Modern Portfolio Theory and Arbitrage Pricing Theory

Introduction

Harry Markowitz pioneered the Modern Portfolio theory. For the first time in history he gave to the investors a formula through which they could invest in assets and maximize their profits at lower risk. The most significant and path breaking proposal that he gave was that of diversification of the portfolio to increase profits and minimize risks. Later Sharpe, Lintner and Mossin developed this theory. In recent times it has been shown that the modern portfolio theory contains quite a few weaknesses and thus cannot be completely relied upon to make investment decisions.

Ross developed a related theory, called the Arbitrage Pricing Theory, to counter these weaknesses and to provide a better indicator of investment in assets. We compare the two theories in this paper, and analyze how the arbitrage pricing theory is better than the modern portfolio theory.

Modern Portfolio Theory

The core idea of modern portfolio theory is the use of diversification of portfolio to reduce risk and maximize returns to the investors.

Harry Markowitz, who proposed this theory in his paper, “Portfolio Selection” (1956), believed that diversification of a portfolio into assets, which are not perfectly correlated, reduces the risk in the portfolio of an investor. Before the proposal of this theory by Markowitz, diversification was considered a cardinal sin by investors. It was thought to increase the risk of investing in stocks. But Markowitz dispelled these notions by showing mathematically how it is possible to reduce overall risk of a portfolio by diversifying into assets, which are not perfectly correlated.

The main premise of this theory is that the correlation coefficient between the assets in the portfolio should never be equal to 1. This offsets the drop in some assets with a rise in others, reducing risk at same returns or alternately, maximizing returns at the same risk.

The concepts introduced by Markowitz in his theory are diversification, beta coefficient, efficient frontier, the capital asset pricing model, capital allocation line, the capital market line, and the securities market line.

The Modern Portfolio Theory model links the risk and returns of a portfolio of securities and proposes that investors choose to select an asset in terms of its risk “relative to the market,” rather than its individual risk reward characteristics. A portfolio of securities or assets is chosen with the aim to diversify and reduce the “overall” risk of the portfolio in comparison with the individual risks of the assets in the portfolio. Thus, the MPT proposes the following (“Modern Portfolio Theory”, Wikipedia):

1.Portfolio return is the mean of the component-weighted returns of all the assets in the portfolio. Return has a linear relation with the “component weightings.”

2. Secondly, the risk in a portfolio is a function of how the component assets in the portfolio are correlated. This function between the risk and component weights of the assets is non-linear in relation to each other.

Markowitz has explained his theory using mathematical notations. He used standard deviation of the portfolio returns to identify the risk in a portfolio. He measures this against the expected (mean) return of the portfolio. This analysis is known as the mean-variance analysis. Thus, mathematically, the relation can be explained thus:

Expected return is the sum of the product of every asset’s weight and their corresponding expected returns.

E(Rp) = ∑Wi E(Ri)

Where E(Rp) = Expected Portfolio Returns

E(Ri) = Expected Returns of each asset

Wi = component weight of each asset

The portfolio risk or variance is the sum of the product of every asset pair’s weight in the portfolio (Wij) ) and their covariance or (σij ). Here this sum will contain the squared weight and variance of each individual asset or (σi2). Here, covariance is (σij ), which is equal to σiσjρij (where ρij is the correlated returns between the two assets) or

σp2 = ∑ ∑WiWj σij = ∑∑WiWjσiσjρij

The portfolio volatility is expressed as σp which is equal to √ σp2 . So, for a two-asset portfolio, the portfolio return will be:

E(Rp) = WA E(RA) + (1-Wa)E(RB) = WAE(RA) + WBE(RB)

And the portfolio variance is

σp2 = W2A σ2A+ W2B σ2B + 2WAWB σAB

Similarly, for a three-asset portfolio the portfolio return can be calculated.

This model propounds that by diversifying his portfolio; an investor can reduce the exposure to the individual asset risk in his portfolio, hence reducing the overall portfolio risk.

This calculation shows that if any two assets in a portfolio are less than perfectly correlated (correlation < 1), then the portfolio risk will be less than the weighted average of the individual asset’s risk.

Markowitz found that certain portfolio dominate the other possible portfolios, in the sense that they present higher returns at lower risk than others. He developed this finding graphically, and gave us the “efficient frontier.” On the efficient frontier any portfolio will get a higher return at lower risk than other possible portfolios with the same or lesser level of risk.

Diagram 1. The Efficient Frontier (Source: Wikipedia)

“Mathematically, the efficient frontier is the intersection of the Set of

Portfolios with Minimum Variance and the Set of Portfolios with Maximum Return.” (WIKI) the efficient frontier is concave in shape as the relation between risk and return is non-linear. This calculation also gives us the Capital Allocation Line (CAL), the Capital Market Line (CML), and the Securities C Line (SCL).

CAL is the locus of all the portfolios, which combine a risky asset and a risk less asset. This line is straight, showing a linear relation between risk and return, as the portfolio risk will only be the same as the risk of the risky asset.

The risk-free asset is defined as an asset, which pays a risk-free rate, e.g. bonds, Treasury bill etcetera. An investor can “leverage” or “deleverage” his portfolio by holding a risk-free asset. Such portfolios give a higher return than those on the efficient frontier. This was given by Sharpe. A Sharpe ratio gives the addition in returns to a portfolio, with the inclusion of a risk free asset, as compared to the risk it adds. The portfolio where the CAL touches the efficient frontier is the “market portfolio.” This is also known as super-efficient portfolio, as it gives the highest (highest Sharpe ratio) on the efficient frontier.

The Capital Market Line (CML) is derived when the “market portfolio” is combined with a risk free asset. All the points along the CML provide better returns than any portfolio on the efficient frontier. This is also the optimal CAL. All points on CML are above the efficient frontier.

The risk contained in such a market portfolio is only the “systematic risk.” In a diversified portfolio, the specific risks “cancel out,” leaving the systematic risk common to all assets in a portfolio.

The MPT as propounded by Markowitz was further developed by Sharpe, Lintner, and Mossin. This came to be known as the Capital Asset Pricing Model (CAPM). This model is the most popular form of MPT that is used today.

The CAPM calculates the required return for an asset in respect of the risk- free rate available to the investors, and the market risk. Mathematically, CAPM is shown to follow the equation (“Modern Portfolio Theory”):

E(Ri) = Rf + βi (E(Rm) – Rf)

where β or beta measures the change in asset price in respect of the change in the market price movements.

An investor can use betas of the assets to find whether a particular stock is overvalued or undervalued, and hence make decisions regarding the stocks to place in his portfolio. A risky stock has a higher beta as compared to a less risky stock. The relation between beta and the market return is plotted in the form of the “Securities market Line.” This is a linear relation.

The CAPM proposes that the risk of an asset is composed of market risk and residual risk. The residual risk is diversifiable, while the former is non-diversifiable. A portfolio, which is highly diversified, is composed mainly of market risk, as the residual risk has been canceled out in diversification. Hence, rewards are better in such portfolios. Where portfolios are dependent on residual risk, there rewards are much less. Thus, the CAPM says that an investor is rewarded for taking the market risk. The higher the beta the better are the returns in a highly diversified portfolio. Any deviation from the market portfolio will have to cover certain costs: a) the extra transaction costs involved; b) the residual or diversifiable risk incurred in the process; c) the cost of analysis in looking up the underpriced assets. A higher risk will bring better returns to the investor.

The Arbitrage Pricing Theory (APT)

Several weaknesses were found to be associated with the Modern Portfolio Theory models. As a substitute to the MPT, Stephen Ross developed the APT in 1976. He described it as a theory which is “related (to MPT) but quite distinct theory” (Mcmenamin, 1999) from the risk-return relationship as found by CAPM.

Ross makes two contentions in his model. These two proposals of APT constitute the main difference of this model from CAPM. The first central contention is that there are no profit-earning arbitrage opportunities in the market equilibrium. He proposes that the expected returns on the assets in a portfolio are in a linear relationship with their betas or “factor loadings.” (Huberman and Wang, 2005) The second contention is that an asset’s returns is dependent on more than one factor, unlike the CAPM. Thus the APT considers multiple market risk factors and their betas, in calculating the asset returns. Each of these factors have their own betas, so the expected rate of return on an asset will be equal to the “risk free rate” and multiple risk free premiums. But the factors should be less in number than the number of assets. The APT is expressed mathematically as follows (“Arbitrage pricing theory,” Wikipedia):

E(rj) = rf + bj1RP1 + bj2RP2 + … + bjnRPn

rj = E(rj) + bj1F1 + bj2F2 + … + bjnFn + εj

Roll and Ross have laid out four factors, which may be included in the model. These are: “unanticipated changes in industrial production; inflation; default risk on premium bonds; and the term structure of interest rates. (Mcmenamin, 1999)

Differences and Similarities of MPT and APT

If we take CAPM as representative of MPT, we find significant differences between the modern portfolio theory and the arbitrage pricing theory. Some of these differences are enumerated below.

The APT is and empirical and explanatory model of asset return, whereas MPT is a statistical model.

APT is a single- period multifactor model as opposed to the single factor CAPM of MPT. The priced factors in APT cannot be identified as their number and nature are subject to change in time or between different economies.

The MPT assumes that each investor holds a similar “market portfolio.” On the other hand, the APT considers the fact that each investor will hold a unique portfolio each of which has its own unique array of betas.

APT is considered a “supply-side” model, while MPT is a “demand-side” model. (“Arbitrage pricing theory,” Wikipedia) The results in APT accrue from the consumer characteristics of its investors, each of which tries to maximize his utility function in an effort to reach the market equiibrium. In the APT model, on the other hand, the economy factor shocks cause changes in the asset’s expected return (its profitability) through the beta coefficients. Thus, it is affected by the supply side factors.

The APT is based on very few assumptions as compared to the MPT, which requires as many as ten assumptions.

Another important difference between APT and MPT is that the former assumes that in the condition of equilibrium in the market, thee are no arbitrage profit opportunities to be taken. So, it does not require an investor’s portfolio to be “mean-variance efficient.” (Mcmenamin, 1999) The MPT, on the other hand, requires its market portfolio to be mean-variance efficient, and offer the best possible risk return combination of assets.

The APT and MPT are similar basically because they formulate the same idea in different forms. They are “related but distinct.”(Mcmenamin, 1999) They are similar in the following ways.

Both models propose a linear relation between “assets’ expected return and their covariance with other random variables.” (Huberman and Wang, 2005)) So, APT is considered as a substitute for CAPM, and CAPM a special case of APT.

Both the models base their calculations of asset price and expected return on betas or the variances of the asset prices in relation to the market price and with each other, or with other factor betas. Both offer a way to investors to value securities while taking other factors into account and the relation between risk and return.

APT better at predicting values than MPT?

The Arbitrage Pricing theory is considered better at predicting values than MPT. There are several reasons for this belief as shown below.

APT makes fewer assumptions than MPT and is less restrictive. The CAPM makes about ten assumptions. But these assumptions are unrealistic. (“Modern Portfolio Theory – Criticisms”) The CAPM assumes that there are no transaction costs in buying and selling securities, which is not realistic. It also assumes that there are no taxes to be paid, and the investor does not consider taxes, dividends or capital gains in making investment decisions. Actually, taxes play an important role in making investment decisions, and dividends and capital gains guide investors on the assets to buy. The model also assumes that liquidity is infinite. But actually it is not, and liquidity prevents people to invest in thinly traded issues. The most significant assumption that MPT makes is that investors are risk adverse. This actually is not true. Investors are actually not rational. The gambling tendency is always present when dealing with stocks. It assumes that investors can lend or borrow at 91-day T-bill rate or the risk free rate. But in actual fact, only the government can borrow at that rate. It also assumes that politics or investor psychology does not affect the market. This has been historically disproved. All these assumptions of MPT are unrealistic. And any value derived from calculations comprising such assumptions cannot be said to be very reliable.

The factors that APT takes into account when calculating asset price, make it much more plausible than the MPT which ignores the real life factors affecting the market at all times.

There is also no imperative need in APT to measure the market portfolio accurately. The MPT, however, requires accurate measurements of market variables. Markowitz in his theory equated risk with volatility, and proposed that higher the risk, higher will be the returns. But a research on risk found that there is no credible correlation between risk and return, and in fact in some cases, there may be no relationship between the two.(“Modern Portfolio Theory – Criticisms”) Another difficulty is to calculate a constant volatility measure over time. Volatility changes very frequently. So volatility cannot be a good measure to make any meaningful changes in a portfolio.

The most important component of MPT is the beta. But beta, as given by Sharpe, Lintner and Mossin, also has been found to possess no predictive value. Beta differs for each period very drastically, and it is difficult to use it for predicting volatility. Fama and French in the study of 9,500 stocks found that beta was not a reliable indicator of performance. (“Modern Portfolio Theory – Criticisms”)

Apart from this, MPT model itself is not easily applied to real life. It assumes that the best portfolio is the market portfolio. But to assemble the market portfolio is a daunting task. Also, MPT connects historic data with future returns in the same equation. This reduces the predictive power of the model.

APT, on the other hand, makes very few assumptions in its model. There is also no requirement to accurately calculate the variance of all the assets returns in a portfolio and the market returns. This reduces the burden of calculations and lessens the expense and effort to calculate. The APT is thus far more effective in predicting the asset prices and hence in making investment decisions. It also incorporates those factors, which affect the stock market, like inflation, industrial production, oil prices etc. these factors have a real role to play in the pricing of stocks and consequent profits. So, APT qualifies much better to make portfolio investment decisions.

Conclusions

The MPT lead the way into making portfolio decisions very early when Markowitz showed how one could maximize their returns from a portfolio of assets at a lower level of risk through the method of diversification. The APT is a relatively recent development. It was proposed to over come some of the weaknesses of MPT and to make a more plausible investment decision regarding investing in assets. There are several differences between the two theories but they are similar on some points basically because APT is a modification of MPT itself. But we find from our analysis that the APT would possess much better predictive values when compared to the MPT. The former is much more realistic in its assumptions and the factors it considers. It is much more reliable as well. Although APT is still in its infancy stage of development, but it can better substitute the modern portfolio theory.

References

“Arbitrage pricing theory.” Wikipedia. Accessed 07 October, 2006. [Online] http://en.wikipedia.org/wiki/Arbitrage_pricing _theory/

Gruber, Martin J. “Modern Portfolio Theory.” [Online] <http://www.law.nyu.edu/ncpl/library/publications/Conf2003_Gruber_Final.pdf>

Huberman, Gur; Wang, Zhenyu. “ Arbitrary Pricing Theory.” Aug 15, 2005. The New Palgrave Dictionary of Economics. Palgrave Macmillan.

Mcmenamin, Jim. “Arbitrary Pricing Theory” AppendixII, pp.235-36. Financial Management: An Introduction. Routledge, 1999.

“Modern Portfolio Theory”. Wikipedia. Accessed 07 October, 2006. [Online]<http://en.wikipedia.org/wiki/Modern_Portfolio_Theory.htm>

“Modern Portfolio Theory – Criticisms.” Accessed 07 October, 2006. [Online]<http://www.travismorien.com/FAQ/portfolios/mptcriticisms.htm>

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Modern Portfolio Theory and Arbitrage Pricing Theory. (2016, Jul 08). Retrieved from https://graduateway.com/modern-portfolio-theory-and-arbitrage-pricing-theory/

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