Chapter 8 The Capital Asset Pricing Model

Chapter 8 The Capital Asset Pricing Model
Keith Pilbeam ©: Finance and Financial Markets 4th Edition

Learning Objectives Keith Pilbeam ©: Finance and Financial Markets 4th Edition

Intro to the Problem A huge number of covariances have to be calculated when assessing the risk to a portfolio 2-assets case is easy (see Markowitz or what we did last week) N-assets case it is necessary to calculate (N2 – N)/2 covariances Thus, with 100 assets one would need to calculate 4950 covariances with 500 securities this increases to 124,750 covariances Keith Pilbeam ©: Finance and Financial Markets 4th Edition

The Market Model Sharpe(1963): Sharpe’s market model suggests that shares tend to move in varying degrees in line with market itself. Market   most shares Keith Pilbeam ©: Finance and Financial Markets 4th Edition

The Market Model For regression purposes the market model becomes:
The characteristic line Keith Pilbeam ©: Finance and Financial Markets 4th Edition

The Market Model Keith Pilbeam ©: Finance and Financial Markets 4th Edition

Portfolio Risk & Return using Market Model
systematic risk unsystematic risk Keith Pilbeam ©: Finance and Financial Markets 4th Edition

Back to intro to the Problem
A major advantage of the market model is that it dramatically reduces the number of variables required to evaluate a portfolio. All we need is 3N + 2 data items; αi, βi and σ2ei Rather than N(N – 1)/2 covariances as required by the Markowtiz model. For 100 securities we need only 302 calculations rather than 4950 under the Markowitz method! However !!! It lacks theoretical base!!! Keith Pilbeam ©: Finance and Financial Markets 4th Edition

The Capital Asset Pricing Model
Was originally devised as an offshoot from the market model by Sharpe (1964) and Litner (1965). CAPM is explaining the relationship between the risk and return on a financial security. If a share helps to stabilize a portfolio, then, it has the similar return with the market. If it makes the portfolio more risky, there will be less demand from risk-averse investors and its rate of return will be above the market. If it reduces the risk of a portfolio, its rate of return will be less than the market. Keith Pilbeam ©: Finance and Financial Markets 4th Edition

The Capital Asset Pricing Model
Another key idea of the CAPM is that in an efficient market all diversifiable risk will be eliminated (given that there are no transaction costs), so that the only risk that will be priced by the market on a portfolio is systematic or market risk. Hence, the CAPM model concentrates only on the pricing of non diversifiable market risk. As we shall see, the CAPM provides a simple measure of the systematic risk attached to a security given by the security’s beta. Keith Pilbeam ©: Finance and Financial Markets 4th Edition

Assumptions of the CAPM
Perfect capital markets All capital assets are perfectly divisible No transaction costs, no taxes, Short selling opportunity Free available information to all market participants Risk averse investors and risk is proxied by standard devs Transactions with the aim of maximizing the utility Common one period ahead time horizon for investment decisions Different Risk Preferences of the customers Identical expectations about risk and return Single risk-free asset (easy borrowing and lending) Keith Pilbeam ©: Finance and Financial Markets 4th Edition

Theory Behind the CAPM In Figure 8.2, there is an efficiency frontier with N risky securities, given by the curve EF. The efficiency frontier is the same for all investors in the economy. A straight line L1 is drawn in the figure from the risk-free rate of interest R* tangential to the efficiency frontier at point M. According to CAPM model, in equilibrium all investors choose to allocate their investment wealth between the same mix of securities given by the portfolio M and risk free asset. Keith Pilbeam ©: Finance and Financial Markets 4th Edition

The capital market line and the market portfolio

Theory Behind the CAPM There are two investors A and B.
Investor A is more risk-averse than the investor B. Investor A prefers less risk for a lower expected rate of return than the investor B. Investor A is on the indifference curve I1. Investor B is on the indifference curve I2. Both investors should choose portfolio M regardless of their risk-return trade off. And borrow or lend funds at the risk-free rate of interest R*. Line L1 : Capital Market Line Capital Market Line: A straight line passing through the risk-free rate of return and the expected rate of return on the market portfolio with risk measured by the standard deviation on the horizontal axis and the rate of return on the vertical axis. Keith Pilbeam ©: Finance and Financial Markets 4th Edition

Theory behind the CAPM Keith Pilbeam ©: Finance and Financial Markets 4th Edition

Theory behind the CAPM Investor A can invest proportion of his money (w) in a risky market portfolio and the remainder (1-w) in the risk-free asset. This means that investor is lending funds. This investor can move onto a higher indifference curve I1*. The closer A* is to M, the greater the proportion invested in the market portfolio. Keith Pilbeam ©: Finance and Financial Markets 4th Edition

Theory behind the CAPM The investor B can move onto a higher indifference curve I2* by borrowing funds at the risk-free rate of interest. The proportion of money invested in the risk free asset is given by (1-w). This is equal to the distance [1-(R*B*)/R*M] By borrowing money, investor B can move onto a higher indifference curve I2*. THUS, both happy to invest in the market regardless of their risk-return preferences. The question is how much? Keith Pilbeam ©: Finance and Financial Markets 4th Edition

Theory behind the CAPM We have established that M is the optimum portfolio. But why is it called the market portfolio? The answer is that since M is the only portfolio that all investors will hold, then all the risky securities in the economy that make up portfolio M must be correctly priced and willingly held by all investors. In consequence, the market portfolio is a portfolio of all the risky assets in the economy weighted by their market value over the market value of all assets in the economy. Since the market portfolio is by definition a value-weighted portfolio of all the risky assets in the economy, it is also a portfolio with no diversifiable risk. The only risk in the market portfolio is market risk; all specific/diversifiable risk has been eliminated. Keith Pilbeam ©: Finance and Financial Markets 4th Edition

Theory behind the CAPM Keith Pilbeam ©: Finance and Financial Markets 4th Edition

CAPM in Risk Premium Form
The expected risk premium on a portfolio E(RPp) can be defined as the difference between the expected return on the portfolio E(Rp) the risk-free rate of return (eq 1). The expected risk premium on the market portfolio E(RPm) can be defined as the difference between the expected return on the market portfolio E(Rm) and the risk-free rate of return (eq 2). Substituting (1) and (2) in the CAPM equation that we have seen before we get equation (3) Keith Pilbeam ©: Finance and Financial Markets 4th Edition

What does this tell us? Excess return above the risk-free rate of interest on a portfolio is a function of the beta of the portfolio and the difference between the market rate of return and the risk-free rate of interest. Keith Pilbeam ©: Finance and Financial Markets 4th Edition

We can conclude that the correct measure of risk for an efficient portfolio, as represented by the capital market line, is the standard deviation of the return, that is the total risk. This must be so because in an efficient portfolio all diversifiable/specific risk has been eliminated, leaving only non diversifiable market risk. By definition an efficient portfolio has no diversifiable/specific risk since it has all been diversified away. Keith Pilbeam ©: Finance and Financial Markets 4th Edition

From the CML to the SML The capital market line is a useful expression for the pricing of efficient portfolios, that is combinations of the market portfolio and the risk-free security, But.... Does not give a clue to the pricing of inefficient portfolios, individual securities or poorly diversified portfolios containing only a few securities. Ideally, we would like to derive an expression for the risk–return trade-off for any individual security or any portfolio, not just efficient portfolios which lie on the capital market line. This is the Securities Market Line

The Securities Market Line
Securities Market Line: A line which measures the relationship between beta (systematic risk) and firm’s expected rate of return Key difference Keith Pilbeam ©: Finance and Financial Markets 4th Edition

The equation for the securities market line can be written below: If beta=1, expected rate of return of the stock coincides with the market rate of return. Defensive security: A security with a beta less than 1, that is expected to earn a lower rate of return than the market. Aggressive security: A security with a beta greater than 1, that is expected to earn a higher rate of return than the market. Keith Pilbeam ©: Finance and Financial Markets 4th Edition

Figure 8.3 The securities market line

The securities market line
Z is underp=riced since its return (RZ) is greater than its expected rate of return E(RZ) as indicated by its beta would merit. Keith Pilbeam ©: Finance and Financial Markets 4th Edition

Figure 8.3 The securities market line
Y is overpriced since its return (RY) is below the expected rate of return E(RY) that its beta would merit. Keith Pilbeam ©: Finance and Financial Markets 4th Edition

Typical defensive stocks are companies in relatively stable sectors of the economy such as food retailers, gas and electricity companies (so-called utilities). Typical aggressive stocks include those that are more volatile than the market such as property, luxury goods manufacturers, technology, fashion industries and so on. Keith Pilbeam ©: Finance and Financial Markets 4th Edition

The CAPM suggest the required rate of return on a security consists of three components The price of time as measured by the risk-free rate of interest The quantity of risk as measured by the beta of the security The market price of risk as measured by the difference between the expected return on the market and the risk-free rate of interest Keith Pilbeam ©: Finance and Financial Markets 4th Edition

Measuring the Beta Coefficient
It is possible to empirically estimate the value of a beta by using historical time-series data. The estimated beta will be affected by a number of variables, such as the time period over which the return is calculated, the frequency of the data (daily, weekly, monthly or quarterly) and the market index used. A study might also estimate the beta coefficient over different time periods to examine its stability over time. Frequency? The data set can consist of daily, weekly, monthly or even quarterly data; the time period and frequency of the data will determine the number of observations used, and it is important that the time period and frequency used for estimation of the beta is the same for both the portfolio and the market. What is the Market? For the market index, the series of returns associated with a broad index such as the FTSE 100 or the Standard & Poor’s 500 are generally used. Keith Pilbeam ©: Finance and Financial Markets 4th Edition

Various agencies conduct regression analyses of the basic CAPM equation to find the relevant coefficients. The analyses are usually based on monthly observations of the last five years’ data. An essential prerequisite is that the estimated betas need to be stable over time (that is, exhibit stationarity). Where ei represents specific risk (is the error of reg). Keith Pilbeam ©: Finance and Financial Markets 4th Edition

Specific risk(ei) is measured by the standard deviation of the error term. The risk-free rate of interest (R*): 7% Market rate of return: 12% Standard Deviation of the Market: 0.20 Question: Does a portfolio made up equally of the five shares A to E have a good risk-return performance? Keith Pilbeam ©: Finance and Financial Markets 4th Edition

We need to calculate three things: The portfolio’s beta The total risk of the portfolio (both systematic and unsystematic risk) The actual return of the portfolio against its systematic return as required by the CAPM Keith Pilbeam ©: Finance and Financial Markets 4th Edition

Empirical Testing of the CAPM
A typical empirical estimation of the CAPM involves looking at portfolio betas. Repeating equation (8.17): E(RPp)=βpE(RPm) The empirical regression will be: Keith Pilbeam ©: Finance and Financial Markets 4th Edition

Key Predictions of the CAPM
The intercept term in equation (8.25) should be equal to zero, that is, α = 0; if it were non-zero then it would mean that the CAPM model is missing something as a complete explanation of a portfolio’s excess return. The beta coefficient should be the sole explanation of the rate of return on the risky portfolio. The estimated slope b should be positive and not differ significantly from the risk premium on the market portfolio, RPm = Rm – R*. There should be a linear relationship given by beta between the average portfolio risk premium and the average market risk premium. Over time, Rm should exceed R*, since a market portfolio is riskier than the risk- free asset. Other explanatory variables such as dividend yield, firm size and price–earnings ratios should not prove to be statistically significant in predicting the required rate of return. Keith Pilbeam ©: Finance and Financial Markets 4th Edition

Applications of the CAPM
Investment in the financial markets Portfolio selection Mispriced shares Measuring portfolio performance Calculating the required rate of return on a firm’s investment projects Keith Pilbeam ©: Finance and Financial Markets 4th Edition

The Empirical Evidence on the CAPM
There have been numerous empirical studies of the CAPM model. Friend and Blume (1970) Black, Jensen and Scholes (1972) Miller and Scholes (1972) Blume and Friend (1973) Fama and MacBeth (1973) Litzenberger and Ramaswamy (1979) Gibbons (1982) Shanken (1985) Keith Pilbeam ©: Finance and Financial Markets 4th Edition

The results of the various studies can be summarised as follows: The overall conclusion of the studies is that the empirical securities market line differs somewhat from the theoretical line as depicted in the next Figure. Estimated intercept a significantly different from zero contrary to prediction 1 Estimated slope b, while positive tends to be less than the difference between market rate of return and risk free rate of interest. Estimated relation is linear with respect to beta and the risky portfolio returns are greater than risk free rate. The general implication is that securities with low betas tend to earn a higher rate of return than the theoretical model would suggest, while securities with high betas tend to earn less. Keith Pilbeam ©: Finance and Financial Markets 4th Edition

The theoretical and empirical securities market lines
The general implication is that securities (and portfolios) with low betas tend to earn a higher rate of return than the theoretical model would suggest, while securities (and portfolios) with high betas tend to earn less than expected. Keith Pilbeam ©: Finance and Financial Markets 4th Edition

Contrary to prediction 5, it is possible to find other factors that can explain a portfolio’s excess return (very low R2 in the characteristic line). Basu (1977) found low price–earnings-ratio portfolios have higher rates of return than predicted by the CAPM; Banz (1981) found that firm size is important, with smaller firms having higher returns than predicted by the CAPM; Litzenberger and Ramaswamy (1979) found that equities with high dividend yields required higher rates of return than predicted by the CAPM. Keith Pilbeam ©: Finance and Financial Markets 4th Edition

Recall CAPM equation Empirically beta SHOULD BE the only determinant of returns Keith Pilbeam ©: Finance and Financial Markets 4th Edition

Is beta sole determinant ?

Is beta sole determinant ?
Some of the further regressors that have proved significant are: Basu (1977) Price/Earnings ratio c<0 i.e. Low P/E stocks have higher returns Banz (1981) Firm Size c<0 i.e small firms have higher returns than predicted by CAPM Litzenberger and Ramaswamy (1979) dividend yield ratio c>0 i.e high dividend Keith Pilbeam ©: Finance and Financial Markets 4th Edition

Black (1993) simulates a portfolio strategy that investors might adopt. The shares of quoted US companies (on the NYSE) are allocated on an annual basis to 10 categories of different beta levels. Each year the betas are recalculated from the returns over the previous 60 months. The first investment portfolio is constructed by hypothetically purchasing all those shares within the top 10 per cent of CAPM-beta values. As each year goes by the betas are recalculated and shares that are no longer in the top 10 per cent are sold and replaced by shares which now have the highest levels of beta. The second portfolio consists of the 10 per cent of shares with the next highest betas and this is reconstituted each year. Keith Pilbeam ©: Finance and Financial Markets 4th Edition

A further blow to the CAPM came with the publication of Eugene Fama and Kenneth French’s (1992) empirical study of US share returns over the period 1963–90. They found ‘no reliable relation between β, and average return’. They continue: “The asset-pricing model of Sharpe (1964), Lintner (1965), and Black (1972) [the CAPM] has long shaped the way academics and practitioners think about average returns and risk In short, our tests do not support the most basic prediction of the SLB model, that average stock returns are positively related to market βs Our bottom-line results are: (a) β does not seem to help explain the cross-section of average stock returns, and (b) the combination of size and book-to-market equity [does].” Fama and French’s later (2006) paper reports that higher beta did not lead to higher returns over the 77 years to 2004. Keith Pilbeam ©: Finance and Financial Markets 4th Edition

Burton Malkiel (1990) found that the returns on US mutual funds in the 1980s were unrelated to their CAPM-betas. Louis Chan and Josef Lakonishok (1993) breathed a little life into the now dying CAPM-beta. They looked at share returns over the period 1926–91 and found a faint pulse of a relationship between CAPM-beta and returns, but were unable to show statistical significance because of the ‘noisy’ data. More vibrant life can be witnessed if the share return data after 1982 are excluded – but, then, shouldn’t it work in all periods? They also argued that beta may be a more valid determinant of return in extreme market circumstances, such as a stock market crash, and therefore should not be written off as being totally ‘dead’. Keith Pilbeam ©: Finance and Financial Markets 4th Edition

Evidence from studies in European share markets. For example, Albert Corhay and co-researchers Gabriel Hawawini and Pierre Michel (1987) found that investors in shares trading in the United States, the United Kingdom and Belgium were not compensated with higher average returns for bearing higher levels of risk (as measured by beta) over the 13-year sample period. Investors in shares trading on the Paris Stock Exchange were actually penalised rather than rewarded, in that they received below-average rates of return for holding shares with above-average levels of beta risk. Rouwenhorst, Heston and Wessels (1999), however, found some relation between beta and returns in 12 European countries. Strong and Xu (1997) show that UK shares during the period 1973–92 displayed evidence consistent with a negative relationship between average returns and CAPM-beta! Keith Pilbeam ©: Finance and Financial Markets 4th Edition

James Montier, analyst at Dresdner Kleinwort, says CAPM has become the financial theory equivalent of Monty Python’s famous dead parrot sketch. He says the model is empirically bogus – it does not work in any way, shape or form. But like the shopkeeper who insists to a customer with a dead parrot in the sketch that the bird is merely resting, financial markets are in denial. ‘The CAPM is, in actual fact, Completely Redundant Asset Pricing (CRAP),’ he says. Keith Pilbeam ©: Finance and Financial Markets 4th Edition

Risk Adjusted Performance Metrics
‘In the last five years our fund grew by 20% per year while the stock market rose only 10% per year’ Would you like that fund? What about the risk that the fund is taking? What about volatility? Sharpe Ratio The Treynor Ratio Jensen’s Alpha Keith Pilbeam ©: Finance and Financial Markets 4th Edition

Sharpe Ratio The Sharpe Ratio is proposed by the Nobel price winner William Sharpe (1966). The idea is excess return in the numerator needs to be compared to risk of the portfolio in the denominator. Reward to variability ratio Keith Pilbeam ©: Finance and Financial Markets 4th Edition

Sharpe Ratio Pros A great advantage of the Sharpe ratio is that it is easy to compute from observed returns. In addition, it can be useful to compare different categories of investments, for example, comparing the Sharpe ratios of various investment classes such as stocks, bonds and commodities. Cons However, the Sharpe ratio can sometimes give very misleading signals about the risk of funds that have employed strategies that for many periods have led to steady positive return, but then eventually blow up disastrously. Also the value of the Sharpe ratio can be quite sensitive to the start and end dates used as well as the specific time periods considered. Keith Pilbeam ©: Finance and Financial Markets 4th Edition

Treynor Ratio Treynor ratio is proposed by Jack Treynor (1965).
It can be a useful measure to compare excess return on a portfolio against the systematic risk of the portfolio. The Treynor ratio can be a useful measure to compare diversified portfolios with relatively little unsystematic risk in them. A fund manager with a higher Treynor ratio than another fund manager with a similar beta is performing relatively well. A fund manager with a higher beta and lower excess return than another fund manager would be underperforming relative to one with a higher excess return and lower beta. Of the two measures, the Sharpe ratio tends to be more widely used than the Treynor ratio because the Sharpe ratio penalizes managers that have relatively undiversified portfolios and therefore high amounts of unsystematic risk. Keith Pilbeam ©: Finance and Financial Markets 4th Edition

Jensen’s Alpha Jensen’s alpha is proposed by Michael Jensen(1968).
If Jensen’s alpha >0, the portfolio manager is beating the market in the sense that the manager is achieving a higher return than the risk profile of the portfolio would merit. If Jensen’s alpha <0, the portfolio manager is underperforming the market in the sense that the manager is achieving a lower return than the risk profile of the portfolio would merit. Keith Pilbeam ©: Finance and Financial Markets 4th Edition

Technical problems with the CAPM
Measuring beta Ex ante theory with ex post testing The market portfolio is unobtainable One period only model Very few government securities are close to being risk free Too many unrealistic assumptions Keith Pilbeam ©: Finance and Financial Markets 4th Edition

Ex ante theory with ex post testing
Applications of the CAPM tend to be focused on the future. Thus, it is investors’ expectations that drive share prices. The CAPM follows this ex ante (before the event) line of reasoning; it describes expected returns and future beta. However, when it comes to testing the theory, we observe what has already occurred (the past) – these are ex post observations. There is usually a large difference between investors’ expectations and the outcome. Keith Pilbeam ©: Finance and Financial Markets 4th Edition

The market portfolio is unobtainable
Roll’s (1977) criticism of the CAPM as untestable, because the benchmark market indices employed, such as the FTSE All-Share Index, are poor substitutes for the true market portfolio, strikes at the heart of the CAPM. If the beta being used to estimate returns is constructed from an inferior proxy then the relationship revealed will not be based on the theoretically true CAPM. Even if all the shares in the world were included in the index this would exclude many other relevant assets, from stamp collections to precious metals. Keith Pilbeam ©: Finance and Financial Markets 4th Edition

One-period model Investments usually involve a commitment for many years, whether the investment is made by a firm in real assets or by investors purchasing financial assets. However, the CAPM is based on parameters measured at one point in time. Key variables such as the risk-free rate of return might, in reality, change. A strict interpretation of the CAPM would insist on the use of the 3-month Treasury bill rate of return sold by a reputable government to investors. A practical solution is to use long-term government bond rates for rf. Keith Pilbeam ©: Finance and Financial Markets 4th Edition

Very few government securities are close to being risk free
For investors and corporate managers in most countries their own government’s bonds and bills (‘sovereign debt’) are regarded as having substantial risk. They are very far from risk free – just ask investors in Greek and Irish government bonds. A couple of options might be: (a) use the interest on top- rated (‘triple A-rated’) bonds issued in that currency, even if that is by a company rather than the government, or (b) while Greece is in the eurozone use the interest rate paid by the safest euro borrower, the German government. But these options present difficulties of their own. Keith Pilbeam ©: Finance and Financial Markets 4th Edition

Unrealistic assumptions
Investors are rational and risk averse. Investors are able to assess returns and standard deviations. Indeed they all have the same forecasts of returns and risk because of the free availability of information. There are no taxes or transaction costs. All investors can borrow or lend at the risk-free rate of interest. All assets are traded and it is possible to buy a fraction of a unit of an asset. However: does it describe market behaviour? Keith Pilbeam ©: Finance and Financial Markets 4th Edition

Alternatives to the CAPM
Factor Models (0ne or multi factor models) The Multi-factor CAPM The Arbitrage Pricing Theory Keith Pilbeam ©: Finance and Financial Markets 4th Edition

Roll’s Critique of the CAPM
In two papers Ross (1976) and Roll and Ross (1980) argued that the CAPM model was empirically untestable. Their basic criticism was that a true ‘market portfolio’ needs to contain all assets, both financial and non-financial, in an economy, and many of these are not empirically observable, such as human capital and so on. Furthermore, the CAPM is based upon microeconomic foundations which require the investors’ utility function to be measured in terms of expected return and risk as measured by the standard deviation of return. Keith Pilbeam ©: Finance and Financial Markets 4th Edition

The Arbitrage Pricing Theory (APT)
The basic postulate of APT is that market risk is itself made up of a number (k) of separate systematic factors. APT says that the return on a security is linearly related to k systematic factors without specifying exactly what these factors are: Keith Pilbeam ©: Finance and Financial Markets 4th Edition

APT & CAPM Proponents of APT argue that it has two main advantages over the CAPM model: The CAPM requires that the investor’s utility function is based upon expected returns and standard deviation risk. The APT does not require standard deviation to be used as a measure of risk. The other main advantage is that the APT does not require an unobservable market index to be compiled. The disadvantage of APT: It does not say what the relevant factors are. It does not say how many relevant factors there are. Keith Pilbeam ©: Finance and Financial Markets 4th Edition

Summary (1) Key Points and Concepts
Risk and return are positively related. Total risk consists of two elements: systematic risk (or market or non-diversifiable risk) unsystematic risk (or specific risk, or diversifiable risk). Beta measures the covariance between the returns on a particular share with the returns on the market as a whole. The security market line (SML) shows the relationship between risk as measured by beta and expected returns. The equation for the capital asset Pricing model is: rj =rf +βj (rm–rf) The slope of the characteristic line represents beta: rj =α+βj rm+e Keith Pilbeam ©: Finance and Financial Markets 4th Edition

Some examples of the CAPM’s application: portfolio selection; identifying mispriced shares; measuring portfolio performance; rate of return on firm’s projects. Technical problems with the CAPM measuring beta; ex ante theory but ex post testing and analysis; unobtainability of the market portfolio; one-period model; few government securities are risk free; unrealistic assumptions. Keith Pilbeam ©: Finance and Financial Markets 4th Edition

Early research seemed to confirm the validity of beta as the measure of risk influencing returns. Later work cast serious doubt on this. Some researchers say beta has no influence on returns. Beta is not the only determinant of return. Multi-factor models allow for a variety of influences on share returns. Factor models refer to diversifiable risk as non-factor risk and non-diversifiable risk as factor risk. Keith Pilbeam ©: Finance and Financial Markets 4th Edition

Chapter 8 The Capital Asset Pricing Model

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