-
Financial Econometrics Short CourseLecture 2 Portfolio Choice
and CAPM
Oliver [email protected]
Renmin University
Oliver Linton [email protected] () Financial Econometrics Short
Course Lecture 2 Portfolio Choice and CAPMRenmin University 1 /
64
-
We have cross section of risky assets with return Ri , i = 1, .
. . ,N, andmarket return Rm , which is a portfolio of risky assets.
We may have a riskfree asset with return Rf .We suppose that
ERi = i
var(Ri ) = ii
cov(Ri ,Rj ) = ij
Portfolio choice. Choose weights wi such that Ni=1 wi = 1 to
findthe mean variance effi cient frontier (the achievable
risk/returntradeoff), either
I Minimize portfolio variance for given portfolio mean, orI
Maximize portfolio mean for given portfolio variance.
CAPM leads to restrictions on the risk/return tradeoff taking
accountof the market portfolio
Oliver Linton [email protected] () Financial Econometrics Short
Course Lecture 2 Portfolio Choice and CAPMRenmin University 2 /
64
-
Let R = (R1, . . . ,RN )be the N 1 vector of returns with mean
vector
and covariance matrix
ER = =
1...N
E[(R ) (R )
]= =
11
. . . jk. . .
NN
.
Oliver Linton [email protected] () Financial Econometrics Short
Course Lecture 2 Portfolio Choice and CAPMRenmin University 3 /
64
-
We consider the problem of mean variance portfolio choice in
this generalsetting. Let Rw denote the random portfolio return
Rw =N
j=1wjRj = w
R,
where w = (w1, . . . ,wN )are weights with Nj=1 wj = 1. The mean
and
variance of the portfolio are
w = w =
N
j=1wjj ;
2w = w
w =
N
j=1
N
k=1
wjwkjk .
There is a trade-off between mean and variance, meaning that as
weincrease the portfolio mean, which is good, we end up increasing
itsvariance, which is bad.
Oliver Linton [email protected] () Financial Econometrics Short
Course Lecture 2 Portfolio Choice and CAPMRenmin University 4 /
64
-
To balance these two effects we choose a portfolio that
minimizes thevariance of the portfolio subject to the mean being a
certain level. Assumethat the matrix is nonsingular, so that 1
exists with1 = 1 = I (otherwise, there exists w such that w = 0).
Wefirst consider the Global Minimum Variance (GMV) portfolio.
DefinitionThe Global Minimum Variance portfolio w is the
solution to the followingminimization problem
minwRN
ww subject to w
i = 1,
where i = (1, . . . , 1).
Oliver Linton [email protected] () Financial Econometrics Short
Course Lecture 2 Portfolio Choice and CAPMRenmin University 5 /
64
-
To solve this problem we form the Lagrangian, which is the
objectivefunction plus the constraint multiplied by the Lagrange
multiplier
L = 12ww + (1 w i).
This has first order condition
Lw
= w i = 0 = w = 1i .
Then premultiplying by the vector i and using the constraint we
have1 = i
w = i
1i , so that = 1/i
1i and the optimal weights are
wGMV =1ii 1i
.
This portfolio has mean and variance
GMV =i1i 1i
; 2GMV =1
i 1i.
Oliver Linton [email protected] () Financial Econometrics Short
Course Lecture 2 Portfolio Choice and CAPMRenmin University 6 /
64
-
The global minimum variance portfolio may sacrifice more mean
returnthan you would like so we consider the more general problem
where we askfor a minimum level m of the mean return.
DefinitionThe portfolio that minimizes variance for a given
level of mean returnsolves
minwRN
ww
subject to the constraints wi = 1 and w
= m.
The Lagrangian is
L = 12ww + (1 w i) + (m w ).
Oliver Linton [email protected] () Financial Econometrics Short
Course Lecture 2 Portfolio Choice and CAPMRenmin University 7 /
64
-
The first order condition isLw
= w i = 0,
which yields
wopt = 1i + 1,
where , R are the two Lagrange multipliers. Then imposing the
tworestrictions: 1 = i
wopt = i
1i + i
1 and
m = wopt =
1i +
1, we obtain a system of two equations
in ,, which can be solved exactly to yield
=C Bm
; =
Am B
A = i1i , B = i
1, C =
1, = AC B2,
provided > 0. This portfolio has mean m and variance
2opt (m) =Am2 2Bm+ C
,
which is a quadratic function of m.Oliver Linton [email protected]
() Financial Econometrics Short Course Lecture 2 Portfolio Choice
and CAPMRenmin University 8 / 64
-
We consider the practical problem of implementing portfolio
choice.Suppose just risky assets vector Rt RN , t = 1, . . . ,T ,
where thepopulation mean and covariance matix is denoted by ,. We
let
=1T
T
t=1Rt , =
1T
T
t=1(Rt ) (Rt )
,
which are consistent estimates of the population quantities as T
forfixed N.Let also A = i
1i , B = i
1, C =
1, = AC B2, and
wopt (m) = 1i + 1,
=C Bm
; =
Am B
,
which is the optimal sample weighting scheme. The
correspondingestimated variance is wopt (m)
wopt (m).
Oliver Linton [email protected] () Financial Econometrics Short
Course Lecture 2 Portfolio Choice and CAPMRenmin University 9 /
64
-
Figure: Effi cient Frontier of Dow StocksOliver Linton
[email protected] () Financial Econometrics Short Course Lecture 2
Portfolio Choice and CAPMRenmin University 10 / 64
-
A necessary condition for to be full rank (and hence invertible)
isthat T > N. In practice, there are many thousands of assets
that areavailable for purchase. There is now an active area of
researchproposing new methods for estimating optimal portfolios
whenthe number of assets is large. Essentially there should be
somestructure on that reduces the number of unknown quantities
fromN(N + 1)/2 to some smaller quantity. The market model and
factormodels are well established ways of doing this and we will
visit themshortly.
Example
The Ledoit and Wolf (2003) shrinkage method replaces by
= + (1 )D,
where D is the diagonal matrix of and R is a tuning parameter.
For (0, 1) the matrix is of full rank and invertible regardless of
therelative sizes of N/T .
Oliver Linton [email protected] () Financial Econometrics Short
Course Lecture 2 Portfolio Choice and CAPMRenmin University 11 /
64
-
The Capital Asset Pricing Model (Risk return tradeoff)
Sharpe-Lintner version with a riskless asset (borrowing or
lending)
E [Ri ] = Rf + i (E [Rm ] Rf ) = Rf + i = Rf + im
for all i .
Relates three quantities
i = E [Ri Rf ] ; m = E [Rm Rf ] ; i =cov(Ri ,Rm)
var(Rm)
all of which can be estimated from time series data
Risk/return tradeoff - more risk, more return. The i is the
relevantmeasure of riskiness of stock i not var(Ri )
Oliver Linton [email protected] () Financial Econometrics Short
Course Lecture 2 Portfolio Choice and CAPMRenmin University 12 /
64
-
Oliver Linton [email protected] () Financial Econometrics Short
Course Lecture 2 Portfolio Choice and CAPMRenmin University 13 /
64
-
Fisher Black version without a riskless asset
Find the (zero beta) portfolio return R0 such that R0 =
argminvar(Rw ) subject to cov(Rw ,Rm) = 0
ThenE [Ri ] = E [R0] + i (E [Rm ] E [R0])
for all i.
Since R0 is not observed, this creates some diffi culties
Oliver Linton [email protected] () Financial Econometrics Short
Course Lecture 2 Portfolio Choice and CAPMRenmin University 14 /
64
-
Testable versions embed within some class of
alternatives.Sharpe-Lintner. Letting Zi = Ri Rf and Zm = Rm Rf
E [Zi ] = i + iE [Zm ]
and testH0 : i = 0 for all i
Black. We have
E [Ri ] = i + iE [Rm ]
and testH0 : i = (1 i )E [R0] for all i .
Here, R0 is the return on the (unobserved) zero beta
portfolio
Oliver Linton [email protected] () Financial Econometrics Short
Course Lecture 2 Portfolio Choice and CAPMRenmin University 15 /
64
-
Market ModelWe have a time series sample on each asset, the
market portfolio, and therisk free rate {Rit ,Rmt ,Rft , i = 1, . .
. ,N, t = 1, . . . ,T}
DefinitionFor Zit = Rit Rft or Zit = Rit and Zmt = Rmt Rft or
Zmt = Rmt :
Zit = i + iZmt + it ,
E [it |Zmt ] = 0 ; var [it |Zmt ] = 2i ; cov(it , is ) = 0.
DefinitionWe may further assume that it are normally
distributed
it N(0, 2i ).
Oliver Linton [email protected] () Financial Econometrics Short
Course Lecture 2 Portfolio Choice and CAPMRenmin University 16 /
64
-
Evidence for Normality
Early proofs of the CAPM often assumed joint normality of
returns,but later it was shown that it can hold under weaker
distributionalassumptions. Neverthelss, much of the literature uses
exact testsbased on assumption of normality.
Measures of non-normality: skewness and excess kurtosis
3 E[(r )3
3
]
4 E[(r )4
4
] 3
For a normal distribution 3, 4 = 0. Daily stock returns
typicallyhave large negative skewness and large positive
kurtosis.
Fama for example argues that monthly returns are closer to
normality
Oliver Linton [email protected] () Financial Econometrics Short
Course Lecture 2 Portfolio Choice and CAPMRenmin University 17 /
64
-
Theorem(Aggregation of (logarithmic) returns). Let A be the
aggregation (e.g.,weekly, monthly) level such that rA = r1 + + rA.
Then under RW1
ErA = AEr var(rA) = Avar(r)
3(rA) =1A
3(r)
4(rA) =1A
4(r).
This says that as you aggregate more (A ), returns become
morenormal, i.e., 3(rA) 0 and 4(rA) 0 as A .
Oliver Linton [email protected] () Financial Econometrics Short
Course Lecture 2 Portfolio Choice and CAPMRenmin University 18 /
64
-
TheoremSuppose that returns follow a stationary martingale
difference sequenceafter a constant mean adjustment and possess
four moments. ThenErA = AEr , var(rA) = Avar(r). Let r t = (rt E
(rt ))/std(rt ). Then
3(rA) =1A
3(r) +3A
A1j=1
(1 j
A
)E (r2t r tj ),
4(rA) =1A
4(r) +4A
A1j=1
(1 j
A
)E (r3t r tj )
+6A
A1j=1
(1 j
A
) [E (r2t r2tj ) 1]
+6A
A1j=1
A1k=1
j 6=k
(1 j
A
)(1 k
A
)E (r2t r tjr tk ).
Oliver Linton [email protected] () Financial Econometrics Short
Course Lecture 2 Portfolio Choice and CAPMRenmin University 19 /
64
-
Oliver Linton [email protected] () Financial Econometrics Short
Course Lecture 2 Portfolio Choice and CAPMRenmin University 20 /
64
-
Oliver Linton [email protected] () Financial Econometrics Short
Course Lecture 2 Portfolio Choice and CAPMRenmin University 21 /
64
-
QQ plots
Oliver Linton [email protected] () Financial Econometrics Short
Course Lecture 2 Portfolio Choice and CAPMRenmin University 22 /
64
-
Oliver Linton [email protected] () Financial Econometrics Short
Course Lecture 2 Portfolio Choice and CAPMRenmin University 23 /
64
-
We next ask how the market model aggregatesSuppose that MM holds
for the highest frequency of data. Then forinteger A we have
t
s=tA
Zis = Ai + it
s=tA
Zmt +t
s=tA
it
for t = A+ 1, . . . ,T , which can be written
ZAit = Ai +
Ai Z
Amt +
Ait .
Suppose that it is independent of Zms for all s, then this is a
validregression model for any A, with
Ai = Ki , Ai = i , var(
Ait ) = Kvar(it ).
If it are iid, then 3(Ait ) = 3(it )/A and 4(Ait ) = 4(
Ait )/A, so
that the aggregated error terms should be closer to normality
thanthe high frequency returns.
Oliver Linton [email protected] () Financial Econometrics Short
Course Lecture 2 Portfolio Choice and CAPMRenmin University 24 /
64
-
Market Model (S&P500-tbill) Daily estimates 1990-2013
se() se() seW () R2
Alcoa Inc. -0.0531 0.0480 1.3598 0.0305 0.0400 0.3929AmEx 0.0170
0.0332 1.4543 0.0211 0.0317 0.6073Boeing -0.0644 0.0492 1.6177
0.0312 0.0667 0.4661Bank of America -0.0066 0.0341 0.9847 0.0217
0.0301 0.4020Caterpillar 0.0433 0.0335 1.0950 0.0213 0.0263
0.4635Cisco Systems 0.0060 0.0264 0.6098 0.0168 0.0272
0.3005Chevron 0.0017 0.0486 1.3360 0.0308 0.0401 0.3793du Pont
-0.0101 0.0358 0.8422 0.0228 0.0314 0.3084Walt Disney -0.0009
0.0281 1.0198 0.0178 0.0241 0.5159General Electric -0.0732 0.0575
1.0650 0.0365 0.0296 0.2169Home Depot -0.0720 0.0534 1.1815 0.0339
0.0418 0.2835HP -0.0083 0.0462 1.0797 0.0294 0.0315 0.3055IBM
-0.0232 0.0483 1.1107 0.0307 0.0349 0.2992Intel 0.0282 0.0280
0.8903 0.0178 0.0242 0.4490Johnson2 -0.0400 0.0539 1.2803 0.0342
0.0360 0.3132
Oliver Linton [email protected] () Financial Econometrics Short
Course Lecture 2 Portfolio Choice and CAPMRenmin University 25 /
64
-
se() se() seW () R2
JP Morgan 0.0041 0.0231 0.5810 0.0147 0.0226 0.3382Coke -0.0380
0.0456 1.5811 0.0290 0.0595 0.4927McD -0.0270 0.0392 0.6012 0.0249
0.0237 0.1598MMM -0.0117 0.0349 0.8093 0.0221 0.0228 0.3032Merck
-0.0783 0.0519 0.7893 0.0329 0.0268 0.1575MSFT -0.0536 0.0521
1.0427 0.0331 0.0408 0.2444Pfizer -0.1121 0.0545 0.7912 0.0346
0.0256 0.1455Proctor & Gamble 0.0221 0.0250 0.5804 0.0159
0.0230 0.3036AT&T -0.0065 0.0303 0.8076 0.0193 0.0264
0.3643Travelers -0.0209 0.0402 0.9750 0.0255 0.0410 0.3224United
Health 0.0173 0.0564 0.8278 0.0358 0.0563 0.1482United Tech -0.0029
0.0452 0.9779 0.0287 0.0298 0.2742Verizon 0.0039 0.0296 0.7606
0.0188 0.0238 0.3472Wall Mart 0.0141 0.0293 0.7555 0.0186 0.0249
0.3495Exxon Mobil 0.0053 0.0349 0.8290 0.0222 0.0292 0.3128
Oliver Linton [email protected] () Financial Econometrics Short
Course Lecture 2 Portfolio Choice and CAPMRenmin University 26 /
64
-
Market Model (S&P500-tbill) Monthly estimates with standard
errors
se() se() seW () R2
Alcoa Inc. -0.0528 0.0499 1.4107 0.1759 0.2185 0.3179AmEx 0.0193
0.0254 1.5594 0.0895 0.1045 0.6875Boeing -0.0675 0.0499 1.6575
0.1759 0.2298 0.3915Bank of America 0.0018 0.0311 1.2767 0.1097
0.1232 0.4955Caterpillar 0.0454 0.0306 1.2212 0.1078 0.1300
0.4819Cisco Systems 0.0134 0.0243 0.8418 0.0856 0.1114
0.4122Chevron 0.0034 0.0472 1.4240 0.1663 0.1505 0.3468du Pont
-0.0191 0.0318 0.5596 0.1120 0.1082 0.1532Walt Disney -0.0048
0.0258 0.9200 0.0908 0.0867 0.4264General Electric -0.0630 0.0568
1.4430 0.2002 0.2678 0.2734Home Depot -0.0715 0.0532 1.1787 0.1877
0.0992 0.2222HP 0.0048 0.0435 1.4292 0.1535 0.1651 0.3858IBM
-0.0270 0.0451 1.1867 0.1592 0.1274 0.2872Intel 0.0299 0.0247
0.9255 0.0872 0.0935 0.4492Johnson2 -0.0449 0.0550 1.1815 0.1940
0.1281 0.2118
Oliver Linton [email protected] () Financial Econometrics Short
Course Lecture 2 Portfolio Choice and CAPMRenmin University 27 /
64
-
se() se() seW () R2
JP Morgan 0.0046 0.0201 0.5912 0.0708 0.0738 0.3358Coke -0.0435
0.0428 1.4111 0.1508 0.1590 0.3880McD -0.0262 0.0364 0.6342 0.1284
0.1218 0.1501MMM -0.0129 0.0317 0.7906 0.1116 0.0814 0.2665Merck
-0.0716 0.0506 0.9235 0.1784 0.2365 0.1627MSFT -0.0529 0.0571
1.1163 0.2012 0.1862 0.1824Pfizer -0.1135 0.0586 0.7811 0.2066
0.1804 0.0938Proctor & Gamble 0.0255 0.0223 0.6626 0.0786
0.0838 0.3401AT&T -0.0111 0.0305 0.6668 0.1075 0.1069
0.2182Travelers -0.0255 0.0361 0.8321 0.1272 0.0977 0.2367United
Health 0.0186 0.0566 0.8556 0.1997 0.1751 0.1174United Tech -0.0047
0.0455 0.9470 0.1606 0.1620 0.2012Verizon 0.0004 0.0277 0.6265
0.0976 0.1200 0.2301Wall Mart 0.0142 0.0255 0.6802 0.0899 0.1026
0.2933Exxon Mobil 0.0025 0.0339 0.7124 0.1196 0.1020 0.2044
Oliver Linton [email protected] () Financial Econometrics Short
Course Lecture 2 Portfolio Choice and CAPMRenmin University 28 /
64
-
Portfolio weights
Tangency portfolio has weights that are proportional to
wTP 1 (ER Rf i)
where i is the N vector of ones.
Under the CAPM, these weights should be the weights of the
marketportfolio and hence should always be positive.
Global Minimum Variance portfolio has weights that are
proportionalto
wMV 1i
Empirically, find many negative weights in both cases (short
selling).
Oliver Linton [email protected] () Financial Econometrics Short
Course Lecture 2 Portfolio Choice and CAPMRenmin University 29 /
64
-
Annualized returns, std, and portfolio weights
wMV wTPAlcoa Inc. -0.0665 0.2151 -0.0665 -0.0346AmEx 0.0009
0.1851 -0.2475 -0.2482Boeing -0.0852 0.2350 -0.0006 0.0097Bank of
America -0.0093 0.1540 0.0369 0.0377Caterpillar 0.0375 0.1595
0.0715 0.0103Cisco Systems 0.0140 0.1103 -0.0584 -0.0732Chevron
-0.0110 0.2151 -0.1038 -0.1016du Pont -0.0088 0.1504 0.1374
0.1503Walt Disney -0.0046 0.1408 0.0820 0.0799General Electric
-0.0782 0.2267 -0.0187 -0.0223Home Depot -0.0803 0.2200 0.0092
0.0220HP -0.0137 0.1937 -0.0978 -0.0907IBM -0.0296 0.2014 -0.0147
0.0065Intel 0.0282 0.1318 0.1769 0.1463Johnson2 -0.0512 0.2268
0.0706 0.0679
Oliver Linton [email protected] () Financial Econometrics Short
Course Lecture 2 Portfolio Choice and CAPMRenmin University 30 /
64
-
wMV wTPJP Morgan 0.0129 0.0991 0.1868 0.1834Coke -0.0577 0.2234
0.0353 0.0331McD -0.0188 0.1491 0.1096 0.1087MMM -0.0094 0.1458
0.0773 0.0903Merck -0.0754 0.1972 -0.0087 0.0127MSFT -0.0579 0.2092
-0.0332 -0.0227Pfizer -0.1093 0.2057 -0.0176 0.0062Proctor &
Gamble 0.0309 0.1045 0.3443 0.2999AT&T -0.0042 0.1327 0.0237
0.0364Travelers -0.0234 0.1703 -0.0043 0.0092United Health 0.0190
0.2132 0.0342 0.0326United Tech -0.0054 0.1852 -0.0092
-0.0197Verizon 0.0075 0.1280 0.0299 0.0071Wall Mart 0.0179 0.1268
0.1877 0.1858Exxon Mobil 0.0070 0.1470 0.0680 0.0770
Oliver Linton [email protected] () Financial Econometrics Short
Course Lecture 2 Portfolio Choice and CAPMRenmin University 31 /
64
-
Chinese Data (Cambridge undergrad Rose Ng did this work in her
thesistesting risk/return in Shanghai/HK). Two markets for same
stocks.
She provides tests of pricing relationships.
Oliver Linton [email protected] () Financial Econometrics Short
Course Lecture 2 Portfolio Choice and CAPMRenmin University 32 /
64
-
Maximum Likelihood Estimation and TestingSuppose that
Zt = + Zmt + t ,
where t N(0,). Do not restrict to be diagonal but requireN
-
The maximum likelihood estimates are the
equation-by-equationtime-series OLS estimates
i =Tt=1(Zmt m)(Zit i )
Tt=1(Zmt m)2=
1
2m
1T
T
t=1(Zmt m)(Zit i )
The maximum likelihood estimates of are
i = i i m
The maximum likelihood estimate of is
=1T
=
(1T
T
t=1
it jt
)i ,j
it = Zit i iZmt
Oliver Linton [email protected] () Financial Econometrics Short
Course Lecture 2 Portfolio Choice and CAPMRenmin University 34 /
64
-
Maximum Likelihood Estimation and Testing
Under the normality assumption we have, conditional on excess
marketreturns Zm1, . . . ,ZmT , the exact distributions:
v N(
,1T
(1+
2m2m
)
)
v N(
,1T1
2m
)Without normality (but under iid) we have for large T
T ( ) = N
(0,(1+
2m2m
)
)
Oliver Linton [email protected] () Financial Econometrics Short
Course Lecture 2 Portfolio Choice and CAPMRenmin University 35 /
64
-
Wald Test Statistic
Wald test statistic for null hypothesis that = 0
W = [var()1] = T
(1+
2m2m
)11
Under null hypothesis for large T
W = 2(N)
provided N < T .The asymptotic approximation is valid
regardless of whether theerrors are normally distributed or not.For
the Dow stocks:Daily W =22.943222 (p-value =0.81758677);Monthly W
=33.615606 (p-value =0.29645969)
Oliver Linton [email protected] () Financial Econometrics Short
Course Lecture 2 Portfolio Choice and CAPMRenmin University 36 /
64
-
Exact Finite-Sample Variant of the Wald Test Statistic
Under normality, we can get an exact test statistic by using an
Fdistribution and a degrees of freedom correction
F =(T N 1)
N((1+
2m2m)1) 1 F (N,T N 1)
This is superior to the Wald test (under the assumption of
normality)
Oliver Linton [email protected] () Financial Econometrics Short
Course Lecture 2 Portfolio Choice and CAPMRenmin University 37 /
64
-
Likelihood Ratio Test
The likelihood ratio test is a natural alternative to a Wald
test:
LR = 2(log `c log `u) = T [log det log det ] = 2(N)
=1T
=
(1T
T
t=1
it jt
)i ,j
where it are the constrained residuals ie the no intercept
residuals
These tests have an exact relationship which allows us to derive
the exactdistribution for the likelihood ratio test under
normality.
Oliver Linton [email protected] () Financial Econometrics Short
Course Lecture 2 Portfolio Choice and CAPMRenmin University 38 /
64
-
CLM Table 5.3. Works with 1965-1994. Monthly returns on ten
valueweighted portfolios based on size. CRSP value weighted index,
1 monthtbill rate.Results for full period show rejection at 5% but
not at 1% level.Five year subperiods: some rejections at 5% some
not.Aggregate assuming independence, ie make use of
2(j) + 2(i) = 2(j + i)
very strong rejections.Likewise for ten year subperiods.The
aggregated results allow for different parameter values across
thesubperiods but hide whether the evidence is getting stronger
agains theCAPM or not
Oliver Linton [email protected] () Financial Econometrics Short
Course Lecture 2 Portfolio Choice and CAPMRenmin University 39 /
64
-
Testing Black Version of the CAPM
Tests for the Black version are more complicated to derive.
Estimate the same unconstrained model as before using total
returnsinstead of excess returns. The constrained model is
Rt = (i )+ Rmt + t
for some scalar unknown , where i is the N vector of ones. There
areN 1 "nonlinear cross-equation" restrictions
11 1
= = N1 N
=
The model to be estimated is nonlinear in the parameters.
Estimatingthe constrained model requires numerical maximization of
thenonlinear (in parameters) system of equations.
Oliver Linton [email protected] () Financial Econometrics Short
Course Lecture 2 Portfolio Choice and CAPMRenmin University 40 /
64
-
Useful trick (profiling or concentration): assume that the
expected returnon the zero-beta portfolio is known exactly (use a
noisy estimate as proxy)
Rt i = (Rmt ) + tso that conditionally on the model is linear in
.With the zero-beta return known, the Black model can be estimated
usingthe same methodology as the Sharpe-Lintner modelThen, relax
the assumption that the zero-beta return is known.
Oliver Linton [email protected] () Financial Econometrics Short
Course Lecture 2 Portfolio Choice and CAPMRenmin University 41 /
64
-
For = (, 1, . . . , N )the (constrained) likelihood function
is
`(,) = c T2log det
12
T
t=1(Rt i (Rmt ))
1 (Rt i (Rmt ))
maximize with respect to . Profile/concentration method.
Define
i () =
Tt=1(Rmt )(Rit )Tt=1(Rmt )2
() =1T
()() =
(1T
T
t=1
it ()jt ()
)i ,j
Oliver Linton [email protected] () Financial Econometrics Short
Course Lecture 2 Portfolio Choice and CAPMRenmin University 42 /
64
-
Then search the profiled likelihood over the scalar
parameter
`P () = c T2log det ()
12
T
t=1(Rt i
()(Rmt ))
()
1
(Rt i ()(Rmt ))
and let be the maximizing value and then let i () and (
) bethe corresponding estimates of i and .
Oliver Linton [email protected] () Financial Econometrics Short
Course Lecture 2 Portfolio Choice and CAPMRenmin University 43 /
64
-
Testing Black Version of the CAPM
The LR statistic compares the relative fit of the constrained
andunconstrained models. As T
LR = T [log det log det ] = 2(N 1)
where is the MLE of in the constrained model. Note onlyN 1
degrees of freedomExact theory much more tricksy, require
simulation methods.
Oliver Linton [email protected] () Financial Econometrics Short
Course Lecture 2 Portfolio Choice and CAPMRenmin University 44 /
64
-
Robustness of MLE and tests to Heteroskedasticity(RW2.5)
Maximum likelihood estimation apparently assumes
multivariatenormal returns. CAPM can hold under weaker
distributionalassumptions (e.g., elliptical symmetry, which
includes multivariatet-distributions with heavy tails).
Actually, the Gaussian MLE of , is robust to
heteroskedasticity,serial correlation, and non-normality since the
estimates are just leastsquares.
The asymptotic test statistics are robust to normality of the
errors,but they are not robust to heteroskedasticity or serial
correlation, andin that case we need to adjust the standard
errors
Oliver Linton [email protected] () Financial Econometrics Short
Course Lecture 2 Portfolio Choice and CAPMRenmin University 45 /
64
-
Suppose that
E (t |Zmt ) = 0E (t
t |Zmt ) = t ,
where t is a potentially random time varying covariance
matrix.This is quite a general assumption, but as we shall see
below it is quitenatural to allow for dynamic heteroskedasticity
for stock return data.
In this case, it is not possible to perform an exact test and
the testswe already defined are unfortunately not properly sized in
this case.
However, we can construct robust Wald tests based on large
sampleapproximations.
Oliver Linton [email protected] () Financial Econometrics Short
Course Lecture 2 Portfolio Choice and CAPMRenmin University 46 /
64
-
Let
T =1T
T
t=1
t t ; T =
1T
T
t=1(Zmt m)
2 t t
V = T +2m4m
T
JH = T V1
Then, under the null hypothesis as T
JH = 2(N).
Oliver Linton [email protected] () Financial Econometrics Short
Course Lecture 2 Portfolio Choice and CAPMRenmin University 47 /
64
-
Cross-Sectional Regression Tests
The CAPM says thati = im ,
where i = E (Ri Rf ) and m = E (Rm Rf ) .Fama and MacBeth (1973)
say embed this in a richer cross-sectionalrelationship
i = 0 + i1.
We should find 0 = 0 and 1 > 0 with 1 = m = E (Rm Rf ) .In
fact we should find 0,2,3 = 0 and 1 > 0 in
i = 0 + i1 + 2i 2 +
2ei3
Oliver Linton [email protected] () Financial Econometrics Short
Course Lecture 2 Portfolio Choice and CAPMRenmin University 48 /
64
-
Problem: We dont observe i (or 2ei ). Solution
First estimate i for each stock or portfolio using time series
data.Then estimate the cross-sectional regression by OLS (or
GLS)
Ri Rf = 0 + i1 + ui
Under the CAPM, 0 = 0 and 1 = E (Rm Rf ) (which can beseparately
estimated by the sample average Rm Rf ). Test 0 = 0by
t-testEmpirically find that there is a positive and linear
relationshipbetween beta risk and return with a high R2, but 0 >
0 and 1significantly lower than market excess return.In fact, they
include additional variables, e.g.,
Ri Rf = 0 + i1 + 2i 2 +
2i3 + ui
where 2i is an estimate of the idiosyncratic error variance.
Test alsoj = 0, j = 2, 3 using t-tests or Wald statistic. FM do not
reject
Oliver Linton [email protected] () Financial Econometrics Short
Course Lecture 2 Portfolio Choice and CAPMRenmin University 49 /
64
-
Standard errorsFM estimate the cross-sectional regressions (for
t = 1, . . . ,T )
Rit Rft = 0t + i1t + uit
Then average the estimates over time (gives the same as the
singleregression of the average returns on the i )
=
[01
]=1T
T
t=1
[0t1t
]They estimate the asymptotic variance matrix of by
V =1T
T
t=1
([0t1t
][
01
])([0t1t
][
01
])Standard errors are widely used, but wrong unless combined
with theportfolio grouping of a large original set of assets.Errors
in variables/generated regressor issue. Shanken (1992) suggests
ananalytical correction necessary for individual stocks.
Oliver Linton [email protected] () Financial Econometrics Short
Course Lecture 2 Portfolio Choice and CAPMRenmin University 50 /
64
-
Actual methodology even more complicated. There are four
stages:
1 Time series estimation of pre-ranking individual stock beta in
period A2 Portfolio formation based on estimated double sorted
individual stockbeta and size
3 Estimate portfolio betas from the time series in period B4
Cross-sectional regressions for each time period in B from which
timeseries of estimated risk premia are obtained t . Test
hypothesis onaverage of time series of risk premia using standard
errors from above
Oliver Linton [email protected] () Financial Econometrics Short
Course Lecture 2 Portfolio Choice and CAPMRenmin University 51 /
64
-
Testing on portfolios (composed of a large number of
individualstocks) rather than individual stocks can mitigate
theerrors-in-variable problem as estimation errors cancel out each
other.
Sorting by beta reduces the shrinkage in beta dispersion
andstatistical power; sorting by size takes into account
correlationbetween size and beta;
performing pre-ranking and estimation in different periods
avoidsselection bias.
Oliver Linton [email protected] () Financial Econometrics Short
Course Lecture 2 Portfolio Choice and CAPMRenmin University 52 /
64
-
Figure: Risk Return Relation
Oliver Linton [email protected] () Financial Econometrics Short
Course Lecture 2 Portfolio Choice and CAPMRenmin University 53 /
64
-
Empirical Evidence in the Literature
Many tests and many rejections of the CAPM!!
Size Effect. Market capitalizationI Firms with a low market
capitalization seem to earn positive abnormalreturns ( > 0),
while large firms earn negative abnormal returns( < 0)
Value effect. Dividend to price ratio (D/P) and book to market
ratio(B/M).
I Value firms (low value metrics relative to market value) have
> 0while growth stocks (high value metrics relative to market
value) have < 0
Momentum effect.I Winner portfolios outperform loser portfolios
over medium term.
Oliver Linton [email protected] () Financial Econometrics Short
Course Lecture 2 Portfolio Choice and CAPMRenmin University 54 /
64
-
Active vs. Passive Portfolio Management
Active portfolio management: attempts to achieve superior
returns through security selection and market timing
I Security selection = picking misspriced individual securities,
trying tobuy low and sell high or short-sell high and buy back
low
I Market timing = trying to enter the market at troughs and
leave atpeaks
Conflicts of interest between owners and managers
Passive portfolio management: tracking a predefined index
ofsecurities with no security analysis whatsoever, just choose
(smartbeta). Index funds, ETFs. No attempt to beat the market, in
linewith EMH, which says this is not possible. Much cheaper than
activemanagement since no costs of information acquisition and
analysis,lower transaction costs (less frequent trading), also
generally greaterrisk diversification (the only free lunch
around).
Oliver Linton [email protected] () Financial Econometrics Short
Course Lecture 2 Portfolio Choice and CAPMRenmin University 55 /
64
-
If an active manager overseeing a 5 billion portfolio could
increasethe annual return by 0.1%, her services would be worth up
to 5million. Should you invest with her?
Role of luckI Imagine 10,000 managers whose strategy is to park
all assets in anindex fund but at the end of every year use a
quarter of it to make(independently) a single bet on red or black
in a casino. After 10 years,many of them no longer keep their jobs
but several survivors have beenvery successful ((1/2)10
1/1000).
I The infinite monkey theorem says that if one had an infinite
numberof monkeys randomly tapping on a keyboard, with probability
one, oneof them will produce the complete works of Shakespeare.
Oliver Linton [email protected] () Financial Econometrics Short
Course Lecture 2 Portfolio Choice and CAPMRenmin University 56 /
64
-
Time Varying Parameters
The starting point of the market model and CAPM testing was
thatwe have a sample of observations independent and
identicallydistributed from a fixed population. This setting was
convivial for thedevelopment of statistical inference. However,
much of the practicalimplementations acknowledge time variation by
working with short,say 5 year or 10 year windows.
A number of authors have pointed out the variation of
estimatedbetas over time. We show estimated betas for IBM (against
theSP500) computed from daily stock returns using a five year
windowover the period 1962-2017. We present the rolling window
estimates.
Oliver Linton [email protected] () Financial Econometrics Short
Course Lecture 2 Portfolio Choice and CAPMRenmin University 57 /
64
-
Figure: Time Varying Betas
Oliver Linton [email protected] () Financial Econometrics Short
Course Lecture 2 Portfolio Choice and CAPMRenmin University 58 /
64
-
Figure: Time Varying Alphas
Oliver Linton [email protected] () Financial Econometrics Short
Course Lecture 2 Portfolio Choice and CAPMRenmin University 59 /
64
-
We next consider a more general framework where time variation
isexplicitly considered.
Suppose thatZit = it + itZmt + it ,
where it , it vary over time, but otherwise the regression
conditionsare satisfied, i.e., E (it |Zmt ) = 0.We may allow 2it =
var(it |Zmt ) to vary over time and asset.
Oliver Linton [email protected] () Financial Econometrics Short
Course Lecture 2 Portfolio Choice and CAPMRenmin University 60 /
64
-
One framework that may work here is that
it = i (t/T ), it = i (t/T ),
where i and i are smooth, i.e., differentiable functions.
We can reconcile this with the CAPM by testing whether i (u) =
0for u [0, 1]. This can be done using the rolling window
framework.Note that it is not necessary to provide a complete
specification for2it as one can construct heteroskedasticity robust
inference.
Oliver Linton [email protected] () Financial Econometrics Short
Course Lecture 2 Portfolio Choice and CAPMRenmin University 61 /
64
-
An alternative framework is the unobserved components model
it = i ,t1 + it , it = i ,t1 + it ,
where it , it are i.i.d. shocks with mean zero and variances
2,
2 ,
Harvey (1990). One may take i0 = 0 and initialize i0 in some
otherway.
In this framework, the issue is to test whether 2 = 0
(whichcorresponds to constant and zero ) versus the general
alternative.
In both of these models, the regression parameters evolve in
somedeterministic or autoregressive way. CCAPM and other
intertemporalmodels drive the evolution of risk premia through
macro and otherstate variables
Oliver Linton [email protected] () Financial Econometrics Short
Course Lecture 2 Portfolio Choice and CAPMRenmin University 62 /
64
-
Criticisms of Mean-Variance analysis and the CAPM
Theoretically, it is claimed that expected utility is invalid,
some of theother assumptions which underline these models are
invalid, and thatthe Mean-Variance criterion may lead to
paradoxical choices. Allais(1953) shows that using EUT in making
choices between pairs ofalternatives, particularly when small
probabilities are involved, maylead to some paradoxes. Roy
(1952).
Even if EUT is intact some fundamental papers question the
validityof the risk aversion assumption: Friedman and Savage
(1948),Markowitz (1952) and Kahneman and Tversky (1979) claim that
thetypical preference must include risk-averse as well as
risk-seekingsegments. Thus, the variance is not a good measure of
risk, whichcasts doubt on the validity of the CAPM.
Oliver Linton [email protected] () Financial Econometrics Short
Course Lecture 2 Portfolio Choice and CAPMRenmin University 63 /
64
-
Roll critique. Cant observe the market portfolio. So rejections
ofCAPM are not valid
Normality (or an Elliptic distribution) is crucial to the
derivation ofthe CAPM. The Normal distribution is statistically
strongly rejectedin the data.
Furthermore, the CAPM has only negligible explanatory power.
Ex ante versus ex post betas. Conditional capm. Time varying
riskpremia. For example recession indicators. Will discuss
later.
Oliver Linton [email protected] () Financial Econometrics Short
Course Lecture 2 Portfolio Choice and CAPMRenmin University 64 /
64