Chapter 7Chapter 7Expected Return Expected Return

and Riskand Risk

• Explain how expected return and risk for securities are determined.

• Explain how expected return and risk for portfolios are determined.

• Describe the Markowitz diversification model for calculating portfolio risk.

• Simplify Markowitz’s calculations by using the single-index model.

Learning ObjectivesLearning Objectives

• Involve uncertainty• Focus on expected returns

– Estimates of future returns needed to consider and manage risk

• Goal is to reduce risk without affecting returns– Accomplished by building a portfolio– Diversification is key

Investment DecisionsInvestment Decisions

• Risk that an expected return will not be realized

• Investors must think about return distributions, not just a single return

• Use probability distributions– A probability should be assigned to each

possible outcome to create a distribution– Can be discrete or continuous

Dealing with UncertaintyDealing with Uncertainty

• Expected value – The single most likely outcome from a

particular probability distribution– The weighted average of all possible return

outcomes– Referred to as an ex ante or expected

return

i

m

1iiprR)R(E

i

m

1iiprR)R(E

Calculating Expected ReturnCalculating Expected Return

• Variance and standard deviation used to quantify and measure risk– Measures the spread in the probability

distribution

– Variance of returns: 2 = (Ri - E(R))2pri

– Standard deviation of returns: =(2)1/2

– Ex ante rather than ex post relevant

Calculating RiskCalculating Risk

• Weighted average of the individual security expected returns– Each portfolio asset has a weight, w,

which represents the percent of the total portfolio value

– The expected return on any portfolio can be calculated as:

n

1iiip )R(Ew)R(E

n

1iiip )R(Ew)R(E

Portfolio Expected ReturnPortfolio Expected Return

• Portfolio risk is not simply the sum of individual security risks

• Emphasis on the risk of the entire portfolio and not on risk of individual securities in the portfolio

• Individual stocks are risky only if they add risk to the total portfolio

Portfolio RiskPortfolio Risk

• Measured by the variance or standard deviation of the portfolio’s return

– Portfolio risk is not a weighted average of the risk of the individual securities in the portfolio

22p 1

ii

wn

i

22

p 1

iiw

n

i

Portfolio RiskPortfolio Risk

• Assume all risk sources for a portfolio of securities are independent

• The larger the number of securities, the smaller the exposure to any particular risk– “Insurance principle”

• Only issue is how many securities to hold

Risk Reduction in PortfoliosRisk Reduction in Portfolios

• Random diversification– Diversifying without looking at relevant

investment characteristics– Marginal risk reduction gets smaller and

smaller as more securities are added

• A large number of securities is not required for significant risk reduction

• International diversification is beneficial

Risk Reduction in PortfoliosRisk Reduction in Portfolios

p %

35

20

0

Number of securities in portfolio10 20 30 40 ...... 100+

Total Portfolio Risk

Market Risk

Portfolio Risk and Portfolio Risk and DiversificationDiversification

• Act of randomly diversifying without regard to relevant investment characteristics

• 15 or 20 stocks provide adequate diversification

Random DiversificationRandom Diversification

p %

35

20

0

Number of securities in portfolio10 20 30 40 ...... 100+

Domestic Stocks onlyDomestic +

International Stocks

International DiversificationInternational Diversification

• Non-random diversification– Active measurement and

management of portfolio risk– Investigate relationships between

portfolio securities before making a decision to invest

– Takes advantage of expected return and risk for individual securities and how security returns move together

Markowitz DiversificationMarkowitz Diversification

• Needed to calculate risk of a portfolio:– Weighted individual security risks

• Calculated by a weighted variance using the proportion of funds in each security

• For security i: (wi i)2

– Weighted co-movements between returns• Return covariances are weighted using the

proportion of funds in each security• For securities i, j: 2wiwj ij

Measuring Co-Movements Measuring Co-Movements in Security Returnsin Security Returns

• Statistical measure of relative co-movements between security returns

mn = correlation coefficient between securities m and n mn = +1.0 = perfect positive correlation

mn = -1.0 = perfect negative (inverse) correlation

mn = 0.0 = zero correlation

Correlation CoefficientCorrelation Coefficient

• When does diversification pay?– Combining securities with perfect positive

correlation provides no reduction in risk• Risk is simply a weighted average of the

individual risks of securities

– Combining securities with zero correlation reduces the risk of the portfolio

– Combining securities with negative correlation can eliminate risk altogether

Correlation CoefficientCorrelation Coefficient

• Absolute measure of association – Not limited to values between -1 and +1– Sign interpreted the same as correlation– The formulas for calculating covariance and

the relationship between the covariance and the correlation coefficient are:

BAABAB

iBi,BA

m

1ii,AAB

pr)]R(ER)][R(ER[

BAABAB

iBi,BA

m

1ii,AAB

pr)]R(ER)][R(ER[

CovarianceCovariance

• Encompasses three factors– Variance (risk) of each security– Covariance between each pair of

securities– Portfolio weights for each security

• Goal: select weights to determine the minimum variance combination for a given level of expected return

Calculating Portfolio RiskCalculating Portfolio Risk

• Generalizations – The smaller the positive correlation

between securities, the better– As the number of securities increases:

• The importance of covariance relationships increases

• The importance of each individual security’s risk decreases

Calculating Portfolio RiskCalculating Portfolio Risk

• Two-Security Case:

2/12222 )2( ABBABBAAp wwww 2/12222 )2( ABBABBAAp wwww

)()( 2/1

1 11

22 jiwww ij

n

i

n

jji

n

iiiP

)()( 2/1

1 11

22 jiwww ij

n

i

n

jji

n

iiiP

• N-Security Case:

Calculating Portfolio RiskCalculating Portfolio Risk

• Markowitz full-covariance model– Requires a covariance between the returns

of all securities in order to calculate portfolio variance

– Full-covariance model becomes burdensome as number of securities in a portfolio grows

• n(n-1)/2 unique covariances for n securities

• Therefore, Markowitz suggests using an index to simplify calculations

Simplifying Markowitz Simplifying Markowitz CalculationsCalculations

• Relates returns on each security to the returns on a common stock index, such as the S&P/TSX Composite Index

• Expressed by the following equation:

iMiii eRR iMiii eRR

• Divides return into two components– a unique part, αi

– a market-related part, βiRMt

The Single-Index ModelThe Single-Index Model

measures the sensitivity of a stock to stock market movements

– If securities are only related in their common response to the market

• Securities covary together only because of their common relationship to the market index

• Security covariances depend only on market risk and can be written as:

2Mjiij 2Mjiij

The Single-Index ModelThe Single-Index Model