2.4 Statistical Foundations Flashcards
Ex ante vs Ex post return distributions
Ex post returns are realized outcomes.
Future possible returns and their probabilities are referred to as expectational or ex ante returns.
Two properties for past return behaviour to predict future potential return
- Return distribution is stationary; expected return and dispersion does not change
- Large sample of past observation
Normal vs Lognormal
- Normal distribution is symmetrical, whereas the lognormal distribution is not.
- Lognormal distributions contain only positive numbers, whereas normal distribution can contain negative values.
Benefit of lognormal over distribution of discretely compounded
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.
Skewness
- 0: Symmetrical
- +: Positive Skew, right tail is larger, mass is left
- -: Negative Skew, left tail is larger, mass is right
Kurtosis
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)
Covariance
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
Correlation Coefficient
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
Covariance vs Variance
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.
Covariance vs Correlation
Covariance measures the DIRECTION of a relationship between two variables, correlation measures the STRENGTH of that relationship
Beta
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
Covariance vs Variance
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.
Relationship between Variance and Standard Deviation
SD = Square root of Variance
Variance = SD squared
Autocorrelation
- Autocorrelation represents the degree of similarity between a given time series and a lagged version of itself over successive time intervals.
- Autocorrelation measures the relationship between a variable’s current value and its past values.
- An autocorrelation of +1 represents a perfect positive correlation, while an autocorrelation of -1 represents a perfect negative correlation.
- Technical analysts can use autocorrelation to measure how much influence past prices for a security have on its future
Relationship between Second-Order vs First-Order Autocorrelation
- 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.
Partial Autocorrelation Coefficient
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.
Durbin Watson test
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
Define and explain standard deviation (volatility)
Typical amount by which actual return deviates from the average
Describe the properties of variance
- 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.
Describe the properties of standard deviation
- 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.
Why are some returns non-normal?
1. A____
2. I____
3. N__-L___
- Autocorrelation causes it to NOT be statistically independent. ST returns can sometimes bee positively auto-correlated if assets not rapidly traded at low cost.
- Observed market price determined by a few market participants. In illquid markets prices estimated by models or professional judgement which shows autocorrelation.
- Nonlinearity, e.g. option where dispersion in return distribution changes through time as underlying asset price changes, even if volatility of underlying remains constant.
Test for Normality
Jarque-Bera
Heteroskedasticity vs Homoskedasticity
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.
Generalized autoregressive conditional heteroskedasticity (GARCH) method:
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.
How is GARCH used to model risk evolution over time
- 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.
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?
The covariance will equal the product of the standard deviations when the correlation coefficient is equal to one.
What is the formula for the beta of an asset using common statistical measures?
Covariance (Return of market, Return of stock) / Variance (Return of Market)
SD (market, stock) / SD2 (market)
[P (market, stock) x SD (stock)] / SD (Market)
How would a large increase in the kurtosis of a return distribution affect its shape?
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.
Using statistical terminology, what does the volatility of a return mean?
standard deviation
Contrast the kurtosis and the excess kurtosis of the normal distribution.
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.
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?
Overall Market: +1
Riskless: 0
Casino: 0
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?
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.
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?
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.
What is the general statistical issue addressed when the GARCH method is used in a time series analysis of returns?
The tendency of an asset’s variance to change through time.
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?
The covariance will equal the product of the standard deviations when the correlation coefficient is equal to one.
