1.Prepared by Ken Hartviksen INTRODUCTION TO CORPORATE FINANCE Laurence BoothW. Sean Cleary Chapter 9 The Capital Asset Pricing Model

2. CHAPTER 9 The Capital Asset Pricing Model (CAPM) 3. Lecture Agenda

Learning Objectives

Important Terms

The New Efficient Frontier

The Capital Asset Pricing Model

The CAPM and Market Risk

Alternative Asset Pricing Models

Summary and Conclusions

• Concept Review Questions

• Appendix 1 Calculating the Ex Ante Beta

• Appendix 2 Calculating the Ex Post Beta

4. Learning Objectives

What happens if all investors are rational and risk averse.

How modern portfolio theory is extended to develop the capital market line, which determines how expected returns on portfolios are determined.

How to assess the performance of mutual fund managers

How the Capital Asset Pricing Models (CAPM) security market line is developed from the capital market line.

How the CAPM has been extended to include other risk-based pricing models.

5. Important Chapter Terms

Arbitrage pricing theory (APT)

Capital Asset Pricing Model (CAPM)

Capital market line (CML)

Characteristic line

Fama-French (FF) model

Insurance premium

Market portfolio

Market price of risk

Market risk premium

New (or super) efficient frontier

No-arbitrage principle

Required rate of return

Risk premium

Security market line (SML)

Separation theorum

Sharpe ratio

Short position

Tangent portfolio

6. Achievable Portfolio Combinations The Capital Asset Pricing Model (CAPM) 7. Achievable Portfolio Combinations The Two-Asset Case

It is possible to construct a series of portfolios with different risk/return characteristics just by varying the weights of the two assets in the portfolio.

Assets A and B are assumed to have a correlation coefficient of -0.379 and the following individual return/risk characteristics

• • Expected Return Standard Deviation

• • Asset A 8% 8.72%

• • Asset B 10% 22.69%

• The following table shows the portfolio characteristics for 100different weighting schemes for just these two securities:

8. Example of Portfolio Combinations and Correlation The first combination simply assumes you invest solely in Asset A The second portfolio assumes 99% in A and 1% in B.Notice the increase in return and the decrease in portfolio risk! You repeat this procedure down until you have determine the portfolio characteristics for all 100 portfolios.Next plot the returns on a graph (see the next slide) 9. Example of Portfolio Combinations and Correlation 10. Two Asset Efficient Frontier

Figure 8 10 describes five different portfolios (A,B,C,D and E in reference to the attainable set of portfolio combinations of this two asset portfolio.

(See Figure 8 -10 on the following slide)

11. Efficient Frontier The Two-Asset Portfolio Combinations Ais not attainable B,Elie on the efficient frontier and are attainable Eis the minimum variance portfolio (lowest risk combination) C, Dare attainable but are dominated by superior portfolios that line on the line aboveE 8 - 10 FIGURE Expected Return % Standard Deviation (%) A E B C D 12. Achievable Set of Portfolio Combinations Getting to the n Asset Case

In a real world investment universe with all of the investment alternatives (stocks, bonds, money market securities, hybrid instruments, gold real estate, etc.) it is possible to construct many different alternative portfolios out of risky securities.

Each portfolio will have its own unique expected return and risk.

Whenever you construct a portfolio, you can measure two fundamental characteristics of the portfolio:

• Portfolio expected return ( ER p )

• Portfolio risk ( p )

13. The Achievable Set of Portfolio Combinations

You could start by randomly assembling ten risky portfolios.

The results (in terms of ERpand p) might look like the graph on the following page:

14. Achievable Portfolio Combinations The First Ten Combinations Created Portfolio Risk( p ) 10 Achievable Risky Portfolio Combinations ER p 15. The Achievable Set of Portfolio Combinations

You could continue randomly assembling more portfolios.

Thirty risky portfoliosmight look like the graph on the following slide:

16. Achievable Portfolio Combinations Thirty Combinations Naively Created Portfolio Risk( p ) 30 Risky Portfolio Combinations ER p 17. Achievable Set of Portfolio Combinations All Securities Many Hundreds of Different Combinations

When you construct many hundreds of different portfolios naively varying the weight of the individual assets and the number of types of assets themselves, you get a set of achievable portfolio combinations as indicated on the following slide:

18. Achievable Portfolio Combinations More Possible Combinations Created Portfolio Risk( p ) ER p E E is the minimum variance portfolio Achievable Set of Risky Portfolio Combinations The highlighted portfolios are efficientin that they offer the highest rate of return for a given level of risk.Rationale investors will choose only from this efficient set. 19. The Efficient Frontier The Capital Asset Pricing Model (CAPM) 20. Achievable Portfolio Combinations Efficient Frontier (Set) Portfolio Risk( p ) Achievable Set of Risky Portfolio Combinations ER p E Efficient frontier is the set of achievable portfolio combinations that offer the highest rate of return for a given level of risk. 21. The New Efficient Frontier Efficient Portfolios Figure 9 1 illustrates three achievable portfolio combinations that are efficient(no other achievable portfolio that offers the same risk, offers a higher return.) Risk 9 - 1 FIGURE Efficient Frontier ER MVP A B 22. Underlying Assumption Investors are Rational and Risk-Averse

We assume investors are risk-averse wealth maximizers.

This means they will not willingly undertake fair gamble.

• A risk-averse investor prefers the risk-free situation.

• The corollary of this is that the investor needs a risk premium to be induced into a risky situation.

• Evidence of this is the willingness of investors to pay insurance premiums to get out of risky situations.

The implication of this, is that investors will only choose portfolios that are members of the efficient set (frontier).

23. The New Efficient Frontier and Separation Theorem The Capital Asset Pricing Model (CAPM) 24. Risk-free Investing

When we introduce the presence of a risk-free investment, a whole new set of portfolio combinations becomes possible.

We can estimate the return on a portfolio made up ofRFasset and a risky assetAletting the weightwinvested in the risky asset and the weight invested in RF as(1 w)

25. The New Efficient Frontier Risk-Free Investing

• Expected return on a two asset portfolio made up of risky assetAandRF :

• The possible combinations of A and RF are found graphed on the following slide.

[9-1] 26. The New Efficient Frontier Attainable Portfolios UsingRFandA 9 - 2 FIGURE Risk ER RF A [9-2] Equation 9 2 illustrates what you can seeportfolio risk increases in direct proportion to the amount invested in the risky asset. [9-3] Rearranging 9 -2 wherew= p/ A and substituting in Equation 1 we get an equation for a straight line with a constant slope. This means you can achieve any portfolio combination along the blue coloured line simply by changing the relative weight ofRFandAin the two asset portfolio. 27. The New Efficient Frontier Attainable Portfolios using theRFandA,andRFandT Which risky portfolio would a rational risk-averse investor choose in the presence of aRFinvestment? PortfolioA ? Tangent PortfolioT ? 9 - 3 FIGURE Risk ER RF A T 28. The New Efficient Frontier Efficient Portfolios using the Tangent PortfolioT 9 - 3 FIGURE Risk ER RF A T ClearlyRFwithT(the tangent portfolio) offers a series of portfolio combinations that dominate those produced byRFandA . Further, they dominate all but one portfolio on the efficient frontier! 29. The New Efficient Frontier Lending Portfolios 9 - 3 FIGURE Risk ER RF A T Portfolios betweenRFandTare lending portfolios, because they are achieved by investing in the Tangent Portfolio and lending funds to the government (purchasing a T-bill, theRF ). Lending Portfolios 30. The New Efficient Frontier Borrowing Portfolios 9 - 3 FIGURE Risk ER RF A T Lending Portfolios The line can be extended to risk levels beyond Tby borrowing at RF and investing it in T.This is a levered investment that increases both risk and expected return of the portfolio. Borrowing Portfolios 31. The New Efficient Frontier The New (Super) Efficient Frontier The optimal risky portfolio (the market portfolio M) 9 - 4 FIGURE ER RF A2 T A B B2 Capital Market Line Clearly RF with T (the market portfolio) offers a series of portfolio combinations that dominate those produced by RF and A. Further, they dominate all but one portfolio on the efficient frontier! This is now called the new (or super) efficient frontier of risky portfolios. Investors can achieve any one of these portfolio combinations by borrowing or investing in RF in combination with the market portfolio. 32. The New Efficient Frontier The Implications Separation Theorem Market Portfolio

All investors will only hold individually-determined combinations of:

• The risk free asset (RF) and

• The model portfolio (market portfolio)

The separation theorem

• The investment decision (how to construct the portfolio of risky assets) is separate from the financing decision (how much should be invested or borrowed in the risk-free asset)

• The tangent portfolio T is optimal for every investor regardless of his/her degree of risk aversion.

The Equilibrium Condition

• The market portfolio must be the tangent portfolio T if everyone holds the same portfolio

• Therefore the market portfolio (M) is the tangent portfolio (T)

33. The New Efficient Frontier The Capital Market Line The optimal risky portfolio (the market portfolio M) ER RF M CML The CML is that set of superior portfolio combinations that are achievable in the presence of the equilibrium condition. 34. The Capital Asset Pricing Model The Hypothesized Relationship between Risk and Return 35. The Capital Asset Pricing Model What is it?

• An hypothesis by Professor William Sharpe

• • Hypothesizes that investors require higher rates of return for greater levels of relevant risk.

• • There are no prices on the model, instead it hypothesizes the relationship between risk and return for individual securities.

• • It is often used, however, the price securities and investments.

36. The Capital Asset Pricing Model How is it Used?

• Uses include:

• • Determining the cost of equity capital.

• • The relevant risk in the dividend discount model to estimate a stocks intrinsic (inherent economic worth) value.(As illustrated below)

Estimate Investments Risk (Beta Coefficient) Determine Investments Required Return Estimate the Investments Intrinsic Value Compare to the actual stock price in the market Is the stock fairly priced? 37. The Capital Asset Pricing Model Assumptions

• CAPM is based on the following assumptions:

• • • All investors have identical expectations about expected returns, standard deviations, and correlation coefficients for all securities.

• • • All investors have the same one-period investment time horizon.

• • • All investors can borrow or lend money at the risk-free rate of return (RF).

• • • There are no transaction costs.

• • • There are no personal income taxes so that investors are indifferent between capital gains an dividends.

• • • There are many investors, and no single investor can affect the price of a stock through his or her buying and selling decisions.Therefore, investors are price-takers.

• • • Capital markets are in equilibrium.

38. Market Portfolio and Capital Market Line

The assumptions have the following implications:

• The optimalrisky portfolio is the one that is tangent to the efficient frontier on a line that is drawn from RF.This portfolio will be the same for all investors.

• This optimal risky portfolio will be themarket portfolio(M) which contains all risky securities.

• (Figure 9 4 illustrates the Market Portfolio M)

39. The Capital Market Line The CML is that set of achievable portfolio combinations that are possible when investing in only two assets (the market portfolio and the risk-free asset (RF). The market portfolio is the optimal risky portfolio, it contains all risky securities and lies tangent (T) on the efficient frontier. The CML has standard deviation of portfolio returns as the independent variable. 9 - 5 FIGURE ER RF M ER M M CML 40. The Capital Asset Pricing Model The Market Portfolio and the Capital Market Line (CML)

• The slope of the CML is the incremental expected return divided by the incremental risk.

• This is calledthe market price for risk. Or

• The equilibrium price of risk in the capital market.

[9-4] 41. The Capital Asset Pricing Model The Market Portfolio and the Capital Market Line (CML)

• Solving for the expected return on a portfolio in the presence of a RF asset and given themarket price for risk :

• Where:

• • ER M= expected return on the market portfolio M

• • M = the standard deviation of returns on the market portfolio

• • P = the standard deviation of returns on the efficient portfolio being considered

[9-5] 42. The Capital Market Line Using the CML Expected versus Required Returns

• In an efficient capital market investors will require a return on a portfolio that compensates them for the risk-free return as well as the market price for risk.

• This means that portfolios should offer returns along the CML.

43. The Capital Asset Pricing Model Expected and Required Rates of ReturnBis a portfolio that offers and expected return equal to the required return. Ais an undervalued portfolio. Expected return is greater than the required return. Demand for Portfolio A will increase driving up the price, and therefore the expected return will fall until expected equals required (market equilibrium condition is achieved.) Required return on A Expected return on A 9 - 6 FIGURE ER RF B C A CML Cis an overvalued portfolio.Expected return is less than the required return. Selling pressure will cause the price to fall and the yield to rise until expected equals the required return. Required Return on C Expected Return on C 44. The Capital Asset Pricing Model Risk-Adjusted Performance and the Sharpe Ratios

• William Sharpe identified a ratio that can be used to assess the risk-adjusted performance of managed funds (such as mutual funds and pension plans).

• It is called the Sharpe ratio:

• Sharpe ratio is a measure of portfolio performance that describes how well an assets returns compensate investors for the risk taken.

• Its value is the premium earned over the RF divided by portfolio riskso it is measuring valued added per unit of risk.

• Sharpe ratios are calculated ex post (after-the-fact) and are used to rank portfolios or assess the effectiveness of the portfolio manager in adding value to the portfolio over and above a benchmark.

[9-6] 45. The Capital Asset Pricing Model Sharpe Ratios and Income Trusts

• Table 9 1 (on the following slide) illustrates return, standard deviation, Sharpe and beta coefficient for four very different portfolios from 2002 to 2004.

• Income Trusts did exceedingly well during this time, however, the recent announcement of Finance Minister Flaherty and the subsequent drop in Income Trust values has done much to eliminate this historical performance.

46. Income Trust Estimated Values 47. CAPM and Market Risk The Capital Asset Pricing Model 48. Diversifiable and Non-Diversifiable Risk

CML applies to efficient portfolios

Volatility (risk) ofindividual security returnsare caused by two different factors:

• Non-diversifiable risk (system wide changes in the economy and markets that affect all securities in varying degrees)

• Diversifiable risk (company-specific factors that affect the returns of only one security)

Figure 9 7 illustrates what happens to portfolio risk as the portfolio is first invested in only one investment, and then slowly invested, naively, in more and more securities.

49. The CAPM and Market Risk Portfolio Risk and Diversification 9 - 7 FIGURE Number of Securities Total Risk ( ) Unique (Non-systematic) Risk Market (Systematic) Risk Market or systematic risk is risk that cannot be eliminated from the portfolio by investing the portfolio into more and different securities. 50. Relevant Risk Drawing a Conclusion from Figure 9 - 7

Figure 9 7 demonstrates that an individual securitiesvolatility of return comes from two factors:

• Systematic factors

• Company-specific factors

When combined into portfolios, company-specific risk is diversified away.

Since all investors are diversifiedthen in an efficient market, no-one would be willing to pay a premiumfor company-specific risk.

Relevant risk to diversified investors then is systematic risk.

Systematic risk is measured using the Beta Coefficient.

51. Measuring Systematic RiskThe Beta Coefficient The Capital Asset Pricing Model (CAPM) 52. The Beta Coefficient What is the Beta Coefficient?

A measure of systematic (non-diversifiable) risk

As a coefficientthe beta is a pure number and has no units of measure.

53. The Beta Coefficient How Can We Estimate the Value of the Beta Coefficient?

There are two basic approaches to estimating the beta coefficient:

• Using a formula(and subjective forecasts)

• Use of regression (using past holding period returns)

• (Figure 9 8 on the following slide illustrates the characteristic line used to estimate the beta coefficient)

54. The CAPM and Market Risk The Characteristic Line for Security A The plotted points are the coincident rates of return earned on the investment and the market portfolio over past periods. 9 - 8 FIGURE 6 4 2 0 -2 -4 -6 Security A Returns ( %) -6 -4 -2 0 2 4 6 8 Market Returns (%) The slope of the regression line is beta. The line of best fit is known in finance as the characteristic line. 55. The Formula for the Beta Coefficient

• Beta is equal to the covariance of the returns of the stock with the returns of the market, divided by the variance of the returns of the market:

[9-7] 56. The Beta Coefficient How is the Beta Coefficient Interpreted?

The beta of the market portfolio is ALWAYS = 1.0

The beta of a security compares the volatility of its returns to the volatility of themarket returns:

• s = 1.0 - the security has the same volatility as the market as a whole

• s > 1.0 - aggressive investment with volatility of returns greater than the market

• s < 1.0 - defensive investment with volatility of returns less than the market

• s < 0.0 - an investment with returns that are negatively correlated with the returns of the market

• Table 9 2 illustrates beta coefficients for a variety of Canadian Investments

57. Canadian BETAS Selected 58. The Beta of a Portfolio

• The beta of a portfolio is simply the weighted average of the betas of the individual asset betas that make up the portfolio.

• Weights of individual assets are found by dividing the value of the investment by the value of the total portfolio.

[9-8] 59. The Security Market Line The Capital Asset Pricing Model (CAPM) 60. The CAPM and Market Risk The Security Market Line (SML)

• The SML is the hypothesized relationship between return (the dependent variable) and systematic risk (the beta coefficient).

• It is a straight line relationship defined by the following formula:

• Where:

• • k i= the required return on security i

• • ER M RF = market premium for risk

• • i= the beta coefficient for security i

• • (See Figure 9 - 9 on the following slide for the graphical representation)

[9-9] 61. The CAPM and Market Risk The Security Market Line (SML) The SML is used to predict required returns for individual securities 9 - 9 FIGURE M= 1 ER RF M ER M The SML uses the beta coefficient as the measure of relevant risk. 62. The CAPM and Market Risk The SML and Security Valuation Similarly, B is an overvalued security. Investors will sell to lock in gains, but the selling pressure will cause the market price to fall, causing the expected return to rise until it equals the required return. 9 - 10 FIGURE AER RF B A BSML Required returns are forecast using this equation. You can see that the required return on any security is a function of its systematic risk ( ) and market factors ( RF and market premium for risk) Ais an undervalued security because its expected return is greater than the required return. Investors will flock to A and bid up the price causing expected return to fall till it equals the required return. Required Return A Expected Return A 63. The CAPM in Summary The SML and CML

• The CAPM is well entrenched and widely used by investors, managers and financial institutions.

• It is a single factor model because it based on the hypothesis that required rate of return can be predicted using one factor systematic risk

• The SML is used to price individual investments and uses the beta coefficient as the measure of risk.

• The CML is used with diversified portfolios and uses the standard deviation as the measure of risk.

64. Alternative Pricing Models The Capital Asset Pricing Model (CAPM) 65. Challenges to CAPM

Empirical tests suggest:

• CAPM does not hold well in practice:

• • Ex post SML is an upward sloping line

• • Ex antey (vertical) intercept is higher that RF

• • Slope is less than what is predicted by theory

• Beta possesses no explanatory power for predicting stock returns (Fama and French, 1992)

CAPM remains in widespread use despite the foregoing.

• Advantages include relative simplicity and intuitive logic.

Because of the problems with CAPM, other models have been developed including:

• Fama-French (FF) Model

• Abitrage Pricing Theory (APT)

66. Alternative Asset Pricing Models The Fama French Model

• A pricing model that uses three factors to relate expected returns to risk including:

• • • A market factor related to firm size.

• • • The market value of a firms common equity (MVE)

• • • Ratio of a firms book equity value to its market value of equity. (BE/MVE)

• This model has become popular, and many think it does a better job than the CAPM in explaining ex ante stock returns.

67. Alternative Asset Pricing Models The Arbitrage Pricing Theory

• A pricing model that uses multiple factors to relate expected returns to risk by assuming that asset returns are linearly related to a set of indexes, which proxy risk factors that influence security returns.

• It is based on the no-arbitrage principle which is the rule that two otherwise identical assets cannot sell at different prices.

• Underlying factors represent broad economic forces which are inherently unpredictable.

[9-10] 68. Alternative Asset Pricing Models The Arbitrage Pricing Theory the Model

• Underlying factors represent broad economic forces which are inherently unpredictable.

• Where:

• • ER i= the expected return on security i

• • a 0 = the expected return on a security with zero systematic risk

• • b i = the sensitivity of security i to a given risk factor

• • F i = the risk premium for a given risk factor

• The model demonstrates that a securitys risk is based on its sensitivity to broad economic forces.

[9-10] 69. Alternative Asset Pricing Models The Arbitrage Pricing Theory Challenges

• Underlying factors represent broad economic forces which are inherently unpredictable.

• Ross and Roll identify five systematic factors:

• • • Changes in expected inflation

• • • Unanticipated changes in inflation

• • • Unanticipated changes in industrial production

• • • Unanticipated changes in the default-risk premium

• • • Unanticipated changes in the term structure of interest rates

• Clearly, something that isnt forecast, cant be used to price securities todaythey can only be used to explain prices after the fact.

70. Summary and Conclusions

In this chapter you have learned:

• How the efficient frontier can be expanded by introducing risk-free borrowing and lending leading to a super efficient frontier called the Capital Market Line (CML)

• The Security Market Line can be derived from the CML and provides a way to estimate a market-based, required return for any security or portfolio based on market risk as measured by the beta.

• That alternative asset pricing models exist including the Fama-French Model and the Arbitrage Pricing Theory.

71. Concept Review Questions The Capital Asset Pricing Model 72. Concept Review Question 1 Risk Aversion

What is risk aversion and how do we know investors are risk averse?

73. Estimating the Ex Ante (Forecast) Beta APPENDIX 1 74. Calculating a Beta Coefficient Using Ex Ante Returns

Ex Ante means forecast

You would use ex ante return data if historical rates of return are somehow not indicative of the kinds of returns the company will produce in the future.

A good example of this is Air Canada or American Airlines, before and after September 11, 2001.After the World Trade Centre terrorist attacks, a fundamental shift in demand for air travel occurred.The historical returns on airlines are not useful in estimating future returns.

75. Appendix 1 Agenda

The beta coefficient

The formula approach to beta measurement using ex ante returns

• Ex ante returns

• Finding the expected return

• Determining variance and standard deviation

• Finding covariance

• Calculating and interpreting the beta coefficient

76. The Beta Coefficient

Under the theory of the Capital Asset Pricing Model total risk is partitioned into two parts:

• Systematic risk

• Unsystematic risk diversifiable risk

Systematic risk is non-diversifiable risk.

Systematic risk is the only relevant risk to the diversified investor

The beta coefficient measures systematic risk

Systematic Risk Unsystematic Risk Total Risk of the Investment 77. The Beta Coefficient The Formula [9-7] 78. The Term Relevant Risk

What does the term relevant risk mean in the context of the CAPM?

• It is generally assumed that all investors are wealth maximizing risk averse people

• It is also assumed that the markets where these people trade are highly efficient

• In a highly efficient market, the prices of all the securities adjust instantly to cause the expected return of the investment to equal the required return

• When E(r) = R(r)then the market price of the stock equals its inherent worth (intrinsic value)

• In this perfect world, the R(r) then will justly and appropriately compensate the investor only for the risk that they perceive as relevant

• Hence investors are only rewarded for systematic risk.

NOTE:The amount of systematic risk varies by investment.High systematic risk occurs when R-square is high, and the beta coefficient is greater than 1.0 79. The Proportion of Total Risk that is Systematic

Every investment in the financial markets vary with respect to the percentage of total risk that is systematic.

Some stocks have virtuallyno systematicrisk.

• Such stocks are not influenced by the health of the economy in generaltheir financial results are predominantly influenced by company-specific factors.

• An example is cigarette companiespeople consume cigarettes because they are addictedso it doesnt matter whether the economy is healthy or notthey just continue to smoke.

Some stocks have a high proportion of their total risk that is systematic

• Returns on these stocks are strongly influenced by the health of the economy.

• Durable goods manufacturers tend to have a high degree of systematic risk.

80. The Formula Approach to Measuring the Beta

You need to calculate the covariance of the returns between the stock and the marketas well as the variance of the market returns.To do this you must follow these steps:

• Calculate the expected returns for the stock and the market

• Using the expected returns for each, measure the variance and standard deviation of both return distributions

• Now calculate the covariance

• Use the results to calculate the beta

81. Ex ante Return DataA Sample

A set of estimates of possible returns and their respective probabilities looks as follows:

By observation you can see the range is much greater for the stock than the market and they move in the same direction. Since the beta relates the stock returns to the market returns, the greater range of stock returns changing in the same direction as the market indicates the beta will be greater than 1 and will be positive.(Positively correlated to the market returns.) 82. The Total of the Probabilities must Equal 100%

This means that we have considered all of the possible outcomes in this discrete probability distribution

83. MeasuringExpected Return on the Stock From Ex Ante Return Data

The expected return is weighted average returns from the given ex ante data

84. MeasuringExpected Return on the MarketFrom Ex Ante Return Data

The expected return is weighted average returns from the given ex ante data

85. Measuring Variances, Standard Deviations of the Forecast Stock Returns

Using the expected return, calculate the deviations away from the mean, square those deviations and then weight the squared deviations by the probability of their occurrence.Add up the weighted and squared deviations from the mean and you have found the variance!

86. Measuring Variances, Standard Deviations of the Forecast Market Returns

Now do this for the possible returns on the market

87. Covariance

From Chapter 8 you know the formula for the covariance between the returns on the stock and the returns on the market is:

Covariance is an absolute measure of the degree of co-movement of returns.

[8-12] 88. Correlation Coefficient

Correlation is covariance normalized by the product of the standard deviations of both securities.It is a relative measure of co-movement of returns on a scale from -1 to +1.

The formula for the correlation coefficient between the returns on the stock and the returns on the market is:

The correlation coefficient will always have a value in the range of+1 to -1.

• • +1 is perfect positive correlation (there is no diversification potential when combining these two securities together in a two-asset portfolio.)

• • - 1 - is perfect negative correlation (there should be a relative weighting mix of these two securities in a two-asset portfolio that will eliminate all portfolio risk)

[8-13] 89. Measuring Covariance from Ex Ante Return Data

Using the expected return (mean return) and given data measure the deviations for both the market and the stock and multiply them together with the probability of occurrencethen add the products up.

90. The Beta Measured Using Ex Ante Covariance (stock, market) and Market Variance

Now you can substitute the values for covariance and the variance of the returns on the market to find the beta of the stock:

• A beta that is greater than 1 means that the investment is aggressiveits returns are more volatile than the market as a whole.

• If the market returns were expected to go up by 10%, then the stock returns are expected to rise by 18%.If the market returns are expected to fall by 10%, then the stock returns are expected to fall by 18%.

91. Lets Prove the Beta of the Market is 1.0

Let us assume we are comparing the possible market returns against itselfwhat will the beta be?

Since the variance of the returns on the market is = .007425 the beta for the market is indeed equal to 1.0 !!! 92. Proving the Beta of Market = 1

If you now place the covariance of the market with itself value in the beta formula you get:

The beta coefficient of the market will always be 1.0 because you are measuring the market returns against market returns. 93. Using the Security Market Line Expected versus Required Return 94. How Do We use Expected and Required Rates of Return?

Once you have estimated the expected and required rates of return, you can plot them on the SML and see if the stock is under or overpriced.

Since E(r)>R(r) the stock is underpriced. % Return Risk-free Rate = 3% B M = 1.0 E(k M )= 4.2% B s= 1.464 R(k s ) = 4.76% E(R s ) = 5.0% SML 95. How Do We use Expected and Required Rates of Return?

The stock is fairly priced if the expected return = the required return.

This is what we would expect to see normally or most of the time in an efficient market where securities are properly priced.

% Return Risk-free Rate = 3% B M = 1.0 E(R M )= 4.2% B S= 1.464 E(R s ) = R(R s ) 4.76% SML 96. Use of the Forecast Beta

We can use the forecast beta, together with an estimate of the risk-free rate and the market premium for risk to calculate the investors required return on the stock using the CAPM:

This is a market-determinedreturn based on the current risk-free rate (RF) as measured by the 91-day, government of Canada T-bill yield, and a current estimate of the market premium for risk (k M RF)

97. Conclusions

Analysts can make estimates or forecasts for the returns on stock and returns on the market portfolio.

Those forecasts can be analyzed to estimate the beta coefficient for the stock.

The required return on a stock can then be calculated using the CAPM but you will need the stocks beta coefficient, the expected return on the market portfolio and the risk-free rate.

The required return is then using in Dividend Discount Models to estimate the intrinsic value (inherent worth) of the stock.

98. Calculating the Beta using Trailing Holding Period Returns APPENDIX 2 99. The Regression Approach to Measuring the Beta

You need to gather historical data about the stock and the market

You can use annual data, monthly data, weekly data or daily data. However, monthly holding period returns are most commonly used.

• Daily data is too noisy (short-term random volatility)

• Annual data will extend too far back in to time

You need at least thirty (30) observations of historical data.

Hopefully, the period over which you study the historical returns of the stock is representative of the normal condition of the firm and its relationship to the market.

If the firm has changed fundamentally since these data were produced (for example, the firm may have merged with another firm or have divested itself of a major subsidiary) there is good reason to believe that future returns will not reflect the pastand this approach to beta estimation SHOULD NOT be used.rather, use the ex ante approach.

100. Historical Beta Estimation The Approach Used to Create the Characteristic Line In this example, we have regressed thequarterlyreturns on the stock against the quarterly returns of a surrogate for the market (TSE 300 total return composite index) and then using Excelused the charting feature to plot the historical points and add a regression trend line. The regression line is a line of best fit that describes the inherent relationship between the returns on the stock and the returns on the market.The slope is the beta coefficient. The cloud of plotted points represents diversifiable or company specificrisk in the securities returns that can be eliminated from a portfolio through diversification.Since company-specific risk can be eliminated, investors dont require compensation for it according to Markowitz Portfolio Theory. 101. Characteristic Line

The characteristic line is a regression line that represents the relationship between the returns on the stock and the returns on the market over a past period of time.(It will be used to forecast the future, assuming the future will be similar to the past.)

Theslope of the Characteristic Lineis the Beta Coefficient.

The degree to which the characteristic line explains the variability in the dependent variable (returns on the stock) is measured by the coefficient of determination. (also known as theR 2 (r-squared or coefficient of determination)).

If the coefficient of determination equals 1.00, this would mean that all of the points of observation would lie on the line.This would mean that the characteristic line would explain 100% of the variability of the dependent variable.

Thealphais the vertical intercept of the regression (characteristic line).Many stock analysts search out stocks with high alphas.

102. Low R 2

An R 2that approaches 0.00 (or 0%) indicates that the characteristic (regression) line explains virtually none of the variability in the dependent variable.

This means that virtually of the risk of the security is company-specific.

This also means that the regression model has virtually no predictive ability.

In this case, you should use other approaches to value the stockdo not use the estimated beta coefficient.

(See the following slide for an illustration of a low r-square)

103. Characteristic Line for Imperial Tobacco An Example of Volatility that is Primarily Company-Specific

High alpha

R-square is very low 0.02

Beta is largely irrelevant

Returns on the Market % (S&P TSX) Returns on Imperial Tobacco % Characteristic Line for Imperial Tobacco 104. High R 2

An R 2that approaches 1.00 (or 100%) indicates that the characteristic (regression) line explains virtually all of the variability in the dependent variable.

This means that virtually of the risk of the security is systematic.

This also means that the regression model has a strong predictive ability. if you can predict what the market will dothen you can predict the returns on the stock itself with a great deal of accuracy.

105. Characteristic Line General Motors A Positive Beta with Predictive Power

Positive alpha

R-square is very high 0.9

Beta is positive and close to 1.0

Returns on the Market % (S&P TSX) Returns on General Motors % Characteristic Line for GM (high R 2 ) 106. An Unusual Characteristic Line A Negative Beta with Predictive Power

Positive alpha

R-square is very high

Beta is negative -1.0

Returns on the Market % (S&P TSX) Returns on a Stock % Characteristic Line for a stock that will provide excellent portfolio diversification (high R 2 ) 107. Diversifiable Risk (Non-systematic Risk)

Volatility in a securitys returns caused by company-specific factors (both positive and negative) such as:

• a single company strike

• a spectacular innovation discovered through the companys R&D program

• equipment failure for that one company

• management competence or management incompetence for that particular firm

• a jet carrying the senior management team of the firm crashes (this could be either a positive or negative event, depending on the competence of the management team)

• the patented formula for a new drug discovered by the firm.

Obviously, diversifiable risk is that unique factor that influences only the one firm.

108. OK lets go back and look at raw data gathering and data normalization

A common source for stock of information is Yahoo.com

You will also need to go to the library a use the TSX Review (a monthly periodical) to obtain:

• Number of shares outstanding for the firm each month

• Ending values for the total return composite index (surrogate for the market)

You want data for at least 30 months.

For each month you will need:

• Ending stock price

• Number of shares outstanding for the stock

• Dividend per share paid during the month for the stock

• Ending value of the market indicator series you plan to use (ie.TSE 300 total return composite index)

109. Demonstration Through Example The following slides will be based on Alcan Aluminum(AL.TO) 110. Five Year Stock Price Chart for AL.TO 111. Spreadsheet Data From Yahoo

Process:

• Go tohttp://ca.finance.yahoo.com

• Use the symbol lookup function to search for the company you are interested in studying.

• Use the historical quotes buttonand get 30 months of historical data.

• Use the download in spreadsheet format feature to save the data to your hard drive.

112. Spreadsheet Data From Yahoo Alcan Example

The raw downloaded datashould look like this:

113. Spreadsheet Data From YahooAlcan Example

The raw downloaded datashould look like this:

Volume of trading done in the stock on the TSE in the month in numbers of board lots The day, month and yearOpening price per share, the highest price per share during the month, the lowest price per share achieved during the month and the closing price per share at the end of the month 114. Spreadsheet Data From YahooAlcan Example

From Yahoo, the only information you can use is the closing price per share and the date.Just delete the other columns.

115. Acquiring the Additional Information You NeedAlcan Example

In addition to the closing price of the stock on a per share basis, you will need to find out how many shares were outstanding at the end of the month and whether any dividends were paid during the month.

You will also want to find the end-of-the-month value of the S&P/TSX Total Return Composite Index (look in the green pages of the TSX Review)

You can find all of this inThe TSX Reviewperiodical.

116. Raw Company DataAlcan Example Number of shares doubled and share price fell by half between January and February 2002 this is indicative of a 2 for 1 stock split. 117. Normalizing the Raw Company DataAlcan Example The adjustment factor is just the value in the issued capital cell divided by 321,400,589. 118. Calculating the HPR on the stock from the Normalized Data Use $59.22 as the ending price, $57.90 as the beginning price and during the month of May, no dividend was declared. 119. Now Put the data from the S&P/TSX Total Return Composite Index in You will find the Total Return S&P/TSX Composite Index values inTSX Reviewfound in the library. 120. Now Calculate the HPR on the Market Index Again, you simply use the HPR formula using the ending values for the total return composite index. 121. Regression In Excel

If you havent alreadygo to the tools menudown to add-ins and check off the VBA Analysis Pac

When you go back to the tools menu, you should now find the Data Analysis bar, under that find regression, define your dependent and independent variable ranges, your output range and run the regression.

122. Regression Defining the Data Ranges The independent variable is the returns on the Market. The dependent variable is the returns on the Stock. 123. Now Use the Regression Function in Excel to regress the returns of the stock against the returns of the market Beta Coefficient is the X-Variable 1 The alpha is the vertical intercept. R-square is the coefficient of determination = 0.0028=.3% 124. Finalize Your ChartAlcan Example

You can use the charting feature in Excel to create a scatter plot of the points and to put a line of best fit (the characteristic line) through the points.

In Excel, you can edit the chart after it is created by placing the cursor over the chart and right-clickingyour mouse.

In this edit mode, you can ask it to add a trendline (regression line)

Finally, you will want to interpret the Beta (X-coefficient) the alpha (vertical intercept) and the coefficient of determination.

125. The BetaAlcan Example

Obviously the beta (X-coefficient) can simply be read from the regression output.

• In this case it was 3.56 making Alcans returns more than 3 times as volatile as the market as a whole.

• Of course, in this simple example with only 5 observations, you wouldnt want to draw any serious conclusions from this estimate.

126. Copyright

Copyright 2007 John Wiley & Sons Canada, Ltd.All rights reserved.Reproduction or translation of this work beyond that permitted by Access Copyright (the Canadian copyright licensing agency) is unlawful.Requests for further information should be addressed to the Permissions Department, John Wiley & Sons Canada, Ltd. The purchaser may make back-up copies for his or her own use only and not for distribution or resale. The author and the publisher assume no responsibility for errors, omissions, or damages caused by the use of these files or programs or from the use of the information contained herein.