This glossary explains some terms that are used on
the pages presented here. This is not a complete dictionary,
but rather a collection of brief descriptions.
Bernoulli distribution. Discrete probability distribution;
corresponding random variable takes only two values, usually 0 and 1.
More.
C
Cauchy distribution. Continuous probability distribution with
heavy tails. This distribution is used to model situations when large
deviations from "mean" value are possible.
More.
Certainty equivalent of a random variable X. In utility theory this means the quantity possessing the same utility
as X. Given
utility function U, certainty equivalent is calculated as
d(X) = U-1(EU(X)),
where U-1 is inverse function for U.
Certainty equivalent may be used as a
risk price in risk exchange transactions.
Concave function. A real function f is called
concave (strictly concave), if a function (-f)
is convex
(strictly convex).
Convex function. A real function f is called
convex, if for any x, y in its domain and any number
a from (0,1) the following inequality is true:
f (ax + (1 - a)y) <= a f(x) + (1 - a) f(y).
If inequality is strict, then f is called
strictly convex. Convexity topics are discussed in
detail in the book
R. Rockafellar. Convex Analysis. Princeton univ. press, Princeton, 1970.
Copula. A multidimensional distribution function on
a hypercube [0,1]n with uniform marginals.
By Sclar theorem any multidimensional dstribution may be
represented by superposition of a copula and marginal
distributions, thus copula completely describes
dependence
of components.
Correlation of random variables
X, Y is a measure of their linear dependence. Correlation
is calculated as
r = E[(X - EX)(Y - EY)] / (s(X) s(Y)),
where s(X), s(Y) are standard deviations of
X, Y. Correlation is especially useful under
normal joint distribution of
X, Y, while under other distributions may be misleading, and
should be used with caution.
D
Distribution density. A charactiristic of a continuous
distribution, that is equal to derivative of a cumulative
distribution function.
More.
Diversification. Distribution of resources, say, investment capital,
among several possibilities or projects (investment tools). Used to
decrease riskiness of a
portfolio obtained. In second order theories diversification
is measured in terms of
expectation and
variance of a portfolio.
Modern theories used to measure riskiness by applying
risk measure concept.
E
Efficient frontier. In a vector optimization problem:
a set of points on criteria plane corresponding to efficient
solutions to a problem. By changing an efficient solution one can
improve value of some criterion only at the expense of worsening
value of another criterion. In a Markowitz problem efficient frontier
is represented by an upper branch of parabola that is a boundary
for admissible solutions on risk-return plane.
Expectation of a random variable -
its mean value. Expectation of number of points in tossing a dice
is equal to 3 1/2.
More.
Expected utility. A generalization of
utility
concept from certain to random objects. Expected utility is calculated as
expectation of utility of random
object of interest.
Let U be an utility function, X be a
random variable, describing, say, a return
of investment instrument. Then expected utility of X
is calculated as
u(X) = EU(X).
Exponential distribution. Continuous distribution on
positive reals. This distribution is often used in reliability
theory to describe life time of technical elements, and sometimes
is used in insurance models to describe human life time.
More.
F
Frechet bounds. Any multidimensional distribution function
FXY...Z(x,y,...,z)
with marginals FX(x),
FY(y), ..., FZ(z) satisfies
inequality
Mode of a random variable is one of
its mean characteristics, represents most probable value.
More.
Monte Carlo method. A general method for solving problems,
if analytic solution is difficult or does not exist. The essence of
the method is representing a solution by probability distribution or
a functional of probability distribution, and then solving the problem
by sequencial sampling from the distribution. Detailed description
of the method is given in the
lecture.
N
Normal distribution. The most widely used probability distribution.
Sometimes assumption of normality may be greatly misleading, so it should
be used with caution.
More.
O
Outcomes space of a probabilistic experiment:
a set X={x,y,z,...} of all possible states that
object of interest may fall into as a result of experiment.
P
Portfolio. A result of distribution of a resource (investment
capital) among several possibilities (projects). For n projects
desribed by a random vector of returns
(X1,...,Xn),
portfolio is defined by weights or portions
y=(y1,...,yn)
of investment capital being invested in each project.
Portfolio return is a
random variable
X=y1X1+...+ynXn,
so portfolio selection problems are usually stated as achieving
the "best" (in some sence) portfolio distribution by choosing
weights subject to rstriction
y1+...+yn=1.
Probability of event. Degree of likelihood of event,
a measure on some probability space. Takes numeric values between
0 and 1. In experiment with tossing a dice the probability of
appearing of each side is equal to 1/6.
Probability distribution. In a simple case the set of all outcomes
X={x,y,z,...} of probabilistic experiment is finite
(consists of n elements). Probability distribution is
a collection of n nonnegative numbers
( probabilities)
px,
py,
pz,...,
that sum to 1.
Q
R
Random variable. (Measurable) mapping from
outcomes space of a fixed probability experiment
into reals. Random variable serves a probabilistic model for
experiments with uncertain number outcome.
Example is number of points in tossing a dice. This random variable takes
values 1, 2, 3, 4, 5, 6 with probability 1/6 each. The more frequently
used parameters of random variable are
expectation and
variance.
Risk. 1. Probabilistic
uncertainty
about future events.
2. Mathematical object - a probability distribution of an abstract
random element, e.g. random variable.
Risk aversion. In utility theory this quantity characterizes
how investor dislikes risk. More risk averse investor would less likely
be involved in highly risky projects. For an
utility function U
the (Pratt) risk aversion is calculated as
- U''(x) / U'(x).
For exponential utility function
U(x) = (1 - exp( - a * x )) / (1 - exp( - a )),
risk aversion coincides with value of parameter a.
Risk measure. "Quantity" of risk contained in a given
risk; a functional on the space of probability
distributions. Examples are: expected utility, Value at risk,
distorted probability. Classic problems of risk theory use
variance and
standard deviation as risk measures.
Partial case of risk measure is
risk price.
Each risk measure generates a notion of
certainty equivalent.
Risk price. A special case of
risk measure. Risk price is used to calculate
risk premium when transferring risk from one carrier to another.
Examples are: insurance, options and other derivative instruments.
Uncertainty. Lack of information about object of interest.
For example there might be known the set of possible states
X={x,y,z,...} of the object, but exact state is unknown.
Uncertainty is called probabilistic if one knowns the
probability distribution
on X that describes probabilities
of "visiting" specific states by the object.
Uniform distribution. A distribution of a random variable X
on an interval [a,b] such that probability of falling X
within subinterval [c,d] of [a,b] is proportional to its length.
In multidimentional case the probability of falling a random vector
X within a specific body is proportional to volume of a body.
This distribution is sometimes called geometric.
More.
Utility function. In utility theory this is a function
U, prescribing a quantity named "utility" to wealth,
outcomes of experiments, results of activities, etc. The latter
are often reals, in which case U is a real function.
It is often thought of as increasing
concave function.
In risk theory a derivative notion of
expected utility is widely used.