CAIA Chapter 3: Statistical Foundations

same kurtosis as the normal distribution (normal-tailed)

Return distribution has positive excess kurtosis, more kurtosis than the normal distribution (fat-tailed)

test for normality. Involves a statistic that is a function of the relative skewness and excess kurtosis of the sample. As a sample exhibits more and more the tendencies of a normal distribution then the JB test statistic will tend to be closer to zero

Down-side risk measures. Uses same formula as variance except that it considers only the negative deviations. Provides a sense of how much variability exists among losses

is to the semivariance as standard deviation is to variance

is simply the probability that the return will be less than the investor's target rate return.

is similar to semivariance except that target semivariance substitutes the investor's target rate of return in place of the mean return.

is simply the square root of the target semivariance

indicates the dispersion of the returns of an investment relative to a benchmark return, when a benchmark return is the contemporaneous realized return on an index or peer group of comparable risk.

is defined as the maximum loss in the value of n asset, usually in percentage return form rather than currency

is the loss figure associated with a particular percentile of a cumulative loss function. VaR is the maximum loss over a specified time period within a specified probability.

aka, expected tail loss. Is the expected loss of the investor, given that VaR has been equaled or exceeded. Provides the investor with information about the potential magnitudes of losses beyond VaR

is a type of simulation in which very many potential paths of the future are projected using an assumed model and the results are analyzed as an approximation to the future probability distribution. It is used in difficult problems (no expected values etc

is when the variance of a variable changes

is when the variance is constant

is when subsequent values to a variable are explained by past values of the same variable

financial asset that have different levels of return variation even when the conditions are similar, for example, when viewed at similar price levels

in the context of financial returns GARCH is a robust methodology that can model return variation through time in a way that allows that variation to change based on the variable's past history &even when some conditions such as price levels havn't chang

for a particular analysis is the smallest time interval for which returns are calculated, such as daily, monthly or even annually.

in investment terminology, the standard deviation of returns is often termed the volatility.

Future possible returns and their probabilities are referred to as ex ante or expectational return probability distributions

meaning from forwards. based on realized outcomes rathern than anticipated outcomes

is the familiar bell-shaped distribution that is also known as Gaussian distribution. The normal distribution tends to approcimate many distributions observed in nature and/or generated as the result of human actions and interactions, incl. financial retu

is the recognition of interest on interest or, more generally, earning on earnings

A computation that does not incorporate compounding

assumes that earning can instantaneously be reinvested to generate additional earnings

any compounding interval other than continuous compounding is known as discrete compounding

the mathematics of returns can be simplified by expression returns as continuously compounded rates

when the arithmetic mean log return is converted into an equivalent simple rate that rate is referred to as the geometric mean return

a variable has a lognormal distribution if the distribution of the logarithm of the variable is normally distributed

referres to the possible correlation of the returns with each other. first-order autocorrelation refers to the correlation between returns in time period t and the return in the previous time period t-1. Positive when moving in the same direction

Autocorrelation, Illiquidity, Nonnormality

many alternative investments are thinly traded. Observed market prices might therefore be heavily influenced by the liquidity needs of the market participants rather than driven toward an efficient price by the actions of numerous well-informed buyers/Sel

e.g. short-term call option: the dispersion in the call option's return distribution changes through time as the underlying assets's price changes, even if the volatility of the underlying asset remains constant

differ from raw moments because they focus on the deviation of the variable from the mean

is the probability weighted average of the deviation squared

the square root of the variance. In investment terminology, the standard deviation of returns is often termed the volatility.

equal to the third central moment divided by the standard deviation of the variable cubed. Provides a numerical measure of the extent to which a distribution flares out in one direction or the other. A positive value indicates that the right tale is large

=capturing the gatness of the tails of a distribution. Normally distributed variable the estimated kurtosis is 3.0. high positive kurtosis (excess kurtosis) means fatter tails (high probability for extreme outcomes)