chapter 12-14

The three basic features that an investment risk measure should be able to take into consideration are

(1) relativity of risk, (2) multidimensionality of risk, and (3) asymmetry of risk.

Five stylized facts have been observed for asset return distributions:

(1) skewness, (2) fat tails, (3) volatility clustering, (4) autoregressive behavior, and (5) temporal behavior of tail thickness.

The stability property is important in portfolio theory and risk management. The stability property means that

(1) the portfolio return (where the portfolio is composed of assets whose returns are normally distributed) will follow a normal distribution and (2) its use allows the aggregation of asset returns over time.

quartiles:

25%, 50%, and 75% quantiles; first quartile, second quartile, and third quartile, respectively

Two measures of kurtosis are

Fisher's kurtosis (also referred to as excess kurtosis) and Pearson's kurtosis.

The two most commonly used measures of skewness are

Fisher's skewness and Pearson's skewness, the latter being equal to the square of Fisher's skewness.

In the past, certain computational aspects made it difficult to use the

Paretian stable distribution for portfolio management (e.g., an infinite variance), but advances in computational finance and modeling (i.e., the tempered stable distribution) have eliminated those problems.

The best-known and most commonly applied reward-risk ratio in finance is the but which is preferred

Sharpe ratio, but because of the Sharpe ratio's drawbacks, the Sortino ratio is preferred.

skewness:

a commonly used measure for the asymmetry of a distribution; third moment of a probability distribution

A random variable is

a function that assigns a numerical value to the potential outcomes of an experiment.

With the exception of a risk-free asset, the return on an asset is

a random variable.

Pearson correlation coefficient (or correlation coefficient):

alternative but related measure to covariance that is not affected by the scale of the two random variables

An average return can be calculated as either

an arithmetic average return or a geometric average return, with the latter being the preferred method.

multidimensionality of risk:

an investor could have multiple investment objectives, which would call for multiple risk benchmarks

Relativity of risk means that risk should be related to an

asset's performing worse than some alternative investment or benchmark.

probability distribution:

assigns a probability to each numerical value of a random variable (four common measures: location, dispersion, asymmetry, and concentration)

CVaR (also called "average value at risk") for a given tail probability is defined as the

average of the VaRs and hence focuses on the losses in the tail that are larger than the corresponding VaR level.

normal distribution (or Guassian distribution):

bell-shaped and symmetric around the mean; dominates portfolio theory and much of financial theory

continuous random variable:

can take on any possible value in the range of possible outcomes

A probability distribution's location is a measure of its

central value (the first moment), and the measures used to describe this moment are the mean (or average), the median, and the mode.

When the random variable of interest is the return on an asset, the random variable is assumed to be

continuous.

A measure of the joint variation of two random variables that are assumed to be in a linear association is the

covariance and its related measure, the correlation.

Covariance/correlation is the commonly used way to measure the

dependence between two random variables.

central value:

described by the mean, the median, and the mode

Reward-risk ratios can be calculated based on

different reward measures and different risk measures.

Variance is a measure of the The square root of the variance is the

dispersion of the outcome that can be realized relative to the mean and is basically the average of the squared deviations from the mean. standard deviation.

A measure of the variability of the outcomes that can be realized is the second moment of a probability distribution: The three most commonly used measures of dispersion are the

dispersion. variance, mean absolute deviation, and range.

Because there is no fundamental theory that can suggest a distributional model for asset returns, alternative distributions must be

empirically tested.

The peakedness of a probability distribution affects how

fat the tails are.

The α-quantile of a probability distribution provides information about where the

first α% of the probability distribution is located.

conditional value-at-risk (CVaR):

for a given tail probability, the average VaRs that are larger than the VaR at that tail probability, thus focusing on the losses in the tail that are larger than the corresponding VaR level; because CVaR indicates the magnitude of such losses, it is also called the average value-at-risk (AvaR) measure

excess kurtosis (or Fisher's kurtosis):

found by subtracting three from Pearson's kurtosis

multivariate probability distribution (or joint probability distribution):

helpful for understanding a portfolio of assets which looks at probability distribution for multiple assets; portfolio return of interest

The tail of a probability distribution is the portion of the distribution that contains

information about extreme outcomes that may arise for the random variable.

dependence of random variables:

interdependence between two return distributions

A limitation of the use of VaR as a measure of risk is that

it ignores the amount of losses larger than the VaR at that tail probability.

The fourth moment of a probability distribution is measured by its kurtosis, which is the

joint measure of peakedness and tail fatness.

kurtosis (α):

joint measure of peakedness and tail fatness; fourth moment of a probability distribution

In portfolio management and risk management, the probability distribution of interest is the

joint probability distribution.

The class of stable distributions is a Because this class includes as a special case the normal distribution, a nonnormal stable distribution is referred to as

large and flexible class of probability distributions, which also allows for skewness and heavy-tailedness for asset returns. a "Paretian stable distribution."

discrete random variable:

limits the outcomes so that the random variable can only take on discrete values

Four commonly used measures to describe a probability distribution (and referred to as the "statistical moments" of a distribution) are

location, dispersion, asymmetry, and concentration in tails.

When constructing a portfolio of assets, it is necessary to

make some assumption about asset return distributions.

Only the first two moments, ___________, are needed to describe a normal distribution. A symmetric nonnormal distribution characterized by a

mean and variance, higher (lower) peak at the mean than the normal distribution is said to be a leptokurtic (platykurtic) distribution and has a fatter (thinner) tail than the normal distribution.

first moment (or expected value):

mean of a probability distribution

The most commonly used measure of location in finance is the

mean or value.

skewness of stock returns:

means that asymmetry exists in the upside and downside potential of price changes; can be negative or positive skewness

volatility clustering:

means that large price changes tend to be followed by large price changes, and small price changes tend to be followed by small price changes

autoregressive behavior:

means that price changes depend on price changes in the past (e.g., positive price changes tend to be followed by positive price changes)

temporal behavior of tail thickness:

means that the probability of extreme price changes through time is smaller in normal markets and much larger in turbulent markets

fat tails:

means that the probability of extreme price movement (up and down) is much larger than predicted by the normal distribution

dispersion:

measure of how spread out the potential outcomes are that can be realized; various measures but variance, mean absolute deviation, and range are the most common

location:

measure of its central value

covariance:

measure of the joint variation of two random variables, where the association is assumed to be a linear one

value-at-risk (VaR):

measure of the minimum level of loss at a given, sufficiently high, confidence level for a predefined time horizon

geometric average return (or time-weighted average return):

measures the compounded rate of growth of the initial value, assuming that all cash distributions are reinvested

variance:

measures the dispersion of the outcomes that can be realized relative to the mean and is referred to as the second moment; variance is the average of the squared deviations from the mean

reward-risk ratios:

metrics measuring the return realized relative to the risk accepted

Because an investor could have multiple investment objectives,

multiple risk benchmarks are necessary, which underscores what is meant by the multidimensionality of risk.

stable Paretian distributions (or Lévy stable distributions):

nonnormal stable distributions

Empirical evidence does not support the assumption that real-world asset return distributions are best described by a instead, they exhibit fatter tails than predicted

normal distribution; by the normal distribution.

tempered stable distributions:

obtained from the class of stable distributions through a process called "tail tempering"; these have been suggested for modeling the distribution of stock returns

mean-variance analysis (or mean-variance optimization):

portfolio theory is often called this because it proposes that investors who select assets for inclusion in a portfolio take into consideration only the first two moments of the probability distribution of return on assets (e.g., mean and variance)

A desirable property that a probability distribution should The class of probability distributions that have this property is known

possess when used in portfolio and risk management is the stability property. as the stable distribution.

Probability distributions are used to describe the

potential outcomes of a random variable.

univariate probability distribution:

probability distribution that involves only a single random variable; helpful for understanding the attributes of the returns for an individual asset

α-quantile:

provides information about where the first α% of the probability distribution is located

asymmetry of risk:

reasonable to expect that risk is an asymmetric concept related to downside outcomes

Reward-risk ratios measure the reward on a

relative or absolute basis.

Sharpe ratio:

reward-risk ratio that measures the reward on an absolute basis

The correlation measure overcomes the limitation of the covariance measure, which depends on the

scale used to measure the random variable. The range of the correlation is −1 to 1.

The asymmetry of a probability distribution around its mean is its An asymmetric distribution can exhibit either

skewness and is the third moment of a probability distribution. negative skewness (skewed to the left) or positive skewness (skewed to the right).

standard deviation:

square root of the variance

stability property:

states that the sum of a number of N random variables that follow a normal distribution will again be a normal distribution, provided that the random variables behave independently of one another

Because of its desirable characteristics with respect to the stability property, the stable distribution has been

suggested to describe asset return distributions.

The stability property that a normal distribution satisfies is that the

sum of N random variables that follow a normal distribution will again be a normal distribution, provided that the random variables behave independently of one another.

The normal distribution or Gaussian distribution is a bell-shaped distribution that is

symmetric around the mean. It is a special case of a stable distribution.

leptokurtic distribution:

symmetric nonnormal distribution characterized by a higher peak at the mean than the normal distribution

tails:

tails of a probability distribution are the portion of the distribution that holds information about extreme outcomes that may arise for the random variable; "fatness" of the tails of a probability distribution is related to the peakedness of the distribution around its mean

relativity of risk:

that risk should be related to an asset's performing worse than some alternative investment or benchmark

stable distribution:

the family of probability distributions that possess the stability property

VaR is defined as

the minimum level of loss at a given, sufficiently high confidence level for a predefined time horizon.

tail probability:

the probability of 1 minus the confidence level

marginal probability distribution:

the return distribution for the common stock of each company

model risk:

the risk that the models are subject to forecasting errors

percentiles:

the1%, 2%, ... , 98%, and 99% quantiles

An asset's return over a given time interval is equal

to the change in the asset's price plus any distributions received from holding the asset, expressed as a fraction of the asset's price at the beginning of the time interval.

arithmetic average return:

unweighted average of the returns achieved during a series of such measurement intervals

sample of size n:

used when computing the central measure for a random variable

statistical moments:

used when referring to the measures of a probability distribution

index of stability:

value for α

Asymmetry of risk means that because it is reasonable to expect that risk is an asymmetric concept related to the downside outcome, any realistic candidate for an investment risk measure has to

value upside and downside outcomes differently.

Two alternative risk measures proposed are

value-at-risk (VaR) and conditional value-at-risk (CvaR).

platykurtic distribution:

when probability has less peakedness than normal distribution

independently distributed:

when there is no dependence between two random variables