Financial Econometrics and Quantitative Risk Managenent Return Properties

Financial Econometrics and Quantitative Risk Managenent Return Properties Eric Zivot Updated: April 1, 2013

Lecture Outline Course introduction Return definitions Empirical properties of returns

Reading FRF chapter 1 FMUND chapter 1 and chapter 2 SDAFE chapter 2 and chapter 4

Discrete Returns Simple Net Return = 1 1 =% Gross Return 1+ = 1

2 Period Return (2) = 2 2 = 2 1 (2) = 1 1 1 2 = (1+ )(1 + 1 ) 1 Period Return 1+ ( ) = (1+ )(1 + 1 ) (1 + +1 ) = 1 Y =0 (1 + )

Adjusting for Dividends (Total Returns) = + 1 1 = 1 1 + 1 Adjusting for Inflation (Real Returns) 1+ Real = 1 1

Portfolio Return = = X =1 X =1 X =1 =1 Excess Returns = = T-bill rate or LIBOR rate

Continuously Compounded Returns Ã! = ln(1+ )=ln = ln( ) ln( 1 ) = 1 = 1+ = 1 = = 1 1 Note: = 1

2 period return (2) = ln(1 + (2)) = ln = ln = ln à 1 1 à 1 = + 1! Ã! 2! 2 Ã! 1 +ln 2 = 2

period return ( ) = ln(1+ ( )) = ln = 1 X =0 Ã! =

Adjusting for Dividends (Total Returns) = ln(1+ )=ln à + 1 = ln( + ) ln( 1 )! Adjusting for Inflation (Real Returns) Real = ln(1+ Real )=ln = Ã! 1 1

Portfolio Return But = ln(1+ ) = ln 1+ 6= X =1 X =1 X =1 if is small

Excess Returns = = ln( )=ln( ) 6= But if is small then

Stylized Facts of Asset Return Distributions Fat tails Asymmetry Aggregated normality Absence of serial correlation Volatility clustering Time-varying cross correlation

Shape Characteristics Let be a random variable with pdf = [ ] : center 2 = var( ) = [( ) 2 ]: spread " # ( ) 3 skew( ) = : symmetry kurt( ) = 3 " ( ) 4 4 # : tail thickness Note: The moment and central moment of is 0 = [ ] = [( ) ]

Normal Distribution ( 2 ) ( ) = Ã! 1 2 2 exp ( )2 2 2 [ ] = var( ) = 2 skew( ) = 0 kurt( ) = 3 = 0 for odd

Sample moments Let { } denote a random sample of size where is a realization of the random variable ˆ = 1 X =1 skew d = ˆ 3 ˆ 3 ˆ = 1 1 ˆ 2 = 1 1 kurt d = ˆ 4 ˆ 3 X =1 ( ˆ ) X =1 ( ˆ ) 2 =ˆ 2 Note: we divide by 1 to get unbiased estimates. Check software to see how moments are computed.

Testing for Normality QQ-plot: plot standardized empirical quantiles vs. theoretical quantiles from specified distribution. Note: Shapiro-Wilks (SW) test for normality: correlation coefficient between values used in QQ-plot Jarque-Bera (JB) test for normality à JB = skew d 2 + ( kurt d 3) 2 6 4 2 (2) Note: if ( 2 ) then d skew (0 6) ( d kurt 3) (0 24)!

Kolmogorov-Smirnov (KS) test compares the empirical CDF of returns with the CDF of the normal distribution (or any other assumed distribution) Sort returns: (1) ( ) and compute empirical CDF ˆ ( ( ) )= Evaluate normal CDF: Φ µ ( ) ˆ ˆ Compute KS statistic: =sup Φ µ ( ) ˆ ˆ

Student s-t distribution Let (0 1), 2 ( ) such that and are independent. Then = q where denotes a (standardized) Student s t distribution with degrees of freedom. Note: [ ] = 0 var( ) = 2 2 skew = 0 kurt 3= 6 4 4 Existence of moments depends on degrees of freedom (df) parameter Cauchy = Student s-t with 1 df. Only density exists.

If then has moments = + q ( 2) [ ]= var( )= 2

Density function ( ; ) = Γ( ) = " Γ{( +1) 2 ( ) 1 2 Γ( 2) Z 0 # 1 {1+( 2 )} ( +1) 2 1 exp( ) = gamma function The d.f. parameter can be estimated by MLE. Note: A simple method of moments estimator for isbasedonkurtosis: kurt 3= 6 4 =6 (kurt 3) + 4

Skew Normal Distribution Azzalini and Capitanio (2002) define ( ) as a skew-normal random variable with density ( ) = 2 ( )Φ( 1 ( )) Ã! 2 Z ( ) = (2 ) 1 2 exp 2 Φ( ) = ( ) = location parameter, = scale parameter, 0 = shape (skew) parameter, Note: Estimation and simulation functions in R package sn

Remarks =0 ( 2 ) 0 positive skewness 0 negative skewness

Skew-t Distribution Azzalini and Capitanio (2002) define ( ) as a skew-t random variable using the transformation = + 1 2 ( ) 2 ( ) The parameters and have the same interpretation as in the skew-normal and = degrees of freedom parameter, 0 Note: Estimation and simulation functions in R package sn

Defn: The stochastic process { } is covariance stationary if [ ] = for all ( ) = [( )( )] = for all and any The parameter is called the order or lag j autocovariance of { } The autocorrelations of { } are defined by = ( ) q ( ) ( ) = 0 and a plot of against is called the autocorrelation function (ACF)

The lag sample autocovariance and lag sample autocorrelation are defined as ˆ = 1 ˆ = ˆ ˆ 0 X = +1 where = 1 P =1 isthesamplemean. ( )( ) ThesampleACF(SACF)isaplotofˆ against.

Example: White noise (GWN) processes Perhaps the most simple stationary time series is the independent Gaussian white noise process { } (0 2 ) (0 2 ). Thisprocesshas = = =0( 6= 0). Two slightly more general processes are the independent white noise (IWN) process, { } (0 2 ),andthewhite noise (WN) process, { } (0 2 ). Both processes have mean zero and variance 2, but the IWN process has independent increments, whereas the WN process has uncorrelated increments.

The SACF is typically shown with 95% confidence limits about zero. These limitsarebasedontheresultthatif{ } (0 2 ) then µ ˆ 0 1 0 The notation ˆ ³ 0 1 means that the distribution of ˆ is approximated by normal distribution with mean 0 and variance 1 and is based on the central limit theorem result ˆ (0 1). The 95% limits about zero are then ± 1 96.

Testing for White Noise Consider testing the null hypothesis 0 : { } (0 2 ) Under the null, all of the autocorrelations for 0 are zero. To test this null, Box and Pierce (1970) suggested the Q-statistic ( ) = X ˆ 2 =1 Under the null, ( ) is asymptotically distributed 2 ( ). In a finite sample, the Q-statistic may not be well approximated by the 2 ( ). Ljung and Box (1978) suggested the modified Q-statistic ( ) = ( +2) X =1 ˆ 2