Prepared byKen Hartviksen

INTRODUCTION TO CORPORATE FINANCELaurence Booth W. Sean Cleary

Chapter 8 Risk, Return and Portfolio Theory

CHAPTER 8 Risk, Return and Portfolio Theory

Lecture AgendaLearning ObjectivesImportant TermsMeasurement of ReturnsMeasuring RiskExpected Return and Risk for PortfoliosThe Efficient FrontierDiversificationSummary and ConclusionsConcept Review Questions

Learning ObjectivesThe difference among the most important types of returnsHow to estimate expected returns and risk for individual securitiesWhat happens to risk and return when securities are combined in a portfolioWhat is meant by an efficient frontierWhy diversification is so important to investors

Important Chapter TermsArithmetic meanAttainable portfoliosCapital gain/lossCorrelation coefficientCovarianceDay traderDiversificationEfficient frontierEfficient portfoliosEx ante returnsEx post returnsExpected returnsGeometric meanIncome yieldMark to marketMarket riskMinimum variance frontierMinimum variance portfolioModern portfolio theoryNave or random diversificationPaper lossesPortfolioRangeRisk averseStandard deviationTotal returnUnique (or non-systematic) or diversifiable riskVariance

Introduction to Risk and ReturnRisk, Return and Portfolio Theory

Introduction to Risk and ReturnRisk 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 isnt 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.

Measuring ReturnsRisk, Return and Portfolio Theory

Measuring ReturnsIntroductionEx Ante ReturnsReturn calculations may be done before-the-fact, in which case, assumptions must be made about the future

Ex Post ReturnsReturn calculations done after-the-fact, in order to analyze what rate of return was earned.

Measuring ReturnsIntroductionIn 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.

Measuring ReturnsIncome YieldIncome 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 receivedP0 = the purchase price

Income Yield Stocks versus BondsFigure 8-1 illustrates the income yields for both bonds and stock in Canada from the 1950s to 2005

The dividend yield is calculated using trailing rather than forecast earns (because next years dividends cannot be predicted in aggregate), nevertheless dividend yields have exceeded income yields on bonds.Reason riskThe 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.)

(See Figure 8 -1 on the following slide)

Ex post versus Ex ante ReturnsMarket Income Yields8-1 FIGUREInsert Figure 8 - 1

Measuring ReturnsCommon Share and Long Canada Bond Yield GapTable 8 1 illustrates the income yield gap between stocks and bonds over recent decadesThe 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.

Sheet1

Table 8-1 Average Yield Gap

Average Yield Gap(%)

1950s0.82

1960s2.35

1970s4.54

1980s8.14

1990s5.51

2000s3.55

Overall4.58

Measuring ReturnsDollar ReturnsInvestors in market-traded securities (bonds or stock) receive investment returns in two different form:Income yieldCapital gain (or loss) yield

The investor will receive dollar returns, for example:$1.00 of dividendsShare 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!)

Measuring ReturnsConverting Dollar Returns to Percentage ReturnsAn investor receives the following dollar returns a stock investment of $25:$1.00 of dividendsShare 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

Measuring ReturnsTotal Percentage ReturnThe investors total return (holding period return) is:

Measuring ReturnsTotal Percentage Return General FormulaThe general formula for holding period return is:

Measuring Average ReturnsEx Post ReturnsMeasurement 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 averageGeometric mean

Measuring Average ReturnsArithmetic Average

Where:ri = the individual returnsn = the total number of observations

Most commonly used value in statisticsSum of all returns divided by the total number of observations

Measuring Average ReturnsGeometric Mean

Measures the average or compound growth rate over multiple periods.

Measuring Average ReturnsGeometric Mean versus Arithmetic AverageIf 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)

Measuring Average ReturnsAverage Investment Returns and Standard Deviations

Sheet1

Table 8 - 2 Average Investment Returns and Standard Deviations, 1938-2005

Annual Arithmetic Average (%)Annual Geometric Mean (%)Standard Deviation of Annual Returns (%)

Government of Canada treasury bills5.205.114.32

Government of Canada bonds6.626.249.32

Canadian stocks11.7910.6016.22

U.S. stocks13.1511.7617.54

Source: Data are from the Canadian Institute of Actuaries

Measuring Expected (Ex Ante) ReturnsWhile past returns might be interesting, investors 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.

Estimating Expected ReturnsEstimating Ex Ante (Forecast) ReturnsThe general formula

Where:ER = the expected return on an investmentRi = the estimated return in scenario iProbi = the probability of state i occurring

Estimating Expected ReturnsEstimating Ex Ante (Forecast) ReturnsExample:This is type of forecast data that are required to make an ex ante estimate of expected return.

Sheet1

State of the EconomyProbability of OccurrencePossible Returns on Stock A in that State

Economic Expansion25.0%30%

Normal Economy50.0%12%

Recession25.0%-25%

Estimating Expected ReturnsEstimating Ex Ante (Forecast) Returns Using a Spreadsheet ApproachExample Solution:Sum the products of the probabilities and possible returns in each state of the economy.

Sheet1

(1)(2)(3)(4)=(2)(1)

State of the EconomyProbability of OccurrencePossible Returns on Stock A in that StateWeighted Possible Returns on the Stock

Economic Expansion25.0%30%7.50%

Normal Economy50.0%12%6.00%

Recession25.0%-25%-6.25%

Expected Return on the Stock =7.25%

Estimating Expected ReturnsEstimating Ex Ante (Forecast) Returns Using a Formula ApproachExample Solution:Sum the products of the probabilities and possible returns in each state of the economy.

Measuring RiskRisk, Return and Portfolio Theory

RiskProbability of incurring harmFor 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.

RiskIllustratedThe 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

RangeThe difference between the maximum and minimum values is called the rangeCanadian common stocks have had a range of annual returns of 74.36 % over the 1938-2005 periodTreasury 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.

Differences in Levels of RiskIllustratedPossible Returns on the StockProbability-30%-20%-10%0%10%20%30%40%The wider the range of probable outcomes the greater the risk of the investment.A is a much riskier investment than BBA

Historical Returns on Different Asset ClassesFigure 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 (when prices fall because of rising interest rates), and the range of returns has been greater than T-bills but less than stocksCommon stock returns have experienced the greatest range of returns(See Figure 8-2 on the following slide)

Measuring RiskAnnual Returns by Asset Class, 1938 - 2005FIGURE 8-2

Refining the Measurement of RiskStandard Deviation ()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)

Measuring RiskEx post Standard Deviation

Measuring RiskExample Using the Ex post Standard DeviationProblemEstimate 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

Ex Post RiskStability of Risk Over TimeFigure 8-3 (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.

Relative UncertaintyEquities versus BondsFIGURE 8-3

Measuring RiskEx ante Standard DeviationA Scenario-Based Estimate of Risk

Scenario-based Estimate of RiskExample Using the Ex ante Standard Deviation Raw DataGIVEN INFORMATION INCLUDES:Possible returns on the investment for different discrete statesAssociated probabilities for those possible returns

Sheet1

State of the EconomyProbabilityPossible Returns on Security A

Recession25.0%-22.0%

Normal50.0%14.0%

Economic Boom25.0%35.0%

Scenario-based Estimate of RiskEx ante Standard Deviation Spreadsheet ApproachThe following two slides illustrate an approach to solving for standard deviation using a spreadsheet model.

Scenario-based Estimate of RiskFirst Step Calculate the Expected ReturnDetermined by multiplying the probability times the possible return.Expected return equals the sum of the weighted possible returns.

Sheet1

State of the EconomyProbabilityPossible Returns on Security AWeighted Possible Returns

Recession25.0%-22.0%-5.5%

Normal50.0%14.0%7.0%

Economic Boom25.0%35.0%8.8%

Expected Return =10.3%

Scenario-based Estimate of RiskSecond Step Measure the Weighted and Squared DeviationsSecond, 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).Now multiply the square deviations by their probability of occurrence.

Sheet1

State of the EconomyProbabilityPossible Returns on Security AWeighted Possible ReturnsDeviation of Possible Return from ExpectedSquared DeviationsWeighted and Squared Deviations

Recession25.0%-22.0%-5.5%-32.3%0.104010.02600

Normal50.0%14.0%7.0%3.8%0.001410.00070

Economic Boom25.0%35.0%8.8%24.8%0.061260.01531

Expected Return =10.3%Variance =0.0420

Standard Deviation =20.50%

Scenario-based Estimate of RiskExample Using the Ex ante Standard Deviation Formula

Sheet1

State of the EconomyProbabilityPossible Returns on Security AWeighted Possible Returns

Recession25.0%-22.0%-5.5%

Normal50.0%14.0%7.0%

Economic Boom25.0%35.0%8.8%

Expected Return =10.3%

Modern Portfolio TheoryRisk, Return and Portfolio Theory

PortfoliosA 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.

DiversificationDiversification has two faces:

Diversification results in an overall reduction in portfolio risk (return volatility over time) with little sacrifice in returns, andDiversification 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.)

Expected Return of a PortfolioModern Portfolio TheoryThe 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 portfolios total value that is invested in that security.

Expected Return of a PortfolioExamplePortfolio value = $2,000 + $5,000 = $7,000rA = 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%

Range of Returns in a Two Asset PortfolioIn 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.

Range of Returns in a Two Asset PortfolioExample 1:

Assume ERA = 8% and ERB = 10%

(See the following 6 slides based on Figure 8-4)

Expected Portfolio ReturnAffect on Portfolio Return of Changing Relative Weights in A and B

Expected Portfolio ReturnAffect on Portfolio Return of Changing Relative Weights in A and B8 - 4 FIGURE

Expected Portfolio ReturnAffect on Portfolio Return of Changing Relative Weights in A and B

Expected Portfolio ReturnAffect on Portfolio Return of Changing Relative Weights in A and B

Expected Portfolio ReturnAffect on Portfolio Return of Changing Relative Weights in A and B

Expected Portfolio ReturnAffect on Portfolio Return of Changing Relative Weights in A and B

Range of Returns in a Two Asset PortfolioExample 1:

Assume ERA = 14% and ERB = 6%

(See the following 2 slides )

Range of Returns in a Two Asset PortfolioE(r)A= 14%, E(r)B= 6% A graph of this relationship is found on the following slide.

Range of Returns in a Two Asset Portfolio E(r)A= 14%, E(r)B= 6%

Chart1

0.06

0.068

0.076

0.084

0.092

0.1

0.108

0.116

0.124

0.132

0.14

Weight Invested in Asset A

Expected Return on Two Asset Portfolio

Range of Portfolio Returns

Sheet1

E(RA) =14.0%

E(RB) =6.0%

Weight of AWeight of BPortfolio Return

0.0%100.0%6.00%

10.0%90.0%6.80%

20.0%80.0%7.60%

30.0%70.0%8.40%

40.0%60.0%9.20%

50.0%50.0%10.00%

60.0%40.0%10.80%

70.0%30.0%11.60%

80.0%20.0%12.40%

90.0%10.0%13.20%

100.0%0.0%14.00%

Expected Portfolio ReturnsExample of a Three Asset PortfolioK. Hartviksen

Sheet1

Relative WeightExpected ReturnWeighted Return

Stock X0.4008.0%0.03

Stock Y0.35015.0%0.05

Stock Z0.25025.0%0.06

Expected Portfolio Return =14.70%

Risk in PortfoliosRisk, Return and Portfolio Theory

Modern Portfolio Theory - MPTPrior to the establishment of Modern Portfolio Theory (MPT), most people only focused upon investment returnsthey 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.

Expected Return and Risk For PortfoliosStandard Deviation of a Two-Asset Portfolio using Covariance[8-11]

Expected Return and Risk For PortfoliosStandard Deviation of a Two-Asset Portfolio using Correlation Coefficient[8-15]

Grouping Individual Assets into PortfoliosThe 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 portfoliothe relative weights of the assets in the portfoliothe degree of comovement of returns of the assets making up the portfolioThe standard deviation of a two-asset portfolio may be measured using the Markowitz model:

Risk of a Three-Asset PortfolioThe 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.ABCa,bb,ca,c

Risk of a Four-asset PortfolioThe 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,ba,db,cc,da,cb,d

CovarianceA statistical measure of the correlation of the fluctuations of the annual rates of return of different investments.

CorrelationThe degree to which the returns of two stocks co-move is measured by the correlation coefficient ().The correlation coefficient () 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

Covariance and Correlation CoefficientSolving for covariance given the correlation coefficient and standard deviation of the two assets:

Importance of CorrelationCorrelation 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.

Affect of Perfectly Negatively Correlated ReturnsElimination of Portfolio Risk

Example of Perfectly Positively Correlated ReturnsNo Diversification of Portfolio RiskTime 012If 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).Returns%10%5%15%20%

Affect of Perfectly Negatively Correlated ReturnsElimination of Portfolio RiskTime 012If 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 variabilityof the portfolios returns over time.Returns%10%5%15%20%

Affect of Perfectly Negatively Correlated ReturnsNumerical Example

Diversification PotentialThe 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.

Example of Portfolio Combinations and CorrelationPerfect Positive Correlation no diversification

Sheet1

AssetExpected ReturnStandard DeviationCorrelation Coefficient

A5.0%15.0%1

B14.0%40.0%

Portfolio ComponentsPortfolio Characteristics

Weight of AWeight of BExpected ReturnStandard Deviation

100.00%0.00%5.00%15.0%

90.00%10.00%5.90%17.5%

80.00%20.00%6.80%20.0%

70.00%30.00%7.70%22.5%

60.00%40.00%8.60%25.0%

50.00%50.00%9.50%27.5%

40.00%60.00%10.40%30.0%

30.00%70.00%11.30%32.5%

20.00%80.00%12.20%35.0%

10.00%90.00%13.10%37.5%

0.00%100.00%14.00%40.0%

Example of Portfolio Combinations and CorrelationPositive Correlation weak diversification potential

Sheet1

AssetExpected ReturnStandard DeviationCorrelation Coefficient

A5.0%15.0%0.5

B14.0%40.0%

Portfolio ComponentsPortfolio Characteristics

Weight of AWeight of BExpected ReturnStandard Deviation

100.00%0.00%5.00%15.0%

90.00%10.00%5.90%15.9%

80.00%20.00%6.80%17.4%

70.00%30.00%7.70%19.5%

60.00%40.00%8.60%21.9%

50.00%50.00%9.50%24.6%

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%

Example of Portfolio Combinations and CorrelationNo Correlation some diversification potential

Sheet1

AssetExpected ReturnStandard DeviationCorrelation Coefficient

A5.0%15.0%0

B14.0%40.0%

Portfolio ComponentsPortfolio Characteristics

Weight of AWeight of BExpected ReturnStandard Deviation

100.00%0.00%5.00%15.0%

90.00%10.00%5.90%14.1%

80.00%20.00%6.80%14.4%

70.00%30.00%7.70%15.9%

60.00%40.00%8.60%18.4%

50.00%50.00%9.50%21.4%

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%

Example of Portfolio Combinations and CorrelationNegative Correlation greater diversification potential

Sheet1

AssetExpected ReturnStandard DeviationCorrelation Coefficient

A5.0%15.0%-0.5

B14.0%40.0%

Portfolio ComponentsPortfolio Characteristics

Weight of AWeight of BExpected ReturnStandard Deviation

100.00%0.00%5.00%15.0%

90.00%10.00%5.90%12.0%

80.00%20.00%6.80%10.6%

70.00%30.00%7.70%11.3%

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%

Example of Portfolio Combinations and CorrelationPerfect Negative Correlation greatest diversification potential

Sheet1

AssetExpected ReturnStandard DeviationCorrelation Coefficient

A5.0%15.0%-1

B14.0%40.0%

Portfolio ComponentsPortfolio Characteristics

Weight of AWeight of BExpected ReturnStandard Deviation

100.00%0.00%5.00%15.0%

90.00%10.00%5.90%9.5%

80.00%20.00%6.80%4.0%

70.00%30.00%7.70%1.5%

60.00%40.00%8.60%7.0%

50.00%50.00%9.50%12.5%

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%

Diversification of a Two Asset Portfolio Demonstrated Graphically

Impact of the Correlation CoefficientFigure 8-7 (see the next slide) illustrates the relationship between portfolio risk () and the correlation coefficientThe 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.

Expected Portfolio ReturnImpact of the Correlation Coefficient

Zero Risk PortfolioWe can calculate the portfolio that removes all risk.When = -1, then

Becomes:

An Exercise to Produce the Efficient Frontier Using Three AssetsRisk, Return and Portfolio Theory

An Exercise using T-bills, Stocks and Bonds

Sheet1

Base Data:StocksT-billsBonds

Expected Return(%)12.73382978726.15170212777.0078723404

Standard Deviation (%)0.1680.0420.102

Correlation Coefficient Matrix:

Stocks1-0.2160.048

T-bills-0.21610.380

Bonds0.0480.3801

Portfolio Combinations:

WeightsPortfolioPortfolio Variance Formula Results - Broken Down by Addend

CombinationStocksT-billsBondsExpected ReturnVarianceStandard DeviationVariance1st2nd3rd4th5th6th

1100.0%0.0%0.0%12.70.028316.8%0.02830.028265844900000

290.0%10.0%0.0%12.10.022615.0%0.02260.02289533440.00001740690-0.000273031800

380.0%20.0%0.0%11.40.017713.3%0.01770.01809014070.00006962740-0.000485389800

470.0%30.0%0.0%10.80.013411.6%0.01340.0138502640.00015666170-0.000637074100

560.0%40.0%0.0%10.10.00979.9%0.00970.01017570420.00027850970-0.000728084700

650.0%50.0%0.0%9.40.00678.2%0.00670.00706646120.00043517140-0.000758421600

740.0%60.0%0.0%8.80.00446.6%0.00440.00452253520.00062664680-0.000728084700

830.0%70.0%0.0%8.10.00285.3%0.00280.0025439260.0008529360-0.000637074100

920.0%80.0%-0.0%7.50.00184.2%0.00180.00113063380.00111403880-0.0004853898-0-0

1010.0%90.0%-0.0%6.80.00143.8%0.00140.00028265840.00140995540-0.0002730318-0-0

110.0%100.0%0.0%6.20.00174.2%0.001700.00174068570000

120.0%0.0%100.0%7.00.010410.2%0.0104000.0103997587000

130.0%10.0%90.0%6.90.00879.3%0.008700.00001740690.0084238045000.0002909556

140.0%20.0%80.0%6.80.00728.5%0.007200.00006962740.0066558456000.0005172544

150.0%30.0%70.0%6.80.00597.7%0.005900.00015666170.0050958818000.0006788965

160.0%40.0%60.0%6.70.00486.9%0.004800.00027850970.0037439131000.0007758817

170.0%50.0%50.0%6.60.00386.2%0.003800.00043517140.0025999397000.0008082101

180.0%60.0%40.0%6.50.00315.5%0.003100.00062664680.0016639614000.0007758817

190.0%70.0%30.0%6.40.00255.0%0.002500.0008529360.0009359783000.0006788965

200.0%80.0%20.0%6.30.00204.5%0.002000.00111403880.0004159903000.0005172544

210.0%90.0%10.0%6.20.00184.2%0.001800.00140995540.0001039976000.0002909556

22100.0%0.0%0.0%12.70.028316.8%0.02830.028265844900000

2390.0%0.0%10.0%12.20.023115.2%0.02310.022895334400.000103997600.00014781690

2480.0%0.0%20.0%11.60.018813.7%0.01880.018090140700.000415990300.00026278560

2570.0%0.0%30.0%11.00.015112.3%0.01510.01385026400.000935978300.00034490610

2660.0%0.0%40.0%10.40.012211.1%0.01220.010175704200.001663961400.00039417840

2750.0%0.0%50.0%9.90.010110.0%0.01010.007066461200.002599939700.00041060250

2840.0%0.0%60.0%9.30.00879.3%0.00870.004522535200.003743913100.00039417840

2930.0%0.0%70.0%8.70.00808.9%0.00800.00254392600.005095881800.00034490610

3020.0%0.0%80.0%8.20.00809.0%0.00800.001130633800.006655845600.00026278560

3110.0%0.0%90.0%7.60.00899.4%0.00890.000282658400.008423804500.00014781690

3233.3%33.3%33.3%8.60.00476.9%0.00470.00314064940.00019340950.0011555287-0.00033707630.000182490.0003592045

3325.0%50.0%25.0%8.00.00305.5%0.00300.00176661530.00043517140.0006499849-0.00037921080.00010265060.000404105

3450.0%25.0%25.0%9.70.00798.9%0.00790.00706646120.00010879290.0006499849-0.00037921080.00020530130.0002020525

3525.0%25.0%50.0%8.20.00497.0%0.00490.00176661530.00010879290.0025999397-0.00018960540.00020530130.000404105

3620.0%60.0%20.0%7.60.00234.8%0.00230.00113063380.00062664680.0004159903-0.00036404240.00006569640.0003879408

3760.0%20.0%20.0%10.30.010610.3%0.01060.01017570420.00006962740.0004159903-0.00036404240.00019708920.0001293136

Achievable PortfoliosResults Using only Three Asset ClassesThe plotted points are attainable portfolio combinations.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.

Chart1

12.7338297872

12.0756170213

11.4174042553

10.7591914894

10.1009787234

9.4427659574

8.7845531915

8.1263404255

7.4681276596

6.8099148936

6.1517021277

7.0078723404

6.9222553191

6.8366382979

6.7510212766

6.6654042553

6.579787234

6.4941702128

6.4085531915

6.3229361702

6.2373191489

12.7338297872

12.1612340426

11.5886382979

11.0160425532

10.4434468085

9.8708510638

9.2982553191

8.7256595745

8.1530638298

7.5804680851

8.6311347518

8.0112765957

9.6568085106

8.2253191489

7.6393617021

10.272212766

Efficient Set

Minimum Variance Portfolio

Standard Deviation of the Portfolio (%)

Portfolio Expected Return (%)

Attainable Portfolio Combinationsand Efficient Set of Portfolio Combinations

Data

Canadian Asset Classes

Historical Data Set

Annual market index returns: 1948 - 94

YearCanadian StocksT-billsLong BondsCPI

194812.250.4-0.088.88

194923.850.455.181.09

195051.690.511.745.91

195125.440.71-7.8910.66

19520.010.955.01-1.38

19532.561.5450

195439.371.6212.230

195527.681.220.130.47

195612.682.63-8.873.24

1957-20.583.767.941.79

195831.252.271.942.64

19594.594.39-5.071.29

19601.783.6612.191.27

196132.752.869.160.42

1962-7.093.815.031.67

196315.63.584.581.64

196425.433.736.162.02

19656.683.790.053.16

1966-7.074.89-1.053.45

196718.094.38-0.484.07

196822.456.222.143.91

1969-0.816.83-2.864.79

1970-3.576.8916.391.31

19718.013.8614.845.16

197227.383.438.114.91

19730.274.781.979.36

1974-25.937.68-4.5312.3

197518.487.058.029.52

197611.029.123.645.87

197710.717.649.049.45

197829.727.94.18.44

197944.7711.04-2.839.69

198030.1312.232.1811.2

1981-10.2519.11-2.0912.2

19825.5415.2745.829.23

198335.499.399.614.51

1984-2.3911.2116.93.77

198525.079.726.684.38

19868.959.3417.214.19

19875.888.21.774.12

198811.088.9411.33.96

198921.3711.9515.175.17

1990-14.813.284.325

199112.029.923.33.78

1992-1.436.6511.572.14

199332.555.6322.091.7

1994-0.184.76-7.390.23

Mean

Canadian StocksT-billsLong BondsCPICovariance (Rs,Rtbills)Covariance (Rs,bonds)Covariance (Rbills,Rbonds)

Mean12.76.27.04.5

Variance282.6617.41104.0012.50

Standard Deviation16.814.1710.203.54-15.178.2116.16

Correlation Coefficient-0.220.050.38

Number of terms47

12.250.4-0.088.882.78284481673.429323766440.767330421

23.850.455.181.09-63.3810913536-20.318940063410.4219836125

51.690.511.745.91-219.7791083748-205.216131552729.7197665912

25.440.71-7.8910.66-69.1431934812-189.294901765581.0697836125

0.010.955.01-1.3866.185572476225.420587596210.392336804

2.561.545046.918672476220.4277514269.2597091444

39.371.6212.230-120.7071892259139.0974812132-23.6651270258

27.681.220.130.47-73.7100594387-102.797850701733.919617655

12.682.63-8.873.240.18957247620.854702489855.917136804

-20.583.767.941.7979.6767575826-31.052742191-2.2293717067

31.252.271.942.64-71.874257311-93.837586871919.6719708465

4.594.39-5.071.2914.347002263598.360136532421.2776133997

1.783.6612.191.2727.2936809869-56.7641443187-12.9123185152

32.752.869.160.42-65.88727007743.0773535536-7.084163196

-7.093.815.031.6746.421504391139.20900461754.6315878678

15.63.584.581.64-7.3709360344-6.95869538256.2437644636

25.433.736.162.02-30.7463424174-10.76473155272.0532942508

6.683.790.053.1614.29734268942.121774830216.4324219104

-7.074.89-1.053.4524.9865341784159.57673227710.1666346763

18.094.38-0.484.07-9.4895381621-40.106318786813.2662793572

22.456.222.143.910.6635937528-47.2970762336-0.3324653237

-0.816.83-2.864.79-9.186750928133.6487833409-6.693356813

-3.576.8916.391.31-12.0370828429-152.96461240386.9268048891

8.013.8614.845.1610.8256107741-36.9976379357-17.9489036215

27.383.438.114.91-39.862512630116.1419492983-2.999663196

0.274.781.979.3617.096661837962.79118334096.9104602082

-25.937.68-4.5312.3-59.0898488004446.098332277-17.6333057492

18.487.058.029.525.16177247625.8158578090.9091921231

11.029.123.645.87-5.0528807153-28.504635808149.0364665912

10.717.649.049.45-3.0120615663-4.11268048893.0244112721

29.727.94.18.4429.6968852422-49.3936145315-5.0838270258

44.7711.04-2.839.69156.602342689-315.1677528293-48.0904504301

30.1312.232.1811.2105.7391043911-83.9864889995-29.3452461747

-10.2519.11-2.0912.2-297.8313126301209.1039492983-117.8929397918

5.5415.2745.829.23-65.5954828429-279.2078400634353.9005410593

35.499.399.614.5173.691257582659.21445993668.4264644636

-2.3911.2116.93.77-76.5008360344-149.60685495750.0373282933

25.079.726.684.3843.7724065188242.678715255869.8025687189

8.959.3417.214.19-12.0639764599-38.603114531532.5274219104

5.888.21.774.12-14.038684970635.8994854685-10.7287227705

11.088.9411.33.96-4.611370077-7.09844857411.967730421

21.3711.9515.175.1750.075087369970.489523766447.3264474423

-14.813.284.325-196.269340289774.0074195111-19.1599546854

12.029.923.33.78-2.6756466727-11.629806020861.0677474423

-1.436.6511.572.14-7.0578062472-64.61719963782.2732985061

32.555.6322.091.7-10.3381381621298.8700088728-7.8683780896

-0.184.76-7.390.2317.9722043911185.931672702620.0375495699

Efficient Set

Efficient Set Data

Portfolio ReturnPortfolio Standard DeviationEfficient Portfolio Combinations from Most Risky (1) to Least (16)

6.83.8

7.54.2

8.15.3

8.86.6

9.48.2

9.78.9

10.19.9

10.310.3

10.411.1

10.811.6

11.012.3

11.413.3

11.613.7

12.115.0

12.215.2

12.716.8

Efficient Set

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

Portfolio Risk (STD, %)

Portfolio Return (%)

Efficient Set

Attainable

Relative Weights

CombinationStocksT-BillsBondsPortfolio ReturnPortfolio Standard DeviationEfficient Portfolio Combinations from Most Risky (1) to Least (16)

1100%0%0%12.716.8

290%10%0%12.115.03

380%20%0%11.413.35

470%30%0%10.811.67

560%40%0%10.19.910

650%50%0%9.48.212

740%60%0%8.86.613

830%70%0%8.15.314

920%80%-0%7.54.215

1010%90%-0%6.83.8Minimum Variance Portfolio Combination

110%100%0%6.24.2

120%0%100%7.010.2

130%10%90%6.99.3

140%20%80%6.88.5

150%30%70%6.87.7

160%40%60%6.76.9

170%50%50%6.66.2

180%60%40%6.55.5

190%70%30%6.45.0

200%80%20%6.34.5

210%90%10%6.24.2

22100%0%0%12.716.81

2390%0%10%12.215.22

2480%0%20%11.613.74

2570%0%30%11.012.36

2660%0%40%10.411.18

2750%0%50%9.910.0

2840%0%60%9.39.3

2930%0%70%8.78.9

3020%0%80%8.29.0

3110%0%90%7.69.4

3233%33%33%8.66.9

3325%50%25%8.05.5

3450%25%25%9.78.911

3525%25%50%8.27.0

3620%60%20%7.64.8

3760%20%20%10.310.39

Attainable

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

Efficient Set

Minimum Variance Portfolio

Standard Deviation of the Portfolio (%)

Portfolio Expected Return (%)

Attainable Portfolio Combinationsand Efficient Set of Portfolio Combinations

Portfolio Combinations

Base Data:StocksT-billsBonds

Expected Return12.73382978726.15170212777.0078723404

Standard Deviation0.1680.0420.102

Correlation Coefficient Matrix:

Stocks1-0.2160.048

T-bills-0.2161.0000.380

Bonds0.0480.3801.000

Portfolio Combinations:

WeightsPortfolioPortfolio Variance Formula Results - Broken Down by Addend

CombinationStocksT-billsBondsExpected ReturnVarianceStandard DeviationVariance1st2nd3rd4th5th6th

1100.0%0.0%0.0%12.70.028316.8%0.02830.028265844900000

290.0%10.0%0.0%12.10.022615.0%0.02260.02289533440.00001740690-0.000273031800

380.0%20.0%0.0%11.40.017713.3%0.01770.01809014070.00006962740-0.000485389800

470.0%30.0%0.0%10.80.013411.6%0.01340.0138502640.00015666170-0.000637074100

560.0%40.0%0.0%10.10.00979.9%0.00970.01017570420.00027850970-0.000728084700

650.0%50.0%0.0%9.40.00678.2%0.00670.00706646120.00043517140-0.000758421600

740.0%60.0%0.0%8.80.00446.6%0.00440.00452253520.00062664680-0.000728084700

830.0%70.0%0.0%8.10.00285.3%0.00280.0025439260.0008529360-0.000637074100

920.0%80.0%-0.0%7.50.00184.2%0.00180.00113063380.00111403880-0.0004853898-0-0

1010.0%90.0%-0.0%6.80.00143.8%0.00140.00028265840.00140995540-0.0002730318-0-0

110.0%100.0%0.0%6.20.00174.2%0.001700.00174068570000

120.0%0.0%100.0%7.00.010410.2%0.0104000.0103997587000

130.0%10.0%90.0%6.90.00879.3%0.008700.00001740690.0084238045000.0002909556

140.0%20.0%80.0%6.80.00728.5%0.007200.00006962740.0066558456000.0005172544

150.0%30.0%70.0%6.80.00597.7%0.005900.00015666170.0050958818000.0006788965

160.0%40.0%60.0%6.70.00486.9%0.004800.00027850970.0037439131000.0007758817

170.0%50.0%50.0%6.60.00386.2%0.003800.00043517140.0025999397000.0008082101

180.0%60.0%40.0%6.50.00315.5%0.003100.00062664680.0016639614000.0007758817

190.0%70.0%30.0%6.40.00255.0%0.002500.0008529360.0009359783000.0006788965

200.0%80.0%20.0%6.30.00204.5%0.002000.00111403880.0004159903000.0005172544

210.0%90.0%10.0%6.20.00184.2%0.001800.00140995540.0001039976000.0002909556

22100.0%0.0%0.0%12.70.028316.8%0.02830.028265844900000

2390.0%0.0%10.0%12.20.023115.2%0.02310.022895334400.000103997600.00014781690

2480.0%0.0%20.0%11.60.018813.7%0.01880.018090140700.000415990300.00026278560

2570.0%0.0%30.0%11.00.015112.3%0.01510.01385026400.000935978300.00034490610

2660.0%0.0%40.0%10.40.012211.1%0.01220.010175704200.001663961400.00039417840

2750.0%0.0%50.0%9.90.010110.0%0.01010.007066461200.002599939700.00041060250

2840.0%0.0%60.0%9.30.00879.3%0.00870.004522535200.003743913100.00039417840

2930.0%0.0%70.0%8.70.00808.9%0.00800.00254392600.005095881800.00034490610

3020.0%0.0%80.0%8.20.00809.0%0.00800.001130633800.006655845600.00026278560

3110.0%0.0%90.0%7.60.00899.4%0.00890.000282658400.008423804500.00014781690

3233.3%33.3%33.3%8.60.00476.9%0.00470.00314064940.00019340950.0011555287-0.00033707630.000182490.0003592045

3325.0%50.0%25.0%8.00.00305.5%0.00300.00176661530.00043517140.0006499849-0.00037921080.00010265060.000404105

3450.0%25.0%25.0%9.70.00798.9%0.00790.00706646120.00010879290.0006499849-0.00037921080.00020530130.0002020525

3525.0%25.0%50.0%8.20.00497.0%0.00490.00176661530.00010879290.0025999397-0.00018960540.00020530130.000404105

3620.0%60.0%20.0%7.60.00234.8%0.00230.00113063380.00062664680.0004159903-0.00036404240.00006569640.0003879408

3760.0%20.0%20.0%10.30.010610.3%0.01060.01017570420.00006962740.0004159903-0.00036404240.00019708920.0001293136

Achievable Two-Security PortfoliosModern Portfolio TheoryThis line represents the set of portfolio combinations that are achievable by varying relative weights and using two non-correlated securities.

DominanceIt is assumed that investors are rational, wealth-maximizing and risk averse.If so, then some investment choices dominate others.

Investment ChoicesThe Concept of Dominance Illustrated

Efficient FrontierThe Two-Asset Portfolio CombinationsA is not attainableB,E lie on the efficient frontier and are attainableE is the minimum variance portfolio (lowest risk combination)C, D are attainable but are dominated by superior portfolios that line on the line above E

Efficient FrontierThe Two-Asset Portfolio CombinationsRational, risk averse investors will only want to hold portfolios such as B.

The actual choice will depend on her/his risk preferences.

DiversificationRisk, Return and Portfolio Theory

DiversificationWe 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 nave 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.

DiversificationDomestic Diversification

DiversificationDomestic Diversification

Sheet1

Table 8-3 Monthly Canadian Stock Portfolio Returns, January 1985 to December 1997

Number of Stocks in PortfolioAverage Monthly Portfolio Return (%)Standard Deviation of Average Monthly Portfolio Return (%)Ratio of Portfolio Standard Deviation to Standard Deviation of a Single StockPercentage of Total Achievable Risk Reduction

11.5113.471.000.00

21.5110.990.8227.50

31.529.910.7439.56

41.539.300.6946.37

51.528.670.6453.31

61.528.300.6257.50

71.517.950.5961.35

81.527.710.5764.02

91.527.520.5666.17

101.517.330.5468.30

141.516.800.5074.19

401.525.620.4287.24

501.525.410.4089.64

1001.514.860.3695.70

2001.514.510.3499.58

2221.514.480.33100.00

Source: Cleary, S. and Copp D. "Diversification with Canadian Stocks: How Much is Enough?" Canadian Investment Review (Fall 1999), Table 1.

Total Risk of an Individual AssetEquals the Sum of Market and Unique RiskThis 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.[8-19]

International DiversificationClearly, 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 Figure 8 -12 found on the following slide.)

DiversificationInternational Diversification

Summary and ConclusionsIn this chapter you have learned:How to measure different types of returnsHow to calculate the standard deviation and interpret its meaningHow 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.

Concept Review QuestionsRisk, Return and Portfolio Theory

Concept Review Question 1Ex Ante and Ex Post ReturnsWhat is the difference between ex ante and ex post returns?

CopyrightCopyright 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.

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