06 portfolio theory capm 2008
TRANSCRIPT
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INSTITUTIONAL FINANCELecture 05: Portfolio Choice, CAPM, Black-Litterman
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OVERVIEW
1. Portfolio Theory in a Mean-Variance world
2. Capital Asset Pricing Model (CAPM)
3. Estimating Mean and CoVariance matrix4. Black-Litterman Model
Taking a view
Bayesian Updating
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EXPECTED RETURNS & VARIANCE
Expected returns (linear)
Variance
recall that correlationcoefficient 2 [-1,1]
2 p := V a r [r p ] = w0V w = ( w 1 w 2 ) 21 1 2 2 1 22
w 1w 2
= ( w 1
2
1+ w 2 2 1 w 1 1 2 + w 2
2
2)
w 1
w 2 = w 21 21 + w 22 22 + 2 w 1 w 2 1 2 0s i n c e 1 2 1 2 :
p := E [r p ] = w j j , where each w j = hj
P j h j
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For certain weights: w 1 and (1-w 1 )mp = w 1 E[r1 ]+ (1-w 1 ) E[r 2 ]s 2 p = w 1 2 s 1 2 + (1-w 1 )2 s 2 2 + 2 w 1 (1-w 1 )s 1 s 2 r 1,2(Specify s 2p and one gets weights and mps)
Special cases [w1 to obtain certain s R]
r 1,2 = 1 ) w1 = (+/- s p s 2 ) / ( s 1 s 2 )r 1,2 = -1 ) w1 = (+/- s p + s 2 ) / ( s 1 + s 2 )
ILLUSTRATION OF 2 ASSET CASE
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2 ASSETS = 1
The Efficient Frontier: Two Perfectly Correlated Risky Assets
Hence,
s 1 s p s 2
mp
Lower part with is irrelevant
E[r2 ]
E[r1 ]
p = E [r 1 ] +E [r 2 ] E [r 1 ]
2 1( R 1 )
p = jw 1 1 + (1 w 1 ) 2 j
p = w 1 1 + (1 w 1 ) 2w 1 =
p 2 1 2
p = 1 + 2 1
2 1 ( p 1)
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2 ASSETS = -1
Efficient Frontier: Two Perfectly Negative Correlated Risky Assets
For r 1,2 = -1:
Hence,
s 1 s 2
E[r2 ]
E[r1 ]
p = jw 1 1 (1 w 1 ) 2 j
p = w 1 1 + (1 w 1 ) 2
2
1 + 2 1 +
1
1 + 2 2
slope: 2 1
1 + 2 p
slope: 2 1
1 + 2 p
w 1 = p + 2
1 + 2
p = 21 + 2 1 +1
1 + 2 2 1
1 + 2 p
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2 ASSETS -1 < < 1
s 1 s 2
E[r2 ]
E[r1 ]
Efficient Frontier: Two Imperfectly Correlated Risky Assets
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2 ASSETS 1 = 0
The Efficient Frontier: One Risky and One Risk Free Asset
s 1 s p s 2
mp
E[r2 ]
E[r1 ]
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EFFICIENT FRONTIER WITH N RISKY ASSETS
A frontier portfolio is one which displays minimum varianceamong all feasible portfolios with the same expected portfolioreturn.
11w
Eew .t.s
T
T
1w
E)r~
(EwN
1ii
N
1i ii
Vww21min T
w
s
E[r]
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The first FOC can be written as:
V w p = e + 1 orw p = V 1 e + V 1 1eT w p = ( eT V 1 e) + ( eT V 1 1)
@ L
@w= V w e 1 = 0
@ L@ = E w
T e = 0@ L@ = 1 w
T 1 = 0
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Noting that e T wp = w Tpe, using the first foc, the second foccan be written as
pre-multiplying first foc with 1 (instead of e T) yields
Solving both equations for and
= CE AD and =B AE
Dwhere D = BC A2 .
1 T w p = w T p 1 = (1 T V 1 e) + (1 T V 1 1) = 1
1 = (1 T V 1 e) | {z }
=: A
+ (1 T V 1 1) | {z }
=: C
E [~r p] = eT w p = ( eT V 1 e)
| {z } := B+ ( eT V 1 1)
| {z } =: A
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E1VAeVCD1
eVA1VBD
1 1111
111
VDAEB
eVDACE
w p
scalar) scalar)
(vector) (vector)
E h g w p(vector) (vector) (scalar)
(6.15)
If E = 0, w p = g If E = 1, w p = g + h
Hence, w p = V-1e + V-11 becomes
Hence, g and g+h are portfolioson the frontier.
linear in expected return E!
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EFFICIENT FRONTIER WITH RISK-FREE ASSET
The Efficient Frontier: One Risk Free and n Risky Assets
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EFFICIENT FRONTIER WITH RISK-FREE ASSET
where is a number
FOC: w p = V 1 ( e r f 1)Multiplying by (e r f 1) T and solving for yields
min w 12 wT V w
s.t. w T e + (1 w T 1) r f = E [r p]
=E [r p] r f
( e r f 1) T V 1 ( e r f 1)
w p = V 1(e r f 1 )
| {z } n 1E [r p ] r f
H 2
H = qB 2Ar f + Cr 2f
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EFFICIENT FRONTIER WITH RISK-FREE ASSET
E [r q ] r f =Cov [r q ; r p]
V ar [r p] | {z }
:= q;p
( E [r p] r f )
Holds for any frontier portfolio p , in particular the market portfolio!
Cov [r q ; r p] = wT q V w p
= wT q (e r f 1 )
| {z } E [r q ] r f E [r p] r f
H 2
=(E [r q ] r f )([E [r p] r f )
H 2
V ar [r p ; r p] =
(E [r p] r f )2
H 2
Result 1: Excess return in frontier excess return
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Result 2: Frontier is linear in (E[r], )-space
EFFICIENT FRONTIER WITH RISK-FREE ASSET
V ar [r p ; r p ] =(E [r p] r f )2
H 2
E [r p] = r f + H p
H =
E [r p ] r f
pwhere H is the Sharpe ratio
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TWO FUND SEPARATION
Doing it in two steps:First solve frontier for n risky asset
Then solve tangency point
Advantage:Same portfolio of n risky asset for different agentswith different risk aversion
Useful for applying equilibrium argument (later)
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Optimal Portfolios of Two Investors with Different Risk Aversion
TWO FUND SEPARATION
Price of Risk == highestSharpe ratio
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MEAN-VARIANCE PREFERENCES
U(mp , s p) with
Example:Also in expected utility framework
quadratic utility function (with portfolio return R)U(R) = a + b R + c R 2
vNM: E[U(R)] = a + b E[R] + c E[R 2 ]= a + b mp + c mp2 + c s p2
= g( mp, s p)
asset returns normally distributed ) R= j w j r j normalif U(.) is CARA ) certainty equivalent = mp - r A /2 s 2 p(Use moment generating function)
@ U @ p
> 0, @ U @ 2p < 0
E [W ] 2 V ar [W ]
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OVERVIEW
1. Portfolio Theory in a Mean-Variance world
2. Capital Asset Pricing Model (CAPM)
3. Estimating Mean and CoVariance matrix
4. Black-Litterman ModelTaking a view
Bayesian Updating
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2. EQUILIBRIUM LEADS TO CAPM
Portfolio theory: only analysis of demandprice/returns are taken as givencomposition of risky portfolio is same for all investors
Equilibrium Demand = Supply (market portfolio)
CAPM allows to deriveequilibrium prices/ returns.risk-premium
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THE CAPM WITH A RISK-FREE BOND
The market portfolio is efficient since it is on the efficient frontier.
All individual optimal portfolios are located on the half-line originating at point (0,r f ).
The slope of Capital Market Line (CML): . M
f M R R E
s
][
p M
f M f p
R R E R R E s
s
][][
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CAPITAL MARKET LINE
s p
M rM
s M
rf
CML
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SECURITY MARKET LINE
bb ibM 1
rf
E(r M)
E(r i)
E(r)
slope SML = (E(r i)-rf) / b i
SML
E [r j ] = j = r f +Cov [r j ; r M ]
V ar [r M ]
| {z } j(E [r M ] r f )
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OVERVIEW
1. Portfolio Theory in a Mean-Variance world
2. Capital Asset Pricing Model (CAPM)
3. Estimating Mean and CoVariance matrix
4. Black-Litterman ModelTaking a view
Bayesian Updating
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3. ESTIMATING MEAN AND CO-VARIANCE
Consider returns as stochastic process (e.g. GBM)Mean return (drift)
For any partition of [0,T] with N points (t=T/N),N*E[r] = Ni=1 rit= p T -p 0 (in log prices)Knowing first p 0 and last price p T is sufficientEstimation is very imprecise!
VarianceVar[r]=1/N Ni=1 (rit-E[r])2 2 as NTheory: Intermediate points help to estimate co-varianceReal world:
time-varying
Market microstructure noise
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3. 1000 ASSETS
Invert a 1000x1000 matrix
Estimate 1000 expected returns
Estimate 1000 variances
Estimate 1000*1001/2 1000 co-variances
Reduce to fewer factors
so far we used past data
(and assumed future will behave the same)
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OVERVIEW
1. Portfolio Theory in a Mean-Variance world
2. Capital Asset Pricing Model (CAPM)
3. Estimating Mean and CoVariance matrix
4. Black-Litterman ModelTaking a view
Bayesian Updating
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4. BLACK-LITTERMAN MODEL
So far we estimated expected returns using historical data.
We ignored statistical priors:
A sector with an unusually high (or low) past returnwas assumed to earn (on average) the same high (orlow) return going forward.
We should have attributed some of this past return to
luck, and only some to the sector being unusualrelative to the population.
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EXPECTED RETURNS
We also ignored economic priors:A sector with a negative past return should not beexpected to have negative expected returns going forward.
A sector that is highly correlated with another sectorshould probably have similar expected returns.
A good deal in the past (i.e. good realized returnrelative to risk) should not persist if everyone isapplying mean-variance optimization.
What is a good starting point from which toupdate based on our analysis?
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BAYESIAN UPDATING
Bayes Rule allows one to update distribution afterobserving some signal/data
from prior to posterior distribution
Recall if all variables are normally distributed with can use
the projection theoremE.g. prior: = N ( , 2 ); signal/view x = + , where = N(0, 2 )
Weights depend on relative precision/confidence of prior vs.signal/view (on portfolio)
. x x E
22
2
22
2
)|(s t
t m
s t s
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BLACK LITTERMAN PRIOR
All expected returns are in proportion to theirrisk.
Expected returns are distributed aroundb i (E[Rm ] Rf )
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PROPERTIES OF A CAPM PRIOR
All expected returns are in proportion to theirrisk.
Expected returns are distributed aroundb i (E[Rm ] Rf )
Is this a good starting point?We can still use optimizationWe dont throw out data (e.g. still can estimatecovariance structure accurately)It is internally consistent if we dont have an
edge, the prior will lead us to holding the market
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BLACK-LITTERMAN
The Black-Litterman model simply takes thestarting point that there are no good deals
And then adjusts returns according to anyviews that the investor has from:
Seeing abnormal returns in the past thatexpected to persist (or reverse)
Fundamental analysisAlphas of active trading strategies
views concern portfolios and not necessarily
individual assets
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BLACK LITTERMAN PRIORS MORE SPECIFIC
Suppose returns of N-assets (in vector/matrix notation)
Equilibrium risk premium,
where risk aversion, weq market portfolio weights
Bayesian prior (with imprecision)
r N (; )
= weq
= +
"0 , where
"0
N (0
; )
See He and Litterman
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VIEWS
View on a single asset affects many weights
Portfolios viewsviews on K portfolios
P: K x N-matrix with portfolio weights
Q: K-vector of expected returns on these portfolios
Investors views
is a off-diagonal values are all zero
and are all orthogonal
P = Q + "v , where "v N (0; - )
"v "0
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BAYESIAN POSTERIOR - REWRITTEN
x
x
x x E
2222
22
2
22
2
22
2
22
2
1111
1
111
111
)|(
s m t s t
s t s
m s t
t
s t t
m s t
s
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BAYESIAN UPDATING
Black Litterman updates returns to reflect views using Bayes Rule.
The updating formula is just the multi-variate (matrix)version of
QPPPQ R E T T 11111]|[ t t
x x E 2222 11111
)|( s m t s t
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BAYESIAN UPDATING
QPPPQ R E T T 11111]|[ t t
x x E 2222 11111
)|( s m t s t
Scaling term Total precision
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BAYESIAN UPDATING
QPPPQ R E T T 11111]|[ t t
x x E 2222 11111
)|( s m t s t
CAPM Priorexpected returns
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BAYESIAN UPDATING IN BLACKLITTERMAN
QPPPQ R E T T 11111]|[ t t
x x E 2222 11111
)|( s m t s t
Weighted byprecision of CAPM Prior
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BAYESIAN UPDATING
QPPPQ R E T T 11111]|[ t t
x x E 2222 11111
)|( s m t s t
Vector of expectedreturn views
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BAYESIAN UPDATING
QPPPQ R E T T 11111]|[ t t
x x E 2222 11111
)|( s m t s t
Weighted byprecision of views
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ADVANTAGES OF BLACK-LITTERMAN
Returns are only adjusted partially towards the investorsviews using Bayesian updating
Recognizes that views may be due to estimation error
Only highly precise/confident views are weighted heavily
Returns are modified in a way that is consistent witheconomic priors
highly correlated sectors have returns modified in the same way
Returns can be modified to reflect absolute or relative views
The resulting weights are reasonable and do not load up onestimation error
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OVERVIEW
1. Portfolio Theory in a Mean-Variance world
2. Capital Asset Pricing Model (CAPM)
3. Estimating Mean and CoVariance matrix
4. Black-Litterman ModelTaking a view
Bayesian Updating