CAPM & APT

CAPM

Capital asset pricing means defining an appropriate risk-adjusted rate of return for a given asset. Capital Asset Pricing Model (CAPM) is a model that helps in this exercise.

William Sharpe, Treynor and Lintner contributed to the development of this model. An important consequence of the modern portfolio theory as introduced by Markowitz was that the only meaningful aspect of total risk to consider for any individual asset is its contribution to the total risk of a portfolio. CAPM extended Harry Markowitz’s portfolio theory to introduce the notions of systematic and unsystematic (or unique) risk.

With the introduction of risk-free lending and borrowing, the efficient frontier of Markowitz was expanded and it was shown that only one risky portfolio (the tangency portfolio) mattered in evaluating the portfolio risk contribution characteristics of any asset. CAPM demonstrated that the tangency portfolio was nothing but the market portfolio consisting of all risky assets in proportion to their market capitalisation.

Since the market portfolio includes all the risky assets in their relative proportions, it is a fully diversified portfolio. The inherent risk of each asset that can be eliminated by belonging to the portfolio has already been eliminated. Only the market risk (also called systematic risk) will remain. This has been discussed in detail in the last unit.

The CAPM is a model for risky asset pricing. Using a statistical technique called linear regression, the total risk of each risky asset is separated into two components:

 variability in its returns (i.e., risk) that is related with the variability of returns in the market portfolio (its contribution to systematic risk)

 variability in its returns that is unrelated with the variability of returns in the market portfolio (called unsystematic risk).

Systematic and Unsystematic Risk

Every investment portfolio has a risk element, which is the investor will always not be certain whether the investment will be able to generate income as per investor’s requirement. The degree of risk defers from industry to industry but also from company to company. It is not possible to eliminate the investment risk altogether but with careful diversification the risk might be minimised. Provided that the investor diversify their investments in a suitably wide portfolio, the investments which perform well and those which perform badly, should tend to cancel each other out, and much risk is diversified away.
Risks that can be reduced through diversification are referred to as unsystematic risk as they are associated with a particular company or sector of the business. Remember investors are supposed to be compensated for any risk assumed, but with unsystematic risk the investor will not be have any extra risk premium as it can be eliminated through diversification.

Some investments are by their nature more risky than others. These risks are called as systematic risk which will always remain in an investment despite holding a well diversified investment opportunities. If an investor would not take systematic risk, then should be prepared to settle for risk free return which has a lower level of return. Where an investor assumes the systematic risk, and then should expect to earn a return which is higher than a risk free rate of return. The amount of systematic risk in an investment varies between different types of investment. The systematic risk in the operating cash flows of a tourism company which is highly sensitive to consumer’s spending power might be greater than the systematic risk for a company which operates a chain of supermarkets. Systematic risk of a risky asset exists in the market portfolio and cannot be eliminated by portfolio diversification. Investors who want to hold the market portfolio must be and are rewarded. Unsystematic risk for any risky asset cannot be eliminated by holding the market portfolio (which includes the risky asset in question).

Thus the major conclusion of CAPM is that expected return on an asset is related to its systematic and not to its total risk or standard deviation. Its systematic risk is given by its beta coefficient (β). An asset’s beta is a measure of its co-movement with the market index.

Assumptions of CAPM

 All investors are assumed to follow the mean-variance approach, i.e. the risk-averse investor will ascribe to the methodology of reducing portfolio risk by combining assets with counterbalancing correlations.

 Assets are infinitely divisible.

 There is a risk-free rate at which an investor may lend or borrow. This risk-free rate is the same for all investors.

 Taxes and transactions costs are irrelevant.

 All investors have same holding period.

 Information is freely and instantly available to all investors.

 Investors have homogeneous expectations i.e. all investors have the same expectations with respect to the inputs that are used to derive the Markowitz efficient portfolios (asset returns, variances and correlations).

 Markets are assumed to be perfectly competitive i.e. the number of buyers and sellers is sufficiently large, and all investors are small enough relative to the market, so that no individual investor can influence an asset’s price.

Consequently all investors are price takers. Market price is determined by matching supply and demand.

Investors are considered to be a homogeneous group. They have the same expectations, same one-period horizon, same risk-free rate and information is freely and instantly available to all investors. This is an extreme case, but it allows the focus to change from how an individual should invest to what would happen to security prices if everyone invested in a similar manner.

Some of these assumptions of CAPM are clearly unrealistic. But relaxing many of these assumptions would have only minor influence on the model and would not change its main implications or conclusions. The primary way to judge a theory is to see how well it explains and helps predict behaviour. While CAPM does not completely explain the variation in stock returns, it remains the most widely used method for calculating the cost of capital.

Market Portfolio

If investors have the same expectations, same one-period horizon, same risk-free rate and if information is freely and instantly available to all investors, it can be shown that the portfolio of risky assets lying on the efficient frontier that the investors hold (the tangency portfolio) is the same for everyone. It is the market portfolio.

Market portfolio consists of all assets available to investors, and each asset is held in proportion to its market value relative to the total market value of all assets.

The tangency portfolio should be the market portfolio. Reason being the tangency portfolio is a portfolio that every rational investor would hold. If a risky asset was not included in this portfolio there would be no demand for it and it would not exist.

As far as the proportion of each risky asset in this market portfolio is concerned, market efficiency and equilibrium ensure that the demand for each asset is reflected in its price so that the relative market capitalisation of each asset (as a percentage of the entire market for risky assets) would be the weight or proportion of each asset in the market portfolio.

The Beta Factor and Risk Free Rate of Return

A share’s beta factor is the measures of measure of its volatility in terms of market risk. The beta factor of the market as a whole is 1.0. Market risk makes market returns volatile and the beta factor is simply a yardstick against which the risk of other investments can be measured. Risk or uncertainty describes a situation where there is not first one possible outcome but array of potential returns. Risk is measured as the beta factor or B.

·         Negative beta - A beta less than 0 - which would indicate an inverse relation to the market - is possible but highly unlikely. Some investors used to believe that gold and gold stocks should have negative betas because they tended to do better when the stock market declined, but this hasn't proved to be true over the long term.

·         Beta of 0 - Basically, cash has a beta of 0. In other words, regardless of which way the market moves, the value of cash remains unchanged (given no inflation).

·         Beta between 0 and 1 - Companies with volatilities lower than the market have a beta of less than 1 (but more than 0). As we mentioned earlier, many utilities fall in this range.

·         Beta of 1 - A beta of 1 represents the volatility of the given index used to represent the overall market, against which other stocks and their betas are measured. is such an index. If a stock has a beta of one, it will move the same amount and direction as the index. So, an index fund that mirrors the S&P 500 will have a beta close to 1.

·         Beta greater than 1 - This denotes a volatility that is greater than the broad-based index. Again, as we mentioned above, many technology companies on the Sensexhave a beta higher than 1.

·         Beta greater than 100 - This is impossible as it essentially denotes a volatility that is 100 times greater than the market. If a stock had a beta of 100, it would be expected to go to 0 on any decline in the stock market. If you ever see a beta of over 100 on a research site it is usually the result of a statistical error, or the given stock has experienced large swings due to low liquidity, such as an over-the-counter stock. For the most part, stocks of well-known companies rarely ever have a beta higher than 4.

·         - The market as a whole has B = 1
- Risk free security has a B = 0
- A security with a B < 1 is lesser risky than average Market
- A security with a B > 1 has risk above market

Essentially, beta expresses the fundamental tradeoff between minimizing risk and maximizing return. Let's give an illustration. Say a company has a beta of 2. This means it is two times as volatile as the overall market. Let's say we expect the market to provide a return of 10% on an investment. We would expect the company to return 20%. On the other hand, if the market were to decline and provide a return of -6%, investors in that company could expect a return of -12% (a loss of 12%). If a stock had a beta of 0.5, we would expect it to be half as volatile as the market: a market return of 10% would mean a 5% gain for the company.)

The Capital Market Line (CML)

The CML says that the expected return on a portfolio is equal to the risk-free rate plus a risk premium.

Where, rf = risk-free rate, rm = return on market portfolio, σm = standard deviation of the return on market portfolio, σp = standard deviation of the return on the portfolio. Graphically, the CML can be drawn as below:

EF is the efficient frontier, M is the market portfolio and the line tangent to the efficient frontier and joining the risk-free rate (rf) with the market portfolio (M) and going beyond is the Capital Market Line (CML).

The risk-free rate compensates investors for the time value of money while the risk premium compensates investors for bearing risk. The risk premium is equal to the market price of risk times the quantity of risk for the portfolio (as measured by the standard deviation of the portfolio).

The term (rm – rf) is the expected return of the market beyond the risk-free return. It is a measure of the reward for holding the risky market portfolio rather than the risk-free asset. The term m is the risk of the market portfolio. Thus, the slope of the CML measures the reward per unit of market risk. It determines the additional return needed to compensate for a unit change in risk. It is also called the market price of risk.

Capital Market Line (CML) leads all investors to invest in the tangency portfolio (M portfolio) which is the investment decision. Individual investors differ in position on the CML depending on risk preferences (which leads to the financing decision). Risk-averse investors will lend part of the portfolio at the risk-free rate (rf) and invest the remainder in the market portfolio (points left of M). Aggressive investors would borrow funds at the risk-free rate and invest everything in the market portfolio (points to the right of M)

The Security Market Line (SML)

For an individual risky asset, the relevant risk measure is the covariance of its returns with the return on the market portfolio. An extension of covariance called beta coefficient of an asset, βi is defined as:

Where σ i,M = covariance of the return on the asset and the return on the market portfolio.

σ2M = variance of return on the market portfolio.

The CAPM equation, whose graphical representation is the Security Market Line (SML), describes a linear relationship between risk and return for an individual asset. Risk, in this case, is measured by beta (β). Required rate of return for a particular asset in a market depends on its sensitivity to the movement of the market portfolio (i.e. the broader market). This sensitivity is known as the asset beta (β) and reflects systematic risk of the asset. For the market portfolio, beta of the portfolio, βM = 1 by definition. More sensitive assets have a higher beta while less sensitive assets have lower beta. Expected return on any security or portfolio is equal to the risk-free rate plus a risk premium.

Thus expected return on a security (ri) depends on the risk-free rate, (rf,) which is the pure time value of money, (rM – rf,) the reward for bearing systematic risk and βi, the amount of unsystematic risk.

CML and SML

The Security Market Line (SML) and the Capital Market Line (CML) are sometimes confused with each other. CML is a straight line on a plot of absolute returns versus risk that begins at the point of the risk-free asset and extends to its point of tangency with the efficient frontier for risky assets that we call the market portfolio and beyond. Along CML, there are only differing proportions of investing in the market portfolio and borrowing or lending at the risk-free rate to either increase or decrease the exposure to the market portfolio.

SML allows us to represent the risk and return characteristics of every asset in the market portfolio. Instead of dealing with the market portfolio as a whole or as a single entity, SML disaggregates the market portfolio into its individual risky assets and plots return against the only meaningful (or rewarded) aspect of total risk for each asset that is rewarded (its beta).

Thus CML shows the relation between the expected return from a portfolio and its standard deviation and helps investors in their capital allocation problem, while SML shows the relation between expected return and beta and helps investors in security selection and individual asset pricing.

Limitations of Capital Asset Pricing Model

 CAPM is a single period model.

 It is a single factor linear model. It defines risky asset returns solely as a function of the asset’s contribution to the systematic risk of the market portfolio.

 The true market portfolio defined by the theory behind the CAPM is unobservable. Therefore, one has to select and use a market portfolio such as Nifty or Sensex as “proxy.”

 If we use historical data to estimate the inputs for the basic CAPM (risk-free rate, beta and market risk premium), we are making the assumption that the past (specifically the period that we select for the historical data) is the best predictor of the future.

Over the years, a lot of research has been done to test the validity of CAPM but there are problems encountered in doing this research. CAPM is a theory about expected returns whereas we can only measure actual returns. This makes it difficult to test the theory as it is conceived. Another problem in testing CAPM is that the market portfolio should include all assets, not just stocks traded in stock exchanges. In practice, most of the tests use stock market indexes such as the S & P 500 as proxies for the market portfolio.

The results of the research, in general, indicate that the model fails a rigorous test of validity. The results do generally indicate that any asset’s returns are, as CAPM asserts, a linear function of its non-diversifiable risk. But these studies, strictly interpreted find a different intercept and a different slope for SML than the one predicted by CAPM—SML seems flatter than that predicted by CAPM.

In spite of its limitations, most observers regard CAPM as the best tool to describe how assets are priced in efficient markets at equilibrium. The model has found its way into the practical tool kit of many security analysts, portfolio designers, financial managers, investors etc. Corporations often use CAPM to help estimate the cost of equity financing, which is in turn an important component of the weighted average cost of capital (WACC).

Single Factor Model and Variance

The simplest factor model, given below, is a one-factor model:

The return on a security ri is given by:

Where F = the factor

ai = the expected return on the security i if the factor has a value of zero

bi = the sensitivity of security i to this factor

εi = the random error term.

The returns on security i are related to two main components. The first of these involves the factor F. Factor F affects all security returns but with different sensitivities. The sensitivity of security i is return to F is bi. Securities that have small values for this parameter will react only slightly as F changes, whereas when bi is large, variations in F cause large movements in the return on security i.

As a concrete example, think of F as the return on a market index (e.g. the Sensex or the Nifty), the variations in which cause variations in individual security returns. Hence, this term causes movements in individual security returns that are related. If two securities have positive sensitivities to the factor, both will tend to move in the same direction.

The second term in the factor model is a random error term, which is assumed to be uncorrelated across different stocks. We denote this term εi and call it the idiosyncratic return component for stock i. An important property of the idiosyncratic component is that it is assumed to be uncorrelated with F, the common factor in stock returns. The expected value of random error term is zero.

According to one factor model, the expected return on security i, can be written as:

Where denotes the expected value of the factor. The random error term drops out as the expected value of the random error term is zero.  F

This equation can be used to estimate the expected return on the security. For example, if the factor F is the GDP growth rate, and the expected GDP growth rate is 5%, ai = 4% and bi = 2, then the expected return is equal to 4% + 2 x 5% = 14%.

The variance of any security in the single factor model is equal to:

Where F2 = the variance of the factor F

εi 2 = the variance of the random error term εi.

Thus if the variance of the factor F2 = 0.0003

Variance of the random error term εi 2 = 0.0015

bi = 2

Variance of the security = 22 x 0.0003 + 0.0015 = 0.0027

Standard deviation of the security = √variance =√0.0027 = 0.0520 = 5.2 %.

In a single factor model, the covariance between any two securities i and j is equal to:

Where bi and bj = the factor sensitivities of the two securities SDsF2 = the variance of the factor F

Arbitrage Pricing Theory (APT)

 Modern portfolio theory helps an investor to identify his optimal portfolio from umpteen number of security portfolios that can be constructed. We have seen in earlier units how the risk-return framework (using expected return and standard deviation of return of securities) along with all the covariances between the securities’ return is used to derive the curved efficient set of Markowitz. For a given risk-free rate, the investor identifies the tangency portfolio and determines the linear efficient set (Capital Market Line). The investor invests in the tangency portfolio and either borrows or lends at the risk-free rate, the amount of borrowing or lending depends on his risk-return preferences.

With a large numbers of securities, the number of statistical inputs required for using the Markowitz model is tremendous. The correlation or covariance between every pair of securities must be evaluated in order to estimate portfolio risk.

The task of identifying the curved Markowitz efficient set can be greatly simplified by introducing a return-generating process. Return generating process is a statistical method that explains how the return on a security is generated. we have studied one type of return-generating model, i.e. the market model. This is a single-factor model which relates a security’s return to a single factor, which is the return on a market index.

However, the return on a security may depend on more than a single factor, necessitating the use of multiple factor models. Multiple factor models relate the return on a security to different factors in the economy, like the expected inflation, GDP growth rate, interest rate, tax rate changes etc.

Factor models or index models assume that the return on a security is sensitive to the movement of multiple factors. To the extent that returns are indeed affected by a variety of factors, the multiple factor models are seen to be more useful than the market model.

Arbitrage Pricing Theory (APT) is a factor model that was developed by Stephen Ross. It starts with the assumption that security returns are related to an unknown number of unknown factors. It does not specify what these factors are. Unlike CAPM, APT does not rely on measuring the performance of the market. Instead, it directly relates the price of the security to fundamental factors. What these factors are is not indicated by the theory, and needs to be empirically determined.

Capital Asset Pricing Model (CAPM), and Arbitrage Pricing Theory (APT) are two of the most commonly used models for pricing risky assets based on their relevant risks.

CAPM calculates the required rate of return for any risky asset based on the security’s beta. Beta is a measure of the movement of the security’s return with the return on the market portfolio, which includes all available securities and where the proportion of each security in the portfolio is its market value as a percentage of total market value of all securities.

The problem with CAPM is that such a market portfolio is hypothetical and not observable and we have to use a market index like the S&P 500 or Sensex as a proxy for the market portfolio.

However, indexes are imperfect proxies for overall market as no single index includes all capital assets, including stocks, bonds, real estate, collectibles, etc. Besides, the indexes do not fully capture the relevant risk factors in the economy.

An alternative pricing theory with fewer assumptions, the Arbitrage Pricing Theory (APT), has been developed by Stephen Ross. It can calculate expected return without taking recourse to the market portfolio. It is a multi-factor model for determining the required rate of return which means that it takes into account economy factors as well. APT calculates relations among expected returns that will rule out arbitrage by investors.

APT requires three assumptions:

1) Returns can be described by a factor model.

2) There are no arbitrage opportunities.

3) There are large numbers of securities that permit the formation of portfolios that diversify the firm-specific risk of individual stocks.

APT starts with the assumption that security returns are related to an unknown number of unknown factors. These factors can be GDP (Gross domestic product), market interest rate, the rate of inflation, or any random variable that impacts security prices. For simplicity, let us assume that there is only one factor (such as the GDP growth rate) that impacts the security price. In this one-factor APT model, the security return is:

Where F1 = Factor

ai = Expected return on the security i if the factor has a value of zero

bi = Sensitivity of security i to this factor (also known as factor loading for security i)

ε I = Random error term.

Imagine an investor holds 3 stocks and the market value of stock 1 is $250,000, of stock 2 is $250,000 and of stock 3 is $1,000,000. Thus the investor’s wealth is equal to $1,500,000. These three stocks have the following returns and sensitivities.

Do these expected returns and factor sensitivities represent an equilibrium condition? If not, what happens to restore equilibrium?

Principle of Arbitrage or Arbitrage Theory

APT shows that for well-diversified portfolios, if the portfolio’s expected return (price) is not equal to the expected return predicted by the portfolio’s sensitivities (bi), then there will be an arbitrage opportunity. According to APT, an investor will explore the possibility of forming an arbitrage portfolio to increase the expected return on his current portfolio without increasing risk. An arbitrage opportunity arises if an investor can construct a zero investment portfolio with no risk, but with a positive profit. Since no investment is required, an investor can create large positions to secure large levels of profits.

An arbitrage portfolio does not require any additional commitment of funds. Let Xi represent the change in the investor’s holding of security i (as a proportion of total wealth. It is therefore, the proportion of security i in the arbitrage portfolio). Thus the requirement of no new investment can be expressed as:

X1 + X2 + X3 =0

An arbitrage portfolio has no sensitivity to any factor. Sensitivity of a portfolio is the weighted average of the sensitivities of the securities in the portfolio to that factor, this requirement can be expressed as b1X1 + b2X2 + b3X3 =0

In the current example,

1.0 X1 + 2.5 X2 + 2.0 X3 =0

At this point we have two equations and three unknowns. As there are more unknowns than equations, an infinite number of combinations of X1, X2 and X3 will satisfy the requirements. As a way of finding one such arbitrage portfolio, arbitrarily assign a value of 0.2 to X1. Thus we have 2 equations and 2 unknowns.

0.2 + X2 + X3 =0

0.2 + 2.5 X2 + 2.0 X3 =0

Solving these two equations gives a value of X2 = 0.4 and X3 = -0.6

Hence, a possible arbitrage portfolio is one with X1= 0.2, X2= 0.4 and X3 = -0.6.

The expected return of an arbitrage portfolio must be greater than 0. Thus to see whether an arbitrage portfolio has actually been identified, its expected return must be determined. If it is positive an arbitrage portfolio has been identified.

Thus the last requirement is:

X1 r1 + X2 r2 + X3 r3 > 0

Or 10% X1 + 18% X2 + 12% X3

Substituting the values:

= 10% x 0.2 + 18% x 0.4 + 12% x (-0.6) = 2%. Since this is positive, an arbitrage portfolio has been identified.

The arbitrage portfolio involves buying 0.2 x $1,500,000 = $300,000 of stock 1, 0.4 x $1,500,000 = $600,000 of stock 2 and selling 0.6 x $1,500,000 = $900,000 of stock 3.

Return on old portfolio

Return on new portfolio

Weights of new portfolio

Stock 1: 0.1667 + 0.2 = 0.3367

Stock 2: 0.1667 + 0.4 = 0.5667

Stock 3: 0.6666 – 0.6 = 0.0666

Thus, we see that the new portfolio gives a return which is 14.67% - 12.67% = 2 % more than the old portfolio as the calculations above have indicated.

Sensitivity of old portfolio: 0.1667 x 1.0 + 0.1667 x 2.5 + 0.6666 x 2.0 = 1.916

Sensitivity of new portfolio: 0.3667 x 1.0 + 0.5667 x 2.5 + 0.0666 x 2.0 = 1.916

Thus, the sensitivity of the old portfolio is the same as that of the new one. The risk would also be approximately the same as the difference in the risk is only due to non-factor risk.

What is the effect of buying stocks 1 and 2 and selling stock 3? As everyone would be doing it to exploit the arbitrage opportunity, the prices of stocks 1 and 2 will rise because of the buying pressure and the price of stock 3 will fall due to the selling pressure. Consequently, the return on stocks 1 and 2 will fall and the return on stock 3 will increase. This buying and selling will continue till all arbitrage possibilities are significantly reduced or eliminated. At this point, there exists a linear relationship between expected returns and sensitivities:

This equation is the asset pricing equation of the APT when returns are generated by a single factor. As an illustration, suppose, 0 = 5 and 1= 4 for the example above,

Thus, the expected returns of stocks 1 and 2 have fallen from 10% and 18% to 9% and 15% respectively, due to buying pressure and the expected return of stock 3 has increased from 12% to 13% because of selling pressure.

Thus, in equilibrium, the expected return on any security is a linear function of the security’s sensitivity to the factor, bi.

is the return on an asset that has no sensitivity to the factor (bi =0). Hence, it is the risk free rate (rf). Thus, we can write the equation for expected return as:

Identifying the Factors in the APT

APT does not identify the factors to be used in the theory. Therefore, they need to be empirically determined. In practice, and in theory, one stock might be more sensitive to one factor than another. For example, the price of ONGC shares will be sensitive to the price of crude oil, but not Colgate shares. In fact, APT leaves it up to the investor or the analyst to identify each of the factors for a particular stock. So the real challenge for the investor is to identify three things:

 the factors affecting a particular stock

 the expected returns for each of these factors

 the sensitivity of the stock to each of these factors.

Identifying and quantifying each of these factors is no trivial matter and is one of the reasons why CAPM remains the dominant theory to describe the relationship between a stock's risk and return.

Ross and others have identified the following macro-economic factors as significant in explaining the return on a stock:

 growth rate in industrial production

 rate of inflation

 spread between long-term and short-term interest rates

 spread between low grade and high grade bonds

 growth rate in GNP (Gross national product)

 growth in aggregate sales in the economy

 rate of return on S&P 500

 investor confidence

 shifts in the yield curve.

With that as guidance, the rest of the work is left to the stock analyst to identify specific factors for a particular stock.

Arbitrage Pricing Theory vs. the Capital Asset Pricing Model

APT and CAPM are the two most influential theories on stock and asset pricing today. The APT model is different from the CAPM in case of less restrictiveness in its assumptions. APT allows the individual investor to develop their model that explains the expected return for a particular asset.

Intuitively, the APT hold true because it removes the CAPM restrictions and basically states that the expected return on an asset is a function of many factors and the sensitivity of stock to these factors. As these factors change, so does the expected return on the stock, and therefore its value to the investor. However, the potentially large number of factors means that more factor sensitivities have to be calculated. There is also no guarantee that all the relevant factors have been identified. This added complexity is the reason APT is less widely used than CAPM.

In the CAPM theory, the expected return on a stock can be described by the movement of that stock relative to the rest of the stock market. The CAPM theory is a simplified version of the APT, where the only factor considered is the risk of a particular stock relative to the rest of the stock market–as described by the stock's beta.

From a practical standpoint, CAPM remains the dominant pricing model used today. When compared to the APT, CAPM is more refined and relatively simpler to calculate.

Factor Models

Single Factor Model and Variance

The simplest factor model, given below, is a one-factor model:

The return on a security ri is given by:

 Where F = the factor

ai = the expected return on the security i if the factor has a value of zero

bi = the sensitivity of security i to this factor

εi = the random error term.

The returns on security i are related to two main components. The first of these involves the factor F. Factor F affects all security returns but with different sensitivities. The sensitivity of security i is return to F is bi. Securities that have small values for this parameter will react only slightly as F changes, whereas when bi is large, variations in F cause large movements in the return on security i.

As a concrete example, think of F as the return on a market index (e.g. the Sensex or the Nifty), the variations in which cause variations in individual security returns. Hence, this term causes movements in individual security returns that are related. If two securities have positive sensitivities to the factor, both will tend to move in the same direction.

The second term in the factor model is a random error term, which is assumed to be uncorrelated across different stocks. We denote this term εi and call it the idiosyncratic return component for stock i. An important property of the idiosyncratic component is that it is assumed to be uncorrelated with F, the common factor in stock returns. The expected value of random error term is zero.

According to one factor model, the expected return on security i, can be written as:

Where denotes the expected value of the factor. The random error term drops out as the expected value of the random error term is zero.  F

This equation can be used to estimate the expected return on the security. For example, if the factor F is the GDP growth rate, and the expected GDP growth rate is 5%, ai = 4% and bi = 2, then the expected return is equal to 4% + 2 x 5% = 14%. The variance of any security in the single factor model is equal to:

Two-factor Models and Variance