11 Chapter Risk and Return Slides Developed by: Terry Fegarty Seneca College

1111Chapt

er

Chapt

er Risk and ReturnRisk and Return

Slides Developed by:

Terry FegartySeneca College

© 2006 by Nelson, a division of Thomson Canada Limited 2

Chapter 11 – Outline (1)

• Why Study Risk and Return? The General Relationship Between Risk and Return Portfolio Theory—Modern Thinking about Risk and

Return The Return on an Investment Returns, Expected and Required Risk—A Preliminary Definition

• Portfolio Theory Review of the Concept of a Random Variable The Return on a Share Investment as a Random

Variable Risk Redefined as Variability Risk Aversion


Chapter 11 – Outline (2)

Decomposing Risk—Systematic (Market) and Unsystematic (Business-Specific) Risk

Portfolios Diversification—How Portfolio Risk Is Affected When

Shares Are Added Measuring Market Risk—The Concept of Beta Using Beta—The Capital Asset Pricing Model (CAPM) The Security Market Line (SML) The Validity and Acceptance of the CAPM and SML


Why Study Risk and Return?

• Returns to equity investments (common shares) historically much higher than return to debt investments Equity returns have averaged more than 10% while

debt returns average between 6% and 7%• Inflation also averaged about 4.5% during same time period

• Returns on equity investments much more volatile than returns on debt instruments in short-run


Why Study Risk and Return?

• Since equity earns much higher return but with higher risk, it would be nice if we could invest and earn high return but reduce risk associated with such investments Investing in portfolios of securities can help manage

risk• Portfolio—collection of financial assets by investors

• We wish to capture the high average returns of equity investing while limiting associated risk as much as possible


The General Relationship Between Risk and Return • Risk in finance is often considered as

probability of losing some or all of money invested Generally investments that offer higher returns

involve higher risks

• Suppose you could invest in a share that would either return you 15% or a loss of everything (-100%) Also, suppose chance of losing everything is 1% and

chance of earning 15% is 99% Risk associated with this investment is 1% chance of

losing everything


The General Relationship Between Risk and Return• Investors more or less expect to receive

positive return but they realize there is risk associated with share investments and the chance that they can lose their money

• How much risk is associated with given level of return? Need to define risk in a measurable way

• The definition has to include all probabilities of loss

Have to relate that measurement to return


Portfolio Theory—Modern Thinking about Risk and Return

• Portfolio theory defines investment risk in a measurable way and relates it to expected level of return from an investment Has had major impact on practical investing

activities


Modern Portfolio Theory

Harry Markowitz William F. Sharpe Merton Miller

Modern portfolio theory was introduced by Harry Markowitz in 1952. Markowitz, Sharpe & Miller were co-recipients of the Nobel Prize in Economics in 1990 for their pioneering work in portfolio theory


The Return on an Investment

• The return is what investor receives divided by amount invested

• One-Year Investments The return on a debt investment is:

The return on a share investment is

Interest paid

Loan amountk

1 1 0

0

D +(P -P )

Pk


Returns, Expected and Required

• Expected return on a share is return investors feel is most likely to occur based on currently available information Anticipated return based on dividends

expected as well as expected future price No rational person makes investment without

some expectation of return


Returns, Expected and Required

• Required return on a share investment is minimum rate at which investors will purchase or hold shares based on their perceptions of its risk People will only invest if they believe expected return

is at least equal to required return• Different people have different levels of both expected and

required return• Significant investment in a company’s shares occurs only

if expected return exceeds required return for substantial number of investors


Risk—A Preliminary Definition

• Preliminary definition of investment risk —probability that return will be less than expected Definition includes both negative returns and

positive returns that are lower than expected


Risk—A Preliminary Definition

• Feelings About Risk Most people have negative feelings about

bearing risk (risk aversion) Risk averse investors prefer lower risk when

expected returns are equal Most people see a trade-off between risk

and return Risk isn't to be avoided, but higher risk

investments must offer higher expect return to encourage investment


Review of the Concept of a Random Variable

• In statistics, a random variable is the outcome of a chance process and has a probability distribution

•Probability distribution—a representation of all possible outcomes and probability of each

Discrete variables can take only specific variables

Continuous variables can take any value within a specified range


Review of the Concept of a Random Variable• The Mean or Expected Value—most

likely outcome for the random variable• For symmetrical probability

distributions, mean is the centre of the distribution

• Statistically, mean is weighted average of all possible outcomes

n

i ii=1

X = XP X


Review of the Concept of a Random Variable—Example

1.0000

0.06254

0.25003

0.37502

0.25001

0.06250

P(X)X

The mean of this distribution is 2, since it is a symmetrical distribution.

Q: If you toss a coin four times what is the chance of receiving heads (x)?

A: The mean of the distribution is:

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Figure 11.1: Discrete Probability Distribution


Review of the Concept of a Random Variable• Standard Deviation

Variability—how far a typical observation of the variable is likely to deviate from the mean

• There's is a great deal of difference in variability around the mean for different distributions

• Telephone poles don't vary much in height from pole to pole—actual pole heights are closely clustered around the mean

• Office buildings do vary a great deal in terms of height—widely dispersed around the mean

• Standard deviation indicates how far from the mean a typical observation is likely to fall


Review of the Concept of a Random Variable• Variance and Standard Deviation

Variance —the average squared deviation from the mean. Variance formula:

n 22

x i ii=1

Var X X X P X

n 2

X x i ii=1

SD X X P X

Standard deviation—the square root of the variance. SD formula:



1.00SD X =

1.00Var X =

0.250.0625424

0.250.2500113

0.000.3750002

0.250.25001-11

0.250.06254-20

(Xi – )2 x P(Xi)P(Xi)(Xi – )2 Xi X

Since the variance is 1.0, the standard

deviation is also 1.0.


A: The variance and standard deviation of the distribution is:

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X(Xi – ) X



• The Coefficient of Variation—the ratio of the standard deviation of a distribution to its mean A relative measure of variation

CV formula:

• For example, if the CV = 0.5, then the typical variation is 50% the size of the mean, or ½

XCVX



0.5CV =

1.00SDX =

0.250.0625424

0.250.2500113

0.000.3750002

0.250.25001-11

0.250.06254-20

(Xi – )2 x P(Xi)P(Xi)Xi X

The standard deviation is 1.0, the mean is 2, thus the CV is 0.5


A: The coefficient of variation of the distribution is:

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(Xi – ) X X(Xi – )2



• Continuous Random Variable Can take on any numerical value within some

range We talk about the probability of an actual

outcome being within a range of values rather than being an exact amount


Figure 11.2: Probability Distribution for a Continuous Random Variable


The Return on a Share Investment as a Random Variable• In portfolio theory, the return on a

share investment is considered a random variable Return is influenced by future price of shares

and expected dividends•Both variables uncertain

• Return is a continuous random variable with a low value of -100% but no limit to the high value


The Return on a Share Investment as a Random Variable• The mean of the distribution of returns is

the share's expected return• The variance and standard deviation

show how likely it is that an actual return will be some distance from the expected value Actual return in a distribution with a large

variance is likely to be different from the mean


Figure 11.3: The Probability Distribution of the Return on an Investment in Shares of X


Risk Redefined as Variability

• In portfolio theory risk is defined as variability in return (= standard deviation)

• A high-risk share has high probability of earning return that differs significantly from mean of distribution Low-risk share is more likely to earn return similar to

the expected return

• In practical terms, risk is probability that return will be less than expected


Figure 11.4: Probability Distributions With Large

and Small Variances


Figure 11.5: Investment Risk Viewed as Variability of Return Over Time

While both shares have the same expected return, the

high risk share has a greater variability in

returns.


Risk Aversion

• Risk aversion means investors prefer lower risk when expected returns are equal

• When expected returns are not equal, choice of investment depends on investor's tolerance for risk


Figure 11.6: Risk Aversion


Example 11.1: Risk Aversion

Q: Michael Okafor is looking for a single company in which he'll make a substantial investment. He's narrowed his search to two firms, Emera Inc. (a public traded utility) and Astro Tech Corp. (a new high-tech company).

Utility companies are low-risk shares, particularly if they are regulated monopolies

High tech firms are high-risk because new technical ideas can be enormously profitable, complete failures, or anything in-between

Michael has made a discrete estimate of the probability distribution of returns for each share as follows:

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Evaluate Michael’s options in terms of statistical concepts of risk and return.

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0.151300.0514

0.20300.1512

0.30150.6010

0.2000.158

0.15-100%0.056%

P(kA)kAP(kE)kE

Astro TechEmera Inc.



A: First calculate the expected return for each company--the mean of each distribution.

15.0%10.0%

130

30

15

0

-100%

kA

0.7

1.8

6.0

1.2

0.3%

kE* P(kE)

19.50.150.0514

6.00.200.1512

4.50.300.6010

0.00.200.158

-15.0%0.150.056%

kA* P(kA)P(kA)P(kE)kE

Astro TechEmera Inc.

Exa

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e


Example 11.1: Risk AversionE

xam

ple

1.7%SD kE =

2.8Var kE =

0.80.0516414

0.60.154212

0.00.600010

0.60.154-28

0.80.0516-4%6%

(kE – )2 x P(kE)P(kE)(kE – )2(kE – )kE Ek Ek Ek

A: Next, calculate the variance and standard deviation of the return on each of the company’s shares

Emera Inc.


Example 11.1: Risk AversionE

xam

ple

63.7SD kA =

4,058Var kA =

1,9840.1513,225115130

450.202251530

00.300015

450.20225-150

1,9840.1513,225-115%-100%

(kA – )2 x P(kA)P(kA)(kA – )2(kA – )kA Ak Ak Ak

E AE A

E A

σ σ1.7 63.7CV = = = 0.17 CV = = = 4.25

10.0 15.0k k

Finally, calculate the coefficient of variation for each share's return.

Astro Tech



A: If Michael considers only expected return, he’ll certainly choose Astro. However, with Emera his investment is relatively safe while with Astro there is a substantial chance he’ll lose everything.

No one but Michael can make the decision as to which investment he should choose. It depends on his degree of risk aversion.

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• Fundamental truth of investment world Returns on shares tend to move up and down

together• Not exactly together or proportionately



• Events and Conditions Causing Movement in Returns Some things influence all shares (market

risk)• Political news, inflation, interest rates, war, etc.

Some things influence only particular firms (business-specific risk)

• Earnings reports, unexpected death of key executive, etc. Some things affect all companies within an

industry• Labour dispute, shortage of raw material


Decomposing Risk—Market and Business-Specific Risk

Comparison of IBM, Boeing and the S&P500

020406080

100120140

10/1

0/20

00

11/1

0/20

00

12/1

0/20

00

1/10

/200

1

2/10

/200

1

3/10

/200

1

4/10

/200

1

5/10

/200

1

6/10

/200

1

7/10

/200

1

8/10

/200

1

9/10

/200

1

10/1

0/20

01

Date

Sto

ck

Pri

ce

02004006008001000120014001600

IBM Boeing S&P500

Ind

ex V

alu

e

Market reopens after World Trade

Center collapses


Decomposing Risk—Market and Business-Specific Risk

• Movement in Return as Risk Total movement in a share's return is the total risk

inherent in the share

• Separating Movement/Risk into Two Parts A share's risk can be separated into systematic or

market risk and unsystematic or business-specific risk


Portfolios

• A portfolio is an investor's total share holdings

• Risk and Return for a Portfolio Each share in portfolio has its own expected return

and its own risk Portfolios have their own risks and returns

• Return on a portfolio is a weighted average of returns of individual shares in portfolio

• The risk of a portfolio is variance or standard deviation of probability distribution of portfolio's return


Portfolios

• The Goal of the Investor/Portfolio Owner Goal of investors: to capture high average

returns of equities while avoiding as much risk as possible• Generally done by constructing diversified

portfolios to minimize portfolio risk for a given return

Investors should be concerned with how shares impact portfolio performance, not with shares' individual performance


Diversification—How Portfolio Risk Is Affected When Shares Are Added

• Diversification means adding different (diverse) shares to a portfolio Can reduce (but not eliminate) risk in

portfolio



• Business-Specific Risk and Diversification Business-specific risk—series of random

events that push returns of individual shares up or down• Effects cancel each other out when added

together over substantial number of shares• Can be diversified away

• Shares within the portfolio must be from fundamentally different industries


Diversification—How Portfolio Risk Is Affected When Shares Are Added• Market Risk and Diversification

If returns of all shares move up and down more or less together, not possible to reduce risk completely• Market risk can be reduced but never entirely

eliminated

The Portfolio• If we have portfolio that is as diversified as the

market, its return will move in tandem with the market



Risk

Business-Specific

Risk

Number of Securities

Portfolio Risk

Market Risk



• The Impact on Portfolio Risk of Adding New Shares If we add a share to portfolio which has

returns positively correlated with portfolio, it will generally add risk to portfolio

If we add a share that is negatively correlated with the portfolio, it will decrease risk of portfolio


Figure 11.7: Risk In and Out of a Portfolio



• The Risk of the New Additions By Themselves and in Portfolios Shares with equal stand-alone risk can have

opposite risk impacts on portfolio because of timing of variation in their returns

A share's risk in a portfolio sense is its market risk


Diversification—How Portfolio Risk Is Affected When Shares Are Added• How do we diversify to reduce market

risk in a portfolio? Add shares that move counter cyclically with

the market• Ex; gold mining shares• Unfortunately it's difficult to find many shares that

are negatively correlated with the market• However numerous shares exist that have returns

that are less than positively correlated with the market

• Adding these shares to the portfolio will generally reduce risk somewhat, but will not eliminate it


Correlation & Risk Reduction

B

AkA

σA σB

kBPerfect Negative Correlation

Perfect Positive Correlation

Less thanPerfect Correlation

To minimize portfolio risk, choose assets that have very low correlations with each other.


Diversification—How Portfolio Risk Is Affected When Shares Are Added• The Importance of Market Risk

Modern portfolio theory is based on the assumption that investors focus on portfolios rather than on individual shares• How shares affect portfolios depends only on

market risk

• Market Return Small investor with limited investment funds

may diversify by buying Index Participation Units (IPUs) or mutual funds


Measuring Market Risk—The Concept of Beta

• Share's beta (coefficient) measures its market risk It measures the variation of share's return

which accompanies market's variation in return

Beta does not measure business-specific risk



Business-Specific or Unsystematic

Risk

Business-Specific or Unsystematic

Risk

Market or Systematic

Risk

Diversifiable Non- Diversifiable

Total Risk /Standard Deviation

The market will compensate us for market risk, measured by

beta


Measuring Market Risk—The Concept of Beta• Developing Beta

Beta is developed by determining historical relationship between share's return and return on a market index, such as the S&P/TSX Composite Index

Share's characteristic line reflects the average relationship between its return and the market

•Beta is slope of characteristic line


Figure 11.8: The Determination of Beta



• Beta > 1.0 means the share tends to move more than the market

• A beta 0 - 1.0 means the share moves with the market, but less

• A beta < 0 means the share tends to move against the market Shares in gold mining companies are a real-world

example of negative beta shares

• Beta for a Portfolio Beta for portfolio is weighted average of betas of

individual shares within portfolio



Characteristic Line for IBM

y = 1.3037x + 0.0009

-0.2

-0.15

-0.1

-0.05

0

0.05

0.1

0.15

-0.06 -0.04 -0.02 0 0.02 0.04 0.06

Return on S&P500

Ret

urn

on

IBM

Characteristic line determined using data from Slide 38. IBM’s

beta



• Projecting Returns with Beta Knowing share's beta enables us to estimate

changes in its return given changes in market's return

However, betas are developed from historical data• May not be accurate if fundamental change in the

business environment occurs


Example 11.2: Measuring Market Risk—The Concept of Beta

Q: Conroy Corp. has a beta of 1.8 and is currently earning a return of 14%. The stock market in general is reacting negatively to a new crisis in the Middle East. Experts estimate that the return on an average share will drop from 12% to 8%.

Estimate the change in the return on Conroy shares and its new price.

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Example 11.2: Measuring Market Risk—The Concept of Beta

A: Beta represents the past average change in Conroy’s return relative to changes in the market’s return.

The new return can be estimated as

kConroy = 14% - 7.2% = 6.8%

Conroy ConroyConroy

M

Conroy

k kb or 1.8

k 4%

k = 7.2%

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Using Beta—The Capital Asset Pricing Model (CAPM)

• CAPM helps us determine how share prices are set in the market

• The CAPM's Approach People won't invest unless a share's expected return

is at least equal to their required return The CAPM attempts to explain how investors'

required returns are determined


Using Beta—The Capital Asset Pricing Model (CAPM)• Rates of Return, The Risk-Free Rate and Risk

Premiums The risk-free rate (kRF)—rate for which there is no

chance of receiving less than what is expected• Current rate of interest on 3-month Treasury bills

Investing in any other investment is risky; thus investors will require return greater than risk-free rate

• Investors want to be compensated for extra risk taken via rate known as the risk premium (kRP)


Using Beta—The Capital Asset Pricing Model (CAPM)• Rates of Return, The Risk-Free Rate and

Risk Premiums Required rate of return is risk-free rate

plus risk premium CAPM attempts to explain how risk premium

in required rates of return is determined


The Security Market Line (SML)

• SML proposes that required rates of return are determined by:

where: kx is the required return on stock X kRF is the risk-free rate

kM is the return on the market x is stock X’s beta coefficient

X RF M RF X

Market Risk Premium

Stock X's Risk Premium

k k k k b



• The SML

Market Risk Premium:• Risk premium for an investment in the market as a whole • Reflects investment community's level of risk aversion

Risk Premium for Stock X:• Beta for stock X times the risk premium of the market• Says that risk premium for a share is determined only by the

share's relationship with the market as measured by beta

X RF M RF X

Market Risk Premium

Stock X's Risk Premium

k k k k b


Figure 11.9: The Security Market Line



• The SML as a Portrayal of the Securities Market • The SML can be viewed as a straight line:

X RF M RF X

y = b + xm

k k k k b

Slope of the SML reflects general level of risk aversion

Vertical intercept of SML represents investment in risk-free short-term government securities

Given a stock’s beta, we can find its required return by moving up to the SML and left to the return axis



• The SML as a Line of Market Equilibrium

If, for every share on the SML, expected return equals required return, SML represents equilibrium

If share's expected return becomes less than required return

• Owners of share would want to sell while potential buyers would no longer be interested

• Share price would drop• Expected return would increase, driving it back toward

equilibrium



• Valuation Using Risk-Return Concepts SML allows us to calculate minimum required rate of

return for a share Return can then be used in the Gordon model to

determine intrinsic value for share

• The Impact of Management Decisions on share Prices Since managers can influence share's beta and

future growth rates, management's decisions impact share price


Example 11.5: The Security Market Line (SML)

Q: The Kelvin Company paid an annual dividend of $1.50 recently, and is expected to grow at 7% into the indefinite future. Short-term Treasury bills are currently yielding 6%, and the S&P/TSX Composite Index is yielding 10%. Kelvin shares are relatively volatile. Their return tends to move in response to political and economic changes about twice as much as does the return on the average stock. What should Kelvin sell for today?

A: The required rate of return using the SML is:kKelvin = 6 + (10 – 6)2.0 = 14%

Plugging this required rate of return along with the growth rate of 7% into the Gordon model gives us the estimated price:

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00

D 1 g $1.5 1.07P $22.93

k g .14 .07



Q: The Kelvin Company has identified a new field into which it can expand using technology it already possesses. The venture promises to increase the firm's growth rate to 9% from the current 7%. However, the project is new and unproven, so there's a chance it will fail and cause a considerable loss. As a result, there's some concern that the stock market won't react favorably to the additional risk. Management estimates that undertaking the venture will raise the firm's beta to 2.3 from its current level of 2.0. Should Kelvin undertake the new project if the firm’s current share price is $22.93?

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A: The new required rate of return will be:

kKelvin = 6 + (10 – 6)2.3 = 15.2%Plugging this rate of return along with the higher growth rate of 9% into the Gordon model gives us the new estimated price:

The new estimated price is higher than the $22.93 price before the project. Thus, the venture looks like a good idea.

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00

D 1 g $1.5 1.09P $26.37

k g .152 .09


The Security Market Line

• Adjustments to Changing Market Conditions Response to change in risk-free rate

• If all else remains the same, increase in the risk-free rate causes corresponding increase in market rate

• Share prices will fall Response to change in risk aversion

• Increase in risk aversion causes increase in risk premium and market return

• Share prices will fall


Figure 11.10: A Shift in the Security Market Line to Accommodate an Increase in the Risk-Free Rate


Figure 11.11: A Rotation of the Security Market Line to Accommodate a

Change in Risk Aversion


The Validity and Acceptance of the CAPM and SML• CAPM is abstraction of reality designed to help

make predictions Its simplicity has led to its popularity

• Relates risk and return in an easy-to-understand concept

• However, CAPM not universally accepted Continuing debate exists as to its relevance and

usefulness• Fama and French found no historical relationship between

returns on shares and their betas

Risk measured by CAPM (beta) is market risk only

Documents

11 Chapter Risk and Return Slides Developed by: Terry Fegarty Seneca College