2201afe vw week 9 return, risk and the sml
TRANSCRIPT
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2201AFE Corporate FinanceWeek 9:
Return, Risk and the Security Market Line
Readings: Chapter 11
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Agenda
Last Lecture
Return, Risk and the Security Market Line Key Concepts and Skills
Real World Application
Estimating Microsofts Beta
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Last Lecture
Returns
Holding Period Returns Averages: Arithmetic Mean & Geometric Mean
Risk
Variance
Standard Deviation
There is a reward for bearing risk
Positive risk-return relationship
Risk Premium
Efficient Market Hypothesis: weak, semi-strong, strong
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Return, Risk and the Security Market Line
Chapter 11
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1. Introduction & Financial
Statements
2. Time Value of Money
3. Valuing Shares & Bonds
7. Mid-semester Exam
8. Some Lessons from Capital
Market History
11. Financial Leverage & Capital
Structure Policy
13. Options & Revision
9. Return, Risk & the Security
Market Line
5. Making Capital Investment
Decisions & Project Analysis
12. Dividends & Dividend Policy
6. Revision for Mid-sem Exam
4. Net Present Value & Other
Investment Criteria
10. Cost of Capital
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Key Concepts and Skills
Expected Returns and Variances
Probabilities
Portfolios
Risk and Returns
The principle of diversification
Risk: Systematic and Unsystematic
The Security Market Line (SML)
Capital Asset Pricing Model (CAPM)
Reward to Risk Ratio
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Expected Returns
Consider an asset which has many possible future returns,
returns that are not equally likely. What is the average
return? What is the expected return?
Average or Expected returns is based on the average of all
possible future returns weighted by their probabilities.
Suppose there are Tpossible returns, and that R1 hasprobability p1 of occurring, R2 has probability p2, , and RT
has probability pT . Then:
7
TT2211
Ti
1i ii
RpRpRpE(R)
RpE(R)
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Example: Expected Returns
Suppose you have predicted the following returns for
stocks C and T in three possible states of nature. What are
the expected returns?
RC = 0.3(0.15) + 0.5(0.10) + 0.2(0.02) = 9.99%
RT = 0.3(0.25) + 0.5(0.20) + 0.2(0.01) = 17.7%
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State Probability Stock C Stock T
Boom 0.3 15% 25%
Normal 0.5 10% 20%Recession ??? 2% 1%
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Variance and Standard Deviation
Variance and standard deviation still measure the volatility
of returns.
Using unequal probabilities for the entire range of
possibilities.
Weighted average of squared deviations.
9
2
2
ii
2
22
2
11
2
T
1i
2
ii
2
VARSD
E(R)][Rp...E(R)][RpE(R)][RpVAR
E(R)][RpVAR
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Example: Variance and Standard Deviation
Consider the previous example. What are the variance and
standard deviation for each stock?
E(R)C = 9.9% and E(R)T = 17.7%
Stock C:
2 = 0.3(0.15-0.099)2 + 0.5(0.10-0.099)2 + 0.2(0.02-0.099)2
= 0.3(0.051)2 + 0.5(0.001)2 + 0.2(-0.079)2
= 0.3(0.002601) + 0.5(0.000001) + 0.2(0.006241)
= 0.0007803 + 0.0000005 + 0.0012482 = 0.002029
Stock T:
2 = 0.3(0.25-0.177)2 + 0.5(0.20-0.177)2 + 0.2(0.01-0.177)2
= 0.3(0.073)2 + 0.5(0.023)2 + 0.2(-0.167)2
= 0.0015987 + 0.0002645 + 0.0055778 = 0.007441
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%5.4045044.0002029.02
%63.8086261.0007441.02
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Portfolios
A portfolio is a collection of assets.
An assets risk and return are important in how they affect
the risk and return of the portfolio.
The risk-return trade-off for a portfolio is measured by the
portfolio expected return and standard deviation, just aswith individual assets.
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Example: Portfolio Weights
Suppose you have $15,000 to invest and you have
purchased securities in the following amounts. What are
your portfolio weights in each security?
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Companies Amount invested Weights
CBA $2,000
WOW $3,000TLS $4,000
BHP $6,000
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Portfolio Expected Returns
The expected return of a portfolio is the weighted averageof the expected returns for each asset in the portfolio (see
example 11.3) Method 1:
Step 1: calculate E(Rasset) based on probability of state
Step 2: calculate E(RP) based on weights of assets
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nn2211asset Rp...RpRp)E(R
)E(Rw...)E(Rw)E(Rw)E(R nn2211P
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Portfolio Expected Returns
You can also find the expected return by finding theportfolio return in each possible state and computing the
expected value as we did with individual securities (seeexample 11.4)
Method 2:
Step 1: calculate E(RP) in each state, eg. boom or bust
Step 2: add the state returns weighted by each probability
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nn2211state,P Rw...RwRw)E(R
)E(Rp...)E(Rp)E(Rp)E(R state nn2state21state1P
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Example: E(RP)
Consider the following information
What are the expected return for a portfolio with an
investment of $6,000 in asset X and $4,000 in asset Z?
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State Probability Asset X Asset Z
Boom 0.25 15% 10%
Normal 0.60 10% 9%
Recession 0.15 5% 10%
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Example: E(RP) Method 1
Weight X = 0.6, Weight Z = 0.4
First way of calculating E(RP):
Step 1: Calculate the expected return of each asset based on
each probability of state occurring:
E(RX) = (0.250.15) + (0.60.10) + (0.150.05) = 0.105
E(RZ) = (0.250.10) + (0.60.09) + (0.150.10) = 0.094
Boom Normal Recession
Step 2: Calculate the E(RP) based on the weights of each
asset:E(RP) = 0.60.105 + 0.40.094 = 0.1006 (10.06%)
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Example: E(RP) Method 2
Weight X = 0.6, Weight Z = 0.4
Second way to calculate E(RP):
Step 1: Calculate the E(RP) in each state based on each asset
weight:
E(RP)Boom = (0.60.15) + (0.40.10) = 0.13
E(RP
)Normal
= (0.60.10) + (0.40.09) = 0.096
E(RP)Recession = (0.60.05) + (0.40.10) = 0.07
Step 2: Calculate total E(RP) using probabilities as weights:
E(RP) = pB E(RBoom) + pN E(RNormal) + pR E(RRecession)
E(RP) = (0.250.13) + (0.60.096) + (0.150.07)
= 0.0325 + 0.05756 + 0.0105 = 0.1006 (10.06%)
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Portfolio Variance with Probabilities
1. Compute the portfolio return for each state (step 1):
E(RP,state) = w1R1 + w2R2
2. Compute the expected portfolio return using probabilities as for asingle asset (step 2):
E(RP) = p1 E(Rstate1) + p2 E(Rstate2) + p3 E(Rstate3)
3. This E(RP) becomes the mean
4. Compute the deviations of each state from the mean, then square thedeviation:
[E(RP,state) E(RP)]2
5. Multiply the squared deviation with probability of each state, then
sum: (pstate [E(RP,state) E(RP)]
2)
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Example: Variance & SD
Variance: (pstate [E(RP,state) E(RP)]2)
Portfolio return in each state (boom, normal, recession) and two-
asset (X and Z) total portfolio return.
E(Rp)boom = 13%
E(Rp)normal = 9.6%
E(Rp)recession = 7%
E(Rp) = 10.06%
Variance:
= 0.25(0.13-0.1006)2 + 0.6(0.096-0.1006)2 + 0.15(0.07-0.1006)2
= 0.00021609 + 0.000012696 + 0.000140454
= 0.00036924
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%92.1019215619.000036924.0SD
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Example 2: E(R), Variance & SD
Consider the following information:
Invest 50% of your money in Asset A
What are the expected return and standard deviation for
each asset?
What are the expected return and standard deviation forthe portfolio?
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State Probability Asset A Asset B Portfolio
Boom 0.40 30% -5% 12.5%
Bust 0.60 -10% 25% 7.5%
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Example 2: E(R), Variance & SD Asset
E(Rasset) = (pstate1 Rasset) + (pstate2 Rasset)
VAR = [pstate (Rasset E(Rasset)2]
Asset A: E(RA) = 0.4(0.30) + 0.6(-0.10) = 6%
Variance(A) = 0.4(0.30-0.06)2 + 0.6(-0.10-0.06)2
= 0.02304 + 0.01536 = 0.0384
Asset B: E(RB) = 0.4(-0.05) + 0.6(0.25) = 13%
Variance(B) = 0.4(-0.05-0.13)2 + 0.6(0.25-0.13)2= 0.01296+0.00864 = 0.0216
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%7.14147.00216.0)B(SD
%6.19196.00384.0)B(SD
2VARSD
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Example 2: E(R), Variance & SD Portfolio
Calculate the Expected return of portfolio in each state:
E(Rp)state = (wasset A RA state) + (wasset B RB state)
E(Rp)boom = 0.5(0.30) + 0.5(-0.05) = 0.125
E(Rp)bust= 0.5(-0.10) + 0.5(0.25) = 0.075
Then the overall Expected portfolio return:
E(Rp) = (pstate1 E(Rp)state1) + (pstate2 E(Rp)state2)
E(Rp) = 0.4(0.125) + 0.6(0.075) = 0.095 Then the Variance of the portfolio:
VARp = [pstate (E(Rp)state E(Rp))2]
VARp = 0.4(0.125 - 0.095)2 + 0.6(0.075 - 0.095)2
= 0.00036 + 0.00024 = 0.0006 Then the Standard Deviation of the portfolio:
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%45.202449.00006.0SD
k d f l h
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Risk and Portfolio Theory
Risk Averse Investors: Require a higher average return to
take on a higher risk.
Portfolio theory assumptions:
Investors prefer the portfolio with the highest expected
return for a given variance or the lowest variance for a given
expected return.
Expected returns and variances of portfolios derived from
historical returns, variances, and co-variances of individual
assets in portfolio.
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C i d C l i C ffi i
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Covariance and Correlation Coefficient
Covariance is an absolute measure of the degree to which
two variables move together over time relative to their
individual mean (average).
Correlation Coefficient, , is a standardised measure of the
relationship between the two variables, ranging between
-1.00 to +1.00
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T
)R)(RR(RCov
2211
2,1
Var
T
)R)(RR(RCov
1111
1,1
212,12,1
21
2,1
1,22,1
Cov
Coventn CoefficiCorrelatio
P tf li VAR & SD f 2 A t P tf li
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Portfolio VAR & SD for a 2-Asset Portfolio
In order to reduce the overall risk, it is best to have assetswith low positive or negative correlation (covariance).
The smaller is the covariance between the assets, the
smaller will be the portfolios variance.
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)nCorrelatio(ww2wwVAR
)iancevarCo(Covww2wwVAR
2,1212122
22
21
21
2p
2,121
2
2
2
2
2
1
2
1
2
p
2
PP SD
E l Ri k f 2 A t P tf li
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Example: Risk of 2-Asset Portfolio
Consider these two assets that have equal weights of 0.50
in the portfolio, and with the following returns and
standard deviation:E(R1) = 30% and 1 = 0.20
E(R2) = 15% and 2 = 0.12
Correlation Coefficient = 0.10
p2 = (0.5)2(0.2)2 + (0.5)2(0.12)2 + 2(0.5)(0.5)(0.2)(0.12)(0.1)
= 0.01 + 0.0036 + 0.0012 = 0.0148
= 12.16% (lower risk for 22.5% portfolio return)0.50.3 + 0.50.15 = 0.225 or 22.5%
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2,12121
2
2
2
2
2
1
2
1
2
p ww2ww
1216.00148.0P
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Diversification
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Diversification
The Principle of Diversification:states that spreading an
investment across many types of assets will eliminate some
but not all of the risk.
Diversification can substantially reduce the variability of
returns without an equivalent reduction in expected
returns.
Size of risk reduction depends on co-variances between
assets in the portfolio.
However, there is a minimum level of risk that cannot bediversified away and that is the systematic portion.
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Two Types of Risk
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Two Types of Risk
Systematic or Non-Diversifiable Risk:
That portion of an assets risk attributed to the market
factors that affect all firms and cannot be eliminated through
the process of diversification.
Unsystematic or Diversifiable Risk:
That portion of an assets risk which is firm specific and can
be eliminated through the process of diversification.
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Total Risk
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Total Risk
Total risk = Systematic risk + Unsystematic risk
The standard deviation of returns is a measure of total risk.
For well-diversified portfolios, unsystematic risk is very
small.
Consequently, the total risk for a diversified portfolio is
essentially equivalent to the systematic risk.
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Systematic Risk Principle
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Systematic Risk Principle
There is a reward for bearing risk.
There is not a reward for bearing risk unnecessarily.
The expected return on a risky asset depends only on that
assets systematic risk since unsystematic risk can bediversified away.
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Measuring Systematic Risk =
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Measuring Systematic Risk =
We use the beta coefficient to measure systematic risk.
Beta measures the responsiveness of a security to
movements in the market.
Market beta m = 1
Therefore if:
A= 1, the asset has the same systematic risk as the overallmarket.
A < 1 implies the asset has less systematic risk than the
overall market.
A > 1 implies the asset has more systematic risk than theoverall market.
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2
m
A,m
A
Cov
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CAPM
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CAPM
E(RA) = Rf+ A[E(RM) Rf]
Where:
E(RA) = expected return on asset A
Rf= risk free rate
A= beta of asset A
E(RM) = expected return on the market
Note: E(RM
) Rf= Market Risk Premium
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Example CAPM
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Example CAPM
If the beta for IBM is 1.15, the risk-free rate is 5%, and the
expected return on market is 12%, what is the required rate
of return for IBM?
Applying the CAPM:
37
%05.13or1305.0
0805.005.0
]07.0[15.105.0
]05.012.0[15.105.0]R)[E(RR)E(R fMIBMfIBM
Example CAPM
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Example CAPM
Consider the betas for each of the assets given earlier. If
the risk-free rate is 2.13% and the market risk premium is
8.6%, what is the expected return for each?
E(RA) = Rf+ A[E(RM) Rf]
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Security Beta Expected Return
CBA 2.685
WOW 0.195
TLS 2.161
BHP 2.434
Security Market Line (SML)
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Security Market Line (SML)
The security market line (SML) is the graphical
representation of CAPM.
Shows the relationship between systematic risk and
expected return.
Positive slope.
The higher the risk, the higher the return.
According to the CAPM, all stocks must lie on the SML,
otherwise they would be under or over-priced.
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Security Market Line (SML)
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y ( )
40
Asset expected
return = E(ri)
E(RM)
SML
Asset beta =iM= 1.0
Market
A undervalued
B overvalued
RF
A B
E(RB)
E(RA) = E(RM) RF
Reward-to-Risk Ratio
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SML slope = Reward-to-Risk Ratio of the Market = Market
Risk Premium
In equilibrium, all assets and portfolios must have the samereward-to-risk ratio and they all must equal the reward-to-
risk ratio for the market.
If not, assets are undervalued or overvalued.
41
)RE(R
)RE(R
R)E(R
R)E(RfM
M
fM
B
fB
A
fA
Reward-to-Risk Ratio: Example
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p
If RM = 12%, Rf= 6%
If Asset A has E(RA) = 15%, A = 1.3, and
Asset B has E(RB) = 10%, B = 0.8 then:
Asset B offers insufficient reward for its level of risk, so B is relatively
overvalued compared to A, or A is relatively undervalued.42
%58.0
%6%10
R)E(R
%9.63.1
%6%15
R)E(R
B
fB
A
fA
%61
%)6%12(Slope
)RE(R
)RE(R
Slope fMM
fM
CAPM and Beta of Portfolio
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Ifw1, w2, , wn are the proportions of the portfolio
invested in nassets 1, 2, , n, the beta of a portfolio (P)
can be written:
Example: If 30% of a portfolio is invested in asset 1 and thebalance in asset 2, and asset 1s beta = 1.7 while asset 2s
beta = 1.2, what is the beta of the portfolio (P).
43
nj
1j
jjnn2211P ww...ww
35.1
)2.1(7.0)7.1(3.0
ww 2211P
Example: Portfolio Beta
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Consider our previous four securities and their betas:
What is the portfolio beta?
= 0.133(2.685) + 0.2(0.195) + 0.167(2.161) + 0.4(2.434)
= 1.731
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Security Weight Beta
CBA 0.133 2.685
WOW 0.200 0.195
TLS 0.167 2.161
BHP 0.400 2.434
Factors Affecting E(R)
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E(RA) = Rf+ A[E(RM) Rf]
Pure time value of money
measured by the risk-free rate, Rf
Reward for bearing systematic risk measured by the market risk premium, E(RM) Rf
Amount of systematic risk
measured by beta,
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Quick Quiz
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What is the difference between systematic and
unsystematic risk?
What type of risk is relevant for determining the expected
return?
Consider an asset with a beta of 1.2, a risk-free rate of 5%and a market return of 13%.
What is the reward-to-risk ratio in equilibrium?
What is the expected return on the asset?
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Real World Application
Estimating Microsofts Beta
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Estimating Microsofts Beta
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To calculate the parameters for an asset, say a share in
Microsoft, we perform a regression of the returns on
Microsoft with the returns on the market.
Historical data for Microsoft and the Market index (S&P
500) are collected and a time series of both returns are
calculated.
An OLS regression is then performed.
The regression provides values for the parameters and a
plot of the observations may be made.
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Example: Microsoft vs S&P500
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Monthly Returns: Microsoft and S&P 500 (n = 60)
-20
-10
0
10
20
30
40
-20 -15 -10 -5 0 5 10
S&P 500 Return
M
icrosoftReturn
Coefficients Standard error t-statistics
2.14 1.21 1.77 1.30 0.27 4.76
Adjusted R2= 0.27
n = 60
Next Week
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We move to calculating a firms cost of capital, termed the
weighted average cost of capital (WACC).
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