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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
Single Financial Assets
Arithmetic Average
year Stock A Stock B
1 6% 20%
2 12% 30%
3 8% 10%
4 -2% -10%
5 18% 50%
6 6% 20%
Arithmetic
Average 8% 20%
Return
year Stock A Stock B
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
Single Financial Assets
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
Single Financial Assets
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
Single Financial Assets
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
Single Financial Assets
Variance
Observed Observed
Return for Return for
year Stock A Stock B
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
Return for Return for
year Stock A Stock B
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
Average =AVERAGE(B4:B9) =AVERAGE(C4:C9)
Variance =VARA(B4:B9) =VARA(C4:C9)
Historical Risk
What you type What you see
Single Financial Assets
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
Single Financial Assets
Standard Deviation
Observed Observed
Return for Return for
year Stock A Stock B
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
Return for Return for
year Stock A Stock B
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
Average =AVERAGE(B4:B9) =AVERAGE(C4:C9)
Standard
Deviation =STDEV(B4:B9) =STDEV(C4:C9)
Historical Risk
What you type What you see
Single Financial Assets
Normal Distribution
Historical Risk
R-2 R-1 R+2R+1R
68%
95%95%
Single Financial Assets
• 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
Single Financial Assets
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
Expected Return & Risk
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 & Risk
Single Financial Assets
Single Financial Assets
Expected Return E(R) = piRi,
where pi = probability of the ith scenario, and
Ri = the forecasted return in the ith scenario.
Expected Return & Risk
Also, the variance of E(R) may be computed as:
and the standard deviation as:
22 )]([ RERipi 2)]([ RERipi 2
n
iiP
1
1
Expected Return & Risk
Single Financial Assets
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)
Risk, Variance, & Standard Deviation
Single Financial Assets
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, Variance, & Standard Deviation
Expected Return & Risk
Risk-Averse
Risk-Indifferent
Risk-Seeking
Averse
Indifferent
Seeking
Risk x1 x2
Expected Return
Coefficient of Variation
Single Financial Assets
• 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
Portfolios of Assets
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)]}
Portfolios of Assets
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%
Portfolios of Assets
Year Weight Return Weight Return Return
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
Portfolio AB(50% in A, 50% in B)
Where the contentsof cell B12 and B13 = 50% in this case.
Here are cellsB12 and B13
Portfolios of Assets
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
Portfolio AB(50% in A, 50% in B)
Portfolios of AssetsPortfolio AB
(40% in A, 60% in B)
Portfolio AB
Percent Percent Percent Percent Weighted
Year Weight Return Weight Return Return
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
Percent Percent Percent Percent Weighted
Year Weight Return Weight Return Return
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
Portfolios of Assets
Portfolio AB
Percent Percent Percent Percent Weighted
Year Weight Return Weight Return Return
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.
Portfolios of AssetsCapital Asset Pricing Model (CAPM)
• 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.
Portfolios of AssetsCapital Asset Pricing Model (CAPM)
• 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.
Portfolios of AssetsCapital Asset Pricing Model (CAPM)
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
Portfolios of AssetsCapital Asset Pricing Model (CAPM)
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
Portfolios of AssetsCapital Asset Pricing Model (CAPM)
Portfolios of AssetsCapital Asset Pricing Model (CAPM)
Portfolios of AssetsCapital Asset Pricing Model (CAPM)
• 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.
Portfolios of AssetsCapital Asset Pricing Model (CAPM)
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.
Portfolios of AssetsCapital Asset Pricing Model (CAPM)
Portfolios of AssetsCapital Asset Pricing Model (CAPM)
• 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.
Portfolios of AssetsCapital Asset Pricing Model (CAPM)
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%
Portfolios of AssetsCapital Asset Pricing Model (CAPM)
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%
Portfolios of AssetsCapital Asset Pricing Model (CAPM)
k%
B
20
15
10
5
1 2MSFTFPL
SML
Portfolios of AssetsCapital Asset Pricing Model (CAPM)
k%
B
20
15
10
5
1 2
Shift due to change in market return from 12% to 15%
FPL MSFT
SML2
SML1
Portfolios of AssetsCapital Asset Pricing Model (CAPM)
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
Portfolios of AssetsCapital Asset Pricing Model (CAPM)
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