Welcome to the course Financial Econometrics II

2 Course objectives Financial data analysis, based on Asset pricing models Econometric methods Active knowledge Try it yourself! Share personal vision Avoid common pitfalls

3 What you already know: Asset pricing models Risk-return trade-off Only systematic risk is priced! Idiosyncratic risk is reduced by diversification What are the risk factors? CAPM: market APT: many other

4 What you already know: Econometric methods Regression Coefficients Standard errors R-squared Potential problems Outliers, heteroscedasticity, endogeneneity, etc. Packages Eviews, Excel

5 Plan for today Specifics of financial data The efficient market hypothesis Tests for return predictability

6 Asset prices: S&P500

7 Asset prices Convenient for graphical analysis, but.. Non-stationary Properties of the stochastic process change over time Must be corrected for dividends, stock splits Should be normalized to compare dynamics over time and across securities

8 Asset returns: S&P500 (monthly)

9 Asset returns: discrete Net return: R t = (P t -P t-1 )/P t-1 Gross return:1+ R t Adjusting for dividends (total returns): R t = (P t +D t -P t-1 )/P t-1 Adjusting for inflation (real returns): 1+R t (real) = (P t /P t-1 )*(CPI t-1 /CPI t ) Excess returns (risk premia): Z t = R t – R F,t

10 Asset returns: discrete k-period return:1+ R t (k) = (1+ R t )*…*( 1+ R t-k+1 ) Annualized return Geometric average (effective return): (1+ R A ) k = 1+ R t (k) Arithmetic average: R A = (1/k) j=0:k-1 R t-j Portfolio return: R P,t = i w i R i,t w i : portfolio weights summing up to 1

11 Asset returns: continuously compounded Log return: r t = ln(1+ R t ) = ln(P t /P t-1 ) = ln(P t ) – ln(P t-1 ) k-period return:r t (k) = ln(P t /P t-k ) = j=0:k-1 r t-j Annualized return:r A = j=0:k-1 r t-j Portfolio return: r P,t = ln(1+ R P,t ) i w i r i,t But r P,t i w i r i,t for small R P,t

12 Asset returns: simple vs. log Simple returns: Exact Easy to aggregate stocks into portfolio Log returns: Easy to aggregate over time Closer to normal distribution –Don't violate limited liability

13 Market microstructure effects Which prices to use to measure returns? Average vs. close Bid vs. ask Can you make real profit out of paper returns? Transaction costs, including bid-ask spread Liquidity Price impact

What do we know about stock returns?

15 Characteristics of the distribution Mean (center): μ = E[R] Variance (spread): 2 = E[(R-μ) 2 ] Skewness (symmetry):S = E[(R-μ) 3 / 3 ] Kurtosis (tail thickness):K = E[(R-μ) 4 / 4 ]

16 Normal distribution X ~ N(μ, 2 ) S = 0 K = 3 QQ-plot: standardized empirical quantiles vs. theoretical quantiles from specified distribution Jarque-Berra test for normality: n : # observations JB has an asymptotic chi-square distribution with 2 degrees of freedom

17 Histogram of S&P500 monthly returns

18 Normal QQ-plot for S&P500 monthly returns

19 S&P500: daily returns

20 Normal QQ-plot for S&P500 daily returns

21 Stylized facts about stock returns Monthly returns Appoximately normal Daily returns Non-normality –Asymmetry: usually, S < 0 for the indices –Thick tails: K > 3 Volatility сlustering –Esp. at daily frequency

22 How can we model stock returns? Traditional approach: Explain asset prices by rational models Only if they fail, resort to irrational investor behavior Behavioral finance models

23 What is an efficient market? Informational efficiency (Fama, 1969): asset prices accommodate all relevant information History of prices / all public variables / all private information Price movements must be random! Otherwise one can forecast future price and make arbitrage profit Prices should immediately respond to new information Why is it important? Stock markets: portfolio management Corporate finance: choice of the capital structure

24 The efficient market hypothesis EMH: stock prices fully and correctly reflect all relevant information P t+1 = E[P t+1 |I t ] + ε t+1 R t+1 = E[R t+1 |I t ] + e t+1 The error has zero expectation and is orthogonal to I t E[R t+1 |I t ] is normal return or opportunity cost implied by some model

25 Different forms of ME Weak (WFE): I includes past prices Semi-strong (SSFE): I includes all public info Strong (SFE): I includes all (also private) info

26 What if the EMH is rejected? The joint hypothesis problem: we simultaneously test market efficiency and the model Either the investors behave irrationally, or the model is wrong One may not necessarily earn on this Ex-ante expected profit is within information acquisition and transaction costs

27 If the EMH is not rejected, then… the underlying model is a good description of the market, fluctuations around the expected price are unforecastable, due to randomly arriving news there is no place for active ptf management… technical analysis (WFE), fundamental analysis (SSFE), or insider trading (SFE) are useless the role of analysts limited to diversification, minimizing taxes and transaction costs or corporate policy: the choice of capital structure has no impact on the firms value still need to correct market imperfections (agency problem, taxes)

28 How realistic is it? The Grossman-Stiglitz paradox: there must be some strong-form inefficiency left to provide incentives for information acquisition Operational efficiency: one cannot make profit on the basis of info, accounting for transaction costs

29 Different types of tests Tests of informational efficiency: Finding variables predicting future returns Statistical significance Tests of operational efficiency: Finding trading rules earning positive profit taking into account transaction costs and risks Economic significance

30 Different types of models Constant expected return: E t [R t+1 ] = μ Tests for return predictability CAPM: E t [R i,t+1 ] – R F = β i (E t [R M,t+1 ] – R F ) Tests for mean-variance efficiency Multi-factor models

31 Topics covered in this course Time series analysis of asset returns Predictability at different horizons Event study analysis –Speed of stock price adjustment in response to news announcements Cross-sectional analysis of asset returns CAPM and market efficiency Return anomalies and multi-factor asset pricing models Performance evaluation of mutual funds Performance persistence, survivorship bias, dynamic strategies, gaming behavior, etc.

How can we model stock returns?

33 Is this market efficient / is price predictable?

34 Martingale and MDS Let {Y t } be a sequence of random variables Let {I t } be a sequence of info sets, I t I (universal info set) (Y t, I t ) is a martingale if I t I t+1 (filtration) Y t I t (Y t is adapted to I t ) E[|Y t |]< E[Y t |I t-1 ] = Y t-1 (martingale property) Example: random walk Y t = Y t-1 + ε t, ε t ~ IID(0, σ 2 ) I t = {ε t, ε t-1, …}

35 Martingale and MDS Law of iterated expectations: E[Y t |I t-2 ] = E[E[Y t |I t-1 ] |I t-2 ] = E[Y t-1 |I t-2 ] = Y t-2 In general, E[Y t |I t-k ] = Y t-k (X t, I t ) is a martingale difference sequence (MDS) if E[X t |I t-1 ] = 0 Construct MDS from a martingale: X t = Y t - E[Y t |I t-1 ] Can we apply it to stock prices/returns? Under EMH, stock prices are martingales after detrending the unexpected stock returns are MDS: E[R t+1 - E[R t+1 |I t ] |I t ] = 0

36 The random walk hypotheses Random walk with drift: Δln(P t ) = μ + ε t or r t = μ + ε t RW1: independent and identically distributed increments, ε t ~IID(0, σ 2 ) Any functions of the increments are uncorrelated Unrealistic RW2: independent increments Allows for heteroskedasticity Test using filter rules, technical analysis RW3: uncorrelated increments, cov(ε t, ε t-k ) = 0, k>0 Allows for dependence on higher moments (e.g., GARCH) Test using autocorrelations, variance ratios, regressions

37 Autocorrelation tests Assume that r t is covariance stationary Test that autocorrelation is 0 For a given lag: Fullers test Asymptotically, T k ~ N(0,1) For m lags: Box-Pierce statistic: Q T Σ k=1:m ρ 2 (k) ~ 2 (m) Ljung-Box statistic (finite-sample correction): Q T(T+2) Σ k=1:m ρ 2 (k)/(T-k) ~ 2 (m)

38 Autocorrelation tests: results CLM, Table 2.4: US, The CRSP stock index has significantly positive first-order autocorrelation at D, W, and M frequency Economic significance: 12% (R 2 = square of 1 ) of the variation in daily value-wtd CRSP index is predictable from the last-day return The equal-wtd index has higher autocorrelation Predictability declines over time

39 Variance ratios Intuition: how to aggregate volatility over time? Usually, T rule for standard deviation Example: compute the variance of a 2-day return: var(r t + r t-1 ) = var(r t ) + var(r t-1 ) + 2 cov(r t, r t-1 ) = {assuming covariance stationarity and no autocorrelation}= 2 var(r t ) Variance ratio: VR(2) var(r t (2)) / 2var(r t ) = In general, VR(q)Var[r t (q)]/(qVar[r t ]) = k=1:q-1 (1-k/q) k Under RW1, VR=1 Test statistic:

40 Variance ratios: results CLM, Tables 2.5, 2.6, 2.7: US, , weekly data Indices: VR(q) rises with time interval (positive autocorrelation), predictability declines over time, is larger for small-caps Individual stocks: weak negative autocorrelation How to reconcile this contradiction? Table 2.8: size-sorted portfolios Large positive cross-autocorrelations, large-cap stocks lead small-caps

41 Regression analysis: ARMA models Testing for long-horizon predictability:, R t+h (h)=a+bR t (h)+u t+h, Non-overlapping horizons: too few observations Overlapping horizons: autocorrelation ρ(k)=h-k => use HAC s.e. Results from Fama&French (1988): US, Negative autocorrelation (mean reversion) for horizons from 2 to 7 years, peak b=-0.5 for 5y Poterba&Summers (1988): similar results based on VR Critique: Small-sample and bias adjustments lower the significance Results are sensitive to the sample period, largely due to (the Great Depression)

42 Interpretation: Behavioral finance Investor overreaction Assume RW with drift, E t [R t+1 ] = μ There is a positive shock at time τ The positive feedback (irrational) traders buying for t=[τ+1:τ+h] after observing R τ >μ SR (up to τ+h): positive autocorrelation, prices overreact LR (after τ+h): negative autocorrelation, prices get back to normal level Volatility increases

43 Interpretation: Market microstructure Non-synchronous trading Low liquidity of some stocks (assuming zero returns for days with no trades) induces negative autocorrelation (and higher volatility) for them positive autocorrelation (and lower volatility) for indices lead-lag cross-autocorrelations Consistent with the observed picture (small stocks are less liquid), but cannot fully explain the magnitude of the autocorrelations

44 Interpretation: More complicated model Time-varying expected returns: E t [R t+1 ] = E t [R F,t+1 ] + E t [RiskPremium t+1 ] Changing preferences / risk-free rate / risk premium Decline in interest rate => increase in prices If temporary, then positive autocorrelation in SR, mean reversion in LR

45 Conclusions Reliable evidence of return predictability at short horizon Mostly among small stocks, which are characterized by low liquidity and high trading costs Weak evidence of return predictability at long horizon May be related to business cycles (i.e., time-varying returns and variances)