Stats Flashcards
Variance
Measures how far a set of numbers are spread out
Variance defined
average squared differences between the mean and each item in the population or in the sample
High variance means
data points are very spread out
Standard deviation defined
measure of dispersion expressed as the square root of the variance
Standard deviation measures
the amount of variability around the average or mean
Advantage of standard deviation
expresses dispersion in the same units as the original values in the sample or population
A low standard deviation indicates
data points gather close to the means
Semi variance measures
data that is below the mean or target value of a data set
Semi variance defined
average of the squared deviations of all values less than the average or mean
Coefficient of variation defined
ration of the standard deviation to the mean
CV formula
standard deviation/mean
CV result
shows the extent of variability of relation to mean of the population
CV application
for comparison of data sets with different units or widely different means, one should use the CV instead of SDEV
Skewness defined
describes asymmetry of data points from a normal distribution
Negative skew
skews to the left
Positive skew
Skews to the right
Non-normal distributions (Skewness applied)
standard mean-variance analysis is limited which means SDEV is less meaningful
For a negative skew - SDEV applications
underestimating the risk
For a positive skew - SDEV applications
overestimating the risk
Kurtosis measures
How fat the tails are on a distribution relative to a normal distribution curve
Positive Kurtosis (Leptokurtic)
will show more extreme outcomes, creating a tendency for the observations around the mean to seem great and appearing to have a higher peak
Low Kurtosis (Platykurtic)
will show thinner tails (fewer extreme outliers), flatter top, less peakedness
Higher kurtosis suggest greater
Risk than reflected in the normal distribution relied upon in the traditional mean-variance framework
Standard deviation is a good measure of risk when returns are
symmetric
What if excess returns are not normally distributed
Standard deviation is no longer a complete measure of risk; Sharpe ratio is not a complete measure of performance; need to consider skew and kurtosis
N vs. N-1
Use N when you are using all of the data available (I.e. Population) then use N-1 when only using a sample (I.e. calculating standard deviation)
z = -1
one SDEV covering about 16% of the mean representing 68% of the outcomes
z = -2
covers two SDEV covering about 95% of the outcomes
z = -3
covers three SDEV covering about 99% of the outcomes
z = -1.65
covers 5% (useful for VaR)
z = -2.33
covers 1 % (useful for VaR)
Monte Carlo Simulation
stat modeling used to approximate the probability of future outcomes through multiple trials using random variables
Investment Consultants Fallacy
shows graphically how portfolio risk decreases over time due to clustering of returns around a long term average
Consultants Graph
the appearance that risk goes over time due to diversification
Samuelson and Merton Graph
the expected terminal (ending) range of values of an investment is much wider with more potential outcomes due to uncertainty over time
Covariance Defined
indicates how two variables are related
Covariance measures
the degree to which the returns of two assets move together
Assets possessing a high covariance with each other
does not offer much diversification
Covariance calculation
Correlation coefficient times standard deviations of both assets
Correlation coefficients defined
indicates the degree of relationship between two variables
Correlation coefficients calculation
covariance divided by sdev * sdev
Multiple regression R squared also known as
coefficient of determination
R squared (coefficient of determination) gives us
the proportion of variation in one variable than can be explained by another variable
R squared (coefficient of determination) metric indicates
the closeness or accuracy of the fit
R squared applications
The higher the number, the more meaningful the relationship; it gives us an indication of the level of diversification; gauges the reliability of alpha as an indicator of the managers return and beta as an indicator of risk
R squared below .6
benchmark is not as helpful as a comparative tool
R squared calculation
Beta Squared times SDEV of market squared divided by SDEV of the portfolio squared
Random Walk defined
event in which a set of events or samples follows a pattern of random unpredictable steps
Random Walk Application
stock prices cannot be accurately predicted
Benefits of multi-period forecasting
improved accuracy, consistency and smoothing of volatility
Variance defined
as the expected value of the difference between the variable and its mean squared
Variance used in
Monte Carlo analysis and portfolio optimization process
Coefficient of variation formula
standard deviation divided by expected return
A normal distribution has a kurtosis of
3
Anything greater than kurtosis of 3 is considered
Fat tailed
One tailed value for 95% percent of the curve tell us
95% of the values are less than 1.64 SDEV above the mean
As the normal distribuiton is symetrical, that 5% of the value are great than
1.6 standard deviations below the mean
The two tailed value tells us that 95% of the mass is
within + or - 1.96 standard deviations from the mean
The two tailed value tell us that 2.5% of the outcomes are
Less than -1.96 standard deviations from the mean, 2.5% are greater than +1.96 Standard deviations from the mean
Null hypothesis
default position that there is no relationship between the variables
Alternative hypothesis
the opposite of the null hypothesis that is being tested for relevance
Regression-based techniques use attribution based on
CAPM and estimating alpha of the manager
A key advantage to Monte Carolo analysis is
being able to factor in a number of data points into one model
An investor reviewing a sample population data set would want to observe
small standard errors
