossin (1966) in what came to be known as the Capital Asset Pricing Model (CAPM), the name coined by Sharpe. The CAPM combined the EMH and Markowitz's mathematical model of portfolio theory into a

FIGURE 2.1 The efficient frontier.

FIGURE 2.1 The efficient frontier.

model of investor behavior based on rational expectations in a general equilibrium framework. In particular, it assumed that investors had homogeneous return expectations—that is, they interpreted information in the same manner. The CAPM was a remarkable advance, arrived at independently by the three developers.

Because the CAPM has been extensively discussed in the literature, the discussion here is limited mostly to aspects that are relevant to the premise that a new paradigm is needed. The CAPM begins by assuming that we live in a world free of transaction costs, commissions, and taxes. These simplifying assumptions were necessary to separate investor behavior from constraints imposed by society. Physicists often do the same thing when they assume friction's nonexistence. Next, CAPM assumes that everyone can borrow and lend at a risk-free rate of interest, which is usually interpreted as the 90-day T-Bill rate. Finally, it assumes that all investors desire Markowitz mean/variance efficiency—that they want the portfolio with the highest level of expected return for a given level of risk, and are risk-averse. Risk is again defined as the standard deviation of returns. Investors are, therefore, rational in the sense of Osborne and Markowitz.

Based on these assumptions, the CAPM goes on to draw a number of conclusions about investor behavior. First, the optimal portfolio for all investors would be combinations of the market portfolio (all risky assets capitalization weighted) and the riskless asset. This type of portfolio is shown in Figure 2.2: a line is tangent to the efficient frontier at the market portfolio (M) and the Y-intercept, which is theffek-free rate (r). Levels of risk can be adjusted by adding to the riskless asset, to reduce the standard deviation of the portfolio, or by borrowing at that rate to lever the market portfolio. The portfolios that lie along this line, called the Capital Market Line (CML), dominate the portfolios on the efficient frontier, investors would prefer these portfolios to all others. In addition, investors are not compensated for assuming nonmarket risk, because the optimal portfolios are along the CML. The model also states that assets with higher risk

FIGURE 2.2 The capital market line.

FIGURE 2.2 The capital market line.

should be compensated for by higher returns. Because risk is now relative to the market portfolio, a linear measure of the sensitivity of the security risk to the market risk is used. The linear measure is called beta. If all risky assets were plotted on a graph of their betas versus their expected returns, the result would be a straight line that intercepts the Y-axis at the risk-free rate of interest and passes through the market portfolio. This result, called the Security Market Line (SML), is shown in Figure 2.3.

This short and necessarily incomplete discussion of the CAPM is intended to show the substantial dependence on standard deviation as the measure of risk. By implication, the CAPM needs efficient markets and normally or log-normally distributed returns, because variances are assumed to be finite.

The CAPM, which made quantitative methods practical, remains the standard for any new model of investor behavior. Markowitz portfolio theory explained why diversification reduced risk. The CAPM explained how investors would behave, if they were rational. Practitioners needed to be

FIGURE 2.3 The security market line.

convinced that the CAPM's underlying assumptions, which were simplifying assumptions, did not detract from the usefulness of the model. The EMH became extensively used as a rationale for the Gaussian assumption of log-normally distributed returns. This struggle for acceptance probably made the early champions of quantitative methods insist that the EMH was true. Their merger of the EMH with the CAPM and its modifications came to be known generally as Modern Portfolio Theory, or MPT. This same struggle for acceptance may have ^TlfflljJhHy""1 of possible mis-specification to be pushed to the backgroutifl^^^Pfi

The EMH reinforced MPT, and the invBStmeat community accepted variance and standard deviation as the measures of risk. Again, the early founders of capital market theory were well aware of these assumptions and their limitations. Samuelson, Sharpe, and Fama (among others) all published work modifying MPT for nonnormal distributions. Empirical evidence continued, through the 1960s, to favor the Stable Paretian Hypothesis of Mandelbrot (1964), which said that, because returns are nonnormal, there was a need for possible revision of the EMH and MPT. (We will discuss the Stable Paretian Hypothesis in detail in Part Two, when we deal with fractals.) The evidence that returns were non-normally distributed was strong when Sharpe (1970), and Fama and Miller (1972) published their texts; both books included sections on needed modifications to standard portfolio theory, to account for Stable Paretian distributions.

By the 1970s, such discussion had ceased, except for a few isolated academic papers, notably by Roll (1977). Advances in financial economics continued, based on the weak-form EMH and its assumption that price changes were independent In addition, the normal distribution, with its Gaussian assumptions to model independence, was convenient to use. Applications of econometrics to capital markets became more complex as the EMH gained wider acceptance and was questioned less and less. Major advances included the option pricing model of Black and Scholes (1973) and the Arbitrage Pricing Theory (APT) of Ross (1976). The APT, a more general pricing model th"" the CAPM, said that price changes came from unexpected changes in factors; the APT could therefore handle nonlinear relationships. However, in practice, standard econometrics (including finite variance assumptions) have been used, in attempts to implement the APT. The APT did present an alternative theoretical pricing model that did not depend on quadratic utility functions.

In recent years, theoretical models have become less frequent. Research in the 1980s generally focused on empirical research and applications of existing models.

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### Responses

• mary martinez
When is the expected rate of return on the SML?
8 years ago