Modern Portfolio Theory and the Capital Asset Pricing Model

1952 was a vintage year for quantitative finance. Chapter 3 has already identified one seminal moment: the publication of Frank Redington’s paper on immunisation theory. A seminal concept in financial economics was introduced in this year by Harry Markowitz in his paper, Portfolio Selection.15 Like Merton Miller, Markowitz spent much of his academic career at the University of Chicago. This paper is the starting point for the branch of financial economics known as modern portfolio theory. It also provided the foundation for major developments in asset pricing theory. It was the first important example of a quantitative, explicitly probabilistic analysis of the impact of financial market risk on rational investor behaviour.

Markowitz considered how rational investors would construct portfolios of risky investments. His analysis was built on two key assumptions: investors were risk-averse; and they cared only about the first two moments of the probability distribution of future investment returns (or that all probability distributions of security returns could be described fully by their first two moments). He also made the more basic assumption that investors were non-satiated (they would always prefer more wealth to less). These assumptions implied that for a given level of expected return on their asset portfolio, an investor would prefer a portfolio return distribution that minimised the standard deviation (or variance) of the return; or equivalently, for a given variance of return, they would prefer the portfolio return distribution that maximised the expected return. Portfolios that produced the minimum variance for a given level of expected return were termed efficient portfolios by Markowitz.

Markowitz used some elementary mathematical statistics to show that, whilst the expected portfolio return was simply a value-weighted average of the underlying expected returns of securities in the portfolio, the variance of the portfolio return was a more complicated expression that was dominated by the covariances between the securities. Markowitz showed that, given these expressions for portfolio expected return and portfolio variance of return, portfolio weights in the universe of available securities could be found that would generate efficient portfolios. Moreover, an efficient frontier of portfolios could be determined where each point on the frontier represented the portfolio with the lowest variance for a given expected return. [1]

The identification of these portfolio weights was a complex computational problem when the investment universe consisted of a large number of securities. But Markowitz was able to generate illustrative explicit solutions for the three-security case, and to make more general observations on the behaviour of the possible portfolio combinations. In particular, under the assumptions that there would be no short-selling of securities or gearing in the portfolio, and that every security’s return was risky (i.e. had non-zero variance), he was able to develop the diagram in Fig. 4.2 below.

Figure 4.2 highlighted the power of the diversification of risk—far better risk/return combinations could be obtained by constructing efficient portfolios than holding individual securities. But Markowitz also showed that diversification was about more than merely holding many different securities —it was driven by the covariances amongst securities, not merely by the number of securities in the portfolio. And whilst imperfect correlation between securities was the mathematical key to diversification, the positive correlations that tended to be empirically observed meant that there was a limit to how much diversification could be generated:

This presumption that the law of large numbers applies to a portfolio of securities cannot be accepted. The returns from securities are too intercorrelated. Diversification cannot eliminate all variance.[2]

Portfolio risk and return (Based on Markowitz (1952) Fig. 1))

Fig. 4.2 Portfolio risk and return (Based on Markowitz (1952) Fig. 1))

Markowitz’s approach echoed the arguments made by the British actuary George May 40 years earlier. In 1912, May urged actuaries and investment managers to give greater consideration to the benefits of diversification and hence to ‘spread these investments over as large an area as possible’.[3] He emphasised that the riskiness of a security should not be measured by ‘the safety of the capital in each investment by itself’.[4] He never used the words, but his argument that the ‘stand-alone’ risk of the security was not the relevant risk measure suggests the security’s risk should be considered by its marginal contribution to the portfolio’s riskiness, which is exactly what Markowitz’s analysis implied.

Of course, May did not develop the mathematical analytical framework of Markowitz that would prove so durable as an investor tool, and so fundamental as a building block of asset pricing theory. For this contribution Markowitz received the 1990 Nobel Prize in Economic Sciences.

The next step in the development of modern portfolio theory was made by James Tobin, a leading US macroeconomist (this time from Yale University). In a 1958 paper he introduced what became known as the Tobin Separation Theorem.[5] Tobin extended Markowitz’s analysis by considering how the presence of a risk-free (zero volatility) security impacted on the efficient frontier of portfolio choices. The risk-free security has zero variance and therefore naturally has zero covariance with all other securities. This seemingly trivial observation has profound implications for the efficient frontier: it results in a new, improved, linear efficient frontier and it implies that all efficient portfolios are a combination of the risk-free security and the risky efficient portfolio that is found at the tangent between the risky efficient frontier and the risk-free return. This linear efficient frontier can be extrapolated to the right of the risky asset portfolio by assuming negative positions in the risk-free security can be obtained (that is, the investor can borrow at the risk-free rate). Figure 4.3 below updates the previous diagram to reflect the impact of the presence of the risk-free asset.

The striking implication of Tobin’s analysis was that all investors should hold the same sub-portfolio of risky assets, irrespective of their risk preferences and wealth. An investor would obtain his preferred level of portfolio risk by choosing the mix of this sub-portfolio of risky assets (labelled P in Fig. 4.3) and the risk-free security that produced their preferred level of portfolio volatility. Thus the choice of risky asset sub-portfolio is separated from the i nvestor’s risk preferences. This is the Tobin Separation Theorem. In 1981, Tobin also collected the Nobel Prize in Economics.

Portfolio risk and return (with risk-free security)

Fig. 4.3 Portfolio risk and return (with risk-free security)

So far, the portfolio theory story has been focused on how rational investors should behave in constructing investment portfolios. The story then moved onto a distinct new phase: financial economists next considered what this behaviour would imply for the pricing of risky assets. If investors followed the behaviour described by Markowitz and Tobin, what type of risk would impact on the price of an asset? In what way? At this time, there was no economic theory of how risk impacted on equilibrium asset prices. A great deal of economic theory explained how the risk-free interest rate was determined. And there was a general recognition that risky assets should offer a risk premium. But no one at this time (circa 1960) had an economic theory for how the risk premium on a particular security should be determined.

Several financial economists in the USA (in particular, Sharpe, Lintner, Mossin and Treynor) took the portfolio risk work of Markowitz and Tobin as a platform and built, independently and simultaneously, theories of risky asset pricing. Somewhat remarkably, they each produced more or less the same theory at more or less the same time, and this theory became known as the Capital Asset Pricing Model (CAPM). Rubinstein’s History summarises this state of affairs:

All four economists adopted nearly the same set of assumptions (mean-variance preferences, perfect and competitive markets, existence of a riskless security, and homogeneous expectations) and reached nearly the same two key conclusions:

(1) all investors, irrespective of differences in preferences and wealth, divide

their wealth between the same two portfolios: cash and the market portfolio,

and (2) equivalent versions of the CAPM pricing equation.[6]

A first glimpse of what was to come was offered by William Sharpe in his 1963 paper, A Simplified Model for Portfolio Analysis’.[7] This paper was still focused solely on portfolio theory rather than asset pricing theory and it addressed the practical application of Markowitz’s portfolio construction optimisation—as Markowitz himself had noted, this would be a computationally intensive process when the universe of risky securities was large in number. Sharpe proposed restricting the covariance relationships between risky securities such that they could be summarised by a single common statistical factor plus a single independent source of risk per security. This ‘Diagonal Model’ was much easier to work with in terms of computing the composition of efficient portfolios. Sharpe also undertook some empirical analysis of 20 securities traded in the New York Stock Exchange between 1940 and 1951 and concluded that ‘the diagonal model may be able to represent the relationship among [the realised returns of] securities rather well’.[8]

Sharpe’s next paper[9] moved decisively into the territory of asset pricing theory: rather than taking asset prices as exogenous inputs as per portfolio theory, he considered how asset prices would behave if all investors rationally pursued the Markowitz/Tobin portfolio construction methods. Lintner’s paper[10] was published the following year, and had already been written when Sharpe’s paper was published, but as the first to publication, Sharpe generally receives the greatest credit for the CAPM’s derivation (Sharpe received the Nobel Prize in Economics in 1990 for CAPM). At a high level, the fundamental insight that drove their asset pricing work was that if all investors held the risky asset portfolio P as implied by Markowitz and Tobin, then P must be the market value-weighted portfolio of all risky assets (generally referred to as the market portfolio). Thus risky asset prices must be determined by a risk pricing process that resulted in this occurring. This observation, together with the observation that the market portfolio is mean-variance efficient and is on the tangent with the linear efficient frontier, facilitated the geometric derivation of the famous CAPM pricing equation:

where:

Г; is the expected return on security i; rf is the risk-free rate of return;

rm is the expected return on the market value-weighted portfolio of risky

assets;

and A = ( / cm )pim

and where oI is the volatility of security i; om is the volatility of the market portfolio return; and pim is the correlation between the returns of security i and the market portfolio.

CAPM delivered a rigorous economic theory of how financial market prices would be impacted by risk. It was also a highly intuitive result. The CAPM stated that the expected return on risky assets would only be impacted by the systematic, non-diversifiable component of its volatility (as measured by its beta). Diversifiable risk would not be rewarded. This was an anticipated result: it was well-understood by this time that diversifiable risk was, by definition, risk whose impact could be easily mitigated by investors, and hence should not command a risk premium. Further, CAPM stated that the expected return would increase linearly in proportion to the security’s risk (as measured by its beta). The linear risk premium was also tentatively anticipated by economists, but prior to CAPM they were unable to show why, or indeed in what component of risk the risk premium would be linearly dependent.

The CAPM implies that the realised return on a given security can be decomposed into two statistically independent components: an expected component of the return (which is conditional on the realised market portfolio return) and an unexpected component. Sharpe’s Diagonal Model of joint realised security returns is consistent with this structure. Recall that in this model all stocks have an exposure to a common factor and an independent idiosyncratic risk (a correlation structure that was used to facilitate computationally efficient portfolio optimisations). However, the Diagonal Model is not the only correlation structure for realised returns that is consistent with CAPM. Empirically, significant correlations between the unexpected components of the realised returns of different securities tend to be observed in returns data (for example, across securities in the same industrial sector). This is not inconsistent with CAPM. CAPM does not restrict the realised return correlation structure such that the unexpected return components (relative to the return component explained by the market portfolio) are uncorrelated across all securities. It simply says that such correlations do not matter to the pricing of the securities. In CAPM, the only correlation that matters to the asset price is the correlation of the security with the market portfolio (which is reflected in the security’s beta). That does not imply it need be the only significant correlation that can be found between realised security returns.

More general asset pricing models were developed following CAPM, such as Stephen Ross’s Arbitrage Pricing Theory (APT).[11] This model allows for the possibility of a broader, indeed arbitrary set of statistical factors to drive realised security returns and to be potentially relevant to the pricing of assets. However, it does not provide a specification of how to determine what risk premiums should be commanded by such factors in equilibrium asset prices— it only states what the relative behaviour of different asset prices must be to avoid arbitrage for a given set of risk premium assumptions for the factors. CAPM can be considered as a special one-factor case of the APT, and one in which the risk premium structure is fully specified.

These key ideas of portfolio theory and asset pricing theory are now long- established cornerstones of investment education and practice: they are found in the syllabi of business schools and finance professions (including actuarial ones) around the world; an asset’s beta is a universally recognised measure of risk (and required return) amongst investment professionals.

  • [1] Markowitz (1952).
  • [2] Markowitz (1952), p. 79.
  • [3] May (1912), p. 153.
  • [4] May (1912), p. 136.
  • [5] Tobin (1958).
  • [6] Rubinstein (2006), p. 172.
  • [7] Sharpe (1963).
  • [8] Sharpe (1963), p. 292.
  • [9] Sharpe (1964).
  • [10] Lintner (1965).
  • [11] Ross (1976).
 
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