Statistical Analysis of Financial Data

Transcription

1 ETH Zürich, WBL Angewandte Statistik 2019 Blockkurs Statistical Analysis of Financial Data Lecture 4 Prof. Dr. Andreas Ruckstuhl Dozent für Statistische Datenanalyse Institut für Datenanalyse und Prozess Design IDP Zürcher Hochschule für Angewandte Wissenschaften ZHAW andreas.ruckstuhl@zhaw.ch 28. Januar 2019 A. Ruckstuhl -- WBL 2019, Lecture 4 of SAoFD -- Page 1

2 Outline of Lecture 4 Lecture 1: Financial Data and Their Properties Lecture 2: Model for Conditional Heteroskedasticity and Risk Measures Lecture 3: Statistical Issues When Applying Portfolio Theory Lecture 4: (Financial) Factor Models The market model Macroeconomic Factor Models Fundamental Factor Model & Example with R Cross-Sectional Fundamental Factor Model Principal Components and Factor Analysis * Factor Models in Return-Based Style Analysis * Our 17 Factor Model Lecture 5: Copulas A. Ruckstuhl -- WBL 2019, Lecture 4 of SAoFD -- Page 2

3 7 (Financial) Factor Models Financial factor analysis explains returns with a small number of fundamental variables called factors or risk factor. A factor model for equity returns (or excess equity returns R j,t - μ f,t ) is R j,t = o,j + 1,j F 1,t + + p,j F p,t + j,t where R j,t is either the return or the excess return on the jth asset at time t, μ f,t is the risk free rate at time t F 1,t,..., F p,t are variables, called factors or risk factors, that represent the state of the financial markets and world economy at time t. 1,t,, n,t are uncorrelated, mean-zero r.v. called the unique risks of the individual stocks Uncorrelated means that all cross-correlation between the returns is due to the factors! The factors are common to all returns j=1,,n k,j is called factor loading and specifies the sensitivity of the j th asset return to the k th factor Introduction of some types of factor models: A. Ruckstuhl -- WBL 2019, Lecture 4 of SAoFD -- Page 3

4 7.1 The market model: The root of factor models The market model R j,t μ f,t = j + j (R m,t μ f,t ) + j,t, where a simple market index R m,t is a proxy for the entire (non-observable) market portfolio. This model is based on the Capital Asset Pricing Model (CAPM) (*), where j would be 0. CAPM is an economic equilibrium model (ex ante, i.e., predictive) whereas the market model is ex post (i.e., explanatory). In CAPM, the market risk factor is the only source of risk besides the unique risk of each asset. Hence, it is the sole source of correlation between any two asset returns Test whether CAPM is valid: Test the hypothesis j =0 in an ex post (!) setting (If is nonzero then the security (=Wertpapier) is mispriced, at least according to CAPM) (*) CAPM was introduced by Jack Treynor, William Sharpe, John Lintner and Jan Mossin independently in the sixties, building on the earlier work of Harry Markowitz. Sharpe, Markowitz and Merton Miller jointly received the Nobel Memorial Prize in Economics for this contribution to the field of financial economics. The market model is a single factor model and there is fairly strong empirical evidence that one factor is not sufficient. A. Ruckstuhl -- WBL 2019, Lecture 4 of SAoFD -- Page 4

5 7.2 Macroeconomic Factor Models Macroeconomic factor models use macroeconomic variables such as Return on the market portfolio Growth rate of the GDP Interest rate on short term Treasury bills or changes in this rate Inflation rate or changes in this rate Interest rate spreads; e.g., difference between long-term Treasury bonds and long-term corporate bonds Base: The efficient market hypothesis (another piece of theory in finance) implies that stock prices change because of new information; i.e., stock returns will be influenced by unpredictable changes in macroeconomic variables. The factors in a macroeconomic model are not the macroeconomic variables themselves, but rather the residuals when changes in the macroeconomic variables are predicted by a times series model, such as, multivariate AR models The R 2 -values of such fits are often very low hence such types of factor models are not used widely in practise. A. Ruckstuhl -- WBL 2019, Lecture 4 of SAoFD -- Page 5

6 7.3 Fundamental Factor Model Fundamental factor models use observable asset characteristics such as Return on portfolio of stocks; e.g., all CH stocks, SMI, KMU,... The difference between returns on two portfolios Example: Fama & French Three-Factor Model ( ) The three factors are Excess return of the market portfolio (from the market model / CAPM) Small minus big (SMB): Difference in returns on a portfolio of small stocks and a portfolio of large stocks (size refer to the size of the market value) Hight minus low (HML): Difference in returns of a portfolio of high book ( ) -to-market value (BE/ME) stocks and a portfolio of low BE/ME stocks ( ) Fama & French (1993, 1995, 1996), Univ. of Chicago 2013, Eugene Fama shared the Nobel Memorial Prize in Economics jointly with Robert Shiller and Lars Peter Hansen ( ) Book value is the net worth of a firm according to its accounting balance sheet A. Ruckstuhl -- WBL 2019, Lecture 4 of SAoFD -- Page 6

7 Example: Fama & French Three-Factor Model applied to monthly returns on three equities GE, IBM and Mobil (Jan 1969 to Dec 1998) Calculate excess returns; e.g., GEf = GE R F > pairs(ffs1) Left: Scatterplot matrix of the variables used in this example Focusing on GEf, GEf is highly correlated with the excess market returns GEf is negatively related with the factor HML GEf behaves as a value stock??!! A. Ruckstuhl -- WBL 2019, Lecture 4 of SAoFD -- Page 7

8 Fitting by Time Series Regression > (FFS.fit <- lm(cbind(gef,ibmf,mobilf) ~ Mkt.RF + SMB + HML, data=ffs1)) Coefficients: GEf IBMf Mobilf (Intercept) Mkt.RF SMB HML Note that GEf now has a positive relationship with HML All three equity returns have negative relationship to SMB; i.e., they behave like large stock as they are suppose to do A. Ruckstuhl -- WBL 2019, Lecture 4 of SAoFD -- Page 8

9 Are the residuals uncorrelated as we assume in the factor model? > cor(ffs1.fit$residuals) GEf IBMf Mobilf GEf IBMf Mobilf > pairs(ffs1.fit$residuals, las=1) The correlation between GEf and Mobilf is rather far from zero > cor.test(ffs1.fit$residuals[,1], FFS1.fit$residuals[,3]) t= , df=358, p-value=1.044e-06 There are a few outliers robust estimation? (covrobust()from R package rrcov) A. Ruckstuhl -- WBL 2019, Lecture 4 of SAoFD -- Page 9

10 Use a time-series factor model to estimate the covariance matrix R : Use factor model R t = β 0 + β T F t +, where β β and t 0, t are unkown The covariance matrix of R t is R = β T F β +, (*) where is a diagonal matrix. F is estimated by the sample covariance of the factors > (h1 <- cov(ffs1[,c("mkt.rf","smb","hml")])) Mkt.RF SMB HML Mkt.RF SMB HML The estimate of β is the matrix of regression coefficients (without intercepts) > (h2 <- coef(ffs1.fit)[-1,]) GEf IBMf Mobilf Mkt.RF SMB HML A. Ruckstuhl -- WBL 2019, Lecture 4 of SAoFD -- Page 10

11 The estimate of is the diagonal matrix of residuals variances > n <- nrow(ffs1) >(h3 <- diag(diag((n-1)/(n-4)* cov(resid(ffs1.fit))))) [,1] [,2] [,3] [1,] [2,] [3,] Note: Devide by (n-4) because we estimated 4 parameters in the factor model Hence, the estimation of β T F β + is > t(h2) %*% h1 %*% h2 + h3 GEf IBMf Mobilf GEf IBMf Mobilf For comparison: the sample covariance matrix R is > cov(ffs1[,c("gef","ibmf","mobilf")]) GEf IBMf Mobilf GEf IBMf Mobilf Largest difference is in the covariance between GEf and Mobilf. Reason: factor model assumes a zero residual correlation, but the data show a correlation of (cf. Slide 9) Note the different usage of regression calculus here compared to that in a standard regression course. A. Ruckstuhl -- WBL 2019, Lecture 4 of SAoFD -- Page 11

12 7.4 Cross-Sectional Fundamental Factor Model Time series factor model do not make use of variables such as dividend yields, book-to-market value, or other variables specific to the kth firm. An alternative is a cross-sectional factor model, which is a regression model using data from many assets but from only a single holding period. Example: Suppose that R k, (B/M) and D k k are the return, book-to-market value and dividend yield for the k the asset at some fixed time t cross-sectional factor model is R k = (B/M) k + 2 D k + ε k The parameter 0, 1 and 2 are unknown values (at each time point t) Hence, in contrast to time series factor models 1 and 2, which are unknown, depend on time they play the role of factors The variables (B/M) k and D k are directly measured they play the role of loadings A. Ruckstuhl -- WBL 2019, Lecture 4 of SAoFD -- Page 12

13 7.5 Statistical Factor Analysis Let R t be a vector of returns from d assets. Let s assume that p latent factors F t, called risk factors, explain the returns up to additive constants and some errors: where is a p x d matrix of unknown loadings is the d dimensional vector of additive constants is the d dimensional vector of unexplained returns (= error ). (*) A. Ruckstuhl -- WBL 2019, Lecture 4 of SAoFD -- Page 13

14 Example: Allocate capital to the equity markets of ten different countries In R: > (R.fa <- factanal(r, factors=3, rotation="none")) Uniquenesses: HongKong Singapore Brazil Argentina UK Germany Canada France Japan USA Loadings: Factor1 Factor2 Factor3 HongKong Singapore Brazil Argentina UK Germany Canada France Japan USA Factor1 Factor2 Factor3 SS loadings Proportion Var Cumulative Var Estimates of β T Test of the hypothesis that 3 factors are sufficient. The chi square statistic is 13.9 on 18 degrees of freedom. The p-value is Estimates of the diagonal of Test whether «approximation is okay» A. Ruckstuhl -- WBL 2019, Lecture 4 of SAoFD -- Page 14

15 Comments on the factor loadings Cutoff is 0.1; i.e. absolute values smaller than 0.1 are indicated by a blank Factor1: Because all its loadings have the same sign, the first factor is an overall index of the 10 countries Factor2: The second factor has negative loadings on Germany and France (the continental European countries), is neutral to UK, Canada and Japan, and has positive loadings on all other countries. Hence it contrasts the continental European countries with the positive group of countries. A. Ruckstuhl -- WBL 2019, Lecture 4 of SAoFD -- Page 15

16 Remarks: NOTE! Since factanal standardizes the variables, β T β + approximates the correlation matrix of R t Let D be the diagonal matrix with the standard deviation of each asset on its diagonal. Then D (β T β + ε ) D approximates the covariance matrix R of R t The latent factors F t are used to model the correlation between the assets factanal would allow to rotate the β which, however, would not change F = β T β. Hence, we leave it. Estimation of R 65 parameters are estimated (=10 (location) + 10 (variances)+10*9/2 (covariances)) Estimation by approximation: 53 (minus constraints) parameters are estimated (= 10 (add. Constant) + 3 (loadings) + 3*10 (Factors) + 10 (diagonal of )) A. Ruckstuhl -- WBL 2019, Lecture 4 of SAoFD -- Page 16

17 Example: Allocate capital to the equity markets of ten different countries Bootstrapping the estimated Sharpe s ratio of the tangency portfolio using the factor analysis approximation (B=250) In R: > EM.TPboot <- BootTP(Dat=100*R, mufree=1/24, nboot=250, NoFAC=3) > boxplot(em.tpboot[,1:2], las=1, ylab="sharpe's ratio") > abline(h=em.sol$tpsharpe, lty=2, lwd=2, col="gray") - «3 factor approx» is as good as sample cov - «1 factor approx» is slightly worse, and would not pass the test on slide 6 as well A. Ruckstuhl -- WBL 2019, Lecture 4 of SAoFD -- Page 17

18 Comment on Principal Component Analysis (PCA): PCA aims to find low-dimensional subspaces containing most of the variation in the data That is, the remaining components will not contain a lot of variation... but may still contain correlation between the assets Hence, PCA cannot be used as a factor model in finance But still as an exploratory tool to gain insights into the return data as does a scatterplot matrix In R: > pairs(r) # see next slide A. Ruckstuhl -- WBL 2019, Lecture 4 of SAoFD -- Page 18

19 Scatterplot matrix A. Ruckstuhl -- WBL 2019, Lecture 4 of SAoFD -- Page 19

20 7.6* Factor Models in Return-Based Style Analysis In return-based style analysis, the factors are returns on representative asset classes, which are usually assimilated to market indices. Sharpe (1988, 1992) Example: Fund of hedge-funds (FoHF) - A less transparent financial instrument Many FoHF will only disclose partial information on their underlying portfolio. The style of FoHF and their underlying investment strategy, which is the crucial characterization of FoHF, is self-declared Return-based style analysis is an instrument to provide a unprejudiced insight into FoHF. Lhabitant (2001) introduced the idea: Use a multifactor model, where the factors are indices of hedge-fund sub-styles This is a very different use of a factor model than in the previous Sections: Not the risk is of its explicit focus but rather the composition (=factors) A. Ruckstuhl -- WBL 2019, Lecture 4 of SAoFD -- Page 20

21 Example from the block Robust Regression Hedge-Fund Sub-Indices of Different Providers (CSFB/T, HFR, EDHEC) CSFB/T: Credit Suisse First Boston / Tremont, nowadays Credit Suisse Hedge Fund Index ( HFR : Hedge Fund Research ( EDEC : EDEC-Risk Institute ( A. Ruckstuhl -- WBL 2019, Lecture 4 of SAoFD -- Page 21

22 Residuals using robust fitting methods ( lmrob in robustbase): FoHF A FoHF B residuals residuals homogeneous shift in the investment strategy? A. Ruckstuhl -- WBL 2019, Lecture 4 of SAoFD -- Page 22

23 Constraints on the β s From an economic perspective, it may seem natural and logical to add constraints on the β s: The β s should not be negative since the short selling strategy is captured by a sub-index FoHF only hold long positions of single hedge funds hence we do not allow short sales on any sub-index The β s should add up to 1 (100%) if the FoHF is invested in hedge funds solely and in reality? If the constraints are enforced in the fitting process a standard residual analysis may not be sensible standard regression inference results cannot be applied A. Ruckstuhl -- WBL 2019, Lecture 4 of SAoFD -- Page 23

24 Constraints on the β s example using the FoHF dataset > round(coef(lm(fohf ~. -1, data=fohf, subset=-(1:28))),2) RV CA FIA EMN ED EDD EDRA LSE GM EM SS > require(quadprog) > X2 <- data.matrix(fohf[-(1:28),-1]) > Y2 <- FoHF$FoHF[-(1:28)] > Dmat2 <- crossprod(x2) > dvec2 <- as.vector(t(as.matrix(y2)) %*% X2) > bvec <- c(1, rep(0, p)) > Amat <- rbind(c(rep(1, p)), diag(p)) > sol2 <- solve.qp(dmat=dmat2, dvec=dvec2, Amat=t(Amat), bvec=bvec, meq=1) > names(sol2$solution) <- colnames(x2) > round(sol2$solution,2) RV CA FIA EMN ED EDD EDRA LSE GM EM SS > sum(sol2$solution) [1] 1 A. Ruckstuhl -- WBL 2019, Lecture 4 of SAoFD -- Page 24

25 But a HoHF may not be invested in HF completely (i.e, sum(β s) < 1) or may be leveraged (i.e, sum(β s) > 1). LASSO (least absolute shrinkage and selection operator) is a regression analysis method that performs both variable selection and regularization in order to enhance the prediction accuracy and interpretability of the statistical model it produces. Example: creinvest AG, May 2000 until December 2005 (b=sum(β s)) CSFB/T (b = 0.91 ) creinvest EDHEC (b = 0.95 ) HFR (b = 1 ) MF SS CA FIA MF SS RV CA FIA SS RV CA FIA EM EMN EM EMN EM EMN GM ED GM ED GM ED LSE EDRA EDD LSE EDRA EDD LSE EDRA EDD The result are different because the index providers use different data bases and different weighting schemes to calculate the sub-indices. A. Ruckstuhl -- WBL 2019, Lecture 4 of SAoFD -- Page 25

26 7.7* Our 17 Factor Model Our extension of Fama & French Three-Factor Model resulted in a 17-Factor Model which is based on 9 factors which should cover the classical investment market. They are actual market indices and 8 factors which should cover the alternative investment market. They are derived from actual market indices Manz, Meier, Ruckstuhl, Stoz, and Weibel, The New 17-Factor Model, Research Report ZHAW, January, 2012 A. Ruckstuhl -- WBL 2019, Lecture 4 of SAoFD -- Page 26

27 Does an actively managed fund (like Fund of Hedge Fund - FoHF) pay off its higher fee? Separate the contribution of the market (=loadings) from that of a single Fund of Hedge Fund (FoHF) manager (=intercept) to the returns since a single FoHF does not need to be invested in every sector, the FoHF may be adequately described just through some of the market factors. Since we are calibrating the factor model for the last 48 months, sometimes more than half of the factors are not statistically significant. This calls for variable selection (i.e., so-called "hard thresholding") Our focus in calibrating the factor model lies in the parameters (especially in the intercept) and its behaviour over time. The least-squares fit and the selection of the factors did not yield stable results due to too many parameters and too short adjustment period, respectively. Way out: A gradual selection of factors based on so-called "'soft thresholding"' methods - Our choice fell on L2 Boosting A. Ruckstuhl -- WBL 2019, Lecture 4 of SAoFD -- Page 27

28 L2 Boosting in our context means: Instead of directly solving the least-squares problem an iterative gradient descent procedure is used Instead of fully iterating, the procedure is stopped earlier A single iteration step consists of fitting the best single regression model (with one of the factors) to the residuals of the updated boosted estimator of the previous step cf., i.e., Bühlmann and Hothorn (2007). Boosting algorithms: Regularization, prediction and model fitting (with discussion). Statistical Science 22, Example: RMF Equity Hedged Strategies (a FoHF) Intercept (=alpha) over time (Jan 2004 to Dec 2008) of a FoHF. The boxplots shows the variability of the alpha estimation over the FoHF universe on hedgegate. Philipp, Ruckstuhl, Manz, Dettling, Dürr and Meier (2009). Performance Rating of Funds of Hedge Funds, Research Report ZHAW A. Ruckstuhl -- WBL 2019, Lecture 4 of SAoFD -- Page 28

29 Take Home Message Lecture 4 Financial factor analysis explains returns with a small number of fundamental variables called factors or risk factor Factor models are used to construct estimators for the covariance matrix ( portfolio optimization) Identify return and risk sources, i.e., Perform style analysis of a portfolio Separating the contribution of the market (=loadings) from that of an active manager If the factors are investable, then you can mimic a less transparent financial instrument by a portfolio of better known instruments Factor models are fitted by Time series regression methods (LS, robust, constrained LS, LASSO, gradient boosting, ) Cross-sectional regression methods (LS, robust, ) Statistical factor analysis A. Ruckstuhl -- WBL 2019, Lecture 4 of SAoFD -- Page 29

30 ᄃ Lecture 4 in the books: Chapter 18 Subsec 4.2.1, Chapter 18 Sections marked by * are not examined A. Ruckstuhl -- WBL 2019, Lecture 4 of SAoFD -- Page 30