2.4 STATISTICAL FOUNDATIONS

Transcription

1 2.4 STATISTICAL FOUNDATIONS Characteristics of Return Distributions Moments of Return Distribution Correlation Standard Deviation & Variance Test for Normality of Distributions Time Series Return Volatility Models 1

2 Characteristics of Return Distributions Ex post Returns Historical returns also called after the fact returns. Ex post Distribution Is a function that assigns relative frequencies to ranges of values of the random variable. Ex ante Returns Future Returns also called before the fact returns. Ex ante distribution Is a function that assigns probabilities to ranges of future possible values of the random variable. Ex post distributions are used to approximate ex ante distributions, so long as i) the distribution is stationary Ii) large number of historical are available from which a proper representation of the distribution can be arrived. 2

3 Characteristics of Return Distributions Normal Distribution(Gaussian Distributions) Is the bell shaped, symmetrical & mesokurtic The tail of the distribution is asymptotic and continuous It can be fully explained by mean and variance Continuous compounding returns (log normal returns) follow normal distribution in contrast to simple returns from discreet compounding. Ex: if monthly returns are normally distributed, the quarterly returns using discreet compounding is not normally distributed, in contrast if monthly log returns are normally distributed then quarterly log returns are also normally distributed. 3

4 Characteristics of Return Distributions Lognormal Distribution It is generated by the function e x, where x is normally distributed It is positively skewed It is bounded from below by zero, hence useful for modelling asset prices which never take negative values As per Central Limit Theorem, even if ln(1+r) does not follow a normal distribution, the sum of the logs can be closely approximated by the normal distribution, as the number of observations increases and uncorrelated over time. 4

5 Characteristics of Return Distributions 5

6 Moments of Return Distribution The shape of probability distributions are described by the moments of the distribution. Raw Moments Are measured relative to an expected value raised to the appropriate power The first raw moment is the mean of the distribution, which is the expected value of the returns k th raw moment: Raw moments for k>1 are not useful for return calculations. 6

7 Moments of Return Distribution Central Moments- (Variance) Are measured relative to the mean 2 nd order central moment is the expected variance k th order central moment: 7

8 Moments of Return Distribution Example 1: Calculate the sample variance and sample standard deviation from the following set of returns. Period Return

9 Moments of Return Distribution Skewness Statistic Is the standardised third central moment It refers to the extent distribution of data is not symmetric about mean 9

10 Moments of Return Distribution 10

11 Moments of Return Distribution Kurtosis Statistic Is the standardized fourth central moment It refers to the degree of peakedness in the distribution of data Normal kurtosis also called mesokurtic is equal to 3, if kurtosis greater than 3 than it is called leptokurtic and less than 3 is called platykurtic. 11

12 Moments of Return Distribution 12

13 Covariance Covariance Is an unscaled statistical measure of how two assets move together The value ranges between - to + 13

14 Correlation Coefficient Correlation Coefficient Also referred as the Pearson correlation coefficient is a relative measure of the strength of relationship between two assets -1 represents perfect negative linear correlation and +1 represents perfect positive linear correlation and zero represent no linear correlation Even for zero correlation there are possibilities of non-linear correlation 14

15 Correlation Coefficient Correlation Coefficient 15

16 Covariance & Correlation Example 2: Calculate the covariance and correlation between two assets. Outcome R i R j

17 Spearman s Rank Correlation ρ s = 1 6 σ d i 2 n(n 2 1) Example: Jason Ganjaleze is examining the cross-sectional relationship between firm size and eps. He has collected an initial sample of 10 firms. Company EPS Market value (in millions of dollars) A B C D E F G H I J

18 The Correlation Coefficient and Diversification 18

19 Portfolio Diversification Portfolio Standard Deviation: σ p = w i 2 σ i 2 + w j 2 σ j 2 + 2w i w j ρ i,j σ i σ j Effects of Correlation on Portfolio Diversification: Expected Return Standard Deviation Domestic Stocks(DS) Domestic Bonds(DB)

20 Beta β i = Cov(R i, R m ) Var(R m ) = ρ i,mσ i σ m σ m 2 = ρ i,m σ i σ m Example: The covariance of returns between the RE Fund and the market portfolio equals 0.20, and the standard deviation of returns equals 0.80 and 0.40 for the RE funds and the market portfolio, respectively. Calculate the correlation between the RE fund and the market portfolio. Next, calculate the beta for the RE fund. 20

21 Autocorrelation k order autocorrelation = E[(R t μ)(r t k μ) σ t σ t k First order autocorrelation(serial correlation) 21

22 Autocorrelation Positive & negative serial correlation 22

23 Durbin-Watson Statistic Hypothesis: H 0 : ρ t,t 1 = 0 H A : ρ t,t 1 0 DW = σ t=2 T (e t e t 1 ) 2 σt t=1 e t 2 If sample is large, approx. Durbin Watson statistic is: DW 2(1 ρ t,t 1 ) 23

24 Durbin-Watson Statistic Example: Assume a large sample of returns is examined for a private equity fund. The correlation od successive returns equals 0.6. Compute and interpret the Durbin Watson statistic. 24

25 Standard Deviation & Variance Confidence Intervals for the Normal Distribution Using Standard Deviation 25

26 Test of Normality Causes of Non-Normality 1. Autocorrelation 2. Illiquidity 3. Nonlinearity Sample Moments Jarque Bera Test where JB is the Jarque-Bera test statistic, n is the number of observations, S is the skewness of the sample, and K is the excess kurtosis of the sample. Example: 26

27 Forecasts of Future Return Volatility Heteroskedastic Conditional Heteroskedastic Auto regression ARCH Model GARCH Model 27