Week 8
Lecture 8
Ross, Westerfield and Jordan 7e
Chapter 13
Return, Risk and the Security Market Line
13-0
Last Lecture..
• Returns
• Holding Period Returns
• Averages: AM, GM
• Risk
• Variance
• Standard Deviation
• There is a reward for bearing risk
• Positive risk-return relationship
• Risk Premium
• EMH: weak, semi-strong, strong
13-1
Chapter 13 Outline
• 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
13-2
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 T possible returns, and that R1 has
probability p1 of occurring, R2 has probability p2 , …, and
RT has probability pT . Then:
i T
E(R)   piRi
i1
E(R)  p1R1  p2R2 ....  pTRT
13-3
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?
State
Boom
Normal
Recession
Probability
0.3
0.5
???
C
15%
10%
2%
T
25%
20%
1%
• 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%
13-4
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
T
Var  σ 2   pi [Ri  E(R)] 2
i1
Var  σ 2  p1[R1  E(R)] 2  p 2 [R 2  E(R)] 2  ...  pi [Ri  E(R)] 2
13-5
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
•  = √σ2 = √0.002029 = 0.045044 = 4.5%
• 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
•  = √σ2 = √0.007441 = 0.086261 = 8.63%
13-6
Quick Quiz
• Consider the following information:
State
Probability
Boom
0.25
Normal
0.50
Slowdown
0.15
Recession
0.10
ABC, Inc. (%)
15
8
4
-3
• What is the expected return?
• What is the variance? (0.00267475)
• What is the standard deviation?
13-7
Portfolios
• A portfolio is a collection of assets
• An asset’s 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 as with
individual assets
13-8
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?
•
•
•
•
$2000 of DCLK
$3000 of KO
$4000 of INTC
$6000 of KEI
DCLK: 2/15 = 0.133
KO: 3/15 = 0.2
INTC: 4/15 = 0.267
KEI: 6/15 = 0.4
13-9
Portfolio Expected Returns
• The expected return of a portfolio is the weighted
average of the expected returns for each asset in the
portfolio (example 13.3)
• Step 1: calculate E(Rasset) based on probability of state
E(R asset )  p1R1  p2R2  ...  pnRn
• Step 2: calculate E(RP) based on weights of assets
E(R P )  w1E(R1 )  w 2E(R 2 )  ...  w nE(R n )
• You can also find the expected return by finding the
portfolio return in each possible state and computing the
expected value as we did with individual securities (ex.13.5)
• Step 1: calculate E(RP) in each state, eg. boom or bust
E(RPstate)  w1R1  w 2R2  ...  wnRn
• Step 2: add the state returns weighted by each probability
E(RP )  p1E(R state1 )  p2E(R state2 )  ...  pnE(R state n )
13-10
Example : E(RP)
• Consider the following information
State
Probability(p)
X
Boom
0.25
15%
Normal
0.60
10%
Recession
0.15
5%
Z
10%
9%
10%
• What are the expected return for a portfolio
with an investment of $6000 in asset X and
$4000 in asset Z?
13-11
Example : E(Rp) continued..(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) = (0.25x0.15) + (0.6x0.1) + (0.15x0.05) = 10.5%
E(RZ) = (0.25x0.1) + (0.6x0.09) + (0.15x0.1) = 9.4%
Boom
Normal
Recession
• Step 2: Calculate the E(RP) based on the weights of
each asset:
E(RP )  w xE(R x )  w zE(R z )
• E(RP) = 0.6x10.5% + 0.4x9.4% = 10.06%
13-12
Example : E(RP) continued..(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
E(RP)Normal
E(RP)Recession
= (0.6 x 0.15) + (0.4 x 0.10) = 13%
= (0.6 x 0.10) + (0.4 x 0.09) = 9.6%
= (0.6 x 0.05) + (0.4 x 0.10) = 7%
• Step 2: Calculate total E(RP) using probabilities
as weights:
E(RP) = pB x E(RB) + pN x E(RN) + pR x E(RR)
Boom
Normal
Recession
E(RP) = (0.25x13%) + (0.6x9.6%) + (0.15x7%) =
3.25% + 5.756% + 1.05% = 10.06%
13-13
Portfolio Variance with Probabilities
• 1) Compute the portfolio return for each state,
boom, bust.. etc. (step 1):
E(RPstate) = w1R1 + w2R2
• 2) Compute the expected portfolio return using
probabilities as for a single asset (step 2):
E(RP) = p1 x E(Rstate1) + p2 x E(Rstate2) + p3 x E(Rstate3)
• 3) This E(RP) becomes the mean
• 4) Compute the deviations of each state from the
mean, then square the deviation: [E(RPstate)-E(RP)]2
• 5) Multiply the squared deviation with probability of
each state, then sum: ∑ (pstate x [E(RPstate)-E(RP)]2)
13-14
Example: Variance & SD
• Variance: ∑ (pstate x [E(RPstate)-E(RP)]2)
• Portfolio return in each state (boom, normal, recession)
and Two-asset (X and Z) total portfolio return (slide 13)
•
•
•
•
E(Rp)boom = 13%
E(Rp)normal = 9.6%
E(Rp)recession = 7%
E(Rp) = 10.06%
• Variance:
Var = 0.25(0.13-0.1006)2 + 0.6(0.096-0.1006)2 +
0.15(0.07-0.1006)2 =
Var = 0.00021609 + 0.000012696 + 0.000140454 =
0.00036924
SD = √0.00036924 = 0.019215619 = 1.92%
13-15
Example 2: E(R), Variance & SD
• Consider the following information:
• Invest 50% of your money in Asset A
State
Probability
A
B
portfolio
Boom 0.4
30%
-5% 12.5%
Bust
0.6
-10%
25% 7.5%
• What are the expected return and standard
deviation for each asset?
• What are the expected return and standard
deviation for the portfolio?
13-16
Example 2: E(R), Var & SD…asset
E(Rasset) = (pstate1 x Rasset) + (pstate2 x Rasset)
Var = ∑[pstate x (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
• Std. Dev.(A) = √0.0384 = 19.6%
• 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
• Std. Dev.(B) = √ 0.0216 = 14.7%
13-17
Example 2: E(R), Var & SD…portf.
• Calculate the Expected return of portfolio in each state
E(Rp)state = (wassetA x RA state) + (wassetB x RB state)
• E(Rp)boom = 0.5(0.30) + 0.5(-0.05) = 12.5%
• E(Rp)bust = 0.5(-0.10) + 0.5(0.25) = 7.5%
• Then the overall Expected portfolio return
E(Rp) = (pstate1 x E(Rp)state1) + (pstate2 x E(Rp)state2)
• E(Rp) = 0.4(0.125) + 0.6(0.075) = 9.5%
• Then the Variance of the portfolio
Varp = ∑[pstate x (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
• SD = √0.0006 = 0.02449 = 2.45%
13-18
Risk and Portfolio Theory
• Risk Averse Investors: require a higher
average return to take on a higher risk
• Portfolio Theory Assumption:
• 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 covariances of individual
assets in portfolio
13-19
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.
Cov 1,2
(R  R )(R  R )


1
1
2
T
2
Cov 1,1
(R  R )(R  R )


 Var
1
1
1
1
T
• Correlation Coefficient, ρ, is a standardised
measure of the relationship between the two
variables, ranging between -1.00 to +1.00
Corr Coeff1,2  ρ1,2
Cov 1,2

σ1σ 2
Cov 1,2  ρ1,2 x σ1σ 2
13-20
Portfolio Variance and Standard
Deviation for a 2-Asset Portfolio
Var  σP2  w12σ12  w 22σ22  2w1w 2Cov 12
SD  σ P  σ P2
Var  σP2  w12σ12  w 22σ22  2w1w 2σ1σ2 (Corr Coeff)
• In order to reduce the overall risk, it is
best to have assets with low positive or
negative correlation (covariance)
• The smaller is the covariance between
the assets, the smaller will be the
portfolio’s variance.
13-21
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 σ1 = 0.20
E(R2) = 15%
and σ2 = 0.12
Corr Coeff = 0.1
σP2  w12σ12  w 22σ22  2w1w 2σ1σ2 (Corr Coeff)
σp2 = (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
σp =
√0.0148 = 0.1216 = 12.16% (lower risk for 22.5% portfolio return)
13-22
Table 13.7
More
assets
Less
risk
13-23
Diversification
• The Principle of Diversification : states that
spreading an investment across many 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 covariances
between assets in the portfolio
• However, there is a minimum level of risk that
cannot be diversified away and that is the
systematic portion
13-24
Two Types of Risk
• Systematic or Non-Diversifiable Risk
• That portion of an asset’s 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 asset’s risk which is firm
specific and can be eliminated through the
process of diversification.
13-25
Figure 13.1
13-26
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
13-27
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 asset’s systematic
risk since unsystematic risk can be
diversified away
13-28
Measuring Systematic Risk = β
• We use the beta coefficient to measure
systematic risk
• Beta measures the responsiveness of a security
to movements in the market.
Cov A,m
• Market beta βm = 1
βA 
2
σm
• Therefore if:
• βA= 1, the asset has the same systematic risk as the
overall market
• βA < 1 implies the asset has less systematic risk than
the overall market
• βA > 1 implies the asset has more systematic risk than
the overall market
13-29
Estimation of Beta
• Two ways:
• Based on the formula calculate the covariance
of the asset with the market, calculate the
variance of the market, then divide the two
• Slope function in excel
• What is the market?
• The index
13-30
The Capital Asset Pricing Model
(CAPM)
• The capital asset pricing model defines the
relationship between risk and return
• If we know an asset’s systematic risk, we
can use the CAPM to determine its
expected return
• This is true whether we are talking about
financial assets or physical assets
13-31
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
13-32
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 :
E(R IBM )  R f  βIBM [E(R M )  R f ]
 0.05  1.15[0.12  0.05]
 0.05  1.15[0.07]
 0.05  0.0805  0.1305  13.05%
13-33
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)
Security
Beta
Expected Return
DCLK
2.685
0.0213 + 2.685(0.086) = 25.22%
KO
0.195
0.0213 + 0.195(0.086) = 3.81%
INTC
2.161
0.0213 + 2.161(0.086) = 20.71%
KEI
2.434
0.0213 + 2.434(0.086) = 23.06%
13-34
Security Market Line
• 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.
13-35
Security Market Line (SML)
Asset expected
return (E (Ri))
SM
L
A - undervalued
= E (RM) – Rf
E (RA)
E (RB)
B - overvalued
E (RM)
market
Rf
M
=
A
B
Asset
beta (i)
1.0
13-36
Reward to Risk Ratio
• SML slope = Reward to Risk Ratio = Market Risk
Premium
E(R A )  Rf E(R B )  Rf
E(R M  Rf )

 ..... 
 E(R M  Rf )
βA
βB
βM
• In equilibrium, all assets and portfolios must have
the same reward-to-risk ratio and they all must
equal the reward-to-risk ratio for the market
• If not, assets are undervalued or overvalued
13-37
Reward-to-Risk Ratio: Example
• If RM=12%, Rf = 6%
• Slope = (E(RM) – Rf)/M = market risk premium
• = (12% - 6%)/1 = 6%
• If asset A has E(RA) = 15%, βA = 1.3, and asset B
has E(RB) = 10%, βB = 0.8
E(R A )  R f 15%  6%

 6.9%
βA
1.3
E(R B )  R f 10%  6%

 5%
βB
0.8
• Asset B offers insufficient reward for its level of
risk, so B is relatively overvalued compared to A, or
A is relatively undervalued
13-38
CAPM and Beta of Portfolio
• If w1, w2, …, wn, are the proportions of the
portfolio invested in n assets 1, 2, …, n, the beta
of a portfolio (P) can be written:
βP  w 1β1  w 2β 2  ...  w nβn
j n
 Σ w jβ j
j1
• Example: If 30% of a portfolio is invested in
asset 1 and the balance in asset 2, and asset 1’s
beta=1.7 while asset 2’s beta =1.2, what is the
beta of the portfolio (P)
βP  w1β1  w 2β2  0.3(1.7)  0.7(1.2)  1.35
13-39
Example: Portfolio Beta
• Consider our previous four securities and their
betas:
Security
Weight
Beta
DCLK
KO
INTC
KEI
.133
.2
.167
.4
2.685
0.195
2.161
2.434
• What is the portfolio beta?
• 0.133(2.685) + 0.2(0.195) + 0.167(2.161) +
0.4(2.434) = 1.731
13-40
Factors Affecting E(R)
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 β
13-41
Quick Quiz
• 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?
13-42
Lecture 8 - Summary
• Expected returns, variances and standard
deviation
• Using probabilities
• Using historical returns
• For a single asset and for a portfolio
•
•
•
•
The principle of diversification
Systematic and Unsystematic risk
SML and CAPM
Reward to Risk ratio
13-43
End Lecture 8
13-44