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Unformatted text preview: Outlines of Lecture 5 Objectives
1. Learn motivations of using multi-factor models
2. Apply multi-factor models in managing risk,
timing factor bets, and creating neutral strategies References
Chapter 16, EGBG; Chapter 10, BKM; Handouts
(Ludwig Chincarini & Daehwan Kim, Quantitative
Equity Portfolio Management, McGraw-Hill ;
Hsiu-lang Chen Portfolio Analysis 1 Factor Models Single Factor Models
Multi Factor Models
Factor Surprises and Risk Premia
Applications of the Factor Model
― Risk Control
― Factor Timing
― Tracking Portfolios
(Long-Short or Neutral Strategies)
A factor is any variable that may predict stock returns.
Hsiu-lang Chen Portfolio Analysis 2 Introduction
Recall the SCL from the CAPM theory: ri,t-rf,t = αi + βi (rm,t-rf,t) + εi,t If the market were the only systematic driver (a.k.a.
factor) you’d expect that εi,t and εj,t would be
Empirically the hypothesis that cov(εi,t,εj,t)=0 if i≠j is
most often rejected for portfolios and especially
This leads us to consider Multi Factor Models.
– Factor models describe asset returns as
dependent on several sources of risk
(factors) apart from “market” risk.
Hsiu-lang Chen Portfolio Analysis 3 Single Factor Models
The single factor model states a return generating process:
ri,t=ai + biF1,t+ εi,t.
where F1 is the single pervasive factor driving all returns.
bi is called the factor loading/sensitivity of asset i.
εi,t is a purely firm specific shock, e.g. cov(εi,t,εj,t)=0.
<Implicit assumption> Two securities covary exclusively
through common reactions to the underlying factor.
The single factor model is agnostic with respect to the
underlying behavior of investors. Instead it relies on the
high-level assumption that F1 is the unique pervasive source
of risk in the economy.
Hsiu-lang Chen Portfolio Analysis 4 What are potential important
COMMON factors? Hsiu-lang Chen Portfolio Analysis 5 Introduction
Our focus will mainly be on portfolios (or asset
classes) rather than individual stocks.
Returns on large portfolios are random variables
conditional mainly upon:
These observations are the main motivation
behind factor models.
The main analysis tool is multivariate regression
Hsiu-lang Chen Portfolio Analysis 6 Introduction
The purpose of factor models in portfolio
allocation is to
Forecast of volatility of portfolio returns,
absolute as well as relative to a benchmark
Characterize investment styles of managers
Attribute portfolio risk and return to the
exposure to a set of underlying systematic
Hsiu-lang Chen Portfolio Analysis 7 Common Risk Factors
In practice you will want to consider more than one
source of pervasive risk (= un-diversifiable risk).
You want to consider types of risk which you think
are being rewarded by the market. 1. Observable Economic/Market Factors:
(Thus, factor risk premium can be measured!)
-the “market” (i.e. your favorite stock index)
-Long/Short Term Inflation
-Default Risk Premium
-Michigan Consumer Sentiment Index (growth
Hsiu-lang Chen Portfolio Analysis 8 Other Candidates for Macro Economic Risk Factors
[Chapter 17, BKM] Hsiu-lang Chen Portfolio Analysis 9 Regular Releases of Indicators [Chapter 17, BKM] Hsiu-lang Chen Portfolio Analysis 10 Reliable Engine of Recovery Loses Steam
Manufacturing Sectors Across the World Slow Down in September; Germany and
Taiwan Shed Pace, U.S. Output Ticks Up ; WSJ; 10/04/2011; A14
Hsiu-lang Chen Portfolio Analysis 11 Common Risk Factors 2. Un-observable Fundamental Factors
(Factor risk premium needs to be estimated!)
An approach of creating a zero-investment portfolio
-HML (distress factor): Ret(Value)-Ret(Growth)
-Momentum (?): Ret(Up)-Ret(Down)
An approach of using observable firm characteristics
-The factor risk premium is estimated from panel
regression of the stock returns on the observed
characteristics. 3. Statistical Factors
Principal Component Analysis
Portfolio Analysis 12 Construction of Fama-French Factors
Year t-1 Year t
Q1 Q2 Q3 Q4
Rank all firms using NYSE breakpoints SIZE: ME June t; BE/ME: BE t-1, ME Dec t-1
Hsiu-lang Chen Portfolio Analysis 13 Multi-Factor Models
The n-factor model:
ri,t = ai + bi,1F1,t + … + bi,nFn,t + εi,t
tries to explain the return on stock i by its sensitivity to
n underlying sources of pervasive risk.
We often assume that the factors are uncorrelated
(this leads to a simpler formula but is otherwise not
The expected return and covariances can be found
analogously to the single factor model using the
assumption that the εi,t are firm specific:
E(ri)= ai + bi,1E(F1) + … + bi,n E(Fn)
σi2=bi,12σf12+…+bi,n2 σfn2 + σεi2
σi,j = bi,1bj,1σf12+…+ bi,nbj,nσfn2
Hsiu-lang Chen Portfolio Analysis 14 Multi-Factor Models K R = β + ∑ β f +e
j = 1 ij jt
⇒ E ( R) N ×1 = B F
N × K K ×1 = BΩ
B′ + D
This is a general expression in which factors may not be independent!
Hsiu-lang Chen Portfolio Analysis 15 ~ E ( R1 ) β10 β11 • • • ~ E ( R ) = β i0 + β i1
i • • •
~ E ( R ) β β N N 0 N1
Hsiu-lang Chen β12
βN2 Portfolio Analysis •
• β1K E ( F1 ) E F • ( 2)
β iK • • • β NK E ( FK ) 16 VN × N = β11 β12 •
• β i1 β i 2 • •
β N1 β N 2 σ 2 ε1
+ 0 0
0 0 •
0 σ ε2
0 0 Hsiu-lang Chen • β1K σ F1F1 σ F1F2 • • σ F2 F1 σ F2 F2 •
• β iK • •
• • •
• β NK σ FK F1 σ FK F2 0 0 0 0 0 0 • 0 0 σ ε2 N •
• • σ F1FK β11
• σ F2 FK β12 •
• • •
• σ FK FK β1K Portfolio Analysis • β i1
• β iK • β N1 • βN2 • • • • • β NK 17 Validity of Multi-Factor Models Use historical data for UAL and GE
• Period: 1990/1 – 1999/12 (10 years)
• Monthly returns Step 1: Compute residual returns using zero, one, and
three factor models Step 2: Regress GE residual return on UAL residual return What R2 in Step 2 would you expect to find? What does the finding of this R2>0 indicate?
Can you replicate this
Hsiu-lang Chen Portfolio Analysis 18 Multi-Factor Models
(Results from the residual regression) Hsiu-lang Chen Portfolio Analysis 19 Multi-Factor Models
Clearly the multifactor model has greater potential for
explaining returns than a single factor model
– Why? What happens to R2 if I add an additional factor? Fit diagnostic
– How do you detect a missing factor? Why not throw everything in?
– R2 caveat
– Number of factor covariances How do we know which model is better empirically?
Hsiu-lang Chen Portfolio Analysis 20 Out-of-Sample Tests To test if a proposed parameter model
works in the future.
By construction, the parameters are
estimated so the model can fit the data
well over the past. |
Parameters estimated here CHECK forecasting ability HERE ˆ
Y = Xβ +ε Yt +1− X t +1β
Hsiu-lang Chen Portfolio Analysis 21 Multi-Factor Models
This discussion raises two crucial questions:
1.How many factors should be used?
2.Which ones? Each firm has its own preferred set of factors
and put a lot of effort into identifying factors and
factor sensitivities. Much like CAPM betas, factor sensitivities can
be time varying and need to be re-estimated
Hsiu-lang Chen Portfolio Analysis 22 Controlling Portfolio Risk Using the Factor Model
Each source of systematic risk has its own
volatility and its own reward .
A well diversified portfolio's long-term return
and its volatility are largely determined by its
By measuring and controlling a portfolio's
relative systematic risk exposures, one can
produce the highest possible return for a
given level of risk.
Hsiu-lang Chen Portfolio Analysis 23 Controlling Portfolio Risk Using the Factor Model Suppose you are the manager of funds of domestic
equity mutual funds.
– You have no stock picking ability
– You do have factor forecasting/timing ability Consider investing in six portfolios, formed by sorting
all stocks according to their market capitalization and
book-to-market ratio. The data is in “fundret.xls.” Hsiu-lang Chen Portfolio Analysis 24 Controlling Portfolio Risk Using the Factor Model
You believe the following four-factor model holds:
ri,t – rf = ai+bi,MFM,t+bi,TSFTS,t+bi,YSFYS,t+bI,OIFOI,t+εi,t
– FM is the excess return on the stock market index
– FTS is the change in the slope of the term structure
– FYS is the change in the yield spread between Baa and
– FOI is Oil inflation (percentage change in Oil price).
Note: Y is an excess-return format when Xs are risk factors!
Hsiu-lang Chen Portfolio Analysis 25 Controlling Portfolio Risk Using the Factor Model
Your analysts have come up with the following sample
estimates on a monthly basis. E(r) is in % while Cov is in 10-4 Sample Period:
Jan 1979 ~ Dec 2010
Hsiu-lang Chen Portfolio Analysis 26 Controlling Portfolio Risk Using the Factor Model
Suppose you form a portfolio
rP = w1 r1 + w2 r2 + … + w6 r6
Then the portfolio factor loadings will be the
weighted average of the factor loadings of the
bP,M = w1 b1,M + w2 b2,M + w3 b3,M + w4 b4,M + w5 b5,M + w6 b6,M
bP,TS =w1 b1,TS +w2 b2,TS +w3 b3,TS +w4 b4,TS +w5 b5,TS +w6 b6,TS
bP,YS =w1 b1,YS +w2 b2,YS +w3 b3,YS +w4 b4,YS +w5 b5,YS +w6 b6,YS
bP,OI = w1 b1,OI +w2 b2,OI +w3 b3,OI + w4 b4,OI + w5 b5,OI + w6 b6,OI Number of equations? Number of variables?
Hsiu-lang Chen Portfolio Analysis 27 Controlling Portfolio Risk Using the Factor Model
Suppose that you want to target at least a 1% expected
return per month, but
1. You would not take oil-price risk.
2. You would like your portfolio to move one-to-one with the
market. This means we would like to set 1.bP,OI=0 or Cov(rP,t,FOI,t) /σOI2 =0 ?
2.bP,M=1 or Cov(rP,t,FM,t)/σM2 =1 ? Why? This can be solved using Markowitz optimization in
Hsiu-lang Chen Portfolio Analysis 28 Controlling Portfolio Risk Using the Factor Model
Portfolio 4 Systematic Risk Factors 1
3 FYS 4 FTS 5
Hsiu-lang Chen X
Portfolio Analysis FOil
29 Controlling Portfolio Risk Using the Factor Model
The covariances above can be expressed in terms of
factor loadings and covariances between factors,
Cov(rP,FOI) = bP,MCov(FM,FOI)+bP,TSCov(FTS, FOI)
Cov(rP,FM) = bP,MCov(FM,FM)+bP,TSCov(FTS, FM)
Your analysts estimate the following factor covariances:
Unit: 10-4 Hsiu-lang Chen Portfolio Analysis 30 Controlling Portfolio Risk Using the Factor Model Is this construction of inputs
consistent with the model? Sample Period: Jan 79 ~ Jun 10
Sample Period: Jan 1979 ~ Dec 2010
Hsiu-lang Chen Portfolio Analysis 31 Controlling Portfolio Risk Using the Factor Model This is a consistent way! Sample Period: Jan 1979 ~ Dec 2010
Hsiu-lang Chen Portfolio Analysis 32 A Common Mistake ri,t – rf = ai+bi,MFM,t+bi,TSFTS,t+bi,YSFYS,t+bI,OIFOI,t+εi,t
The risk of security i is determined by the factor
loadings on FM, FTS, FYS, FOI simultaneously. ri,t – rf =αi+βI,YSFYS,t+εi,t
The risk of security i is determined by βYS, the
factor loading on FYS only. Note: bI,YS ≠ βI,YS
Hsiu-lang Chen Portfolio Analysis 33 You believe the CAPM and you are a passive
manager. Since your benchmark is Russell 1000
Index, you prefer your risky portfolio perfectly vary
with Russell 1000 Index. In other words, you prefer
both of your portfolio and Russell 1000 Index have
the same risk sensitivity to the market factor. After
running regressions, your assistant has provided
you the following information:
RRussell 1000 = -2% + 0.9 RMarket + eRussell 1000 R2 = 0.8
RStock Fund = -2% + 1.2 RMarket + eStock Fund R2=0.64
RBond Fund = 3% + 0.2 RMarket + eBond Fund
σMarket = 20%
How do you construct such a portfolio based on the
stock fund and the bond fund you can select? Hsiu-lang Chen Portfolio Analysis 34 Factor Surprises and Risk Premia
Consider rewriting the n-factor model as ri,t = E(ri) + bi,1 f1,t + … + bi,n fn,t + εi,t (*).
where fi,t = Fi,t – E[Fi,t] is the realized ith factor
surprise at time t. How did we arrive at (*) from the n-factor model (**) below? ri,t = αi + bi,1 F1,t + … + bi,n Fn,t + εi,t (**). Thus the realized return on an asset equals the ex-ante
expected return plus factor loadings time factor surprises plus
idiosyncratic noise. If the n-factor model holds, then (*) holds for all stocks and
cov(εi,t, εj,t)=0 for j≠i.
Hsiu-lang Chen Portfolio Analysis 35 Factor Surprises and Risk Premia
We require that, for each of the factors, E(fk)=0, (why?).
For example: fk is the deviation of economic growth from
what was expected rather than economic growth itself.
bi,k denotes the loading of the i’th asset on the k'th factor.
The εi,t is non-systematic, idiosyncratic,
or residual risk, which is the security
movement that is not associated with
any of the systematic factors.
For example, εi,t will be negative when a
firm's CEO dies (assuming he was any
good), or a firm loses a big contract.
WSJ, 09/05/2007, A1
Hsiu-lang Chen Portfolio Analysis 36 Test on Arbitrage Pricing Theory (APT) Step 1: Run the time-series regression on individual stocks ri,t = E(ri) + bi,1 f1,t + … + bi,n fn,t + εi,t Step 2: Run the cross sectional regression: E(ri) = λ0 + bi,1 λ1 + … + bi,n λn + ui (**). λj is called the market price of factor j risk since it tells us
how much extra expected return the average investor (i.e. the
market) requires to take on an extra unit of factor j risk.
In a one factor model where the factor is the realized excess
return on the market, (**) is just the CAPM SML regression
If the n-factor model is correctly specified and only
systematic risk (e.g. factor risk) is priced, then λ0=rf and ui=0.
Hsiu-lang Chen Portfolio Analysis 37 Factor Surprises and Risk Premia
Example 1: Factor models → Expected returns. Suppose that two factors have been identified for the U.S.
– the growth rate of industrial production, IP.
– the inflation rate, IR.
– IP is expected to be 4%, and IR 6%. A stock with a factor loading of 1.0 on IP and 0.4 on IR
currently is expected to provide a rate of return of 14%. If industrial production actually grows by 5%, while the
inflation rate turns out to be 7%, what is your revised estimate
of the return on the stock?
Hsiu-lang Chen Portfolio Analysis 38 Factor Surprises and Risk Premia
Example 1: Factor models → Expected returns. We know E(IP) = 4% and bIP = 1, E(IR) = 6%, bIR = 0.4,
and E(ri) = 14%
The two factor surprises are therefore:
fIP=(0.05-0.04)=0.01 and fIR=(0.07-0.06)=0.01
Plug these into the return generating process gives the
expected return conditional on these realization of the
industrial production growth rate (IP) and the inflation
E(ri|fIP=0.01,fIR=0.01) = 0.14 + 1x0.01 + 0.4x0.01 = 0.154
Hsiu-lang Chen Portfolio Analysis 39 Example 2 Hsiu-lang Chen Portfolio Analysis 40 Factor Surprises and Risk Premia
Example 3: Factor Timing Your analyst gives you the following information
on three securities that are correctly priced
according to a 2 factor model
rA = 0.06 + 1 f1 + 1 f2 + eA
rB = 0.04 + 1 f1 + 2 f2 + eB
rC = 0.10 + 3 f1 + 2 f2 + eC
– Here, the E[r]’s (constants) are what the market
expects – not what we expect!
Hsiu-lang Chen Portfolio Analysis 41 Factor Surprises and Risk Premia Example 3: Factor Timing
Factor 1 is a foreign income factor
Factor 2 is a U.S. earnings price ratio factor. Assumptions:
– The way the model is constructed these factors are uncorrelated.
– You believe very strongly that Japan will finally come out of its
recession in the next few months and therefore exports of U.S.
produced goods will rise more than the market expects.
– Moreover, you believe the earnings price ratio factor will not change
at all in this time period, consistent with what analysts expect. Using the above three securities, you wish to construct a portfolio
that takes advantage of all of these facts.
(i) the composition of the portfolio?
(ii) the b's of the portfolio?
(iii) the expected return on the portfolio?
Hsiu-lang Chen Portfolio Analysis 42 Factor Surprises and Risk Premia
Example 3: Factor Timing
We want to construct a portfolio with a lot of factor 1 exposure and no
factor 2 exposure.
– Let’s assume we want a loading of 10 on factor 1 and 0 on factor 2.
– Therefore we solve the three equations.
1 wA + 1 wB + 3 wC = 10. (Is 10 reasonable?)
in excel this is solved as:
1 wA + 2 wB + 2 wC = 0.
1 wA + 1 wB + 1 wC = 1. so wA=2, wB=-5.5, wC=4.5 and the portfolio
rp = 2 rA – 5.5 rB + 4.5 rC.
will have the desired factor loading: bp,1=10, bp,2=0.
Hsiu-lang Chen Portfolio Analysis 43 Factor Surprises and Risk Premia
Example 3: Factor Timing Assuming that you believe that the foreign income factor
will rise by 2%, the expected return on this portfolio is:
2 · 0.06 – 5.5 · 0.04 + 4.5 · 0.1 + 10 · 0.02 = 55%
Is this the highest Sharpe-Ratio portfolio possible? No reason why this portfolio should be the optimal one.
Need to know the idiosyncratic variances, then we can
solve Markowitz with
E(rA)=0.06+1·0.02, E(rB)=0.04+1·0.02, E(rC)=0.10+3·0.02
Hsiu-lang Chen Portfolio Analysis 44 Tracking Benchmarks; Hedging Liability
Most institutional investors, e.g. pension funds, insurance
companies etc. have to worry about their liabilities when
Mutual funds have to worry about their benchmarks too.
– Losing money when the benchmark is down is OK.
– Losing money when markets are up can be fatal for a
This suggests that the relevant return measure is the
tracking error vis á vis the benchmark, not the actual
Similarly, the relevant risk measure is the standard
deviation of the tracking error.
Hsiu-lang Chen Portfolio Analysis 45 Discussion on Tracking Error
Consider a BENCHMARK called B. You can invest in
two risky assets stock1 and stock2.
The aim is to construct a portfolio of stock1 and
stock2 that mimics the benchmark B as closely as
possible subject to achieving a target level of
Define the TRACKING ERROR = Assets - Liabilities
Your expected TE: E(rTE)=wE(r1)+(1-w)E(r2)-E(rB)
The variance of the TE: σ2TE=w2σ12+(1-w)2σ22+σ2B
+2w(1-w) cov(r1,r2)-2w cov(r1,rB)-2(1-w) cov(r2,rB)
Hsiu-lang Chen Portfolio Analysis 46 Discussion on Tracking Error
Setting up the Tracking Error Problem using Markowitz
If you look closely at the equations above, it is really as if
– there are three assets
– the weight on the last asset (B) is fixed at -1
– the weights on the first two assets (stock1&2) add up to 1
– We want to minimize σTE for each given level of E(r)
Let’s do an example:
– Consider the inputs – And lets draw the best possible E(r) vs. σTE trade-offs
This is just like an efficient frontier!
Hsiu-lang Chen Portfolio Analysis 47 Discussion on Tracking Error
Note the change in the constraints, reflecting
– Fixed weight of -1 on the benchmark
– The sum of all weights equal 0 instead of 1 [Excel] Formula? – As in the standard setup, we minimize the std.dev. (here
the TE std.dev) for each level of expected return (here the
Hsiu-lang Chen Portfolio Analysis 48 Apply Tracking Error Discussion to
the Creation of a Long-Short Strategy
B Minimize σ2 (rTE) ≡ σ2 (rP – rB )
rP,t = αP + bP,1 F1,t + … + bP,n Fn,t + εP,t
rB,t = αB + bB,1 F1,t + … + bB,n Fn,t + εB,t Exposure
Strategy The Aggregated Portfolio (AP)
B Minimize σ2 (rAP ) ≡ σ2 (rP – rB )
Hsiu-lang Chen Portfolio Analysis 49 Tracking Benchmarks Hsiu-lang Chen Portfolio Analysis 50 Tracking Benchmarks
- Hedging systematic component of tracking error risk - Suppose you want use the factor model to create a
portfolio of A, B and C which tracks the Market with no
systematic tracking error risk What should the factor loadings of your tracking portfolio
be? You want your portfolio to react to factor surprises in the
same way as the market.
– This implies that your portfolio should have the same
factor loadings as the market: Hsiu-lang Chen Portfolio Analysis 51 Tracking Benchmarks
- Minimize total tracking error risk - Suppose you create an optimal tracking portfolio earning
1% above the expected return on the market, i.e. 9%
– What is the optimal portfolio?
– What is the factor sensitivity of your tracking error? This is a standard tracking error problem
– We now care about the entire tracking error, not
just the systematic part
– We wish to know how the tracking error will
respond to changes in the factors
– This will enable us to predict in which scenarios the
tracking error will be exacerbated
– We may want to eliminate one type of factor risk
from the tracking error if we have a view.
Hsiu-lang Chen Portfolio Analysis 52 Tracking Benchmarks
- Minimize total tracking error risk - We solve using Markowitz. Q. How does the surprise of systematic risk factors affect
your portfolio performance relative to the benchmark? Formula? Hsiu-lang Chen Portfolio Analysis 53 Tracking Benchmarks
S&P500 Index published in 1957
captures 75% coverage of U.S.
equities. It has over US$ 4.83 trillion
Number of Constituents
benchmarked, with index assets
comprising approximately US$ 1.1 Adjusted Market Cap ($ Billion)
trillion of this total.
Constituent Mkt. Cap(Adjusted $ Billion)
- Average 20.47 - Largest 354.11 - Smallest .70 - Median 9.45 % Weight Largest Constituent 3.46% Top 10 Holdings(% Market Cap Share) 20.52% As of 8/31/2011, Vanguard 500 Index Fund (VFINX) has total
asset $97.9b (504 stocks) with expense ratio of 0.17%, while
Vantagepoint 500 Index I Fund (VPFIX) has total asset $367m
(8/31/2011) with expense ratio of 0.43%. How to effectively mimic
S&P 500 Index? Discussions on Tracking Portfolios When the factors are correlated, will this
affect the construction of tracking
portfolios in slides 50~53? The example in slides 50~53 has three
underlying assets and two factors, we
cannot neutralize the factor exposures and
achieve higher returns at the same time.
Can you accomplish both if there are more
underlying assets? Conclusion
Factor models introduce multiple sources of systematic risk.
Each source of risk has a “market price”.
The market price tends to be negative on counter cyclical
factors and positive on pro cyclical factors.
Factor models can be used to decompose volatility as well as
Factor models can be used to take bets on future factor
realizations and to manage risk.
It is easy to incorporate factor models in creating tracking
portfolios into the Markowitz framework.
Hsiu-lang Chen Portfolio Analysis 56 Further References in Factor Models Chen, Nai-Fu, Richard Roll, and Stephen Ross,
1986, Economic Forces and the Stock Market,
Journal of Business 59, 383-403.
Fama, Eugene F., and K. R. French, 1996,
Multifactor explanations of asset pricing
anomalies, Journal of Finance 51, 55−84.
Chan, Louis K.C., Jason Karceski, and Josef
Lakonishok, 1999, On portfolio optimization:
Forecasting covariances and choosing the risk
model, Review of Financial Studies 12, 937-974.
Patton, Andrew J., 2009, Are “market neutral”
hedge funds really market netural? Review of
Financial Studies 22, 2295-2330. Hsiu-lang Chen Portfolio Analysis 57 ...
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This note was uploaded on 10/07/2011 for the course FIN 512 taught by Professor Hengchen during the Fall '11 term at Ill. Chicago.
- Fall '11