Deviation risk measure
Encyclopedia
In financial mathematics, a deviation risk measure is a function to quantify financial risk
(and not necessarily downside risk
) in a different method than a general risk measure
. Deviation risk measures generalize the concept of standard deviation
.
, where 
is the L2 space of random
portfolio returns
, is a deviation risk measure if
R where for any
R is expectation bounded if
for any nonconstant X and 
for any constant X.
If
for every X (where 
is the essential infimum), then there is a relationship between D and a coherent risk measure
.
Financial risk
Financial risk an umbrella term for multiple types of risk associated with financing, including financial transactions that include company loans in risk of default. Risk is a term often used to imply downside risk, meaning the uncertainty of a return and the potential for financial loss...
(and not necessarily downside risk
Downside risk
Downside risk is the financial risk associated with losses. That is, the risk of difference between the actual return and the expected return , or the uncertainty of that return....
) in a different method than a general risk measure
Risk measure
A Risk measure is used to determine the amount of an asset or set of assets to be kept in reserve. The purpose of this reserve is to make the risks taken by financial institutions, such as banks and insurance companies, acceptable to the regulator...
. Deviation risk measures generalize the concept of standard deviation
Standard deviation
Standard deviation is a widely used measure of variability or diversity used in statistics and probability theory. It shows how much variation or "dispersion" there is from the average...
.
Mathematical definition
A function, where is the L2 space of randomRandom variable
In probability and statistics, a random variable or stochastic variable is, roughly speaking, a variable whose value results from a measurement on some type of random process. Formally, it is a function from a probability space, typically to the real numbers, which is measurable functionmeasurable...
portfolio returns
Portfolio (finance)
Portfolio is a financial term denoting a collection of investments held by an investment company, hedge fund, financial institution or individual.-Definition:The term portfolio refers to any collection of financial assets such as stocks, bonds and cash...
, is a deviation risk measure if
- Shift-invariant:
for any 
- Normalization:
- Positively homogeneous:
for any 
and 
- Sublinearity:
for any 
- Positivity:
for all nonconstant X, and 
for any constant X.
Relation to risk measure
There is a one-to-one relationship between a deviation risk measure D and an expectation-bounded risk measureRisk measure
A Risk measure is used to determine the amount of an asset or set of assets to be kept in reserve. The purpose of this reserve is to make the risks taken by financial institutions, such as banks and insurance companies, acceptable to the regulator...
R where for any
-
-
.
R is expectation bounded if
for any nonconstant X and for any constant X.If
for every X (where is the essential infimum), then there is a relationship between D and a coherent risk measureCoherent risk measure
In the field of 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...
.