principles of managerial finance 9th edition chapter 6 risk and return

Principles of Managerial Finance

9th Edition

Chapter 6

Risk and Return

Learning Objectives• Understand the meaning and fundamentals of risk,

return, and risk aversion.

• Describe procedures for measuring the risk of a single

asset.

• Discuss the measurement of return and standard

deviation for a portfolio and the various types of

correlation that can exist between series of numbers.

• Understand the risk and return characteristics of a

portfolio in terms of correlation and diversification and

the impact of international assets on a portfolio.

Learning Objectives

• Review the two types of risk and the derivation and

role of beta in measuring the relevant risk of both an

individual security and a portfolio.

• Explain the capital asset pricing model (CAPM) and its

relationship to the security market line (SML), and

shifts in the SML caused by changes in inflationary

expectations and risk aversion.

Introduction• If everyone knew ahead of time how much a stock

would sell for some time in the future, investing would

be simple endeavor.

• Unfortunately, it is difficult -- if not impossible -- to

make such predictions with any degree of certainty.

• As a result, investors often use history as a basis for

predicting the future.

• We will begin this chapter by evaluating the risk and

return characteristics of individual assets, and end by

looking at portfolios of assets.

Risk Defined

• In the context of business and finance, risk is defined

as the chance of suffering a financial loss.

• Assets (real or financial) which have a greater chance

of loss are considered more risky than those with a

lower chance of loss.

• Risk may be used interchangeably with the term

uncertainty to refer to the variability of returns

associated with a given asset.

Return Defined• Return represents the total gain or loss on an investment.

• The most basic way to calculate return is as follows:

kt = Pt - Pt-1 + Ct

Pt-1

• Where kt is the actual, required or expected return

during period t, Pt is the current price, Pt-1 is the price

during the previous time period, and Ct is any cash

flow accruing from the investment

Chapter Example

year Stock A Stock B

1 6% 20%

2 12% 30%

3 8% 10%

4 -2% -10%

5 18% 50%

6 6% 20%

Return

Risk and Return

Single Financial Assets

Arithmetic Average

• The historical average (also called arithmetic average

or mean) return is simple to calculate.

• The accompanying text outlines how to calculate this

and other measures of risk and return.

• All of these calculations were discussed and taught in

your introductory statistics course.

• This slideshow will demonstrate the calculation of

these statistics using EXCEL.

Historical Return


Arithmetic Average


1 6% 20%

2 12% 30%

3 8% 10%

4 -2% -10%

5 18% 50%

6 6% 20%

Arithmetic

Average 8% 20%

Return


1 0.06 0.2

2 0.12 0.3

3 0.08 0.1

4 -0.02 -0.1

5 0.18 0.5

6 0.06 0.2

Arithmetic

Average =AVERAGE(B6:B11) =AVERAGE(C6:C11)

Return

Historical Return

What you type What you see


Variance

• Historical risk can be measured by the variability of its

returns in relation to its average.

• Variance is computed by summing squared deviations

and dividing by n-1.

• Squaring the differences ensures that both positive and

negative deviations are given equal consideration.

• The sum of the squared differences is then divided by

the number of observations minus one.

Historical Risk


Variance

Observed Observed Difference

year Return for - Mean Squared Variance

1 6% -2% 0.00040

2 12% 4% 0.00160

3 8% 0% -

4 -2% -10% 0.01000

5 18% 10% 0.01000

6 6% -2% 0.00040

Average 8.00% Sum of Dif 0.02240 0.00448

Stock A

Historical Risk


Variance

Observed Observed Difference

year Return for - Mean Squared Variance

1 20% 0% -

2 30% 10% 0.01000

3 10% -10% 0.01000

4 -10% -30% 0.09000

5 50% 30% 0.09000

6 20% 0% -

Average 20.00% Sum of Dif 0.20000 0.04000

Stock B

Historical Risk


Variance

Observed Observed

Return for Return for


1 6% 20%

2 12% 30%

3 8% 10%

4 -2% -10%

5 18% 50%

6 6% 20%

Average 8.00% 20.00%

Variance 0.45% 4.00%

Observed Observed



1 0.06 0.2

2 0.12 0.3

3 0.08 0.1

4 -0.02 -0.1

5 0.18 0.5

6 0.06 0.2


Variance =VARA(B4:B9) =VARA(C4:C9)

Historical Risk



Standard Deviation• Squaring the deviations makes the variance difficult to

interpret.• In other words, by squaring percentages, the resulting

deviations are in percent squared terms.• The standard deviation simplifies interpretation by

taking the square root of the squared percentages.• In other words, standard deviation is in the same units

as the computed average.• If the average is 10%, the standard deviation might be

20%, whereas the variance would be 20% squared.

Historical Risk


Standard Deviation

Observed Observed



1 6% 20%

2 12% 30%

3 8% 10%

4 -2% -10%

5 18% 50%

6 6% 20%

Average 8.00% 20.00%

Standard

Deviation 6.69% 20.00%

Observed Observed



1 0.06 0.2

2 0.12 0.3

3 0.08 0.1

4 -0.02 -0.1

5 0.18 0.5

6 0.06 0.2


Standard

Deviation =STDEV(B4:B9) =STDEV(C4:C9)

Historical Risk



Normal Distribution

Historical Risk

R-2 R-1 R+2R+1R

68%

95%95%


• Investors and analysts often look at historical returns

as a starting point for predicting the future.

• However, they are much more interested in what the

returns on their investments will be in the future.

• For this reason, we need a method for estimating

future or “ex-ante” returns.

• One way of doing this is to assign probabilities for

future states of nature and the returns that would be

realized if a particular state of nature would occur.

Expected Return & Risk


State Probability Stock A Stock B

Boom 30% 17% 29%

Normal 50% 12% 15%

Bust 20% 5% -2%

Expected Return 12.1% 15.8%

Expected Return


optimistic

Most likely

pessimistic

State Probability Stock A Stock B

Boom 0.3 0.17 0.29

Normal 0.5 0.12 0.15

Bust 0.2 0.05 -0.02

Expected Return =(B12*C12)+(B13*C13)+(B14*C14) =(B12*D12)+(B13*D13)+(B14*D14)

Expected Return

Single Financial AssetsExpected Return & Risk

State Pi Stock A pi[Ai - E(R)]2

Boom 0.30 0.170 0.00072

Normal 0.50 0.120 0.00000

Bust 0.20 0.050 0.00101

Expected Return 0.121

Variance = Sum of pi[Ai - E(R)]2 0.00173

Standard Deviation = (Var)1/2 0.04158

Risk, Variance, & Standard Deviation




Expected Return E(R) = piRi,

where pi = probability of the ith scenario, and

Ri = the forecasted return in the ith scenario.


Also, the variance of E(R) may be computed as:

and the standard deviation as:

22 )]([ RERipi 2)]([ RERipi 2

n

iiP

1

1



State Pi Stock A pi[Ai - E(R)]2

Boom 0.3 0.17 =B3*(C3-$C$6)^2

Normal 0.5 0.12 =B4*(C4-$C$6)^2

Bust 0.2 0.05 =B5*(C5-$C$6)^2

Expected Return =(B3*C3)+(B4*C4)+(B5*C5)

Variance = Sum of pi[Ai - E(R)]2 =SUM(D3:D5)

Standard Deviation = (Var)1/2 =(D7)^(1/2)



State Pi Stock B pi[Ai - E(R)]2

Boom 0.30 0.290 0.00523

Normal 0.50 0.150 0.00003

Bust 0.20 -0.020 0.00634

Expected Return 0.158

Variance = Sum of pi[Ai - E(R)]2 0.01160

Standard Deviation = (Var)1/2 0.10768



Risk-Averse

Risk-Indifferent

Risk-Seeking

Averse

Indifferent

Seeking

Risk x1 x2

Expected Return

Coefficient of Variation


• One problem with using standard deviation as a

measure of risk is that we cannot easily make risk

comparisons between two assets.

• The coefficient of variation overcomes this problem by

measuring the amount of risk per unit of return.

• The higher the coefficient of variation, the more risk per

return.

• Therefore, if given a choice, an investor would select

the asset with the lower coefficient of variation.

C.V.

State Pi Stock A Stock B

Boom 0.3 0.17 0.29

Normal 0.5 0.12 0.15

Bust 0.2 0.05 -0.02

Expected Return 0.121 0.158

Standard Deviation 0.04158 0.108

Coefficient of Variation 0.344 0.684

Coefficient of Variation

Single Financial AssetsCoefficient of Variation

CV = Standard Deviation / Expected Return

Portfolios of Assets• An investment portfolio is any collection or

combination of financial assets.

• If we assume all investors are rational and therefore risk averse, that investor will ALWAYS choose to invest in portfolios rather than in single assets.

• Investors will hold portfolios because he or she will diversify away a portion of the risk that is inherent in “putting all your eggs in one basket.”

• If an investor holds a single asset, he or she will fully suffer the consequences of poor performance.

• This is not the case for an investor who owns a diversified portfolio of assets.

Portfolios of Assets• Diversification is enhanced depending upon the extent

to which the returns on assets “move” together.

•This movement is typically measured by a statistic known as “correlation” as shown in Figure 6.5 and 6.6.

Portfolios of Assets

Negative correlation between F and G


year Stock A kAi-kA Stock B kBI-kB 相乘1 6% -2% 20% 0% 0

2 12% 4% 30% 10% 0.004

3 8% 0% 10% -10% 0

4 -2% -10% -10% -30% 0.03

5 18% 10% 50% 30% 0.03

6 6% -2% 20% 0% 0

0.064

Return

Recall Stocks A and B

kA=8% kB=20%

96.02.0067.0

0128.00128.0

5

064.0

BA

ABABAB

σAB=∑Pi[ RAi-E(RAi)]*[ RBi-E(RBi)]

= E{[RA-E(RA)]*[RB-E(RB)]}


Portfolio AB

Percent Percent Percent Percent Weighted

Year Weight Return Weight Return Return

1 50% 6 50% 20 13

2 50% 12 50% 30 21

3 50% 8 50% 10 9

4 50% -2 50% -10 -6

5 50% 18 50% 50 34

6 50% 6 50% 20 13

Weight A 50% Sum of Weighted Returns 84

Weight B 50% Portfolio Average Return 14

Stock A Stock B

Portfolio AB(50% in A, 50% in B)

kA=8% kB=20% Kp=0.5×8%+0.5×20%=14%



1 =B$12 6 =B$13 20 =(B6*C6)+(D6*E6)

2 =B$12 12 =B$13 30 =(B7*C7)+(D7*E7)

3 =B$12 8 =B$13 10 =(B8*C8)+(D8*E8)

4 =B$12 -2 =B$13 -10 =(B9*C9)+(D9*E9)

5 =B$12 18 =B$13 50 =(B10*C10)+(D10*E10)

6 =B$12 6 =B$13 20 =(B11*C11)+(D11*E11)

Weight A 0.5 Sum of Weighted Returns=SUM(F6:F11)

Weight B =(1-B12) Portfolio Average Return=F12/6


Where the contentsof cell B12 and B13 = 50% in this case.

Here are cellsB12 and B13


Investment Returns

-20

-10

0

10

20

30

40

50

60

1 2 3 4 5 6

Year

Pe

rce

nt Stock A

Stock B

Portfolio AB


Portfolios of AssetsPortfolio AB

(40% in A, 60% in B)

Portfolio AB



1 40% 6 60% 20 14.4

2 40% 12 60% 30 22.8

3 40% 8 60% 10 9.2

4 40% -2 60% -10 -6.8

5 40% 18 60% 50 37.2

6 40% 6 60% 20 14.4

Weight A 40% Sum of Weighted Returns 91.2

Weight B 60% Portfolio Average Return 15.2

Stock A Stock B

Changing theweights

Kp=0.4×8%+0.6×20%=15.2%

Portfolios of AssetsPortfolio AB

(20% in A, 80% in B)

Portfolio AB



1 20% 6 80% 20 17.2

2 20% 12 80% 30 26.4

3 20% 8 80% 10 9.6

4 20% -2 80% -10 -8.4

5 20% 18 80% 50 43.6

6 20% 6 80% 20 17.2

Weight A 20% Sum of Weighted Returns 105.6

Weight B 80% Portfolio Average Return 17.6

Stock A Stock B

And Again

Kp=0.2×8%+0.8×20%=17.6%

Weight A Return A (%) Return B (%) Return AB (%) SD-A (%) SD-B (%) SD-AB (%)

100% 8.0 20.0 8.0 6.7 20.0 6.7

80% 8.0 20.0 10.4 6.7 20.0 9.3

60% 8.0 20.0 12.8 6.7 20.0 11.9

40% 8.0 20.0 15.2 6.7 20.0 14.6

20% 8.0 20.0 17.6 6.7 20.0 17.3

0% 8.0 20.0 20 6.7 20.0 20.0

Portfolios of AssetsPortfolio Risk & Return

Summarizing changes in risk and return as the composition of the portfolio

changes.

ABBABBAAp WWWW 22222

0128.0,2.0,067.0 ABBA

ρAB*σA* σB = σAB , ρAB=0.96

若投資組合有 3 個資產，則

σp2 = WA

2*σA2+WB

2*σB2+WC

2*σC2+2WA*WB*σAB

2WA*WC*σAC+2WB*WC*σBC

若投資組合有 N 個資產，則

σp2= ∑ Wi

2*σi2+ ∑∑Wi*Wj*σij for i≠j ,If Wi=1/N

= ∑(1/N)2*σi2+ ∑∑ (1/N)*(1/N)*σij for i≠j

= (1/N)*∑[(1/N)*σi2]+[(N-1)/N]*∑∑{1/[N*(N-1)]*σij} for i≠j

= (1/N)* σi2 + [(N-1)/N]* σij

= (1/N)*(σi2- σij)+σij

σp2 =lim [(1/N)*(σi

2- σij)+ σij] = σij (N ∞)

σij= 系統風險 , (1/N)*(σi2- σij) = 非系統風險


Portfolio AB



1 50% 20 50% 0 10

2 50% 16 50% 4 10

3 50% 12 50% 8 10

4 50% 8 50% 12 10

5 50% 4 50% 16 10

6 50% 0 50% 20 10

Weight A 50% Sum of Weighted Returns 60

Weight B 50% Portfolio Average Return 10

Stock A Stock B

Portfolio Risk & Return(Perfect Negative Correlation)

%10%105.0%105.0%10%10 pBA kkk

1AB

Portfolios of AssetsPortfolio Risk & Return

(Perfect Negative Correlation)

Weight A Return A (%) Return B (%) Return AB (%) SD-A (%) SD-B (%) SD-AB (%)

100% 10.0 10.0 10.0 7.5 7.5 7.5

80% 10.0 10.0 10.0 7.5 7.5 4.5

60% 10.0 10.0 10.0 7.5 7.5 1.5

50% 10.0 10.0 10.0 7.5 7.5 0.0

40% 10.0 10.0 10.0 7.5 7.5 1.5

20% 10.0 10.0 10.0 7.5 7.5 4.5

0% 10.0 10.0 10.0 7.5 7.5 7.5

Notice that if we weight the portfoliojust right (50/50 in this case), we can

completely eliminate risk.

075.0,075.0,%,10%,10 BABBAApBA kWkWkkk BBAAp

AB

WW

1

Portfolios of AssetsPortfolio Risk

(Adding Assets to a Portfolio)

0 # of Stocks

Systematic (non-diversifiable) Risk

Unsystematic (diversifiable) Risk

Portfolio Risk (SD)

SDM

Firm-specific risk

market

Market risk

Portfolios of AssetsPortfolio Risk

(Adding Assets to a Portfolio)

0 # of Stocks

Portfolio of both Domestic and International Assets

Portfolio of Domestic Assets Only

Portfolio Risk (SD)

SDM

Portfolios of AssetsCapital Asset Pricing Model (CAPM)

• If you notice in the last slide, a good part of a portfolio’s

risk (the standard deviation of returns) can be

eliminated simply by holding a lot of stocks.

• The risk you can’t get rid of by adding stocks

(systematic) cannot be eliminated through

diversification because that variability is caused by

events that affect most stocks similarly.

• Examples would include changes in macroeconomic

factors such interest rates, inflation, and the business

cycle.


• In the early 1960s, researchers (Sharpe, Treynor, and Lintner) developed an asset pricing model that measures only the amount of systematic risk a particular asset has.

• In other words, they noticed that most stocks go down when interest rates go up, but some go down a whole lot more.

• They reasoned that if they could measure this variability -- the systematic risk -- then they could develop a model to price assets using only this risk.

•The unsystematic (company-related) risk is irrelevant because it could easily be eliminated simply by diversifying.


• To measure the amount of systematic risk an asset

has, they simply regressed the returns for the

“market portfolio” -- the portfolio of ALL assets --

against the returns for an individual asset.

• The slope of the regression line -- beta -- measures an

assets systematic (non-diversifiable) risk.

• In general, cyclical companies like auto companies

have high betas while relatively stable

companies, like public utilities,have low betas.

• Let’s look at an example to see how this works.


Market Stock B

Year Return Return

1 10 20

2 16 30

3 9 10

4 -4 -10

5 28 50

6 13 20• We will demonstrate the calculation using the

regression analysis feature in EXCEL.

Bi=σim/σm2 , Bp=∑ Wi*Bi

SUMMARY OUTPUT

Regression StatisticsMultiple R 0.993698R Square 0.987435Adjusted R Square 0.983246Standard Error 2.894265Observations 5

ANOVAdf SS MS F Significance F

Regression 1 1974.87 1974.87 235.7556 0.0006Residual 3 25.13031 8.376768Total 4 2000

CoefficientsStandard Error t Stat P-value Lower 95%Upper 95%Lower 95.0%Upper 95.0%Intercept -3.77513 2.018166 -1.87057 0.158163 -10.1978 2.64758 -10.1978 2.64758

10 1.917349 0.124873 15.35433 0.0006 1.519946 2.314753 1.519946 2.314753


This slide is the result of aregression using the Excel.The slope of the regression(beta) in this case is 1.92.Apparently, this stock hasa considerable amount of systematic risk

Market Stock B

Year Return Return

1 10 20

2 16 30

3 9 10

4 -4 -10

5 28 50

6 13 20

在 Excel 計算 β( 可使用統計函數 )

=SLOPE(C3:C8,B3:B8)

=1.92

Graphic Derivation of Beta for Asset B

-10

-5

0

5

10

15

20

25

30

-20 -10 0 10 20 30 40 50 60

Market Return (%)

As

se

t B

Re

turn

(%

)

Y

Predicted Y


• The required return for all assets is

composed of two parts: the risk-free rate and

a risk premium.

The risk-free rate (rf) is usually estimated from the return on US T-bills

The risk premium is a function of both market conditions and the asset

itself.


rf

fmi rrE )(

• The risk premium for a stock is composed of

two parts:

– The Market Risk Premium which is the

return required for investing in any risky asset

rather than the risk-free rate

– Beta, a risk coefficient which measures the

sensitivity of the particular stock’s return to

changes in market conditions.



• After estimating beta, which measures a specific asset’s systematic risk, relatively easy to estimate variables may be obtained to calculate an asset’s required return..

E(Ri) = Rf + Bi [E(Rm) - Rf], where

E(Ri) = an asset’s expected or required return,

RF = the risk free rate of return,

Bi = an asset or portfolio’s beta

E(Rm) = the expected return on the market portfolio.


Example

Calculate the required return for Federal Express assuming it has a beta of 1.25, the rate on US T-bills is 5.07%, and the expected return for the S&P 500 is 15%.

E(Ri) = 5.07 + 1.25 [15% - 5.07%]

E(Ri) = 17.48%


GraphicallyE(Ri)

beta

rf = 5.07%

1.251.0

15.0%

17.48%0507.015.0

1

)(斜率

fm rrE

Required Return

k = rf + (rm - Rf)B

Example: If the rate of return on U.S. T-blls is 5%, and the expected returnfor the S&P 500 is 15%, what would be the required returnfor Microsoft with a beta 1.5, and Florida Power and Light witha beta of 0.8?

MSFT FPL

rf 5.0% 5.0%

rm 15.0% 15.0%

B 1.5 0.8Answer k? 20.0% 13.0%


k%

B

20

15

10

5

1 2MSFTFPL

SML


k%

B

20

15

10

5

1 2

Shift due to change in market return from 12% to 15%

FPL MSFT

SML2

SML1


若市場所有投資人變得更risk averse ，則 E(rm)-rf 會增加，而 rf 不變，所以斜率增加，而截距不變

k%

B

20

15

10

5

1 2

Shift due to change in risk-free rate from 5% to 8%. Note that all returns

will increase by 3%

MSFTFPL

SML2SML1


若因 expected inflation 使得 rf 3% ，則 E(rm) 也會 3% ，所以 E(rm)-rf 不會變， i.e. 斜率不變，但截距

歷史資料 ( 假設有 N 期資料 )

單一資產：

平均報酬率： r = ∑ ri/N

變異數： σ2 = ∑(ri-r)2 / (N-1)

兩個資產 A 和 B ：

共變數： σAB = ∑(rAi-rA)*(rBi-rB) / (N-1)

相關係數： ρAB = σAB /(σA*σB) 且 ρ 介於正負 1 之間

預期未來 ( 假設未來有 N 種狀態 )

單一資產：平均報酬率： E (r) = ∑ Pi*ri 變異數： σ2 = ∑ [(ri-E(r)]2 / Pi

兩個資產 A 和 B ：共變數： σAB = ∑Pi * [(rAi- E (rA)]*[(rBi- E (rB)]

相關係數： ρAB = σAB / (σA*σB) 且 ρ 介於正負 1 之間

n 個資產所形成之投資組合：歷史資料 rP = ∑Wi* ri ,

預期未來 E(rP) = ∑ Wi*E(ri)

σp2 = ∑ Wi

2*σi2 + ∑ ∑ Wi * Wj * σij , i ≠ j

If n=2 →σp2 = W1

2*σ12+W2

2*σ22+2*W1*W2*σ12

σp=W1σ1+W2σ2 if ρ12=1

σp=W1 σ1- W2σ2 if ρ12= -1

If n=3 →σp2 = W1

2*σ12+W2

2*σ22+W3

2*σ32+2*W1*W2*σ12

+ 2*W1*W3*σ13+ 2*W2*W3*σ23

principles of managerial finance 9th edition chapter 6 risk and return

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