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FINS5513 Lecture 3
CAPM and SIM
2❑ 3.1 General Equilibrium: Derivation of the CAPM
➢ Capital Asset Pricing Model (CAPM) Assumptions
➢ The Market Portfolio and Capital Market Line
➢ Derivation of the CAPM
❑ 3.2 Interpreting the CAPM
➢ Beta
➢ Security Market Line
➢ Applications and Extensions of the CAPM
❑ 3.3 The Single Index Model (SIM)
➢ Limitations of the CAPM
➢ SIM
➢ Alpha
➢ Decomposition of the total risk
Lecture Outline
3.1 General
Equilibrium: Derivation
of the CAPM
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4Capital Asset Pricing Model
(CAPM) Assumptions
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5❑ Markowitz established modern portfolio theory through important contributions in 1952
and 1959
❑ Based on this work, Sharpe, Lintner, Mossin and others published papers between 1964-
1966 which collectively became known as the Capital Asset Pricing Model (CAPM)
❑ CAPM is a model for deriving expected returns on risky assets under equilibrium
conditions
❑ CAPM is derived under a general equilibrium framework.The assumptions under
which the CAPM is derived simplify the world with regard to:
➢ Individual behaviour
➢ Market structure
❑ The CAPM assumptions are often considered to be stylised and not reflective of reality
Capital Asset Pricing Model (CAPM) Evolution
6❑ Individual behaviour
➢ Investors are rational, mean-variance optimisers (as per Markowitz)
➢ Investors are price takers - no investor is large enough to influence equilibrium prices
➢ Investors common planning horizon is a single period
➢ Investors have homogeneous expectations on the statistical properties of all assets (i.e.
same expected returns and covariances and all relevant information is publicly
available)
❑ Market structure
➢ Investors can borrow and lend at a common risk-free rate with no borrowing constraints
➢ All assets are publicly held and traded on public exchanges
➢ Perfect capital markets - there are no financial frictions such as short selling
constraints, transaction costs, taxes etc
❑ Under these assumptions, all investors derive the same efficient frontier, CAL, and ∗
CAPM Assumptions
7The Market Portfolio and
Capital Market Line
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8❑ The foundations of the CAPM come from Markowitz and separation theorem:
➢ Sharpe began with the question “What if everyone is optimising a la Markowitz?”
❑ Under separation theorem, EVERY rational investor invests along the CAL, regardless of
risk aversion
➢ because all portfolio combinations on the CAL have the highest Sharpe ratio
❑ Therefore, in market equilibrium all investors hold the same optimal risky portfolio ∗
❑ If, in equilibrium, all investors are holding the same ∗, this must be the market portfolio
, and must comprise all assets
➢ The weight of each asset in is the asset’s market value divided by the total value of
➢ Every investor holds some portion of this market portfolio
❑ Since ∗ is the market portfolio , also has the highest possible Sharpe ratio
Separation Theorem Implications
9❑ The rational way to increase return (and risk) is to invest more in (rather than deviating
from and buying risky assets in different weightings to their weightings in )
❑ Since every investor invests in , the common CAL associated with is called the Capital
Market Line (CML)
The Capital Market Line
❑ The CML is equivalent to the optimal CAL. It is
the aggregation of EVERY investors’ CAL
which are all the same
❑ is equivalent to ∗. It is the aggregation of
EVERY investor’s optimal risky portfolio ∗
which are all the same
❑ Every investor invests along the CML.
Movements along the CML represent different
allocations between and the risk-free asset
10
❑ An investor can implement an entirely passive strategy requiring no security analysis
❑ Requires two assets:
➢ Risk-free: short-term T-Bills or money market mutual fund
➢ Risky: mutual fund or an ETF tracking a broad based market index (eg the S&P500)
❑ Then, draw a line joining the two assets. This represents is a practical version of the CML
❑ Example: According to BKM Table 6.7, the U.S. equity market returned 11.72% with a
standard deviation of 20.36% (assume for simplicity this represents the Market portfolio
although it is only a proxy) and 1-month T-bills (the risk-free asset) returned 3.38% between
1926 and 2018.
a) Draw the Capital Market Line. What is the slope of this line and what does it represent?
b) Aggressive Investor A and Conservative Investor C target maximum portfolio risk of 25%
and 14% respectively. For investors A and C: calculate their allocation to the equity market
(∗) and T-bills (1 − ∗), the expected return on their complete portfolio (∗), the Sharpe
ratio of their complete portfolio, and their implied risk aversion coefficient .
Example: A Practical CML
11
➢ The slope of the CML is the market’s Sharpe ratio
= 0.1172 − 0.0338 = 8.34%
ℎ =
0.0834
0.2036
= 0.41
Example: A Practical CML
0.0%
2.0%
4.0%
6.0%
8.0%
10.0%
12.0%
14.0%
16.0%
0.000 0.050 0.100 0.150 0.200 0.250 0.300
P
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rt
fo
lio
E
xp
e
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e
d
R
e
tu
rn
E
(r
)
Portfolio Standard Deviation σ
Capital Market Line
Investor A
Investor C
Market
– T-bills
Investor A Investor C
% in market 0.25
0.2036
= 122.8%
0.14
0.2036
= 68.8%
% in T-bills 1 − 1.228 = −22.8%
i.e., Borrow 22.8% at
1 − 0.688 = 31.2%
i.e., Invest 31.2% in T-
Bills
Expected Return
()
0.0338 + 1.228 × 0.0834
= 13.62%
0.0338 + 0.688 × 0.0834
= 9.11%
Risk 25% 14%
Sharpe Ratio 0.1362 − 0.0338
0.25
= 0.41
0.0911 − 0.0338
0.14
= 0.41
Risk Aversion
()
0.0834
1.228 × 0.20362
= 1.64
0.0834
0.688 × 0.20362
= 2.93
12
❑ We have already concluded that under the CAPM assumptions, every investor would
hold for their risky asset allocation. Therefore:
➢ The attractiveness of an individual asset should be assessed based on its
contribution to market return and market risk
• Individual asset contribution to market return simply depends on its return (and
weight)
• However, its contribution to market risk depends on its covariance with all
other assets in the market
❑ So the risk of an individual asset is no longer measured just by its own variance, but
rather its covariance with the market
➢ In equilibrium, this is now our key measure of risk
➢ This is a key insight of the CAPM
The Market Portfolio
13
❑ Let’s restate the previous slide mathematically (using risk premium instead or raw return):
❑ The risk premium on the market portfolio (RM ) is: = σ=1
. So, we have
() =
=1
()
ℎ =
σ=1
ℎ ℎ
➢ So, asset ’s contribution to ’s risk premium is ()
❑ The risk on the market portfolio (
2 ) is:
2 = , =
=1
, =
=1
,
➢ So, asset ’s contribution to ’s risk (variance) is ,
Under CAPM Risk is Measured by Covariance
14
Derivation of the CAPM
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15
❑ Where have the CAPM assumptions taken us so far?:
1. In equilibrium, all rational investors would hold
2. If investors are holding , the risk measure for an individual asset is its covariance with
❑ From our previous slide, given individual asset risk is measured as covariance with the
market, let’s define a reward-to-risk ratio for any asset as:
′
′
=
()
( , )
=
()
( , )
❑ Next, consider this: in a world of homogeneous expectations, there would be no reason for an
investor to buy an asset with a lower reward-to-risk ratio than another asset
➢ If one asset had a higher reward-to-risk ratio than another, all investors would buy it, until
it’s reward-to-risk ratio was in parity with all other assets
• Therefore, in equilibrium, all assets should have the same reward-to-risk ratio:
()
(, )
=
()
(, )
(, …)
Deriving the CAPM
16
❑ In equilibrium, would have the highest Sharpe ratio and would therefore also have the
highest reward-to-risk ratio given by:
()
( , )
=
()
2
❑ In equilibrium, any individual asset’s reward-to-risk ratio should equal ’s reward-to-risk ratio
(the highest possible reward-to-risk ratio). If this were not the case, the price of the asset
would adjust until its ratio is in parity with all other assets within and with itself:
()
( , )
=
2
Rearranging: =
,
2 ()
Defining: =
,
2 Then: =
Often stated as the CAPM equation: − = [ − ]
Deriving the CAPM
3.2 Interpreting the
CAPM
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18
❑ Let’s review the CAPM equation: − = [ − ]
❑ So, asset ’s risk premium equals: ×
➢ Risk: measured by its relative contribution to the variance of (. . , ):
• =
,
2 which tells us how asset moves with
➢ Price of Risk: measured by the market risk premium (i.e., − )
• There is one price of risk in the market
❑ In equilibrium, more risky assets are compensated with higher expected returns to
make them equally favourable to less risky ones
➢ One of the reasons the CAPM is widely used is because it is simple and intuitive
➢ It basically states that asset ’s risk premium is linearly related to the market risk
premium according to it’s
Interpretation of the CAPM
19
❑ The market risk premium − can be regarded as the average risk premium among all
assets in the market
❑ The market risk premium depends on the average risk aversion of all market participants
➢ Recall that each individual investor chooses an allocation to the optimal portfolio (in a
general equilibrium) such that
=
−
2 ,
ℎ ℎ ℎ
➢ In the CAPM economy, all borrowing is offset by lending, so on average = 1
Therefore: =
2
➢ Therefore, the market risk premium tends to expand when average risk aversion in
the market rises (to encourage risk taking) and contract when average market risk
aversion falls
What determins Market Risk Premium?
20
Beta
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21
❑ An asset’s measures how much risk the assets contributes in the market portfolio,
relative to the contribution of the average asset (i.e., the market portfolio – which is
the weighted average of all risky assets)
➢ > 1 means that , >
2 - the asset contributes more risk than the
average asset (or is riskier than the market)
➢ < 1 means that , <
2 - the asset contributes less risk than the
average asset (or is less risky than the market)
❑ One of the most important concepts in finance and widely used in industry
What is ?
22
❑ Assume in a CAPM equilibrium the market’s expected return () = 11%, the risk-free
rate = 3% and the market’s variance
2 = 0.0256. Applying CAPM, Asset A has
expected return of () = 12.6% and Asset B has expected return of () = 8.6%.
a) What is the market risk premium?
b) What is the of Asset A and B
c) What is the covariance of Asset A and B with the market return (respectively)?
➢ = 0.11 – 0.03 = 8.0%
➢ = 0.126 – 0.03 = 9.6%; so =
()−
()−
=
0.096
0.08
= .
= 0.086 – 0.03 = 5.6%; so =
()−
()−
=
0.056
0.08
= .
➢ Recall that =
,
2 so:
Asset A: , = 1.2 × 0.0256 = 0.03072
Asset B: , = 0.7 × 0.0256 = 0.01792
Example
23
❑ For the Market Portfolio itself, the CAPM simplifies to:
=
( , )
2 = 1
= + − = + 1 × − = ()
❑ For the risk-free asset, the CAPM simplifies to:
=
( , )
2 = 0
= + − = + 0 × − =
❑ () and are, of course, set by the market - we take them as given. The CAPM
then prices all other assets relative to them
Market and Risk-Free Asset
24
Security Market Line
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25
❑ The Security Market Line (SML) is captured in the − space
➢ Investors will demand higher expected returns of assets that contribute more risk in the
portfolio
Security Market Line
rf
M
i
( )irE
SML
1=M0=f
fM rrESlope −= )(
26
❑ The SML shows a positive relationship between systematic risk and expected excess
return above . The slope of the SML is the market risk premium (the “MRP”) – the
market price of risk
❑ We can interpret the SML as follows:
➢ An asset with = 0.5 contributes half the risk of the average asset, and thus gets
half reward in terms of excess returns
➢ An asset with = 2 contributes twice the risk of the average asset, and thus gets
twice the reward in terms of excess returns
Interpreting the SML
27
Applications and Extensions
of the CAPM
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❑ The CAPM is an equilibrium model
➢ In equilibrium, the CAPM predicts that all rational investors have diversified and eliminated
unsystematic risk
❑ The CAPM therefore only prices systematic risk
➢ If all investors hold and includes all assets, by definition all investors have eliminated
unsystematic risk
➢ Investors are therefore not compensated with a higher return for taking on unsystematic risk
(because it should be diversified away in equilibrium)
➢ Systematic risk is measured by
❑ The implication is that we should hold well-diversified portfolios
❑ The CAPM recommends a passive strategy – i.e., hold the market portfolio (“meet the market,
don’t try to beat the market”)
CAPM Prices Systematic Risk
29
Despite its limitations, the CAPM is widely used in a number of applications:
❑ The CAPM has implications for portfolio construction:
➢ Avoid unsystematic risk by holding a well-diversified portfolio
➢ Monitor the amount of systematic risk (beta) in the portfolio
❑ The CAPM can be used to evaluate investment performance:
➢ Relate returns to a benchmark after adjusting for systematic risk
➢ Examine whether return performance relative to the benchmark is due to systematic or
unsystematic factors (which assists in determining if good or bad returns are due to
luck)
❑ The CAPM is increasingly used in corporate finance:
➢ Estimate the for assets and projects using comparable listed asset ’s
➢ Use the resulting cost of equity or WACC as the discount rate to discount estimated
future cash flows and derive the asset/project NPV
CAPM Applications
30
❑ Heterogeneous expectations
➢ Investors with different expectations offset - although non-symmetry of shorting vs buying,
and taxes complicates models (there are extensions of the CAPM which incorporate taxes)
❑ Zero-beta model and borrowing constraints
➢ Borrowing constraints mean investors cannot borrow at the risk-free rate
➢ A portfolio can be constructed which has no covariance with the market (zero beta)
❑ Labour income and other nontraded assets (Mayers)
❑ Multiperiod models and hedge portfolios
➢ Merton’s Intertemporal CAPM derives similar conclusions in a multiperiod model where
investors are less concerned with dollar wealth than the stream of consumption it can buy
❑ Liquidity
➢ The speed with which assets can be turned into cash has a material impact on value.
Therefore, there may also be a “liquidity beta”
Extensions of the CAPM
31
❑ In a multi-period framework, what matters to investors is their lifetime flow of consumption
❑ Rather than measuring risk as the covariance with market return, risk is measured as the
covariance of returns with aggregate consumption
➢ Replace the market portfolio with a portfolio that tracks changes in aggregate consumption
➢ Assets with positive covariance with consumption will underperform in recessions and
outperform in booms
• These assets will be viewed as riskier and have higher risk premiums
• That is, they will have higher “consumption betas” which result in higher risk premiums
➢ The consumption portfolio can be constructed while the true market portfolio cannot
❑ One problem with this model is that there is little variance in consumption and data is
released infrequently, unlike markets
➢ Therefore studies tend to use factor portfolios rather than consumption directly
Consumption CAPM
3.3 The Single Index
Model (SIM)
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33
Limitations of the CAPM
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34
❑ CAPM relies on several strong assumptions that may not hold in reality:
➢ Are all investors rational mean-variance optimizers?
➢ Some investors may exhibit behavioural bias, and make irrational investment decisions
❑ Homogeneous expectations and public tradability of all assets are not realistic
➢ The existence of one optimal risky portfolio which all investors agree on is unrealistic (if we
had homogeneous expectations investors would almost never trade)
➢ As investors have different views there will be multiple optimal risky portfolios and SMLs
❑ Beta estimates are problematic and may be time-varying (which the CAPM does not preclude)
➢ Betas measured over 5 years can be dramatically different to betas measured over 1 year
➢ Betas measured using daily data may differ to monthly
❑ Use of historical data to predict future returns can be particularly misleading (or as Sharpe
himself says “take historic data with a lot of grains of salt”)
Limitations of the CAPM
35
❑ Under the CAPM, the “true” market portfolio includes all risky assets, encompassing any
asset with marketable value eg collectibles, an individuals’ human capital, privately owned
small businesses etc and all these assets are tradeable
➢ The “true” market portfolio does not exist in reality because not all assets are tradeable and
there is no such market index or portfolio which encompasses all assets
❑ Roll’s critique showed that if no such observable exists, the CAPM itself cannot be tested
❑ Researchers accept is not observable, but still try to test whether the CAPM predictions (a
linear relationship between risk premium and beta) can be validated in a particular market:
➢ Researchers pick a proxy for when testing the linearity of the beta-risk premium
relationship (as predicted by the SML) – using APT rather than CAPM
➢ A common proxy is a broad, value-weighted market index e.g., S&P 500, ASX 200 etc
➢ Regardless of the results of these tests, they are not tests of the validity of the
CAPM itself (based on Roll’s critique) because the true market portfolio is not being
used
The CAPM is Untestable
36
The Single Index Model (SIM)
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37
❑ The true market portfolio is not observable
❑ We try to pick a proxy for M when using the CAPM in practice
➢ A common proxy for is a broad value-weighted market index, like the S&P 500
for the US, the ASX 200 for Australia, or the MSCI Global Index for global stocks
❑ To emphasise that we are dealing with a proxy – a single market using a single index
(and not the true market portfolio ) - we refer to the resulting empirical model as the
Single Index Model, or the SIM
❑ CAPM is an equilibrium model while SIM is an empirical model
Using a Proxy for
38
❑Recall the CAPM equation (i.e., Theoretical Model): () − = [() – ]. In
practice, we are interested in applying CAPM (e.g., to derive the implied beta based
on historical time-series returns). So, let’s express CAPM in a regression format (i.e.,
Empirical Model), which we called Single Index Model:
− = + − + , ℎ = 0; , = 0
➢ =
➢ =
➢ =
➢ =
➢ =
➢ =
❑We are interested in variations in both the cross section and over time. That
is why we have notation assets () over time () in SIM.
Single Index Model (SIM)
39
❑About regression assumptions:
➢ = 0 – the residuals would offset each other and sum to ≈0 (negative
residuals cancel out positive residuals); and
➢ , = 0 – the residuals would have no covariance with the market return
(or else they would be captured in the beta ). This is the key assumption to make
sure the unbiasedness of the regression estimate
Single Index Model (SIM)
40
❑ SIM equation: − = + − + , ℎ = 0; , = 0
❑ Let’s take Expectation on both side of SIM equation to “convert” SIM back to CAPM:
− = [+ − + ]
❑ Recall Expectation is a linear operator (i.e., + = + ), thus we have:
− = + − + ()
❑ Recall , , are constant, and = 0, thus we have:
− = + [ − ]
= + + −
❑ Recall CAPM equation: = + − , thus we have:
= +
❑ So, what is the meaning of ????
Relation between CAPM and SIM
41
Alpha
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42
❑ So, what is alpha ?
➢ We refer to as Jensen’s alpha or simply as the “alpha”
➢ If ≠ 0, the asset is mispriced based on the CAPM, and is a measure of this
mispricing
❑ Notice: There could be two interpretation on a non-zero :
1. The asset is correctly priced, but the model is wrong (i.e., model does not describe
the full relationship between the variables)
2. The model is correct, but the asset is mispriced
❑ Unfortunately, we do not know what the “correct” model is. Often, a non-zero indicates
that there’s something wrong with the model. However, let’s assume that the model is
correct and a non-zero indicates asset mispricing:
• When > 0 the asset is under-priced (the asset has greater return than predicted)
• When < 0 the asset is over-priced (the asset has less return than predicted)
❑ We will discuss how to explore the mispriced assets in the next lecture.
Jensen’s Alpha
43
❑ Assets correctly priced by the CAPM plot on the SML in equilibrium
❑ In the graph below, assets A and C are mispriced i.e., ≠ 0
➢ In a CAPM equilibrium, this should not occur
➢ If includes mispriced assets, it is no longer the portfolio with the highest Sharpe ratio (we
will discuss more about exploring mispriced assets in next lecture)
❑ If alpha persists either our single factor model does not capture all variables impacting price
(add more factors/independent variables) or our model is correct, and the asset is mispriced
Mispriced Assets
rf
M
( )irE
SML
A
B
C
αA < 0
αC > 0
For Asset A
< 0 ∴ it is
overpriced
For Asset C
> 0 ∴ it is
underpriced
44
❑ The average annualised return for Warren Buffet’s Berkshire Hathaway A shares
(BRK.A) between October 1976 and March 2017 was 23.14% p.a. with = 0.698.
The average annualised market return in this period was 12.04% and the average T-
Bill rate was = 4.56%. What has been BRK.A’s in this period? Plot the Security
Market Line and show where BRK.A sits in relation to it. Should this be happening in
efficient markets?
➢ = − + − = 0.2314 − [
]
0.0456 + 0.698 × (
)
0.1204 −
0.0456 = 13.36%
➢ This should not be happening in efficient markets and is inconsistent with a CAPM
equilibrium, as alpha has been persistent for over 40 years for BRK
Example: Alpha
45
Decomposition of the total risk
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46
❑ Recall SIM: − = + − + . Let’s calculate the variance of asset :
= + + − +
❑ Recall, , , are constant, thus we have:
= − + =
2 − + + 2( − , )
=
2 + + 2(, )
❑ Recall regression assumption: , = 0, thus we have
=
2 +
❑ Therefore, the total risk (i.e.,
2) is decomposed into two components:
➢ Systematic risk/market risk:
2
2
➢ Unsystematic risk/idiosyncratic risk:
2
❑ The ratio of systematic risk / total risk is the R-Squared given by: 2 =
2
2
2 = , 2
➢ The proportion of the movement in asset which is caused by market movements
Decomposition of the total risk
47
❑ The CAPM/SIM framework allows us to bifurcate our analysis of risk
❑ Systematic Risk / Market Risk / Non-diversifiable Risk
➢ Risk attributable to market-wide (macro) factors which remain even after diversification
➢ The part of an asset’s risk that is common with the market, and thus unable to be
diversified
➢ Examples: market-wide events such as an oil embargo, interest rate rise, recession,
pandemic, natural disaster, political crisis, exchange rate collapse
❑ Unsystematic Risk / Firm-Specific Risk / Diversifiable Risk / Idiosyncratic Risk
➢ The part of an asset’s risk that is specific to the asset itself
➢ Since this risk comes from sources that do not affect the whole market, it can be
reduced to zero or negligible levels in a diversified portfolio
➢ Examples: a retailer launches a poor product, misses sale expectations, or loses a
major customer. This would be bad for the retailer’s stock, but it’s competitors may
actually benefit
Revisit Diversification and Portfolio Risk
48
❑ As we add more assets to our portfolio through diversification, we eliminate unsystematic
(firm-specific) risk and we converge toward the systematic (market) risk level – which cannot
be diversified away
Systematic and Unsystematic Risk
Standard
Deviation
Number of stocks
Systematic (market) risk
Non-systematic (firm-specific) risk
49
❑ Under SIM, we can estimate covariance between any two assets through their ′s:
, =
2 , =
❑ Correlation between two assets is simply the product of their correlation with the
market and is given by: , =
2
=
2
2
= , × ,
➢ All correlation can be related back to the market under SIM
❑ SIM/CAPM imposes additional assumption on the variance-covariance structure,
which is the key difference between SIM framework and Markowitz framework.
Covariance calculation under SIM
50
❑ Using the information in the table below from the 1976-2017 study:
a) Calculate BRK’s systematic and unsystematic risk and R2?
b) What is BRK’s covariance and correlation with the market return?
c) Using the information in the table below, calculate the covariance and correlation of
BRK with each of these stocks
➢ Systematic risk:
2
2 = (0.698)2× (0.153)2= 0.011113
➢ Unsystematic risk:
2 = (0.24)2− 0.698 2 × (0.153)2= 0.046487
➢
2 =
0.011113
0.242
= 19.29%
Example: SIM Risk Measures
Stock Beta Standard Deviation
Market 1.00 15.3%
BRK (BRK.A) 0.698 24.0%
Apple (AAPL) 1.15 24.0%
Microsoft (MSFT) 1.28 22.8%
Amazon (AMZN) 1.35 30.6%
51
➢ Covariance: ( , ) = ×
2 = (0.698) (0.153)2 = 0.0163
➢ Correlation: , =
(,)
=
0.0163
[0.24×0.153]
= 0.445
Example: SIM Risk Measures
Stock Beta Covariance/Correlation BRK
Apple (AAPL) 1.15 • , = 0.698 × 1.15 × 0.153
2 = 0.01879
• , =
0.01879
0.24×0.24
= 0.3262
Microsoft
(MSFT)
1.28 • , = 0.698 × 1.28 × 0.153
2 = 0.02091
• , =
0.02091
0.24× 0.228
= 0.3821
Amazon
(AMZN)
1.35 • , = 0.698 × 1.35 × 0.153
2 = 0.02206
• , =
0.02206
0.24×0.306
= 0.3004
52
❑ Both the CML and SML start from , but the CML accounts for total risk () while the SML only
accounts for systematic risk ()
SML vs CML
E(r)
M
CML
Efficient
Frontier
rf
E(r)
SML
β
βM=1
A •
P
rf
A
M
P
βPβA
53
❑ CML’s -axis is but SML’s -axis is
❑ The of the market portfolio is always 1 but may change. The SML does not
show the level of as it scales risk to 1
➢ A change in has no effect on the SML – except where it is accompanied by a
change in () – but it would change the slope of the CML
❑ () is the same for the CML and SML
❑ The two graphs have different purposes:
➢ CML (and CAL) is more of a portfolio construction tool
➢ SML is more of a valuation tool, separating fairly priced assets (which plot on the
SML) from mispriced assets (which plot over or under the SML) in a CAPM
equilibrium
SML vs CML
54
❑ BKM Chapter 8
❑ 4.1 SIM Regression: The Security Characteristic Line
❑ 4.2 Portfolio Construction Under SIM vs Markowitz
❑ 4.3 What is Active Investing?
❑ 4.4 Active Portfolio Construction
Next Lecture
CAPM and SIM
2❑ 3.1 General Equilibrium: Derivation of the CAPM
➢ Capital Asset Pricing Model (CAPM) Assumptions
➢ The Market Portfolio and Capital Market Line
➢ Derivation of the CAPM
❑ 3.2 Interpreting the CAPM
➢ Beta
➢ Security Market Line
➢ Applications and Extensions of the CAPM
❑ 3.3 The Single Index Model (SIM)
➢ Limitations of the CAPM
➢ SIM
➢ Alpha
➢ Decomposition of the total risk
Lecture Outline
3.1 General
Equilibrium: Derivation
of the CAPM
FINS5513
4Capital Asset Pricing Model
(CAPM) Assumptions
FINS5513
5❑ Markowitz established modern portfolio theory through important contributions in 1952
and 1959
❑ Based on this work, Sharpe, Lintner, Mossin and others published papers between 1964-
1966 which collectively became known as the Capital Asset Pricing Model (CAPM)
❑ CAPM is a model for deriving expected returns on risky assets under equilibrium
conditions
❑ CAPM is derived under a general equilibrium framework.The assumptions under
which the CAPM is derived simplify the world with regard to:
➢ Individual behaviour
➢ Market structure
❑ The CAPM assumptions are often considered to be stylised and not reflective of reality
Capital Asset Pricing Model (CAPM) Evolution
6❑ Individual behaviour
➢ Investors are rational, mean-variance optimisers (as per Markowitz)
➢ Investors are price takers - no investor is large enough to influence equilibrium prices
➢ Investors common planning horizon is a single period
➢ Investors have homogeneous expectations on the statistical properties of all assets (i.e.
same expected returns and covariances and all relevant information is publicly
available)
❑ Market structure
➢ Investors can borrow and lend at a common risk-free rate with no borrowing constraints
➢ All assets are publicly held and traded on public exchanges
➢ Perfect capital markets - there are no financial frictions such as short selling
constraints, transaction costs, taxes etc
❑ Under these assumptions, all investors derive the same efficient frontier, CAL, and ∗
CAPM Assumptions
7The Market Portfolio and
Capital Market Line
FINS5513
8❑ The foundations of the CAPM come from Markowitz and separation theorem:
➢ Sharpe began with the question “What if everyone is optimising a la Markowitz?”
❑ Under separation theorem, EVERY rational investor invests along the CAL, regardless of
risk aversion
➢ because all portfolio combinations on the CAL have the highest Sharpe ratio
❑ Therefore, in market equilibrium all investors hold the same optimal risky portfolio ∗
❑ If, in equilibrium, all investors are holding the same ∗, this must be the market portfolio
, and must comprise all assets
➢ The weight of each asset in is the asset’s market value divided by the total value of
➢ Every investor holds some portion of this market portfolio
❑ Since ∗ is the market portfolio , also has the highest possible Sharpe ratio
Separation Theorem Implications
9❑ The rational way to increase return (and risk) is to invest more in (rather than deviating
from and buying risky assets in different weightings to their weightings in )
❑ Since every investor invests in , the common CAL associated with is called the Capital
Market Line (CML)
The Capital Market Line
❑ The CML is equivalent to the optimal CAL. It is
the aggregation of EVERY investors’ CAL
which are all the same
❑ is equivalent to ∗. It is the aggregation of
EVERY investor’s optimal risky portfolio ∗
which are all the same
❑ Every investor invests along the CML.
Movements along the CML represent different
allocations between and the risk-free asset
10
❑ An investor can implement an entirely passive strategy requiring no security analysis
❑ Requires two assets:
➢ Risk-free: short-term T-Bills or money market mutual fund
➢ Risky: mutual fund or an ETF tracking a broad based market index (eg the S&P500)
❑ Then, draw a line joining the two assets. This represents is a practical version of the CML
❑ Example: According to BKM Table 6.7, the U.S. equity market returned 11.72% with a
standard deviation of 20.36% (assume for simplicity this represents the Market portfolio
although it is only a proxy) and 1-month T-bills (the risk-free asset) returned 3.38% between
1926 and 2018.
a) Draw the Capital Market Line. What is the slope of this line and what does it represent?
b) Aggressive Investor A and Conservative Investor C target maximum portfolio risk of 25%
and 14% respectively. For investors A and C: calculate their allocation to the equity market
(∗) and T-bills (1 − ∗), the expected return on their complete portfolio (∗), the Sharpe
ratio of their complete portfolio, and their implied risk aversion coefficient .
Example: A Practical CML
11
➢ The slope of the CML is the market’s Sharpe ratio
= 0.1172 − 0.0338 = 8.34%
ℎ =
0.0834
0.2036
= 0.41
Example: A Practical CML
0.0%
2.0%
4.0%
6.0%
8.0%
10.0%
12.0%
14.0%
16.0%
0.000 0.050 0.100 0.150 0.200 0.250 0.300
P
o
rt
fo
lio
E
xp
e
ct
e
d
R
e
tu
rn
E
(r
)
Portfolio Standard Deviation σ
Capital Market Line
Investor A
Investor C
Market
– T-bills
Investor A Investor C
% in market 0.25
0.2036
= 122.8%
0.14
0.2036
= 68.8%
% in T-bills 1 − 1.228 = −22.8%
i.e., Borrow 22.8% at
1 − 0.688 = 31.2%
i.e., Invest 31.2% in T-
Bills
Expected Return
()
0.0338 + 1.228 × 0.0834
= 13.62%
0.0338 + 0.688 × 0.0834
= 9.11%
Risk 25% 14%
Sharpe Ratio 0.1362 − 0.0338
0.25
= 0.41
0.0911 − 0.0338
0.14
= 0.41
Risk Aversion
()
0.0834
1.228 × 0.20362
= 1.64
0.0834
0.688 × 0.20362
= 2.93
12
❑ We have already concluded that under the CAPM assumptions, every investor would
hold for their risky asset allocation. Therefore:
➢ The attractiveness of an individual asset should be assessed based on its
contribution to market return and market risk
• Individual asset contribution to market return simply depends on its return (and
weight)
• However, its contribution to market risk depends on its covariance with all
other assets in the market
❑ So the risk of an individual asset is no longer measured just by its own variance, but
rather its covariance with the market
➢ In equilibrium, this is now our key measure of risk
➢ This is a key insight of the CAPM
The Market Portfolio
13
❑ Let’s restate the previous slide mathematically (using risk premium instead or raw return):
❑ The risk premium on the market portfolio (RM ) is: = σ=1
. So, we have
() =
=1
()
ℎ =
σ=1
ℎ ℎ
➢ So, asset ’s contribution to ’s risk premium is ()
❑ The risk on the market portfolio (
2 ) is:
2 = , =
=1
, =
=1
,
➢ So, asset ’s contribution to ’s risk (variance) is ,
Under CAPM Risk is Measured by Covariance
14
Derivation of the CAPM
FINS5513
15
❑ Where have the CAPM assumptions taken us so far?:
1. In equilibrium, all rational investors would hold
2. If investors are holding , the risk measure for an individual asset is its covariance with
❑ From our previous slide, given individual asset risk is measured as covariance with the
market, let’s define a reward-to-risk ratio for any asset as:
′
′
=
()
( , )
=
()
( , )
❑ Next, consider this: in a world of homogeneous expectations, there would be no reason for an
investor to buy an asset with a lower reward-to-risk ratio than another asset
➢ If one asset had a higher reward-to-risk ratio than another, all investors would buy it, until
it’s reward-to-risk ratio was in parity with all other assets
• Therefore, in equilibrium, all assets should have the same reward-to-risk ratio:
()
(, )
=
()
(, )
(, …)
Deriving the CAPM
16
❑ In equilibrium, would have the highest Sharpe ratio and would therefore also have the
highest reward-to-risk ratio given by:
()
( , )
=
()
2
❑ In equilibrium, any individual asset’s reward-to-risk ratio should equal ’s reward-to-risk ratio
(the highest possible reward-to-risk ratio). If this were not the case, the price of the asset
would adjust until its ratio is in parity with all other assets within and with itself:
()
( , )
=
2
Rearranging: =
,
2 ()
Defining: =
,
2 Then: =
Often stated as the CAPM equation: − = [ − ]
Deriving the CAPM
3.2 Interpreting the
CAPM
FINS5513
18
❑ Let’s review the CAPM equation: − = [ − ]
❑ So, asset ’s risk premium equals: ×
➢ Risk: measured by its relative contribution to the variance of (. . , ):
• =
,
2 which tells us how asset moves with
➢ Price of Risk: measured by the market risk premium (i.e., − )
• There is one price of risk in the market
❑ In equilibrium, more risky assets are compensated with higher expected returns to
make them equally favourable to less risky ones
➢ One of the reasons the CAPM is widely used is because it is simple and intuitive
➢ It basically states that asset ’s risk premium is linearly related to the market risk
premium according to it’s
Interpretation of the CAPM
19
❑ The market risk premium − can be regarded as the average risk premium among all
assets in the market
❑ The market risk premium depends on the average risk aversion of all market participants
➢ Recall that each individual investor chooses an allocation to the optimal portfolio (in a
general equilibrium) such that
=
−
2 ,
ℎ ℎ ℎ
➢ In the CAPM economy, all borrowing is offset by lending, so on average = 1
Therefore: =
2
➢ Therefore, the market risk premium tends to expand when average risk aversion in
the market rises (to encourage risk taking) and contract when average market risk
aversion falls
What determins Market Risk Premium?
20
Beta
FINS5513
21
❑ An asset’s measures how much risk the assets contributes in the market portfolio,
relative to the contribution of the average asset (i.e., the market portfolio – which is
the weighted average of all risky assets)
➢ > 1 means that , >
2 - the asset contributes more risk than the
average asset (or is riskier than the market)
➢ < 1 means that , <
2 - the asset contributes less risk than the
average asset (or is less risky than the market)
❑ One of the most important concepts in finance and widely used in industry
What is ?
22
❑ Assume in a CAPM equilibrium the market’s expected return () = 11%, the risk-free
rate = 3% and the market’s variance
2 = 0.0256. Applying CAPM, Asset A has
expected return of () = 12.6% and Asset B has expected return of () = 8.6%.
a) What is the market risk premium?
b) What is the of Asset A and B
c) What is the covariance of Asset A and B with the market return (respectively)?
➢ = 0.11 – 0.03 = 8.0%
➢ = 0.126 – 0.03 = 9.6%; so =
()−
()−
=
0.096
0.08
= .
= 0.086 – 0.03 = 5.6%; so =
()−
()−
=
0.056
0.08
= .
➢ Recall that =
,
2 so:
Asset A: , = 1.2 × 0.0256 = 0.03072
Asset B: , = 0.7 × 0.0256 = 0.01792
Example
23
❑ For the Market Portfolio itself, the CAPM simplifies to:
=
( , )
2 = 1
= + − = + 1 × − = ()
❑ For the risk-free asset, the CAPM simplifies to:
=
( , )
2 = 0
= + − = + 0 × − =
❑ () and are, of course, set by the market - we take them as given. The CAPM
then prices all other assets relative to them
Market and Risk-Free Asset
24
Security Market Line
FINS5513
25
❑ The Security Market Line (SML) is captured in the − space
➢ Investors will demand higher expected returns of assets that contribute more risk in the
portfolio
Security Market Line
rf
M
i
( )irE
SML
1=M0=f
fM rrESlope −= )(
26
❑ The SML shows a positive relationship between systematic risk and expected excess
return above . The slope of the SML is the market risk premium (the “MRP”) – the
market price of risk
❑ We can interpret the SML as follows:
➢ An asset with = 0.5 contributes half the risk of the average asset, and thus gets
half reward in terms of excess returns
➢ An asset with = 2 contributes twice the risk of the average asset, and thus gets
twice the reward in terms of excess returns
Interpreting the SML
27
Applications and Extensions
of the CAPM
FINS5513
28
❑ The CAPM is an equilibrium model
➢ In equilibrium, the CAPM predicts that all rational investors have diversified and eliminated
unsystematic risk
❑ The CAPM therefore only prices systematic risk
➢ If all investors hold and includes all assets, by definition all investors have eliminated
unsystematic risk
➢ Investors are therefore not compensated with a higher return for taking on unsystematic risk
(because it should be diversified away in equilibrium)
➢ Systematic risk is measured by
❑ The implication is that we should hold well-diversified portfolios
❑ The CAPM recommends a passive strategy – i.e., hold the market portfolio (“meet the market,
don’t try to beat the market”)
CAPM Prices Systematic Risk
29
Despite its limitations, the CAPM is widely used in a number of applications:
❑ The CAPM has implications for portfolio construction:
➢ Avoid unsystematic risk by holding a well-diversified portfolio
➢ Monitor the amount of systematic risk (beta) in the portfolio
❑ The CAPM can be used to evaluate investment performance:
➢ Relate returns to a benchmark after adjusting for systematic risk
➢ Examine whether return performance relative to the benchmark is due to systematic or
unsystematic factors (which assists in determining if good or bad returns are due to
luck)
❑ The CAPM is increasingly used in corporate finance:
➢ Estimate the for assets and projects using comparable listed asset ’s
➢ Use the resulting cost of equity or WACC as the discount rate to discount estimated
future cash flows and derive the asset/project NPV
CAPM Applications
30
❑ Heterogeneous expectations
➢ Investors with different expectations offset - although non-symmetry of shorting vs buying,
and taxes complicates models (there are extensions of the CAPM which incorporate taxes)
❑ Zero-beta model and borrowing constraints
➢ Borrowing constraints mean investors cannot borrow at the risk-free rate
➢ A portfolio can be constructed which has no covariance with the market (zero beta)
❑ Labour income and other nontraded assets (Mayers)
❑ Multiperiod models and hedge portfolios
➢ Merton’s Intertemporal CAPM derives similar conclusions in a multiperiod model where
investors are less concerned with dollar wealth than the stream of consumption it can buy
❑ Liquidity
➢ The speed with which assets can be turned into cash has a material impact on value.
Therefore, there may also be a “liquidity beta”
Extensions of the CAPM
31
❑ In a multi-period framework, what matters to investors is their lifetime flow of consumption
❑ Rather than measuring risk as the covariance with market return, risk is measured as the
covariance of returns with aggregate consumption
➢ Replace the market portfolio with a portfolio that tracks changes in aggregate consumption
➢ Assets with positive covariance with consumption will underperform in recessions and
outperform in booms
• These assets will be viewed as riskier and have higher risk premiums
• That is, they will have higher “consumption betas” which result in higher risk premiums
➢ The consumption portfolio can be constructed while the true market portfolio cannot
❑ One problem with this model is that there is little variance in consumption and data is
released infrequently, unlike markets
➢ Therefore studies tend to use factor portfolios rather than consumption directly
Consumption CAPM
3.3 The Single Index
Model (SIM)
FINS5513
33
Limitations of the CAPM
FINS5513
34
❑ CAPM relies on several strong assumptions that may not hold in reality:
➢ Are all investors rational mean-variance optimizers?
➢ Some investors may exhibit behavioural bias, and make irrational investment decisions
❑ Homogeneous expectations and public tradability of all assets are not realistic
➢ The existence of one optimal risky portfolio which all investors agree on is unrealistic (if we
had homogeneous expectations investors would almost never trade)
➢ As investors have different views there will be multiple optimal risky portfolios and SMLs
❑ Beta estimates are problematic and may be time-varying (which the CAPM does not preclude)
➢ Betas measured over 5 years can be dramatically different to betas measured over 1 year
➢ Betas measured using daily data may differ to monthly
❑ Use of historical data to predict future returns can be particularly misleading (or as Sharpe
himself says “take historic data with a lot of grains of salt”)
Limitations of the CAPM
35
❑ Under the CAPM, the “true” market portfolio includes all risky assets, encompassing any
asset with marketable value eg collectibles, an individuals’ human capital, privately owned
small businesses etc and all these assets are tradeable
➢ The “true” market portfolio does not exist in reality because not all assets are tradeable and
there is no such market index or portfolio which encompasses all assets
❑ Roll’s critique showed that if no such observable exists, the CAPM itself cannot be tested
❑ Researchers accept is not observable, but still try to test whether the CAPM predictions (a
linear relationship between risk premium and beta) can be validated in a particular market:
➢ Researchers pick a proxy for when testing the linearity of the beta-risk premium
relationship (as predicted by the SML) – using APT rather than CAPM
➢ A common proxy is a broad, value-weighted market index e.g., S&P 500, ASX 200 etc
➢ Regardless of the results of these tests, they are not tests of the validity of the
CAPM itself (based on Roll’s critique) because the true market portfolio is not being
used
The CAPM is Untestable
36
The Single Index Model (SIM)
FINS5513
37
❑ The true market portfolio is not observable
❑ We try to pick a proxy for M when using the CAPM in practice
➢ A common proxy for is a broad value-weighted market index, like the S&P 500
for the US, the ASX 200 for Australia, or the MSCI Global Index for global stocks
❑ To emphasise that we are dealing with a proxy – a single market using a single index
(and not the true market portfolio ) - we refer to the resulting empirical model as the
Single Index Model, or the SIM
❑ CAPM is an equilibrium model while SIM is an empirical model
Using a Proxy for
38
❑Recall the CAPM equation (i.e., Theoretical Model): () − = [() – ]. In
practice, we are interested in applying CAPM (e.g., to derive the implied beta based
on historical time-series returns). So, let’s express CAPM in a regression format (i.e.,
Empirical Model), which we called Single Index Model:
− = + − + , ℎ = 0; , = 0
➢ =
➢ =
➢ =
➢ =
➢ =
➢ =
❑We are interested in variations in both the cross section and over time. That
is why we have notation assets () over time () in SIM.
Single Index Model (SIM)
39
❑About regression assumptions:
➢ = 0 – the residuals would offset each other and sum to ≈0 (negative
residuals cancel out positive residuals); and
➢ , = 0 – the residuals would have no covariance with the market return
(or else they would be captured in the beta ). This is the key assumption to make
sure the unbiasedness of the regression estimate
Single Index Model (SIM)
40
❑ SIM equation: − = + − + , ℎ = 0; , = 0
❑ Let’s take Expectation on both side of SIM equation to “convert” SIM back to CAPM:
− = [+ − + ]
❑ Recall Expectation is a linear operator (i.e., + = + ), thus we have:
− = + − + ()
❑ Recall , , are constant, and = 0, thus we have:
− = + [ − ]
= + + −
❑ Recall CAPM equation: = + − , thus we have:
= +
❑ So, what is the meaning of ????
Relation between CAPM and SIM
41
Alpha
FINS5513
42
❑ So, what is alpha ?
➢ We refer to as Jensen’s alpha or simply as the “alpha”
➢ If ≠ 0, the asset is mispriced based on the CAPM, and is a measure of this
mispricing
❑ Notice: There could be two interpretation on a non-zero :
1. The asset is correctly priced, but the model is wrong (i.e., model does not describe
the full relationship between the variables)
2. The model is correct, but the asset is mispriced
❑ Unfortunately, we do not know what the “correct” model is. Often, a non-zero indicates
that there’s something wrong with the model. However, let’s assume that the model is
correct and a non-zero indicates asset mispricing:
• When > 0 the asset is under-priced (the asset has greater return than predicted)
• When < 0 the asset is over-priced (the asset has less return than predicted)
❑ We will discuss how to explore the mispriced assets in the next lecture.
Jensen’s Alpha
43
❑ Assets correctly priced by the CAPM plot on the SML in equilibrium
❑ In the graph below, assets A and C are mispriced i.e., ≠ 0
➢ In a CAPM equilibrium, this should not occur
➢ If includes mispriced assets, it is no longer the portfolio with the highest Sharpe ratio (we
will discuss more about exploring mispriced assets in next lecture)
❑ If alpha persists either our single factor model does not capture all variables impacting price
(add more factors/independent variables) or our model is correct, and the asset is mispriced
Mispriced Assets
rf
M
( )irE
SML
A
B
C
αA < 0
αC > 0
For Asset A
< 0 ∴ it is
overpriced
For Asset C
> 0 ∴ it is
underpriced
44
❑ The average annualised return for Warren Buffet’s Berkshire Hathaway A shares
(BRK.A) between October 1976 and March 2017 was 23.14% p.a. with = 0.698.
The average annualised market return in this period was 12.04% and the average T-
Bill rate was = 4.56%. What has been BRK.A’s in this period? Plot the Security
Market Line and show where BRK.A sits in relation to it. Should this be happening in
efficient markets?
➢ = − + − = 0.2314 − [
]
0.0456 + 0.698 × (
)
0.1204 −
0.0456 = 13.36%
➢ This should not be happening in efficient markets and is inconsistent with a CAPM
equilibrium, as alpha has been persistent for over 40 years for BRK
Example: Alpha
45
Decomposition of the total risk
FINS5513
46
❑ Recall SIM: − = + − + . Let’s calculate the variance of asset :
= + + − +
❑ Recall, , , are constant, thus we have:
= − + =
2 − + + 2( − , )
=
2 + + 2(, )
❑ Recall regression assumption: , = 0, thus we have
=
2 +
❑ Therefore, the total risk (i.e.,
2) is decomposed into two components:
➢ Systematic risk/market risk:
2
2
➢ Unsystematic risk/idiosyncratic risk:
2
❑ The ratio of systematic risk / total risk is the R-Squared given by: 2 =
2
2
2 = , 2
➢ The proportion of the movement in asset which is caused by market movements
Decomposition of the total risk
47
❑ The CAPM/SIM framework allows us to bifurcate our analysis of risk
❑ Systematic Risk / Market Risk / Non-diversifiable Risk
➢ Risk attributable to market-wide (macro) factors which remain even after diversification
➢ The part of an asset’s risk that is common with the market, and thus unable to be
diversified
➢ Examples: market-wide events such as an oil embargo, interest rate rise, recession,
pandemic, natural disaster, political crisis, exchange rate collapse
❑ Unsystematic Risk / Firm-Specific Risk / Diversifiable Risk / Idiosyncratic Risk
➢ The part of an asset’s risk that is specific to the asset itself
➢ Since this risk comes from sources that do not affect the whole market, it can be
reduced to zero or negligible levels in a diversified portfolio
➢ Examples: a retailer launches a poor product, misses sale expectations, or loses a
major customer. This would be bad for the retailer’s stock, but it’s competitors may
actually benefit
Revisit Diversification and Portfolio Risk
48
❑ As we add more assets to our portfolio through diversification, we eliminate unsystematic
(firm-specific) risk and we converge toward the systematic (market) risk level – which cannot
be diversified away
Systematic and Unsystematic Risk
Standard
Deviation
Number of stocks
Systematic (market) risk
Non-systematic (firm-specific) risk
49
❑ Under SIM, we can estimate covariance between any two assets through their ′s:
, =
2 , =
❑ Correlation between two assets is simply the product of their correlation with the
market and is given by: , =
2
=
2
2
= , × ,
➢ All correlation can be related back to the market under SIM
❑ SIM/CAPM imposes additional assumption on the variance-covariance structure,
which is the key difference between SIM framework and Markowitz framework.
Covariance calculation under SIM
50
❑ Using the information in the table below from the 1976-2017 study:
a) Calculate BRK’s systematic and unsystematic risk and R2?
b) What is BRK’s covariance and correlation with the market return?
c) Using the information in the table below, calculate the covariance and correlation of
BRK with each of these stocks
➢ Systematic risk:
2
2 = (0.698)2× (0.153)2= 0.011113
➢ Unsystematic risk:
2 = (0.24)2− 0.698 2 × (0.153)2= 0.046487
➢
2 =
0.011113
0.242
= 19.29%
Example: SIM Risk Measures
Stock Beta Standard Deviation
Market 1.00 15.3%
BRK (BRK.A) 0.698 24.0%
Apple (AAPL) 1.15 24.0%
Microsoft (MSFT) 1.28 22.8%
Amazon (AMZN) 1.35 30.6%
51
➢ Covariance: ( , ) = ×
2 = (0.698) (0.153)2 = 0.0163
➢ Correlation: , =
(,)
=
0.0163
[0.24×0.153]
= 0.445
Example: SIM Risk Measures
Stock Beta Covariance/Correlation BRK
Apple (AAPL) 1.15 • , = 0.698 × 1.15 × 0.153
2 = 0.01879
• , =
0.01879
0.24×0.24
= 0.3262
Microsoft
(MSFT)
1.28 • , = 0.698 × 1.28 × 0.153
2 = 0.02091
• , =
0.02091
0.24× 0.228
= 0.3821
Amazon
(AMZN)
1.35 • , = 0.698 × 1.35 × 0.153
2 = 0.02206
• , =
0.02206
0.24×0.306
= 0.3004
52
❑ Both the CML and SML start from , but the CML accounts for total risk () while the SML only
accounts for systematic risk ()
SML vs CML
E(r)
M
CML
Efficient
Frontier
rf
E(r)
SML
β
βM=1
A •
P
rf
A
M
P
βPβA
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❑ CML’s -axis is but SML’s -axis is
❑ The of the market portfolio is always 1 but may change. The SML does not
show the level of as it scales risk to 1
➢ A change in has no effect on the SML – except where it is accompanied by a
change in () – but it would change the slope of the CML
❑ () is the same for the CML and SML
❑ The two graphs have different purposes:
➢ CML (and CAL) is more of a portfolio construction tool
➢ SML is more of a valuation tool, separating fairly priced assets (which plot on the
SML) from mispriced assets (which plot over or under the SML) in a CAPM
equilibrium
SML vs CML
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❑ BKM Chapter 8
❑ 4.1 SIM Regression: The Security Characteristic Line
❑ 4.2 Portfolio Construction Under SIM vs Markowitz
❑ 4.3 What is Active Investing?
❑ 4.4 Active Portfolio Construction
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