Chapter 15 the equity market (cross-section patterns and time-series patterns) Fan Longzhen Introduction ? In this class, we again look at the stock return data, but with a very different view point; ? Previously, we examined the data through the “eyes” of CAPM. We had a noble intension, although it didn’t work very well; ? Now we are going to get our hands “dirty”, and plunge right into the data, without a formal model; ? In particular, we will look at some well-established patterns---size,value, and momentum—that have been successful in explaining the cross-sectional stock returns. Cross-section vs time-series ? For a publicly traded firm i, the following information can be readily obtained: ? ---the stock price at any time t; ? ---the cash dividend paid between t-1 and t; ? At any time t, we can calculate the realized stock return for firm i: ? ---percentage returns: ? ---log-returns: ? Cross-section of stock returns: ? Time series of stock returns: i t P i t D 1? i t i t i t i t i t P PDP R 1 1 ? ? ?+ = i t i t i t i t PDPr 1 ln)ln( ? ?+= NiR i t ,...,2,1, = Ttr i t ,...,2,1, = Multifactor-regressions ? For each asset i, we use a multi-factor time-series regression to quantify the asset’s tendency to move with multiple risk factors: ? 1. Systematic factors: ?: risk premium ?：risk premium ? 2 idiosyncratic factors: ? : no risk premium ? 3. Factor loadings: ? beta(i): sensitivity to market risk; ? : sensitivity to the factor risk. i tti f t M tii f t i t eFfrrrr ++?+=? )(βα M t r )( f t M t M rrE ?=λ t F )( t F FE=λ i t e 0)( = i t eE i f The pricing relation ? Given the risk premia of the systematic factors, the determinants of expected returns: ? What are the additional systematic factors? F i M i f t i t frrE λλβ +=? )( Size: small or big ? We sort the socks by their market capitalization: share price* number of shares outstanding Value or growth ? We can sort the cross-section of stocks by their book-to-market ratios: growth stocks:firm with low book-to-market ratios; ? Value stocks:firms with high book-to-market ratios. Other factors ? Price-to-earining ratios, ? The market skewness fcator ? ---Havey and Siddique(JF,2000) report that systematic skewness is economically important and commands an average risk premium of 3.6% per year. Time-series behavior ? For the time-series behavior of stock returns, we focus in particular on the time-varying nature of expected return and volatility; ? We are interested in building dynamic models that explicitly incorporate conditioning information to best describe the behavior of future stock returns. Conditional information ? Let Y be the price of a single family house, let X be the zipcode; ? 450000 35000 X=450052 X=200433 65000 )())(( YEXYEE = Building a dynamic model for stock prices ? Let be the log of stock price, we build a model for stock prices simply ? We simply say stock has a constant expected return and a constant variance, is unpredictable at time time t; ? In this model stock prices are always positive ? Let be the collection of information at time t, with this information , stock returns and prices next period are not predictable ? This model is called random work model. ? How can we test it? tt Pp ln= 0)(, 111 =+=? +++ ttttt Ewithpp εσεμ 1+t ε 1 1 + + + == tt ePePP t R tt σεμ t I μ= + )( 1 tt IRE Test the random walk ? One version random walk model hypothesis was tested by Professor Andrew Lo in the late 1980’s, the paper is titled “stock market prices do not follow random walks:evidence from a simple test. ? Autocorrelation: autocorrelation is a important concept in time-series analysis, it measures the persistence of the time-series data: ? The sample autocorrelation : ? What is the implication of the random walk hypothesis on autocorrelation? ),( 1+ = tt RRcorrρ ∑∑ ???= + t t t tt mR N mRmR N 2 1 )?( 1 /)?)(?( 1 ? ρ Sample autocorrelation results ? Weekly returns from 1962 to 1987 0.08-0.03 Individual shocks 6.4 1.9 0.05 0.04 0.30 0.07 Equal- weighted Value weighted T-statS.E. ρ ? Weak negative autocorrelation of individual returns and strong positive autocorrelation of portfolio returns? An auto regression model ? An auto regression model explicitly incorporates the positive autocorrelation ? In financial time series analysis, this model is called AR(1). ? 1. The conditional expected return ? 2. The conditional volatility ? 3. The autocorrelation coefficient ? 4 what are about the unconditional expected return and unconditional volatility? 11 ++ ++= ttt RaR σερ tttt RRRE ραμ +== + )( 1 σσ == + )var( 1 ttt RR ρ= + ),( 1 tt RRcorr Time-varying ? More generally, we can build a dynamic model with time-varying expected return and volatility ? Given the public information at time t, ? 1. : conditional expectation is our best guess of the return ? 2. : conditional volatility:our best guess of how volatile would be. ? 3. Is unpredictable at time t: ? σμ, 11 ++ += tttt R εσμ )( 1 ttt IRE + =μ 1+t R )( 1 ttt IRstd + =σ 1+t R 1+t ε 1)()( 0)()( 2 1 2 1 11 == == ++ ++ ttt ttt EIE EIE εε εε Predictability and market efficiency ? In addition to the random walk test, there is mounting evidence that stock returns are predictable; ? Some argue that predictability implies market inefficiency. What do you think? ? Others contend it is simple a result of rational variation in expected returns; ? Suppose this is true. Can we find a coherent story that relates the variation through time of expected returns to business conditions What cause expected returns to vary? ? Using the intuition of CAPM, expected returns can vary for two reasons: ? 1. Varying risk aversion; ? 2. Varying exposure to market risk, or varying market risk. ? When income is high, investor want save more, higher saving lead to lower expected returns; ? Empirical implication? Variables related to business condition ? Default spreads: difference in yields between defaultable bonds and treasury bonds with similar maturities. When the business condition is bad, the systematic default risk increases, widening the default spread. ? Term premiums: difference in yields between long- and short-term treasury bonds. This is a forward-looking variable predictive of future inflation, and is found to be important in forecasting real economic activity. ? Financial ratios: book-to-market, dividend yields, ect. Variables that are important in fundamental valuation Empirical findings ? Kothrori and Shanken, “Book-to-market, dividend yield, and expected market returns: a time-series analysis”,Journal of financial Economics, 1997 ttttttt BtMbDYLTbTERMbDEFbaIRE 43211 )( ++++== + μ Time-varying volatility--- volatility also changes with time ? If the rate of information arrival is time-varying, so is the rate of price adjustment, causing the volatility to change over time; ? The time-varying volatility of the market return is related to the time- varying volatility of a variety of economic variables, including inflation, money growth, and industrial production; ? Stock market volatility increases with financial leverage: a decrease in stock price causes an increase in financial leverage, cause volatility to increase; ? Investors’s sudden changes of risk attitudes, changes in market liquidity, and temporary imbalance of supply and demand could all cause market volatility to change over time. ? What are the testable empirical implications? The ARCH and GARCH models ? To focus on the time-varying volatility , let us assume constant expected return: ? Where is our best guess of tommorrow’s market volatility using all of the information available to us today. ? The ARCH model, was proposed by Rob Engle in 1982. The GARCH is a generalized version of ARCH: t σ 11 ++ += ttt R εσμ )var( 1 ttt IR + =σ 2 12 2 10 2 )( ? +?+= ttt aRaa σμσ Stochastic volatility ? More generally, volatility is a stochastic process of its own, this is an active area of research in academics and in industry. ? Some well-known facts about stochastic volatility: ? 1. It is persistent: volatile periods are followed by volatile periods; ? 2. It is mean-reverting: over time, volatility converges to its long-run mean. ? 3. There is a negative correlation between volatility and return: large negative price jumps are coupled with large positive volatility jumps.