In
financial mathematics
Mathematical finance, also known as quantitative finance and financial mathematics, is a field of applied mathematics, concerned with mathematical modeling of financial markets.
In general, there exist two separate branches of finance that require ...
, a conditional risk measure is a
random variable
A random variable (also called random quantity, aleatory variable, or stochastic variable) is a mathematical formalization of a quantity or object which depends on random events. It is a mapping or a function from possible outcomes (e.g., the po ...
of the
financial risk
Financial risk is any of various types of risk associated with financing, including financial transactions that include company loans in risk of default. Often it is understood to include only downside risk, meaning the potential for financial ...
(particularly the
downside risk
Downside risk is the financial risk associated with losses. That is, it is the risk of the actual return being below the expected return, or the uncertainty about the magnitude of that difference.
Risk measures typically quantify the downside ris ...
) as if measured at some point in the future. A
risk measure
In financial mathematics, a risk measure is used to determine the amount of an asset or set of assets (traditionally currency) to be kept in reserve. The purpose of this reserve is to make the risks taken by financial institutions, such as banks ...
can be thought of as a conditional risk measure on the trivial
sigma algebra
Sigma (; uppercase Σ, lowercase σ, lowercase in word-final position ς; grc-gre, σίγμα) is the eighteenth letter of the Greek alphabet. In the system of Greek numerals, it has a value of 200. In general mathematics, uppercase Σ is used as ...
.
A dynamic risk measure is a risk measure that deals with the question of how evaluations of risk at different times are related. It can be interpreted as a sequence of conditional risk measures.
A different approach to dynamic risk measurement has been suggested by Novak.
Conditional risk measure
Consider a
portfolio's returns
Return may refer to:
In business, economics, and finance
* Return on investment (ROI), the financial gain after an expense.
* Rate of return, the financial term for the profit or loss derived from an investment
* Tax return, a blank document or t ...
at some terminal time
as a
random variable
A random variable (also called random quantity, aleatory variable, or stochastic variable) is a mathematical formalization of a quantity or object which depends on random events. It is a mapping or a function from possible outcomes (e.g., the po ...
that is
uniformly bounded
In mathematics, a uniformly bounded family of functions is a family of bounded functions that can all be bounded by the same constant. This constant is larger than or equal to the absolute value of any value of any of the functions in the family.
...
, i.e.,
denotes the payoff of a portfolio. A mapping
is a conditional risk measure if it has the following properties for random portfolio returns
:
; Conditional cash invariance
:
; Monotonicity
:
; Normalization
:
If it is a conditional
convex risk measure then it will also have the property:
; Conditional convexity
:
A conditional
coherent risk measure
In the fields of actuarial science and financial economics there are a number of ways that risk can be defined; to clarify the concept theoreticians have described a number of properties that a risk measure might or might not have. A coherent risk ...
is a conditional convex risk measure that additionally satisfies:
; Conditional positive homogeneity
:
Acceptance set
The
acceptance set In financial mathematics, acceptance set is a set of acceptable future net worth which is acceptable to the regulator. It is related to risk measures.
Mathematical Definition
Given a probability space (\Omega,\mathcal,\mathbb), and letting L^p = L ...
at time
associated with a conditional risk measure is
:
.
If you are given an acceptance set at time
then the corresponding conditional risk measure is
:
where
is the
essential infimum
In mathematics, the concepts of essential infimum and essential supremum are related to the notions of infimum and supremum, but adapted to measure theory and functional analysis, where one often deals with statements that are not valid for ''a ...
.
Regular property
A conditional risk measure
is said to be ''regular'' if for any
and
then
where
is the
indicator function
In mathematics, an indicator function or a characteristic function of a subset of a set is a function that maps elements of the subset to one, and all other elements to zero. That is, if is a subset of some set , one has \mathbf_(x)=1 if x\i ...
on
. Any normalized conditional convex risk measure is regular.
The financial interpretation of this states that the conditional risk at some future node (i.e.