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Risk and Rates of Return

What does it mean to take risk when investing?

How are risk and return of an investment measured?

For what type of risk is an average investor rewarded?

How can investors reduce risk?

What actions do investors take when the return they require to purchase an investment is different from the return the investment is expected to produce?

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RISK AND RATES OF RETURN Definitions and General ConceptsProbability DistributionsExpected returnStandard deviation,

Risk AttitudesCoefficient of VariationPortfolio Risk and ReturnDiversificationRelevant riskBeta coefficients

Determining Return—Capital Asset Pricing ModelReal (Physical) Assets Versus Financial Assets

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What is Risk?

Dictionary definition—chance of loss In finance we define risk as the chance

that something other than what is expected occurs—that is, variability of returns

Risk can be considered “bad”—that is, when the results are worse than expected (lower-than-expected returns)—or “good”—that is, when the results are better than expected (higher-than-expected returns)

Dictionary definition—chance of loss In finance we define risk as the chance

that something other than what is expected occurs—that is, variability of returns

Dictionary definition—chance of loss In finance we define risk as the chance

that something other than what is expected occurs—that is, variability of returns

Dictionary definition—chance of loss

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Risk

Stand-alone risk—risk of an investment if it was held by itself, or alone

Portfolio risk—risk of an investment when it is combined in a portfolio with other investments

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Risk

We know that an investment is risky if more than one future outcome is possible—that is, there are two or more possible payoffs associated with the investmentA probability distribution summarizes each possible outcome along with the chance, or probability, that the outcome will occur

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Probability Distributions—Example

Economy Probability PayoffBooming 0.2 18.0%Normal 0.5 8.0Recession 0.3 -2.0

Economy Probability PayoffBooming 0.2 18.0%Normal 0.5 8.0Recession 0.3 -2.0

1.0

Risky Risk-FreeEconomy Probability Asset AssetBooming 0.2 18.0% 5.0%Normal 0.5 8.0 5.0Recession 0.3 -2.0 5.0

1.0

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Probability Distributions

Probability

Return (%)

0.1

0.2

0.3

0.4

0.5

-5 0 5 10 15-2 8 18

Probability

Return (%)

0.1

0.2

0.3

0.4

0.5

-5 0 5 10 15-2 8 18

Investment A

Investment B

Discrete Distributions Continuous Distributions

Risk-Free Asset

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Expected Return

Weighted average of the various possible outcomes based on the probability that each outcome will occur

Average of the outcomes if the action—for example, an investment—was continued over and over again and the probability for each outcome remained the same—that is, the probability distribution does not change

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Expected Return

n

1iii r Pr

nn2211 rrrr̂return ofrate Expected PrPrPr

ri = the result of outcome i

Pri = the probability that outcome i will occur

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Expected Return

Economy Probability, Pri Payoff, ri Pri x rBoom 0.2 18.0% 3.6%

Normal 0.5 8.0 4.0Recession 0.3 -2.0 -0.6

Boom 0.2 18.0% 3.6%Normal 0.5 8.0 4.0Recession 0.3 -2.0 -0.6

7.0%= r̂

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Measuring Stand-Alone RiskStandard Deviation,

Measures the tightness, or variability, of a set of outcomes, or a probability distribution

The tighter the distribution, the less the variability of the outcomes and the less risk associated with the event

Measures risk for a single investment—that is an investment held by itself (standing alone)

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Measuring Stand-Alone RiskStandard Deviation,

n

i2

i1=i

)r̂r( Prn

2n2

221

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2 )r̂ r()r̂ r()r̂ r(= PrPrPr

Variance, 2—measures the variability of outcomes

Standard deviation,

i2

i

n

1=i

2 ) r̂ - r( = σ = σ ∑ Pr

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Standard Deviation,

18.0%

ri – = ri– (ri– )2 x Pri (ri– )2Prir̂ r̂ r̂ r̂

18.0% – 7.0%18.0% – 7.0% =11.0%18.0% – 7.0% =11.0% 121.018.0% – 7.0% =11.0% 121.0 x 0.218.0% – 7.0% =11.0% 121.0 x 0.2 = 24.28.0 – 7.0 = 1.0 1.0 x 0.5 = 0.5

-2.0 – 7.0 = -9.0 81.0 x 0.3 = 24.38.0 – 7.0 = 1.0 1.0 x 0.5 = 0.5

-2.0 – 7.0 = -9.0 81.0 x 0.3 = 24.32 = 49.0

7.0% 49.0 σ

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Risk Attitudes

Risk Aversion—all else equal, risk averse investors prefer higher returns to lower returns as well as less risk to more risk

Risk averse investors demand higher returns for investments with higher risk

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Risk Aversion

0 Risk

Return

rRF

Risk-Free Return = rRF

Risk Premium = RP r = rRF + RP

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Coefficient of Variation

Measures the relationship between risk and return

Allows for comparisons among various investments that have different risks and different returns

rσ

ReturnRisk

Variationof

tCoefficienˆ

==

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Coefficient of Variation

Economy Probability ABoom 0.2 18.0%Normal 0.5 8.0Recession 0.3 -2.0

PayoffsEconomy Probability A B CBoom 0.2 18.0% -5.0% 55.0%Normal 0.5 8.0 7.0 14.0Recession 0.3 -2.0 15.0 -10.0

PayoffsEconomy Probability A B CBoom 0.2 18.0% -5.0% 55.0%Normal 0.5 8.0 7.0 14.0Recession 0.3 -2.0 15.0 -10.0

Expected return, 7.0%r̂

PayoffsEconomy Probability A B CBoom 0.2 18.0% -5.0% 55.0%Normal 0.5 8.0 7.0 14.0Recession 0.3 -2.0 15.0 -10.0

Expected return, 7.0% 7.0%r̂

PayoffsEconomy Probability A B CBoom 0.2 18.0% -5.0% 55.0%Normal 0.5 8.0 7.0 14.0Recession 0.3 -2.0 15.0 -10.0

Expected return, 7.0% 7.0% 15.0%r̂

PayoffsEconomy Probability A B CBoom 0.2 18.0% -5.0% 55.0%Normal 0.5 8.0 7.0 14.0Recession 0.3 -2.0 15.0 -10.0

Expected return, 7.0% 7.0% 15.0%

Standard deviation, 7.0% 6.9% 22.5%

r̂

PayoffsEconomy Probability A B CBoom 0.2 18.0% -5.0% 55.0%Normal 0.5 8.0 7.0 14.0Recession 0.3 -2.0 15.0 -10.0

Expected return, 7.0% 7.0% 15.0%

Standard deviation, 7.0% 6.9% 22.5%Coeff of variation, CV 1.00 0.99 1.50

r̂

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Portfolio Risk

By combining investments to form a portfolio, or collection of investments, diversification can be achievedWhen evaluated in a portfolio, the performance of a single investment is not very important, because some investments will perform better than expected while others will perform worse than expectedThe performance of the portfolio as a whole is important

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Portfolio Return

Expected return of a portfolio = weighted average of the expected returns of the individual investments in the portfolio

∑N

1=jjj

NN2211P

r̂w=

r̂w++r̂w+r̂w=r̂

wj = proportion of funds invested in Asset j

j Investmentfor return expected rj=ˆ

0.1wN

1jj

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Portfolio Return

r̂

PayoffsProbability A B

Boom 0.2 18.0% -5.0%Norm 0.5 8.0 7.0Recess0.3 -2.0 15.0

7.0% 7.0% 7.0% 6.9%CV 1.00 0.99

Payoffs PortfolioProbability A B wA=0.6; wB=0.4

0.2 18.0% -5.0%0.5 8.0 7.00.3 -2.0 15.0

7.0% 7.0% 7.0% 6.9%CV 1.00 0.99

Payoffs PortfolioProbability A B wA=0.6; wB=0.4

0.2 18.0% -5.0% 18(0.6) + (-5)(0.4)= 8.8

0.5 8.0 7.00.3 -2.0 15.0

7.0% 7.0% 7.0% 6.9%CV 1.00 0.99

Payoffs PortfolioProbability A B wA=0.6; wB=0.4

0.2 18.0% -5.0% 18(0.6) + (-5)(0.4)= 8.8

0.5 8.0 7.0 8(0.6) + 7(0.4) = 7.6

0.3 -2.0 15.07.0% 7.0%

7.0% 6.9%CV 1.00 0.99

Payoffs PortfolioProbability A B wA=0.6; wB=0.4

0.2 18.0% -5.0% 18(0.6) + (-5)(0.4)= 8.8

0.5 8.0 7.0 8(0.6) + 7(0.4) = 7.6

0.3 -2.0 15.0 (-2)(0.6) + 15(0.4)= 4.8

7.0% 7.0% 7.0% 6.9%CV 1.00 0.99

Payoffs PortfolioProbability A B wA=0.6; wB=0.4

0.2 18.0% -5.0% 18(0.6) + (-5)(0.4)= 8.8

0.5 8.0 7.0 8(0.6) + 7(0.4) = 7.6

0.3 -2.0 15.0 (-2)(0.6) + 15(0.4)= 4.8

7.0% 7.0% 7(0.6) + 7(0.4)= 7.0 7.0% 6.9%CV 1.00 0.99

1.5%27.0)0.3(4.827.0)0.5(7.627.0)0.2(8.8σ

Payoffs PortfolioProbability A B wA=0.6; wB=0.4

0.2 18.0% -5.0% 18(0.6) + (-5)(0.4)= 8.8

0.5 8.0 7.0 8(0.6) + 7(0.4) = 7.6

0.3 -2.0 15.0 (-2)(0.6) + 15(0.4)= 4.8

7.0% 7.0% 7(0.6) + 7(0.4)= 7.0 7.0% 6.9% 1.5CV 1.00 0.99 0.22

%0.7)8.4(3.0%)6.7(5.0%)8.8(2.0r̂

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Portfolio Risk—Diversification

When investments that are not perfectly correlated—that is, do not mirror each others’ movements on a relative basis—are combined to form a portfolio, the risk of the portfolio can be reduced (diversification)The amount of the risk reduction depends on how the investments in a portfolio are relatedThe smaller (greater) the positive (negative) relationship among the various investments included in a portfolio, the greater the diversificationDiversification—investing in a combination of stocks generally reduces risk overall

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Risk

Stand-alone risk

=

Stand-alone risk

= = total risk

= firm-specific risk

Stand-alone risk

= = total risk

Stand-alone risk

= = total risk

= firm-specific risk + market risk

Stand-alone risk

= = total risk

= firm-specific risk + market risk

= diversifiable

Stand-alone risk

= = total risk

= firm-specific risk + market risk

= diversifiable + nondiversifiable

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Firm-Specific Risk

Caused by actions that are specific to the firm—management decisions, labor characteristics, etc.The impact of this type of risk on the expected return associated with an investment is generally fairly random This risk component is often called unsystematic risk This risk is also called diversifiable risk, because this portion of total risk can be reduced in a portfolio of investments

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Market Risk

Results from movements in factors that affect the economy as a whole—interest rates, employment, etc.

This risk affects all companies, thus all investments; it is a system wide risk that cannot be diversified away

This risk is called systematic, or nondiversifiable, risk

Even though all investments are affected by systematic risk, they are not all affected to the same degree

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Relevant Risk

Risk that cannot be reduced or diversified awayRelevant risk = systematic, or market risk“Irrelevant” risk = firm-specific, or unsystematic risk, because this portion of total risk can be reduced through diversificationInvestors should not be rewarded for taking “irrelevant” risk—that is, for not diversifyingRisk premiums are based on the amount of systematic risk associated with an investment

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Relevant Risk

0 Risk (systematic)

Return

rRF

Risk-Free Return

Risk Premium based on

systematic risk

r = rRF + RP

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Concept of Beta

Market, or systematic, risk is measured by comparing the return on an investment with the return on the market in general, or an average stockThe market is very well diversified so that any movements should be the result on systematic risk only Beta coefficient, β—measures the relationship between an individual investment’s returns and the market’s returns, thus the systematic risk of the investment

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Concept of Beta

slope

Return on theMarket, r M

Return on the Stock, r j

..

..

.

..

.. .

.

..

.

..

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Portfolio Beta Coefficients

A portfolio’s beta, βp is a function of the betas of the individual

investments in the portfolio

A portfolio beta is the weighted average of the betas associated with the individual investments contained in the portfolio

N

1jjj

NN2211P

w

www

wj = % of total funds invested in asset jβj = asset j’s beta coefficient

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Portfolio Beta Coefficients—Example

Stock A $ 30,000Stock B 20,000Stock C 10,000Stock D 40,000

100,000

AmountInvestment Invested Beta, j

Stock A $ 30,000 2.0Stock B 20,000Stock C 10,000Stock D 40,000

100,000

Stock A $ 30,000 2.0Stock B 20,000 1.5Stock C 10,000Stock D 40,000

100,000

Stock A $ 30,000 2.0Stock B 20,000 1.5Stock C 10,000 1.0Stock D 40,000

100,000

Stock A $ 30,000 2.0Stock B 20,000 1.5Stock C 10,000 1.0Stock D 40,000 0.5

100,000

Stock A $ 30,000 2.0 0.3Stock B 20,000 1.5Stock C 10,000 1.0Stock D 40,000 0.5

100,000

Stock A $ 30,000 2.0 0.3Stock B 20,000 1.5 0.2Stock C 10,000 1.0 0.1Stock D 40,000 0.5 0.4

100,000

Stock A $ 30,000 2.0 0.3Stock B 20,000 1.5 0.2Stock C 10,000 1.0 0.1Stock D 40,000 0.5 0.4

100,000 1.0

AmountInvestment Invested Beta, j Weight, wj

j x wjStock A $ 30,000 2.0 0.3 0.6Stock B 20,000 1.5 0.2Stock C 10,000 1.0 0.1Stock D 40,000 0.5 0.4

100,000 1.0

Stock A $ 30,000 2.0 0.3 0.6Stock B 20,000 1.5 0.2 0.3Stock C 10,000 1.0 0.1 0.1Stock D 40,000 0.5 0.4 0.2

100,000 1.0

Stock A $ 30,000 2.0 0.3 0.6Stock B 20,000 1.5 0.2 0.3Stock C 10,000 1.0 0.1 0.1Stock D 40,000 0.5 0.4 0.2

100,000 1.0 p= 1.2

AmountInvestment Invested

AmountInvestment Invested Beta, j Weight, wj

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Relationship between Risk and Rates of Return

Market risk premium = RPM = rM - rRF

RPM = return associated with the riskiness of a portfolio that contains all the investments in the marketRPM is based on how risk averse investors are on average Because an investment’s beta coefficient indicates volatility relative to the market, we can use β to determine the risk premium for an investmentInvestment risk premium = RPInvest = RPM x βInvest

A more volatile investment—that is, an investment with a higher β—will earn a higher return than a less volatile investment

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Relationship between Risk and Rates of Return

Return = Risk-free rate + Risk Premium

rInvest = rRF + RPInvest

= rRF + ( RPM ) βInvest

= rRF + ( rM – rRF ) βInvest

Capital Asset Pricing Model (CAPM)

5.05.0

5.0

rRF = 5.0% rM = 9%

9.0 5.0

4.0

j = 1.5

1.5

1.5

6.0 = 11.0

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CAPM Graph—SML

0 Risk—Measured by

Return, %

rRF

Risk-Free Return

Risk Premium based on

1.0

rM

Security Market Line, SML

RPM

rj = rRF + RPMj

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CAPM—Inflation Effects

rRF2 = 8

0 Risk—Measured by

Return, %

rRF1 = 6

1.0

rM1 = 10RPM1 = 4

Risk-Free Return

Risk Premium based on

0 Risk—Measured by

Return, %

rRF1 = 6 Risk-Free Return

Risk Premium based on

1.0

rM1 = 10rRF2 = 8

rM2 = 12

r1= 6 + 4(1.5) = 12r2= 8 + 4(1.5) = 14

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CAPM—Changes in Risk Aversion

0 Risk—Measured by

Return, %

rRF = 6

1.0

rM1 = 10RPM1 = 4

Risk-Free Return

Risk Premium based on

rM2 = 11

Risk—Measured by 0

Return, %

rRF = 6

1.0

rM1 = 10

Risk-Free Return

Risk Premium based on RPM2 = 5

r1= 6 + 4(1.5) = 12.0r2= 6 + 5(1.5) = 13.5

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CAPM—Changes in Beta

0 Risk—Measured by

Return, %

rRF = 6Risk-Free

Return

Risk Premium based on

1.0

rM = 10rB1 = 12

1.5

RPM = 4

1.25

rB2 = 11

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Changes in Equilibrium Stock Prices

Stock prices are not constant due to changes in rRF, RPM, x, and so forth.

If the required rate of return, rs, and the

expected rate of return, , are not equal, then the price of the investment will change until .

sr̂

ss rr ˆ

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Risk and Rates of Return

What does it mean to take risk when investing?More than one outcome is possible

How are risk and return of an investment measured?Variability of its possible outcomes; greater

variability = greater risk

How can investors reduce risk? Risk can be reduced through diversification

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For what type of risk is an average investor rewarded? Investors should only be rewarded for risks

they must take

What actions do investors take when the return they require to purchase an investment is different from the return the investment is expected to produce? Investors will purchase a security only when its

expected return is greater than or equal to its required return

Risk and Rates of Return