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F520 Asset Valuation and Strategy. Overview Risk and Return. Overview of Market Participants and Financial Innovation. What Types of Risk does a Corporation or a Financial Intermediary Encounter?. Overview (Cont.). How can Financial Products or Intermediaries reduce these risks. - PowerPoint PPT Presentation
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F520 – Portfolio Concepts 1
F520 Asset Valuation and Strategy
Overview
Risk and Return
F520 – Portfolio Concepts 2
Overview of Market Participants and Financial Innovation
• What Types of Risk does a Corporation or a Financial Intermediary Encounter?
F520 – Portfolio Concepts 3
Overview (Cont.)
• How can Financial Products or Intermediaries reduce these risks
F520 – Portfolio Concepts 4
Risk and Return - Outline
• How is the return on an asset affected by the risk of the asset?
• How do we measure risk and return on an asset?– Unique Risk
(diversifiable, unsystematic, residual, or specific)
– Market Risk(undiversifiable, systematic, or covariance)
• Constructing Portfolios -- How do we measure risk and return on a portfolio of assets?
• Choosing Stocks -- Development of the Efficient Frontier and use of Indifference Curves
F520 – Portfolio Concepts 5
Outline - Cont.
• More on Systematic Risk Beta The Capital Asset Pricing Model (CAPM) Security Market Line (SML)
• Obtaining Estimates of Beta
• Uses of Beta
• Tests of the Capital Asset Pricing Line and Beta.
• Arbitrage Pricing Theory (APT), an alternative to CAPM
F520 – Portfolio Concepts 6
Measuring Risk - Single Period
PPPr
D
0
011
P1 = the market value at the end of the intervalP0 = the market value at the beginning of the intervalD = the cash distributions during the interval
F520 – Portfolio Concepts 7
Measuring Return - Multiple PeriodsArithmetic
• Assumes no reinvestment of cash flows at the end of each period
N
R
a
N
ii
R 1
^
F520 – Portfolio Concepts 8
Measuring Return - Multiple PeriodsGeometric
• Also referred to as Time-Weighted Rate of Return
• Assumes reinvestment of cash flows at the end of each period.
1)]1)...(1)(1)(1[( /1
321 N
pNpppt RRRRR
F520 – Portfolio Concepts 9
Measuring Return - Multiple PeriodsInternal Rate of Return
• Also referred to as Dollar-Weighted Rate of Return
• Allows additions and withdrawals
• When no further additions or withdrawals occur and all dividends are reinvested, the Geometric and the IRR will yield the same
ND
NN
DDD R
VC
R
C
R
C
R
CV
)1(...
)1()1()1( 33
22
11
0
F520 – Portfolio Concepts 10
Example of Return Calculations: Growth of$1 investment
assumingPeriod Price Dividend Return reinvestment
0 10 1.00 1 20 0 100% 2.00 2 10 -50% 1.00
IRR Cash FlowsArithmetic Return: 25.00% Cash flows Shares Cash flowsGeometric Return 0.00% for IRR no owned with for IRR withIRR without reinvestment 0.00% reinvestment reinvestment reinvestmentIRR with reinvestment 0.00% -10 1 -10
0 1.00 010 1.00 10
Comparing Return CalculationsWithout Dividend (Income) Cash Flows
F520 – Portfolio Concepts 11
Example of Return Calculations: Growth of$1 investment
assumingPeriod Price Dividend Return reinvestment
0 10 1.00 1 18 2 100% 2.00 2 9 -50% 1.00
IRR Cash FlowsArithmetic Return: 25.00% Cash flows Shares Cash flowsGeometric Return 0.00% for IRR no owned with for IRR withIRR without reinvestment 5.39% reinvestment reinvestment reinvestmentIRR with reinvestment 0.00% -10 1 -10
2 1.11 09 1.11 10
Comparing Return CalculationsWith Dividend (Income) Cash Flows
F520 – Portfolio Concepts 12
Measuring Total RiskVariance of actual returns
• Measures of the dispersion of returns
• Standard Deviation (STD)Standard deviation measures dispersion in percents
VarianceN tr r r
t
n
( )
1
1
2
1
Variance
F520 – Portfolio Concepts 13
Historical Returns, Standard Deviations, and Frequency Distributions: 1926-2009
• Frequency distribution is a histogram of yearly returns
Example Frequency Distribution
Goal: Select the lowest risk portfolio
• 0% stock, 100% bond
• 20% stock, 80% bond
• 40% stock, 60% bond
• 60% stock, 40% bond
• 80% stock, 20% bond
• 100% stock, 0% bond
F520 – Portfolio Concepts 15
F520 – Portfolio Concepts 16
Constructing Portfolios
• Investors seek to maximize the expected return from their investment given some level of risk, or
• Investors seek to minimize the risk they are exposed to given some target expected return.
F520 – Portfolio Concepts 17
Constructing PortfoliosPortfolio Return
• Expected Return of a Portfolio equals the weighted average return on the portfolio
Rp = wa * Ra + wb * Rb wa = weight of asset a
wb = weight of asset b
Ra = Expected return of asset a
Rb = Expected return of asset b
• General Formula
– Weights must add to 1w1 + w2 + ... + wn = 1
RwR ip
n
ii
1
F520 – Portfolio Concepts 18
Constructing PortfoliosPortfolio Variance
• Two Asset CaseVar(Rp) = Var(wa * Ra + wb * Rb )
• General Case
– for h g – since 12 = 21, each covariance term is included in this equation twice. i is the variance of asset i gh is the covariance between asset g and asset h
where
abbabbaa wwww 22222
ghg
G
hhg
G
g
G
gg www
111
2 2
hggh
G
hhg
G
g
G
gg pwww g
111
2 2
hg
gh
ghp
F520 – Portfolio Concepts 19
Portfolio VarianceUsing Correlation
• Correlation is the covariance standardized by the standard deviation of the two variables.– p = 1, perfect positive correlation
– p = -1, perfect negative correlation
– p = 0, no correlation
• Two Asset Case
• General Case
hggh
G
hhg
G
g
G
ggVAR pwwwR gp
111
2)(2
hg
gh
ghp
baabbabbaapxxxxRVAR p 2)( 2222
F520 – Portfolio Concepts 20
Efficient FrontierCorrelation = 1
10.0%
11.0%
12.0%
13.0%
14.0%
15.0%
16.0%
17.0%
0.0% 3.0% 6.0% 9.0% 12.0% 15.0% 18.0% 21.0%
Exp
ect
ed
Re
turn
Standard Deviation
Efficeint Frontier (Corr = 1)
Correlation = 1
Input Data A B
Return 12% 16%
Std. Dev. 10% 20%
Correlation 1.00
bbaa
bbaa
xx
xx
pxxxx
p
p
baabbabbaap
22
22222
2
F520 – Portfolio Concepts 21
Efficient FrontierCorrelation = -1
10.0%
11.0%
12.0%
13.0%
14.0%
15.0%
16.0%
17.0%
0.0% 3.0% 6.0% 9.0% 12.0% 15.0% 18.0% 21.0%
Exp
ect
ed
Re
turn
Standard Deviation
Efficeint Frontier (Corr = -1)
Correlation = -1
Input Data A B
Return 12% 16%
Std. Dev. 10% 20%
Correlation -1.00
bbaa
bbaa
xx
xx
xxxx
pxxxx
p
p
bababbaap
baabbabbaap
22
22222
22222
2
2
F520 – Portfolio Concepts 22
Efficient FrontierCorrelation = 0
10.0%
11.0%
12.0%
13.0%
14.0%
15.0%
16.0%
17.0%
0.0% 3.0% 6.0% 9.0% 12.0% 15.0% 18.0% 21.0%
Exp
ect
ed
Re
turn
Standard Deviation
Efficeint Frontier (Corr = 0)
Correlation = -1 Correlation = 1 Correlation = 0
Input Data A B
Return 12% 16%
Std. Dev. 10% 20%
Correlation 0.00
2222
22222
22222
2
bbaap
bbaap
baabbabbaap
xx
xx
pxxxx
F520 – Portfolio Concepts 23
Portfolio DiversificationAverage annualstandard deviation (%)
Number of stocksin portfolio
Diversifiable risk
Nondiversifiablerisk
49.2
23.9
19.2
1 10 20 30 40 1000
F520 – Portfolio Concepts 24
Efficient Frontier Conclusions
• The covariance of two assets is important in determining the variance of a portfolio
• As long as assets are not perfectly correlated, combining them in a portfolio reduces risk
• Systematic risk cannot be eliminated by diversification because it is the covariance risk. Also called non-diversifiable or market risk, since it is primarily from economy wide factors.
• Unsystematic risk (also called diversifiable risk, unique risk, or firm specific risk) comes from circumstances unique to the firm. This is why in a well diversified portfolio, unique risk is unimportant.
F520 – Portfolio Concepts 25
Covariance – the key to diversificationMathematical Example
• Assume a Special Case: Cov(i,h) = 0
• As our portfolio gets large, the variances of the portfolio gets vary small if all the covariances are 0.
• If all assets have weight Yn then x = 1 / n
• If the largest variance is V
• As n gets large, this goes to zero.• Therefore, our portfolio choices are dominated by concern over the covariance terms. In
other words, well diversified investors need only price the risk associated with the covariance of assets.
p i ii
n
x2 2 2
1
n
i
j
j
n
ip
nn 1
2
22
1
2
21
pi
n V nV V
nn n2
2 21
ihi
G
hhi
G
i
G
ii www
111
2 2
F520 – Portfolio Concepts 26
Covariance the key to diversification- Intuitive Example
# of Assets in the Portfolio
# of Variance Terms
# of Covariance terms
1 1 02 2 13 3 34 4 65 5 10
10 10 4520 20 19050 50 1225
100 100 4950
F520 – Portfolio Concepts 27
Conclusions on Covariance
• QuestionWhat will the addition of this asset to my portfolio do to my level of risk?
• Answer:Look at the covariance of the asset with my portfolio, rather than the variance.
F520 – Portfolio Concepts 28
Choosing Stocks
• Investors maximize their welfare by choosing the:
– Set of securities (investments) that maximize return for a given level of risk.
– Set of securities (investments) that minimize risk for a given level of return.
F520 – Portfolio Concepts 29
Efficient FrontierCorrelation = 0
0.0%
2.0%
4.0%
6.0%
8.0%
10.0%
12.0%
0.0% 3.0% 6.0% 9.0% 12.0% 15.0% 18.0%
Exp
ecte
d R
etur
n
Standard Dev iation
Efficeint Frontier
Correlation = 0
Input Data
A BReturn 6.5% 12%
Std. Dev. 7.1% 16%
Correlation 0.00
QU: How do Investors Choose a Portfolio on the Efficient Frontier?
F520 – Portfolio Concepts 30
Use Indifference Curves – measures of investor risk aversion
0.0%
2.0%
4.0%
6.0%
8.0%
10.0%
12.0%
0.0% 3.0% 6.0% 9.0% 12.0% 15.0% 18.0%
Exp
ecte
d R
etur
n
Standard Dev iation
Efficeint Frontier
Correlation = 0
QU: How Does this Change when a Risk-free asset is offered?
F520 – Portfolio Concepts 31
Investors can move to a higher indifference curve – greater utility.
0.0%
2.0%
4.0%
6.0%
8.0%
10.0%
12.0%
0.0% 3.0% 6.0% 9.0% 12.0% 15.0% 18.0%
Exp
ecte
d R
etu
rn
Standard Deviation
Efficeint Frontier
Correlation = 0
QU: Can you identify the important parts in the graph.
F520 – Portfolio Concepts 32
Important points on the graph.
0.0%
2.0%
4.0%
6.0%
8.0%
10.0%
12.0%
0.0% 3.0% 6.0% 9.0% 12.0% 15.0% 18.0%
Expe
cted
Ret
urn
Standard Deviation
Efficeint Frontier
Correlation = 0
Market Portfolio
Borrowing
Lending
AAL – Asset Allocation orCML – Capital Market Line
Risk-freerate
QU: What is meant by two-fund separation?
F520 – Portfolio Concepts 33
Measuring Risk and Return for the CML
• The risk free asset has no variance and its return is known with certainty (proxy – T-bill)
• Portfolio Return on CML
• Portfolio Risk on CML
RxRxR mmFRFp
bbp
bbp
bbp
baabbabbaap
baabbabbaap
wRwR
wRwwwwR
pwwwwR
STD
STD
VAR
VAR
VAR
)(
)(
)(
)(
)(
22
22
2222
2222
0020
2
Standard Deviation is a linear function of the STD of the market portfolio
F520 – Portfolio Concepts 34
Conclusions from Efficient Frontier and CML
• As long as there are only risky assets, it makes sense for investors to hold a portfolio on the efficient frontier. The existence of a risk-free asset changes this. The new efficient frontier (called the capital market line) will connect the risk free asset to some risky portfolio.
• The market portfolio (Rm) should be chosen because any other security will lead to a lower return for a given level of risk (Tangent portfolio).
• All investors will hold some combination of the risk-free asset and the market portfolio, since this will maximize their risk-return trade-off. (called two-fund separation)
• The CML portfolio chosen by an investor depends upon their risk aversion
F520 – Portfolio Concepts 35
• The Capital Market Line (CML) is Rp = Rf + slope (Standard Deviation)
• The CML is a linear relationship between the efficient portfolio’s standard deviation and its expected return.
pFp
M
FM RRRR
QU: Can we transform the CML to another measure of risk which only accounts for systematic risk?
F520 – Portfolio Concepts 36
SML, Beta, and CAPM
• The CML shows that all investors must hold a combination of the risk-free asset and the market portfolio to maximize their utility. Furthermore, it shows that their is a linear relationship between risk and return. Knowing that two points make a line, let’s form the SML by plotting these points.
Return
Beta1.00.0
Rf
Rm
F520 – Portfolio Concepts 37
• Ri = Rf + (Rm - Rf)
• Where (Rm - Rf) is the slope of the line
• Beta measures the risk of a stock in regards to the market portfolio (similar to the average stock).
Return
Beta1.00.0
Rf
Rm
Security Market Line
F520 – Portfolio Concepts 38
Understanding Beta and Calculating Portfolio Betas
• Beta measures the relative volatility of stock i with the market portfolio.
• The beta of a portfolio is the market value weighted average of the betas in the portfolio.
n
iiip BxB
1
F520 – Portfolio Concepts 39
Example: Portfolio Beta Calculations
Market PortfolioStock Value Weights Beta
(1) (2) (3) (4) (3) x (4)
Haskell Mfg. $ 6,000 50% 0.90 0.450
Cleaver, Inc. 4,000 33% 1.10 0.367
Rutherford Co. 2,000 17% 1.30 0.217
Portfolio $12,000 100% 1.034
F520 – Portfolio Concepts 40
Beta, Expected Return and the Choice of Projects (Stock)
• The concept that all assets must lie on the SML can also be Shown through an arbitrage argument. Consider Assets A, B, C, and D below. What will happen to the prices and expected returns of these assets in a competitive market using diversification techniques to eliminate all unsystematic risk?
Return
Beta 1.0 0.0
Rf
Rm
B
A C
D QU: How do I set up a trade to take advantage of this “mis-pricing”?
F520 – Portfolio Concepts 41
Hedge Fund Example
• How should I invest in these securities to take advantage of my expectations in returns relative to the required return. (Think about a hedge fund.)
Return
Beta 1.0 0.0
Rf
Rm
B
A C
D
CML = 5+B(6)Beta E(Return) Req. Ret
A 0.6 8.6 5+.6*6 = 8.6
B 0.8 12.0 5+.8*6 = 9.8
C 1.4 10 5+1.4*6 = 13.4
D 0.6 4 5+.6*6 = 8.6
F520 – Portfolio Concepts 42
Hedge Fund Example
• Some may think of having a net investment of zero, but look at the returns with market movements. None of our securities moved closer to efficiency in the example below. They each just followed the market as their risk would suggest.
Beta E(Return)Req. Ret Invest
Portfolio Beta
Market +10%
Profit (Loss)
Market -10%
Profit (Loss)
A 0.6 8.6 8.6 0 - 6.00% -$ -6.00% -$ B 0.8 12 9.8 2000 0.40 8.00% 160$ -8.00% (160)$ C 1.4 10 13.4 -1000 (0.35) 14.00% (140)$ -14.00% 140$ D 0.6 4 8.6 -1000 (0.15) 6.00% (60)$ -6.00% 60$
4000 (0.10) -1.00% (40)$ 1.00% 40$ Absolute
• How can we reduce our market risk while still taking a position on our expectations?
F520 – Portfolio Concepts 43
Hedge Fund Example
Beta E(Return)Req. Ret Invest
S=-1 L=+1
Net Position
Portfolio Beta
Market +10%
Profit (Loss)
Market -10%
A 0.6 8.6% 8.6 0 - - 0.00% -$ 0.00%B 0.8 12.0% 9.8 2000 1 2,000 0.40 8.00% 160$ -8.00%C 1.4 10.0% 13.4 500 -1 (500) (0.17) -14.00% (70)$ 14.00%D 0.6 4.0% 8.6 1500 -1 (1,500) (0.23) -6.00% (90)$ 6.00%
4000 0 0.00 0.00% 0$ 0.00%
• How can we reduce our market risk while still taking a position on our expectations?
• Wb*Bb + Wc*Bc + Wd*Bd = 0 [no market risk]• Wb + Wc + Wd = 0 [no investment for arbitrage]• Having a portfolio beta of zero immunizes the portfolio from the
market changes, and allows us to profit only from the unsystematic movements in prices, which is where one would find “mis-pricing”.
• Remember this still has risk (betas could be incorrect, our estimates of over- and under-pricing could be incorrect).
• Controlling for market movements, you expect prices of securities with expected returns that are higher relative to the required return to increase and lower expected returns to decrease.
F520 – Portfolio Concepts 44
Hedge Fund Example
Beta E(Return)Req. Ret Invest
S=-1 L=+1
Net Position
Portfolio Beta
Exp Ret, Mkt +10%
Actual Return
Profit (Loss)
A 0.6 8.6% 8.6 0 - - 0.00% 0.00% -$ B 0.8 12.0% 9.8 2000 1 2,000 0.40 8.00% 10.00% 200$ C 1.4 10.0% 13.4 500 -1 (500) (0.17) -14.00% -12.00% (60)$ D 0.6 4.0% 8.6 1500 -1 (1,500) (0.23) -6.00% -4.00% (60)$
4000 0 0.00 0.00% 80$
• The prior example showed no profit, because we assume that the returns on the stock were exactly equal to their expected return based on the market return and their beta. What is the hedge fund correctly predicted over and undervalued stocks?
• Stock B is undervalued (Exp Ret > Req Ret), so we purchased a long position. Based on a market return of 10%, we expected it to increase 8% (market * beta), but our hedge fund model prediction was correct, adding 2%, so we made a net 10%.
• Stock C is overvalued (Exp Ret < Req Ret), so we took a short position. Based on a market return of 10%, we expected it to increase 14% (market * beta), but our hedge fund model prediction was correct, reducing it by 2% for a net increase of 12%. Since we were short, we lost 12%.
• Stock D is overvalued (Exp Ret < Req Ret), so we took a short position. Based on a market return of 10%, we expected it to increase 6% (market * beta), but our hedge fund model prediction was correct, reducing it by 2% for a net increase of 4%. Since we were short, we lost 4%.
• Our portfolio has 0 beta and made money.
F520 – Portfolio Concepts 45
Hedge Fund Example
Beta E(Return)Req. Ret Invest
S=-1 L=+1
Net Position
Portfolio Beta
Exp Ret, Mkt -10%
Actual Return
Profit (Loss)
A 0.6 8.6% 8.6 0 - - 0.00% 0.00% -$ B 0.8 12.0% 9.8 2000 1 2,000 0.40 -8.00% -6.00% (120)$ C 1.4 10.0% 13.4 500 -1 (500) (0.17) 14.00% 16.00% 80$ D 0.6 4.0% 8.6 1500 -1 (1,500) (0.23) 6.00% 8.00% 120$
4000 0 0.00 0.00% 80$
• What is the market had decreased in value?
• Stock B is undervalued (Exp Ret > Req Ret), so we purchased a long position. Based on a market return of -10%, we expected it to decrease 8% (market * beta), but our hedge fund model prediction was correct, adding 2%, so we made a lost 6%.
• Stock C is overvalued (Exp Ret < Req Ret), so we took a short position. Based on a market return of -10%, we expected it to decrease 14% (market * beta), but our hedge fund model prediction was correct, reducing it by 2% for a net decrease of 16%. Since we were short, we made 16%.
• Stock D is overvalued (Exp Ret < Req Ret), so we took a short position. Based on a market return of -10%, we expected it to decrease 6% (market * beta), but our hedge fund model prediction was correct, reducing it by 2% for a net decrease of 8%. Since we were short, we made 8%.
• Our portfolio has 0 beta and made money.• As long as our hedge fund model to predict over and under-valued stocks is correct, we
make money in either an up or a down market.
F520 – Portfolio Concepts 46
Uses of Beta
• Discount rates in capital budgeting
• Discount rates for pricing assets (stocks)
• Utilities often base rates on the rate of return investors demand.
• Cost of capital calculations
• QU: What does the SML tell about the risk that managers should be concerned with when choosing a real asset investment (specifically a capital budgeting decision)?
F520 – Portfolio Concepts 47
Estimating Beta – Characteristic Line• Ri = Rf + (Rm - Rf)• rearranging terms
Ri = Rf + *Rm - *Rf
Ri = (1- ) Rf + * Rm
• Characteristic Line (also called market model)Ri = ά + * Rm + eit
• Where
= covariance (Ri, Rm) / Var (Rm)• Based on the market model, we can also break down an
assets total risk into systematic and unsystematic components.Total Risk = 2
i = 2i 2
m + 2ei
Biim
m
2
F520 – Portfolio Concepts 48
Differences in Beta Calculations
• Merrill Lynch – 5 years of monthly returns
• Value Line – 5 years of weekly returns
• Historic Beta – Calculated with only the raw return data
• Adjusted Beta – Begins with a firms historic beta and makes an adjustment for the expected future movement towards one. (Beta has been found to gradually approach 1 over time)
• Fundamental Beta – Adjusts historic betas for variables such as financial leverage, sale volatility, etc.
F520 – Portfolio Concepts 49
Data For Beta Calculation – Lilly StockCalculations in yellow, WRETD = Value weighted return,
DATE DIVAMT PRC CFACPR Adj Prc Adj Div Return VWRETD31-Jan-95 65.875 4 16.46875 028-Feb-95 0.645 67 4 16.75 0.16125 0.026869 0.039622831-Mar-95 73.125 4 18.28125 0 0.091418 0.026982328-Apr-95 74.75 4 18.6875 0 0.022222 0.024882831-May-95 0.645 74.625 4 18.65625 0.16125 0.006957 0.034146530-Jun-95 78.5 4 19.625 0 0.051926 0.030840731-Jul-95 78.25 4 19.5625 0 -0.003185 0.0406674
31-Aug-95 0.645 81.875 4 20.46875 0.16125 0.054569 0.009342729-Sep-95 89.875 4 22.46875 0 0.09771 0.036390531-Oct-95 96.625 4 24.15625 0 0.075104 -0.01114530-Nov-95 0.685 99.5 4 24.875 0.17125 0.036843 0.04297129-Dec-95 0 56.25 2 28.125 0 0.130653 0.015399931-Jan-96 57.25 2 28.625 0 0.017778 0.028087429-Feb-96 0.3425 60.625 2 30.3125 0.17125 0.064934 0.016058229-Mar-96 65 2 32.5 0 0.072165 0.011202130-Apr-96 59.125 2 29.5625 0 -0.090385 0.025125331-May-96 0.3425 64.25 2 32.125 0.17125 0.092474 0.026722128-Jun-96 65 2 32.5 0 0.011673 -0.00765931-Jul-96 56 2 28 0 -0.138462 -0.05339
30-Aug-96 0.3425 57.25 2 28.625 0.17125 0.028438 0.032222430-Sep-96 64.5 2 32.25 0 0.126638 0.052991831-Oct-96 70.5 2 35.25 0 0.093023 0.013937729-Nov-96 0.3425 76.5 2 38.25 0.17125 0.089965 0.065729631-Dec-96 73 2 36.5 0 -0.045752 -0.01136231-Jan-97 87.125 2 43.5625 0 0.193493 0.053039528-Feb-97 0.36 87.375 2 43.6875 0.18 0.007001 -0.00088931-Mar-97 82.25 2 41.125 0 -0.058655 -0.04439630-Apr-97 87.875 2 43.9375 0 0.068389 0.042480230-May-97 0.36 93 2 46.5 0.18 0.062418 0.07126330-Jun-97 109.3125 2 54.65625 0 0.175403 0.044199431-Jul-97 113 2 56.5 0 0.033734 0.0763158
29-Aug-97 0.36 104.625 2 52.3125 0.18 -0.070929 -0.03645630-Sep-97 121 2 60.5 0 0.156511 0.058009431-Oct-97 0 67.0625 1 67.0625 0 0.108471 -0.034116
F520 – Portfolio Concepts 50
Data For Beta Calculation – Lilly StockSUMMARY OUTPUT
Total PercentRegression Statistics Systematic Risk 7.6811E-05 0.00806 R-squared
Multiple R 0.089765 Unsystematic Risk 0.00945575 0.99194 1- R-squaredR Square 0.008058 0.008057732 Total Risk 0.00953256Adjusted R Square -0.006318 0.00953256 1 Should add to 1
Standard Error 0.09864 due to the degrees of freedom inObservations 71 regression these are not exactly
equal to each otherANOVA
df SS MS F Significance FRegression 1 0.00545357 0.00545357 0.56049986 0.456603Residual 69 0.671358409 0.009729832Total 70 0.676811978
Coefficients Standard Error t Stat P-value Lower 95% Upper 95% Lower 95.0% Upper 95.0%Intercept 0.027862 0.01243276 2.241034391 0.02824458 0.00306 0.05266491 0.00305957 0.052664912X Variable 1 0.196282 0.262175336 0.748665388 0.45660343 -0.326744 0.71930692 -0.3267437 0.719306923
RESIDUAL OUTPUT
Observation Predicted Y Residuals SS(residuals)1 0.035639 -0.008770407 7.692E-05
Beta estimate
Alpha
Percent of total variation explained =MS(reg)/SS(total)
F520 – Portfolio Concepts 51
Assumptions of the CAPM (SML)
• Assumptions about investor behavior– Investors use only two measures to determine their strategy,
expected return and risk,
– Investors will choose portfolios as a risk reduction technique,
– Investors make investment decisions over some single-period investment horizon,
– Homogenous expectations with respect to asset returns, variances, and correlations
• Assumptions about capital markets– Perfect competition,
– No transaction costs • -No bid-ask spreads, -No commissions, -No information costs, -No
taxes, -No regulation, and -all assets are marketable
– Investors can borrow and lend at the risk-free rate.
F520 – Portfolio Concepts 52
Test of the CAPM (SML)• Clearly the assumptions are unrealistic, but the true test of a
model comes from answering two questions – Does the model change when the assumptions are changed? – How well does the model predict?
• Empirical Findings– There is a significant positive relationship between realized returns and
systematic risk. However, the slope is usually less than predicted by the CAPM.
– The relationship between risk and return appears to be linear. No evidence of curvature has been found.
– Tests assessing the importance of company specific risk after controlling for market risk are inconclusive. Econometrically controlling for market risk given its high correlation with total risk is difficult.
– The CAPM should be valid for all assets; however, bonds do not track along the SML.
– Betas of individual stocks are not stable over time; however, betas for portfolios are stable over time.
F520 – Portfolio Concepts 53
Anomalies with using the CAPM
• Small firm effect
• Price-to-Book Ratios (Growth versus value stocks)
• January effect
Common Question:
When using CAPM [Ri=Rf+i(Rm – Rf)],
what is the Risk Premium (Rm – Rf)
F301_CH12-54
What is the Rf you are using?
Should you use Large or Small Stocks?
Should you use arithmetic or geometric returns?
Can CAPM be used for bond?(August 9, 2013 data)
• Lehman Index (ticker = AGG)http://www.ishares.com/product_info/fund/overview/AGG.htm
Effective Duration 5.05 years Average Yield to Maturity 2.16% http://finance.yahoo.com/q/rk?s=AGG+Risk
Beta (against Standard Index) 1.01 Yahoo R-squared (against Standard Index) 98.88
Yahoo betas are 5-years• Lehman 1-3 year Treasury Bond Fund (ticker = SHY)
http://www.ishares.com/product_info/fund/overview/SHY.htm
Effective Duration 1.86 Average Yield to Maturity 0.32% http://finance.yahoo.com/q/rk?s=SHY+Risk
Beta (against Standard Index) 0.14 YahooR-squared (against Standard Index) 24.31What is the standard index in this case?
– So what is beta in this case?1.86 / 5.05 = 0.36, compare to Beta?
F520 – Portfolio Concepts 55
Can CAPM be used for bond?• Lehman 7-10 Year Treasury Bond Fund (ticker = IEF)
http://www.ishares.com/product_info/fund/overview/IEF.htm
Effective Duration 7.48Average Yield to Maturity 2.29% http://finance.yahoo.com/q/rk?s=IEF+Risk
Beta (against Standard Index) 1.70 YahooR-squared (against Standard Index) 69.52
– So what is beta in this case?7.48 / 5.05 = 1.48, compare to Beta?
• Lehman 20+ Year Treasury Bond Fund (ticker = TLT)http://www.ishares.com/product_info/fund/overview/TLT.htm
Effective Duration 16.43Average Yield to Maturity 3.61% http://finance.yahoo.com/q/rk?s=TLT
Beta (against Standard Index) 3.37 Yahoo R-squared (against Standard Index) 53.81
– So what is beta in this case?16.43 / 5.05 = 3.25, compare to Beta?
F520 – Portfolio Concepts56
Cont.• The concept of Beta, used by Yahoo Finance and MSN Money for bonds is
not the same concept of beta referred to in stocks. When a bond index is used as the standard index, we obtain a relative measure of duration. When a stock index is used, we obtain the traditional measure of systematic risk.
• When using public betas, identify the index used to interpret the concept of beta reported. For many companies/funds, they state a “Standard Index”, to properly interpret the measures, you must clearly identify the index. (MSN Money provides identification, Yahoo does not.)
– For Ishare Austria Fund: http://investing.money.msn.com/investments/etf-management?symbol=ewo For Ishare Japan Fund: http://investing.money.msn.com/investments/etf-management?symbol=ewj Standard Index is MSCI EAFE NDTR_DEAFE stands for Europe, Australasia, and Far East. The index has stocks from 21 developed markets, excluding the U.S. and Canada.
– For Ishare S&P Small Cap 600 Index, (uses S&P500)http://investing.money.msn.com/investments/etf-management?symbol=ijr For Ishare NAREIT Industrial/Office Index Fund (uses MSCI World)http://investing.money.msn.com/investments/etf-management?symbol=fnio
F520 – Portfolio Concepts 57
F520 – Portfolio Concepts 58
Multifactor CAPM
• Multi-Factor CAPME(Ri) = Rf + i,M[E(RM) - Rf] + i,f1[E(Rf1) - Rf]+
i,f2[E(Rf2) - Rf] +…+ i,fn[E(Rfn) - Rf]
• By rearranging terms we get the multiple regression typically used.E(Ri) = Rf + i,M*E(RM) - i,M*Rf + i,f1*E(Rf1) - i,f1*Rf +
i,f2*E(Rf2) - i,f2*Rf +…+ i,fn*E(Rfn) - i,fn*Rf
E(Rit) = + i,Mt*E(RMt) + i,f1*E(Rf1t) + i,f2*E(Rf2t) +…+
i,fn*E(Rfnt) + eit
where = Rf - i,M*Rf - i,f1*Rf - i,f2*Rf -…- i,fn*Rf
Rf = Riskfree Rate
Rf1 = Expected Return on factor 1
F520 – Portfolio Concepts 59
Arbitrage Pricing Theory (APT), an alternative to the CAPM
• E(Ri) = Rf + i,f1[E(Rf1) - Rf] + i,f2[E(Rf2) - Rf] +…+
i,fn[E(Rfn) - Rf]
• By rearranging terms we get the multiple regression typically used.E(Rit) = + i,f1*E(Rf1t) + i,f2*E(Rf2t) +…+ i,fn*E(Rfnt) + eit
where = Rf - i,f1*Rf - i,f2*Rf -…- i,fn*Rf
Rf = Risk-free Rate
Rf1 = Expected Return on factor 1
F520 – Portfolio Concepts 60
Assumptions of APT
• APT assumes returns are a function of several factors, not just one as in the CAPM
• Suggested factors (Roll & Ross 1983) – Index of Industrial Production,
– Changes in the default risk premium on bonds,
– Changes in the yield curve,
– Unanticipated inflation
• Other factors frequently considered– Factors for size
– Factors for book-to-market value
F520 – Portfolio Concepts 61
Principles to Take Away from the APT and CAPM
• Investing has two dimensions, risk and return.
• It is inappropriate to look at the risk of an individual asset when deciding whether it should be included in a portfolio. What is important is how the inclusion of an asset into a portfolio will affect risk of the portfolio (covariance and/or beta must be considered).
• Risk can be divided into two categories, systematic and unsystematic
• Investors should only be concerned about systematic risks.
F520 – Portfolio Concepts 62
Commonly used Portfolio Performance Criteriaare based on the Efficient Frontier or CAPM concepts
Global Tech Fund: Return: +37.2%, Beta: 1.29, Std. Dev 25%, Riskfree = 5.0%, RiskPremium = 6.0%
• Sharpe Ratio
= (Rp – Rf)/σp
= (37.2 – 5)/25 = 1.29
• Treynor Ratio
= (Rp – Rf)/Bp
= (37.2 – 5)/1.29 = 25.0%
• Jensen’s alpha (αp)
= Rp – CAPM
= Rp – [Rf+Bp(RM – Rf)]= 37.2 – [5+1.29(6)] = 24.5%
Return
Beta1.00.0
Rf
Rm
0.0%
2.0%
4.0%
6.0%
8.0%
10.0%
12.0%
0.0% 3.0% 6.0% 9.0% 12.0% 15.0% 18.0%
Exp
ecte
d R
etu
rn
Standard Deviation
Efficeint Frontier
Correlation = 0
F520 – Portfolio Concepts 63
This fund has a beta of 1.29, substantially greater than the market beta of 1.0. To compare it to the market, we must determine what portion can be invested in the risk-free rate and what portion invested in the Global Tech Fund to have the same risk as the market.
Let x = the percent invested in the Global Tech Fund, subsequently (1-x) is the percent in the risk-free asset.Note that the beta of the risk-free asset is equal to zero.(1-x)(0) + x(1.29) = 1.0Solve for x.x = 1/1.29 = .78 portion of the portfolio in the Global Tech Fund1-x = .22 portion invested in the risk-free asset
Now calculate your risk-adjusted return:The Global Tech Fund earned 37.2% and the risk-free asset over this 3-year
period earned 5%. The proportions in each asset are calculated above..22(5%) + .78(37.2%) = 30.1%This value can be compared to what the market earned during this period, since it
has a beta of 1.
M-Squared Measure (Modigliani and Modigliani)Global Tech Fund: Return: +37.2%, Beta: 1.29, Std. Dev 25%, Riskfree = 5.0%, RiskPremium = 6.0%