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**Unformatted text preview: **BFF3121 Investments and
Portfolio Management
Lecture 3 – Asset Pricing Models
Lecturer: Dr. Manapon Limkriangkrai, CFA Learning Outcomes
Capital Market Line (CML)
Capital Asset Pricing Model (CAPM) and Security Market
Line (SML)
CAPM: Assumptions & Applications
APT
Fama-French (FF) Three-Factor Model
Carhart Four-Factor Model
ᩒ૱ᕚҁCML҂
ᩒᩒԾਧհཛྷࣳҁCAPM҂ᦤڭ૱
ᤈҁSML҂
CAPMғᦡଫአ
APT
ဩሸ - ဩҁFF҂ӣࢩᔰཛྷࣳ
Carhartࢥࢩᔰཛྷࣳ 2 The Capital Market Line (CML)
Separation Theorem (Tobin, 1958) ‘breaking down the portfolio selection problem into stages at different levels of aggregation-allocation first among, and
then within, asset categories’ (Tobin, 1958, p.85)
There is an optimal portfolio for all investors.
Investment and Financing [or Leverage] decisions
made separately. ړᐶਧቘҁಓ҅1958҂
'ਖ਼ಭᩒᕟݳೠᳯ᷌ړᥴԅӧݶᕆڦጱᘸړݳᯈ - ḒࣁضᩒԾᔄڦԏᳵԏݸᬰᤈړᯈ“ҁಓ҅1958ଙ҅ᒫ85ᶭ҂
ಅํಭᩒᘏ᮷ํ๋֯ጱಭᩒᕟ̶ݳ
ಭᩒᣟᩒ[]٬ᒽړᬰᤈ̶ 3 The Capital Market Line (CML)
Optimal risky portfolio with access to a risk-free asset
• the Optimal / Market [M] risky portfolio is at the
tangency point between the efficient frontier and the
original opportunity set.
• Due to the linear relationship between expected return
and standard deviation, the efficient set is now a range
of possible combinations of the risk-free asset and
Market portfolio [M].
ݢզ឴෫ᷚᴾᩒԾጱ๋֯ᷚᴾಭᩒᕟݳ
•๋ս/૱[M]ᷚᴾಭᩒᕟ॒ݳԭํපᬟኴܻতտᵞԏᳵጱڔᅩ̶
•ኧԭᶼ๗තፅӨຽٵ૧ԏᳵጱᕚىᔮ҅ํපᵞݳሿࣁฎ෫ᷚᴾᩒԾ૱ಭᩒᕟݳጱݢᚆᕟݳࢱ[M]̶ 4 The Capital Market Line (CML)
Optimal portfolio with access to a risk-free asset
• The investment decision involves choosing a particular
portfolio combination of risky and risk-free assets
• Once the investor decides upon the optimal risky
portfolio the financing decision involves a choice
between borrowing or lending the risk-free asset to
maximise utility
- points to the left (right) of the tangency portfolio
(i.e. Rp) represent lending (borrowing) at the risk-free
rate.
ݢզ឴෫ᷚᴾᩒԾጱ๋֯ಭᩒᕟݳ
•ಭᩒ٬ᒽၿ݊ೠٍํᷚᴾ෫ᷚᴾᩒԾጱᇙਧಭᩒᕟݳ
•Ӟ෮ಭᩒᘏ٬ਧ๋սᷚᴾಭᩒᕟ҅ݳᣟᩒ٬ᒽ੪տೠفᬮฎڊ෫ᷚᴾᩒԾզਫሿපአ๋य़۸
5
- ڔݻᅩಭᩒᕟݳҁܨRp҂ጱૢᬟҁݦᬟ҂ᤒᐏ෫ᷚᴾڥሲጱᩅྃҁྃ҂̶ The Capital Market Line (CML)
Efficient frontier with Rf asset -> Optimal portfolio
Opportunity set expanded -> CML
The allocation depends on investors’ preferences.
ٍํRfᩒԾጱํපᬟኴ - >๋֯ಭᩒᕟݳ
տᵞ - > CML
ړᯈݐ٬ԭಭᩒᘏጱ؇অ̶ Investors can vary the risk of their portfolio investment by
changing weights in the Rf asset and the Market portfolio M.
ಭᩒᘏݢզ᭗ᬦදݒRfᩒԾ૱ಭᩒᕟݳጱ᯿දݒಭᩒᕟݳಭᩒጱᷚᴾ̶
໑ഝCMLᬯӻಭᩒᕟࢧݳಸғ This portfolio return according to the CML: E Rp Rf E RM Rf
p
M
6 The Capital Market Line (CML) Source: (Business Finance 9th Ed., Peirson et al.) 7 The Capital Market Line (CML)
With reference to Figure 7.11: • Any portfolios on the line RfT are inefficient.
• Investors can achieve a portfolio with the best possible
return for any level of risk by holding Portfolio M.
• The line RfMN tangents with the efficient frontier.
• The line RfMN represents different combinations of risk-free
asset and the Portfolio M. RfTᕚӤጱձ֜ಭᩒᕟݳ᮷පሲ֗ӥ̶
•᭗ᬦ೮ํಭᩒᕟݳM҅ಭᩒᘏݢզ᭗ᬦձ֜ᷚᴾଘ឴๋֯ࢧಸ̶
•RfMNᕚӨṛපᬟڔᕚ̶
•RfMNᕚᤒᐏ෫ᷚᴾᩒԾಭᩒᕟݳMጱӧݶᕟ̶ݳ 8 The Capital Market Line (CML)
With reference to Figure 7.11:
• Investor 2 (with U2) prefers to invest 100% of the wealth in
Portfolio M.
• Investor 1 (with U1) prefers to invest < 100% of the wealth
in Portfolio M and lend at the risk-free rate.
• Investor 3 (with U3) prefers to invest > 100% of the wealth
in Portfolio M and borrow at the risk-free rate.
ಭᩒᘏ2ҁU2҂ݻԭਖ਼100ѾጱᨰಭᩒԭಭᩒᕟݳM.
•ಭᩒᘏ1ҁU1҂ݻԭಭᩒᕟݳMӾጱ<100Ѿጱᨰଚզ෫ᷚᴾڥሲ׀ᩅ̶ྃ
•ಭᩒᘏ3ҁU3҂ݻԭಭᩒᕟݳMӾ100ѾզӤጱᨰଚզ෫ᷚᴾڥሲᬰᤈᩅ̶ 9 The Capital Market Line (CML)
Because the market is in equilibrium, all assets are included
in this portfolio in proportion to their market value.
In the Market Portfolio, unsystematic risk is diversified away
-> Only systematic risk remains. The decision to borrow or lend to obtain a point on the CML
is a separate decision based on risk preferences.
Separation theorem: separation of the investment decision
from the financing decision. That is, there is only one
Optimal Portfolio for all investors.
ኧԭ૱॒ԭᤍᇫா҅ಅํᩒԾ᮷ೲٌ૱հ꧊ᦇفಭᩒᕟ̶ݳ
ࣁ૱ಭᩒᕟݳӾ҅ᶋᔮᕹᷚᴾग़ز۸ - >ํݝᔮᕹᷚᴾׁᆐਂࣁ̶
አڊCMLጱ٬ਧฎचԭᷚᴾ؇অጱᇿᒈ٬ਧ̶
ړᐶਧቘғਖ਼ಭᩒ٬ᒽӨᣟᩒ٬ᒽ̶ړԞ੪ฎ᧔҅ಅํಭᩒᘏํݝӞӻ๋֯ಭᩒᕟ̶ݳ 10 Capital Asset Pricing Model (CAPM)
What determines the E(r) on an individual asset?
Relationship between returns and systematic risk.
Investors sufficiently compensated for taking on the risk.
Determine:
How the aggregate of investors will behave
How market equilibrium returns are set
Determine the relevant measure of risk
CAPM: Developed by Sharpe, Lintner and Mossin (SLM)
Ջԍ٬ਧԧܔӻᩒԾጱEҁr҂Ҙ තፅӨᔮᕹᷚᴾԏᳵጱىᔮ̶
ಭᩒᘏಥ᪃ड़ጱᷚᴾᤑ̶ؑ Ꮯਧғ
ಭᩒᘏහਖ਼ই֜ᤒሿ ই֜ᦡਧ૱ᤍතፅ Ꮯਧፘىጱᷚᴾଶᰁ
CAPMғኧSharpe҅LintnerMossinݎҁSLM҂ 11 CAPM
E ( Ri ) Rf E ( RM ) Rf 2
M cov i , M • The covariance term is the only explanatory factor in the equation that is specific to asset i. •协⽅方差项是唯⼀一的解释性因素
⽅方程式是特定于资产i的⽅方程式。
• • As Cov(Ri,RM) is the risk of an asset held as part of the market portfolio, and M is the risk of the market portfolio,
beta
measures the risk of i relative to the risk of the
market as a whole. COVi , M i 2
M 由于Cov（Ri，RM）是资产作为市场投资组合的⼀一部分⽽而持有的⻛风险，且
所以β衡量量i相对于⻛风险的⻛风险整个市场。 2M是市场投资组合的⻛风险， 12 CAPM & Security Market Line (SML)
CAPM: E Ri Rf i E RM Rf SML is the Graphical Form of CAPM 13 CAPM Assumptions
Assumptions:
Investors are risk averse and maximize utility
Access to unlimited borrowing and lending at Rf
Homogeneous expectations
One period investment horizon 假设：
投资者是⻛风险规避者，并将效⽤用最⼤大化
在Rf均匀期望下获得⽆无限制的借贷
⼀一期投资期限
投资是⽆无限可分的没有交易易成本或税收 Investments are infinitely divisible
No transaction costs or taxes 14 CAPM and Risk
βi measures the risk of an individual asset held as part of
the market portfolio.
The market will only reward investors for bearing systematic
risk.
Unsystematic risk can be diversified away by holding a
sufficiently large number of securities in a portfolio.
However, CAPM states the reward for bearing systematic
risk is a higher expected return, consistent with the returnrisk tradeoff.
βi衡量量作为市场组合⼀一部分持有的个⼈人资产的⻛风险。
市场只会奖励投资者承担系统⻛风。
通过在投资组合中持有⾜足够多的证券可以使⾮非系统性⻛风险多元化。
然⽽而，CAPM表示，承担系统性⻛风险的回报是较⾼高的预期回报，与回报 - ⻛风险权衡⼀一致。 15 CML vs SML
Comparing the Security Market Line (SML) with the
Capital Market Line (CML)
CML SML 在CML下，B效率低下，B的额外开发者是可以分散的
在SML下，CAPM仅对不不可分散的⻛风险A和B具有相同的Beta，因此相同的E（R） Under CML, B is inefficient, the extra std dev of B is
diversifiable
Under SML, CAPM only prices non-diversifiable risk
A and B have the same Beta and hence same E(R) 16 CAPM Application
Given the CAPM equation;
E(Ri) = Rf + i(E(Rm)-Rf) In order to apply the CAPM, inputs are needed for:
Rf: risk-free rate
E(Rm)-Rf : market risk premium
i : beta specific to the asset 17 CAPM Application
Example: Asset A has an actual return of 12%. Its
covariance with the market return is 0.15 and the market
variance is 0.10. The market return is 9% and the Rf is
5%. Is this asset fairly priced?
COVi , M
i i E Ri 2
M = 0.15/0.10 = 1.5 Rf i E RM Rf E(Ri) = 5% + 1.5[9% - 5%] = 11%
18 CAPM Application
Example:
E(R) Underpriced SML
12%
11%
Fairly Priced Rf = 5% β
1.5
19 CAPM Application
In equilibrium, all assets should plot on the SML
Any security with an estimated return that plots above
the SML is underpriced
Any security with an estimated return that plots below
the SML is overpriced
在平衡中，所有资产都应该在SML上绘制
任何证券估算的回报都在上⾯面绘制
SML被低估了了
估计回报低于SML的任何证券都被⾼高估 20 CAPM Predictions
CAPM预测：
资产的预期收益与系统⻛风险之间存在线性关系。和;
测试版是衡量量⻛风险的唯⼀一指标。根据超额收益重新编写CAPM： The CAPM predicts that:
There is a linear relationship between asset’s expected
return and its systematic risk. And;
Beta is the only measure of risk.
Re-write CAPM in terms of Excess Return: E ( Ri ) Rf This implies that: i i i 0 RM Rf 21 Empirical CAPM Test
Early empirical tests of CAPM
R*i = 0 + 1 i CAPM的早期实证测试 + ei where R*i = Ri - Rf Tests focused on:
0 = 0
i explains return variation in total, and a linear
relationship in beta
1 = risk premium
Market risk premium is positive. ie. E(Rm)>Rf Evidence generally found:
1 was statistically significant
0 also statistically significant 通常发现的证据是：
1有统计学意义
0也有统计学意义 22 Extensions of the CAPM
No risk-free asset (Source: Bodie, et al., 2007)
When borrowing is restricted, the market portfolio is no
longer the common optimal portfolio for all investors.
Investors can choose risky portfolios on the efficient
frontier, which will depend on their risk preferences.
CAPM can be derived using a Zero Beta portfolio (Black,
1972).
当借款受到限制时，市场组合是否定的
所有投资者的共同最优投资组合更更⻓长。
投资者可以选择有效边界上的⻛风险投资组合，这将取决于其⻛风险偏好。
CAPM可以使⽤用Zero Beta组合（Black，1972）推导出来。 23 Extensions of the CAPM
No risk-free asset
Black’s model is based on the 3 properties:
[1] Any portfolio constructed by combining efficient
portfolios is itself on the efficient frontier.
[2] Every portfolio on the efficient frontier has a
‘companion’ portfolio on the inefficient part on the
frontier with which it is uncorrelated -> Zero Beta
portfolio.
[3] The E(r) of any asset can be expressed as an exact,
linear function of E(r) on any two frontier portfolios.
Black的模型基于3个属性：
[1]通过组合有效投资组合构建的任何投资组合本身就是有效的边界。
24
[2]有效边界上的每个投资组合都有⼀一个“伴侣”投资组合，位于边界上效率不不⾼高的部分 - “零投资组合”。
[3]任何资产的E（r）都可以表示为任意两个边界投资组合的E（r）的精确线性函数。 Extensions of the CAPM
No risk-free asset
E(R) M
Rz A
z z S.D.
25 Extensions of the CAPM
Unequal borrowing/lending rates:
E(R) IC(b) D IC(l)
Rb C B Rl A S.D.
26 Critiques on the CAPM Tests
Roll’s Critiques:
CAPM is an ex-ante model: But it is tested using
ex-post data.
The proxy used for the market return is not a true
market portfolio.
The tests of the CAPM essentially test the
efficiency of the market portfolio and not the
prediction of the CAPM itself.
CAPM是⼀一个事前模型：但它使⽤用事后数据进⾏行行测试。
⽤用于市场回报的代理理不不是真正的市场投资组合。
CAPM的测试基本上测试了了市场投资组合的效率，⽽而不不是CAPM本身的预测。 27 Arbitrage Pricing Theory (APT)
Arbitrage
Generally defined as where positive returns can be made
with no risk and with zero net investment.
Buy an asset at a low price and sell an identical asset at a
higher price.
套利利
⼀一般定义为在没有⻛风险和净投资为零的情况下可以获得正回报。
以较低的价格购买资产并以较⾼高的价格出售相同的资产。 Investors may have to sell an asset that they don't already
own in order to make arbitrage, in the process known as
short-selling.
The arbitrage process will effectively restore equilibrium.
投资者可能不不得不不出售他们尚未拥有的资产以进⾏行行套利利，这个过程称为卖空。
套利利过程将有效恢复平衡。 28 Arbitrage Pricing Theory (APT)
Developed primarily as a response to the shortcomings of
the CAPM.
Less restrictive assumptions.This model does not require:
- quadratic preference functions
主要是为了了回应CAPM的缺点⽽而开发的。
限制较少的假设。该模型不不要求： - ⼆二次偏好函数
- normally distributed returns
- 正常分配的回报
- identification of market model
- 确定市场模式
Empirically testable. A model of asset pricing that describes the risk premium for
a risky asset as a linear combination of various risk factors.
经验可检验。
描述⻛风险溢价的资产定价模型
作为各种⻛风险因素的线性组合的⻛风险资产。 29 Arbitrage Pricing Theory (APT)
The APT Model
E(Ri)=λ0+ λ1bi1+ λ2bi2+…+ λkbik
where:
λ0 = expected return on an asset with zero systematic risk
λj = risk premium related to the j th common risk factor
bij = pricing relationship between the risk premium and the
asset; that is, how responsive asset i is to the j th common
factor
哪⾥里里：
λ0=没有系统⻛风险的资产的预期收益
λj=与第j个共同⻛风险因素相关的⻛风险溢价
bij =⻛风险溢价与资产之间的定价关系;也就是说，我对第j常⻅见资产的反应如何
因⼦子 30 Arbitrage Pricing Theory (APT)
A Comparison with CAPM
In CAPM, the relationship is as follows: E(Ri)=RFR + βi[(E(RM-RFR)]
Comparing CAPM and APT (Exhibit 9.1)
CAPM APT Form of Equation Linear Linear Number of Risk Factors
Factor Risk Premium
Factor Risk Sensitivity
“Zero-Beta” Return 1
[E(RM) – RFR]
βi K (≥ 1) RFR {λj}
{bij}
λ0 31 APT: The Factors
What factors should be included in the model?
How many factors should be included in the model?
It is only with retrospective factor analysis that these
questions can be addressed.
These questions are best answered empirically as the theory
does not explicitly specify what the risk factors are.
模型中应包含哪些因素？
该模型应包含多少因素？
只有回顾性因素分析才能解决这些问题。
这些问题最好根据经验来回答，因为理理论没有明确说明⻛风险因素是什什么。 32 APT Empirical Evidence APT的经验证据
1）确定不不可观察因素的统计技术Faff（1988-1992）确定了了5个因素
因素结构⼀一般在3-5个因素的范围内 Empirical evidence on APT
1) Statistical techniques to identify unobservable factors
Faff (1988-1992) identify 5 factors
Factor structure generally in the range of 3-5 factors 2) Arbitrary selection of economic variables
Chen, Roll and Ross (1986) VWNY = Return on the value-weighted New York Stock Index,
MP = monthly growth rate in industrial production,
2）任意选择经济变量量
DEI = change in expected inflation,
陈，罗和罗斯（1986年年）
VWNY =纽约股票价值指数收益率，
UI = unanticipated inflation,
MP =⼯工业⽣生产⽉月增⻓长率，
UPR = unanticipated change in risk premium DEI =预期通货膨胀率的变化，
UI =意外通货膨胀，
UTS = unanticipated change in term structure UPR =⻛风险溢价未预料料到的变化
UTS =期限结构的意外变化
33 Problems with APT 缺乏接受的原因：复杂性
应⽤用困难
怀疑模型理理论局限性的表现
APT没有确定哪些因素 Reasons for lack of acceptance:
complexity
application difficulty
doubts about performance of model
theoretical limitations APT does not identify what the factors are 34 Fama–French Three-Factor Model
Fama and French (1992) provide evidence on factors that
explain asset returns — no support for CAPM Fama and French (1993, 1995 & 1996) leads to the
Three-Factor Model:
E(Ri) = Rf + bi [E(RM) – Rf] + si E(SMB) + hi E(HML)
where
• Rf is the risk-free rate
• MRP is the market risk-premium
• SMB is the size premium
• HML is the book-to-market premium.
35 FF Model: Application FF Model: Empirical Evidence
Results indicate that the three-factor model is able to fully
explain security returns. However, the three-factor model is derived empirically.
FF模型：经验证据
结果表明三因素模型能够充分解释安全回报。
但是，三因素模型是凭经验推导出来的。 36 FF Model: Carhart’s Extension
Momentum effect, where winners outperform losers, is well
documented.
The finding has led to the incorporation of the momentum
factor into the model (Carhart, 1997):
动⼒力力效应，赢家胜过输家，有据可查。
这⼀一发现导致将动量量因⼦子纳⼊入模型（Carhart，1997）： E(Ri) = Rf + bi [E(RM) – Rf] + si E(SMB) + hi E(HML) + pi E(UMD)
For the purpose of fund performance evaluation.
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