微信客服：xiaoxionga100
微信客服：ITCS521
comp5216代写-2022S2
时间：2022-11-17
2022S2
Tutor: Mandy
FINC3017
Investment and Portfolio Management
Weekly Class 3 (Week 5-6)
Schedule:
Ø Week 5 Factor Models
1. Factor Models Overview
1) Intuition of Factor Model / Why factor model
2) Factor model introduction
2. SIM Introduction
1) What is SIM?
2) SIM !"#<$%&'(>
3) Diversification & Portfolio construction with SIM
4) SIM summary
5) SIM Implementation
3. Treynor-Black Model (actively managed portfolio)
4. Issues and practice of Factor models
Ø Week 6 Arbitrage Pricing Theory (APT)
1. Preliminary Knowledge
1) Intuition of APT
2) Arbitrage
2. APT Overview
1) Assumptions
2) Single Factor
3) Multiple Factors
4) APT & Risk Premium
5) Test of APT
3. Summary
1) Relationship with CAPM
2) Stock v.s. Portfolio
3) Issues with APT
!
Ø Week 5 Factor Models
1. Factor Models Overview
1) Intuition of Factor Model / Why factor model
l CAPM/Markowitz Model)*+,-./.0123456789
l Estimating all these parameters accurately is an extremely difficult task (:;)<-.
=>,?@ABCD3)
l Errors in these estimates can have a significant impact on the composition of the
portfolio which can lead to poor performance (-.,=>EFGHIJKLMNO,
PF4
à
l There are a relatively small number of factors that are the main determinants of risk
and hence return (YZ)
l Firms returns may be influenced by other underlying factors
Aim: to understand how to link these factors to returns and understand the implications
for portfolio construction
2) Factor model introduction
l Definition: A factor model is a statistical specification of how a dependent variable
(asset returns) is linked to independent variables (factors) which are usually identified
with the return on an index (or multiple indices)
l Assumptions used
* dRefghi 0jdReklVm4nopRi 0j
* Factorf qdReopRi 0jr factorVkst4nuvopRi 0j
!
l Expected return/ Variance/ Cov expression in common mathematic equation
ü Expected return (as excess return)
ü Variance
ü Covariance
l Expected return/ Variance/ Cov expression in matrix
ü Notation [wxyz{|}~!,]
* n: nK
* k: k| factor
ü Expected return in matrix
ü Variance in matrix
l Application in portfolio construction
p#56, return vector covariance matrix NOKLM
!
2. SIM Introduction
1) What is SIM?
ü A return-generating model which provides the estimation of the expected return of
a security given certain parameters. (`-.4W;,=gbc=/
>5,)
ü A single-factor linear model. (YA)
ü The factor is the return on a market index.
2) SIM !"#<$%&'(>
l SIMp!( i t ¡(, excess return)
Ri,t = ai + βiRM,t + ei,t
ü : extra-market excess return
ü : sensitivity to the market index M (W¢£LM,¤¥C)
ü RM: excess return of the index
ü : unexpected return (firm-specific) or residual/error term. (unexpected event
Vm,bc¦)
l § Risk-Returnx¨ SIM(bc[W\],$©)
ü Total risk = Systematic risk + Non-systematic risk
= Market risk + Firm-specific (idiosyncratic) risk
wª«¬4®A\]l¯°±²³JK´µ¶4a·il¯²³\]4¸
¹º»¼½¾,¿À/»¼Ág,ÂÿÀÄÅÆ®A\]ÇÈ,¯É°±²
³JK´ÊËÅ
ü Total Variance = Systematic variance + Non-systematic variance
[wx̲Í\]]
σi2 = βi2σm2 + σ2(ei)
² βi2σm2!®A\](systematic/market risk).
WÎ,mti market related return.
² σ2(ei) !ÏÐ<\]4ÑÆ®A\].
WÎ, ei ÏÐ<,l@`YZÒ«, unexpected return.
ü ri q rj, Covariance
!
3) Diversification & Portfolio construction with SIM [wxÓ]
ü Assumption: portfolio is equally weighted (xi = 1/n)
ü Portfolio return
ü Portfolio variance
04n¬ÞØlßàáKâJKLMã 4ÏÐ<\]G
ÙÚ 04"ä²³\],åæ(diversification).>
çè portfolio variance¯!i
4) SIM summary
l SIM estimates
a) Expression/formula
ü Expected return for portfolios/assets (as excess return)
ü Volatility for portfolios/assets
ü Covariance
b) Total number of estimated parameters (3n+2)
- n estimates of ai
- n estimates of bi
- n estimates of firm specific risk sei
- 1 estimate of market variance sm
- 1 estimate of the market risk premium µ
!
l Advantages & Disadvantages of SIM
a) Advantages
ü It allows analysts from different sectors to combine their information in a
simple way to construct portfolios
ü All the information required are the respective stocks alpha and beta
b) Disadvantages
s It assumes that there is only one common source of risk (captured by the
market index) and thus omits risk for a particular industry
s It ignores correlations among firm specific risks and thus may not provide the
best risk minimising weights in portfolio theory
l Markowitz Mean-Variance Model V.S. Single Index Model <$%&'(>
Markowitz Mean-Variance Model Single Index Model
Number of Input Data
(初始需要的数据量)
• N return estimates
• N Variance estimates
• (N2 – N)/2 Covariance estimates
Total = (N2 + 3N)/2
• N α estimates
• N β estimates
• N Variance estimates
• 1 estimate of market return
• 1 estimate of market variance
Total = 3N + 2
Assumptions of return
(假设条件)
• No assumption on return generating process.
• Parameter values obtained by calculating
sample estimates from historical data.
• SIM makes assumptions about the return generating
process.
• The parameters are obtained through the imposed
assumption on data generating process.
Computation issue • More data involved as n increases à difficult to
calculation.
• Easier than Markowitz model.
• Vastly reduced the number of estimates. ! and Covariance Matrix • Estimated from historical data • Calculated based on the estimates obtained from
regression model.
!
5) Single Index Model Implementation
l 假设 Portfolio中有 n类资产，每类资产的收益满足：
ri,t = ai + βimt + ei,t
Y X
l 利用 SIM求解 minimum variance portfolio
解读：
ü 回归方程求解出的截
距项(intercept)即为方程中
的 ai；
ü 斜率(X variable 1对
应的相关系数)即为 βi；
ü 残差项(residual)对应
的mean squared error
(MS)为 σ2(ei).
函数解读：该函数计算的参数返回值如下：
Column 1 Column 2 Column … Column n Column n+1
Row 1 系数m.n 系数m.n-1 … 系数m.1 b
Row 2 se_m.n se_m.n-1 … se_m.1 se_b
Row 3 R-square se_y
Row 4 F df
Row 5 SSR (regression) SSE (residual)
Step 1. 分别对每个 asset 进行线性回归(asset excess return & market excess return)à求得并记录 ai、βi和
σ2(ei)
Ø ai、βi
Ø σ2(ei) = σi2 - βi2σm2
注意参数：Input Y 为 asset excess return； Input X 为 market excess return
!
Step 3. 依据以上两步计算出的数据计算出和 Covariance，并以矩阵形式表示.
Ø
Step 4. 计算 portfolio整体的 return和 standard deviation(同 Markowitz中讲解的公式)，但是使
用的数据用上述方式求解的和 Covariance，通过 Solver求解得到 minimum variance portfolio.
注意区别：SIM 中的 Covariance Matrix 不是像Markowitz Mean-Variance Model直接用 Excel 的
Covariance 求解，而是需要根据 Step 3的 matrix中的公式计算.
Step 2. 依据 market excess return 历史数据求出 E(m)和 σm2.
Ø
!
3. Treynor-Black Model (actively managed portfolio)
l Intuition of Treynor-Black Model: construct an optimal actively managed portfolio
l Methodology
ü Composition: 1 active portfolio (< n-1|K) + 1 passive portfolio (market
index;ë n|K)
ü Assumption: built with SIM (only one factor: market index)
l Properties/notations for Market index /Passive portfolio
l Properties/notations for active portfolio
l Weight of each asset in the portfolio (ìíJK,îï4ðKñò!"#)
* xi: PortfolioãK i ,ñò/weight
* ai: PortfolioãK i ,óôbc/reward
* !!" : dRepR/residual variance
* aI /!!: information ratio
l Weight of active portfolio and passive portfolio/market
• #$i optimal portfolioã^ÃJKLM(active portfolio),õöñò(initial
position)4n% = 1;
• W# i÷øbAâùú.û 4active portfolio,ñòj
• ^ÃJKLM(active portfolio)üÃJKLM(passive portfolio),býþ 4Ä#
7t4ØbA.ûÿ94^ÃJKLMüÃJKLMÈ¡,VmA!34Ó"Q§
üÃJKLMã#$,%þÖ/²³AbcDX4YvGÊËüÃJKLM,&
'4àá^ÃJKLM,&'4nW#Gàá>
l Economic intuition
ü The benefits of the active portfolio are extra returns in the form of a (NO active
portfolio,Ó[()ô*bc4Éa!)
ü The downside is to take on idiosyncratic risk to achieve the extra return (+[,
*+-.ƮA\]4n|/|K,/A\])
!
4. Issues and practice of Factor models
1) Estimation Issues [Ôx]
l Beta is not constant (and neither is alpha) what we estimate may not always reconcile
with what happens in the future (ù0ã betaG124l[ý7l2,)
ü There is research on identifying variables to forecast beta (Rosenberg and Guy,
1976)
ü Forward looking betas may also be extracted from options prices
• Buss and Vilkov (2012)
• Chang, Christoffersen, Jacobs, Vainberg (2012)
l Some other econometric issues exist
2) Do Factor Models Work? [Ôx; 3Vm paper45]
l The efficacy of factor models is difficult to ascertain as it comes down to how well one
can estimate alpha and beta (,<åA63C[7_W alpha betaû,=
)
l Ledoit and Wolf (2003) showed that using the SIM to help compute the covariance
matrix (shrinkage estimation) resulted in portfolios with significantly lower out of
sample variance
l This was better than sample covariance estimators and even better than multifactor
estimators (8þ8,opR-.>5É3%YZ>5p9)
l Perhaps a simple model is better able to capture market dynamics/is more robust to
error (5E¦!9)
3) Factor model in practice
l Fama-French 3-factor model [wx]
* RM: excess return of the market portfolio relative to risk free rate.
* SMB(small minus big): size risk factor; excess return of small stocks relative to
large stocks
* HML(high minus low): value risk factor; excess return of value stocks relative to
growth stocks
!
l Some research paper and results [Ôx]
a) Paper1: Chen, Roll and Ross (1986)
• Major objective: Examine factor models where factors are drawn from the
macroeconomy.
• Findings: Some of the factors found to provide explanatory power include:
ü spread between long and short interest rates
ü expected and unexpected inflation
ü changes in industrial production
ü spread between high- and low-grade bonds
b) Paper2: Harvey, Liu and Zhu (2016)
• Findings: find that many factors studied in the literature do not hold
statistical significance in more general settings
4) Application: Market Tracking [wx]
l Objective: create a tracking portfolio to remove excess market risk.(°±NO tracking
portfolio4:;<Ç<&'4É=>¢£Ë?Ò«,\]4"äW@åæ4:A
alpha)
l Examplef
ü BCD
ü NO tracking portfolio RTf1.4 JK¢£LM4-0.4 T-bill (+:;øñ
òi 1)
ü , ;< Rp (long position) RT(short position)4¯$f
l Implications:
ü Popular strategy in hedge fund
ü Allows to extract the a while eliminating all market risk
!
Ø Week 6 Arbitrage Pricing Theory (APT)
1. Preliminary Knowledge
1) Intuition of APT (compare with CAPM)
l Relatively less assumptions than CAPM <¶Ö/EFÔýGBC>
ü CAPM: assume investors preferences for risk as quadratic utility
ü APT: none regarding investor preferences
l Core mechanism
ü CAPM: equilibrium as the pricing mechanism (HI,K`J)
ü APT: arbitrage as the pricing mechanism (KLÔM)
2) Arbitrage
l Definition: constructing a net zero investment portfolio which has no possibility of loss
and a strictly positive probability of making a positive return.
(Ûõö784Û\]4
l Characteristics:
ü Results from securities being mispriced relative to each other (EF`JO)
ü Exploiting the arbitrage returns prices back to arbitrage free values via strong
buying/selling; arbitrage is a riskless profit, this buying/selling pressure is very
strong and prices hence revert to fair values quickly (°±M4KJPGQR
âÛMJP; ´ equilibrium/HI,±ýSGTDUV"ä)
l Various arbitrage concepts [wx]
ü Pure arbitrage: A truly risk free investment which has positive probability of return;
This is the definition used in derivatives pricing models and is usually theoretical.
(WM4nÛ\],M#4X
ü Statistical arbitrage: Assets are mispriced on average and a net zero investment
which exploits this can profit in expectation. This strategy is used by sophisticated
investors and exploited via large portfolios and (usually) high frequency trading.
(®5M4ýY¢£ãA¾Z44°[\3]^31KÉ39_]^)
ü Merger arbitrage: Betting on the outcome of a merger. If it goes through then the
price of target and acquirer should converge, if not the price gap widens. Not pure
arbitrage but a risky bet. Sometimes called risk arbitrage (M:M4`WaO:
b]^,cp)
ü Asymptotic arbitrage: when an arbitrage opportunity may be reached by
building portfolios of a large number of assets (lim n→∞) which can be used to
diversify all risks away (dQJKLMãK.1,àá4JKLM,\]eP,
îï)
!
2. APT Overview
1) Law of one price v.s. Arbitrage
l Law of one price: equivalent securities or bundles of securities must sell at equal
prices to preclude arbitrage opportunities. / Same security, delivering the same cash
flows, must have the same initial price. (ýJ`f4n,Ä.1,,;4JPÎý
IXlêM,g¡)
l Arbitrage: h¸4 ,Kl,¢£,l,iJM7$N
2) Assumptions
l Security returns can be described by a factor model
l There are sufficient securities to diversify away idiosyncratic risk
l Well functioning securities markets do not permit the prolonged presence of arbitrage
(ýS«¬4jN[,;¢£4MilGkgê)
3) Single Factor
l Factor structure/expression
* f: factor, random variable; nl,,Y4dl21
* b: factor loading/factor sensitivity (WÎY,¤¥C)
* mnÄ#fdQKLMãK.1àá4/A\]/|<\]ü%²²³
l Example 1_Portfolio with same b [wxÓ]
opfBCÍ|\]%²²³,JKLM(well diversified portfolio) A B4,b4+[KLMbcl,
x¨f
s qri factor
s sribc
s t¦ib
s B,bcË A4uMÏ#4Rôi% − &
s »¼vÓfwKLM BxLKLM
A4nl*+õö&'4¯É#$N
!
l Example 2_Portfolio with different b [wxÓ]
opfBCKLM C4 AKLM D
x¨f
s qrib4sribc
s zÝ{,KLM4¯wxiV, factor
,l,b,KLMbc
s BCKLM C, b = 0.54bc Erc= 0.06
s KLM A, b = 14bc ErA= 0.10
s NOKLM D: °±;< 50% , A, 50%, rf,
n D,bü÷øi 0.5
s »¼vÓf°±w CxL DM
4) Multiple Factors
l Factor structure/expression_Two Factor Model
l Tracking portfolio / serves as benchmark
s b1 = 1
s b2 = 0
l Example _ Multi-Factor Model
s BCKLM A¯!if
s NOJKLM Qf
* bA1f1 : ¯wxiÉbA1,Th/ñòJ f1
* bA2f2 : ¯wxiÉbA2,Th/ñòJ f2
* (1-bA1-bA2)rf: ¯wxiÉ(1-bA1-bA2),Th/ñòJÛ\]K rf
s MlGfw AxL Q4¯!i
/Ä#ã,Û\]K rf4|4O214n¯4}_`j>
*~fYi E[fi] = 04)É E[rQ]= (1-bA1-bA2)rfj
!
Ø Example 1. There are two well diversified portfolios P, Q and the estimated single factor model are
as follows:
R' = 0.07 + 0.8 Rm + ' R( = 0.02 + 0.8 Rm + (
Please identify whether there’s an arbitrage opportunity. If yes, explain how you will trade to obtain it.
Ø Example 2. Jack observed and estimated single factor model for two well diversified portfolio S, T:
R) = 0.06 + 0.4 f* + ) R+ = 0.02 + 0.2 f* + +
Assume you can invest by borrowing/lending at the risk free rate. Please identify whether there’s an
arbitrage opportunity. If yes, explain how you will trade to obtain it.
Ø Example 3. Assume there are two well diversified portfolios and excess return described as two-
factor model as follows:
R# = 0.05 + 1.2f* + 0.8f" R& = 0.05 + 0.6f* + 0.4f"
The factors are not directly tradeable. Please explain whether the arbitrage opportunity exists, if yes,
please clarify how you will obtain it.
!
5) APT & Risk Premium
l °± APT¯&4ØrK, returnâÛMlGê,4nØK/KLM,bV, 4b
cÎÝýI
l Expression in premium
s allows to define whether a factor is relevant or not.
s A factor that has a risk premium of 0 has no systematic impact on expected
returns
à Examine a statistical test to determine if risk premia l are statistically different from 0
6) Test with Fama-MacBeth Regressions (çõ[Î CAPM)
l Objective/,f; risk premium/-.l [li 0
l Methodology/p9: Fama-MacBeth Regressions
l Core idea/f
a) Running a time series regression to compute b
b) Running a cross-sectional regression to compute risk premia
c) Test the estimated risk premia for statistical significance
l Detailed Procedures
a) Step 1:
s Find estimates of the betas from time series regressions
r,,. − r/,. = , + 0(r1,2 − r/,.) + ,,. t = 1,2,3, … T, for each tji = 1,2,3,…,N
0: Excel function: = SLOPE(Y,X) à
b) Step 2:
s With the estimated betas from the first stage, running a cross-sectional
regression at each time point t. (,ý ¡WÎ,l,) à$ä
TL λ0,t λM,t r,,. − r/,. = $,2 + 1,20 + v,,., t = 1,2,3, … T, for each tji = 1,2,3,…,N
Y: Excess return of the same item throughout different time
X: Excess return of the market throughout different time
!
c) Step 3: test the λM,t
s Compute the t-statistics for λM,t
s °±TD56, t-stat ®51ûTD4ßA
l Test methodology – t-test <$%&'(>
a) Null hypothesis(DBC): ¯wxi-.i 0
b) t-stat formula
s t-stat = 456789.6:;0; <=>?:@"#$!#%&'% = 45$@"#$!#%&'(
c) ßp9 1: t-stat <ÿ3ÿ>
s ä`A(significant level), tû
ü t-table
ü Excel function: = TINV(Probability, deg_freedom)
ü 1.96Ñ 2 (significant level=5%)
s WT-.WÎ, t-stat `A(significant level), tû
s ßuª
ü t-stat > tû à L4DBC4WÎ-.li 0.
ü t-stat < t ûà lDBC4WÎ-.i 0.
d) ßp9 2: p-value <ÿÿ>
s WT p-value `A(significant level)
s ßuª
ü p-value < significant level (5%) à DBC4WÎ-.li 0.
ü p-value > significant level (5%) à lDBC4WÎ-.i 0.
l Problems/issues with Fama-MacBeth Regression [Ôx]
ü Suffer from the error-in-variables problem because thebetas used in the second
pass regression must be estimated. (are not observed)
ü The tests assume that the time-series residuals (e) are uncorrelated though time
(+[lý`?@)
Coefficients Standard Error t Stat P-value Lower 95% Upper 95%
Intercept -0.0011341 0.006276525 -0.1806863 0.85724367 -0.0136979 0.01142975
Mkt - rf 0.97007904 0.189292106 5.12477282 3.5697E-06 0.59116972 1.34898836
!
3. Summary
1) Relationship with CAPM
àBCý| well diversified portfolio4éÉ market cap weights/market portfolioi_`b
c,ýVmYZ4:É excess returni APTã, factor4¯!if
x¨f§Ï#!" #«¡4# CAPM,8!"#ýIj+[ APT CAPM,B
Cl,jX APTã, market portfolio CAPMã,¢£l,4 APTã4market
portfolio¯É[ý well diversified portfolio,´ CAPMã,`[¢¤¥¢£)<¯
JK,K/;,¦MÅn§Ï#§C4¯Éwxi CAPM[ýY¨, APT4´ APT
VØ[ CAPM,©£/ª«Å
2) Stock v.s. Portfolio
¬f¸æl®¯ return-beta relationship/APT ,KLOKLM4ÝKLM
[°®¯±
uªf²KLM[}M APT,4ãN7LM,KaH®¯/}MVÎm.
3) Issues with APT [Ôx]
l Issues: provides no guidance on what factors are appropriate
l Solutions/ suggestions
a) Chen Roll and Ross: suggested using factors which represent risk based on the
macro-economy (¹º»¼YZ)
b) Fama-French Three Factor Model <-³Week 5>
s Market return: Excess return on a market cap weighted portfolio
s Small – Big: return of a portfolio of small stocks in excess of returns of a
portfolio of big stocks (size premium). Small stocks are fundamentally riskier
than large ones so a premium is required to invest in them
s High – Low: return of stocks with a high book to market ratio minus return on
portfolio of stocks with low book to market ratio (value premium). Low BM
implies undervalued stock or poorly performing.
c) Carhart (1997) Model: Fama-French Three Factor Model´µáÔ
momentum factor/Ã1Y (!±¶!U·¸,YZ)
Tutor: Mandy
FINC3017
Investment and Portfolio Management
Weekly Class 3 (Week 5-6)
Schedule:
Ø Week 5 Factor Models
1. Factor Models Overview
1) Intuition of Factor Model / Why factor model
2) Factor model introduction
2. SIM Introduction
1) What is SIM?
2) SIM !"#<$%&'(>
3) Diversification & Portfolio construction with SIM
4) SIM summary
5) SIM Implementation
3. Treynor-Black Model (actively managed portfolio)
4. Issues and practice of Factor models
Ø Week 6 Arbitrage Pricing Theory (APT)
1. Preliminary Knowledge
1) Intuition of APT
2) Arbitrage
2. APT Overview
1) Assumptions
2) Single Factor
3) Multiple Factors
4) APT & Risk Premium
5) Test of APT
3. Summary
1) Relationship with CAPM
2) Stock v.s. Portfolio
3) Issues with APT
!
Ø Week 5 Factor Models
1. Factor Models Overview
1) Intuition of Factor Model / Why factor model
l CAPM/Markowitz Model)*+,-./.0123456789
l Estimating all these parameters accurately is an extremely difficult task (:;)<-.
=>,?@ABCD3)
l Errors in these estimates can have a significant impact on the composition of the
portfolio which can lead to poor performance (-.,=>EFGHIJKLMNO,
PF4
à
l There are a relatively small number of factors that are the main determinants of risk
and hence return (
l Firms returns may be influenced by other underlying factors
Aim: to understand how to link these factors to returns and understand the implications
for portfolio construction
2) Factor model introduction
l Definition: A factor model is a statistical specification of how a dependent variable
(asset returns) is linked to independent variables (factors) which are usually identified
with the return on an index (or multiple indices)
l Assumptions used
* dRefghi 0jdReklVm4nopRi 0j
* Factorf qdReopRi 0jr factorVkst4nuvopRi 0j
!
l Expected return/ Variance/ Cov expression in common mathematic equation
ü Expected return (as excess return)
ü Variance
ü Covariance
l Expected return/ Variance/ Cov expression in matrix
ü Notation [wxyz{|}~!,]
* n: nK
* k: k| factor
ü Expected return in matrix
ü Variance in matrix
l Application in portfolio construction
p#56, return vector covariance matrix NOKLM
!
2. SIM Introduction
1) What is SIM?
ü A return-generating model which provides the estimation of the expected return of
a security given certain parameters. (`-.4W;,=gbc=/
>5,)
ü A single-factor linear model. (YA)
ü The factor is the return on a market index.
2) SIM !"#<$%&'(>
l SIMp!( i t ¡(, excess return)
Ri,t = ai + βiRM,t + ei,t
ü : extra-market excess return
ü : sensitivity to the market index M (W¢£LM,¤¥C)
ü RM: excess return of the index
ü : unexpected return (firm-specific) or residual/error term. (unexpected event
Vm,bc¦)
l § Risk-Returnx¨ SIM(bc[W\],$©)
ü Total risk = Systematic risk + Non-systematic risk
= Market risk + Firm-specific (idiosyncratic) risk
wª«¬4®A\]l¯°±²³JK´µ¶4a·il¯²³\]4¸
¹º»¼½¾,¿À/»¼Ág,ÂÿÀÄÅÆ®A\]ÇÈ,¯É°±²
³JK´ÊËÅ
ü Total Variance = Systematic variance + Non-systematic variance
[wx̲Í\]]
σi2 = βi2σm2 + σ2(ei)
² βi2σm2!®A\](systematic/market risk).
WÎ,mti market related return.
² σ2(ei) !ÏÐ<\]4ÑÆ®A\].
WÎ, ei ÏÐ<,l@`YZÒ«, unexpected return.
ü ri q rj, Covariance
!
3) Diversification & Portfolio construction with SIM [wxÓ]
ü Assumption: portfolio is equally weighted (xi = 1/n)
ü Portfolio return
ü Portfolio variance
ÙÚ 04"ä²³\],åæ(diversification).>
çè portfolio variance¯!i
4) SIM summary
l SIM estimates
a) Expression/formula
ü Expected return for portfolios/assets (as excess return)
ü Volatility for portfolios/assets
ü Covariance
b) Total number of estimated parameters (3n+2)
- n estimates of ai
- n estimates of bi
- n estimates of firm specific risk sei
- 1 estimate of market variance sm
- 1 estimate of the market risk premium µ
!
l Advantages & Disadvantages of SIM
a) Advantages
ü It allows analysts from different sectors to combine their information in a
simple way to construct portfolios
ü All the information required are the respective stocks alpha and beta
b) Disadvantages
s It assumes that there is only one common source of risk (captured by the
market index) and thus omits risk for a particular industry
s It ignores correlations among firm specific risks and thus may not provide the
best risk minimising weights in portfolio theory
l Markowitz Mean-Variance Model V.S. Single Index Model <$%&'(>
Markowitz Mean-Variance Model Single Index Model
Number of Input Data
(初始需要的数据量)
• N return estimates
• N Variance estimates
• (N2 – N)/2 Covariance estimates
Total = (N2 + 3N)/2
• N α estimates
• N β estimates
• N Variance estimates
• 1 estimate of market return
• 1 estimate of market variance
Total = 3N + 2
Assumptions of return
(假设条件)
• No assumption on return generating process.
• Parameter values obtained by calculating
sample estimates from historical data.
• SIM makes assumptions about the return generating
process.
• The parameters are obtained through the imposed
assumption on data generating process.
Computation issue • More data involved as n increases à difficult to
calculation.
• Easier than Markowitz model.
• Vastly reduced the number of estimates. ! and Covariance Matrix • Estimated from historical data • Calculated based on the estimates obtained from
regression model.
!
5) Single Index Model Implementation
l 假设 Portfolio中有 n类资产，每类资产的收益满足：
ri,t = ai + βimt + ei,t
Y X
l 利用 SIM求解 minimum variance portfolio
解读：
ü 回归方程求解出的截
距项(intercept)即为方程中
的 ai；
ü 斜率(X variable 1对
应的相关系数)即为 βi；
ü 残差项(residual)对应
的mean squared error
(MS)为 σ2(ei).
函数解读：该函数计算的参数返回值如下：
Column 1 Column 2 Column … Column n Column n+1
Row 1 系数m.n 系数m.n-1 … 系数m.1 b
Row 2 se_m.n se_m.n-1 … se_m.1 se_b
Row 3 R-square se_y
Row 4 F df
Row 5 SSR (regression) SSE (residual)
Step 1. 分别对每个 asset 进行线性回归(asset excess return & market excess return)à求得并记录 ai、βi和
σ2(ei)
Ø ai、βi
Ø σ2(ei) = σi2 - βi2σm2
注意参数：Input Y 为 asset excess return； Input X 为 market excess return
!
Step 3. 依据以上两步计算出的数据计算出和 Covariance，并以矩阵形式表示.
Ø
Step 4. 计算 portfolio整体的 return和 standard deviation(同 Markowitz中讲解的公式)，但是使
用的数据用上述方式求解的和 Covariance，通过 Solver求解得到 minimum variance portfolio.
注意区别：SIM 中的 Covariance Matrix 不是像Markowitz Mean-Variance Model直接用 Excel 的
Covariance 求解，而是需要根据 Step 3的 matrix中的公式计算.
Step 2. 依据 market excess return 历史数据求出 E(m)和 σm2.
Ø
!
3. Treynor-Black Model (actively managed portfolio)
l Intuition of Treynor-Black Model: construct an optimal actively managed portfolio
l Methodology
ü Composition: 1 active portfolio (< n-1|K) + 1 passive portfolio (market
index;ë n|K)
ü Assumption: built with SIM (only one factor: market index)
l Properties/notations for Market index /Passive portfolio
l Properties/notations for active portfolio
l Weight of each asset in the portfolio (ìíJK,îï4ðKñò!"#)
* xi: PortfolioãK i ,ñò/weight
* ai: PortfolioãK i ,óôbc/reward
* !!" : dRepR/residual variance
* aI /!!: information ratio
l Weight of active portfolio and passive portfolio/market
position)4n% = 1;
• W# i÷øbAâùú.û 4active portfolio,ñòj
• ^ÃJKLM(active portfolio)üÃJKLM(passive portfolio),býþ 4Ä#
7t4ØbA.ûÿ94^ÃJKLMüÃJKLMÈ¡,VmA!34Ó"Q§
üÃJKLMã#$,%þÖ/²³AbcDX4YvGÊËüÃJKLM,&
'4àá^ÃJKLM,&'4nW#Gàá>
l Economic intuition
ü The benefits of the active portfolio are extra returns in the form of a (NO active
portfolio,Ó[()ô*bc4Éa!)
ü The downside is to take on idiosyncratic risk to achieve the extra return (+[,
*+-.ƮA\]4n|/|K,/A\])
!
4. Issues and practice of Factor models
1) Estimation Issues [Ôx]
l Beta is not constant (and neither is alpha) what we estimate may not always reconcile
with what happens in the future (ù0ã betaG124l[ý7l2,)
ü There is research on identifying variables to forecast beta (Rosenberg and Guy,
1976)
ü Forward looking betas may also be extracted from options prices
• Buss and Vilkov (2012)
• Chang, Christoffersen, Jacobs, Vainberg (2012)
l Some other econometric issues exist
2) Do Factor Models Work? [Ôx; 3Vm paper45]
l The efficacy of factor models is difficult to ascertain as it comes down to how well one
can estimate alpha and beta (,<åA63C[7_W alpha betaû,=
)
l Ledoit and Wolf (2003) showed that using the SIM to help compute the covariance
matrix (shrinkage estimation) resulted in portfolios with significantly lower out of
sample variance
l This was better than sample covariance estimators and even better than multifactor
estimators (8þ8,opR-.>5É3%YZ>5p9)
l Perhaps a simple model is better able to capture market dynamics/is more robust to
error (5E¦!9)
3) Factor model in practice
l Fama-French 3-factor model [wx]
* RM: excess return of the market portfolio relative to risk free rate.
* SMB(small minus big): size risk factor; excess return of small stocks relative to
large stocks
* HML(high minus low): value risk factor; excess return of value stocks relative to
growth stocks
!
l Some research paper and results [Ôx]
a) Paper1: Chen, Roll and Ross (1986)
• Major objective: Examine factor models where factors are drawn from the
macroeconomy.
• Findings: Some of the factors found to provide explanatory power include:
ü spread between long and short interest rates
ü expected and unexpected inflation
ü changes in industrial production
ü spread between high- and low-grade bonds
b) Paper2: Harvey, Liu and Zhu (2016)
• Findings: find that many factors studied in the literature do not hold
statistical significance in more general settings
4) Application: Market Tracking [wx]
l Objective: create a tracking portfolio to remove excess market risk.(°±NO tracking
portfolio4:;<Ç<&'4É=>¢£Ë?Ò«,\]4"äW@åæ4:A
alpha)
l Examplef
ü BCD
ü NO tracking portfolio RTf1.4 JK¢£LM4-0.4 T-bill (+:;øñ
òi 1)
ü , ;< Rp (long position) RT(short position)4¯$f
l Implications:
ü Popular strategy in hedge fund
ü Allows to extract the a while eliminating all market risk
!
Ø Week 6 Arbitrage Pricing Theory (APT)
1. Preliminary Knowledge
1) Intuition of APT (compare with CAPM)
l Relatively less assumptions than CAPM <¶Ö/EFÔýGBC>
ü CAPM: assume investors preferences for risk as quadratic utility
ü APT: none regarding investor preferences
l Core mechanism
ü CAPM: equilibrium as the pricing mechanism (HI,K`J)
ü APT: arbitrage as the pricing mechanism (KLÔM)
2) Arbitrage
l Definition: constructing a net zero investment portfolio which has no possibility of loss
and a strictly positive probability of making a positive return.
(Ûõö784Û\]4
l Characteristics:
ü Results from securities being mispriced relative to each other (EF`JO)
ü Exploiting the arbitrage returns prices back to arbitrage free values via strong
buying/selling; arbitrage is a riskless profit, this buying/selling pressure is very
strong and prices hence revert to fair values quickly (°±M4KJPGQR
âÛMJP; ´ equilibrium/HI,±ýSGTDUV"ä)
l Various arbitrage concepts [wx]
ü Pure arbitrage: A truly risk free investment which has positive probability of return;
This is the definition used in derivatives pricing models and is usually theoretical.
(WM4nÛ\],M#4X
ü Statistical arbitrage: Assets are mispriced on average and a net zero investment
which exploits this can profit in expectation. This strategy is used by sophisticated
investors and exploited via large portfolios and (usually) high frequency trading.
(®5M4ýY¢£ãA¾Z44°[\3]^31KÉ39_]^)
ü Merger arbitrage: Betting on the outcome of a merger. If it goes through then the
price of target and acquirer should converge, if not the price gap widens. Not pure
arbitrage but a risky bet. Sometimes called risk arbitrage (M:M4`WaO:
b]^,cp)
ü Asymptotic arbitrage: when an arbitrage opportunity may be reached by
building portfolios of a large number of assets (lim n→∞) which can be used to
diversify all risks away (dQJKLMãK.1,àá4JKLM,\]eP,
îï)
!
2. APT Overview
1) Law of one price v.s. Arbitrage
l Law of one price: equivalent securities or bundles of securities must sell at equal
prices to preclude arbitrage opportunities. / Same security, delivering the same cash
flows, must have the same initial price. (ýJ`f4n,Ä.1,,;4JPÎý
IXlêM,g¡)
l Arbitrage: h¸4 ,Kl,¢£,l,iJM7$N
2) Assumptions
l Security returns can be described by a factor model
l There are sufficient securities to diversify away idiosyncratic risk
l Well functioning securities markets do not permit the prolonged presence of arbitrage
(ýS«¬4jN[,;¢£4MilGkgê)
3) Single Factor
l Factor structure/expression
* f: factor, random variable; nl,,Y4dl21
* b: factor loading/factor sensitivity (WÎY,¤¥C)
* mnÄ#fdQKLMãK.1àá4/A\]/|<\]ü%²²³
l Example 1_Portfolio with same b [wxÓ]
opfBCÍ|\]%²²³,JKLM(well diversified portfolio) A B4
x¨f
s qri factor
s sribc
s t¦ib
s B,bcË A4uMÏ#4Rôi% − &
s »¼vÓfwKLM BxLKLM
A4nl*+õö&'4¯É#$N
!
l Example 2_Portfolio with different b [wxÓ]
opfBCKLM C4 A
x¨f
s qrib4sribc
s zÝ{,KLM4¯wxiV, factor
,l,b,KLMbc
s BCKLM C, b = 0.54bc Erc= 0.06
s KLM A, b = 14bc ErA= 0.10
s NOKLM D: °±;< 50% , A, 50%, rf,
n D,bü÷øi 0.5
s »¼vÓf°±w CxL DM
4) Multiple Factors
l Factor structure/expression_Two Factor Model
l Tracking portfolio / serves as benchmark
s b1 = 1
s b2 = 0
l Example _ Multi-Factor Model
s BCKLM A¯!if
s NOJKLM Qf
* bA1f1 : ¯wxiÉbA1,Th/ñòJ f1
* bA2f2 : ¯wxiÉbA2,Th/ñòJ f2
* (1-bA1-bA2)rf: ¯wxiÉ(1-bA1-bA2),Th/ñòJÛ\]K rf
s MlGfw AxL Q4¯!i
*~fYi E[fi] = 04)É E[rQ]= (1-bA1-bA2)rfj
!
Ø Example 1. There are two well diversified portfolios P, Q and the estimated single factor model are
as follows:
R' = 0.07 + 0.8 Rm + ' R( = 0.02 + 0.8 Rm + (
Please identify whether there’s an arbitrage opportunity. If yes, explain how you will trade to obtain it.
Ø Example 2. Jack observed and estimated single factor model for two well diversified portfolio S, T:
R) = 0.06 + 0.4 f* + ) R+ = 0.02 + 0.2 f* + +
Assume you can invest by borrowing/lending at the risk free rate. Please identify whether there’s an
arbitrage opportunity. If yes, explain how you will trade to obtain it.
Ø Example 3. Assume there are two well diversified portfolios and excess return described as two-
factor model as follows:
R# = 0.05 + 1.2f* + 0.8f" R& = 0.05 + 0.6f* + 0.4f"
The factors are not directly tradeable. Please explain whether the arbitrage opportunity exists, if yes,
please clarify how you will obtain it.
!
5) APT & Risk Premium
l °± APT¯&4Ø
cÎÝýI
l Expression in premium
s allows to define whether a factor is relevant or not.
s A factor that has a risk premium of 0 has no systematic impact on expected
returns
à Examine a statistical test to determine if risk premia l are statistically different from 0
6) Test with Fama-MacBeth Regressions (çõ[Î CAPM)
l Objective/,f; risk premium/-.l [li 0
l Methodology/p9: Fama-MacBeth Regressions
l Core idea/f
a) Running a time series regression to compute b
b) Running a cross-sectional regression to compute risk premia
c) Test the estimated risk premia for statistical significance
l Detailed Procedures
a) Step 1:
s Find estimates of the betas from time series regressions
r,,. − r/,. = , + 0(r1,2 − r/,.) + ,,. t = 1,2,3, … T, for each tji = 1,2,3,…,N
0: Excel function: = SLOPE(Y,X) à
b) Step 2:
s With the estimated betas from the first stage, running a cross-sectional
regression at each time point t. (,ý ¡WÎ,l,) à$ä
TL λ0,t λM,t r,,. − r/,. = $,2 + 1,20 + v,,., t = 1,2,3, … T, for each tji = 1,2,3,…,N
Y: Excess return of the same item throughout different time
X: Excess return of the market throughout different time
!
c) Step 3: test the λM,t
s Compute the t-statistics for λM,t
s °±TD56, t-stat ®51ûTD4ßA
l Test methodology – t-test <$%&'(>
a) Null hypothesis(DBC): ¯wxi-.i 0
b) t-stat formula
s t-stat = 456789.6:;0; <=>?:@"#$!#%&'% = 45$@"#$!#%&'(
c) ßp9 1: t-stat <ÿ3ÿ>
s ä`A(significant level), tû
ü t-table
ü Excel function: = TINV(Probability, deg_freedom)
ü 1.96Ñ 2 (significant level=5%)
s WT-.WÎ, t-stat `A(significant level), tû
s ßuª
ü t-stat > tû à L4DBC4WÎ-.li 0.
ü t-stat < t ûà lDBC4WÎ-.i 0.
d) ßp9 2: p-value <ÿÿ>
s WT p-value `A(significant level)
s ßuª
ü p-value < significant level (5%) à DBC4WÎ-.li 0.
ü p-value > significant level (5%) à lDBC4WÎ-.i 0.
l Problems/issues with Fama-MacBeth Regression [Ôx]
ü Suffer from the error-in-variables problem because thebetas used in the second
pass regression must be estimated. (are not observed)
ü The tests assume that the time-series residuals (e) are uncorrelated though time
(+[lý`?@)
Coefficients Standard Error t Stat P-value Lower 95% Upper 95%
Intercept -0.0011341 0.006276525 -0.1806863 0.85724367 -0.0136979 0.01142975
Mkt - rf 0.97007904 0.189292106 5.12477282 3.5697E-06 0.59116972 1.34898836
!
3. Summary
1) Relationship with CAPM
àBCý| well diversified portfolio4éÉ market cap weights/market portfolioi_`b
c,ýVmYZ4:É excess returni APTã, factor4¯!if
x¨f§Ï#!" #«¡4# CAPM,8!"#ýIj+[ APT CAPM,B
Cl,jX APTã, market portfolio CAPMã,¢£l,4 APTã4market
portfolio¯É[ý well diversified portfolio,´ CAPMã,`[¢¤¥¢£)<¯
JK,K/;,¦MÅn§Ï#§C4¯Éwxi CAPM[ýY¨, APT4´ APT
VØ[ CAPM,©£/ª«Å
2) Stock v.s. Portfolio
¬f¸æl®¯ return-beta relationship/APT ,KLOKLM4ÝKLM
[°®¯±
uªf²KLM[}M APT,4ãN7LM,KaH®¯/}MVÎm.
3) Issues with APT [Ôx]
l Issues: provides no guidance on what factors are appropriate
l Solutions/ suggestions
a) Chen Roll and Ross: suggested using factors which represent risk based on the
macro-economy (¹º»¼YZ)
b) Fama-French Three Factor Model <-³Week 5>
s Market return: Excess return on a market cap weighted portfolio
s Small – Big: return of a portfolio of small stocks in excess of returns of a
portfolio of big stocks (size premium). Small stocks are fundamentally riskier
than large ones so a premium is required to invest in them
s High – Low: return of stocks with a high book to market ratio minus return on
portfolio of stocks with low book to market ratio (value premium). Low BM
implies undervalued stock or poorly performing.
c) Carhart (1997) Model: Fama-French Three Factor Model´µáÔ
momentum factor/Ã1Y (!±¶!U·¸,YZ)
