Topic 5. The Capital Asset Pricing Model
(CAPM)
ECON30024 Economics of Finanical Markets
Shuyun May Li
Department of Economics
The University of Melbourne
Semester 1, 2025
1
Outline
1. Assumptions of the CAPM
2. Derivation of the CAPM
• A necessary condition for efficient portfolios
• Market clearing and capital market line
• The CAPM and its implications
• The security market line
3. Properties of beta
4. CAPM in asset prices
5. Extensions of the CAPM
Required reading: Chap. 6 of Bailey
2
1. Assumptions of the CAPM
• A core objective of asset pricing theory is to explain the risk
premium, µj − r0, for each asset. One of the most widely
discussed explanations is provided by the CAPM.
• The CAPM extends the mean-variance analysis of portfolio
selection for an individual investor to the market as a whole.
• It addresses the question: if all investors behave according to a
mean-variance objective and if they all have the same beliefs,
then what can be inferred about the pattern of expected
returns on assets when asset markets are in equilibrium?
• The CAPM is based on a long list of assumptions, condensed
into assumptions on markets and assumptions on investors.
3
• Assumptions on markets
– Markets are in equilibrium: prices of assets adjust so that
aggregate demand for each asset equals its supply.
– Frictionless markets
· Neutral transaction costs and taxes
· No institutional restrictions on assets trade
· Divisible assets
– Unlimited risk-free borrowing and lending
– Competitive asset markets (investors are price takers)
4
• Assumptions on investors
– All investors behave according to a single-period investment
horizon.
– All investors select their portfolios according to a
mean-variance objective.
– All investors have homogeneous beliefs, i.e., use the same
estimates of the expectations, variances and covariances of
asset returns
(µj , σj , ρij values are the same for all investors)
• Why these assumptions are needed in the derivation of CAPM?
(Tutorial 5 discussion)
5
2. Derivation of the CAPM
• To derive the CAPM prediction, we first characterise an
individual investor’s portfolio selection problem, then impose
market clearing.
– Recall the basic analytical framework in Topic 1.
2.1 A necessary condition for investors’ efficient portfolios
• Recall how the PF with n risky assets and a risk-free asset for
an individual investor is defined (Topic 4 slide 34):
(∗) min
(a0,a1,a2,...,an)
σ
2
P =
nX
i=1
nX
j=1
aiajσij
s.t. µP = a0r0 +
nX
j=1
ajµj, a0 +
nX
j=1
aj = 1
An efficient portfolio is a solution to problem (∗).
6
– Note that a0 = 1−
∑n
j=1 aj so that
µP =

1−
nX
j=1
aj
!
r0 +
nX
j=1
ajµj = r0 +
nX
j=1
(µj − r0)aj (1)
– So problem (∗) can be rewritten as
min
(a1,a2,...,an)
σ2P =
n∑
i=1
n∑
j=1
aiajσij , s.t. (1)
• Form the Lagrangian for this minimisation problem:
L =
n∑
i=1
n∑
j=1
aiajσij + γ

µP − r0 − n∑
j=1
(µj − r0)aj

 ,
where γ is the Lagrange multiplier on the constraint.
– The first-order conditions: partially differentiate L with
respect to a1, a2, . . ., an, γ and set the expressions to zero.
7
∂L
∂aj
= 2
nX
i=1
aiσij − γ(µj − r0) = 0, j = 1, 2, . . . , n (2)
∂L
∂γ
= µP − r0 −
nX
j=1
(µj − r0)aj = 0 (this is (1))
– Eqn. (2) implies that
n∑
i=1
aiσij =
γ
2
(µj − r0), j = 1, 2, . . . , n
Note that
∑n
i=1 aiσij is the co-variance of rP and rj :
cov(rP , rj) = cov(a0r0 + a1r1 + . . .+ anrn, rj)
= a1cov(r1, rj) + a2cov(r2, rj) + . . .+ ancov(rn, rj)
=
n∑
i=1
aiσij ≡ σjP
8
So we have
σjP ≡
n∑
i=1
aiσij =
γ
2
(µj − r0), j = 1, 2, . . . , n (3)
– Now, multiply (3) by aj on both sides:
n∑
i=1
aiajσij =
γ
2
(µj − r0)aj
Sum over j:
n∑
j=1
n∑
i=1
aiajσij =
γ
2
n∑
j=1
(µj − r0)aj ,
which is equivalent to, by the definition of σ2P and (1),
σ2P =
γ
2
(µP − r0) (4)
9
– Combining (3) and (4) yields
µj − r0
σjP
=
µP − r0
σ2P
, for j = 1, 2, . . . , n. (5)
• Any efficient portfolio P must satisfy (5).
• See the Appendix for the economic intuition behind (5).
– For P to be an efficient portfolio, the marginal increase in
the portfolio’s expected return per unit of risk resulting
from a small change in aj must be the same for all assets.
– Eqn. (5) can also be interpreted as a no arbitrage
condition: the risk and return tradeoffs must be equalised
across assets.
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2.2 Market clearing and capital market line
• Recall the EF with n risky assets and a risk-free asset (Figure
10 in Topic 4). The tangent portfolio Z is an efficient
portfolio consisting of only n risky assets.
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• Hence, the risky asset portfolio Z must satisfy (5):
µj − r0
σjZ
=
µZ − r0
σ2Z
, for j = 1, 2, . . . , n.
Define βjZ =
σjZ
σ2
Z
, the equation above can be rewritten as
µj − r0 = βjZ(µZ − r0), for j = 1, 2, . . . , n (6)
• Next, we impose market clearing and show that Z is the
market portfolio.
• The assumption of homogeneous beliefs implies that all
investors have the same EF.
– All investors’ optimum portfolios are portfolios of the
risk-free asset and Z.
– Because investors can differ by their mean-variance
objectives, their optimum portfolios can be different.
12
• Let i = 1, 2, . . . ,m index the investors in the market, each with
initial wealth Ai and holding an optimum portfolio given by
(1− aiZ , a
i
Z),
where aiZ is the proportion of A
i invested in portfolio Z.
• Let Z ≡ (z1, z2, . . . , zn), where zj is the proportion of asset j’s
value in the total value of portfolio Z,
∑n
j=1 zj = 1
• What is investor i’s demand for risky asset j, j = 1, 2, . . . , n?
– Investor i invests AiaiZ in portfolio Z.
– Hence, investor i invests AiaiZzj in asset j.
– Her holding of asset j, i.e., her demand for asset j is
xij =
AiaiZzj
pj
,
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• What is the total demand for risky asset j by all investors?
m∑
i=1
xij =
m∑
i=1
[
AiaiZzj
pj
]
=
(
m∑
i=1
AiaiZ
)
zj
pj
≡ B
zj
pj
,
where B ≡
∑m
i=1A
iaiZ is the total investment in risky assets
portfolio Z, i.e., total market value of risky assets.
• Suppose the exogenous fixed supply of risky asset j is Xj , then
market clearing for asset j implies:
Xj =
m∑
i=1
xij = B
zj
pj
⇒ zj =
pjXj
B
, for j = 1, 2, . . . , n, (7)
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• Note that
pjXj
B
is the market value of asset j relative to the
total market value of all risky assets, i.e., the market share of
asset j, denoted as mj
– In equilibrium, the tangent portfolio Z is the market
portfolio, M ≡ (m1,m2, . . . ,mn).
– The market portfolio is a portfolio of risky assets, with the
proportion of each risky asset given by its market share in
the risky assets market.
• Replacing Z with M in the previous figure, we obtain the EF
for all investors in equilibrium, which is called the capital
market line (CML), as shown in Figure 1.
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Figure 1. The capital market line
16
– The CML is a straight line represented by
µP = r0 +
(
µM − r0
σM
)
σP . (8)
– In equilibrium, all investors have the same EF, which is the
CML.
– All investors’ optimum portfolios are located on the CML,
and hence are portfolios of the risk-free asset and the
market portfolio.
– However, the optimum portfolios of investors are not
necessarily the same; different investors can choose to hold
different efficient portfolios on the CML.
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2.3 The CAPM and its implications
• In equilibrium, equation (6) becomes
µj − r0 = βj(µM − r0), for j = 1, 2, . . . , n
where
βj ≡ βjM =
σjM
σ2M
.
Equivalently,
µj = r0 + βj(µM − r0), for j = 1, 2, . . . , n. (9)
This is the basic form of the CAPM!
• Implications of the CAPM
– The expected return on an asset equals the risk-free rate
plus a risk premium which depends on the asset’s beta
coefficient.
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– The risk premium on asset j, commonly defined as the
expected excess return µj − r0, is given by:
µj − r0 = βj(µM − r0),
where µM − r0 is the market risk premium.
– Recall that βj ≡
σjM
σ2
M
, so βj measures the comovement of
asset j’s return with the market return.
– βj measures the systematic risk of asset j, arising from
fluctuations in the market return which is a systematic
and undiversifiable risk inherent to the market as a whole.
– The CAPM implies that asset j’s risk premium is
determined by βj rather than σj (the std. of rj).
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2.4 The security market line (SML)
• The CAPM equation (9) implies a linear relationship between
an asset’s expected return and its beta:
µj = r0 + (µM − r0)βj , for all j
Plotting µj against βj in the (βj , µj) space, we get the SML.
• The intercept of SML equals the risk-free rate, r0, and the
slope equals µM − r0.
• The CAPM predicts that the expected returns and betas of all
assets and portfolios of assets lie on the SML.
• Difference between CML and SML (Tutorial 5 Discussion)
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Figure 2. The security market line
21
Interim Summary
• The CAPM assumes a world in which (a) asset markets are in
equilibrium; (b) investors choose portfolios to maximise
mean-variance objectives according to common values of the
means and variances of asset returns.
• The CAPM predicts that in equilibrium all investors have the
same EF, which is the CML, and they hold efficient portfolios
of the risk-free asset and the market portfolio.
• The CAPM implies that expected asset returns can be
predicted from their betas. The linear relationship forms the
security market line.
• The CAPM implies that each asset’s beta is a more appropriate
measure of its risk than its standard deviation of return.
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3. Properties of beta
• Recall the CAPM as summarised in (9):
µj = r0 + βj(µM − r0),
where βj ≡
σjM
σ2
M
measures the systematic risk of asset j.
• Rewrite β, using ρjM ≡
σjM
σjσM
:
βj =
ρjM σjσM
σ2M
= ρjM
σj
σM
.
– The market portfolio M has a beta equal to 1, βM = 1
– What is the beta of an efficient portfolio E on the CML?
βE =
σE
σM
, because ρEM = 1
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– An asset with βj = 0 is called a zero-beta asset:
βj = 0 ⇒ µj = r0 + βj(µM − r0) = r0.
Where is this asset on the SML? Is it risk free?
• We can classify all assets in terms of their betas:
– βj = 1: its risk premium equals the market risk premium.
– βj > 1: “aggressive asset” with σj > σM
– βj < 1: “defensive asset”
· Assets with negative betas offer a form of insurance
against volatility in the market return, and hence their
expected reutrns are lower than the risk-free rate.
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• Beta is “linearly additive”.
– Suppose asset 1 and asset 2 have betas of β1 and β2.
– Consider a portfolio P with a proportion a in asset 1 and
1− a in asset 2, then
βP =
σPM
σ2M
=
cov(rP , rM )
σ2M
=
cov(ar1 + (1− a)r2, rM )
σ2M
=
a cov(r1, rM ) + (1− a)cov(r2, rM )
σ2M
=
a σ1M + (1− a)σ2M
σ2M
= a
(
σ1M
σ2M
)
+ (1− a)
(
σ2M
σ2M
)
= aβ1 + (1− a)β2
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4. The CAPM in asset prices
• The CAPM is a model of asset prices, though it is typically
expressed in terms of expected rates of return rather than
prices.
• To see this, recall the definition of the rate of return on asset j:
rj =
vj − pj
pj
=
vj
pj
− 1.
Taking expectations on both sides:
µj =
E(vj)
pj
− 1.
• The CAPM prediction for µj is given by (9):
µj = r0 + βj(µM − r0)
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• Combining the two equations above gives
pj =
E(vj)
1 + µj
=
E(vj)
1 + r0 + βj(µM − r0)
, j = 1, 2, . . . , n. (10)
This is the CAPM prediction for asset prices.
– In the absence of uncertainty, no arbitrage implies that
µM = r0, so price is simply given by the present value of the
payoff: pj =
vj
1+r0
.
– With uncertainty, vj is replaced by E(vj); and the
discount factor includes the risk premium βj(µM − r0) to
capture the risk aversion of investors.
• Back to the SML, suppose an asset locates above the SML. Is
this asset over-priced or under-priced, according to the CAPM?
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5. Extensions of the CAPM
5.1 The Black CAPM
• Thus far it has been assumed that investors can borrow or lend
without restriction.
• Now remove this opportunity and assume that all assets are
risky. All other assumptions remain. This defines what is
called the ‘Black CAPM’ (Black, 1972).
• CML in the Black CAPM
– As investors have homogeneous beliefs, they all have the
same portfolio frontier (a hyperbola).
– The market portfolio M is efficient, lying on the upward
sloping arm of the portfolio frontier, as shown in Figure 3.
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CML in Black CAPM
29
Figure 3. Zero-beta portfolio
30
– Now draw a line tangent to M , which meets the vertical
axis at some point, ω.
– The ray from ω passing through M is the capital market line
in the Black CAPM model (is CML the EF of investors? )
• Zero-beta portfolios
– All the portfolios or assets located on the horizontal line
starting at W have an expected return equal to ω.
– It can be shown that for any portfolio on this line, its rate
of return has zero correlation with the market return.
– All portfolios on this line are ‘zero-beta portfolios’.
31
• The Black CAPM:
µj = ω + βj(µM − ω), for j = 1, 2, . . . , n. (11)
– Intuitively, ω plays the same role as the risk-free rate r0 in
the standard CAPM.
– A security market line can be constructed from (11) in the
same way as when a risk-free asset is available.
• Applications of the model are much the same as when a
risk-free asset is available, except that now ω must be
estimated or calculated separately.
– ω can be estimated by constructing a zero-beta portfolio.
– We delay the testing and application of CAPM to Topic 7.
32
5.2 Other extensions
• Following the introduction of the CAPM, a range of extensions
were proposed and they continue to be.
• They all boil down to relaxing one or more of the assumptions
listed at the outset.
• Some of the modifications are routine - for example, allowing
for different tax rates on dividend income and capital gains.
• Others, such as allowing for heterogeneous beliefs, involve
much more complexity.
• One important extension involves generalising the single-period
mean-variance framework to allow for intertemporal
decision making with expected utility. We’ll briefly discuss it
in Topic 9.
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Review questions
1. What are the assumptions of the CAPM?
2. Understand the procedure to derive the CAPM. What does the
necessary condition capture and what does market clearing do?
3. Understand how we showed portfolio Z is the market portfolio M in
equilibrium. Why all investors hold the same portfolio of risky
assets?
4. Be able to draw the capital market line, and understand why it is
the same for all investors, why all efficient portfolios of investors
locate on the CML, and why the optimal portfolios of investors can
be different points on the CML.
5. Why all efficient portfolios have the same Sharpe ratio as the market
portfolio? Why all efficient portfolios are perfectly correlated?
6. Write down the CAPM in its basic form. Understand each notation
in the equation.
7. Understand the definition of the security market line and
understand why all assets or portfolios should lie on the SML if
CAPM is correct. Be able to draw the SML.
34
8. Understand the definition and meaning of beta in the CAPM. What
kind of risk it measures? Be able to calculate an asset’s beta, using
given information on mean, variance and covariance.
9. According to the CAPM, how is the risk premium of an asset
determined?
10. How to use the linearly additive property of beta to calculate the
beta of a portfolio, given betas of individual assets in the portfolio?
11. Be able to derive the CAPM prediction for asset prices and
understand how the expected payoff is discounted to yield today’s
price under the CAPM.
12. If you observe an asset located above or below the SML, whether
this asset is over-priced or under-priced by the CAPM?
13. Represent a zero-beta portfolio on the (βj , µj) space and on the
(σP , µP ) space. In the standard CAPM model, is the risk-free asset
the only zero-beta asset?
14. Understand how the Black CAPM is constructed, in particular, the
interpretation of ω.
15. Write down the Black CAPM equation.
35
Appendix: the derivation of eqn. (2)
• When we differentiate
∑n
i=1
∑n
j=1 aiajσij with respect to a
particular aj , we only need to consider items that contain aj .
• Take a1 as an example. Items in
∑n
i=1
∑n
j=1 aiajσij that
contain a1:
a1a1σ11 + a2a1σ21 + a3a1σ31 · · ·+ ana1σn1
a1a2σ12 + a1a3σ13 · · ·+ a1anσ1n
= a21σ11 + 2[a2a1σ21 + a3a1σ31 · · ·+ ana1σn1]
⇒
∂
[∑n
i=1
∑n
j=1 aiajσij
]
∂a1
= 2[a1σ11+a2σ21+a3σ31 · · ·+anσn1]
= 2
n∑
i=1
aiσi1
• Similarly,
∂[
Pn
i=1
Pn
j=1 aiajσij]
∂aj
= 2
∑n
i=1 aiσij , as in eqn. (2).
36
Appendix: Economic intuition behind (5)
• Suppose an investor holds a feasible portfolio P . Denote its
expected return and risk by µP and σP , respectively.
• Now suppose that a small increase is made in the holding of
asset j and the necessary funds needed is borrowed at the
risk-free interest rate. The holding of other risky assets remains
unchanged.
– Denote the increase in the proportion of j in the portfolio
by ∆aj (and ∆a0 = −∆aj by construction).
• The change in the portfolio’s expected return:
µP = a0r0 +
n∑
i=1
aiµi
⇒ (∆µP )
j = ∆a0r0 +∆ajµj = ∆aj(µj − r0)
37
• The change in the portfolio’s risk:
σP =
[∑n
i=1
∑n
j=1 aiajσij
] 1
2
⇒
∂σP
∂aj
=
1
2
"
nX
i=1
nX
j=1
aiajσij
#
−
1
2

2
nX
i=1
aiσij
!
=
ˆ
σ
2
P
˜
−
1
2 σjP =
σjP
σP
So
(∆σP )
j
≈
(
∂σP
∂aj
)
∆aj =
σjP
σP
∆aj
• For P to be an efficient portfolio, it must be the case that a
small change in the portfolio proportion of any asset must
disturb the expected return per unit of risk by the same
amount for all assets.
38
• That is, (∆µP )
j
(∆σP )j
=
µj−r0
σjP
σP
must be equal for all assets, i.e.,
µj−r0
σjP
must be equal for all assets j = 1, 2, . . . , n:
µ1 − r0
σ1P
=
µ2 − r0
σ2P
= . . . =
µn − r0
σnP
• The portfolio P can itself be interpreted as a single composite
asset, so
µP − r0
σ2P
=
µj − r0
σjP
for all assets j = 1, 2, . . . , n. This is equation (5).
39
• When we differentiate
∑n
i=1
∑n
j=1 aiajσij with respect to a
particular aj , we only need to consider items that contain aj .
• Take a1 as an example. Items in
∑n
i=1
∑n
j=1 aiajσij that
contain a1:
a1a1σ11 + a2a1σ21 + a3a1σ31 · · ·+ ana1σn1
a1a2σ12 + a1a3σ13 · · ·+ a1anσ1n
= a21σ11 + 2[a2a1σ21 + a3a1σ31 · · ·+ ana1σn1]
⇒
∂
[∑n
i=1
∑n
j=1 aiajσij
]
∂a1
= 2[a1σ11+a2σ21+a3σ31 · · ·+anσn1]
= 2
n∑
i=1
aiσi1
• Similarly,
∂[
Pn
i=1
Pn
j=1
aiajσij]
∂aj
= 2
∑n
i=1 aiσij , as in eqn. (2).
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