Section I.A. Statistics Flashcards
What is the difference between N vs. N-1?
N is used for an entire population
N-1 is used for a sample
What is Variance?
- It is defined as the average squared difference between the mean and each item in the population or in the sample.
- it is always non-negative
- measures how far a set of numbers is spread out
- high variance means all the data points are very spread out
- variance of zero means all values with the set are identical
- standard deviation is easier to use and understand
What is Standard Deviation?
- a measure of dispersion expressed as the square roots of the variance
- measures the amount of variability around the average or mean
- an advantage of using standard deviation is that it expressed dispersion in the same units as the original values in the sample or population
- low standard deviation indicates data points gather close to the mean
- high standard deviation indicates that data points are spread far apart from the mean
- represented by Greek letter sigma (σ)
- considered a measure of “total risk”
- used in Sharpe Ratio, M^2, information ratio and Capital Allocation Line (CAL)
- good measure of risk when returns are symmetric
What is annualized standard deviation?
σ(annualized) = σ √t
Multiply the observed standard deviation by the square root of the number of periods in one year
What is Semi-Variance?
- Measures data that is below the mean or target value of a data set
- considered a better measurement of downside risk
- is the average of the squared deviations of all values less than the average or mean
What is Coefficient if Variation (CV)?
- the ratio of the standard deviation to the mean
- known as relative standard deviation
- shows the extent of variability in relation to mean of the population
What is positive and negative skewness?
- this describes the asymmetry of datat points from a normal distribution
- data points skewed to the left is negative skew. The standard deviation may be underestimating the risk because the possibility of that extreme left tail event is not captured by the stat.
- data points skewed to the right is positive skew. The standard deviation may be overestimating the risk.
What is Kirtosis?
- measures how fat the tas are on a distribution relative to a normal distribution curve
- positive (leptokurtic) - higher peak
- low (platykurtic) - thinner tails and flatter top
- higher Kirtosis suggests greater risk than reflected in the normal distribution
Difference between Normal and non-normal distributions?
- Normal distribution is considered foundational in the development of Modern Portfolio Theory
- if excess returns are not normally distributed, then standard deviation is no longer a complete measure of risk, Sharpe ratio is not a complete measure of portfolio performance and you need to consider skew and kurtosis
How do you calculate expected return?
Sum of probability of a state times the return if a state occurs.
E(r)=Σp(s)r(s)
P(s) = probability of a state
R(s) = return if a state occurs
S= state
How do you calculate expected return, risky asset with a risk-free asset?
E(Rp) = Wa(E(Ra))+WfRf
Uses, advantages and disadvantages of Monte Carlo Simulations?
Statistical modeling method used to approximate the probability of future outcomes through mutiple trials (Simulations) using random variables.
Advantages: can help one visualized and understand variability of future growth (and returns); offers a way to analyze risk; power tool for illustrating a variety of possible outcomes that could be useful in planning
Disadvantages: generates a normal distribution where the most likely scenario is found in the middle of the events - this is not always realistic and can create overconfidence which may lead to developing overly aggressive, risky portfolios; these models are not built to allow for a wide range of inputs: factors, expectations, etc.; Model assumes efficient markets.
What is coveriance?
- indicates how two variables are related
- a measure of the degree to which the returns of two assets move together
- positive covariance indicates that assets move together while a negative covariance indicates that assets move inversly
- assets possessing a high covariance with each other do not offer much diversification
What is discrete probability distribution and how do you go about calculating correlation coefficient?
Occurs when there are different probabilities related to different specific outcomes. That fact impacts the calculation mean, variance, covariance and correlation coefficient.
For Variance, you have to apply the probabilities as well to the difference of ead outcome to it’s mean (the squared) and, as a result, do not have to divide by the number of data points n (or n-1) because you’ve already weighted it by the probabilities.
You would apply the same principle to calculating correlation coefficient. That requires you to first determine the covariance between assets A & B. As you would in calculating standard deviation, you would apply the probability of each outcome state to the product of the differences of each asset’s outcome less its mean.
Again, you would not have to divide by n or n-1 because you’re already weighting each outcome by it’s probability.
What is seasonality?
Adjusting past performance or forecasts for demonstrated or expected impact if factors suchs as seasonality may be beneficial when analyzing data including speicifc company and industry stcok returns and volatility.
