Returns and stylized facts

Returns Objective: Use available information to say something about future returns

Frequencies – moments computation Lottery Mean Variance R1R1 R2R2 R n-1 RnRn p1p1 p2p2 P n-1 pnpn TodayTomorrow R – random variable R i - outcomes p i – probabilities of the outcomes

Probability density function We have too many outcomes – not practical. Instead we will use the continuous distribution. Mean Variance Standard Deviation

Probability density function (cont’d) Probability of future outcomes – intervals under the bell shaped curve

Standard Normal Distribution Using Standard Normal Tables Past evolution Future evolution Normal random variable STANDARD Normal Random Variable Probability Density

Measuring Interest Rates The compounding frequency used for an interest rate is the unit of measurement The difference between quarterly and annual compounding is analogous to the difference between miles and kilometers

Continuous Compounding In the limit as we compound more and more frequently we obtain continuously compounded interest rates $100 grows to $ 100e RT when invested at a continuously compounded rate R for time T $100 received at time T discounts to $ 100e -RT at time zero when the continuously compounded discount rate is R

Conversion Formulas Define R c : continuously compounded rate R m : same rate with compounding m times per year

Nonstationary time series

Stationary time series

Stylized facts of asset returns The daily geometric or “log” return on an asset: ln(x) ~ x – 1 when x~1

Stylized facts of asset returns One advantage of the log return is that we can easily calculate the compounded return at the K- day horizon simply as the sum of the daily returns Second advantage: Regular returns are bounded by -1 (we cannot lose more than what we invested)

Distributions of daily returns Day 1 Day 2 Day 3 Day 4

Conditional returns R t+1 is conditional on R t Realizations of R t+1 Realizations of R t

Stylized facts of asset returns Daily returns have very little autocorrelation. We can write It means that returns are almost impossible to predict from their own past. We will take this as evidence that the conditional mean is roughly constant. The unconditional distribution of daily returns has fatter tails than the normal distribution.

Autocorrelation Function for Raw Return Series – BET-C

Fat tails – BET

Stylized facts of asset returns The stock market exhibits occasional, very large drops but not equally large up-moves. It is asymmetric, or negatively skewed. The standard deviation of returns completely dominates the mean of returns at short horizons such as daily. It is not possible to reject a zero mean return. Variance, measured for example by squared returns, displays positive correlation with its own past. This is most evident at short horizons such as daily or weekly.

Autocorrelation Function of the Squared Returns BET-C

Stylized facts of asset returns Equity and equity indices display negative correlation between variance and returns. This is often termed the leverage effect, arising from the fact that a drop in a stock price will increase the leverage of the firm as long as debt stays constant. Correlation between assets appears to be time varying. Correlation appears to increase in highly volatile down markets and extremely so during market crashes.

Stylized facts of asset returns Based on these facts our model of individual asset returns will take the generic form: