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FINS5513 Derivation of the CAPM
创建日期：2024-08-09 17:46:14
所属分类：经济Economics
标签： FINS5513
Derivation of the CAPM

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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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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
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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)?
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COMP2013 IB9X60 INVP001 EMATM0051 COMP220 BLAW1002 E20 TCP8001 ECON61001 EART40005 ECONM1022 MATH 365 MATH 367 ELEC6218 COMP 5812M EDUC 5061M ELEC ENG 1101 equation MATH6174 ST334 C4 EC9570 A33648 EBUS603 ECON47815 stata WFMS0001 PEM102 INF6060 ionically ELECTRICAL MATH 375 MMW 12 MKTG860 ECON60401 ALY6010 SOST20131 Introduction COM3524 MME2 GGR305 JNB 538 EECS101 COMM5030 IEOR E4601 2B CSC336 EN2315 LAW 432 CSC263 MATH2001 OLET1143 CSCB63 SIT772 COMM1170 STAT 441 MS930 INFT3050 GLBH0030 Stat 120B MAST10007 MGEC61 Machine GGR 252H1S ECON7030 Financial Marketing MGMT30019 ECON5120 SDSC 8009 CSC165H1S DSCI 551 3081W ACM41000 ECA 5304 MATH 1064 ECSE444 DATA 604 COMP7105 MIE1615 ECON4004 ENV200H1S EC481 MKT 306 EC1B1 MATH08051 ALY6140 CS260C CSE30 MSIN0226 CS4103 CSC3064 AM16 COMP11212 CSE4101 PSTAT 174 Math 380W CNT 5106C STAT 310 FIN2028 CS5014 STAT 425 COMP 4102A ELEC2140 MANF9544 QRM 9 QRM 8 ECON 457 COMPSCI 4091 MATH 523 DNSC-4211 PEAS 6000 MSIN0180 COMP2119 A2A HT 2022 IB93F0 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