INTRODUCTION TO CORPORATE FINANCE Laurence Booth • W. Sean Cleary Prepared by Ken Hartviksen

CHAPTER 8 Risk, Return, and Portfolio Theory

CHAPTER 8 – Risk, Return and Portfolio Theory Lecture Agenda Learning Objectives Important Terms Measurement of Returns Measuring Risk Expected Return and Risk for Portfolios The Efficient Frontier Diversification Summary and Conclusions Concept Review Questions CHAPTER 8 – Risk, Return and Portfolio Theory

CHAPTER 8 – Risk, Return and Portfolio Theory 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 CHAPTER 8 – Risk, Return and Portfolio Theory

Important Chapter Terms Arithmetic mean Attainable portfolios Capital gain/loss Correlation coefficient Covariance Day trader Diversification Efficient frontier Efficient portfolios Ex ante returns Ex post returns Expected returns Geometric mean Income yield Mark to market Market risk Minimum variance frontier Minimum variance portfolio Modern portfolio theory Naïve or random diversification Paper losses Portfolio Range Risk averse Standard deviation Total return Unique (or non-systematic) or diversifiable risk Variance CHAPTER 8 – 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 markets and that generally, higher returns can only be achieved by taking on greater risk. Risk isn’t just the potential loss of return, it is the potential loss of the entire investment itself (loss of both principal and interest). Consequently, taking on additional risk in search of higher returns is a decision that should not be taking lightly. CHAPTER 8 – Risk, Return and Portfolio Theory

Introduction to Risk and Return RF Risk Risk Premium Real Return Expected Inflation Rate CHAPTER 8 – 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 8 – Risk, Return and Portfolio Theory

Measuring Returns Introduction In Chapter 7 you learned that the constant growth DDM can be decomposed into the two forms of income that equity investors may receive, dividends and capital gains. 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 8 – Risk, Return and Portfolio Theory

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. Where CF1 = the expected cash flow to be received P0 = the purchase price [8-1] CHAPTER 8 – Risk, Return and Portfolio Theory

Income Yield Stocks versus Bonds The dividend yield is calculated using trailing rather than forecast earns (because next year’s dividends cannot be predicted in aggregate), nevertheless dividend yields have exceeded income yields on bonds Reason – risk The risk of earning bond income is much less than the risk incurred in earning dividend income (Remember, bond investors, as secured creditors of the first have a legally-enforceable contractual claim to interest.) CHAPTER 8 – Risk, Return and Portfolio Theory

Measuring Returns Common Share and Long Canada Bond Yield Gap Table 8-1 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. CHAPTER 8 – Risk, Return and Portfolio Theory

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 8 – Risk, Return and Portfolio Theory

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 [8-2] CHAPTER 8 – Risk, Return and Portfolio Theory

Measuring Returns Total Percentage Return The investor’s total return (holding period return) is: [8-3] CHAPTER 8 – Risk, Return and Portfolio Theory

Measuring Returns Total Percentage Return – General Formula The general formula for holding period return is: [8-3] CHAPTER 8 – Risk, Return and Portfolio Theory

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 8 – Risk, Return and Portfolio Theory

Measuring Average Returns Arithmetic Average 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 [8-4] CHAPTER 8 – Risk, Return and Portfolio Theory

Measuring Average Returns Geometric Mean Measures the average or compound growth rate over multiple periods. [8-5] CHAPTER 8 – Risk, Return and Portfolio Theory

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 8-2 illustrates this principle on major asset classes 1938 – 2005) CHAPTER 8 – Risk, Return and Portfolio Theory

CHAPTER 8 – Risk, Return and Portfolio Theory Measuring Average Returns Average Investment Returns and Standard Deviations The greater the difference, the greater the volatility of annual returns. CHAPTER 8 – Risk, Return and Portfolio Theory

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 8 – Risk, Return and Portfolio Theory

Estimating Expected Returns Estimating Ex Ante (Forecast) Returns The general formula Where: ER = the expected return on an investment Ri = the estimated return in scenario i Probi = the probability of state i occurring [8-6] CHAPTER 8 – Risk, Return and Portfolio Theory

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. CHAPTER 8 – Risk, Return and Portfolio Theory

CHAPTER 8 – Risk, Return and Portfolio Theory 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. CHAPTER 8 – Risk, Return and Portfolio Theory

CHAPTER 8 – Risk, Return and Portfolio Theory 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. CHAPTER 8 – Risk, Return and Portfolio Theory

CHAPTER 8 – Risk, Return and Portfolio Theory 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 8 – Risk, Return and Portfolio Theory

CHAPTER 8 – Risk, Return and Portfolio Theory Risk Illustrated Possible Returns on the Stock Probability -30% -20% -10% 0% 10% 20% 30% 40% Outcomes that produce harm The range of total possible returns on the stock A runs from -30% to more than +40%. If the required return on the stock is 10%, then those outcomes less than 10% represent risk to the investor. A CHAPTER 8 – Risk, Return and Portfolio Theory

CHAPTER 8 – Risk, Return and Portfolio Theory 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 8 – Risk, Return and Portfolio Theory

Differences in Levels of Risk Illustrated The wider the range of probable outcomes the greater the risk of the investment. A is a much riskier investment than B Outcomes that produce harm Probability B A -30% -20% -10% 0% 10% 20% 30% 40% Possible Returns on the Stock CHAPTER 8 – Risk, Return and Portfolio Theory

Historical Returns on Different Asset Classes Figure 8-2 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, 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 8-2 on the following slide) CHAPTER 8 – Risk, Return and Portfolio Theory

Measuring Risk Annual Returns by Asset Class, 1938 - 2005 FIGURE 8-2 CHAPTER 8 – Risk, Return and Portfolio Theory

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 CHAPTER 8 – Risk, Return and Portfolio Theory

Measuring Risk Ex post Standard Deviation [8-7] CHAPTER 8 – Risk, Return and Portfolio Theory

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 Step 2 – Calculate the Standard Deviation CHAPTER 8 – Risk, Return and Portfolio Theory

Ex Post Risk Stability of Risk Over Time Figure 8-3 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 Scenario-based estimates of risk is done through ex ante estimates and calculations CHAPTER 8 – Risk, Return and Portfolio Theory

Relative Uncertainty Equities versus Bonds FIGURE 8-3 CHAPTER 8 – Risk, Return and Portfolio Theory

Measuring Risk Ex ante Standard Deviation A Scenario-Based Estimate of Risk [8-8] CHAPTER 8 – Risk, Return and Portfolio Theory

CHAPTER 8 – Risk, Return and Portfolio Theory 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 CHAPTER 8 – Risk, Return and Portfolio Theory

CHAPTER 8 – Risk, Return and Portfolio Theory Scenario-Based Estimate of Risk Ex ante Standard Deviation Spreadsheet Approach: First Step – Calculate the Expected Return Determined by multiplying the probability times the possible return. Expected return equals the sum of the weighted possible returns. CHAPTER 8 – Risk, Return and Portfolio Theory

CHAPTER 8 – Risk, Return and Portfolio Theory Scenario-Based Estimate of Risk Second Step – Measure the Weighted and Squared Deviations Now multiply the square deviations by their probability of occurrence. First calculate the deviation of possible returns from the expected. Second, square those deviations from the mean. The sum of the weighted and square deviations is the variance in percent squared terms. The standard deviation is the square root of the variance (in percent terms). CHAPTER 8 – Risk, Return and Portfolio Theory

CHAPTER 8 – Risk, Return and Portfolio Theory Scenario-Based Estimate of Risk Example Using the Ex ante Standard Deviation Formula CHAPTER 8 – Risk, Return and Portfolio Theory

CHAPTER 8 – 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 8 – Risk, Return and Portfolio Theory

CHAPTER 8 – Risk, Return and Portfolio Theory Diversification Diversification has two faces: Diversification results in an overall reduction in portfolio risk (return volatility over time) with little sacrifice in returns 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 8 – Risk, Return and Portfolio Theory

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: The portfolio weight of a particular security is the percentage of the portfolio’s total value that is invested in that security. [8-9] CHAPTER 8 – Risk, Return and Portfolio Theory

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% CHAPTER 8 – Risk, Return and Portfolio Theory

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 Example 1: Assume ERA = 8% and ERB = 10% (See the following 6 slides based on Figure 8-4) CHAPTER 8 – Risk, Return and Portfolio Theory

CHAPTER 8 – Risk, Return and Portfolio Theory Expected Portfolio Return Affect on Portfolio Return of Changing Relative Weights in A and B Expected Return % Portfolio Weight 10.50 10.00 9.50 9.00 8.50 8.00 7.50 7.00 0 0.2 0.4 0.6 0.8 1.0 1.2 FIGURE 8-4 ERA=8% ERB= 10% CHAPTER 8 – Risk, Return and Portfolio Theory

CHAPTER 8 – Risk, Return and Portfolio Theory Expected Portfolio Return Affect on Portfolio Return of Changing Relative Weights in A and B FIGURE 8-4 Expected Return % Portfolio Weight 10.50 10.00 9.50 9.00 8.50 8.00 7.50 7.00 0 0.2 0.4 0.6 0.8 1.0 1.2 ERA=8% ERB= 10% 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) CHAPTER 8 – Risk, Return and Portfolio Theory

CHAPTER 8 – Risk, Return and Portfolio Theory Expected Portfolio Return Affect on Portfolio Return of Changing Relative Weights in A and B Expected Return % Portfolio Weight 10.50 10.00 9.50 9.00 8.50 8.00 7.50 7.00 0 0.2 0.4 0.6 0.8 1.0 1.2 FIGURE 8-4 ERA=8% ERB= 10% The potential returns of the portfolio are bounded by the highest and lowest returns of the individual assets that make up the portfolio. CHAPTER 8 – Risk, Return and Portfolio Theory

CHAPTER 8 – Risk, Return and Portfolio Theory Expected Portfolio Return Affect on Portfolio Return of Changing Relative Weights in A and B Expected Return % Portfolio Weight 10.50 10.00 9.50 9.00 8.50 8.00 7.50 7.00 0 0.2 0.4 0.6 0.8 1.0 1.2 FIGURE 8-4 ERA=8% ERB= 10% The expected return on the portfolio if 100% is invested in Asset A is 8%. CHAPTER 8 – Risk, Return and Portfolio Theory

CHAPTER 8 – Risk, Return and Portfolio Theory Expected Portfolio Return Affect on Portfolio Return of Changing Relative Weights in A and B FIGURE 8-4 Expected Return % Portfolio Weight 10.50 10.00 9.50 9.00 8.50 8.00 7.50 7.00 0 0.2 0.4 0.6 0.8 1.0 1.2 ERA=8% ERB= 10% The expected return on the portfolio if 100% is invested in Asset B is 10%. CHAPTER 8 – Risk, Return and Portfolio Theory

CHAPTER 8 – Risk, Return and Portfolio Theory Expected Portfolio Return Affect on Portfolio Return of Changing Relative Weights in A and B FIGURE 8-4 Expected Return % Portfolio Weight 10.50 10.00 9.50 9.00 8.50 8.00 7.50 7.00 0 0.2 0.4 0.6 0.8 1.0 1.2 ERA=8% ERB= 10% The expected return on the portfolio if 50% is invested in Asset A and 50% in B is 9%. CHAPTER 8 – Risk, Return and Portfolio Theory

CHAPTER 8 – Risk, Return and Portfolio Theory Range of Returns in a Two Asset Portfolio Example 1: (r)A= 14%, E(r)B= 6% A graph of this relationship is found on the following slide. CHAPTER 8 – Risk, Return and Portfolio Theory

Range of Returns in a Two-Asset Portfolio E(r)A= 14%, E(r)B= 6% CHAPTER 8 – Risk, Return and Portfolio Theory

Expected Portfolio Returns Example of a Three Asset Portfolio CHAPTER 8 – Risk, Return and Portfolio Theory K. Hartviksen 5

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 8 – Risk, Return and Portfolio Theory

CHAPTER 8 – Risk, Return and Portfolio Theory Expected Return and Risk For Portfolios Standard Deviation of a Two-Asset Portfolio using Covariance [8-11] Risk of Asset A adjusted for weight in the portfolio Risk of Asset B adjusted for weight in the portfolio Factor to take into account co-movement of returns. This factor can be negative. CHAPTER 8 – Risk, Return and Portfolio Theory

CHAPTER 8 – Risk, Return and Portfolio Theory Expected Return and Risk For Portfolios Standard Deviation of a Two-Asset Portfolio using Correlation Coefficient [8-15] Factor that takes into account the degree of co-movement of returns. It can have a negative value if correlation is negative. CHAPTER 8 – Risk, Return and Portfolio Theory

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 co-movement of returns of the assets making up the portfolio The standard deviation of a two-asset portfolio may be measured using the Markowitz model: CHAPTER 8 – Risk, Return and Portfolio Theory

Risk of a Three-Asset Portfolio The data requirements for a three-asset portfolio grows dramatically if we are using Markowitz Portfolio selection formulae Needed: Three correlation coefficients between A and B; A and C; and B and C A ρa,b ρa,c B C ρb,c CHAPTER 8 – Risk, Return and Portfolio Theory

Risk of a Four-asset Portfolio The data requirements for a four-asset portfolio grows dramatically if we are using Markowitz Portfolio selection formulae Needed: Six correlation coefficients between A and B; A and C; A and D; B and C; C and D; and B and D A C B D ρa,b ρa,d ρa,c ρb,d ρb,c ρc,d CHAPTER 8 – Risk, Return and Portfolio Theory

CHAPTER 8 – Risk, Return and Portfolio Theory Covariance A statistical measure of the correlation of the fluctuations of the annual rates of return of different investments. [8-12] CHAPTER 8 – Risk, Return and Portfolio Theory

CHAPTER 8 – Risk, Return and Portfolio Theory 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 [8-13] CHAPTER 8 – Risk, Return and Portfolio Theory

Covariance and Correlation Coefficient Solving for covariance given the correlation coefficient and standard deviation of the two assets: [8-14] CHAPTER 8 – Risk, Return and Portfolio Theory

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 8 – Risk, Return and Portfolio Theory

CHAPTER 8 – Risk, Return and Portfolio Theory Affect of Perfectly Negatively Correlated Returns Elimination of Portfolio Risk Time 0 1 2 If returns of A and B are perfectly negatively correlated, a two-asset portfolio made up of equal parts of Stock A and B would be riskless. There would be no variability of the portfolios returns over time. Returns on Stock A Returns on Stock B Returns on Portfolio Returns % 10% 5% 15% 20% CHAPTER 8 – Risk, Return and Portfolio Theory

CHAPTER 8 – Risk, Return and Portfolio Theory Example of Perfectly Positively Correlated Returns No Diversification of Portfolio Risk Returns % If returns of A and B are perfectly positively correlated, a two-asset portfolio made up of equal parts of Stock A and B would be risky. There would be no diversification (reduction of portfolio risk). 20% 15% Returns on Stock A Returns on Stock B Returns on Portfolio 10% 5% Time 0 1 2 CHAPTER 8 – Risk, Return and Portfolio Theory

CHAPTER 8 – Risk, Return and Portfolio Theory Affect of Perfectly Negatively Correlated Returns Elimination of Portfolio Risk Returns % If returns of A and B are perfectly negatively correlated, a two-asset portfolio made up of equal parts of Stock A and B would be riskless. There would be no variability of the portfolios returns over time. 20% 15% Returns on Stock A 10% Returns on Stock B Returns on Portfolio 5% Time 0 1 2 CHAPTER 8 – Risk, Return and Portfolio Theory

Affect of Perfectly Negatively Correlated Returns Numerical Example Perfectly Negatively Correlated Returns over time CHAPTER 8 – Risk, Return and Portfolio Theory

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 8 – Risk, Return and Portfolio Theory

Example of Portfolio Combinations and Correlation Perfect Positive Correlation – no diversification Both portfolio returns and risk are bounded by the range set by the constituent assets when ρ=+1 CHAPTER 8 – Risk, Return and Portfolio Theory

Example of Portfolio Combinations and Correlation Positive Correlation – weak diversification potential When ρ=+0.5 these portfolio combinations have lower risk – expected portfolio return is unaffected. CHAPTER 8 – Risk, Return and Portfolio Theory

Example of Portfolio Combinations and Correlation No Correlation – some diversification potential Portfolio risk is lower than the risk of either asset A or B. CHAPTER 8 – Risk, Return and Portfolio Theory

Example of Portfolio Combinations and Correlation Negative Correlation – greater diversification potential Portfolio risk for more combinations is lower than the risk of either asset CHAPTER 8 – Risk, Return and Portfolio Theory

Example of Portfolio Combinations and Correlation Perfect Negative Correlation – greatest diversification potential Risk of the portfolio is almost eliminated at 70% invested in asset A CHAPTER 8 – Risk, Return and Portfolio Theory

The Effect of Correlation on Portfolio Risk: Diversification of a Two Asset Portfolio Demonstrated Graphically The Effect of Correlation on Portfolio Risk: The Two-Asset Case Expected Return Standard Deviation 0% 10% 4% 8% 20% 30% 40% 12% B rAB= +1 A rAB = 0 rAB = -0.5 rAB = -1 CHAPTER 8 – Risk, Return and Portfolio Theory

Impact of the Correlation Coefficient Figure 8-7 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 8 – Risk, Return and Portfolio Theory

Expected Portfolio Return Impact of the Correlation Coefficient FIGURE 8-7 15 10 5 Standard Deviation (%) of Portfolio Returns Correlation Coefficient (ρ) -1 -0.5 0 0.5 1 CHAPTER 8 – Risk, Return and Portfolio Theory

CHAPTER 8 – Risk, Return and Portfolio Theory Zero Risk Portfolio We can calculate the portfolio that removes all risk. When ρ = -1, then Becomes: [8-15] [8-16] CHAPTER 8 – Risk, Return and Portfolio Theory

An Exercise using T-bills, Stocks and Bonds Historical averages for returns and risk for three asset classes Each achievable portfolio combination is plotted on expected return, risk (σ) space, found on the following slide. Historical correlation coefficients between the asset classes Portfolio characteristics for each combination of securities CHAPTER 8 – Risk, Return and Portfolio Theory

Results Using only Three Asset Classes The efficient set is that set of achievable portfolio combinations that offer the highest rate of return for a given level of risk. The solid blue line indicates the efficient set. The plotted points are attainable portfolio combinations. CHAPTER 8 – Risk, Return and Portfolio Theory

Two-Security Portfolios Modern Portfolio Theory FIGURE 8-9 Expected Return % Standard Deviation (%) 13 12 11 10 9 8 7 6 0 10 20 30 40 50 60 This line represents the set of portfolio combinations that are achievable by varying relative weights and using two non-correlated securities. CHAPTER 8 – Risk, Return and Portfolio Theory

CHAPTER 8 – Risk, Return and Portfolio Theory Dominance It is assumed that investors are rational, wealth-maximizing and risk averse If so, then some investment choices dominate others CHAPTER 8 – Risk, Return and Portfolio Theory

Investment Choices The Concept of Dominance Illustrated B C Return % Risk 10% 5% To the risk-averse wealth maximizer, the choices are clear, A dominates B, A dominates C. A dominates B because it offers the same return but for less risk. A dominates C because it offers a higher return but for the same risk. 20% CHAPTER 8 – Risk, Return and Portfolio Theory

Efficient Frontier Modern Portfolio Theory FIGURE 8-10 Expected Return % Standard Deviation (%) A E B C D A is not attainable B,E lie on the efficient frontier and are attainable E is the minimum variance portfolio C, D are attainable but are dominated by superior portfolios CHAPTER 8 – Risk, Return and Portfolio Theory

Efficient Frontier Modern Portfolio Theory 8 - 10 FIGURE Expected Return % Standard Deviation (%) A E B C D Rational, risk averse investors will only want to hold portfolios such B. The actual choice will depend on her/his risk preferences. CHAPTER 8 – Risk, Return and Portfolio Theory

CHAPTER 8 – Risk, Return and Portfolio Theory Diversification Risk of a portfolio can be reduced by spreading the value of the portfolio across, two, three, four or more assets The key 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 Figure 8-11 and Table 8-3 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 8 – Risk, Return and Portfolio Theory

Diversification Domestic Diversification FIGURE 8-11 14 12 10 8 6 4 2 Standard Deviation (%) Number of Stocks in Portfolio 0 50 100 150 200 250 300 Average Portfolio Risk January 1985 to December 1997 CHAPTER 8 – Risk, Return and Portfolio Theory

Diversification Domestic Diversification CHAPTER 8 – Risk, Return and Portfolio Theory

Average Portfolio Risk Total Risk of an Individual Asset Equals the Sum of Market and Unique Risk Standard Deviation (%) Number of Stocks in Portfolio Average Portfolio Risk This graph illustrates that total risk of a stock is made up of market risk (that cannot be diversified away because it is a function of the economic ‘system’) and unique, company-specific risk that is eliminated from the portfolio through diversification. Diversifiable (unique) risk [8-19] Nondiversifiable (systematic) risk [8-19] CHAPTER 8 – Risk, Return and Portfolio Theory

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 to 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 Figure 8 -12 on the following slide.) CHAPTER 8 – Risk, Return and Portfolio Theory

Diversification International Diversification FIGURE 8-12 100 80 60 40 20 Percent risk Number of Stocks 0 10 20 30 40 50 60 International stocks U.S. stocks 11.7 CHAPTER 8 – Risk, Return and Portfolio Theory

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 CHAPTER 8 – Risk, Return and Portfolio Theory

CHAPTER 8 – Risk, Return and Portfolio Theory Copyright Copyright © 2007 John Wiley & Sons Canada, Ltd. All rights reserved. Reproduction or translation of this work beyond that permitted by Access Copyright (the Canadian copyright licensing agency) is unlawful. Requests for further information should be addressed to the Permissions Department, John Wiley & Sons Canada, Ltd. The purchaser may make back-up copies for his or her own use only and not for distribution or resale. The author and the publisher assume no responsibility for errors, omissions, or damages caused by the use of these files or programs or from the use of the information contained herein. CHAPTER 8 – Risk, Return and Portfolio Theory