Owais Husain PhD. – Let's Learn….in a easy way



PORTFOLIO MODELS

Beta and risk:

The measure of an asset’s return sensitivity to the market’s return, its market risk, is referred to as that asset’s beta, ß. Market risk is measured by beta, which is another measure of investment risk that is based on the volatility of returns. In contrast to standard deviation, beta measures volatility relative to a relevant baseline rather than to the mean of the asset that is being evaluated. Beta is the appropriate measure of an asset's contribution to your portfolio's risk, as it measures only systematic risk, i.e., market risk.

The beta of a an asset, such as a stock, measures the market risk, a nondiversifiable risk, of that particular asset as compared to the rest of the market—hence, it also measure volatility of the asset compared to the general market. The beta is calculated by comparing the historical return of an asset compared to the market return using statistical techniques to calculate their covariance:

Formula for the Beta Coefficient of a Stock

Βeta Coefficient of Stock = Cov(rs, rm)

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σ2m

rs = Stock Return

rm = Market Return

σ2m = Market Variance

Betas are mostly used to compare return/risk ratios for stocks and mutual funds, because the stock market, or funds composed of stocks, have a greater diversity of volatility than other asset classes. However, stock betas don’t have to be calculated, since most are published in detailed stock quotations offered by major online financial services. Mutual funds also have published betas.

The beta of the S&P 500 stock index market is considered to be 1. Most stocks have a positive beta, which means that most stocks move in the same direction as the general market. Hence, a stock with a beta of greater than 1 is riskier than the general market, but potentially more profitable; a beta of less than 1 is generally less risky than the general market. Most stocks have betas than range from 0.5 – 1.75. Some stocks have a negative beta because they have a negative correlation to the general market—they move in the opposite direction to the general market. For instance, a stock with a beta of -1 will decrease in value by 1% for each increase of 1% in the general stock market, and vice versa.

Security Betas of U.S. Companies’ Common Stock

Source: Value Line Investment Survey, September 20, 2002

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The observation that beta is related to risk leads to the following interpretations which are given for market model (they can be written equally for the general single-index model):

• If βiM > 1 then the asset is more volatile (or risky) than the market. In this case it is termed “aggressive“. An increase (or decrease) in the return on the market is magnified in the increase (or decrease) in the return on the asset.

• If βiM < 1 then the asset is less volatile than the market. In this case it is termed “defensive“. An increase (or decrease) in the return on the market is diminished in the increase (or decrease) in the return on the asset.

SML (Security market line)

Security market line (SML) is the graphical representation. It displays the expected rate of return of an individual security as a function of systematic, nondiversifiable risk (its beta). When the relative risk premium, represented by beta, is plotted in a graph against the required return, it yields a straight line known as the security market line (SML). This line begins at the risk-free rate and rises with beta. The graph below assumes a market return of 12% and a risk-free rate of 4%. Note that a beta of zero is equal to the risk-free rate while a beta of 1 has a relative risk equal to the market.

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When used in portfolio management, the SML represents the investment's opportunity cost (investing in a combination of the market portfolio and the risk free asset). All the correctly priced securities are plotted on the SML. The assets above the line are undervalued because for a given amount of risk (beta), they yield a higher return. The assets below the line are overvalued because for a given amount of risk, they yield a lower return.

There is a question what the SML is when beta is negative. A rational investor should reject all the assets yielding sub-risk-free returns, so beta-negative returns have to be higher than the risk-free rate. Therefore, the SML should be V-shaped.

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Single Index Model

This is a standard modeling technique in all sciences. A model is now provided that much reduces the information needed to calculate the variance of the return on a portfolio and provides the investor with an appealingly direct way of thinking about the riskiness of assets.

The basis of the model is the specification of a process for generating asset returns. This process relates the returns on all the assets that are available to a single underlying variable. This ties together the returns on different assets and by doing so simplifies the calculation of covariance. The single variable can be thought of for now as a summary of financial conditions.

Let there be N assets, indexed by i = 1,2,3,...,N. The single index model assumes that the return any asset i can be written as ri = α iI + β iI rI + ε iI

where ri is the return on asset i

α iI- and β iI are constants

rI is the return on an index.

ε iI is a random error term.

What this model is saying is that the returns on all assets can be linearly related to a single common influence and that this influence is summarized by the return on an index. Furthermore, the return on the asset is not completely determined by the index so that there is some residual variation unexplained by the index – the random error.

Before proceeding to describe the further assumptions that are made, some discussion of what is meant by the index will be helpful. The index can be an aggregate of assets such as a portfolio of stocks for all the firms in an industry or sector. Frequently the index is taken to be the market as a whole. The single index model is usually called the market model.

The single-index model is completed by adding to the specification three assumptions on the structure of the errors, ε iI

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The first assumption ensures that there is no general tendency for the model to over- or under-predict the return on the asset. The second ensures that the random errors are unexplained by the return on the index. The third assumption requires that there is no other influence that systematically affects the assets. It is possible in an implementation of the model for some of these assumptions to be true and others false.

Capital Asset Pricing Model

Capital Asset Pricing Model (CAPM) allows the evaluation of portfolio performance. The model generates an equilibrium relationship between expected return and risk. If a portfolio delivers a lower level of expected return than predicted by this relationship for its degree of risk then it is a poor portfolio. The CAPM model in the area of corporate finance is used as a tool in capital budgeting and project analysis. The CAPM provides an explanation of asset returns uses the concept of financial market equilibrium. A position of equilibrium is reached when the supply of assets is equal to the demand. This position is achieved by the adjustment of asset prices and hence the returns on assets. This adjustment occurs through trading behavior.

If the expected return on an asset is viewed as high relative to its risk then demand for the asset will exceed supply. The price of the asset will rise, and the expected return will fall until equilibrium is achieved. The particular assumption about investors’ preferences and information made by a model then determines additional features of the equilibrium.

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The Expected Return and Risk for All Possible Portfolios of Assets, Including a Risk-Free Asset

The CAPM determines very precise equilibrium relationships between the returns on different assets. Investors construct the efficient set and choose the portfolio that makes the value of their mean variance expected utility as high as possible. Some additional assumptions are then added and the implications are then traced. It is shown that this model leads to especially strong conclusions concerning the pricing of assets in equilibrium. If the model is correct, these can be very useful in guiding investment and evaluating investment decisions.

Assumptions

The set of assumptions upon which the CAPM is based upon are now described. The interpretation of each assumption is also discussed.

All assets are marketable. This is the basic idea that all assets can be traded so that all investors can buy anything that is available. For the vast majority of assets this an acceptable assumption. How easily an asset can be traded depends upon the extent to which an organized market exists. There are some assets cannot be easily traded. An example is human capital. It can be rented as a labor service but cannot be transferred from one party to another.

All assets are infinitely divisible The consequence of this assumption is that it is possible to hold any portfolio no matter what are the portfolio proportions. In practice assets are sold in discrete units. It is possible to move close to this assumption by buying a fraction of a mutual funds. For instance, treasury bills may have denominations of $100,000 but a fraction of one can be bought if it is shared between several investors. The second set of assumptions characterize the trading environment.

No transaction costs Transactions costs are the costs of trading. Brokers charge commission for trade and there is a spread between the buying and selling prices. The role of the assumption is to allow portfolios to be adjusted costless to continually ensure optimality.

Short sales are allowed The role of short sales has already been described in the extension of the efficient frontier. They are permitted in actual financial markets. Where the CAPM diverges from practice is that it is assumed there are no charges for short selling. In practice margin must be deposited with the broker which is costly to the investor since it earns less than the market return.

No taxes: taxes affect the returns on assets and tax rules can alter the benefit of capital gains relative to dividends and coupons. The assumption that there are no taxes removes this distortion from the system. The next pair of assumptions imply that the market is perfect.

Lending and borrowing can be undertaken at the risk-less rate Investors face a single rate of interest. This is the assumption of a perfect capital market. There are no asymmetries of information that prevent lending and borrowing at a fair rate of interest.

No individual can affect an asset price This is idea of a competitive market where each trader is too small to affect price. It takes away any market power and rules out attempts to distort the market. The next set of assumptions describe the trading behavior of investors.

All investors have mean/variance preferences This allows us to set the model in mean variance space and analyze choice through the efficient frontier.

All investors have a one period horizon This simplifies the investment decision. Final assumption ties together all the individual investors.

All investors hold same expectations This makes the investors identical in some sense. Note that the investors are not assume identical because they can differ in their risk aversion. Some may be very risk averse some may be less risk averse.

7.5 Arbitrage Pricing Theory

Arbitrage Pricing Theory (APT) is an alternative to CAPM as a theory of equilibrium in the capital market. It works under much weaker assumptions. Basically, all that is required is that the returns on assets are linearly related to a set of indices and that investors succeed in finding all profitable opportunities. The equilibrium is then obtained by asserting that there can be no unrealized returns. This results from investors arbitraging away all possible excess profits.

Price of Risk

There is one final point to be made about the APT. The coefficient 1i is the price of risk associated with factor i. That is, an extra unit of bki will be rewarded with an increase in expected return equal to 1i. This is just a reflection again of the fact that an investor will only accept greater variability (measured by a higher value of bki) if more return is gained. In equilibrium, the 1is determine just how much greater this risk has to be.

The final term to consider is λ0. The asset with bki = 0, k = 1,….,n is the risk-free asset. Hence λ0 is return on the risk-free asset.

The Arbitrage Pricing Model

An alternative to CAPM in relating risk and return is the arbitrage pricing model, which was developed by Stephen Ross. The arbitrage pricing model (APM) is an asset pricing model that is based on the idea that identical assets in different markets should be priced identically. While the CAPM is based on a market portfolio of assets, the APM doesn’t mention a market portfolio at all. Instead, the APM states that an asset’s returns should compensate the investor for the risk of the asset where the risk is due to a number of economic influences or company factors. Therefore, the expected return on the asset i, ri, is:

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where each of the δ’s reflect the asset’s return sensitivity to the corresponding economic factor. The APM looks much like the CAPM, but the CAPM has one factor—the market portfolio. There are many factors in the APM. What if an asset’s price is out of line with what is expected? That’s where arbitrage comes in. Any time an asset’s price is out of line with how market participants feel it should be priced—based on the basic economic influences—investors will enter the market and buy or sell the asset until its price is in line with what they think it should be. The APM provides theoretical support for an asset pricing model where there is more than one risk factor. Consequently, models of this type are referred to as multifactor risk models. There are three types of multifactor risk models: statistical factor models, macroeconomic factor models, and fundamental factor models. In a statistical factor model a statistical technique called factor analysis is used to derive risk factors that best explain observed asset returns. Let’s suppose that there are six “factors” identified by the model that are statistically found to best explain common stock returns. These “factors” are statistical artifacts. The objective in a statistical factor model then becomes to determine the economic meaning of each of these statistically derived factors. Because of the problem of interpretation, it is difficult to use the factors from a statistical factor model. Instead, practitioners prefer the two other models described below, which allow them to prespecify meaningful factors, and thus produce a more intuitive model. In a macroeconomic factor model, observable macroeconomic variables are used to try to explain observed asset returns. An example of a proprietary macroeconomic factor model is the Burmeister, Ibbotson, Roll, and Ross model. In this model, there are five macroeconomic factors that have been found that do a good job of explaining common stock returns. They are unanticipated changes in the following acroeconomic variables: investor confidence (confidence risk); interest rates (time horizon risk); inflation (inflation risk); real business activity (business cycle risk); and market index (market timing risk). The most common model used by practitioners is the fundamental factor model. It uses company and industry attributes and market data to determine the factors that best explain observed asset returns. The most often used fundamental factor model for explaining common stock returns is the one developed by the firm of Barra. In the Barra model the risk factors, referred to as risk indexes, are indexes of stock price volatility, stock price momentum, market capitalization (size) of the firm, earnings growth, earnings yield, book-to-value ratio, earnings variability, exposure to foreign currencies, dividend yield, and leverage. In addition, the Barra model indicates that the industry that a firm is in is another factor that explains the return on common stock. In the Barra model there are 55 industry groups.

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