Modelling Returns: the CER and the CAPM
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1 Modelling Returns: the CER and the CAPM Carlo Favero Favero () Modelling Returns: the CER and the CAPM 1 / 20
2 Econometric Modelling of Financial Returns Financial data are mostly observational data: they are not generated by well-designed experiment to test hypothesis, they are given to the econometrician. These data can be used to construct non-causal predictive models and to evaluate treatment effects. The second exercise involves a deeper understanding of causation while the implementation of non-causal predictive modelling requires understanding conditional expectations. Favero () Modelling Returns: the CER and the CAPM 2 / 20
3 Non experimental data What do we do with non-experimental data? When you try and explain returns you do not control how the data are generated Think of estimating a CAPM model Favero () Modelling Returns: the CER and the CAPM 3 / 20
4 Non experimental data: CAPM r i t r m t ui,t u m,t r rf t r rf t = β 0,i + β 1,i r m t r rf t + u i,t = µ m + u m,t u i,t s n.i.d. 0, σ 2 i 0 σii 0 s n.i.d., 0 0 σ mm you have a time-series on excess returns of a given stock on the safe asset and on the market on the safe asset (you cannot control these data) you specify a linear model in which the excess returns on asset i are function of the excess returns on the market, and the excess returns on the market follow a CER. you introduce hypothesis on the error terms, to estimate parameters, to implement tests, and to use the model for simulation Favero () Modelling Returns: the CER and the CAPM 4 / 20
5 Econometric Modelling of Financial Returns Econometric models of financial returns specify the distribution of a vector of variables y t conditional upon other variables z t that are helpful in predicting them. The mapping between y t and z t is determined by some functional relation and some unknown parameters. All the relevant variables are stochastic and they are therefore characterized by a density function. Linear Econometric Models specify conditional means of the y t as linear functions of the z t. Favero () Modelling Returns: the CER and the CAPM 5 / 20
6 Econometric Modelling of Financial Returns the data a general multivariate model D (y t, z t, w t j Y t 1, Z t 1, W t 1, θ) D (y t, z t j Y t 1, Z t 1, β) decomposing a multivariate into conditional and marginal D (y t j z t, Y t 1, Z t 1, β 1 ) D (z t j Y t 1, Z t 1, β 2 ) a general linear univariate conditional model y t = β 0 z t + u 1t z t = β 2 z t 1 + u 2t Favero () Modelling Returns: the CER and the CAPM 6 / 20
7 Econometric Modelling of Financial Returns There are many ways in which the CAPM can go wrong: other factors beyond the market are relevant in determining excess returns on asset i the excess returns on the market do depend on excess returns on asset i the model is non-linear the residuals are non-normal and their variance-covariance matrix does not fit the assumptions made Favero () Modelling Returns: the CER and the CAPM 7 / 20
8 Reduction Process Conditional densities are best interpreted as the outcome of a reduction process that allows a simplified representation of reality. Of course such a simplified representation omits an enormous amount of information. The validity of the model adopted is crucially affected by the importance of the omitted information in determining the density of y t. Favero () Modelling Returns: the CER and the CAPM 8 / 20
9 Reduction Process To understand the reduction process partiton the set of all variables into three types of variables: x t = (w t, y t, z t ), w t identifies variables which are ignored in the specification of the econometric model. Exclusion is obtained by factorizing the joint density and integrating it with respect to w t.in formal terms, we have no information loss only if D (y t, z t j Y t 1, Z t 1, β) = D (y t, z t, w t j Y t 1, Z t 1, W t 1, θ). This is the statistical model considered by the econometrician, this is technically called i.e. the reduced form of the structure of interest. In general this reduced form is a more general model than the one estimated. It is constructed by parameterizing E (y t, z t j Y t 1, Z t 1, β) and by deriving a vector of innovations from the difference between the vector of observed variables and the vector of their means. Favero () Modelling Returns: the CER and the CAPM 9 / 20
10 The CAPM Reduced Form In the case of the CAPM the general specification of the reduced form is the following one: r i t r rf t r m t r rf t ui,t u m,t = µ i + β i u m,t + u i,t = µ m + u m,t 0 s n.i.d. 0 σii, σ im σ im σ mm Favero () Modelling Returns: the CER and the CAPM 10 / 20
11 From the Statistical Model to the Conditional Model: the CAPM Statistical model r i t r rf t r m t r rf t ui,t u m,t = µ i + β i u m,t + u i,t = µ m + u m,t 0 s n.i.d. 0 σii, σ im σ im σ mm Estimated Equation E r i t j r rf t r m t r rf t, Y t 1, Z t 1, β i = α i + β i r m t r rf t if σ im = 0,then the estimated equation is a valid approximation to the statistical model for the estimation of β i Favero () Modelling Returns: the CER and the CAPM 11 / 20
12 Modelling returns Favero () Modelling Returns: the CER and the CAPM 12 / 20
13 Modelling returns The (naive) log random walk (LRW) hypothesis on the evolution of prices states that, prices evolve approximately according to the stochastic difference equation: ln P t = µ + ln P t + ɛ t where the innovations ɛ t are assumed to be uncorrelated across time (cov(ɛ t ; ɛ t 0) = 0 8t 6= t 0 ), with constant expected value 0 and constant variance σ 2. Consider what happens over a time span of, say, 2. ln P t = 2µ + ln P t 2 + ɛ t + ɛ t = ln P t 2 + u t having set u t = ɛ t + ɛ t. Favero () Modelling Returns: the CER and the CAPM 12 / 20
14 Modelling returns Consider now the case in which the time interval is of the length of 1-period. If we take prices as inclusive of dividends we can write the following model for log-returns r t,t+1 = µ + σɛ t ɛ t = i.i.d.(0, 1) E(r t,t+n ) = E( Var(r t,t+n ) = Var( n i=1 n i=1 r t+i,t+i 1 ) = r t+i,t+i 1 ) = n i=1 n i=1 E(r t+i,t+i 1 ) = nµ Var(r t+i,t+i 1 ) = nσ 2 Favero () Modelling Returns: the CER and the CAPM 13 / 20
15 Monte-Carlo simulation given some estimates of the unknown parameters in the model (µ σ in our case). an assumption is made on the distribution of ɛ t. The an artificial sample for ɛ t of the length matching that of the available can be computer simulated. The simulated residuals are then mapped into simulated returns via µ, σ. This exercise can be replicated N times (and therefore a Monte-Carlo simulation generates a matrix of computer simulated returns whose dimension are defined by the sample size T and by the number of replications N). The distribution of model predicted returns can be then constructed and one can ask the question if the observed data can be considered as one draw from this distribution. Favero () Modelling Returns: the CER and the CAPM 14 / 20
16 do exactly like in Monte-Carlo but rather than using a theoretical distribution for ɛ t use their empirical distribution and resample from it with reimmission. Favero () Modelling Returns: the CER and the CAPM 15 / 20
17 Simulation can be used for several tasks, provide statistical evidence of the capability of the model to replicate the data derive the distribution of returns to implement VaR assess statistical properties of estimators Favero () Modelling Returns: the CER and the CAPM 16 / 20
18 Stocks for the long-run The fact that, under the LRW, the expected value grows linearly with the length of the time period while the standard deviation (square root of the variance) grows with the square root of the number of observations, has created a lot of discussion We have three flavors of the stocks for the long run argument. The first and the second are a priori arguments depending on the log random walk hypothesis or something equivalent to it, the third is an a posteriori argument based on historical data. Favero () Modelling Returns: the CER and the CAPM 17 / 20
19 Stocks for the long-run Assume that single period (log) returns have (positive) expected value µ and variance σ 2. Moreover, assume for simplicity that the investor requires a Sharpe ratio of say S out of his-her investment. Under the above hypotheses, plus the log random walk hypothesis, the Sharpe ratio over n time periods is given by S = nµ p nσ = p n µ σ so that, if n is large enough, any required value can be reached. Favero () Modelling Returns: the CER and the CAPM 18 / 20
20 Stocks for the long-run Another way of phrasing the same argument, when we add the hypothesis of normality on returns, is that, there any given probability α and any given required return C there is always an horizon for which the probability for n period return less than C is less than α. Pr (R p < C) = α. R Pr (R p p nµ < C) = α () Pr p < C nµ p = α nσ nσ () Φ C µ! p = α, σ p C = nµ + Φ 1 (α) p nσ But nµ + Φ 1 (α) p nσ, for p n > 1 2 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 that from that n onward, the probability for an n period return less than C is less than α. Favero () Modelling Returns: the CER and the CAPM 19 / 20
21 Stocks for the long-run Note, however, that the value of n for which this lower bound crosses a given C level is the solution of nµ + Φ 1 (α) p nσ C In particular, for C = 0 the solution is p n Φ 1 (α) σ µ Consider now the case of a stock with σ/µ ratio for one year is of the order of 6. Even allowing for a large α,say 0.25, so that Φ 1 (α) is near minus one, the required n shall be in the range of 36 which is only slightly shorter than the average working life. As a matter of fact, based on the analysis of historical prices and risk adjusted returns, stocks have been almost always a good long run investment. Favero () Modelling Returns: the CER and the CAPM 20 / 20
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