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Asset Allocation with CER

Carlo Favero

2013-14

Favero () Asset Allocation with CER 2013-14 1 / 26

The Constant Expected Returns Model

The CER model assumes that an asset’s return over time isnormally distributed with a constant mean and constant variance.The model allows for the returns on different assets to becontemporaneously correlated with correlations constant overtime.Returns are independent over time both across assets and withinthe same asset.

ri,t = µi + σiεit

εit ∼ NID (0, 1)

cov(εit, εjs

)=

{σij t = s0 t 6= s


The Constant Expected Returns Model

The CAPM provides a good measure of risk and thus a goodexplanation for why some stocks earn higher average returns thanothers according to the simple model

µ−rf e = β� [(µM−rf )e],

Excess returns are close to unpredictable; any predictability is astatistical artifact or cannot be exploited after transaction costs areimputed to actual trades based on such alleged predictability, i.e.,whatever is our information set It,E[rt+1−rf e|It] = E[rt+1−rf e] = µ−rf e, there is nothing to be learntfrom It for practical purposes;Volatility and covariances are approximately constant over time,i.e., Σt ≡ Var[rt+1−rf e|It] = Var[rt+1−rf e] = Σ.asset prices behave as a (log) random walk with drift


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Regression Model Representation

y+ = X+δ+ + u+, (1)

where y+ is a (NT× 1) vector, X+ is a NT×N∑

i=1Ki matrix (Ki is the

number of regressors available at each point in time), δ+ is aN∑

i=1Ki × 1

vector of unknown parameters, and u+ is a (NT× 1) vector ofresiduals:

y+ =

y1......

yN

, X+ =

X1 0 0 0

0 X2 · · · ...

0...

. . ....

0 · · · · · · XN

,

δ+ =

δ1......

δN

, u+ =

u1......

uN

.Favero () Asset Allocation with CER 2013-14 5 / 26


Each equation can be represented as follows:

yi = Xiδi + ui,

i = 1, 2, ..., N. (??) is then easy to use to derive inferences on means,variance and covariances:

E[y+] = X+δ+

Var[y+] = Var[u+]



If ui is assumed to have standard white noise properties, then thefollowing properties hold for u+:

E(u+)= 0NT

E[u+(u+)′] =

E (u1u′1) E (u1u′2) · · · E (u1u′N)

E (u2u′1) E (u2u′2) · · ·...

... · · · . . ....

E (uNu′1) · · · · · · E (uNu′N)

=

σ11IT σ12IT · · · σ1NIT

σ21IT σ22IT · · · ...... · · · . . .

...σN1IT · · · · · · σNNIT

= ˚⊗ IT.

where each block of the covariance matrix E[u+(u+)′] is T× T byconstruction. Here ⊗ denotes a standard Kronecker product. ˚ isnon-singular covariance matrix.



The simplest case of the CER also assumes that all residuals are bothcontemporaneously and serially uncorrelated, with diagonalcovariance matrix ∑d ≡ diag{σ11, σ22, ...., σNN}. Then, because thediagonal structure of ˚d, classical OLS equation by equation can beapplied. Consider for example the observations on the ith return:

yi = eTδi + ui,

where

yi =

yi1yi2...

yiT

, Xi = eT =

11...1

.

The OLS estimates of the relevant parameters are then simply

δi =1T

T

∑t=1

rit = ri σ11 = σ21 =

1T

T

∑t=1(rit − ri)

2 ,

which are sample mean and sample variance.Favero () Asset Allocation with CER 2013-14 8 / 26

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A Static Asset Allocation Problem with ConstantExpected Returns

Let’s denote with r the random vector of linear total returns from timet to time T from a given menu of N risky assets for interval [t, T],r ∼ D (µ, Σ)Given a degree of risk aversion λ, a standard mean-variance descriptionof this allocation problem is the following:

maxw

(1−w′e

)rf +w′µ− 1

2λ(w′Σw)

where E[r] = (1−w′e) rf +w′µ =rf +w′(µ−rf e) and Var[r] = w′Σw.



first-order conditions (FOCs) are necessary and sufficient and definethe following system of N linear equations in N unknowns, theportfolio weights w ∈RN:

(µ−rf e)−λΣw = 0.

Solving the FOCs yields:

w =1λ

Σ−1(

µ−rf e)

,

Consider now the special case in which w′e = 1, that is no investmentin the riskfree bond is allowed. The optimal portfolio in this case is thefamous tangency portfolio:

e′w =1λ

e′Σ−1(

µ−rf e)= 1 =⇒ λ = e′Σ−1

(µ−rf e

)wT =

Σ−1 (µ−rf e)

e′Σ−1(µ−rf e

) ,Favero () Asset Allocation with CER 2013-14 11 / 26


(1) The weights in the tangency portfolio do not depend on the riskaversion parameter λ.(2) Given that the optimal risky portfolio is uniquely determined, thetangency portfolio must then coincide with the market portfolio.Agents maximize their utility by taking a linear combination of themarket portfolio and and the risk-free securities. Note that in this casewe can express the return on any portfolio in the following way:

rp = (1− β) rf + βrM

rp − rf = β(

rp − rM)

and the CAPM holds.



(3) Efficient portfolios are those with the highest expected return for agiven level of risk.If we summarize the expected return-risk(mean-variance) properties of the feasible portfolios in a plot withportfolio expected return, µP, on the vertical axis and portfoliostandard-deviation, σP, on the horizontal axis, then all efficientportfolios can be represented as points in the space

(σP, µP) and the

efficient frontier is the line that connects all these points. given theproperties of the tangency portfolios, weights for all portfolios on theefficient frontier are obtained by inputing different values for therisk-free rate in the expression for optimal weights(4) the CER implies that the tangency portfolios and the efficientfrontier do not depend on the horizon at which returns are defined.


What Happens in Practice ?

First, estimates/forecasts of µ and Σ are not constant over time:think of an investor who as avalaible a sample from time t0 to timetT to decide optimal asset allcation over period the T+ 1...T+ k.The estimates of µ and Σ based on the sample t0 ...tT are usuallyvery different from those based on the sample T+ 1...T+ k andoptimal asset allocation ex-ante does not coincide with theoptimal asset allocation ex-post.Moreover, the CER approach to portfolio allocation can lead todramatic swings in optimal portfolio weights for small changes ininvestment views and conditions, as given by theestimates/forecasts of µ and Σ. There is a simple reason for thesecommon findings: too much sampling error in the estimation ofthe vector of expected returns and, due to this, an asset allocationwhich is idiosyncratic to the specific estimation sample.


Solutions

The are several solutions to the above problem

explicitly allow for sample uncertaintystabilize weights by combining a portfolio based on views with amarket portfolioAbandon the CER andwork with more complicated predictivemodels


The resampled optimal mean-variance portfolio

Implement bootstrap methods to derive the optimal portfolio allocation.Consider the estimation of a simple multivariate model, in which theonly regressor is a constant for the returns ri

t on N assets, i = 1, 2, ..., N:

r1t = µ1 + u1t

r2t = µ2 + u2t

...rNt = µN + uNt[

u1t u2t ... uNt]′ ∼ N (0, Σ

).


Implement the following algorithm.1) Collect of the residuals from estimation in the following T×Nmatrix:

U ≡

u11 u21 ... uN1u12 u22 ... uN2

... .... . .

...u1T u2T ... uNT

.

At this point, draw a new sample of size T of residuals by extractingrandomly T rows from U.2) Given these new, re-sampled residuals collected in a vector u1

t(t = 1, 2, ..., T) and the estimates µ, we proceed to generate a newartificial sample of returns using

r1t = µ+u1

t ,

where the subscript “1” alludes to the fact that this represents the firstiteration of the algorithm. At this point, a new OLS estimation of themodel is performed on this artificial data, obtaining as an outcome apair of new, bootstrapped estimates, µ1 and Σ1 and, using the classicalformula, w1.


3) Iterate the algorithm B times, where B is in general a large number(let’s say 5,000 or 10,000 times), using the fact that at the bth iterationone simply draws a new sample of size T of residuals by extractingrandomly T rows from U, generate a new artificial sample of returnsusing

rbt = µ+ub

t ,

perform OLS estimation of µb and ^b to obtain wb, for b = 1, 2, ..., B.This total number B of replications of this procedure will generate Boptimal portfolio allocations {wb}B

b=1.The desidered vector of re-sampled, optimized portfolio weights maybe represented by the average, across the B bootstraps, of the weightsin {wb}B

b=1:

wboot =1B

B

∑b=1

wb.


Black and Litterman’s approach

Given the knowledge of the market portfolio weights wmkt and someestimates of the variance-covariance matrix of returns, we can use theoptimal portfolio allocation condition to derive the expected returnsconsistent with the market capitalization:

wmkt =1λ

Σ−1(

µ−rf e)=⇒ µmkt = λΣwmkt + rf e.

Assume now that the portfolio manager holds some (normallydistributed, for simplicity) views on a subset of size Q ≤ N of the Nexpected returns included in the market portfolio:

Pµr ∼ NQ (v, Γ) ,

where NQ denotes a q-variate multivariate normal distribution, µr is avector of N expected returns, and P is an appropriate Q×N selectionmatrix that selects the subset of returns on which there are subjectiveviews expressed by the investor.



The views are expressed as a vector of mean expected returns V and adiagonal variance-covariance matrix Γ, expressing the confidence onthe views. Such subjective views have to be balanced against thedistribution of returns implied by the market portfolio:

µr ∼ N (µmkt, τΣ) ,

where τ is a scalar smaller than one (and conventionally set to 1/3 byBlack and Littermann and most of the subsequent literature) to filterout of the estimated covariance matrix of returns the impact of theirrandom variation (i.e., to take into account the effect of noise in smallsamples).Black and Littermann’s approach aims then at generating a value forthe expected return vector µBL by optimally combining the distributionof returns implied in the market capitalization and the subjectiveviews of the portfolio manager.



This is obtained by solving the following optimization problem:

µBL = arg minµ

(µ− µmkt)′ (τΣ)−1 (µ− µmkt) + (Pµ− v)′ Γ−1 (Pµ− v) .

This is a weighted least squares problems, where the weights dependon covariance matrix.When the diagonal elements of Γ all approach zero, that is, when thereis infinite confidence in the subjective views by the investor, theproblem becomes a constrained least squares problem where therelevant constraint is PµBL = v. On the other hand, when Γ hasdiagonal elements diverging to infinity (no confidence in the views),the solution to the problem is simply µBL = µmkt.



The first order conditions for the solution of the problem can bewritten as follows:

2 (τΣ)−1 (µBL − µmkt) + 2P′Γ−1 (PµBL − v) = 0

from which we can derive:

µBL =((τΣ)−1 + P′Γ−1P

)−1 ((τΣ)−1 µmkt + P′Γ−1v

)

=

((τΣ)−1 + P′Γ−1P)−1

(τΣ)−1︸︷︷︸=Ψ

µmkt +

((τΣ)−1 + P′Γ−1P)−1

P′Γ−1︸︷︷︸=IN−Ψ

v

This expression emphasizes that µBL is obtained by optimallycombining market views (µmkt) with the investor’s views (v), through arather complex weighting matrices given by Ψ and IN −Ψ,respectively.



Also note that µBL can be equivalently written as:

µBL = µmkt + K (v− Pµmkt) K = (τΣ)P′(PτΣP′ + Γ

)−1 .

At this point, given µBL the optimal BL portfolio weights are obtainedby the usual formula:

wBL =1λ

Σ−1(

µBL−rf e)

or wTBL =

Σ−1 (µBL − rf e)

e′Σ−1(µBL − rf e

) .

Similarly to how optimal portfolio weights in the tangency portfolioare computed, the BL efficient frontier can also be computed using µBLand Σ as inputs.


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