Tail value at risk
Encyclopedia
Tail value at risk also known as tail conditional expectation (TCE), is a risk measure
associated with the more general value at risk
. It is equivalent to expected shortfall
when the underlying distribution function
is continuous
at
. This is not a coherent risk measure
in general, however it is coherent if the underlying distribution is continuous. TVaR accounts for the severity of the failure, not only the chance of failure. The TVaR is a measure of the expectation
only in the tail of the distribution.
is the payoff of a portfolio at some future time and given a parameter 
then we can define the tail value at risk by
where
is the upper 
-quantile
given by
.
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...
associated with the more general value at risk
Value at risk
In financial mathematics and financial risk management, Value at Risk is a widely used risk measure of the risk of loss on a specific portfolio of financial assets...
. It is equivalent to expected shortfall
Expected shortfall
Expected shortfall is a risk measure, a concept used in finance to evaluate the market risk or credit risk of a portfolio. It is an alternative to value at risk that is more sensitive to the shape of the loss distribution in the tail of the distribution...
when the underlying distribution function
Cumulative distribution function
In probability theory and statistics, the cumulative distribution function , or just distribution function, describes the probability that a real-valued random variable X with a given probability distribution will be found at a value less than or equal to x. Intuitively, it is the "area so far"...
is continuous
Continuous function
In mathematics, a continuous function is a function for which, intuitively, "small" changes in the input result in "small" changes in the output. Otherwise, a function is said to be "discontinuous". A continuous function with a continuous inverse function is called "bicontinuous".Continuity of...
at
. This is not 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...
in general, however it is coherent if the underlying distribution is continuous. TVaR accounts for the severity of the failure, not only the chance of failure. The TVaR is a measure of the expectation
Expectation
In the case of uncertainty, expectation is what is considered the most likely to happen. An expectation, which is a belief that is centered on the future, may or may not be realistic. A less advantageous result gives rise to the emotion of disappointment. If something happens that is not at all...
only in the tail of the distribution.
Mathematical definition
If is the payoff of a portfolio at some future time and given a parameter then we can define the tail value at risk by
where
is the upper -quantileQuantile
Quantiles are points taken at regular intervals from the cumulative distribution function of a random variable. Dividing ordered data into q essentially equal-sized data subsets is the motivation for q-quantiles; the quantiles are the data values marking the boundaries between consecutive subsets...
given by
.