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Modelling Returns: the CER and the CAPM

Carlo Favero

Favero () Modelling Returns: the CER and the CAPM 1 / 20

Econometric Modelling of Financial Returns

Financial data are mostly observational data: they are notgenerated by well-designed experiment to test hypothesis, theyare given to the econometrician.These data can be used to construct non-causal predictive modelsand to evaluate treatment effects.The second exercise involves a deeper understanding of causationwhile the implementation of non-causal predictive modellingrequires understanding conditional expectations.


Non experimental data

What do we do with non-experimental data ?When you try and explain returns you do not control how thedata are generatedThink of estimating a CAPM model


Non experimental data: CAPM

�ri

t � rrft

�= β0,i + β1,i

�rm

t � rrft

�+ ui,t�

rmt � rrf

t

�= µm + um,t

ui,t s n.i.d.�

0, σ2i

��

ui,tum,t

�s n.i.d.

��00

�,�

σii 00 σmm

��you have a time-series on excess returns of a given stock on thesafe asset and on the market on the safe asset (you cannot controlthese data)you specify a linear model in which the excess returns on asset iare function of the excess returns on the market, and the excessreturns on the market follow a CER.you introduce hypothesis on the error terms, to estimateparameters, to implement tests, and to use the model forsimulationThis is a very simplified model, the crucial issue is the relevanceof your simplifying assumptions



Econometric models of financial returns specify the distribution ofa vector of variables yt conditional upon other variables ztthat arehelpful in predicting them.The mapping between yt and zt is determined by some functionalrelation and some unknown parameters.All the relevant variables are stochastic and they are thereforecharacterized by a density function.Linear Econometric Models specify conditional means of the yt aslinear functions of the zt.



the data

D (yt, zt, wt j Yt�1, Zt�1, Wt�1,θ)

a general multivariate model

D (yt, zt j Yt�1, Zt�1, β)

decomposing a multivariate into conditional and marginal

D (yt j zt, Yt�1, Zt�1, β1)D (zt j Yt�1, Zt�1, β2)

a general linear univariate conditional model

yt = β0zt + u1t

zt = β2zt�1 + u2t



There are many ways in which the CAPM can go wrong:

other factors beyond the market are relevant in determining excessreturns on asset ithe excess returns on the market do depend on excess returns onasset ithe model is non-linearthe residuals are non-normal and their variance-covariance matrixdoes not fit the assumptions made


Reduction Process

Conditional densities are best interpreted as the outcome of areduction process that allows a simplified representation of reality.Of course such a simplified representation omits an enormousamount of information.The validity of the model adopted is crucially affected by theimportance of the omitted information in determining the densityof yt.


Reduction Process

To understand the reduction process partiton the set of allvariables into three types of variables:

xt = (wt, yt, zt) ,

wt identifies variables which are ignored in the specification ofthe econometric model.Exclusion is obtained by factorizing the joint density andintegrating it with respect to wt.In formal terms, we have noinformation loss only if

D (yt, zt j Yt�1, Zt�1, β) = D (yt, zt, wt j Yt�1, Zt�1, Wt�1,θ) .

This is the statistical model considered by the econometrician, this istechnically called i.e. the reduced form of the structure of interest. Ingeneral this reduced form is a more general model than the oneestimated. It is constructed by parameterizing E (yt, zt j Yt�1, Zt�1, β)and by deriving a vector of innovations from the difference betweenthe vector of observed variables and the vector of their means.


The CAPM Reduced Form

In the case of the CAPM the general specification of the reduced formis the following one:

�ri

t � rrft

�= µi + βium,t + ui,t�

rmt � rrf

t

�= µm + um,t�

ui,tum,t

�s n.i.d.

��00

�,�

σii σimσim σmm

��


From the Statistical Model to the Conditional Model:the CAPM

Statistical model�ri

t � rrft

�= µi + βium,t + ui,t�

rmt � rrf

t

�= µm + um,t�

ui,tum,t

�s n.i.d.

��00

�,�

σii σimσim σmm

��Estimated Equation

E��

rit � rrf

t

�j�

rmt � rrf

t

�, Yt�1, Zt�1, βi

�= αi + βi

�rm

t � rrft

�if σim = 0,then the estimated equation is a valid approximation to thestatistical model for the estimation of βi


Modelling returns


Modelling returns

The (naive) log random walk (LRW) hypothesis on the evolution ofprices states that, prices evolve approximately according to thestochastic difference equation:

ln Pt = µ∆+ ln Pt�∆ + εt

where the ’innovations’ εt are assumed to be uncorrelated across time(cov(εt; εt0) = 0 8t 6= t0), with constant expected value 0 and constantvariance σ2∆.Consider what happens over a time span of, say, 2∆.

ln Pt = 2µ∆+ ln Pt�2∆ + εt + εt�∆ = ln Pt�2∆ + ut

having set ut = εt + εt�∆.


Modelling returns

Consider now the case in which the time interval is of the length of1-period. If we take prices as inclusive of dividends we can write thefollowing model for log-returns

rt,t+1 = µ+ σεt

εt = i.i.d.(0, 1)

E(rt,t+n) = E(n

∑i=1

rt+i,t+i�1) =n

∑i=1

E(rt+i,t+i�1) = nµ

Var(rt,t+n) = Var(n

∑i=1

rt+i,t+i�1) =n

∑i=1

Var(rt+i,t+i�1) = nσ2


Monte-Carlo simulation

given some estimates of the unknown parameters in the model (µσ in our case).an assumption is made on the distribution of εt.The an artificial sample for εt of the length matching that of theavailable can be computer simulated.The simulated residuals are then mapped into simulated returnsvia µ, σ.This exercise can be replicated N times (and therefore aMonte-Carlo simulation generates a matrix of computersimulated returns whose dimension are defined by the samplesize T and by the number of replications N).The distribution of model predicted returns can be thenconstructed and one can ask the question if the observed data canbe considered as one draw from this distribution.


do exactly like in Monte-Carlo but rather than using a theoreticaldistribution for εt use their empirical distribution and resample from itwith reimmission.


Simulation can be used for several tasks,

provide statistical evidence of the capability of the model toreplicate the dataderive the distribution of returns to implement VaRassess statistical properties of estimators


Stocks for the long-run

The fact that, under the LRW, the expected value grows linearly withthe length of the time period while the standard deviation (square rootof the variance) grows with the square root of the number ofobservations, has created a lot of discussionWe have three flavors of the “stocks for the long run” argument. Thefirst and the second are a priori arguments depending on the lograndom walk hypothesis or something equivalent to it, the third is ana posteriori argument based on historical data.



Assume that single period (log) returns have (positive) expected valueµ and variance σ2. Moreover, assume for simplicity that the investorrequires a Sharpe ratio of say S out of his-her investment. Under theabove hypotheses, plus the log random walk hypothesis, the Sharperatio over n time periods is given by

S =nµpnσ=p

nµ

σ

so that, if n is large enough, any required value can be reached.



Another way of phrasing the same argument, when we add thehypothesis of normality on returns, is that, there any given probabilityα and any given required return C there is always an horizon forwhich the probability for n period return less than C is less than α.

Pr (Rp < C) = α.

Pr (Rp < C) = α () Pr�

Rp � nµpnσ

<C� nµp

nσ

�= α

() Φ

C� µp

σp

!= α,

C = nµ+Φ�1 (α)p

nσ

But nµ+Φ�1 (α)p

nσ, forp

n > 12

Φ�1(α)µ σ is an increasing function in

n so that for any α and any chosen value C, there exists a n such thatfrom that n onward, the probability for an n period return less than Cis less than α.



Note, however, that the value of n for which this lower bound crossesa given C level is the solution of

nµ+Φ�1 (α)p

nσ � C

In particular, for C = 0 the solution is

pn � �Φ�1 (α) σ

µ

Consider now the case of a stock with σ/µ ratio for one year is of theorder of 6. Even allowing for a large α,say 0.25, so that Φ�1 (α) is nearminus one , the required n shall be in the range of 36 which is onlyslightly shorter than the average working life.As a matter of fact, based on the analysis of historical prices and riskadjusted returns, stocks have been almost always a good long runinvestment.


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