06 portfolio theory capm 2008

8/8/2019 06 Portfolio Theory CAPM 2008

1/45

INSTITUTIONAL FINANCELecture 05: Portfolio Choice, CAPM, Black-Litterman


2/45

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


3/45

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


4/45

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


5/45

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)


6/45

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


7/45

2 ASSETS -1 < < 1

s 1 s 2

E[r2 ]

E[r1 ]

Efficient Frontier: Two Imperfectly Correlated Risky Assets


8/45

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 ]


9/45

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]


10/45

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


11/45

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


12/45

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!


13/45

EFFICIENT FRONTIER WITH RISK-FREE ASSET

The Efficient Frontier: One Risk Free and n Risky Assets


14/45


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


15/45


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


16/45

Result 2: Frontier is linear in (E[r], )-space


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


17/45

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)


18/45

Optimal Portfolios of Two Investors with Different Risk Aversion

TWO FUND SEPARATION

Price of Risk == highestSharpe ratio


19/45

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 ]


20/45

OVERVIEW



3. Estimating Mean and CoVariance matrix

4. Black-Litterman ModelTaking a view

Bayesian Updating


21/45

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


22/45

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

][][


23/45

CAPITAL MARKET LINE

s p

M rM

s M

rf

CML


24/45

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 )


25/45

OVERVIEW





Bayesian Updating


26/45

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


27/45

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)


28/45

OVERVIEW





Bayesian Updating


29/45

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.


30/45

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?


31/45

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


32/45

BLACK LITTERMAN PRIOR

All expected returns are in proportion to theirrisk.

Expected returns are distributed aroundb i (E[Rm ] Rf )


33/45

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


34/45

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


35/45

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


36/45

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


37/45

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


38/45

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


39/45

BAYESIAN UPDATING


x x E 2222 11111

)|( s m t s t

Scaling term Total precision


40/45

BAYESIAN UPDATING


x x E 2222 11111

)|( s m t s t

CAPM Priorexpected returns


41/45

BAYESIAN UPDATING IN BLACKLITTERMAN


x x E 2222 11111

)|( s m t s t

Weighted byprecision of CAPM Prior


42/45

BAYESIAN UPDATING


x x E 2222 11111

)|( s m t s t

Vector of expectedreturn views


43/45

BAYESIAN UPDATING


x x E 2222 11111

)|( s m t s t

Weighted byprecision of views


44/45

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


45/45

OVERVIEW





Bayesian Updating

06 portfolio theory capm 2008

Documents