The Econometrics of Financial Returns

Transcription

1 The Econometrics of Financial Returns Carlo Favero December 2017 Favero () The Econometrics of Financial Returns December / 55

2 The Econometrics of Financial Returns Predicting the distribution of returns of financial assets is a task of primary importance for identifying desirable investments, performing optimal asset allocation within a portfolio, as well as measuring and managing portfolio risk. Horizon at which returns are defined matters for financial and statistical reasons Portfolio allocation, i.e. the choice of optimal weights to be attributed to the different financial assets in a portfolio, is typically based on a long horizon perspective, the measurement of risk of a given portfolio takes typically a very short-horizon perspective. A long-run investor decides her optimal portfolio allocation on the basis of the (joint) distribution of the returns of the relevant financial assets at low frequency. However, the monitoring of the daily risk of her portfolio depends on the statistical properties of the distribution of returns at high frequency. Favero () The Econometrics of Financial Returns December / 55

3 The relevant dimensions of the data The statistical properties of the distribution of returns vary with the horizon at which they are defined. There are three relevant dimensions in financial returns time-series cross-section the horizon at which returns are defined Favero () The Econometrics of Financial Returns December / 55

4 The relevant dimensions of the data In general, we shall define r i t,t+k as the returns realized by holding between time t and time t + k, the asset i. the t index captures the time-series dimension the i index the cross-sectional dimension the k index the horizon dimension Favero () The Econometrics of Financial Returns December / 55

5 Financial data In general financial data are not generated by experiments, we have only "observational data" Special issues arise routinely in economic and financial data (special days, seasonality, trends, cycles) We look at the data through "empirical models" Favero () The Econometrics of Financial Returns December / 55

6 The Process of Econometric Modelling exploratory analysis of the data (graphics and descriptive statistics) model specification model estimation model simulation model validation and applications Favero () The Econometrics of Financial Returns December / 55

7 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 and do not depend on 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 () The Econometrics of Financial Returns December / 55

8 Econometric Modelling of Financial Returns the data a general multivariate model D (y t, z t, w t Y t 1, Z t 1, W t 1, θ) D (y t, z t Y t 1, Z t 1, β) decomposing a multivariate into conditional and marginal D (y t z t, Y t 1, Z t 1, β 1 ) D (z t Y t 1, Z t 1, β 2 ) a general linear univariate conditional model y t = β z t + u 1t z t = β 2 z t 1 + u 2t Favero () The Econometrics of Financial Returns December / 55

9 Econometric Modelling of Financial Returns Econometric Models y t = β 1 z t + u 1t z t = β 2 z t 1 + u 2t Probabilistic Structure (the reduced form) y t = β 1 β 2 z t 1 + β 1 u 2t + u 1t z t = β 2 z t 1 + u 2t Favero () The Econometrics of Financial Returns December / 55

10 A simple example ( ) r i t r rf t ( ) r m t r rf t ] [ ut ɛ t = β 0,i + β 1,i ɛ t + u t = β 0,m + ɛ t u i,t n.i.d. ([ 0 0 ] [ σ 2, i 0 0 σ 2 m ]) Favero () The Econometrics of Financial Returns December / 55

11 To do list estimate β 0,i, β 1,i, β 0,m, σ 2 i, σ2 m simulate the model to predict future returns and their distribution validate the model Favero () The Econometrics of Financial Returns December / 55

12 Exogeneity and Identification Reduced forms and their validation are not the common first stage of the analysis as researchers tend to estimate directly a structure consistent with the theoretical model of interest. Think about the following generalized version of the CAPM ( ) r i t r rf t ( ) r m t r rf t ] [ ut ɛ t = β 0,i + β 1,i ( r m t = β 0,m + ɛ t u i,t n.i.d. ) r rf t + u t ([ 0 0 ] [ σ 2, i σ im σ im σ 2 m ]) Favero () The Econometrics of Financial Returns December / 55

13 Exogeneity and Identification The estimated CAPM equation implies that (( ) ( ) ) ( E r i t r rf t r m t r rf t, β i = β 0,i + β 1,i r m t ) r rf t (1) Consider now as DGP the generalized version of the CAPM ( and ask ) yourself under which conditions the conditional mean of r i t rrf t derived by considering the full system coincides with (1)? Favero () The Econometrics of Financial Returns December / 55

14 Exogeneity and Identification To derive the conditional mean from the full system consider the following theorem: Consider a partitioning of an n-variate normal vector in two sub-vectors of dimensions n 1 and n n 1 : we then have: ( x1 x 2 1 x 1 N (µ 1, Σ 11 ); ) N (( µ1 µ 2 ), ( Σ11 Σ 12 Σ 21 Σ 22 )). 2 (x 1 x 2 ) N ( µ 1 + Σ 12 Σ 1 22 (x 2 µ 2 ), Σ 11 Σ 12 Σ 1 22 Σ 21), Favero () The Econometrics of Financial Returns December / 55

15 Exogeneity and Identification By applying 2. to CAPM we have: (( ) E r i t r rf t ( r m t ) ) r rf t, β i = µ i + = ( µ i + ( β1,i σ 2 m + σ im σ 2 m ( β 1,i + σ im σ 2 m ( β 1,i + σ ) ( im σ 2 r m t m ) ( r m t ) µ m ) + ) r rf t r rf t β 0,m ) µ i = β 0,i + β 1,i µ m so if σ im = 0,then the conditional mean derived by the model coincide with that of the DGP. Favero () The Econometrics of Financial Returns December / 55

16 Exogeneity and Identification We can say that the market excess returns are weakly exogenous for the estimation of the parameters of interest in our CAPM equation. In the case weak exogeneity is not satisfied the data will deliver estimated parameters that reflect the full system, so the estimated slope of the CAPM will be the more distant from β i the higher the absolute value of σ im. if the weak exogeneity condition is satisfied, then the structural model that is compatible with the reduced form system is unique. In this case we say that the structural model of interest is identified. Favero () The Econometrics of Financial Returns December / 55

17 Why the model can be wrong? There are many ways in which the model 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 Favero () The Econometrics of Financial Returns December / 55

18 This course The objective of this course is to lead students to learn the econometrics of financial returns by developing skills along four different, but highly interrelated, dimensions: knowledge of the relevant theory knowledge of the relevant data; knowledge of the relevant econometric methods; capability of implementing empirical applications (coding). Favero () The Econometrics of Financial Returns December / 55

19 The Exam Students will be assessed in a final computer based exam. The objective of the exam will be to evaluate the individual capability of students of using the inputs given to build the relevant output During the exam students will be required to modify the R codes that they have built during the course to generate answers to the questions posed in the exercises. Working on the exercise step by step and using all the inputs given is the best preparation strategy for the exam. The exams will be open books. Favero () The Econometrics of Financial Returns December / 55

20 The Data We will consider time series data of monthly observations of different portfolios made available by Ken French from his website: data are available in EXCEL format in the file FF_Data.xls. This files combines 3 data files from the website to make available monthly observation on 32 time series from July 1927 onward. We will also consider data made available by Bob Shiller from his website, that consists of an updated version of the time-series used in the book Irrational Exuberance Favero () The Econometrics of Financial Returns December / 55

21 Ken French Data The dataset contains the risk-free rate,rf, and the five Fama-French factors : EXRET_MKT, SMB, and HML, RMW and CMA. SMB (Small Minus Big) is the average return on the small stock portfolios minus the average return on the big stock portfolios HML (High Minus Low) is the average return on the value portfolios minus the average return on the growth portfolios RMW (Robust Minus Weak) is the average return on the robust operating profitability portfolios minus the average return on the weak operating profitability portfolios CMA (Conservative Minus Aggressive) is the average return on the conservative investment portfolios minus the average return on the aggressive investment portfolios details on the construction of these portfolios are available at f_5_factors_2x3.html Favero () The Econometrics of Financial Returns December / 55

22 Ken French Data The momentum factor, MOM, that comes from the file Momentum factor. It contains a momentum factor, constructed from six value-weight portfolios formed using independent sorts on size and prior return of NYSE, AMEX, and NASDAQ stocks. MOM is the average of the returns on two (big and small) high prior return portfolios minus the average of the returns on two low prior return portfolios. The portfolios are constructed monthly. Big means a firm is above the median market cap on the NYSE at the end of the previous month; small firms are below the median NYSE market cap. Prior return is measured from month -12 to - 2. Firms in the low prior return portfolio are below the 30th NYSE percentile. Those in the high portfolio are above the 70th NYSE percentile. Favero () The Econometrics of Financial Returns December / 55

23 Ken French Data The last 25 time series come from the file 25_portfolios_5x5. This file contains value- and equal-weighted returns for the intersections of 5 ME portfolios and 5 BE/ME portfolios. The 25 time series we consider are to equal weighted returns. The portfolios are constructed at the end of Jun. ME is market cap at the end of Jun. BE/ME is book equity at the last fiscal year end of the prior calendar year divided by ME as of 6 months before formation. Firms with negative BE are not included in any portfolio. PR11 are the returns on the portfolio made with the smallest firm and the lowest book equity, PR12 are the returns on the portfolio made with the smallest firm and the second lower book to equity and so on until PR55, which are the returns on the portfolio of the largest firms with the highest book to equity. Favero () The Econometrics of Financial Returns December / 55

24 An Historycal Perspective The plan of our journey is determined by the evolution of the understanding and empirical modelling of asset prices and financial returns from the 1960s onwards. We shall start from the classical Constant Expected Returns (CER) model, when a simple econometric model serves the purpose of modelling returns at all horizon, to move to Time-Varying Expected Returns (TVER) model where different econometric models for returns are to be adopted according to the different horizon at which they are defined. Favero () The Econometrics of Financial Returns December / 55

25 The view from the 1960: Efficient Markets and CER The history of empirical finance starts with the "efficient market hypothesis"(fama(1970):). expected returns are constant, CAPM is a good measure of risk and thus a good explanation of why some stocks earn higher average returns than others; excess returns are close to unpredictable: any predictability is a statistical artifact or cannot be exploited after transaction costs; volatility of returns is constant. Favero () The Econometrics of Financial Returns December / 55

26 Time-Series Implications r i t,t+1 = µi + σ i ɛ it ɛ it NID (0, 1) Cov ( { ) σij t = s ɛ it, ɛ js = 0 t = s. Note that the absence of predictability of excess returns is not a consequence of market efficiency per se but it instead results from a joint hypothesis: market efficiency plus some assumptions on the process generating returns (i.e., the Contant Expected Returns model). Favero () The Econometrics of Financial Returns December / 55

27 Returns at different horizons In this world, the horizon k does not matter for the prediction of returns because once µ i and σ i are estimated, expected retrurns at all horizons and the variance of returns at all horizon are derived deterministically. n E(r i t,t+n) = E( k=1r i n t+k,t+k 1 ) = E(r i t+k,t+k 1 ) = nµ k=1 n Var(r i t,t+n) = Var( i=1r i n t+k,t+k 1 ) = Var(r i t+k,t+k 1 ) = nσ2 i=1 Favero () The Econometrics of Financial Returns December / 55

28 The cross-section of returns Cross-sectional heterogeneity is related to a single factor, the market factor, and the CAPM determines all the cross-sectional variation in returns. The statistical model that determines all returns r i t and the market return r m t,is: ( ( r i t r rf t r m t t ( ui,t ) ) r rf u m,t ) = µ i + β i u m,t + u i,t = µ m + u m,t [( 0 n.i.d. 0 ) ( σii σ, im σ im σ mm )] where r rf t is the return on the risk-free asset. We shall see that σ im = 0 is a crucial assumption for the valid estimation of the CAPM betas, and that assumption that risk adjusted excess returns are zero (usually known as zero alpha assumption) requires that µ i = β i µ m. Favero () The Econometrics of Financial Returns December / 55

29 The Volatility of Returns The volatility of returns is constant in the CER model which therefore is not capable of explaining time-varying volatility in the markets and the presence of alternating period of high and low volatility. Favero () The Econometrics of Financial Returns December / 55

30 Implications for Asset Allocation Optimal asset allocation can be achieved by utility maximization that uses as inputs the historical moments of the distribution of returns, optimal portfolio weights are constant through the investment horizon, optimal portfolio is always a combination between the market portfolio and the risk free asset. The risk associated to any given asset or portfolio of assets is constant over time. Favero () The Econometrics of Financial Returns December / 55

31 Measuring Risk Think of measuring the risk of a portfolio with its Value-at-Risk (VaR). The VaR is the percentage loss obtained with a probability at most of α percent: Pr (R p < VaR α ) = α. where R p are the returns on the portfolio. Favero () The Econometrics of Financial Returns December / 55

32 Measuring Risk If the distribution of returns is normal, then α-percent VaR α is obtained as follows (assume α (0, 1)): Pr (R p < VaR α ) = ( R p µ p α Pr < VaR ) α + µ p = α σ p σ p ( Φ VaR ) α + µ p = α, σ p where Φ ( ) is the cumulative density of a standard normal. At this point, defining Φ 1 ( ) as the inverse CDF function of a standard normal, we have that VaR α + µ p σ p = Φ 1 (α) VaR α = µ p σ p Φ 1 (α). and, given that µ p and σ p are constant over time, VaR α is also constant over-time. Favero () The Econometrics of Financial Returns December / 55

33 Measuring Risk Consider the case of a researcher interested in the one per cent value at risk. Because Φ 1 (0.01) = 2.33 under the normal distribution, we can easily obtain VaR if we have available estimates of the first and second moments of the distribution of portfolio returns: VaR 0.01 = ˆµ p 2.33 ˆσ p Favero () The Econometrics of Financial Returns December / 55

34 Empirical Challenges to the CER model The tenet that expected returns are constant is not compatible with the observed volatility of stock prices. Stock prices in fact are "too volatile" to be determined only by expected dividends; There is evidence of returns predictability that increases with the horizon at which returns are defined. There are anomalies that make returns predictable on occasion of special events. The CAPM is rejected when looking at the cross-section of returns and multi factor models are needed to explain the cross-sectional variability of returns high frequency returns are non-normal and heteroscedastic, therefore risk is not constant over time. Favero () The Econometrics of Financial Returns December / 55

35 Empirical Challenges to the CER model: the DDG model Practictioners implementing portfolio allocation based on the CER model experienced rather soon a number of problems that stressed limitations of this model but it was the work of Robert Shiller and co-authors that led te profession to go beyond the CER model. The basic empirical evidence against the CER model is the excessive volatility of asset prices and returns which is clearly illustrated in Shiller(1981). Favero () The Econometrics of Financial Returns December / 55

36 A simple log price-dividend ratio framework Total returns to a stock i can be satisfactorily approximated as follows: r s t+1 = κ + ρ (p t+1 d t+1 ) + d t+1 (p t d t ) where P t is the stock price at time t and D t is the dividend paid at time t, p t = ln(p t ), d t = ln(d t ), κ is a constant and ρ = P/D 1+P/D, P/D is the average price to dividend ratio. In practice ρ can be interpreted as a discount parameter(0 < ρ < 1). Favero () The Econometrics of Financial Returns December / 55

37 A simple log price-dividend ratio framework By forward recursive substitution one obtains: (p t d t ) = κ m 1 ρ + j=1ρ j 1 ( ) m ( ) d t+j ρ j 1 r s t+j + ρ m (p t+m+1 d t+m+1 ) j=1 which shows that the (p t d t ) measures the value of a very long-term investment strategy (buy and hold) which is equal to the stream of future dividend growth discounted at the appropriate rate, which reflects the risk free rate plus risk premium required to hold risky assets. Favero () The Econometrics of Financial Returns December / 55

38 Rejection of the CER Under the null of the CER the price-dividend ratio should be completely determined by the process generating dividends and determining their expectations. But stock prices are too volatile to be determined only by expected dividends (see Shiller(1981) and Campbell-Shiller(1987)). The price-dividend ratio is useful to predict returns, the more so the longer the horizon at which returns are defined. Favero () The Econometrics of Financial Returns December / 55

39 The empirical evidence Favero () The Econometrics of Financial Returns December / 55

40 The empirical evidence Favero () The Econometrics of Financial Returns December / 55

41 Anomalies:an example Lucca Moench(2014) Document large average excess returns on U.S. equities in anticipation of monetary policy decisions made at scheduled meetings of the Federal Open Market Committee (FOMC) in the past few decades. Favero () The Econometrics of Financial Returns December / 55

42 Multiple Factor Models The CAPM has implication for the cross section of assets if ( ) E r i r f = β i E (r ) M r f then heterogeneity in excess returns to different assets should be totally explained by the different exposure to a single common risk factor, the market excess returns. Favero () The Econometrics of Financial Returns December / 55

43 Given a sample of observations on r i t, rf t, rm t,the β i can be estimated first by OLS regression over the time series of returns, then the following second-pass equations can be estimated over the cross-section of returns: _ r i = γ 0 + γ 1 β i + u i Where _ r i are the average returns in the period over which the β i have been computed. If the CAPM is valid, then γ 0 and γ 1 should satisfy: γ 0 = _ r f, γ 1 = _ r M where _ r M is the mean market excess return. When the model is estimated with appropriate methods, the restrictions are strongly rejected (Fama-French(1992), Fama-McBeth) Favero () The Econometrics of Financial Returns December / 55

44 The different exposure to a single factor model cannot explain the observed cross-sectional behaviour of returns. This evidence paved the way to the estimation of multi-factor models of returns. Fama-French(1993) introduced a three-factor model based on the integration of the CAPM with a small-minus-big market value (SMB) and high-minus-low book-to-market ratio (HML). These factors are equivalent to zero-cost arbitrage portfolio that takes a long position in high book-to-market (small-size) stocks and finances this with a short position in low book-to-market (large-size) stocks. Favero () The Econometrics of Financial Returns December / 55

45 Jegadeesh and Titman(1993) dis-covered the importance of a further additional factor in explaining excess returns: momentum(mom). An investment strategy that buys stocks that have performed well and sells stocks that have performed poorly over the past 3-to 12-month period generates significant excess returns over the following year. It is interesting to note that augmenting the CAPM with SMB and HML does not challenge per se the CER model, which still hold as valid if the constant expected return model can be applied to the two aditional factors. However, momentum provides direct evidence against the CER model as it indicates that the conditional expections of future returns is not constant. Favero () The Econometrics of Financial Returns December / 55

46 Non-normality and heteroscedasticity At small horizon (i.e. when k is small: infra-daily, daily, weekly or at most monthly returns) the following framework is supported by the data : R t,t+k = σ k,t u t+k σ 2 k,t = f (I t ) u t+k IID D(0, 1). The following features of the model at high frequency are noteworthy: 1 The distribution of returns is centered around a mean of zero, and the zero mean model dominates any alternative model based on predictors. 2 The variance is time-varying and predictable, given the information set, I t, available at time t. 3 The distribution of returns at high frequency is not normal, i.e., D(0, 1) may often differ from N (0, 1) Favero () The Econometrics of Financial Returns December / 55

47 The implications of the new evidence The new evidence has brought about importnat implications both for asset pricing and risk management Favero () The Econometrics of Financial Returns December / 55

48 Asset Pricing with Predictable Returns Consider a situation in which in each period k state of nature can occur and each state has a probability π(k), in the absence of arbitrage opportunities the price of an asset i at time t can be written as follows: P i,t = k s=1 π t+1 (s)m t+1 (s) X i,t+1 (s) m t+1 (s) is the discounting weight attributed to future pay-offs, which (as the probability π) is independent from the asset i, X i,t+1 (s) are the payoffs of the assets (we have seen that in case of stocks we have X i,t+1 = P t+1 + D t+1 ), and therefore returns on assets are defined as 1 + R s,t+1 = X i,t+1 P i,t. Favero () The Econometrics of Financial Returns December / 55

49 Asset Pricing with Predictable Returns For the safe asset, whose payoffs do not depend on the state of nature, we have: P s,t = X s,t+1 m j=1π t+1 (s)m t+1 (s) 1 + R s,t+1 = m j=1 1 π t+1 (s)m t+1 (s) so we can write: P s,t = X s,t+1 E t (m t+1 ) 1 + R s,t+1 = 1 E t (m t+1 ) Favero () The Econometrics of Financial Returns December / 55

50 Asset Pricing with Predictable Returns For risky assets E t (m t+1 (1 + R i,t+1 )) = 1 P i,t = E t (m t+1 X i,t+1 ) Cov (m t+1 R i,t+1 ) = 1 E t (m t+1 ) E t (1 + R i,t+1 ) E t (1 + R i,t+1 ) = Cov (m t+1r i,t+1 ) + (1 + R s,t+1 ) E t (m t+1 ) Favero () The Econometrics of Financial Returns December / 55

51 Asset Pricing with Predictable Returns Consider now the case where the period t is made by two points in time very close to each other (a short holding horizon), in this case m t+1 can be safely approximated by a constant (very close to one) and excess returns are not predictable. As the point in time that define the period becomes further and furthes separated, then time variation in m cannot be discounted anymore and future excess returns becomes predictable if their covariance with m is predictable. In fact, we can write: E t (R i,t+1 R s,t+1 ) = (1 + R s,t+1 ) cov (m t+1 R i,t+1 ) Assets whose returns are low when the stochastic discount factor is high (i.e. when agents values payoffs more) require an higher risk premium, i.e. an higher excess return on the risk-free rate. Favero () The Econometrics of Financial Returns December / 55

52 Risk Management Once the portfolio weights (^w) are chosen, possibly exploiting the predictability of the distribution of the relevant future returns, the distribution of a portfolio returns can be described as follows: ( ) R p D µ p, σ 2 p µ p = µ ^w σ2 p = ^w ^w Having solved the portfolio problem and having committed to a given allocation described by ^w, there is a different role that econometrics can play at high frequencies: measuring volatility and providing information on portfolio risk. Altough noise is not predictable, its volatility is. Favero () The Econometrics of Financial Returns December / 55

53 Predictive Models in Finance Predictive models are special cases of this general specification: r t,t+k = f ( X µ t, ) Θµ t + Ht+k ɛ t+k (2) Σ t+k = H t+k H t+k. Σ t+k = g (X σ t, Θ σ t ) + q j=1 B jσ t+k j B j, (3) ɛ t+k D (0, I) where r t,t+k is the vector of returs between time t and time t+k in which we are interested, X µ t is the vector of predictors for the mean of our returns that we observe at time t, f specifiies the functional relation (that is potentially time-varying) between the mean returns and the predictors that depends also on a set of parameters Θ µ t, the matrix H t+k determines the potentially time varying variance-covariance of the vector of returns..the process for the variance is predictable as there is a functional relation determining the relationship between H t+k and a vector of predictors Xt σ that is driven by a vector of unknown Favero parameters () Θ σ The. Econometrics of Financial Returns December / 55

54 Predictive Models in Finance Our first look at the data clearly show that the appropriate specification of the general predicitive model depends on the horizon at which returns are defined. When k is small and high-frequency returns r t,t+k = 0 + σ t+k u t+k u t+k IID D(0, 1), σ 2 t+k = ω + ασ 2 t+k 1 + βu2 t+k 1, α + β < 1 This is a model that feautres no predictability in the mean of r returns (the expected future return at any horizon is constant at zero), but there is predictability in the variance of returns that it is mean reverting towards a long-term value of ω/ (1 α β). No assumption of normality is made for the innovation innovation in the process generating returns. Favero () The Econometrics of Financial Returns December / 55

55 Predictive Models in Finance Consider now the case of large k, i.e. long-horizon returns (note that in the continuously compounded case, r t,t+k k j=1 r t,t+j), in this case the relevant predictive model can be written as follows: r t,t+k = α + β X t + σu t+k u t+k IID N (0, 1), where X t is a set of predictors observed at time t. In this case we have that returns feature predictability in mean, constant variance and the innovations are normally distributed. Favero () The Econometrics of Financial Returns December / 55