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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
➢ Systematic and Unsystematic Risk
➢ Security Market Line
➢ Applications and Extensions of the CAPM
❑ 3.3 The Single Index Model (SIM)
➢ Limitations of the CAPM
➢ SIM
➢ Alpha
➢ SIM Risk Measures
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
❑ 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
Risk Premium = 0.1172 - 0.0338 = 8.34%
Market Sharpe ratio =
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
Excel: “CML and SML”
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%
Borrow 22.8% at
1 − 0.688 = 31.2%
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 know from week 1 (1.4) that:
➢ An individual asset’s effect on portfolio return is simply proportional to its weight
➢ However, its effect on portfolio risk depends on its covariance with other portfolio assets
❑ 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 of raw return):
❑ The risk premium on the market portfolio (RM ) is:
= σ=1
() = σ=1
()
ℎ =
σ=1
ℎ ℎ
➢ So, asset ’s contribution to ’s risk premium is ()
❑ The risk on the market portfolio (
2 ) is:
2 = , = σ=1
,
= σ=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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❑ The market risk premium − or 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
Market Risk Premium
19
Beta
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20
❑ Let’s review the CAPM equation: − = [ − ]
Which can be restated as: =
❑ So, asset ’s expected return above the risk-free rate (its “risk premium”) equals:
×
➢ Risk is measured by its relative contribution to the variance of , which is it’s :
• =
,
2 which tells us how asset moves with
➢ Price of Risk is the market risk premium: −
• 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
21
❑ Rather than running covariances between each individual asset in the portfolio, as we do
under the Markowitz model, we estimate each asset’s return as a function of its covariance
with one common factor - the market return under CAPM
➢ This significantly reduces the calculations we need to make
➢ Importantly, the relates each asset’s return to the market return
❑ can be < 0 and can be > 1:
➢ > 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)
❑ Derived by regression – we will see how in lecture 4 and the iLab
❑ 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 () = 11.0%, the risk-free rate
= 3.0% 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%
() = () then =
()
()
→ so =
()
()
=
0.096
0.08
= 1.2
= 0.086 – 0.03 = 5.6% → so =
()−
()−
=
0.056
0.08
= 0.7
➢ Recall that =
,
2 so for:
Asset A: , = 1.2 × 0.0256 = 0.03072
Asset B: , = 0.7 × 0.0256 = 0.01792
Example
23
❑ For the Market Portfolio the CAPM simplifies to:
❑ For the risk-free asset, the CAPM simplifies to:
❑ () 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
( )
( ) ( ) ( ) ( )MfMffMMfM
M
MM
M
rErrErrrErrE
rrCov
=−+=−+=
==
*1
1
,
2
( )
( ) ( ) ( ) ( )MfMffMMfM
M
M
M
rErrErrrErrE
rrCov
=−+=−+=
==
*1
1
,
2
( )
( ) ( ) ( ) ffMffMfff
M
Mf
f
rrrErrrErrE
rrCov
=−+=−+=
==
*0
0
,
2
( )
( ) ( ) ( ) ffMffMfff
M
Mf
f
rrrErrrErrE
rrCov
=−+=−+=
==
*0
0
,
2
24
Systematic and Unsystematic
Risk
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25
❑ The CAPM 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
➢ As is a measure of market covariance, systematic risk is measured by (“beta risk”)
➢ Examples: market-wide events such as an oil embargo, interest rate rise, recession,
pandemic, natural disaster, political crisis, exchange rate collapse
• As these events affect all companies, their returns would move similarly
❑ 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
Diversification and Portfolio Risk
26
❑ 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
27
Security Market Line
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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
❑ The Security Market Line (SML) is captured in the − space because systematic risk
(measured by ) is the only risk that is rewarded under a CAPM equilibrium
❑ The − relationship is linear as indicated by the CAPM equation
Security Market Line
rf
M
i
( )irE
SML
1=M0=f
fM rrESlope −= )(
❑ Movements up and
down the SML
represent the reward of
higher return at
the cost of higher
systematic risk
Video: “How to Plot the SML”
30
❑ The SML shows a positive relationship between systematic risk and expected excess return
above (“risk premium”)
➢ The slope of the SML is the market risk premium (the “MRP”) – the market price of risk
❑ The CAL is plotted in the − space because it is measuring total risk
➢ The SML is plotted in the − space because, in a CAPM equilibrium, only systematic
risk is rewarded
➢ For , all unsystematic risk is diversified away, and therefore all of ’s risk is systematic
❑ As an example, we can interpret movements along the SML as follows:
➢ An asset with = 0.5 has half the risk of the average asset, and its risk premium above is
therefore half the MRP (the market price of risk)
➢ An asset with = 2 has twice the risk of the average asset, and its risk premium above is
therefore twice the MRP (the market price of risk)
Interpreting the SML
31
❑ The market’s expected return () = 11.0% and the risk-free rate = 3.0%. Three assets in
the market have the following risk/return characteristics
a) Plot the Security Market Line
b) Plot the individual assets C, D and E
c) Determine whether each asset is fairly priced and whether you would buy, sell or hold it
Example: Plotting the SML
Asset Actual Return Beta
Asset C 10.9% 1.4
Asset D 7.8% 0.6
Asset E 18.6% 1.7
0.0%
2.0%
4.0%
6.0%
8.0%
10.0%
12.0%
14.0%
16.0%
18.0%
20.0%
0.00 0.50 1.00 1.50 2.00 2.50
Ex
pe
ct
ed
R
et
ur
n
E(
r)
Beta β
Security Market Line
Market
Asset D:
Fair price
Asset C:
Overpriced
Asset E:
Underpriced ➢ Asset C = 1.4 x .08 + .03 = 14.2%. Actual return
is 10.9% so overpriced (below SML) – Sell
➢ Asset D = 0.6 x .08 + .03 = 7.8%. Actual return
is 7.8% so fairly priced (on SML) – Hold
➢ Asset E = 1.7 x .08 + .03 = 16.6%. Actual return
is 18.6% so under-priced (above SML) – Buy
Excel: “CML and SML”
32
❑ 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
33
❑ 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
34
Applications and Extensions
of the CAPM
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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
Mini Case Study: “MPT and CAPM in Sharpe’s Own Words”
36
❑ 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
37
❑ 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 market indices
➢ Therefore, studies tend to use factor portfolios rather than consumption directly
Consumption CAPM
3.3 The Single Index
Model (SIM)
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Limitations of the CAPM
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❑ 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
➢ We will discuss these biases in week 7
❑ 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 statistics (ex post) to predict future returns (ex ante) can be particularly
misleading (or as Sharpe himself says “take historic data with a lot of grains of salt”)
Limitations of the CAPM
41
❑ 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
42
The Single Index Model (SIM)
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❑ The true market portfolio is not observable
❑ Therefore, we 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 (>1650 large cap
stocks worldwide)
❑ 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
44
❑ The CAPM equation: () − = [() – ] is forward-looking (ex-ante) – it attempts to
predict future returns
❑ If we were to use this equation based on historical data (ex-post) – for example to derive the
implied beta based on historical returns – we would use the following:
− = (rM − f)
➢ can be estimated by regressing asset ’s excess returns on the market excess returns
➢ In Excel - the SLOPE command derives
❑ Even if we were in a CAPM equilibrium, the actual observed excess return is unlikely to equal
the expected excess return at every single sample point
➢ For example, if we are using historical monthly data, it is unlikely the actual observed
excess return for every single month will equal that month’s predicted excess return
➢ Denote “residual” unpredicted returns (the difference between actual and predicted return)
Regression Formulation of the CAPM
Actual observed
excess return
Predicted excess
return
Excel: “Simple Regression Analysis”
45
❑ Therefore, we can restate the equation as:
− = (rM − ) +
where are unpredicted, unsystematic deviations of actual return from expected return
❑ But if the CAPM holds and the market is in equilibrium, should be random without any
specific pattern. In other words:
➢ = 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 )
❑ Under SIM, we would regress an asset, a cross-section of assets, or even a portfolio of
assets over time (not just at a single point in time). Therefore, we have:
− = (rMt − ) +
The notation reflects that we are testing a cross-section of assets () over time ()
Single Index Model (SIM) Regression
46
❑ But what if the residuals do not sum to 0?
➢ We would need to add a constant into the equation, to capture that the actual excess return
is on average higher (a positive constant) or lower (a negative constant) than predicted
➢ This constant is the alpha and can be considered the average of the residuals over time:
− = + (rMt − ) +
❑ Using our excess return notation, we can restate this as:
= + RMt +
➢ This is the academic version of the SIM which is often stated in terms of excess returns
➢ Note that the finance industry often runs the SIM on absolute (raw) returns:
= + rMt +
We will generally focus on excess return
Single Index Model (SIM) Regression
47
❑ Note the key differences between the SIM and CAPM equations:
➢ The CAPM uses the Market Portfolio , the SIM uses a broad-based index as a proxy for
➢ SIM adds unpredicted, unsystematic (firm-specific) risk components – the residuals
➢ SIM adds alpha , the average unpredicted return due to unsystematic factors over time
❑ Under CAPM, an investor is rewarded with higher expected return only for bearing additional
systematic risk. Why? Because in equilibrium, rational investors have well diversified
portfolios which have eliminated unsystematic risk. In other words, the unsystematic residual
components are random, without any specific pattern, and converge towards zero
➢ So, in a CAPM equilibrium, we would expect that:
• = 0
• () = 0
• ( , ) = 0
• ( , ) = 0
Under the SIM there is a possibility that ≠ 0
Recall Some CAPM Equilibrium Conditions
48
Alpha
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❑ So, what is alpha (often referred to as “Jensen’s alpha”)?
➢ It is the average return which is not explained by market movements (i.e., unsystematic)
➢ 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 applied is incomplete (i.e., needs more variables)
2. The single index model is correct, but the asset is mispriced
➢ Unfortunately, we do not know what the “correct” model is. 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)
❑ The is the y-intercept when regressing an asset’s returns against the market’s returns
➢ In other words, In Excel - the INTERCEPT command derives
➢ For a single asset , alpha can be derived as:
= − RM
Jensen’s Alpha
Actual observed excess return
Predicted excess return
Excel: “Simple Regression Analysis”
Video: “How Alpha is Calculated”
50
❑ 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
❑ 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
51
0.0%
2.0%
4.0%
6.0%
8.0%
10.0%
12.0%
14.0%
16.0%
18.0%
20.0%
0.00 0.50 1.00 1.50 2.00 2.50
Ex
pe
ct
ed
R
et
ur
n
E(
r)
Beta β
Conservative Fund Security Market Line
❑ The average annualised return for Conservative Fund between 2000 and 2020 was 9.3% p.a.
with = 0.5. The average annualised return for the market in this period was 10.7% and the
average T-Bill rate was = 2.5%. What has been Conservative Fund’s in this period? Plot
the Security Market Line and show where Conservative Fund’s return sits in relation to it.
Should this be happening in efficient markets?
➢ = 0.093 – 0.025 = 0.068
➢ = 0.107 – 0.025 = 0.082
➢ Calculate alpha: = − Con RM
= 0.068 – 0.5 × 0.082 = 2.7%
➢ This should not be happening in efficient
markets. This is inconsistent with a CAPM
equilibrium, as alpha has been persistent
for 20 years for the Conservative Fund
Example: Alpha
Conservative
Fund Alpha
Market
Excel: “Berkshire Hathaway SCL”
52
SIM Risk Measures
FINS5513
53
❑ We have split the return into systematic ( × ) and unsystematic (, ) components. Now
we can do the same for risk
❑ Given the introduction of unsystematic (firm-specific) deviations we can deconstruct the
total variance (i.e. total risk) of an asset into its component parts:
Total Variance = Systematic risk + Unsystematic risk:
2 =
2
2 +
2 Assumes , = 0
❑ The ratio of systematic risk / total risk is the R-Squared given by:
2 =
=
2 2
2 = , 2
➢ The 2 indicates that proportion of the movement in asset which is caused by market
movements (also equal to the correlation between and squared)
SIM Risk Measures
Variance or
“Total Risk”
54
❑ Unlike Markowitz, we no longer need to estimate covariance between any asset and in a
portfolio. But if we want to know their covariance, we can estimate it through their ′s
, =
2 Assumes , =
➢ Covariance between assets in the portfolio can therefore be derived as an output (rather
than as an input as per Markowitz)
➢ The additional assumption of the SIM that the residuals have no covariance differentiates it
from the Markowitz framework
❑ 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
SIM Risk Measures
55
❑ Use the table below describing Invest Co’s returns between 2010-2021:
a) Calculate Invest Co’s systematic and unsystematic risk and R2?
b) What is Invest Co’s covariance and correlation with the market return?
c) Using the information in the table below, calculate the covariance and correlation of Invest
Co with each of these stocks
➢ Systematic risk: (1.8)2× (0.153)2 = 27.5%
➢ Unsystematic risk: (0.3)2−(0.275)2 = 11.9%
➢ R2 =
0.2752
0.32
= 84.2%
Example: SIM Risk Measures
Stock Beta Standard Deviation
Market 1.0 15.3%
Invest Co 1.8 30.0%
BRK (BRK.A) 0.7 24.0%
World Co 1.3 21.0%
Hedge Co 0.6 34.0%
56
➢ Covariance: (, ) = (1.8) × (0.153)
2 = 0.042
➢ Correlation: , =
0.042
[0.3×0.153]
= 0.917
Example: SIM Risk Measures
Stock Beta Covariance/Correlation BRK
BRK (BRK.A) 0.7 Cov(rIC, rBRK) = 1.8 x 0.7 x 0.153
2 = 0.02946
, = 0.02946 / [0.30 x 0.24] = 0.4092
World Co 1.3 Cov(rIC, rWC) = 1.8 x 1.3 x 0.153
2 = 0.05471
, = 0.05471 / [0.30 x 0.21] = 0.8685
Hedge Co 0.6 Cov(rIC, rHC) = 1.8 x 0.6 x 0.153
2 = 0.02525
, = 0.02525 / [0.30 x 0.34] = 0.2476
57
❑ 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
➢ Systematic and Unsystematic Risk
➢ Security Market Line
➢ Applications and Extensions of the CAPM
❑ 3.3 The Single Index Model (SIM)
➢ Limitations of the CAPM
➢ SIM
➢ Alpha
➢ SIM Risk Measures
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
❑ 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
Risk Premium = 0.1172 - 0.0338 = 8.34%
Market Sharpe ratio =
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
Excel: “CML and SML”
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%
Borrow 22.8% at
1 − 0.688 = 31.2%
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 know from week 1 (1.4) that:
➢ An individual asset’s effect on portfolio return is simply proportional to its weight
➢ However, its effect on portfolio risk depends on its covariance with other portfolio assets
❑ 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 of raw return):
❑ The risk premium on the market portfolio (RM ) is:
= σ=1
() = σ=1
()
ℎ =
σ=1
ℎ ℎ
➢ So, asset ’s contribution to ’s risk premium is ()
❑ The risk on the market portfolio (
2 ) is:
2 = , = σ=1
,
= σ=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
❑ The market risk premium − or 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
Market Risk Premium
19
Beta
FINS5513
20
❑ Let’s review the CAPM equation: − = [ − ]
Which can be restated as: =
❑ So, asset ’s expected return above the risk-free rate (its “risk premium”) equals:
×
➢ Risk is measured by its relative contribution to the variance of , which is it’s :
• =
,
2 which tells us how asset moves with
➢ Price of Risk is the market risk premium: −
• 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
21
❑ Rather than running covariances between each individual asset in the portfolio, as we do
under the Markowitz model, we estimate each asset’s return as a function of its covariance
with one common factor - the market return under CAPM
➢ This significantly reduces the calculations we need to make
➢ Importantly, the relates each asset’s return to the market return
❑ can be < 0 and can be > 1:
➢ > 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)
❑ Derived by regression – we will see how in lecture 4 and the iLab
❑ 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 () = 11.0%, the risk-free rate
= 3.0% 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%
() = () then =
()
()
→ so =
()
()
=
0.096
0.08
= 1.2
= 0.086 – 0.03 = 5.6% → so =
()−
()−
=
0.056
0.08
= 0.7
➢ Recall that =
,
2 so for:
Asset A: , = 1.2 × 0.0256 = 0.03072
Asset B: , = 0.7 × 0.0256 = 0.01792
Example
23
❑ For the Market Portfolio the CAPM simplifies to:
❑ For the risk-free asset, the CAPM simplifies to:
❑ () 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
( )
( ) ( ) ( ) ( )MfMffMMfM
M
MM
M
rErrErrrErrE
rrCov
=−+=−+=
==
*1
1
,
2
( )
( ) ( ) ( ) ( )MfMffMMfM
M
M
M
rErrErrrErrE
rrCov
=−+=−+=
==
*1
1
,
2
( )
( ) ( ) ( ) ffMffMfff
M
Mf
f
rrrErrrErrE
rrCov
=−+=−+=
==
*0
0
,
2
( )
( ) ( ) ( ) ffMffMfff
M
Mf
f
rrrErrrErrE
rrCov
=−+=−+=
==
*0
0
,
2
24
Systematic and Unsystematic
Risk
FINS5513
25
❑ The CAPM 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
➢ As is a measure of market covariance, systematic risk is measured by (“beta risk”)
➢ Examples: market-wide events such as an oil embargo, interest rate rise, recession,
pandemic, natural disaster, political crisis, exchange rate collapse
• As these events affect all companies, their returns would move similarly
❑ 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
Diversification and Portfolio Risk
26
❑ 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
27
Security Market Line
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
❑ The Security Market Line (SML) is captured in the − space because systematic risk
(measured by ) is the only risk that is rewarded under a CAPM equilibrium
❑ The − relationship is linear as indicated by the CAPM equation
Security Market Line
rf
M
i
( )irE
SML
1=M0=f
fM rrESlope −= )(
❑ Movements up and
down the SML
represent the reward of
higher return at
the cost of higher
systematic risk
Video: “How to Plot the SML”
30
❑ The SML shows a positive relationship between systematic risk and expected excess return
above (“risk premium”)
➢ The slope of the SML is the market risk premium (the “MRP”) – the market price of risk
❑ The CAL is plotted in the − space because it is measuring total risk
➢ The SML is plotted in the − space because, in a CAPM equilibrium, only systematic
risk is rewarded
➢ For , all unsystematic risk is diversified away, and therefore all of ’s risk is systematic
❑ As an example, we can interpret movements along the SML as follows:
➢ An asset with = 0.5 has half the risk of the average asset, and its risk premium above is
therefore half the MRP (the market price of risk)
➢ An asset with = 2 has twice the risk of the average asset, and its risk premium above is
therefore twice the MRP (the market price of risk)
Interpreting the SML
31
❑ The market’s expected return () = 11.0% and the risk-free rate = 3.0%. Three assets in
the market have the following risk/return characteristics
a) Plot the Security Market Line
b) Plot the individual assets C, D and E
c) Determine whether each asset is fairly priced and whether you would buy, sell or hold it
Example: Plotting the SML
Asset Actual Return Beta
Asset C 10.9% 1.4
Asset D 7.8% 0.6
Asset E 18.6% 1.7
0.0%
2.0%
4.0%
6.0%
8.0%
10.0%
12.0%
14.0%
16.0%
18.0%
20.0%
0.00 0.50 1.00 1.50 2.00 2.50
Ex
pe
ct
ed
R
et
ur
n
E(
r)
Beta β
Security Market Line
Market
Asset D:
Fair price
Asset C:
Overpriced
Asset E:
Underpriced ➢ Asset C = 1.4 x .08 + .03 = 14.2%. Actual return
is 10.9% so overpriced (below SML) – Sell
➢ Asset D = 0.6 x .08 + .03 = 7.8%. Actual return
is 7.8% so fairly priced (on SML) – Hold
➢ Asset E = 1.7 x .08 + .03 = 16.6%. Actual return
is 18.6% so under-priced (above SML) – Buy
Excel: “CML and SML”
32
❑ 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
33
❑ 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
34
Applications and Extensions
of the CAPM
FINS5513
35
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
Mini Case Study: “MPT and CAPM in Sharpe’s Own Words”
36
❑ 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
37
❑ 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 market indices
➢ Therefore, studies tend to use factor portfolios rather than consumption directly
Consumption CAPM
3.3 The Single Index
Model (SIM)
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Limitations of the CAPM
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❑ 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
➢ We will discuss these biases in week 7
❑ 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 statistics (ex post) to predict future returns (ex ante) can be particularly
misleading (or as Sharpe himself says “take historic data with a lot of grains of salt”)
Limitations of the CAPM
41
❑ 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
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The Single Index Model (SIM)
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❑ The true market portfolio is not observable
❑ Therefore, we 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 (>1650 large cap
stocks worldwide)
❑ 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
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❑ The CAPM equation: () − = [() – ] is forward-looking (ex-ante) – it attempts to
predict future returns
❑ If we were to use this equation based on historical data (ex-post) – for example to derive the
implied beta based on historical returns – we would use the following:
− = (rM − f)
➢ can be estimated by regressing asset ’s excess returns on the market excess returns
➢ In Excel - the SLOPE command derives
❑ Even if we were in a CAPM equilibrium, the actual observed excess return is unlikely to equal
the expected excess return at every single sample point
➢ For example, if we are using historical monthly data, it is unlikely the actual observed
excess return for every single month will equal that month’s predicted excess return
➢ Denote “residual” unpredicted returns (the difference between actual and predicted return)
Regression Formulation of the CAPM
Actual observed
excess return
Predicted excess
return
Excel: “Simple Regression Analysis”
45
❑ Therefore, we can restate the equation as:
− = (rM − ) +
where are unpredicted, unsystematic deviations of actual return from expected return
❑ But if the CAPM holds and the market is in equilibrium, should be random without any
specific pattern. In other words:
➢ = 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 )
❑ Under SIM, we would regress an asset, a cross-section of assets, or even a portfolio of
assets over time (not just at a single point in time). Therefore, we have:
− = (rMt − ) +
The notation reflects that we are testing a cross-section of assets () over time ()
Single Index Model (SIM) Regression
46
❑ But what if the residuals do not sum to 0?
➢ We would need to add a constant into the equation, to capture that the actual excess return
is on average higher (a positive constant) or lower (a negative constant) than predicted
➢ This constant is the alpha and can be considered the average of the residuals over time:
− = + (rMt − ) +
❑ Using our excess return notation, we can restate this as:
= + RMt +
➢ This is the academic version of the SIM which is often stated in terms of excess returns
➢ Note that the finance industry often runs the SIM on absolute (raw) returns:
= + rMt +
We will generally focus on excess return
Single Index Model (SIM) Regression
47
❑ Note the key differences between the SIM and CAPM equations:
➢ The CAPM uses the Market Portfolio , the SIM uses a broad-based index as a proxy for
➢ SIM adds unpredicted, unsystematic (firm-specific) risk components – the residuals
➢ SIM adds alpha , the average unpredicted return due to unsystematic factors over time
❑ Under CAPM, an investor is rewarded with higher expected return only for bearing additional
systematic risk. Why? Because in equilibrium, rational investors have well diversified
portfolios which have eliminated unsystematic risk. In other words, the unsystematic residual
components are random, without any specific pattern, and converge towards zero
➢ So, in a CAPM equilibrium, we would expect that:
• = 0
• () = 0
• ( , ) = 0
• ( , ) = 0
Under the SIM there is a possibility that ≠ 0
Recall Some CAPM Equilibrium Conditions
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Alpha
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❑ So, what is alpha (often referred to as “Jensen’s alpha”)?
➢ It is the average return which is not explained by market movements (i.e., unsystematic)
➢ 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 applied is incomplete (i.e., needs more variables)
2. The single index model is correct, but the asset is mispriced
➢ Unfortunately, we do not know what the “correct” model is. 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)
❑ The is the y-intercept when regressing an asset’s returns against the market’s returns
➢ In other words, In Excel - the INTERCEPT command derives
➢ For a single asset , alpha can be derived as:
= − RM
Jensen’s Alpha
Actual observed excess return
Predicted excess return
Excel: “Simple Regression Analysis”
Video: “How Alpha is Calculated”
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❑ 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
❑ 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
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0.0%
2.0%
4.0%
6.0%
8.0%
10.0%
12.0%
14.0%
16.0%
18.0%
20.0%
0.00 0.50 1.00 1.50 2.00 2.50
Ex
pe
ct
ed
R
et
ur
n
E(
r)
Beta β
Conservative Fund Security Market Line
❑ The average annualised return for Conservative Fund between 2000 and 2020 was 9.3% p.a.
with = 0.5. The average annualised return for the market in this period was 10.7% and the
average T-Bill rate was = 2.5%. What has been Conservative Fund’s in this period? Plot
the Security Market Line and show where Conservative Fund’s return sits in relation to it.
Should this be happening in efficient markets?
➢ = 0.093 – 0.025 = 0.068
➢ = 0.107 – 0.025 = 0.082
➢ Calculate alpha: = − Con RM
= 0.068 – 0.5 × 0.082 = 2.7%
➢ This should not be happening in efficient
markets. This is inconsistent with a CAPM
equilibrium, as alpha has been persistent
for 20 years for the Conservative Fund
Example: Alpha
Conservative
Fund Alpha
Market
Excel: “Berkshire Hathaway SCL”
52
SIM Risk Measures
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❑ We have split the return into systematic ( × ) and unsystematic (, ) components. Now
we can do the same for risk
❑ Given the introduction of unsystematic (firm-specific) deviations we can deconstruct the
total variance (i.e. total risk) of an asset into its component parts:
Total Variance = Systematic risk + Unsystematic risk:
2 =
2
2 +
2 Assumes , = 0
❑ The ratio of systematic risk / total risk is the R-Squared given by:
2 =
=
2 2
2 = , 2
➢ The 2 indicates that proportion of the movement in asset which is caused by market
movements (also equal to the correlation between and squared)
SIM Risk Measures
Variance or
“Total Risk”
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❑ Unlike Markowitz, we no longer need to estimate covariance between any asset and in a
portfolio. But if we want to know their covariance, we can estimate it through their ′s
, =
2 Assumes , =
➢ Covariance between assets in the portfolio can therefore be derived as an output (rather
than as an input as per Markowitz)
➢ The additional assumption of the SIM that the residuals have no covariance differentiates it
from the Markowitz framework
❑ 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
SIM Risk Measures
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❑ Use the table below describing Invest Co’s returns between 2010-2021:
a) Calculate Invest Co’s systematic and unsystematic risk and R2?
b) What is Invest Co’s covariance and correlation with the market return?
c) Using the information in the table below, calculate the covariance and correlation of Invest
Co with each of these stocks
➢ Systematic risk: (1.8)2× (0.153)2 = 27.5%
➢ Unsystematic risk: (0.3)2−(0.275)2 = 11.9%
➢ R2 =
0.2752
0.32
= 84.2%
Example: SIM Risk Measures
Stock Beta Standard Deviation
Market 1.0 15.3%
Invest Co 1.8 30.0%
BRK (BRK.A) 0.7 24.0%
World Co 1.3 21.0%
Hedge Co 0.6 34.0%
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➢ Covariance: (, ) = (1.8) × (0.153)
2 = 0.042
➢ Correlation: , =
0.042
[0.3×0.153]
= 0.917
Example: SIM Risk Measures
Stock Beta Covariance/Correlation BRK
BRK (BRK.A) 0.7 Cov(rIC, rBRK) = 1.8 x 0.7 x 0.153
2 = 0.02946
, = 0.02946 / [0.30 x 0.24] = 0.4092
World Co 1.3 Cov(rIC, rWC) = 1.8 x 1.3 x 0.153
2 = 0.05471
, = 0.05471 / [0.30 x 0.21] = 0.8685
Hedge Co 0.6 Cov(rIC, rHC) = 1.8 x 0.6 x 0.153
2 = 0.02525
, = 0.02525 / [0.30 x 0.34] = 0.2476
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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
Next Lecture
