# Portfolio Theory and Capital Asset Pricing Model

## Introduction

Capital Asset Pricing Model offers investors or fund managers a method of calculating risk when one wants to diversify his portfolio. This is because the expected return from individual securities carries some degree of risk. Risk is the standard deviation of returns from the mean of an asset. In effect, I equate a security risk with the variability of its return. More dispersion or variability about a security’s expected return means the security is riskier than one with less dispersion wrote Amihud, and Mendelson (1986).

The simple fact is that securities carry different degrees of expected risk, and this makes most investors have the notion of holding more than one security at a time. This is to spread risk according to Brealey, Myres, and Marcus (2001). Diversification of one holding is intended to reduce risk in an economy in which every asset’s returns are subject to some degree of uncertainty. Most investors hope that if they hold several assets, even if one goes bad the others will provide some protection from an extreme loss.

## Literature

Stock market returns of individual stocks covering relatively short periods closely approximate the normal distribution. This allows describing the entire distribution of returns in terms of mean and standard deviation according to Markowitz, (1991) this mean-variance preference can be regarded as particular cases of expected utility. Individual investors are assumed to be risk-averse – they like higher returns and lower risk. This can be quantified with the means of a quadratic utility function. This implicitly assumes that the standard deviation measures risk, all assets are tradable, and no transaction costs are present.

This simple portfolio theory gives the basic principles of selecting the optimal portfolio in terms of risk and return, analogous to an insurance contract against the uncertainty of a particular event happening. Agent B would be compensated for increased risk and would demand a risk premium. The reasons for a trade would be to insure against potential losses, motivated by differences in risk exposure or speculation – very similar to the stock market.

Agents use diversification to get a lower standard deviation of the combined portfolio according to Brealey, Myres, and Marcus (2001). This gives an infinite number of possible combinations which forms the efficient frontier of portfolios. Portfolios are considered efficient if, for a given standard deviation, they give the highest return, or for a given return they have the lowest standard deviation. One can extend the idea of an efficient portfolio to the universe of available securities.

All the efficient portfolios can be identified and depending on one’s risk-return preference, an investor can choose the best portfolio. Every investor will only choose a portfolio on this curve. Any choice not on the efficient set will be dominated by a portfolio on the efficient set. The curve represents all the available efficient portfolios, i.e., the optimal portfolio will be the same for all investors (Amihud, and Mendelson, 1986).

According to Brealey, Myres, and Marcus (2001), the expected return from individual securities carries some degree of risk. Risk is defined as the standard deviation around the expected return. In effect, it is equated a security risk with the variability of its return. More dispersion or variability about a security’s return meant the security was riskier than one with less dispersion.

The simple fact that securities carry differing degrees of expected risk leads most investors to the notion of holding more than one security at a time, to spread risks by not putting all their eggs into one basket. Diversification of one holding is intended to reduce risk in an economy in which every asset returns are subject to some degree of uncertainty. Even the value of cash suffers from the inroads of inflation. Most investors hope that if they hold several assets, even if one goes bad, the others will provide some protection from an extreme loss.

Diversification: – Efforts to spread land minimize risk take the form of diversification. The more forms of diversification have concentrated upon holding several security types across industries the reasons are related to inherent differences in bond and equity contracts, coupled with the notion that an investment in firms in dissimilar industries would most likely do better than in firms within the same industry. Holding one stock each from mining, utility, manufacturing groups is superior to holding three mining stocks according to Brealey, Myres, and Marcus (2001).

Carried to its extreme, this approach leads to the conclusion that the best diversification comes through holding large numbers of securities is five times more diversified than holding ten scattered stocks. Most people would agree that a portfolio consisting of two stocks is probably less risky than one holding either stock alone. However, experts disagree concerning the right king of diversification and the right reason. This is not at odds with traditional approaches in concept. The key differences lie in Markowitz assumption that investor attitudes toward portfolios depend exclusively upon

- expected return and risk, and
- quantification of risk and risk is by proxy the statistical notion of variance or standard deviation of return (Amihud, and Mendelson, 1986).

Many traditional approaches to diversification stress that the more securities one holds in a portfolio, the better. Markowitz type diversification stresses not the number of securities but the right kinds; the right kinds of securities are those that exhibit less than perfect positive correlation.

An unfortunate fact is that nearly all securities are positively correlated with each other and the market. King noted that about half the variance in typical stock results from elements that affect the whole market (systematic risk) the upshot of this is that risk cannot be reduced to zero in portfolios of any size. The one half of total risk that is not related to market forces (unsystematic) can be reduced by proper diversification, but once unsystematic risk is reduced or eliminated, we are left with systematic risk, which no one can escape (Amihud, and Mendelson, 1986).

Thus, beyond some finite number of securities adding more is expensive in time and money spent to search them out and monitor their performance, and this cost is not balanced by any benefits in the form of additional reduction of risk! Evans and Archer’s work suggest that unsystematic risk can be reduced naively by holding as few as ten to fifteen stocks according to Brealey, Myres, and Marcus (2001). (in fact, risk can be increased by duplicating within industries) this results from simply allowing unsystematic risk on these stocks to average out to near zero. With Markowitz type diversification, risk can technically be reduced below the systematic level if securities can be found whose returns have correlations. Negative correlations are ideal.

The risk involved in individual securities can be measured by standard deviation or variance. When two securities are combined, we need to consider their interactive risk or covariance. If the rates of return of two securities move together, we say their interactive risk or covariance is positive (Amihud, and Mendelson, 1986).

## Relevance to the investor

The modern portfolio theory uses diversification to optimize portfolios and how a risky asset should be priced. For the investor to know the risk of a share, he must have its beta which measures the volatility of the shares of a company. The beta of the company chosen has not been provided and this makes it difficult to measure the volatility of shares. Beta measures the sensitivity of the stock’s price to market forces in the market.

It is more sensitive to the changes in the market since the beta is more than one. It is always assumed that the overall market beta is equal to one. Therefore, the inclusion of this company in the portfolio will increase the risk of the company at the same time increase the profitability since a higher risk has a higher return. A decreasing market return generates decreasing security returns. This means that if it is a positive return then they will have higher profits, in consideration of these facts the inclusion of these in the overall portfolio will increase the risk consequently increasing the profit near future.

The crucial point is that these approaches explicitly – not subjectively – consider expected values, standard deviations, and correlation between projects to select the projects which best fulfil management’s objectives according to Brealey, Myres, and Marcus (2001).

## Capital Asset Pricing Model

The Capital Asset Pricing Model (CAPM) proposes a linear relationship between a portfolio’s risk and its return. Its reliability can only be tested by comparing empirical evidence to the results anticipated by mathematical proof. So firstly, we will establish the model’s degree of reliability. This will lead us to the problem of identifying empirical evidence. Once the shortcomings have been identified, the question arises whether there is any room for improvements of the existing model and what the possible alternatives are (Amihud, and Mendelson, 1986).

The Capital Asset Pricing Model (CAPM) is an equilibrium model between expected rates of return and return covariance for all assets implicitly assuming portfolio theory and mean-variance efficiency. It essentially asks: If everyone holds an efficient portfolio, how securities should be priced such that demand equals supply. The CAPM is relying on the following assumptions: no transaction costs, tradable, infinitely divisible assets, no taxes, and perfect competition, mean-variance criteria (either returns normally distributed or quadratic utility functions), unlimited short sales/borrowing and lending at the risk-free rate, rationality, and homogeneous expectations according to Brealey, Myres, and Marcus (2001).

In the CAPM, investing in risky assets such as the market portfolio, S should carry a premium compared to the risk-free rate. Otherwise, investors would not take the risk. The market portfolio is the basic benchmark of risk in the CAPM. The risk premium is determined by measuring the difference between market return and the risk-free rate. The beta for the market portfolio is 1. The CAPM postulates a linear relationship between risk and returns called s*ecurity market line *(SML).

The risk of each stock is measured by its beta (un-diversifiable risk) and the risk premium varies in direct proportion to beta. All stocks will lie along the SML and the expected return on a stock can be calculated by adding the risk premium to the risk-free rate according to Brealey, Myres, and Marcus (2001).

The key implication is it is the *risk of a stock that matters in its contribution to portfolio risk*. This is what is measured by beta, and it depends on the stock’s sensitivity to changes in the value of the market portfolio. The combination of any asset and market must be efficient. Returns reflect risk.

The mathematical proof of the CAPM has attracted the attention of many scholars; an extensive overview of the existing literature on this topic and on the mathematical proof itself was given by Roll (1977). He stresses the present impossibility to deliver a mathematical test of the CAPM, because the restrictions we would have to impose would be unrealistic according to Brealey, Myres, and Marcus (2001).

There is of course the possibility of using a proxy that is perfectly correlated to the market portfolio; the resulting cross-sectional model would yield a ŷ* exactly proportional to the market’s y*. Tests of such a correlation are, however, highly prone to type II errors, i.e., being rejected even though the hypothesis is correct because the prediction about the position of the market portfolio is so specific (Amihud, and Mendelson, 1986).

Hence, when it comes to evaluating empirical evidence in this regard, one is faced with the problem of identifying the “true” market portfolio. Furthermore, it is open to doubt, whether we live in a world of perfect capital markets – where every asset (such as human capital for example) is valued appropriately. If these conditions are not met, there is no way of coming to any certain conclusion. There were many empirical studies made by different scholars, but it became obvious that even when using the same data, different people came to different conclusions about the validity of the CAPM according to Brealey, Myres, and Marcus (2001).

This leads us to the question of whether it might be necessary to introduce other factors than beta to determine the risk premium. Fama & French decided to extent the CAPM after their study showed that small size and low market-to-book ratios provide higher returns, no matter what the beta is like. They came to the following equation according to Brealey, Myres, and Marcus (2001):

*Risk premium = b (market) r (market factor) + b (size) r (size factor) + b (market-to-book) r (market-to-book factor)*

Even though there is empirical evidence supporting this model, it is still open to the same critique as the CAPM: If the mean-variance efficiency of the market portfolio is not sustained, the CAPM and any of its extensions will remain controversial.

## Relevance to investor

CAPM performs a balancing act between marketability and accuracy. Even though there are many convincing reasons why the CAPM might not be precise, it appeals through its simplicity and the fact that it is easily remembered. With beta as its catchword the model was so memorable and the intuition behind it so straightforward, that it established itself quickly all over the world. And when we apply Farrell & Saloner’s study (1985) on the effects that standards have on the acceptance of improvements, we can explain that once a generally recognized standard such as the CAPM is established, no comparatively small parties in segmented markets, such as the financial markets, has an incentive to pioneer a new model.

The risk/return profile of all assets should lie somewhere along the line RFT, which is known as the security market line (SML). It is important to recognize the difference between the capital market line and the security risk is measured in terms of the standard deviation of returns (Amihud, and Mendelson, 1986).

## Benefits of SML and CML

There are many benefits of SML and CML. These two lines are used by investors to choose an optimal portfolio. This is appropriate because the CML represents the risk/return trade-off for efficient portfolios, i.e., the risk is measured by β i.e., only by the systematic risk elements the individual security. No individual security’s risk/return profile is shown by the CML because all individual securities have an element of specific risk, i.e., they are inefficient. Thus, all individual securities (and indeed all inefficient portfolios) lie to the bottom-right of the efficient frontier according to Brealey, Myres, and Marcus (2001).

## Critical analysis of CAPM and portfolio

CAPM is used with the assistance of financial analysts as an ordinary member of the public will not use these models. During periods of high volatility, CAPM can give conflicting solutions to an investment decision.

Another weakness of CAPM is its assumption. The model assumes that with an expected return, investors would likely prefer lower risk than higher risks. Given a certain level of risk will prefer higher returns to lower ones. Investors are not allowed to accept lower returns for higher risk. The model assumes that all investors agree about the risk and possible returns of all assets. The model also assumes that there are no taxes or transaction costs. No country in the world does not charge taxes except one. There the model is realistic.

## Application

Application of the CAPM can be demonstrated. Assume a security with a beta of 1.2 is being considered at a time when the risk-free is 4percent and the market return is expected to be 12%. Substituting these data into the CAPM equation, we get

Rs = 4% + (1.20 x (12% – 4%)

=4% + (1.20 x 8%)

=4% +9.6% = 13.6%

The investor should therefore require a 13.6 per cent return on this investment compensation for the non-diversifiable risk assumed, given the security’s beta of 1.2. if the beta were lower, say 1.00 the required return would be 12% (4% +(1.00 x (12% – 4%); and if the beta had been higher, say 1.50, the required return would be 16///5 (4% +(1.50 x (12% – 4%) CAPM reflects a positive mathematical relationship between risk and return, since the higher the risk (beta) the higher the required return.

## Conclusion

CAPM model is usually used to price portfolio and individual securities in a portfolio to determine the riskiness of the security in question. The investor has a higher rate of risk since he invests in the form of assets. An asset or portfolio may be in the form of bonds, stocks, options, warrants, real estate and all its other forms. In this kind of model, investors were risk averse (Amihud, and Mendelson, 1986). Given two assets with the same expected return, it is expected that they rather choose the less risky one. Investors can lower their risks by holding a non-diversifiable risk portfolio. When an investor diversifies the same portfolio returns it results in reduced risks.

This means that the investor will wish to choose lower risk than higher risks as projected returns under this model. According to this model, investors are assumed not to take an asset with low risk because it has low returns. This assumption which implies that all investors have the same preference towards risk is not possible in the real world. However, this model is important to an investor because it uses diversification; it allows them to limit their risk in investing through this process (Amihud, and Mendelson, 1986).

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