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Portfolio Construction, Management & Protection Robert Strong Chapter 5 – The Mathematics of Diversification Prepared by Ken Hartviksen CHAPTER 5 The Mathematics of Diversification Lecture Agenda • Learning Objectives • Important Terms • Measurement of Returns • Measuring Risk • Expected Return and Risk for Portfolios • The Efficient Frontier • Diversification • The New Efficient Frontier • The Capital Asset Pricing Model • The CAPM and Market Risk • Alternative Asset Pricing Models • Summary and Conclusions CHAPTER 5 – The Mathematics of Diversification 8-3 Learning Objectives • The difference among the most important types of returns • How to estimate expected returns and risk for individual securities • What happens to risk and return when securities are combined in a portfolio • What is meant by an “efficient frontier” • Why diversification is so important to investors • How to measure risk and return in portfolios CHAPTER 5 – The Mathematics of Diversification 8-4 Important Chapter Terms • Arithmetic mean • Mark to market • Attainable portfolios • Market risk • Capital gain/loss • Minimum variance frontier • Correlation coefficient • Minimum variance portfolio • Covariance • Modern portfolio theory • Day trader • Naïve or random diversification • Diversification • Paper losses • Efficient frontier • Portfolio • Efficient portfolios • Range • Ex ante returns • Risk averse • Ex post returns • Standard deviation • Expected returns • Total return • Geometric mean • Unique (or non-systematic) or • Income yield diversifiable risk • Variance CHAPTER 5 – The Mathematics of Diversification 8-5 Introduction to Risk and Return Risk, Return and Portfolio Theory Introduction to Risk and Return Risk and return are the two most important attributes of an investment. Research has shown that the two are linked in the capital Return markets and that generally, % higher returns can only be achieved by taking on greater risk. Risk Premium Risk isn‟t just the potential loss of return, it is the potential loss of RF the entire investment itself Real Return (loss of both principal and Expected Inflation Rate interest). Risk Consequently, taking on additional risk in search of higher returns is a decision that should not be taking lightly. CHAPTER 5 – The Mathematics of Diversification 8-7 Measuring Returns Risk, Return and Portfolio Theory Measuring Returns Introduction Ex Ante Returns • Return calculations may be done „before-the- fact,‟ in which case, assumptions must be made about the future Ex Post Returns • Return calculations done „after-the-fact,‟ in order to analyze what rate of return was earned. CHAPTER 5 – The Mathematics of Diversification 8-9 Measuring Returns Introduction You know that the constant growth DDM can be decomposed into the two forms of income that equity investors may receive, dividends and capital gains. D1 kc g Income / Dividend Yield Capital Gain (or loss) Yield P0 WHEREAS Fixed-income investors (bond investors for example) can expect to earn interest income as well as (depending on the movement of interest rates) either capital gains or capital losses. CHAPTER 5 – The Mathematics of Diversification 8 - 10 Measuring Returns Income Yield • Income yield is the return earned in the form of a periodic cash flow received by investors. • The income yield return is calculated by the periodic cash flow divided by the purchase price. CF1 [8-1] Income yield P0 Where CF1 = the expected cash flow to be received P0 = the purchase price CHAPTER 5 – The Mathematics of Diversification 8 - 11 Measuring Returns Common Share and Long Canada Bond Yield Gap – Table on this slide illustrates the income yield gap between stocks and bonds over recent decades. – The main reason that this yield gap has varied so much over time is that the return to investors is not just the income yield but also the capital gain (or loss) yield as well. Average Yield Gap between Stocks and Bonds Average Yield Gap (%) 1950s 0.82 1960s 2.35 1970s 4.54 1980s 8.14 1990s 5.51 2000s 3.55 Overall 4.58 CHAPTER 5 – The Mathematics of Diversification 8 - 12 Measuring Returns Dollar Returns Investors in market-traded securities (bonds or stock) receive investment returns in two different form: • Income yield • Capital gain (or loss) yield The investor will receive dollar returns, for example: • $1.00 of dividends • Share price rise of $2.00 To be useful, dollar returns must be converted to percentage returns as a function of the original investment. (Because a $3.00 return on a $30 investment might be good, but a $3.00 return on a $300 investment would be unsatisfactory!) CHAPTER 5 – The Mathematics of Diversification 8 - 13 Measuring Returns Converting Dollar Returns to Percentage Returns An investor receives the following dollar returns a stock investment of $25: • $1.00 of dividends • Share price rise of $2.00 The capital gain (or loss) return component of total return is calculated: ending price – minus beginning price, divided by beginning price P P0 $27 - $25 [8-2] Capital gain (loss) return 1 .08 8% P0 $25 CHAPTER 5 – The Mathematics of Diversification 8 - 14 Measuring Returns Total Percentage Return • The investor‟s total return (holding period return) is: Total return Income yield Capital gain (or loss) yield CF1 P P0 1 [8-3] P0 CF P P 1 1 0 P0 P0 $1.00 $27 $25 $25 0.04 0.08 0.12 12% $25 CHAPTER 5 – The Mathematics of Diversification 8 - 15 Measuring Returns Total Percentage Return – General Formula • The general formula for holding period return is: Total return Income yield Capital gain (or loss) yield CF1 P P0 1 [8-3] P0 CF1 P P0 1 P0 P0 CHAPTER 5 – The Mathematics of Diversification 8 - 16 Measuring Average Returns Ex Post Returns • Measurement of historical rates of return that have been earned on a security or a class of securities allows us to identify trends or tendencies that may be useful in predicting the future. • There are two different types of ex post mean or average returns used: – Arithmetic average – Geometric mean CHAPTER 5 – The Mathematics of Diversification 8 - 17 Measuring Average Returns Arithmetic Average n [8-4] r i Arithmetic Average (AM) i 1 n Where: ri = the individual returns n = the total number of observations • Most commonly used value in statistics • Sum of all returns divided by the total number of observations CHAPTER 5 – The Mathematics of Diversification 8 - 18 Measuring Average Returns Geometric Mean 1 [8-5] Geometric Mean (GM) [( 1 r1 )( 1 r2 )( 1 r3 )...(1 rn )] -1 n • Measures the average or compound growth rate over multiple periods. CHAPTER 5 – The Mathematics of Diversification 8 - 19 Measuring Average Returns Geometric Mean versus Arithmetic Average If all returns (values) are identical the geometric mean = arithmetic average. If the return values are volatile the geometric mean < arithmetic average The greater the volatility of returns, the greater the difference between geometric mean and arithmetic average. (Table on the following slide illustrates this principle on major asset classes 1938 – 2005) CHAPTER 5 – The Mathematics of Diversification 8 - 20 Measuring Average Returns Average Investment Returns and Standard Deviations Average Investment Returns and Standard Deviations, 1938-2005 Annual Annual Standard Deviation Arithmetic Geometric of Annual Returns Average (%) Mean (%) (%) Government of Canada treasury bills 5.20 5.11 4.32 Government of Canada bonds 6.62 6.24 9.32 Canadian stocks 11.79 10.60 16.22 U.S. stocks 13.15 11.76 17.54 So urce: Data are fro m the Canadian Institute o f A ctuaries The greater the difference, the greater the volatility of annual returns. CHAPTER 5 – The Mathematics of Diversification 8 - 21 Measuring Expected (Ex Ante) Returns • While past returns might be interesting, investor‟s are most concerned with future returns. • Sometimes, historical average returns will not be realized in the future. • Developing an independent estimate of ex ante returns usually involves use of forecasting discrete scenarios with outcomes and probabilities of occurrence. CHAPTER 5 – The Mathematics of Diversification 8 - 22 Estimating Expected Returns Estimating Ex Ante (Forecast) Returns • The general formula n [8-6] Expected Return (ER) (ri Probi ) i 1 Where: ER = the expected return on an investment Ri = the estimated return in scenario i Probi = the probability of state i occurring CHAPTER 5 – The Mathematics of Diversification 8 - 23 Estimating Expected Returns Estimating Ex Ante (Forecast) Returns Example: This is type of forecast data that are required to make an ex ante estimate of expected return. Possible Returns on Probability of Stock A in that State of the Economy Occurrence State Economic Expansion 25.0% 30% Normal Economy 50.0% 12% Recession 25.0% -25% CHAPTER 5 – The Mathematics of Diversification 8 - 24 Estimating Expected Returns Estimating Ex Ante (Forecast) Returns Using a Spreadsheet Approach Example Solution: Sum the products of the probabilities and possible returns in each state of the economy. (1) (2) (3) (4)=(2)×(1) Possible Weighted Returns on Possible Probability of Stock A in that Returns on State of the Economy Occurrence State the Stock Economic Expansion 25.0% 30% 7.50% Normal Economy 50.0% 12% 6.00% Recession 25.0% -25% -6.25% Expected Return on the Stock = 7.25% CHAPTER 5 – The Mathematics of Diversification 8 - 25 Estimating Expected Returns Estimating Ex Ante (Forecast) Returns Using a Formula Approach Example Solution: Sum the products of the probabilities and possible returns in each state of the economy. n Expected Return (ER) (ri Probi ) i 1 (r1 Prob1 ) (r2 Prob2 ) (r3 Prob3 ) (30% 0.25) (12% 0.5) (-25% 0.25) 7.25% CHAPTER 5 – The Mathematics of Diversification 8 - 26 Measuring Risk Risk, Return and Portfolio Theory Risk • Probability of incurring harm • For investors, risk is the probability of earning an inadequate return. – If investors require a 10% rate of return on a given investment, then any return less than 10% is considered harmful. CHAPTER 5 – The Mathematics of Diversification 8 - 28 Risk Illustrated The range of total possible returns on the stock A runs from -30% to Probability more than +40%. If the required return on the stock is 10%, then those outcomes less than 10% Outcomes that produce harm represent risk to the investor. A -30% -20% -10% 0% 10% 20% 30% 40% Possible Returns on the Stock CHAPTER 5 – The Mathematics of Diversification 8 - 29 Range • The difference between the maximum and minimum values is called the range – Canadian common stocks have had a range of annual returns of 74.36 % over the 1938-2005 period – Treasury bills had a range of 21.07% over the same period. • As a rough measure of risk, range tells us that common stock is more risky than treasury bills. CHAPTER 5 – The Mathematics of Diversification 8 - 30 Differences in Levels of Risk Illustrated Outcomes that produce harm The wider the range of probable outcomes the greater the risk of the Probability investment. B A is a much riskier investment than B A -30% -20% -10% 0% 10% 20% 30% 40% Possible Returns on the Stock CHAPTER 5 – The Mathematics of Diversification 8 - 31 Historical Returns on Different Asset Classes • Figure on the next slide illustrates the volatility in annual returns on three different assets classes from 1938 – 2005. • Note: – Treasury bills always yielded returns greater than 0% – Long Canadian bond returns have been less than 0% in some years (when prices fall because of rising interest rates), and the range of returns has been greater than T-bills but less than stocks – Common stock returns have experienced the greatest range of returns (See Figure on the following slide) CHAPTER 5 – The Mathematics of Diversification 8 - 32 Measuring Risk Annual Returns by Asset Class, 1938 - 2005 CHAPTER 5 – The Mathematics of Diversification 8 - 33 Refining the Measurement of Risk Standard Deviation (σ) • Range measures risk based on only two observations (minimum and maximum value) • Standard deviation uses all observations. – Standard deviation can be calculated on forecast or possible returns as well as historical or ex post returns. (The following two slides show the two different formula used for Standard Deviation) CHAPTER 5 – The Mathematics of Diversification 8 - 34 Measuring Risk Ex post Standard Deviation n _ (ri r ) 2 [8-7] Ex post i 1 n 1 Where : the standard deviation _ r the average return ri the return in year i n the number of observatio ns CHAPTER 5 – The Mathematics of Diversification 8 - 35 Measuring Risk Example Using the Ex post Standard Deviation Problem Estimate the standard deviation of the historical returns on investment A that were: 10%, 24%, -12%, 8% and 10%. Step 1 – Calculate the Historical Average Return n r i 10 24 - 12 8 10 40 Arithmetic Average (AM) i 1 8.0% n 5 5 Step 2 – Calculate the Standard Deviation n _ (r r ) i 2 (10 - 8) 2 (24 8) 2 (12 8) 2 (8 8) 2 (14 8) 2 Ex post i 1 n 1 5 1 2 2 16 2 20 2 0 2 2 2 4 256 400 0 4 664 166 12.88% 4 4 4 CHAPTER 5 – The Mathematics of Diversification 8 - 36 Ex Post Risk Stability of Risk Over Time Figure on the next slide demonstrates that the relative riskiness of equities and bonds has changed over time. Until the 1960s, the annual returns on common shares were about four times more variable than those on bonds. Over the past 20 years, they have only been twice as variable. Consequently, scenario-based estimates of risk (standard deviation) is required when seeking to measure risk in the future. (We cannot safely assume the future is going to be like the past!) Scenario-based estimates of risk is done through ex ante estimates and calculations. CHAPTER 5 – The Mathematics of Diversification 8 - 37 Relative Uncertainty Equities versus Bonds CHAPTER 5 – The Mathematics of Diversification 8 - 38 Measuring Risk Ex ante Standard Deviation A Scenario-Based Estimate of Risk n [8-8] Ex ante (Probi ) (ri ERi ) 2 i 1 CHAPTER 5 – The Mathematics of Diversification 8 - 39 Scenario-based Estimate of Risk Example Using the Ex ante Standard Deviation – Raw Data GIVEN INFORMATION INCLUDES: - Possible returns on the investment for different discrete states - Associated probabilities for those possible returns Possible State of the Returns on Economy Probability Security A Recession 25.0% -22.0% Normal 50.0% 14.0% Economic Boom 25.0% 35.0% CHAPTER 5 – The Mathematics of Diversification 8 - 40 Scenario-based Estimate of Risk Ex ante Standard Deviation – Spreadsheet Approach • The following two slides illustrate an approach to solving for standard deviation using a spreadsheet model. CHAPTER 5 – The Mathematics of Diversification 8 - 41 Scenario-based Estimate of Risk First Step – Calculate the Expected Return Determined by multiplying the probability times the possible return. Possible Weighted State of the Returns on Possible Economy Probability Security A Returns Recession 25.0% -22.0% -5.5% Normal 50.0% 14.0% 7.0% Economic Boom 25.0% 35.0% 8.8% Expected Return = 10.3% Expected return equals the sum of the weighted possible returns. CHAPTER 5 – The Mathematics of Diversification 8 - 42 Scenario-based Estimate of Risk Second Step – Measure the Weighted and Squared Deviations Now multiply the square deviations by First calculate the deviation of their probability of occurrence. possible returns from the expected. Deviation of Weighted Possible Weighted Possible and State of the Returns on Possible Return from Squared Squared Economy Probability Security A Returns Expected Deviations Deviations Recession 25.0% -22.0% -5.5% -32.3% 0.10401 0.02600 Normal 50.0% 14.0% 7.0% 3.8% 0.00141 0.00070 Economic Boom 25.0% 35.0% 8.8% 24.8% 0.06126 0.01531 Expected Return = 10.3% Variance = 0.0420 Standard Deviation = 20.50% Second, square those deviations The sum of the weighted and square deviations from deviation The standardthe mean. is the square root of the variance squared terms. is the variance in percent (in percent terms). CHAPTER 5 – The Mathematics of Diversification 8 - 43 Scenario-based Estimate of Risk Example Using the Ex ante Standard Deviation Formula Possible Weighted State of the Returns on Possible Economy Probability Security A Returns Recession 25.0% -22.0% -5.5% Normal 50.0% 14.0% 7.0% Economic Boom 25.0% 35.0% 8.8% Expected Return = 10.3% n Ex ante (Prob ) (r ER ) i 1 i i i 2 P (r1 ER1 ) 2 P2 (r2 ER2 ) 2 P (r3 ER3 ) 2 1 1 .25(22 10.3) 2 .5(14 10.3) 2 .25(35 10.3) 2 .25(32.3) 2 .5(3.8) 2 .25(24.8) 2 .25(.10401) .5(.00141) .25(.06126) .0420 .205 20.5% CHAPTER 5 – The Mathematics of Diversification 8 - 44 Modern Portfolio Theory Risk, Return and Portfolio Theory Portfolios • A portfolio is a collection of different securities such as stocks and bonds, that are combined and considered a single asset • The risk-return characteristics of the portfolio is demonstrably different than the characteristics of the assets that make up that portfolio, especially with regard to risk. • Combining different securities into portfolios is done to achieve diversification. CHAPTER 5 – The Mathematics of Diversification 8 - 46 Diversification Diversification has two faces: 1. Diversification results in an overall reduction in portfolio risk (return volatility over time) with little sacrifice in returns, and 2. Diversification helps to immunize the portfolio from potentially catastrophic events such as the outright failure of one of the constituent investments. (If only one investment is held, and the issuing firm goes bankrupt, the entire portfolio value and returns are lost. If a portfolio is made up of many different investments, the outright failure of one is more than likely to be offset by gains on others, helping to make the portfolio immune to such events.) CHAPTER 5 – The Mathematics of Diversification 8 - 47 Expected Return of a Portfolio Modern Portfolio Theory The Expected Return on a Portfolio is simply the weighted average of the returns of the individual assets that make up the portfolio: n [8-9] ER p ( wi ERi ) i 1 The portfolio weight of a particular security is the percentage of the portfolio‟s total value that is invested in that security. CHAPTER 5 – The Mathematics of Diversification 8 - 48 Expected Return of a Portfolio Example Portfolio value = $2,000 + $5,000 = $7,000 rA = 14%, rB = 6%, wA = weight of security A = $2,000 / $7,000 = 28.6% wB = weight of security B = $5,000 / $7,000 = (1-28.6%)= 71.4% n ER p ( wi ERi ) (.286 14%) (.714 6% ) i 1 4.004% 4.284% 8.288% CHAPTER 5 – The Mathematics of Diversification 8 - 49 Range of Returns in a Two Asset Portfolio In a two asset portfolio, simply by changing the weight of the constituent assets, different portfolio returns can be achieved. Because the expected return on the portfolio is a simple weighted average of the individual returns of the assets, you can achieve portfolio returns bounded by the highest and the lowest individual asset returns. CHAPTER 5 – The Mathematics of Diversification 8 - 50 Range of Returns in a Two Asset Portfolio Example 1: Assume ERA = 8% and ERB = 10% (See the following 6 slides based on Figure 8-4) CHAPTER 5 – The Mathematics of Diversification 8 - 51 Expected Portfolio Return Affect on Portfolio Return of Changing Relative Weights in A and B 10.50 10.00 ERB= 10% Expected Return % 9.50 9.00 8.50 8.00 ERA=8% 7.50 7.00 0 0.2 0.4 0.6 0.8 1.0 1.2 Portfolio Weight CHAPTER 5 – The Mathematics of Diversification 8 - 52 Expected Portfolio Return Affect on Portfolio Return of Changing Relative Weights in A and B A portfolio manager can select the relative weights of the two assets in the portfolio to get a desired return between 8% (100% invested in A) and 10% (100% invested in B) 10.50 10.00 ERB= 10% Expected Return % 9.50 9.00 8.50 8.00 ERA=8% 7.50 7.00 0 0.2 0.4 0.6 0.8 1.0 1.2 Portfolio Weight CHAPTER 5 – The Mathematics of Diversification 8 - 53 Expected Portfolio Return Affect on Portfolio Return of Changing Relative Weights in A and B 10.50 ERB= 10% 10.00 Expected Return % 9.50 The potential returns of the portfolio are bounded by the highest 9.00 and lowest returns of the individual assets 8.50 that make up the portfolio. 8.00 ERA=8% 7.50 7.00 0 0.2 0.4 0.6 0.8 1.0 1.2 Portfolio Weight CHAPTER 5 – The Mathematics of Diversification 8 - 54 Expected Portfolio Return Affect on Portfolio Return of Changing Relative Weights in A and B 10.50 ERB= 10% 10.00 Expected Return % 9.50 9.00 The expected return on the portfolio if 100% is 8.50 invested in Asset A is 8%. 8.00 ERp wA ERA wB ERB (1.0)(8%) (0)(10%) 8% ERA=8% 7.50 7.00 0 0.2 0.4 0.6 0.8 1.0 1.2 Portfolio Weight CHAPTER 5 – The Mathematics of Diversification 8 - 55 Expected Portfolio Return Affect on Portfolio Return of Changing Relative Weights in A and B 10.50 The expected return on the portfolio if 100% is invested in Asset B is ERB= 10% 10.00 10%. Expected Return % 9.50 9.00 8.50 ERp wA ERA wB ERB (0)(8%) (1.0)(10%) 10% 8.00 ERA=8% 7.50 7.00 0 0.2 0.4 0.6 0.8 1.0 1.2 Portfolio Weight CHAPTER 5 – The Mathematics of Diversification 8 - 56 Expected Portfolio Return Affect on Portfolio Return of Changing Relative Weights in A and B 10.50 The expected return on the portfolio if 50% is invested in Asset A and ERB= 10% 10.00 50% in B is 9%. Expected Return % 9.50 ER p wA ERA wB ERB 9.00 (0.5)(8%) (0.5)(10%) 8.50 4 % 5 % 9% 8.00 ERA=8% 7.50 7.00 0 0.2 0.4 0.6 0.8 1.0 1.2 Portfolio Weight CHAPTER 5 – The Mathematics of Diversification 8 - 57 Range of Returns in a Two Asset Portfolio Example 1: Assume ERA = 14% and ERB = 6% (See the following 2 slides ) CHAPTER 5 – The Mathematics of Diversification 8 - 58 Range of Returns in a Two Asset Portfolio E(r)A= 14%, E(r)B= 6% Expected return on Asset A = 14.0% Expected return on Asset B = 6.0% Expected Weight of Weight of Return on the Asset A Asset B Portfolio 0.0% 100.0% 6.0% 10.0% 90.0% 6.8% 20.0% 80.0% 7.6% 30.0% 70.0% 8.4% 40.0% 60.0% 9.2% A graph of this 50.0% 50.0% 10.0% relationship is 60.0% 40.0% 10.8% found on the 70.0% 30.0% 11.6% following slide. 80.0% 20.0% 12.4% 90.0% 10.0% 13.2% 100.0% 0.0% 14.0% CHAPTER 5 – The Mathematics of Diversification 8 - 59 Range of Returns in a Two Asset Portfolio E(r)A= 14%, E(r)B= 6% Range of Portfolio Returns Expected Return on Two 16.00% 14.00% Asset Portfolio 12.00% 10.00% 8.00% 6.00% 4.00% 2.00% 0.00% 0% % % % % % % % % % 0% .0 .0 .0 .0 .0 .0 .0 .0 .0 0. 0. 10 20 30 40 50 60 70 80 90 10 Weight Invested in Asset A CHAPTER 5 – The Mathematics of Diversification 8 - 60 Expected Portfolio Returns Example of a Three Asset Portfolio Relative Expected Weighted Weight Return Return Stock X 0.400 8.0% 0.03 Stock Y 0.350 15.0% 0.05 Stock Z 0.250 25.0% 0.06 Expected Portfolio Return = 14.70% CHAPTER 5 – The Mathematics of Diversification K. Hartviksen 8 - 61 Risk in Portfolios Risk, Return and Portfolio Theory Modern Portfolio Theory - MPT • Prior to the establishment of Modern Portfolio Theory (MPT), most people only focused upon investment returns…they ignored risk. • With MPT, investors had a tool that they could use to dramatically reduce the risk of the portfolio without a significant reduction in the expected return of the portfolio. CHAPTER 5 – The Mathematics of Diversification 8 - 63 Expected Return and Risk For Portfolios Standard Deviation of a Two-Asset Portfolio using Covariance [8-11] p ( wA ) 2 ( A ) 2 ( wB ) 2 ( B ) 2 2( wA )( wB )(COV A, B ) Risk of Asset A Risk of Asset B Factor to take into adjusted for weight adjusted for weight account comovement in the portfolio in the portfolio of returns. This factor can be negative. CHAPTER 5 – The Mathematics of Diversification 8 - 64 Expected Return and Risk For Portfolios Standard Deviation of a Two-Asset Portfolio using Correlation Coefficient [8-15] p ( wA ) 2 ( A ) 2 ( wB ) 2 ( B ) 2 2( wA )( wB )( A, B )( A )( B ) Factor that takes into account the degree of comovement of returns. It can have a negative value if correlation is negative. CHAPTER 5 – The Mathematics of Diversification 8 - 65 Grouping Individual Assets into Portfolios • The riskiness of a portfolio that is made of different risky assets is a function of three different factors: – the riskiness of the individual assets that make up the portfolio – the relative weights of the assets in the portfolio – the degree of comovement of returns of the assets making up the portfolio • The standard deviation of a two-asset portfolio may be measured using the Markowitz model: p w w 2 wA wB A, B A B 2 A 2 A 2 B 2 B CHAPTER 5 – The Mathematics of Diversification 8 - 66 Risk of a Three-Asset Portfolio The data requirements for a three-asset portfolio grows dramatically if we are using Markowitz Portfolio selection formulae. We need 3 (three) correlation coefficients between A and B; A and C; and B and C. A ρa,b ρa,c B C ρb,c p A wA B wB C wC 2wA wB A, B A B 2wB wC B ,C B C 2wA wC A,C A C 2 2 2 2 2 2 CHAPTER 5 – The Mathematics of Diversification 8 - 67 Risk of a Four-asset Portfolio The data requirements for a four-asset portfolio grows dramatically if we are using Markowitz Portfolio selection formulae. We need 6 correlation coefficients between A and B; A and C; A and D; B and C; C and D; and B and D. A ρa,b ρa,d ρa,c B D ρb,d ρb,c ρc,d C CHAPTER 5 – The Mathematics of Diversification 8 - 68 Covariance • A statistical measure of the correlation of the fluctuations of the annual rates of return of different investments. n _ _ [8-12] COVAB Probi ( x A,i xi )( xB ,i - xB ) i 1 CHAPTER 5 – The Mathematics of Diversification 8 - 69 Correlation • The degree to which the returns of two stocks co-move is measured by the correlation coefficient (ρ). • The correlation coefficient (ρ) between the returns on two securities will lie in the range of +1 through - 1. +1 is perfect positive correlation -1 is perfect negative correlation COV AB [8-13] AB A B CHAPTER 5 – The Mathematics of Diversification 8 - 70 Covariance and Correlation Coefficient • Solving for covariance given the correlation coefficient and standard deviation of the two assets: [8-14] COVAB AB A B CHAPTER 5 – The Mathematics of Diversification 8 - 71 Importance of Correlation • Correlation is important because it affects the degree to which diversification can be achieved using various assets. • Theoretically, if two assets returns are perfectly positively correlated, it is possible to build a riskless portfolio with a return that is greater than the risk-free rate. CHAPTER 5 – The Mathematics of Diversification 8 - 72 Affect of Perfectly Negatively Correlated Returns Elimination of Portfolio Risk Returns If returns of A and B are % 20% perfectly negatively correlated, a two-asset portfolio made up of equal parts of Stock A and B would be riskless. There would 15% be no variability of the portfolios returns over time. 10% Returns on Stock A Returns on Stock B 5% Returns on Portfolio Time 0 1 2 CHAPTER 5 – The Mathematics of Diversification 8 - 73 Example of Perfectly Positively Correlated Returns No Diversification of Portfolio Risk Returns If returns of A and B are % 20% perfectly positively correlated, a two-asset portfolio made up of equal parts of Stock A and B would be risky. There would be 15% no diversification (reduction of portfolio risk). 10% Returns on Stock A Returns on Stock B 5% Returns on Portfolio Time 0 1 2 CHAPTER 5 – The Mathematics of Diversification 8 - 74 Affect of Perfectly Negatively Correlated Returns Elimination of Portfolio Risk Returns If returns of A and B are % 20% perfectly negatively correlated, a two-asset portfolio made up of equal parts of Stock A and B would be riskless. There would 15% be no variability of the portfolios returns over time. 10% Returns on Stock A Returns on Stock B 5% Returns on Portfolio Time 0 1 2 CHAPTER 5 – The Mathematics of Diversification 8 - 75 Affect of Perfectly Negatively Correlated Returns Numerical Example Weight of Asset A = 50.0% Weight of Asset B = 50.0% n Expected ER p ( wi ERi ) (.5 5%) (.5 15% ) i 1 Return on Return on Return on the 2.5% 7.5% 10% Year Asset A Asset B Portfolio xx07 5.0% 15.0% 10.0% xx08 10.0% 10.0% 10.0% xx09 15.0% 5.0% 10.0% n ER p ( wi ERi ) (.5 15%) (.5 5% ) i 1 7.5% 2.5% 10% Perfectly Negatively Correlated Returns over time CHAPTER 5 – The Mathematics of Diversification 8 - 76 Diversification Potential • The potential of an asset to diversify a portfolio is dependent upon the degree of co-movement of returns of the asset with those other assets that make up the portfolio. • In a simple, two-asset case, if the returns of the two assets are perfectly negatively correlated it is possible (depending on the relative weighting) to eliminate all portfolio risk. • This is demonstrated through the following series of spreadsheets, and then summarized in graph format. CHAPTER 5 – The Mathematics of Diversification 8 - 77 Example of Portfolio Combinations and Correlation Perfect Expected Standard Correlation Positive Asset Return Deviation Coefficient Correlation – A 5.0% 15.0% 1 no B 14.0% 40.0% diversification Portfolio Components Portfolio Characteristics Expected Standard Both Weight of A Weight of B Return Deviation portfolio 100.00% 0.00% 5.00% 15.0% returns and 90.00% 10.00% 5.90% 17.5% risk are 80.00% 20.00% 6.80% 20.0% bounded by 70.00% 30.00% 7.70% 22.5% the range set 60.00% 40.00% 8.60% 25.0% by the 50.00% 50.00% 9.50% 27.5% constituent 40.00% 60.00% 10.40% 30.0% assets when 30.00% 70.00% 11.30% 32.5% ρ=+1 20.00% 80.00% 12.20% 35.0% 10.00% 90.00% 13.10% 37.5% 0.00% 100.00% 14.00% 40.0% CHAPTER 5 – The Mathematics of Diversification 8 - 78 Example of Portfolio Combinations and Correlation Positive Expected Standard Correlation Correlation – Asset Return Deviation Coefficient weak A 5.0% 15.0% 0.5 diversification B 14.0% 40.0% potential Portfolio Components Portfolio Characteristics Expected Standard When ρ=+0.5 Weight of A Weight of B Return Deviation these portfolio 100.00% 0.00% 5.00% 15.0% combinations 90.00% 10.00% 5.90% 15.9% have lower 80.00% 20.00% 6.80% 17.4% risk – 70.00% 30.00% 7.70% 19.5% expected 60.00% 40.00% 8.60% 21.9% portfolio return 50.00% 50.00% 9.50% 24.6% is unaffected. 40.00% 60.00% 10.40% 27.5% 30.00% 70.00% 11.30% 30.5% 20.00% 80.00% 12.20% 33.6% 10.00% 90.00% 13.10% 36.8% 0.00% 100.00% 14.00% 40.0% CHAPTER 5 – The Mathematics of Diversification 8 - 79 Example of Portfolio Combinations and Correlation No Expected Standard Correlation Correlation – Asset Return Deviation Coefficient some A 5.0% 15.0% 0 diversification B 14.0% 40.0% potential Portfolio Components Portfolio Characteristics Expected Standard Weight of A Weight of B Return Deviation Portfolio 100.00% 0.00% 5.00% 15.0% risk is 90.00% 10.00% 5.90% 14.1% lower than 80.00% 20.00% 6.80% 14.4% the risk of 70.00% 30.00% 7.70% 15.9% either 60.00% 40.00% 8.60% 18.4% asset A or 50.00% 50.00% 9.50% 21.4% B. 40.00% 60.00% 10.40% 24.7% 30.00% 70.00% 11.30% 28.4% 20.00% 80.00% 12.20% 32.1% 10.00% 90.00% 13.10% 36.0% 0.00% 100.00% 14.00% 40.0% CHAPTER 5 – The Mathematics of Diversification 8 - 80 Example of Portfolio Combinations and Correlation Negative Expected Standard Correlation Correlation – Asset Return Deviation Coefficient greater A 5.0% 15.0% -0.5 diversification B 14.0% 40.0% potential Portfolio Components Portfolio Characteristics Expected Standard Weight of A Weight of B Return Deviation Portfolio risk for more 100.00% 0.00% 5.00% 15.0% combinations 90.00% 10.00% 5.90% 12.0% is lower than 80.00% 20.00% 6.80% 10.6% the risk of 70.00% 30.00% 7.70% 11.3% either asset 60.00% 40.00% 8.60% 13.9% 50.00% 50.00% 9.50% 17.5% 40.00% 60.00% 10.40% 21.6% 30.00% 70.00% 11.30% 26.0% 20.00% 80.00% 12.20% 30.6% 10.00% 90.00% 13.10% 35.3% 0.00% 100.00% 14.00% 40.0% CHAPTER 5 – The Mathematics of Diversification 8 - 81 Example of Portfolio Combinations and Correlation Perfect Negative Expected Standard Correlation Correlation – Asset Return Deviation Coefficient greatest A 5.0% 15.0% -1 diversification B 14.0% 40.0% potential Portfolio Components Portfolio Characteristics Expected Standard Weight of A Weight of B Return Deviation 100.00% 0.00% 5.00% 15.0% 90.00% 10.00% 5.90% 9.5% Risk of the 80.00% 20.00% 6.80% 4.0% portfolio is almost 70.00% 30.00% 7.70% 1.5% eliminated at 60.00% 40.00% 8.60% 7.0% 70% invested in 50.00% 50.00% 9.50% 12.5% asset A 40.00% 60.00% 10.40% 18.0% 30.00% 70.00% 11.30% 23.5% 20.00% 80.00% 12.20% 29.0% 10.00% 90.00% 13.10% 34.5% 0.00% 100.00% 14.00% 40.0% CHAPTER 5 – The Mathematics of Diversification 8 - 82 Diversification of a Two Asset Portfolio Demonstrated Graphically The Effect of Correlation on Portfolio Risk: The Two-Asset Case Expected Return B AB = -0.5 12% AB = -1 8% AB = 0 AB= +1 A 4% 0% 0% 10% 20% 30% 40% Standard Deviation CHAPTER 5 – The Mathematics of Diversification 8 - 83 Impact of the Correlation Coefficient • Figure on the next slide illustrates the relationship between portfolio risk (σ) and the correlation coefficient – The slope is not linear a significant amount of diversification is possible with assets with no correlation (it is not necessary, nor is it possible to find, perfectly negatively correlated securities in the real world) – With perfect negative correlation, the variability of portfolio returns is reduced to nearly zero. CHAPTER 5 – The Mathematics of Diversification 8 - 84 Expected Portfolio Return Impact of the Correlation Coefficient 15 Standard Deviation (%) of Portfolio Returns 10 5 0 -1 -0.5 0 0.5 1 Correlation Coefficient (ρ) CHAPTER 5 – The Mathematics of Diversification 8 - 85 Zero Risk Portfolio • We can calculate the portfolio that removes all risk. • When ρ = -1, then [8-15] p ( wA ) 2 ( A ) 2 ( wB ) 2 ( B ) 2 2( wA )( wB )( A, B )( A )( B ) • Becomes: [8-16] p w A (1 w) B CHAPTER 5 – The Mathematics of Diversification 8 - 86 An Exercise to Produce the Efficient Frontier Using Three Assets Risk, Return and Portfolio Theory An Exercise using T-bills, Stocks and Bonds Base Data: Stocks T-bills Bonds Historical Expected Return(%) 12.73383 6.151702 7.0078723 averages for Standard Deviation (%) 0.168 0.042 0.102 returns and risk for Correlation Coefficient Matrix: three asset Stocks 1 -0.216 0.048 Each achievable classes T-bills -0.216 1 0.380 portfolio Bonds 0.048 0.380 1 combination is Historical Portfolio Combinations: on plotted correlation expected return, coefficients Weights Portfolio Expected Standard risk between the asset (σ) space, classes Combination Stocks T-bills Bonds Return Variance Deviation found on the 1 100.0% 0.0% 0.0% 12.7 0.0283 16.8% 2 90.0% 10.0% 0.0% 12.1 0.0226 15.0% following slide. 3 80.0% 20.0% 0.0% 11.4 0.0177 13.3% Portfolio 4 70.0% 30.0% 0.0% 10.8 0.0134 11.6% characteristics for 5 60.0% 40.0% 0.0% 10.1 0.0097 9.9% each combination 6 50.0% 50.0% 0.0% 9.4 0.0067 8.2% of securities 7 40.0% 60.0% 0.0% 8.8 0.0044 6.6% 8 30.0% 70.0% 0.0% 8.1 0.0028 5.3% 9 20.0% 80.0% 0.0% 7.5 0.0018 4.2% 10 10.0% 90.0% 0.0% 6.8 0.0014 3.8% CHAPTER 5 – The Mathematics of Diversification 8 - 88 Achievable Portfolios Results Using only Three Asset Classes Attainable Portfolio Combinations The efficient set is that set of and Efficient Set of Portfolio Combinations achievable portfolio combinations that offer the highest rate of return for a 14.0 Efficient Set given level of risk. The solid Portfolio Expected Return (%) 12.0 blue line indicates the efficient Minimum Variance Portfolio set. 10.0 8.0 The plotted points are 6.0 attainable portfolio 4.0 combinations. 2.0 0.0 0.0 5.0 10.0 15.0 20.0 Standard Deviation of the Portfolio (%) CHAPTER 5 – The Mathematics of Diversification 8 - 89 Achievable Two-Security Portfolios Modern Portfolio Theory This line represents 13 the set of 12 portfolio combinations Expected Return % 11 that are 10 achievable by 9 varying relative 8 weights and 7 using two non- 6 0 10 20 30 40 50 60 correlated Standard Deviation (%) securities. CHAPTER 5 – The Mathematics of Diversification 8 - 90 Dominance • It is assumed that investors are rational, wealth- maximizing and risk averse. • If so, then some investment choices dominate others. CHAPTER 5 – The Mathematics of Diversification 8 - 91 Investment Choices The Concept of Dominance Illustrated Return A dominates B % because it offers A B the same return 10% but for less risk. A dominates C C because it offers a 5% higher return but for the same risk. 5% 20% Risk To the risk-averse wealth maximizer, the choices are clear, A dominates B, A dominates C. CHAPTER 5 – The Mathematics of Diversification 8 - 92 Efficient Frontier The Two-Asset Portfolio Combinations A is not attainable B,E lie on the efficient frontier and are attainable A B E is the minimum Expected Return % variance portfolio C (lowest risk combination) C, D are E attainable but are D dominated by superior portfolios that line on the line Standard Deviation (%) above E CHAPTER 5 – The Mathematics of Diversification 8 - 93 Efficient Frontier The Two-Asset Portfolio Combinations Rational, risk averse investors will only want to A B hold Expected Return % portfolios C such as B. E The actual D choice will depend on her/his risk Standard Deviation (%) preferences. CHAPTER 5 – The Mathematics of Diversification 8 - 94 Diversification Risk, Return and Portfolio Theory Diversification • We have demonstrated that risk of a portfolio can be reduced by spreading the value of the portfolio across, two, three, four or more assets. • The key to efficient diversification is to choose assets whose returns are less than perfectly positively correlated. • Even with random or naïve diversification, risk of the portfolio can be reduced. – This is illustrated in the Figure and Table found on the following slides. • As the portfolio is divided across more and more securities, the risk of the portfolio falls rapidly at first, until a point is reached where, further division of the portfolio does not result in a reduction in risk. • Going beyond this point is known as superfluous diversification. CHAPTER 5 – The Mathematics of Diversification 8 - 96 Diversification Domestic Diversification Average Portfolio Risk January 1985 to December 1997 14 12 10 8 Standard Deviation (%) 6 4 2 0 0 50 100 150 200 250 300 Number of Stocks in Portfolio CHAPTER 5 – The Mathematics of Diversification 8 - 97 Diversification Domestic Diversification Monthly Canadian Stock Portfolio Returns, January 1985 to December 1997 Number of Average Standard Deviation Ratio of Portfolio Percentage of Stocks in Monthly of Average Standard Deviation to Total Achievable Portfolio Portfolio Monthly Portfolio Standard Deviation of a Risk Reduction Return (%) Return (%) Single Stock 1 1.51 13.47 1.00 0.00 2 1.51 10.99 0.82 27.50 3 1.52 9.91 0.74 39.56 4 1.53 9.30 0.69 46.37 5 1.52 8.67 0.64 53.31 6 1.52 8.30 0.62 57.50 7 1.51 7.95 0.59 61.35 8 1.52 7.71 0.57 64.02 9 1.52 7.52 0.56 66.17 10 1.51 7.33 0.54 68.30 14 1.51 6.80 0.50 74.19 40 1.52 5.62 0.42 87.24 50 1.52 5.41 0.40 89.64 100 1.51 4.86 0.36 95.70 200 1.51 4.51 0.34 99.58 222 1.51 4.48 0.33 100.00 So urce: Cleary, S. and Co pp D. "Diversificatio n with Canadian Sto cks: Ho w M uch is Eno ugh?" Canadian Investment Review (Fall 1999), Table 1. CHAPTER 5 – The Mathematics of Diversification 8 - 98 Total Risk of an Individual Asset Equals the Sum of Market and Unique Risk Average Portfolio Risk • This graph illustrates that total risk of a stock is made up of market risk (that Standard Deviation (%) cannot be diversified Diversifiable away because it is a (unique) risk function of the [8-19] economic „system‟) and unique, company- Nondiversifiable specific risk that is (systematic) risk eliminated from the portfolio through Number of Stocks in Portfolio diversification. [8-19] Total risk Market (systematic) risk Unique (non - systematic) risk CHAPTER 5 – The Mathematics of Diversification 8 - 99 International Diversification • Clearly, diversification adds value to a portfolio by reducing risk while not reducing the return on the portfolio significantly. • Most of the benefits of diversification can be achieved by investing in 40 – 50 different „positions‟ (investments) • However, if the investment universe is expanded to include investments beyond the domestic capital markets, additional risk reduction is possible. (See the following slide.) CHAPTER 5 – The Mathematics of Diversification 8 - 100 Diversification International Diversification 100 80 Percent risk 60 40 U.S. stocks 20 International stocks 11.7 0 0 10 20 30 40 50 60 Number of Stocks CHAPTER 5 – The Mathematics of Diversification 8 - 101 Achievable Portfolio Combinations The Capital Asset Pricing Model (CAPM) 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 100 different weighting schemes for just these two securities: CHAPTER 5 – The Mathematics of Diversification 8 - 103 Example of Portfolio Combinations and Correlation You repeat this procedure Expected Standard Correlation down until you Asset Return Deviation Coefficient have determine A 8.0% 8.7% -0.379 the portfolio B 10.0% 22.7% characteristics The first for second The all 100 Portfolio Components Portfolio Characteristics combination portfolios. portfolio Expected Standard simply assumes 99% Weight of A Weight of B Return Deviation Next plot the A and 1% inassumesin 100% 0% 8.00% 8.7% returns on a B. Notice the 99% 1% 8.02% 8.5% you invest increase the graph (see in 98% 2% 8.04% 8.4% solely in next slide) return and the 97% 3% 8.06% 8.2% Asset A decrease in 96% 4% 8.08% 8.1% portfolio risk! 95% 5% 8.10% 7.9% 94% 6% 8.12% 7.8% 93% 7% 8.14% 7.7% 92% 8% 8.16% 7.5% 91% 9% 8.18% 7.4% 90% 10% 8.20% 7.3% 89% 11% 8.22% 7.2% CHAPTER 5 – The Mathematics of Diversification 8 - 104 Example of Portfolio Combinations and Attainable Portfolio Combinations for a Two Asset Portfolio Correlation 12.00% Expected Return of the 10.00% 8.00% Portfolio 6.00% 4.00% 2.00% 0.00% 0.0% 5.0% 10.0% 15.0% 20.0% 25.0% Standard Deviation of Returns CHAPTER 5 – The Mathematics of Diversification 8 - 105 Two Asset Efficient Frontier • Figure on the next slide 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 on the following slide) CHAPTER 5 – The Mathematics of Diversification 8 - 106 Efficient Frontier The Two-Asset Portfolio Combinations A is not attainable B,E lie on the efficient frontier and are attainable A B E is the minimum Expected Return % variance portfolio C (lowest risk combination) C, D are E attainable but are D dominated by superior portfolios that line on the line Standard Deviation (%) above E CHAPTER 5 – The Mathematics of Diversification 8 - 107 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 (ERp) – Portfolio risk (σp) CHAPTER 5 – The Mathematics of Diversification 8 - 108 The Achievable Set of Portfolio Combinations • You could start by randomly assembling ten risky portfolios. • The results (in terms of ER p and σp )might look like the graph on the following page: CHAPTER 5 – The Mathematics of Diversification 8 - 109 Achievable Portfolio Combinations The First Ten Combinations Created ERp 10 Achievable Risky Portfolio Combinations Portfolio Risk (σp) CHAPTER 5 – The Mathematics of Diversification 8 - 110 The Achievable Set of Portfolio Combinations • You could continue randomly assembling more portfolios. • Thirty risky portfolios might look like the graph on the following slide: CHAPTER 5 – The Mathematics of Diversification 8 - 111 Achievable Portfolio Combinations Thirty Combinations Naively Created ERp 30 Risky Portfolio Combinations Portfolio Risk (σp) CHAPTER 5 – The Mathematics of Diversification 8 - 112 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: CHAPTER 5 – The Mathematics of Diversification 8 - 113 Achievable Portfolio Combinations More Possible Combinations Created The highlighted portfolios are ERp ‘efficient’ in that they offer the highest rate of E is the return for a given minimum level of risk. variance Rationale investors portfolio Achievable Set of will choose only Risky Portfolio from this efficient Combinations set. E Portfolio Risk (σp) CHAPTER 5 – The Mathematics of Diversification 8 - 114 The Efficient Frontier The Capital Asset Pricing Model (CAPM) Achievable Portfolio Combinations Efficient Frontier (Set) Efficient ERp frontier is the set of achievable portfolio combinations Achievable Set of that offer the Risky Portfolio Combinations highest rate of return for a given level of E risk. Portfolio Risk (σp) CHAPTER 5 – The Mathematics of Diversification 8 - 116 The New Efficient Frontier Efficient Portfolios Figure 9 – 1 illustrates Efficient Frontier three ER achievable portfolio combinations B that are A „efficient‟ (no other achievable MVP portfolio that offers the same risk, Risk offers a higher return.) CHAPTER 5 – The Mathematics of Diversification 8 - 117 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). CHAPTER 5 – The Mathematics of Diversification 8 - 118 The New Efficient Frontier and Separation Theorem The Capital Asset Pricing Model (CAPM) 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 of RF asset and a risky asset A letting the weight w invested in the risky asset and the weight invested in RF as (1 – w) CHAPTER 5 – The Mathematics of Diversification 8 - 120 The New Efficient Frontier Risk-Free Investing – Expected return on a two asset portfolio made up of risky asset A and RF: ER p RF w (ER A - RF) The possible combinations of A and RF are found graphed on the following slide. CHAPTER 5 – The Mathematics of Diversification 8 - 121 The New Efficient Frontier Attainable Portfolios Using RF and A This means you can 9 – 2 Equation Rearranging 9 ER achieve any illustrates w=σ -2 where portfolio can p / σA and what you combination see…portfolio substituting in [9-2] E(RpA - w A ) RF along the blue risk increases [9-3] ER P RF P Equation 1 we A A coloured line in direct get an simply by to proportion a equation for changing the the amount RF straight line relative weight invested in the with a risky and A of RFasset. in constant the two asset slope. portfolio. Risk CHAPTER 5 – The Mathematics of Diversification 8 - 122 The New Efficient Frontier Attainable Portfolios using the RF and A, and RF and T Which risky portfolio ER would a rational risk- T averse investor A choose in the presence of a RF RF investment? Portfolio A? Tangent Risk Portfolio T? CHAPTER 5 – The Mathematics of Diversification 8 - 123 The New Efficient Frontier Efficient Portfolios using the Tangent Portfolio T Clearly RF with T (the tangent portfolio) offers ER a series of portfolio combinations T that dominate A those produced by RF and A. Further, they RF dominate all but one portfolio on the efficient Risk frontier! CHAPTER 5 – The Mathematics of Diversification 8 - 124 The New Efficient Frontier Lending Portfolios Portfolios between RF and T are ER Lending Portfolios „lending‟ portfolios, because they T are achieved by A investing in the Tangent Portfolio and RF lending funds to the government (purchasing a T-bill, the RF). Risk CHAPTER 5 – The Mathematics of Diversification 8 - 125 The New Efficient Frontier Borrowing Portfolios The line can be extended to risk levels beyond ER Lending Portfolios Borrowing Portfolios „T‟ by borrowing at RF and investing it T in T. This is a A levered investment that increases both RF risk and expected return of the portfolio. Risk CHAPTER 5 – The Mathematics of Diversification 8 - 126 The New Efficient Frontier The New (Super) Efficient Frontier This is now called the with Clearly RFnew Capital Market Line (or super) T (the market The optimal efficient offers portfolio)frontier ER risky portfolio B2 of risky a series of (the market portfolios. portfolio ‘M’) portfolio T B combinations Investors can that dominate A2 achieve any those produced one of these by RF and A. A portfolio RF combinations Further, they by borrowing or dominate all but σρ investing in RF one portfolio on in combination the efficient frontier! market with the portfolio. CHAPTER 5 – The Mathematics of Diversification 8 - 127 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) CHAPTER 5 – The Mathematics of Diversification 8 - 128 The New Efficient Frontier The Capital Market Line The CML is that CML set of superior The optimal portfolio ER risky portfolio combinations (the market that are ‘M’) portfolio M achievable in the presence of the equilibrium condition. RF σρ CHAPTER 5 – The Mathematics of Diversification 8 - 129 The Capital Asset Pricing Model The Hypothesized Relationship between Risk and Return 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. CHAPTER 5 – The Mathematics of Diversification 8 - 131 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 stock‟s intrinsic (inherent economic worth) value. (As illustrated below) Estimate Investment’s Determine Investment’s Estimate the Compare to the actual Risk (Beta Coefficient) Required Return Investment’s Intrinsic stock price in the market Value COVi,M D1 i σM ki RF ( ERM RF ) i P0 Is the stock kc g 2 fairly priced? CHAPTER 5 – The Mathematics of Diversification 8 - 132 The Capital Asset Pricing Model Assumptions – CAPM is based on the following assumptions: 1. All investors have identical expectations about expected returns, standard deviations, and correlation coefficients for all securities. 2. All investors have the same one-period investment time horizon. 3. All investors can borrow or lend money at the risk-free rate of return (RF). 4. There are no transaction costs. 5. There are no personal income taxes so that investors are indifferent between capital gains an dividends. 6. 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. 7. Capital markets are in equilibrium. CHAPTER 5 – The Mathematics of Diversification 8 - 133 Market Portfolio and Capital Market Line • The assumptions have the following implications: 1. The “optimal” risky 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. 2. This optimal risky portfolio will be the market portfolio (M) which contains all risky securities. (Figure on the next slide illustrates the Market Portfolio ‘M’) CHAPTER 5 – The Mathematics of Diversification 8 - 134 The Capital Market Line ER CML The CML is that ERM RF of achievable setThe market k P RF P ERM M The CML the portfolio portfolio ishas M optimal risky standard combinations deviation of portfolio, it that are possible portfolio returns when investing contains all risky in as the only two securities and RF assets (the independent lies tangent (T) on variable. the portfolio market efficient frontier. and the risk-free σρ asset (RF). σM CHAPTER 5 – The Mathematics of Diversification 8 - 135 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. ER M - RF Slope of the CML M – This is called the market price for risk. Or – The equilibrium price of risk in the capital market. CHAPTER 5 – The Mathematics of Diversification 8 - 136 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 the market price for risk : ERM - RF E ( RP ) RF P σM – Where: • ERM = expected return on the market portfolio M • σM = the standard deviation of returns on the market portfolio • σP = the standard deviation of returns on the efficient portfolio being considered CHAPTER 5 – The Mathematics of Diversification 8 - 137 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. CHAPTER 5 – The Mathematics of Diversification 8 - 138 The Capital Asset Pricing Model Expected and Required Rates of Return A is an overvalued C B a portfolio that offers andExpected undervalued portfolio. expected Required return equal than is Expected portfolio.lessto the Return on C ER CML the required return. required return. return is greater than the required Expected A Selling pressure will return on A return. cause the price to Demand foryield to fall and the C Portfolio A will rise until expected Required increase driving up equals the required return on A B return. the price, and therefore the Expected Return on C expected return will RF fall until expected equals required (market equilibrium condition is achieved.) σρ CHAPTER 5 – The Mathematics of Diversification 8 - 139 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: ER P - RF Sharpe ratio P – Sharpe ratio is a measure of portfolio performance that describes how well an asset‟s returns compensate investors for the risk taken. – It‟s value is the premium earned over the RF divided by portfolio risk…so 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. CHAPTER 5 – The Mathematics of Diversification 8 - 140 The Capital Asset Pricing Model Sharpe Ratios and Income Trusts – Table (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. CHAPTER 5 – The Mathematics of Diversification 8 - 141 Income Trust Estimated Values Income Trusts Estimated Values Return σP Sharpe β Median income trusts 25.83% 18.66% 1.37 0.22 Equally weighted trust portfolio 29.97% 8.02% 3.44 0.28 S&P/TSX Composite Index 8.97% 13.31% 0.49 1.00 Scotia Capital government bond index 9.55% 6.57% 1.08 20.02 Source: Adapted from L. Kryzanowski, S. Lazrak, and I. Ratika, " The True Cost of Income Trusts," Canadian Investment Review 19, no. 5 (Spring 2006), Table 3, p. 15. CHAPTER 5 – The Mathematics of Diversification 8 - 142 CAPM and Market Risk The Capital Asset Pricing Model Diversifiable and Non-Diversifiable Risk • CML applies to efficient portfolios • Volatility (risk) of individual security returns are 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. CHAPTER 5 – The Mathematics of Diversification 8 - 144 The CAPM and Market Risk Portfolio Risk and Diversification Total Risk (σ) Market or systematic Unique (Non-systematic) Risk risk is risk that cannot be eliminated from the portfolio by investing the Market (Systematic) Risk portfolio into more and different securities. Number of Securities CHAPTER 5 – The Mathematics of Diversification 8 - 145 Relevant Risk Drawing a Conclusion from Figure • Figure demonstrates that an individual securities‟ volatility 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 „diversified‟ then in an efficient market, no-one would be willing to pay a „premium‟ for company-specific risk. • Relevant risk to diversified investors then is systematic risk. • Systematic risk is measured using the Beta Coefficient. CHAPTER 5 – The Mathematics of Diversification 8 - 146 Measuring Systematic Risk The Beta Coefficient The Capital Asset Pricing Model (CAPM) The Beta Coefficient What is the Beta Coefficient? • A measure of systematic (non-diversifiable) risk • As a „coefficient‟ the beta is a pure number and has no units of measure. CHAPTER 5 – The Mathematics of Diversification 8 - 148 The Beta Coefficient How Can We Estimate the Value of the Beta Coefficient? • There are two basic approaches to estimating the beta coefficient: 1. Using a formula (and subjective forecasts) 2. 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) CHAPTER 5 – The Mathematics of Diversification 8 - 149 The CAPM and Market Risk The Characteristic Line for Security A Security A Returns (%) 6 4 The plotted The slope of Market Returns (%) points are the the regression coincident line is beta. 2 rates of return The line the earned on of 0 best fit is investment -6 -4 -2 0 2 4 6 8 known in and the market -2 portfolio over finance as the past periods. characteristic line. -4 -6 CHAPTER 5 – The Mathematics of Diversification 8 - 150 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: COVi,M i , M i i σM 2 M CHAPTER 5 – The Mathematics of Diversification 8 - 151 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 the market returns: βs = 1.0 - the security has the same volatility as the market as a whole βs > 1.0 - aggressive investment with volatility of returns greater than the market βs < 1.0 - defensive investment with volatility of returns less than the market βs < 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 CHAPTER 5 – The Mathematics of Diversification 8 - 152 Canadian BETAS Selected Canadian BETAS Company Industry Classification Beta Abitibi Consolidated Inc. Materials - Paper & Forest 1.37 Algoma Steel Inc. Materials - Steel 1.92 Bank of Montreal Financials - Banks 0.50 Bank of Nova Scotia Financials - Banks 0.54 Barrick Gold Corp. Materials - Precious Metals & Minerals 0.74 BCE Inc. Communications - Telecommunications 0.39 Bema Gold Corp. Materials - Precious Metals & Minerals 0.26 CIBC Financials - Banks 0.66 Cogeco Cable Inc. Consumer Discretionary - Cable 0.67 Gammon Lake Resources Inc. Materials - Precious Metals & Minerals 2.52 Imperial Oil Ltd. Energy - Oil & Gas: Integrated Oils 0.80 Source: Research Insight, Compustat North American database, June 2006. CHAPTER 5 – The Mathematics of Diversification 8 - 153 Risk-Based Models and the Cost of Common Equity Estimating the Cost of Equity Using the CAPM Risk-Based Models and the Cost of Common Equity Using the CAPM to Estimate the Cost of Common Equity • CAPM can be used to estimate the required return by common shareholders. • It can be used in situations where DCF methods will perform poorly (growth firms) • CAPM estimate is a „market determined‟ estimate because: – The RF (risk-free) rate is the benchmark return and is measured directly, today as the yield on 91-day T-bills – The market premium for risk (MRP) is taken from current market estimates of the overall return in the market place less RF (ERM –RF) CHAPTER 5 – The Mathematics of Diversification 8 - 155 Risk-Based Models and the Cost of Common Equity Using the CAPM to Estimate the Cost of Common Equity • As a single-factor model, we estimate the common shareholder‟s required return based on an estimate of the systematic risk of the firm (measured by the firm‟s beta coefficient) K e RF MRP e • Where: Ke = investor‟s required rate of return βe = the stock‟s beta coefficient Rf = the risk-free rate of return MRP = the market risk premium (ERM - Rf ) CHAPTER 5 – The Mathematics of Diversification 8 - 156 Risk-Based Models and the Cost of Common Equity Estimating the Market Risk Premium K e RF MRP e • Rf is „observable‟ (yield on 91-day T-bills) • Getting an estimate of the market risk premium is one of the more difficult challenges in using this model. – We really need a „forward‟ looking of MRP or a „forward‟ looking estimate of the ERM • One approach is to use an estimate of the current, expected MRP by examining a long-run average that prevailed in the past. • Table illustrates the % returns on S&P/TSX Composite annually for the first five years of this century. CHAPTER 5 – The Mathematics of Diversification 8 - 157 Risk-Based Models and the Cost of Common Equity Using the CAPM to Estimate the Cost of Common Equity Returns on the S&P/TSX Composite Index Investors are It would be unlikely to expect Returns better to use negative returns average on the stock 2000 market. If they 7.5072% realized did, no one would 2001 -12.572% returns over hold shares! 2002 -12.438% an entire 2003 Who would have business/mar 26.725% guessed before 2004 14.480% ket cycle. hand, there would 2005 be two 24.127% consecutive years of aggregate market losses? Such is the reality of investing since none of us are clairvoyant. CHAPTER 5 – The Mathematics of Diversification 8 - 158 Risk-Based Models and the Cost of Common Equity Using the CAPM to Estimate the Cost of Common Equity Long-run average rates of return are more reliable. Average Investment Returns and Standard Deviations (1938 to 2005) Annual Annual Standard Arithmetic Geometric Deviation of Average (%) Mean (%) Annual Returns (%) Government of Canada Treasury Bills 5.20 5.11 4.32 Government of Canada Bonds 6.62 6.24 9.32 Canadian Stocks 11.79 10.60 16.22 U.S. Stocks 13.15 11.76 17.54 Source: Data from Canadian Institute of Actuaries The consensus Canadian Average risk premium ofis that the Canadian MRP over the long-term stocks over bonds was 5.17% bond yield (an observable yield) is between 4.0 and 5.5%. CHAPTER 5 – The Mathematics of Diversification 8 - 159 Risk-Based Models and the Cost of Common Equity Using the CAPM to Estimate the Cost of Common Equity Long-Run Financial Projections Financial Forecasts Average Annual Percent Return Bank of Canada Overnight Rate 4.50 Cash: 3-Month T-bills 4.40 Income: Scotia Universe Bond Index 5.60 Canadian Equities: S&P/TSX Composite Index 7.30 U.S. Equities: S&P 500 Index 7.80 International Non-U.S. Equities: MSCI EAFE Index 7.50 Source: TD Economics The Scotia Universeis very Index is a long-term bond index that An estimate of ERM Bond important. contains Canada‟s and corporate bonds with default risk. TD Economicson a risk-adjusted basis,above estimates for„forward‟ Nevertheless, recently generated the the TD forecast of MRP is looking rates. an arithmetic risk premium of 4.3% consistent with CHAPTER 5 – The Mathematics of Diversification 8 - 160 Risk-Based Models and the Cost of Common Equity Estimating Betas • After obtaining estimates of the two important market rates (Rf and MRP), an estimate for the company beta is required. • Figure on the following slide illustrates that estimated betas for major sub-indexes of the S&P/TSX have varied widely over time: CHAPTER 5 – The Mathematics of Diversification 8 - 161 Risk-Based Models and the Cost of Common Equity Estimated Betas for Sub Indexes of the S&P/TSX Composite Index CHAPTER 5 – The Mathematics of Diversification 8 - 162 Risk-Based Models and the Cost of Common Equity Estimated Betas for Sub Indexes of the S&P/TSX Composite Index • Actual data for the preceding Figure is presented in Table on the following slide: • You should note: – IT sub index shows rapidly increasing betas – Other sub index betas show constant or decreasing trends. • Reasons: – The weighted average of all betas = 1.0 (by definition they are the market) – If one sub index is changing…that change alone affects all others in the opposite direction. • What Happened in the 1995 – 2005 decade? – The internet bubble of the late 1990s resulted in rapid growth in the IT sector till it burst in the early 2000s. CHAPTER 5 – The Mathematics of Diversification 8 - 163 Risk-Based Models and the Cost of Common Equity Estimating Betas IT Bubble Table 20-15 S&P/TSX Sub Index Beta Estimates Energy Materials Industrials ConsDisc ConsStap Health Fin IT Telco Utilities 1995 0.93 1.41 1.19 0.82 0.68 0.36 0.92 1.25 0.53 0.67 1996 0.93 1.28 1.10 0.83 0.66 0.39 1.02 1.36 0.61 0.65 1997 0.98 1.33 0.97 0.82 0.62 0.60 0.93 1.56 0.62 0.53 1998 0.85 1.12 0.94 0.80 0.60 1.02 1.11 1.40 0.92 0.55 1999 0.91 1.04 0.78 0.73 0.43 1.00 1.00 1.55 1.11 0.30 2000 0.67 0.74 0.73 0.69 0.23 1.10 0.79 1.78 0.92 0.14 2001 0.50 0.60 0.82 0.68 0.10 0.98 0.67 2.12 0.94 -0.03 2002 0.43 0.57 0.86 0.73 0.11 0.99 0.67 2.27 0.92 -0.06 2003 0.27 0.42 0.91 0.74 -0.04 0.85 0.39 2.75 0.82 -0.26 2004 0.17 0.42 1.04 0.81 -0.02 0.84 0.41 2.89 0.55 -0.14 2005 0.48 0.78 1.12 0.84 0.14 0.74 0.58 2.71 0.71 -0.01 Source: Data from Financial Post Corporate Analyzer Database CHAPTER 5 – The Mathematics of Diversification 8 - 164 Risk-Based Models and the Cost of Common Equity Nortel Stock Price • Nortel‟s stock price reflects the IT bubble and crash. (See Figure on the following slide for Nortel Stock Price history) CHAPTER 5 – The Mathematics of Diversification 8 - 165 Risk-Based Models and the Cost of Common Equity Nortel Stock Price CHAPTER 5 – The Mathematics of Diversification 8 - 166 Risk-Based Models and the Cost of Common Equity IT Bubble effect on Betas of Other Companies Outside the Sector • The bubble in IT stocks has driven down the betas in other sectors. • This is demonstrated in Rothman‟s beta over the 1966 – 2004 period. • Remember, Rothman‟s is a stable company and it‟s beta should be expected to remain constant. (See Figure on the following slide for Rothman‟s beta history) CHAPTER 5 – The Mathematics of Diversification 8 - 167 Risk-Based Models and the Cost of Common Equity Rothman‟s Beta Estimates CHAPTER 5 – The Mathematics of Diversification 8 - 168 Risk-Based Models and the Cost of Common Equity Adjusting Beta Estimates and Establishing a Range • When betas are measured over the period of a sector bubble or crash, it is necessary to adjust the beta estimates of firms in other sectors. • Take the industry grouping as a major input, plus the individual company beta estimate. – Using current MRP and Rf Develop estimates of Ke using the range of Company betas prior to the bubble or crash CHAPTER 5 – The Mathematics of Diversification 8 - 169 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. [9-8] P wA A wB B ... wn n Weights of individual assets are found by dividing the value of the investment by the value of the total portfolio. CHAPTER 5 – The Mathematics of Diversification 8 - 170 The Security Market Line The Capital Asset Pricing Model (CAPM) 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: [9-9] ki RF ( ERM RF ) i – Where: ki = the required return on security ‘i’ ERM – RF = market premium for risk Βi = the beta coefficient for security ‘i’ (See Figure 9 - 9 on the following slide for the graphical representation) CHAPTER 5 – The Mathematics of Diversification 8 - 172 The CAPM and Market Risk The Security Market Line (SML) ER ki RF ( ERM RF ) i M The SML The SML is ERM uses the used to beta predict coefficient as required the measure returns for of relevant individual RF risk. securities βM = 1 β CHAPTER 5 – The Mathematics of Diversification 8 - 173 The CAPM and Market Risk The SML and Security Valuation Required B is an Similarly, A is an returns ER ki RF ( ERM RF ) i overvalued are forecast using undervalued security. security because this equation. SML its expected Investor‟s will sell You can see that return is greater to lock in gains, the required than the required Expected A but the selling return on any Return A return. security is a pressure will Required Return A B Investors will function ofmarket cause the its RF „flock‟to fall,risk (β) systematicand bid price to A causing the and market up the price causing expected factors (RF and to expected return return premium marketto it equals rise until fall till it βA βB β equals the required for risk)the required return. return. CHAPTER 5 – The Mathematics of Diversification 8 - 174 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. CHAPTER 5 – The Mathematics of Diversification 8 - 175 Alternative Pricing Models The Capital Asset Pricing Model (CAPM) Challenges to CAPM • Empirical tests suggest: – CAPM does not hold well in practice: • Ex post SML is an upward sloping line • Ex ante y (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) CHAPTER 5 – The Mathematics of Diversification 8 - 177 Alternative Asset Pricing Models The Fama – French Model – A pricing model that uses three factors to relate expected returns to risk including: 1. A market factor related to firm size. 2. The market value of a firm‟s common equity (MVE) 3. Ratio of a firm‟s 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. CHAPTER 5 – The Mathematics of Diversification 8 - 178 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. [9-10] ERi a0 bi1 F1 bi1 F1 ... bin Fn – 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. CHAPTER 5 – The Mathematics of Diversification 8 - 179 Alternative Asset Pricing Models The Arbitrage Pricing Theory – the Model – Underlying factors represent broad economic forces which are inherently unpredictable. ERi a0 bi1 F1 bi1 F1 ... bin Fn – Where: • ERi = the expected return on security i • a0 = the expected return on a security with zero systematic risk • bi = the sensitivity of security i to a given risk factor • Fi = the risk premium for a given risk factor – The model demonstrates that a security‟s risk is based on its sensitivity to broad economic forces. CHAPTER 5 – The Mathematics of Diversification 8 - 180 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: 1. Changes in expected inflation 2. Unanticipated changes in inflation 3. Unanticipated changes in industrial production 4. Unanticipated changes in the default-risk premium 5. Unanticipated changes in the term structure of interest rates • Clearly, something that isn‟t forecast, can‟t be used to price securities today…they can only be used to explain prices after the fact. CHAPTER 5 – The Mathematics of Diversification 8 - 181 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. CHAPTER 5 – The Mathematics of Diversification 8 - 182 Estimating the Ex Ante (Forecast) Beta APPENDIX 1 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. CHAPTER 5 – The Mathematics of Diversification 8 - 184 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 CHAPTER 5 – The Mathematics of Diversification 8 - 185 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 Total Risk of the Investment Systematic Risk Unsystematic Risk • Systematic risk is non-diversifiable risk. • Systematic risk is the only relevant risk to the diversified investor • The beta coefficient measures systematic risk CHAPTER 5 – The Mathematics of Diversification 8 - 186 The Beta Coefficient The Formula Covariance of Returns between stock ' i' returns and the market Beta Variance of the Market Returns COVi,M i , M i [9-7] i σM 2 M CHAPTER 5 – The Mathematics of Diversification 8 - 187 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 CHAPTER 5 – The Mathematics of Diversification 8 - 188 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 virtually no systematic risk. – Such stocks are not influenced by the health of the economy in general…their financial results are predominantly influenced by company-specific factors. – An example is cigarette companies…people consume cigarettes because they are addicted…so it doesn‟t matter whether the economy is healthy or not…they 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. CHAPTER 5 – The Mathematics of Diversification 8 - 189 The Formula Approach to Measuring the Beta Cov(k i k M ) Beta Var(k M ) You need to calculate the covariance of the returns between the stock and the market…as 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 CHAPTER 5 – The Mathematics of Diversification 8 - 190 Ex ante Return Data A Sample A set of estimates of possible returns and their respective probabilities looks as follows: Possible Since the beta Future State Possible Possible relates the stock By observation returns to the of the Returns on Returns on market returns, you can see the Economy Probability the Stock the Market the greater range range is much of stock returns Boom 25.0% 28.0% 20.0% the greater for the changing in stockdirection as same than the Normal 50.0% 17.0% 11.0% market and they the market Recession 25.0% -14.0% -4.0% indicates the beta move in the will be direction. same greater than 1 and will be positive. (Positively correlated to the market returns.) CHAPTER 5 – The Mathematics of Diversification 8 - 191 The Total of the Probabilities must Equal 100% This means that we have considered all of the possible outcomes in this discrete probability distribution Possible Future State Possible Possible of the Returns on Returns on Economy Probability the Stock the Market Boom 25.0% 28.0% 20.0% Normal 50.0% 17.0% 11.0% Recession 25.0% -14.0% -4.0% 100.0% CHAPTER 5 – The Mathematics of Diversification 8 - 192 Measuring Expected Return on the Stock From Ex Ante Return Data The expected return is weighted average returns from the given ex ante data (1) (2) (3) (4) Possible Future State Possible of the Returns on Economy Probability the Stock (4) = (2)*(3) Boom 25.0% 28.0% 0.07 Normal 50.0% 17.0% 0.085 Recession 25.0% -14.0% -0.035 Expected return on the Stock = 12.0% CHAPTER 5 – The Mathematics of Diversification 8 - 193 Measuring Expected Return on the Market From Ex Ante Return Data The expected return is weighted average returns from the given ex ante data (1) (2) (3) (4) Possible Future State Possible of the Returns on Economy Probability the Market (4) = (2)*(3) Boom 25.0% 20.0% 0.05 Normal 50.0% 11.0% 0.055 Recession 25.0% -4.0% -0.01 Expected return on the Market = 9.5% CHAPTER 5 – The Mathematics of Diversification 8 - 194 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! (1) (2) (3) (4) (5) (6) (7) Possible Weighted Future State Possible and of the Returns on Squared Squared Economy Probability the Stock (4) = (2)*(3) Deviations Deviations Deviations Boom 25.0% 0.28 0.07 0.16 0.0256 0.0064 Normal 50.0% 0.17 0.085 0.05 0.0025 0.00125 Recession 25.0% -0.14 -0.035 -0.26 0.0676 0.0169 Expected return (stock) = 12.0% Variance (stock)= 0.02455 Standard Deviation (stock) = 15.67% CHAPTER 5 – The Mathematics of Diversification 8 - 195 Measuring Variances, Standard Deviations of the Forecast Market Returns Now do this for the possible returns on the market (1) (2) (3) (4) (5) (6) (7) Possible Weighted Future State Possible and of the Returns on Squared Squared Economy Probability the Market (4) = (2)*(3) Deviations Deviations Deviations Boom 25.0% 0.2 0.05 0.105 0.011025 0.002756 Normal 50.0% 0.11 0.055 0.015 0.000225 0.000113 Recession 25.0% -0.04 -0.01 -0.135 0.018225 0.004556 Expected return (market) = 9.5% Variance (market) = 0.007425 Standard Deviation (market)= 8.62% CHAPTER 5 – The Mathematics of Diversification 8 - 196 Covariance From Chapter 8 you know the formula for the covariance between the returns on the stock and the returns on the market is: n _ _ [8-12] COVAB Probi (k A,i ki )(k B ,i - k B ) i 1 Covariance is an absolute measure of the degree of „co- movement‟ of returns. CHAPTER 5 – The Mathematics of Diversification 8 - 197 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: COV AB AB A B [8-13] 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) CHAPTER 5 – The Mathematics of Diversification 8 - 198 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 occurrence…then add the products up. (1) (2) (3) (4) (5) (6) (7) (8) "(9) Possible Possible Deviations Deviations Future Returns Possible from the from the State of the on the (4) = Returns on mean for mean for Economy Prob. Stock (2)*(3) the Market (6)=(2)*(5) the stock the market (8)=(2)(6)(7) Boom 25.0% 28.0% 0.07 20.0% 0.05 16.0% 10.5% 0.0042 Normal 50.0% 17.0% 0.085 11.0% 0.055 5.0% 1.5% 0.000375 Recession 25.0% -14.0% -0.035 -4.0% -0.01 -26.0% -13.5% 0.008775 E(kstock) = 12.0% E(kmarket ) = 9.5% Covariance = 0.01335 CHAPTER 5 – The Mathematics of Diversification 8 - 199 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: Cov S, M .01335 Beta 1.8 VarM .007425 • A beta that is greater than 1 means that the investment is aggressive…its 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%. CHAPTER 5 – The Mathematics of Diversification 8 - 200 Lets Prove the Beta of the Market is 1.0 Let us assume we are comparing the possible market returns against itself…what will the beta be? (1) (2) (3) (4) (5) (6) (6) (7) (8) Possible Possible Possible Deviations Deviations Future Returns Cov Returns .007425 from the from the State of the Beta (4) = on the `M,M on the 1.0 mean for mean for (8)=(2)(6)(7 Economy Prob. Market Var (2)*(3) .007425 Market (6)=(2)*(5) M the stock the market ) Boom 25.0% 20.0% 0.05 20.0% 0.05 10.5% 10.5% 0.002756 Normal 50.0% 11.0% 0.055 11.0% 0.055 1.5% 1.5% 0.000113 Recession 25.0% -4.0% -0.01 -4.0% -0.01 -13.5% -13.5% 0.004556 E(kM) = 9.5% E(kM) = 9.5% Covariance = 0.007425 Since the variance of the returns on the market is = .007425 …the beta for the market is indeed equal to 1.0 !!! CHAPTER 5 – The Mathematics of Diversification 8 - 201 Proving the Beta of Market = 1 If you now place the covariance of the market with itself value in the beta formula you get: Cov MM .007425 Beta 1.0 Var(R M ) .007425 The beta coefficient of the market will always be 1.0 because you are measuring the market returns against market returns. CHAPTER 5 – The Mathematics of Diversification 8 - 202 Using the Security Market Line Expected versus Required Return 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. % Return E(Rs) = 5.0% R(ks) = 4.76% SML E(kM)= 4.2% Risk-free Rate = 3% BM= 1.0 Bs = 1.464 Since E(r)>R(r) the stock is underpriced. CHAPTER 5 – The Mathematics of Diversification 8 - 204 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 E(Rs) = R(Rs) 4.76% SML E(RM)= 4.2% Risk-free Rate = 3% BM= 1.0 BS = 1.464 CHAPTER 5 – The Mathematics of Diversification 8 - 205 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 investor‟s required return on the stock using the CAPM: Required Return RF βi [E (k M ) RF] • This is a „market-determined‟ return 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 (kM – RF) CHAPTER 5 – The Mathematics of Diversification 8 - 206 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 stock‟s 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. CHAPTER 5 – The Mathematics of Diversification 8 - 207 Calculating the Beta using Trailing Holding Period Returns APPENDIX 2 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 past…and this approach to beta estimation SHOULD NOT be used….rather, use the ex ante approach. CHAPTER 5 – The Mathematics of Diversification 8 - 209 Historical Beta Estimation The Approach Used to Create the Characteristic Line In this example, we have regressed the quarterly returns on the stock against the quarterly returns of a surrogate for the market (TSE 300 total return composite index) and then using Excel…used the charting feature to plot the historical points and add a regression trend line. The ‘cloud’ of plotted points represents ‘diversifiable or company Period HPR(Stock) HPR(TSE 300) Ch a r a c te r istic L in e (Re gr e ssio n ) in the specific’ risk-4.0% securities returns 2006.4 1.2% 30.0% a portfolio that can be eliminated from-7.0% 2006.3 -16.0% 25.0% 2006.2through diversification. Since 32.0% 12.0% 20.0% company-specific risk can be Returns on Stock 2006.1 16.0% 8.0% eliminated, investors don’t require 15.0% 2005.4 -22.0% -11.0% compensation for it according to 2005.3 15.0% 16.0% 10.0% 2005.2Markowitz Portfolio Theory. 28.0% 13.0% 5.0% 2005.1 19.0% 7.0% 0.0% 2004.4 -16.0% -4.0% -40.0% -20.0% -5.0%0.0% 20.0% 40.0% The regression line is a line of ‘best 2004.3 8.0% 16.0% fit’ that describes the inherent -10.0% relationship-3.0% 2004.2 between the-11.0% on returns -15.0% 25.0% 2004.1 stock 34.0% returns on the the and the Returns on TSE 300 market. The slope is the beta coefficient. CHAPTER 5 – The Mathematics of Diversification 8 - 210 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.) • The slope of the Characteristic Line is 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 the R2 (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. • The alpha is the vertical intercept of the regression (characteristic line). Many stock analysts search out stocks with high alphas. CHAPTER 5 – The Mathematics of Diversification 8 - 211 Low R2 • An R2 that 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 stock…do not use the estimated beta coefficient. (See the following slide for an illustration of a low r-square) CHAPTER 5 – The Mathematics of Diversification 8 - 212 Characteristic Line for Imperial Tobacco An Example of Volatility that is Primarily Company-Specific Characteristic Returns on Line for Imperial Imperial Tobacco Tobacco % • High alpha • R-square is very low ≈ 0.02 • Beta is largely irrelevant Returns on the Market % (S&P TSX) CHAPTER 5 – The Mathematics of Diversification 8 - 213 High R2 • An R2 that 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 do…then you can predict the returns on the stock itself with a great deal of accuracy. CHAPTER 5 – The Mathematics of Diversification 8 - 214 Characteristic Line General Motors A Positive Beta with Predictive Power Characteristic Returns on Line for GM General Motors % (high R2) • Positive alpha • R-square is very high ≈ 0.9 • Beta is positive and close to 1.0 Returns on the Market % (S&P TSX) CHAPTER 5 – The Mathematics of Diversification 8 - 215 An Unusual Characteristic Line A Negative Beta with Predictive Power Returns on a Characteristic Line for a stock Stock % that will provide excellent portfolio diversification • Positive alpha (high R2) • R-square is very high • Beta is negative <0.0 and > -1.0 Returns on the Market % (S&P TSX) CHAPTER 5 – The Mathematics of Diversification 8 - 216 Diversifiable Risk (Non-systematic Risk) • Volatility in a security‟s returns caused by company- specific factors (both positive and negative) such as: – a single company strike – a spectacular innovation discovered through the company‟s 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. CHAPTER 5 – The Mathematics of Diversification 8 - 217 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) CHAPTER 5 – The Mathematics of Diversification 8 - 218 Demonstration Through Example The following slides will be based on Alcan Aluminum (AL.TO) Five Year Stock Price Chart for AL.TO CHAPTER 5 – The Mathematics of Diversification 8 - 220 Spreadsheet Data From Yahoo Process: – Go to http://ca.finance.yahoo.com – Use the symbol lookup function to search for the company you are interested in studying. – Use the historical quotes button…and get 30 months of historical data. – Use the download in spreadsheet format feature to save the data to your hard drive. CHAPTER 5 – The Mathematics of Diversification 8 - 221 Spreadsheet Data From Yahoo Alcan Example The raw downloaded data should look like this: Date Open High Low Close Volume 01-May-02 57.46 62.39 56.61 59.22 753874 01-Apr-02 62.9 63.61 56.25 57.9 879210 01-Mar-02 64.9 66.81 61.68 63.03 974368 01-Feb-02 61.65 65.67 58.75 64.86 836373 02-Jan-02 57.15 62.37 54.93 61.85 989030 03-Dec-01 56.6 60.49 55.2 57.15 833280 01-Nov-01 49 58.02 47.08 56.69 779509 CHAPTER 5 – The Mathematics of Diversification 8 - 222 Spreadsheet Data From Yahoo Alcan Example The raw downloaded data should look like this: Date Open High Low Close Volume 01-May-02 57.46 62.39 56.61 59.22 753874 01-Apr-02 62.9 63.61 56.25 57.9 879210 Volume of Opening price per share, the trading done The day, highest price per share during the in the stock month and month, the lowest price per share on the TSE in year achieved during the month and the the month in closing price per share at the end numbers of of the – The Mathematics of Diversification CHAPTER 5 month lots board 8 - 223 Spreadsheet Data From Yahoo Alcan Example From Yahoo, the only information you can use is the closing price per share and the date. Just delete the other columns. Date Close 01-May-02 59.22 01-Apr-02 57.9 01-Mar-02 63.03 01-Feb-02 64.86 02-Jan-02 61.85 CHAPTER 5 – The Mathematics of Diversification 8 - 224 Acquiring the Additional Information You Need Alcan 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 in The TSX Review periodical. CHAPTER 5 – The Mathematics of Diversification 8 - 225 Raw Company Data Alcan Example Closing Price Cash Issued for Alcan Dividends Date Capital AL.TO per Share 01-May-02 321,400,589 $59.22 $0.00 01-Apr-02 321,400,589 $57.90 $0.15 01-Mar-02 321,400,589 $63.03 $0.00 01-Feb-02 321,400,589 $64.86 $0.00 02-Jan-02 160,700,295 $123.70 $0.30 01-Dec-01 160,700,295 $119.30 $0.00 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. CHAPTER 5 – The Mathematics of Diversification 8 - 226 Normalizing the Raw Company Data Alcan Example Closing Price for Cash Issued Alcan Dividends Adjustment Normalized Normalized Date Capital AL.TO per Share Factor Stock Price Dividend 01-May-02 321,400,589 $59.22 $0.00 1.00 $59.22 $0.00 01-Apr-02 321,400,589 $57.90 $0.15 1.00 $57.90 $0.15 01-Mar-02 321,400,589 $63.03 $0.00 1.00 $63.03 $0.00 01-Feb-02 321,400,589 $64.86 $0.00 1.00 $64.86 $0.00 02-Jan-02 160,700,295 $123.70 $0.30 0.50 $61.85 $0.15 01-Dec-01 145,000,500 $111.40 $0.00 0.45 $50.26 $0.00 The adjustment factor is just the value in the issued capital cell divided by 321,400,589. CHAPTER 5 – The Mathematics of Diversification 8 - 227 Calculating the HPR on the stock from the Normalized Data Normalized Normalized ( P P0 ) D1 HPR 1 P0 Date Stock Price Dividend HPR $59.22 - $57.90 $0.00 01-May-02 $59.22 $0.00 2.28% $57.90 01-Apr-02 $57.90 $0.15 -7.90% 2.28% 01-Mar-02 $63.03 $0.00 -2.82% 01-Feb-02 $64.86 $0.00 4.87% 02-Jan-02 $61.85 $0.15 23.36% 01-Dec-01 $50.26 $0.00 Use $59.22 as the ending price, $57.90 as the beginning price and during the month of May, no dividend was declared. CHAPTER 5 – The Mathematics of Diversification 8 - 228 Now Put the data from the S&P/TSX Total Return Composite Index in Ending Normalized Normalized TSX Date Stock Price Dividend HPR Value 01-May-02 $59.22 $0.00 2.28% 16911.33 01-Apr-02 $57.90 $0.15 -7.90% 16903.36 01-Mar-02 $63.03 $0.00 -2.82% 17308.41 01-Feb-02 $64.86 $0.00 4.87% 16801.82 02-Jan-02 $61.85 $0.15 23.36% 16908.11 01-Dec-01 $50.26 $0.00 16881.75 You will find the Total Return S&P/TSX Composite Index values in TSX Review found in the library. CHAPTER 5 – The Mathematics of Diversification 8 - 229 Now Calculate the HPR on the Market Index ( P P0 ) HPR 1 P0 16,911.33 - 16,903.36 Ending Normalized Normalized 16,903.36 TSX HPR on 0.05% Date Stock Price Dividend HPR Value the TSX 01-May-02 $59.22 $0.00 2.28% 16911.33 0.05% 01-Apr-02 $57.90 $0.15 -7.90% 16903.36 -2.34% 01-Mar-02 $63.03 $0.00 -2.82% 17308.41 3.02% 01-Feb-02 $64.86 $0.00 4.87% 16801.82 -0.63% 02-Jan-02 $61.85 $0.15 23.36% 16908.11 0.16% 01-Dec-01 $50.26 $0.00 16881.75 Again, you simply use the HPR formula using the ending values for the total return composite index. CHAPTER 5 – The Mathematics of Diversification 8 - 230 Regression In Excel • If you haven‟t already…go to the tools menu…down 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. CHAPTER 5 – The Mathematics of Diversification 8 - 231 Regression Defining the Data Ranges Ending Normalized Normalized TSX HPR on Date Stock Price Dividend HPR Value the TSX 01-May-02 $59.22 $0.00 2.28% 16911.33 0.05% 01-Apr-02 $57.90 $0.15 -7.90% 16903.36 -2.34% 01-Mar-02 $63.03 $0.00 -2.82% 17308.41 3.02% 01-Feb-02 $64.86 $0.00 4.87% 16801.82 -0.63% 02-Jan-02 $61.85 $0.15 23.36% 16908.11 0.16% 01-Dec-01 $50.26 $0.00 16881.75 dependent variable is the returns on the Stock. The independent variable is the returns on the Market. CHAPTER 5 – The Mathematics of Diversification 8 - 232 Now Use the Regression Function in Excel to regress the returns of the stock against the returns of the market SUMMARY OUTPUT Regression Statistics R-square is the Multiple R 0.05300947 R Square 0.00281 coefficient of Adjusted R Square -0.2464875 determination = Standard Error 5.79609628 Observations 6 0.0028=.3% ANOVA df SS MS F Significance F Regression 1 0.3786694 0.37866937 0.011271689 0.920560274 Residual 4 134.37893 33.5947321 Total 5 134.7576 CoefficientsStandard Error t Stat P-value Lower 95% Upper 95% Lower 95.0%Upper 95.0% Intercept 59.3420816 2.8980481 20.4765686 3.3593E-05 51.29579335 67.38836984 51.2957934 67.38837 X Variable 1 3.55278937 33.463777 0.10616821 0.920560274 -89.35774428 96.46332302 -89.3577443 96.46332 Beta The alpha is the Coefficient is vertical intercept. the X- Variable 1 CHAPTER 5 – The Mathematics of Diversification 8 - 233 Finalize Your Chart Alcan 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-clicking‟ your 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. CHAPTER 5 – The Mathematics of Diversification 8 - 234 The Beta Alcan Example • Obviously the beta (X-coefficient) can simply be read from the regression output. – In this case it was 3.56 making Alcan‟s returns more than 3 times as volatile as the market as a whole. – Of course, in this simple example with only 5 observations, you wouldn‟t want to draw any serious conclusions from this estimate. CHAPTER 5 – The Mathematics of Diversification 8 - 235 Summary and Conclusions In this chapter you have learned: – How to measure different types of returns – How to calculate the standard deviation and interpret its meaning – How to measure returns and risk of portfolios and the importance of correlation in the diversification process. – How the efficient frontier is that set of achievable portfolios that offer the highest rate of return for a given level of risk. CHAPTER 5 – The Mathematics of Diversification 8 - 236 Appendix 3

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