Financial Returns: Stylized Features and Statistical Models Qiwei Yao Department of Statistics London School of Economics q.yao@lse.ac.uk p.1
Definitions of returns Empirical evidence: daily prices in Jan 1985 Feb 2011 of S&P500 index Apple Inc stock Efficient markets hypothesis Statistical models for returns p.2
Simple returns and gross returns Let P t be the price of an asset at time t. One-period simple (net) return: R t = (P t P t 1 )/P t 1. Often we use: 100R t, as R t = 100R t % One period gross return: P t /P t 1 = R t +1 k-period simple (net) return: R t (k) = (P t P t k )/P t k k-period gross return: P t /P t k = R t (k)+1. Multiperiod returns expressed in terms of one-period returns: P t P t k = P t P t 1 P t 1 P t 2 Pt k+1 P k k R t (k) = P t P t k 1 = (R t +1)(R t 1 +1) (R t k+1 +1) 1 R t +R t 1 + +R t k+1, provided all R j small p.3
Log returns One period log return: r t = logp t logp t 1 = log(p t /P t 1 ) = log(1+r t ) k period log return: r t (k) = log(p t /P t k ) = r t +r t 1 + +r t k+1 An investment of $A at time t-k yields the capital at time t: Aexp{r t (k)} = Aexp(r t +r t 1 + +r t k+1 ) = Ae k r, where r = 1 k k j=1 r t j+1 is the average one-period log returns. p.4
It holds always that R t r t. However when returns are small, r t = log(p t /P t 1 ) = log ( 1+ P t P t 1 P t 1 ) P t P t 1 P t 1 = R t. log return 0.6 0.2 0.2 Apple Daily return log return 0.6 0.2 0.2 Apple Weekly return log return 0.8 0.4 0.0 0.4 Apple Monthly return 0.4 0.0 0.2 simple return 0.4 0.0 0.2 0.4 simple return 0.6 0.2 0.2 simple return Plots of log returns against simple returns of the Apple Inc share prices in January 1985 February 2011. The blue straight lines mark the positions where the two returns are identical. p.5
Continuously compounding Log return is also called continuously compounded return Example. For a bank deposit account, an interest rate of 5% payable every 6 months will be quoted as a simple interest of 10% per annual. The gross return for 12 months is 1 (1+.05) 1 1 (1+.05) (1+.05) 1 (1+.05) = (1+.05) 2 = 1.1025, i.e. the annual simple return is 1.1025 1 = 10.25% > 10% due to the interest-on-interest in the second 6 months. p.6
Simple interest rate Simple interest rate: r No. of payments per annual: m (at the rate of r/m each time) The gross annual return: (1+r/m) m e r, as m. Two interpretations for interest rate : the annual simple return if the interest is paid once the annual log return if the interest is compounded continuously. p.7
Behavior of financial return data Two data sets: Daily (adjusted) close prices in January 1985 February 2011 of S&P500 index Apple Inc stock Only consider log returns from now on p.8
P500: time series of daily indices, and daily, weekly and monthly return Daily price Daily log return Weekly log return Monthly log return 200 800 1400 0.20 0.05 0.10 0.20 0.05 0.10 0.2 0.0 1985 1988 1991 1994 1997 2000 2003 2006 2009 1985 1988 1991 1994 1997 2000 2003 2006 2009 1985 1988 1991 1994 1997 2000 2003 2006 2009 1985 1988 1991 1994 1997 2000 2003 2006 2009 The index peaked on 24/5/00 during the dot-com bubble, then lost about 50% of its value in 2002. It peaked again on 9/10/07 before suffered in 2008-2010. High volatilities in 08-10 did not show off in monthly returns Volatility clustering? p.9
pple: time series of daily prices, and daily, weekly and monthly returns Daily price 0 100 300 Daily log return 0.6 0.2 0.2 Weekly log return 0.6 0.2 0.2 1985 1988 1991 1994 1997 2000 2003 2006 2009 1985 1988 1991 1994 1997 2000 2003 2006 2009 On 29/9/00, Apple s value sliced in half, due to the earning warning in the last quarter The recent sharp increase were due to successes with ipot, iphone and ipat 1985 1988 1991 1994 1997 2000 2003 2006 2009 Monthly log return 0.8 0.2 0.4 1985 1988 1991 1994 1997 2000 2003 2006 2009 Stationary or not? p.10
S&P500 returns: Normality? Daily returns Weekly returns 0.20 0.05 0.10 0.20 0.05 0.10 Monthly returns 0.2 0.0 0.1 Density 0 10 20 30 40 Density 0 5 10 15 20 Density 0 2 4 6 8 12 Normal Q Q Plot Normal Q Q Plot 4 2 0 2 4 Normal quantile 3 1 1 2 3 Normal quantile Normal Q Q Plot 3 1 1 2 3 Normal quantile p.11 Quantile of daily returns 0.20 0.05 0.05 Quantile of weekly returns 0.20 0.10 0.00 0.10 Quantile of monthly returns 0.2 0.1 0.0 0.1
Apple returns: Normality? Daily returns Weekly returns 0.6 0.2 0.2 0.6 0.2 0.2 Monthly returns 0.8 0.4 0.0 0.4 Density 0 2 4 6 8 Density 0 2 4 6 8 Density 0.0 1.0 2.0 3.0 Normal Q Q Plot Normal Q Q Plot 4 2 0 2 4 Normal quantile 3 1 1 2 3 Normal quantile Normal Q Q Plot 3 1 1 2 3 Normal quantile p.12 Quantile of daily returns 0.6 0.2 0.2 Quantile of weekly returns 0.6 0.2 0.2 Quantile of monthly returns 0.8 0.4 0.0 0.4
S&P500 returns: Serially correlated? Daily returns Weekly returns Monthly returns 0 10 20 30 0 5 10 20 30 0 5 10 15 20 Squared daily returns Squared weekly returns Squared monthly returns 0 10 20 30 0 5 10 20 30 0 5 10 15 20 Absolute daily returns Absolute weekly returns Absolute monthly returns 0 10 20 30 0 5 10 20 30 0 5 10 15 20 p.13
Apple returns: Serially correlated? Daily returns Weekly returns Monthly returns 0 10 20 30 0 5 10 20 30 0 5 10 15 20 Squared daily returns Squared weekly returns Squared monthly returns 0 10 20 30 0 5 10 20 30 0 5 10 15 20 Absolute daily returns Absolute weekly returns Absolute monthly returns 0 10 20 30 0 5 10 20 30 0 5 10 15 20 p.14
Stylized Features of financial returns (i) Stationarity. Asset prices are not stationary, due to e.g. expansion of economy, technology innovation, economic recessions or financial crisis. Returns can be modelled as a stationary process (ii) Heavy tails. Distribution of high-frequency (such as daily) returns r t exhibits heavier (than normal) tails. But it is accepted that E(r 2 t ) <. (iii) Asymmetry. Distribution of high-frequency (such as daily) returns r t is often negatively skewed: financial markets go down much faster! (iv) Volatility clustering. Large price changes (i.e. returns with large absolute values) occur in clusters, and periods of tranquillity alternate with periods of high volatility. p.15
Stylized Features of financial returns (continue) (v) Aggregational Gaussianity When sampling frequency decreases, the CLT sets in. The distribution of the returns over a long time-horizon (such as a month) tends toward a normal distribution. (vi) No serial correlations. The overwhelming evidence to support the claim that financial returns are white noise, i.e. showing no serial correlations. (vii) Long range dependence. Daily squared and absolute returns often exhibit small and significant autocorrelations. Those autocorrelations are persistent for absolute returns; indicating possible long-memory properties. However the serial correlation become weaker and less persistent when the sampling interval increases. p.16
prices are fair Efficient markets hypothesis (EMH) information is accessible for everybody, and is assimilated rapidly to adjust prices people (traders) are rational Hence P t incorporates all relevant info upto time t the change P t+1 P t is only due to the arrival of news between t and t+1 no arbitrage opportunities p.17
Models for returns under EMH Under EMH, a basic model for returns is of the form where r t+1 = µ t +ε t+1, ε t (0, σ 2 t), µ t is the rational expectation of of r t+1 at time t, ε t+1 is unpredictable based on P t, and therefore also on {P t,p t 1, }. Hence no serial correlations! Eε t+1 = 0; on average the actual change of log price equals the expectation. As r t+1 = log(p t+1 /P t ), logp t+1 = µ t +logp t +ε t Hence log prices follow a random walk, provided µ t µ and ε t p.18
Statistical models for returns Most statistical models for returns are of the form where µ = Er t. r t = µ+ε t, ε t WN(0,σ 2 ) Further hypotheses on ε t : martingale difference or i.i.d. ε t is WN: future returns cannot be predicted based on linear relationships ε t is MD: future returns cannot be predicted, but some nonlinear functions of future returns can be predicted (e.g. ARCH, GARCH structures) ε t is i.i.d.: nothing in the future can be predicted. Note i.i.d. martingale difference white noise p.19