Risk

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CHAPTER 5CHAPTER 5Risk and Rates of Risk and Rates of

ReturnReturn Stand-alone riskStand-alone risk Portfolio riskPortfolio risk Risk & return: CAPM / SMLRisk & return: CAPM / SML

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Investment returnsInvestment returns

The rate of return on an investment can be The rate of return on an investment can be calculated as follows:calculated as follows:

(Amount received – Amount invested)(Amount received – Amount invested)

Return =Return = ________________________________________________

Amount investedAmount invested

For example, if $1,000 is invested and $1,100 is For example, if $1,000 is invested and $1,100 is returned after one year, the rate of return for returned after one year, the rate of return for this investment is: this investment is:

($1,100 - $1,000) / $1,000 = 10%.($1,100 - $1,000) / $1,000 = 10%.

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ReturnsReturns

Expected ReturnExpected Return - the return that an - the return that an investor expects to earn on an asset, given investor expects to earn on an asset, given its price, growth potential, etc. its price, growth potential, etc. kk

Required ReturnRequired Return - the return that an - the return that an investor requires on an asset given itsinvestor requires on an asset given its riskrisk. . kk

Realized Return – Realized Return – the return that was the return that was _ actually earned during some past period kactually earned during some past period k

Average Return - Average Return - _ the average annual the average annual realized return earned during the last n realized return earned during the last n years years _

symbolsymbol k k avgavg

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Probability distributionsProbability distributions

A listing of all possible outcomes, and A listing of all possible outcomes, and the probability of each occurrence.the probability of each occurrence.

Can be shown graphically.Can be shown graphically.

Expected Rate of Return

Rate ofReturn (%)100150-70

Firm X

Firm Y

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Expected ReturnExpected Return

State of Probability ReturnState of Probability Return

Economy (P) Economy (P) X X YY

Recession .20 4% Recession .20 4% -10%-10%

Normal .50 10% Normal .50 10% 14% 14%

Boom .30 14% Boom .30 14% 30% 30%

For each firm, the expected return on For each firm, the expected return on the stock is just a the stock is just a weighted averageweighted average::

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Recession .20 4% -Recession .20 4% -10%10%

Normal .50 10% Normal .50 10% 14%14%

Boom .30 14% Boom .30 14% 30%30%

For each firm, the expected return on the For each firm, the expected return on the stock is just a stock is just a weighted averageweighted average::

k = P(kk = P(k11)*k)*k11 + P(k + P(k22)*k)*k22 + ...+ P(k + ...+ P(knn)*kn)*kn

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Normal .50 10% Normal .50 10% 14%14%

Boom .30 14% Boom .30 14% 30%30%


k k (X) (X) = .2 (4%) + .5 (10%) + .3 (14%) = 10%= .2 (4%) + .5 (10%) + .3 (14%) = 10%

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Normal .50 10% Normal .50 10% 14%14%

Boom .30 14% Boom .30 14% 30%30%


k k (Y) (Y) = .2 (-10%)+ .5 (14%) + .3 (30%) = 14%= .2 (-10%)+ .5 (14%) + .3 (30%) = 14%

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Based only on your Based only on your expected return expected return

calculations, which calculations, which stock would you stock would you

prefer?prefer?

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RISK?RISK?

Have you considered

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RISKRISK

How to How to measuremeasure risk risk

(variance, standard (variance, standard deviation, beta)deviation, beta)

How to How to reducereduce risk risk

(diversification)(diversification)

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What is investment risk?What is investment risk? Risk Risk is an uncertain outcome or chance of is an uncertain outcome or chance of

an adverse outcome.an adverse outcome. Concerned with the riskiness of cash flows Concerned with the riskiness of cash flows

from financial assets.from financial assets. Two types of investment riskTwo types of investment risk

Stand-alone riskStand-alone risk Portfolio risk Portfolio risk

Investment risk is related to the probability Investment risk is related to the probability of earning a low or negative actual return.of earning a low or negative actual return.

The greater the chance of lower than The greater the chance of lower than expected or negative returns, the riskier expected or negative returns, the riskier the investment.the investment.

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Stand Alone Risk: Single AssetStand Alone Risk: Single Asset relevant risk measure is the relevant risk measure is the total risktotal risk

of expected cash flows measured by of expected cash flows measured by standard deviationstandard deviation . .

Portfolio Context: A group of Portfolio Context: A group of assets. Total risk consists of:assets. Total risk consists of: Diversifiable Risk (company-specific, Diversifiable Risk (company-specific,

unsystematic)unsystematic) Market Risk (non-diversifiable, Market Risk (non-diversifiable,

systematic)systematic)

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How do we Measure Risk?How do we Measure Risk?

A more scientific approach is to A more scientific approach is to examine the stock’s examine the stock’s STANDARD STANDARD DEVIATIONDEVIATION of returns. of returns.

Standard deviation is a measure of Standard deviation is a measure of the dispersion of possible the dispersion of possible outcomes. outcomes.

The greater the standard deviation, The greater the standard deviation, the greater the uncertainty, and the greater the uncertainty, and therefore , the greater the RISK.therefore , the greater the RISK.

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Standard DeviationStandard Deviation

n

i=1

= (k= (kii - k) P(k - k) P(kii))2

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Company X. Company X. Company X. Company X.

= (k= (kii - k) P(k - k) P(kii))2 n

i=1

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Company XCompany X

( 4% - 10%)( 4% - 10%)22 (.2) = 7.2 (.2) = 7.2

Company XCompany X

( 4% - 10%)( 4% - 10%)22 (.2) = 7.2 (.2) = 7.2

n

i=1= (k= (kii - k) P(k - k) P(kii))2

n

i=1

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Company X Company X

( 4% - 10%)( 4% - 10%)22 (.2) = 7.2 (.2) = 7.2

(10% - 10%)(10% - 10%)22 (.5) = 0 (.5) = 0

Company X Company X

( 4% - 10%)( 4% - 10%)22 (.2) = 7.2 (.2) = 7.2

(10% - 10%)(10% - 10%)22 (.5) = 0 (.5) = 0

= (k= (kii - k) P(k - k) P(kii))2 n

i=1

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Company XCompany X

( 4% - 10%)( 4% - 10%)22 (.2) = 7.2 (.2) = 7.2

(10% - 10%)(10% - 10%)22 (.5) = 0 (.5) = 0

(14% - 10%)(14% - 10%)22 (.3) (.3) = = 4.84.8

Company XCompany X

( 4% - 10%)( 4% - 10%)22 (.2) = 7.2 (.2) = 7.2

(10% - 10%)(10% - 10%)22 (.5) = 0 (.5) = 0

(14% - 10%)(14% - 10%)22 (.3) (.3) = = 4.84.8

= (k= (kii - k) P(k - k) P(kii))2 n

i=1

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Company X Company X

( 4% - 10%)( 4% - 10%)22 (.2) = 7.2 (.2) = 7.2

(10% - 10%)(10% - 10%)22 (.5) = 0 (.5) = 0

(14% - 10%)(14% - 10%)22 (.3) (.3) = = 4.84.8

Variance = 12Variance = 12

Company X Company X

( 4% - 10%)( 4% - 10%)22 (.2) = 7.2 (.2) = 7.2

(10% - 10%)(10% - 10%)22 (.5) = 0 (.5) = 0

(14% - 10%)(14% - 10%)22 (.3) (.3) = = 4.84.8


= (k= (kii - k) P(k - k) P(kii))2 n

i=1

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Company X Company X

( 4% - 10%)( 4% - 10%)22 (.2) = 7.2 (.2) = 7.2

(10% - 10%)(10% - 10%)22 (.5) = 0 (.5) = 0

(14% - 10%)(14% - 10%)22 (.3) (.3) = = 4.84.8


Stand. dev. = 12 = Stand. dev. = 12 = 3.46%3.46%

Company X Company X

( 4% - 10%)( 4% - 10%)22 (.2) = 7.2 (.2) = 7.2

(10% - 10%)(10% - 10%)22 (.5) = 0 (.5) = 0

(14% - 10%)(14% - 10%)22 (.3) (.3) = = 4.84.8



= (k= (kii - k) P(k - k) P(kii))2 n

i=1

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Company YCompany Y

(-10% - 14%)(-10% - 14%)22 (.2) = (.2) = 115.2115.2

= (k= (kii - k) P(k - k) P(kii))2 n

i=1

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Company Y Company Y

(-10% - 14%)(-10% - 14%)22 (.2) = (.2) = 115.2115.2

(14% - 14%)(14% - 14%)22 (.5) = (.5) = 00

= (k= (kii - k) P(k - k) P(kii))2 n

i=1

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Company Y Company Y

(-10% - 14%)(-10% - 14%)22 (.2) = (.2) = 115.2115.2

(14% - 14%)(14% - 14%)22 (.5) = (.5) = 00

(30% - 14%)(30% - 14%)22 (.3) (.3) = = 76.876.8

= (k= (kii - k) P(k - k) P(kii))2 n

i=1

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Company YCompany Y

(-10% - 14%)(-10% - 14%)22 (.2) = 115.2 (.2) = 115.2

(14% - 14%)(14% - 14%)22 (.5) = 0 (.5) = 0

(30% - 14%)(30% - 14%)22 (.3) (.3) = = 76.8 76.8


= (k= (kii - k) P(k - k) P(kii))2 n

i=1

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Company yCompany y

(-10% - 14%)(-10% - 14%)22 (.2) = 115.2 (.2) = 115.2

(14% - 14%)(14% - 14%)22 (.5) = 0 (.5) = 0

(30% - 14%)(30% - 14%)22 (.3) (.3) = = 76.8 76.8



= (k= (kii - k) P(k - k) P(kii))2 n

i=1

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Which stock would you prefer?Which stock would you prefer?

How would you decide?How would you decide?

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Company

Company X Y

Expected Return 10% 14%

Standard Deviation 3.46% 13.86%

Company

Company X Y

Expected Return 10% 14%

Standard Deviation 3.46% 13.86%

SummarySummarySummarySummary

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It depends on your tolerance for risk! It depends on your tolerance for risk!

Remember there’s a tradeoff between risk and return.Remember there’s a tradeoff between risk and return.

Return

Risk

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Coefficient of variationCoefficient of variation

Coefficient of variation (CV): A Coefficient of variation (CV): A standardized measure of dispersion about standardized measure of dispersion about the expected value, that shows the the expected value, that shows the amount of risk per unit of return.amount of risk per unit of return.

r̂

return Expected

deviation Standard CV

s

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Portfolio construction:Portfolio construction:Risk and returnRisk and return

Assume a two-stock portfolio is created with Assume a two-stock portfolio is created with $50,000 invested in both HT and Collections.$50,000 invested in both HT and Collections.

Expected return of a portfolio is a Expected return of a portfolio is a weighted average of each of the weighted average of each of the component assets of the portfolio.component assets of the portfolio.

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Calculating portfolio expected Calculating portfolio expected returnreturn

9.6% (1.7%) 0.5 (17.4%) 0.5 k

kw k

:average weighted a is k

p

^

n

1i

i

^

ip

^

p

^

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PortfoliosPortfolios

Combining several Combining several securities in a securities in a portfolioportfolio can can actually actually reducereduce overall risk. overall risk.

How does this work?How does this work?

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DiversificationDiversification

Investing in Investing in more than onemore than one security to security to reduce riskreduce risk..

If two stocks are If two stocks are perfectly perfectly positivelypositively correlated correlated, , diversification has diversification has no effectno effect on on risk.risk.

If two stocks are If two stocks are perfectly perfectly negativelynegatively correlated correlated, the , the portfolio is portfolio is perfectlyperfectly diversified. diversified.

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Some risk can be diversified Some risk can be diversified away and some can not.away and some can not.

Market RiskMarket Risk is also called is also called Nondiversifiable risk. Nondiversifiable risk. This This type of risk can not be type of risk can not be diversified away.diversified away.

Firm-Specific riskFirm-Specific risk is also called is also called diversifiable riskdiversifiable risk. This type of . This type of risk can be reduced through risk can be reduced through diversification.diversification.

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Market RiskMarket Risk

Unexpected changes in Unexpected changes in interest rates.interest rates.

Unexpected changes in cash Unexpected changes in cash flows due to tax rate changes, flows due to tax rate changes, foreign competition, and the foreign competition, and the overall business cycle.overall business cycle.

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Firm-Specific RiskFirm-Specific Risk

A company’s labor force goes A company’s labor force goes on strike.on strike.

A company’s top management A company’s top management dies in a plane crash.dies in a plane crash.

A huge oil tank bursts and A huge oil tank bursts and floods a company’s floods a company’s production area.production area.

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Portfolio RiskPortfolio Risk

# Stocks in Portfolio10 20 30 40 2,000+

Diversifiable Risk

Market Risk

20

0

Stand-Alone Risk, sp

sp (%)35

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Investor Attitude Investor Attitude Towards RiskTowards Risk

Investors are assumed to be Investors are assumed to be risk averserisk averse.. Risk aversion – assumes investors dislike Risk aversion – assumes investors dislike

risk and require higher rates of return to risk and require higher rates of return to encourage them to hold riskier securities.encourage them to hold riskier securities.

Risk premium – the difference between Risk premium – the difference between the return on a risky asset and a riskless the return on a risky asset and a riskless asset, which serves as compensation for asset, which serves as compensation for investors to hold riskier securities.investors to hold riskier securities.

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Investor attitude towards Investor attitude towards riskrisk

Risk aversion – assumes investors Risk aversion – assumes investors dislike risk and require higher dislike risk and require higher rates of return to encourage them rates of return to encourage them to hold riskier securities.to hold riskier securities.

Risk premium – the difference Risk premium – the difference between the return on a risky between the return on a risky asset and less risky asset, which asset and less risky asset, which serves as compensation for serves as compensation for investors to hold riskier investors to hold riskier securities.securities.

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Capital Asset Pricing Model Capital Asset Pricing Model (CAPM)(CAPM)

Model based upon concept that a Model based upon concept that a stock’s required rate of return is equal stock’s required rate of return is equal to the risk-free rate of return plus a risk to the risk-free rate of return plus a risk premium that reflects the riskiness of premium that reflects the riskiness of the stock after diversification.the stock after diversification.

Primary conclusion: The relevant Primary conclusion: The relevant riskiness of a stock is its contribution to riskiness of a stock is its contribution to the riskiness of a well-diversified the riskiness of a well-diversified portfolio.portfolio.

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Well-diversified PortfolioWell-diversified Portfolio Large Portfolio (10-15 assets) eliminates Large Portfolio (10-15 assets) eliminates

diversifiable risk for the most part.diversifiable risk for the most part. Interested in Interested in Market RiskMarket Risk which is the which is the

risk that cannot be diversified away.risk that cannot be diversified away. The relevant risk measure is The relevant risk measure is BetaBeta which which

measures the riskiness of an individual measures the riskiness of an individual asset in relation to the market portfolio.asset in relation to the market portfolio.

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Failure to diversifyFailure to diversify

If an investor chooses to hold a one-stock If an investor chooses to hold a one-stock portfolio (exposed to more risk than a portfolio (exposed to more risk than a diversified investor), would the investor be diversified investor), would the investor be compensated for the risk they bear?compensated for the risk they bear? NO!NO! Stand-alone risk is not important to a well-Stand-alone risk is not important to a well-

diversified investor.diversified investor. Rational, risk-averse investors are concerned Rational, risk-averse investors are concerned

with with σσpp, which is based upon market risk., which is based upon market risk. There can be only one price (the market There can be only one price (the market

return) for a given security.return) for a given security. No compensation should be earned for No compensation should be earned for

holding unnecessary, diversifiable risk.holding unnecessary, diversifiable risk.

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BetaBeta

Beta: a measure of market risk.Beta: a measure of market risk.Measures a stock’s market risk, and Measures a stock’s market risk, and shows a stock’s volatility relative to the shows a stock’s volatility relative to the market. market.

ororIt’s a measure of the It’s a measure of the “sensitivity”“sensitivity” of an individual stock’s returns of an individual stock’s returns to changes in the market.to changes in the market.

Indicates how risky a stock is if the stock Indicates how risky a stock is if the stock is held in a well-diversified portfolio.is held in a well-diversified portfolio.

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A firm that has a A firm that has a beta = 1beta = 1 has has average market risk. The stock is average market risk. The stock is no more or less volatile than the no more or less volatile than the market.market.

A firm with a A firm with a beta > 1beta > 1 is more is more volatile than the market (ex: volatile than the market (ex: computer firms).computer firms).

A firm with a A firm with a beta < 1beta < 1 is less volatile is less volatile than the market (ex: utilities).than the market (ex: utilities).

The market’s beta is 1The market’s beta is 1

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What is the market risk What is the market risk premium?premium?

Additional return over the risk-free rate Additional return over the risk-free rate needed to compensate investors for needed to compensate investors for assuming an average amount of risk.assuming an average amount of risk.

Its size depends on the perceived risk Its size depends on the perceived risk of the stock market and investors’ of the stock market and investors’ degree of risk aversion.degree of risk aversion.

Varies from year to year, but most Varies from year to year, but most estimates suggest that it ranges estimates suggest that it ranges between 4% and 8% per year.between 4% and 8% per year.

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RequiredRequired

rate of rate of

returnreturn==

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RequiredRequired

rate of rate of

returnreturn==

Risk-freeRisk-freerate ofrate ofreturnreturn

++

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RequiredRequired

rate of rate of

returnreturn==


++RiskRisk

PremiumPremium

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RequiredRequired

rate of rate of

returnreturn==


++RiskRisk

PremiumPremium

MarketMarketRiskRisk

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RequiredRequired

rate of rate of

returnreturn==


++RiskRisk

PremiumPremium


Firm-specificFirm-specificRiskRisk

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== ++

RequiredRequired

rate of rate of

returnreturn


RiskRiskPremiumPremium


Firm-specificFirm-specificRiskRisk

can be diversifiedcan be diversifiedawayaway

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The CAPM equation:The CAPM equation:

kkjj = k = krfrf + + jj (k (kmm - k - krfrf))

where:where:

kkjj = the Required Return on = the Required Return on security j,security j,

kkrfrf = the risk-free rate of = the risk-free rate of interest,interest,

jj = the beta of security j, and = the beta of security j, and

kkmm = the return on the market = the return on the market index.index.

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This linear relationship This linear relationship between risk and required between risk and required

return is known as the return is known as the Capital Capital Asset Pricing ModelAsset Pricing Model (CAPM). (CAPM).

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RequiredRequired

rate of rate of

returnreturn

Beta

Let’s try to graph thisrelationship!

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RequiredRequired

rate of rate of

returnreturn

Risk-freeRisk-freerate ofrate ofreturnreturn(6%)(6%)

Beta

12%12% .

11

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RequiredRequired

rate of rate of

returnreturn


Beta

12%12% .

11

securitysecuritymarketmarket

line line (SML)(SML)

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RequiredRequired

rate of rate of

returnreturn


BetaBeta

12%12% .

11

SMLSML

00

Is there a risklessIs there a riskless(zero beta) security?(zero beta) security?

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RequiredRequired

rate of rate of

returnreturn

Beta

12%12% .

11

SMLSML

00

Is there a risklessIs there a riskless(zero beta) security?(zero beta) security?

TreasuryTreasurysecurities aresecurities are

as close to risklessas close to risklessas possible. as possible. Risk-freeRisk-free

rate ofrate ofreturnreturn(6%)(6%)

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Can the beta of a security Can the beta of a security be negative?be negative?

Yes, if the correlation between Stock i Yes, if the correlation between Stock i and the market is negative (i.e., and the market is negative (i.e., ρρi,mi,m < < 0).0).

If the correlation is negative, the If the correlation is negative, the regression line would slope downward, regression line would slope downward, and the beta would be negative.and the beta would be negative.

However, a negative beta is highly However, a negative beta is highly unlikely.unlikely.

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Factors that change the Factors that change the SMLSML

What if investors raise inflation expectations What if investors raise inflation expectations by 3%, what would happen to the SML?by 3%, what would happen to the SML?

SML1

ki (%)SML2

0 0.5 1.0 1.5

1815

11 8

I = 3%

Risk, βi

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Factors that change the Factors that change the SMLSML

What if investors’ risk aversion increased, What if investors’ risk aversion increased, causing the market risk premium to increase causing the market risk premium to increase by 3%, what would happen to the SML?by 3%, what would happen to the SML?

SML1

ki (%) SML2

0 0.5 1.0 1.5

1815

11 8

RPM = 3%

Risk, βi

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Verifying the CAPM Verifying the CAPM empiricallyempirically

The CAPM has not been verified The CAPM has not been verified completely.completely.

Statistical tests have problems that Statistical tests have problems that make verification almost impossible.make verification almost impossible.

Some argue that there are additional Some argue that there are additional risk factors, other than the market risk factors, other than the market risk premium, that must be risk premium, that must be considered.considered.

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More thoughts on the More thoughts on the CAPMCAPM

Investors seem to be concerned with both Investors seem to be concerned with both market risk and total risk. Therefore, the market risk and total risk. Therefore, the SML may not produce a correct estimate SML may not produce a correct estimate of kof kii..

kkii = k = kRF RF + (k+ (kM M – k– kRFRF) ) ββii + ??? + ???

CAPM/SML concepts are based upon CAPM/SML concepts are based upon expectations, but betas are calculated expectations, but betas are calculated using historical data. A company’s using historical data. A company’s historical data may not reflect investors’ historical data may not reflect investors’ expectations about future riskiness.expectations about future riskiness.

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Example:Example:

Suppose the Treasury bond Suppose the Treasury bond rate is rate is 6%6%,, the average return the average return on the S&P 500 index is on the S&P 500 index is 12%12%,, and Walt Disney has a beta of and Walt Disney has a beta of 1.21.2..

According to the According to the CAPMCAPM, what , what should be the should be the required rate of required rate of returnreturn on Disney stock? on Disney stock?

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kkjj = k = krfrf + (k + (kmm - k - krfrf))

kj = .06 + 1.2 (.12 - .06)kj = .06 + 1.2 (.12 - .06)

kj = .132 = kj = .132 = 13.2%13.2%

According to the CAPM, According to the CAPM, Disney stock should be Disney stock should be priced to give a priced to give a 13.2%13.2% return.return.

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An example:An example:Equally-weighted two-stock Equally-weighted two-stock

portfolioportfolio Create a portfolio with 50% invested in Create a portfolio with 50% invested in

HT and 50% invested in Collections.HT and 50% invested in Collections. The beta of a portfolio is the weighted The beta of a portfolio is the weighted

average of each of the stock’s betas.average of each of the stock’s betas.

ββPP = w = wHTHT ββHTHT + w + wCollColl ββCollColl

ββPP = 0.5 (1.30) + 0.5 (-0.87) = 0.5 (1.30) + 0.5 (-0.87)

ββPP = 0.215 = 0.215

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Calculating portfolio required Calculating portfolio required returnsreturns

The required return of a portfolio is the The required return of a portfolio is the weighted average of each of the stock’s weighted average of each of the stock’s required returns.required returns.

kkPP = w = wHTHT k kHTHT + w + wCollColl k kCollColl

kkPP = 0.5 (17.1%) + 0.5 (1.9%) = 0.5 (17.1%) + 0.5 (1.9%)

kkPP = 9.5% = 9.5%

Or, using the portfolio’s beta, CAPM can be Or, using the portfolio’s beta, CAPM can be used to solve for expected return.used to solve for expected return.

kkPP = k = kRF RF + (k+ (kM M – k– kRFRF) ) ββPP

kkPP = 8.0% = 8.0% + (15.0%+ (15.0% – 8.0%) (0.215)– 8.0%) (0.215)

kkPP = 9.5% = 9.5%

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