2.4 Statistical Foundations

Question 1

Ex ante vs Ex post return distributions

Answer 1

Ex post returns are realized outcomes.

Future possible returns and their probabilities are referred to as expectational or ex ante returns.

Question 2

Two properties for past return behaviour to predict future potential return

Answer 2

1. Return distribution is stationary; expected return and dispersion does not change
2. Large sample of past observation

Question 3

Normal vs Lognormal

Answer 3

• Normal distribution is symmetrical, whereas the lognormal distribution is not.
• Lognormal distributions contain only positive numbers, whereas normal distribution can contain negative values.

Question 4

Benefit of lognormal over distribution of discretely compounded

Answer 4

Modeling the distribution of discretely compounded returns as being normally distributed over a particular time interval –> model will not be valid for any other choice of time interval.

Normal distribution replicates additively; thus, if the log returns over one time interval can be modeled as being normally distributed, then the log returns over all time intervals will be lognormal as long as they are statistically independent through time.

Question 5

Skewness

Answer 5

• 0: Symmetrical
• +: Positive Skew, right tail is larger, mass is left
• -: Negative Skew, left tail is larger, mass is right

Question 6

Kurtosis

Answer 6

Captures the fatness of tail in distribution. High value = fatter tails.

Shapes
1. Leptokurtic (Tallest): positive excess kurtosis –> fat tail (higher probabilities of extreme outcomes)
2. Mesokurtic (Normal)
3. Platykurtic (Lowest): negative excess kurtosis –> skinny tail (lower probabilities of extreme outcomes)

Question 7

Covariance

Answer 7

Directional relationship between the returns on two assets.

Covariance is calculated by
1. analyzing at-return surprises (standard deviations from the expected return) OR
2. multiplying the correlation between the two random variables by the standard deviation of each variable.

• Positive: same direction
• Negative: opp direction
• 0: move independently

Question 8

Correlation Coefficient

Answer 8

Strength of association between two variables

Perfect Linear Negative Correlation (-1): 2 assets move in exact opposite direction in same proportion

Perfect Linear Positive Correlation (+1): 2 assets move in exact same direction in the proportion

0: no linear association between returns of two assets

Question 9

Covariance vs Variance

Answer 9

Both variance and covariance measure how data points are distributed around a calculated mean. However, variance measures the spread of data along a SINGLE AXIS, while covariance examines the DIRECTIONAL RELATIONSHIP between TWO VARIABLES.

Question 10

Covariance vs Correlation

Answer 10

Covariance measures the DIRECTION of a relationship between two variables, correlation measures the STRENGTH of that relationship

Question 11

Beta

Answer 11

Beta is a measure of a stock’s volatility in relation to the overall market.

Textbook: covariance between asset return and index return, divided by variance of index return

Question 12

Covariance vs Variance

Answer 12

Both variance and covariance measure how data points are distributed around a calculated mean. However, variance measures the spread of data along a SINGLE AXIS, while covariance examines the DIRECTIONAL RELATIONSHIP between TWO VARIABLES.

Question 13

Relationship between Variance and Standard Deviation

Answer 13

SD = Square root of Variance

Variance = SD squared

Question 14

Autocorrelation

Answer 14

1. Autocorrelation represents the degree of similarity between a given time series and a lagged version of itself over successive time intervals.
2. Autocorrelation measures the relationship between a variable’s current value and its past values.
3. An autocorrelation of +1 represents a perfect positive correlation, while an autocorrelation of -1 represents a perfect negative correlation.
4. Technical analysts can use autocorrelation to measure how much influence past prices for a security have on its future

Question 15

Relationship between Second-Order vs First-Order Autocorrelation

Answer 15

• Correlation between T and T-2 would be a function of the shared correlation of returns from (T and T-1) and (T and T-2).

E.g. First order correlation: 0.7. If no further causality beyond one period, then the Second order correlation will be 0.7x0.7=0.49. If more than 0.49, then returns between T and T-2 is positive, beyond the correlation of T and T-1.

Question 16

Partial Autocorrelation Coefficient

Answer 16

Adjusts autocorrelation coefficients to isolate the portion of the correlation in a time series attributable directly to a particular higher-order relation.

E.g. “removes” the effect of first order autocorrelation from second order to isolate the marginal affect of T-2 return on T.

Question 17

Durbin Watson test

Answer 17

Test to detect autocorrelation in the residuals from a stats or regression model.

Always has a value between 0 and 4

• Value of 2 indicates no correlation
• 0 to 2: Positive correlation
• 2 to 4: Negative correlation

Question 18

Define and explain standard deviation (volatility)

Answer 18

Typical amount by which actual return deviates from the average

Question 19

Describe the properties of variance

Answer 19

• Covariance of any variable with itself is Variance
• If uncorrelated, then covariance is 0
• Multiperiod: compounded rate of return is the sum of the return of each period
• If we assume returns through time has no autocorrelation, therefore no covariance, so variance of weekly retune is SUM of variance of daily return.

Question 20

Describe the properties of standard deviation

Answer 20

• Perfectly correlated cross sectional return
• One variable can be expressed as a linear combination of another variable
• Returns of a levered position can be well approximated as linear function of the returns of unlevered in same asset
• SD of a multiperiod return can be approximated by sum of SD of each sub period.

Question 21

Why are some returns non-normal?
1. A____
2. I____
3. N__-L___

Answer 21

1. Autocorrelation causes it to NOT be statistically independent. ST returns can sometimes bee positively auto-correlated if assets not rapidly traded at low cost.
2. Observed market price determined by a few market participants. In illquid markets prices estimated by models or professional judgement which shows autocorrelation.
3. Nonlinearity, e.g. option where dispersion in return distribution changes through time as underlying asset price changes, even if volatility of underlying remains constant.

Question 22

Test for Normality

Answer 22

Jarque-Bera

Question 23

Heteroskedasticity vs Homoskedasticity

Answer 23

Heteroskedasticity happens when the standard errors of a variable, monitored over a specific amount of time, are non-constant.

Homoskedasticity is when the variance of a variable is constant.

With heteroskedasticity, the tell-tale sign upon visual inspection of the residual errors is that they will tend to fan out over time. IMPACT: violation of the assumptions for linear regression modeling, like CAPM.

Question 24

Generalized autoregressive conditional heteroskedasticity (GARCH) method:

Answer 24

Generalized AutoRegressive Conditional Heteroskedasticity (GARCH) is a statistical model used in analyzing time-series data where the variance error is believed to be serially autocorrelated. GARCH models assume that the variance of the error term follows an autoregressive moving average process.

Question 25

How is GARCH used to model risk evolution over time

Answer 25

• GARCH is a statistical modeling technique used to help predict the volatility of returns on financial assets.
• GARCH is appropriate for time series data where the variance of the error term is serially autocorrelated following an autoregressive moving average process.
• GARCH is useful to assess risk and expected returns for assets that exhibit clustered periods of volatility in returns.

Question 26

The COVARIANCE between the returns of two financial assets is equal to the product of the STANDARD DEVIATION of the returns of the two assets. What is the primary statistical terminology for this relationship?

Answer 26

The covariance will equal the product of the standard deviations when the correlation coefficient is equal to one.

Question 27

What is the formula for the beta of an asset using common statistical measures?

Answer 27

Covariance (Return of market, Return of stock) / Variance (Return of Market)

SD (market, stock) / SD2 (market)

[P (market, stock) x SD (stock)] / SD (Market)

Question 28

How would a large increase in the kurtosis of a return distribution affect its shape?

Answer 28

Kurtosis is typically viewed as capturing the fatness of the tails of a distribution, with high values of kurtosis, or positive values of excess kurtosis, indicating fatter tails (i.e., higher probabilities of extreme outcomes) than is found in the case of a normally distributed variable.

Kurtosis can also be viewed as indicating the peakedness of a distribution, with a sharp narrow peak in the center being associated with high values of kurtosis, or positive values of excess kurtosis.

Question 29

Using statistical terminology, what does the volatility of a return mean?

Answer 29

standard deviation

Question 30

Contrast the kurtosis and the excess kurtosis of the normal distribution.

Answer 30

Kurtosis serves as an indicator of the peaks and tails of a distribution. In the case of a normally—distributed variable the kurtosis is 3.

Excess kurtosis is equal to kurtosis minus 3. Thus a normally distributed variable has an excess kurtosis of 0.

Excess kurtosis provides a more intuitive measure of kurtosis relative to the normal distribution since it varies around zero to indicate kurtosis that is larger (positive) or smaller (negative) than the case of the normal distribution.

Question 31

What is the value of the beta of the following three investments: a fund that tracks the overall market index, a riskless asset, and a bet at a casino table?

Answer 31

Overall Market: +1
Riskless: 0
Casino: 0

Question 32

In the case of a financial asset with returns that have ZERO CORRELATION, what is the relationship between the variance of the asset’s daily returns and the variance of the asset’s monthly return?

Answer 32

The variance of the monthly returns are T times the variance of the daily returns where T is the number of trading days in the month.

Question 33

In the case of a financial asset with returns that have autocorrelation approaching +1, what is the relationship between the standard deviation of the asset’s monthly returns and the standard deviation of the asset’s annual return?

Answer 33

In the perfectly correlated case the standard deviation of a multiperiod return is proportional to T. In this case the annual volatility is 12 times the monthly volatility.

Question 34

What is the general statistical issue addressed when the GARCH method is used in a time series analysis of returns?

Answer 34

The tendency of an asset’s variance to change through time.

Question 35

The covariance between the returns of two financial assets is equal to the product of the standard deviations of the returns of the two assets. What is the primary statistical terminology for this relationship?

Answer 35

The covariance will equal the product of the standard deviations when the correlation coefficient is equal to one.