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84-472: Expected shortfall is also called conditional value at risk ( CVaR ), average value at risk ( AVaR ), expected tail loss ( ETL ), and superquantile . ES estimates the risk of an investment in a conservative way, focusing on the less profitable outcomes. For high values of q {\displaystyle q} it ignores the most profitable but unlikely possibilities, while for small values of q {\displaystyle q} it focuses on

168-515: A coherent spectral measure of financial portfolio risk. It is calculated for a given quantile -level q {\displaystyle q} and is defined to be the mean loss of portfolio value given that a loss is occurring at or below the q {\displaystyle q} -quantile. If X ∈ L p ( F ) {\displaystyle X\in L^{p}({\mathcal {F}})} (an L )

252-505: A cumulative probability of q {\displaystyle q} and then calculate an expectation over those cases. In general, the last row selected may not be fully used (for example in calculating − ES 0.20 {\displaystyle -\operatorname {ES} _{0.20}} we used only 10 of the 30 cases per 100 provided by row 2). As a final example, calculate − ES 1 {\displaystyle -\operatorname {ES} _{1}} . This

336-416: A distribution that minimizes expected squared error while the median minimizes expected absolute error. Least absolute deviations shares the ability to be relatively insensitive to large deviations in outlying observations, although even better methods of robust regression are available. The quantiles of a random variable are preserved under increasing transformations, in the sense that, for example, if m

420-502: A generalization that can cover as special cases the continuous distributions. For discrete distributions the sample median as defined through this concept has an asymptotically Normal distribution, see Ma, Y., Genton, M. G., & Parzen, E. (2011). Asymptotic properties of sample quantiles of discrete distributions. Annals of the Institute of Statistical Mathematics, 63(2), 227–243. Computing approximate quantiles from data arriving from

504-417: A linear loss function ℓ ( w , x j ) = − w T x j {\displaystyle \ell (w,x_{j})=-w^{T}x_{j}} turns the optimization problem into a linear program. Using standard methods, it is then easy to find the portfolio that minimizes expected shortfall. Closed-form formulas exist for calculating the expected shortfall when

588-787: A portfolio X {\displaystyle X} follows the generalized Student's t-distribution with p.d.f. f ( x ) = Γ ( ν + 1 2 ) Γ ( ν 2 ) π ν σ ( 1 + 1 ν ( x − μ σ ) 2 ) − ν + 1 2 {\displaystyle f(x)={\frac {\Gamma \left({\frac {\nu +1}{2}}\right)}{\Gamma \left({\frac {\nu }{2}}\right){\sqrt {\pi \nu }}\sigma }}\left(1+{\frac {1}{\nu }}\left({\frac {x-\mu }{\sigma }}\right)^{2}\right)^{-{\frac {\nu +1}{2}}}} then

672-581: A portfolio with profit X s {\displaystyle X_{s}} for s ∈ S {\displaystyle s\in S} . If X 1 : S , . . . , X S : S {\displaystyle X_{1:S},...,X_{S:S}} is the order statistic the discounted maximum loss is simply − δ X 1 : S {\displaystyle -\delta X_{1:S}} , where δ {\displaystyle \delta }

756-404: A stream can be done efficiently using compressed data structures. The most popular methods are t-digest and KLL. These methods read a stream of values in a continuous fashion and can, at any time, be queried about the approximate value of a specified quantile. Both algorithms are based on a similar idea: compressing the stream of values by summarizing identical or similar values with a weight. If

840-531: Is where Q α {\displaystyle {\mathcal {Q}}_{\alpha }} is the set of probability measures which are absolutely continuous to the physical measure P {\displaystyle P} such that d Q d P ≤ α − 1 {\displaystyle {\frac {dQ}{dP}}\leq \alpha ^{-1}} almost surely . Note that d Q d P {\displaystyle {\frac {dQ}{dP}}}

924-425: Is a loss function for a set of portfolio weights w ∈ R p {\displaystyle w\in \mathbb {R} ^{p}} to be applied to the returns. Rockafellar/Uryasev proved that F α ( w , γ ) {\displaystyle F_{\alpha }(w,\gamma )} is convex with respect to γ {\displaystyle \gamma } and

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1008-430: Is also used in peer-reviewed scientific research articles. The meaning used can be derived from its context. If a distribution is symmetric, then the median is the mean (so long as the latter exists). But, in general, the median and the mean can differ. For instance, with a random variable that has an exponential distribution , any particular sample of this random variable will have roughly a 63% chance of being less than

1092-479: Is an extension beyond traditional statistics definitions. The following two examples use the Nearest Rank definition of quantile with rounding. For an explanation of this definition, see percentiles . Consider an ordered population of 10 data values [3, 6, 7, 8, 8, 10, 13, 15, 16, 20]. What are the 4-quantiles (the "quartiles") of this dataset? So the first, second and third 4-quantiles (the "quartiles") of

1176-842: Is equivalent to the expected shortfall at the minimum point. To numerically compute the expected shortfall for a set of portfolio returns, it is necessary to generate J {\displaystyle J} simulations of the portfolio constituents; this is often done using copulas . With these simulations in hand, the auxiliary function may be approximated by: F ~ α ( w , γ ) = γ + 1 ( 1 − α ) J ∑ j = 1 J [ ℓ ( w , x j ) − γ ] + {\displaystyle {\widetilde {F}}_{\alpha }(w,\gamma )=\gamma +{1 \over {(1-\alpha )J}}\sum _{j=1}^{J}[\ell (w,x_{j})-\gamma ]_{+}} This

1260-600: Is equivalent to the formulation: min γ , z , w γ + 1 ( 1 − α ) J ∑ j = 1 J z j , s.t.  z j ≥ ℓ ( w , x j ) − γ , z j ≥ 0 {\displaystyle \min _{\gamma ,z,w}\;\gamma +{1 \over {(1-\alpha )J}}\sum _{j=1}^{J}z_{j},\quad {\text{s.t. }}z_{j}\geq \ell (w,x_{j})-\gamma ,\;z_{j}\geq 0} Finally, choosing

1344-409: Is known to lead to a generally non-convex optimization problem. However, it is possible to transform the problem into a linear program and find the global solution. This property makes expected shortfall a cornerstone of alternatives to mean-variance portfolio optimization , which account for the higher moments (e.g., skewness and kurtosis) of a return distribution. Suppose that we want to minimize

1428-499: Is more common to consider the distribution of losses L = − X {\displaystyle L=-X} , the expected shortfall in this case corresponds to the right-tail conditional expectation above VaR α ⁡ ( L ) {\displaystyle \operatorname {VaR} _{\alpha }(L)} and the typical values of α {\displaystyle \alpha } are 95% and 99%: Since some formulas below were derived for

1512-576: Is the Radon–Nikodym derivative of Q {\displaystyle Q} with respect to P {\displaystyle P} . Expected shortfall can be generalized to a general class of coherent risk measures on L p {\displaystyle L^{p}} spaces ( Lp space ) with a corresponding dual characterization in the corresponding L q {\displaystyle L^{q}} dual space . The domain can be extended for more general Orlicz Hearts. If

1596-495: Is the discount factor . Given a general probability space ( Ω , F , P ) {\displaystyle (\Omega ,{\mathcal {F}},\mathbb {P} )} , let X {\displaystyle X} be a portfolio with discounted return δ X ( ω ) {\displaystyle \delta X(\omega )} for state ω ∈ Ω {\displaystyle \omega \in \Omega } . Then

1680-486: Is the lower incomplete gamma function , y {\displaystyle y} is the Euler-Mascheroni constant . Discounted maximum loss Discounted maximum loss , also known as worst-case risk measure , is the present value of the worst-case scenario for a financial portfolio . In investment, in order to protect the value of an investment, one must consider all possible alternatives to

1764-1599: Is the upper incomplete gamma function , l i ( x ) = ∫ d x ln ⁡ x {\displaystyle \mathrm {li} (x)=\int {\frac {dx}{\ln x}}} is the logarithmic integral function . If the loss of a portfolio L {\displaystyle L} follows the GEV , then the expected shortfall is equal to ES α ⁡ ( X ) = { μ + σ ( 1 − α ) ξ [ γ ( 1 − ξ , − ln ⁡ α ) − ( 1 − α ) ] if  ξ ≠ 0 , μ + σ 1 − α [ y − li ( α ) + α ln ⁡ ( − ln ⁡ α ) ] if  ξ = 0. {\displaystyle \operatorname {ES} _{\alpha }(X)={\begin{cases}\mu +{\frac {\sigma }{(1-\alpha )\xi }}{\bigl [}\gamma (1-\xi ,-\ln \alpha )-(1-\alpha ){\bigr ]}&{\text{if }}\xi \neq 0,\\\mu +{\frac {\sigma }{1-\alpha }}{\bigl [}y-{\text{li}}(\alpha )+\alpha \ln(-\ln \alpha ){\bigr ]}&{\text{if }}\xi =0.\end{cases}}} , where γ ( s , x ) {\displaystyle \gamma (s,x)}

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1848-2108: Is the upper incomplete gamma function . If the payoff of a portfolio X {\displaystyle X} follows the GEV with p.d.f. f ( x ) = { 1 σ ( 1 + ξ x − μ σ ) − 1 ξ − 1 exp ⁡ [ − ( 1 + ξ x − μ σ ) − 1 / ξ ] if  ξ ≠ 0 , 1 σ e − x − μ σ e − e − x − μ σ if  ξ = 0. {\displaystyle f(x)={\begin{cases}{\frac {1}{\sigma }}\left(1+\xi {\frac {x-\mu }{\sigma }}\right)^{-{\frac {1}{\xi }}-1}\exp \left[-\left(1+\xi {\frac {x-\mu }{\sigma }}\right)^{-{1}/{\xi }}\right]&{\text{if }}\xi \neq 0,\\{\frac {1}{\sigma }}e^{-{\frac {x-\mu }{\sigma }}}e^{-e^{-{\frac {x-\mu }{\sigma }}}}&{\text{if }}\xi =0.\end{cases}}} and c.d.f. F ( x ) = { exp ⁡ ( − ( 1 + ξ x − μ σ ) − 1 / ξ ) if  ξ ≠ 0 , exp ⁡ ( − e − x − μ σ ) if  ξ = 0. {\displaystyle F(x)={\begin{cases}\exp \left(-\left(1+\xi {\frac {x-\mu }{\sigma }}\right)^{-{1}/{\xi }}\right)&{\text{if }}\xi \neq 0,\\\exp \left(-e^{-{\frac {x-\mu }{\sigma }}}\right)&{\text{if }}\xi =0.\end{cases}}} then

1932-413: Is the expectation over all cases, or The value at risk (VaR) is given below for comparison. The expected shortfall ES q {\displaystyle \operatorname {ES} _{q}} increases as q {\displaystyle q} decreases. The 100%-quantile expected shortfall ES 1 {\displaystyle \operatorname {ES} _{1}} equals negative of

2016-562: Is the lower α {\displaystyle \alpha } - quantile and 1 A ( x ) = { 1 if  x ∈ A 0 else {\displaystyle 1_{A}(x)={\begin{cases}1&{\text{if }}x\in A\\0&;{\text{else}}\end{cases}}} is the indicator function . Note, that the second term vanishes for random variables with continuous distribution functions. The dual representation

2100-708: Is the payoff of a portfolio at some future time and 0 < α < 1 {\displaystyle 0<\alpha <1} then we define the expected shortfall as where VaR γ {\displaystyle \operatorname {VaR} _{\gamma }} is the value at risk . This can be equivalently written as where x α = inf { x ∈ R : P ( X ≤ x ) ≥ α } = − VaR α ⁡ ( X ) {\displaystyle x_{\alpha }=\inf\{x\in \mathbb {R} :P(X\leq x)\geq \alpha \}=-\operatorname {VaR} _{\alpha }(X)}

2184-402: Is the standard normal p.d.f., Φ ( x ) {\displaystyle \Phi (x)} is the standard normal c.d.f., so Φ − 1 ( α ) {\displaystyle \Phi ^{-1}(\alpha )} is the standard normal quantile. If the loss of a portfolio L {\displaystyle L} follows the normal distribution,

2268-445: Is the standard t-distribution p.d.f., T ( x ) {\displaystyle \mathrm {T} (x)} is the standard t-distribution c.d.f., so T − 1 ( α ) {\displaystyle \mathrm {T} ^{-1}(\alpha )} is the standard t-distribution quantile. If the loss of a portfolio L {\displaystyle L} follows generalized Student's t-distribution,

2352-444: Is the value of the p -quantile for 0 < p < 1 (or equivalently is the k -th q -quantile for p = k / q ), where μ is the distribution's arithmetic mean , and where σ is the distribution's standard deviation . In particular, the median ( p = k / q = 1/2) is never more than one standard deviation from the mean. The above formula can be used to bound the value μ + zσ in terms of quantiles. When z ≥ 0 ,

2436-916: The GPD with p.d.f. and the c.d.f. then the expected shortfall is equal to and the VaR is equal to If the loss of a portfolio L {\displaystyle L} follows the Weibull distribution with p.d.f. f ( x ) = { k λ ( x λ ) k − 1 e − ( x / λ ) k if  x ≥ 0 , 0 if  x < 0. {\displaystyle f(x)={\begin{cases}{\frac {k}{\lambda }}\left({\frac {x}{\lambda }}\right)^{k-1}e^{-(x/\lambda )^{k}}&{\text{if }}x\geq 0,\\0&{\text{if }}x<0.\end{cases}}} and

2520-411: The cumulative distribution function ) to the values {1/ q , 2/ q , …, ( q − 1)/ q }. As in the computation of, for example, standard deviation , the estimation of a quantile depends upon whether one is operating with a statistical population or with a sample drawn from it. For a population, of discrete values or for a continuous population density, the k -th q -quantile is the data value where

2604-449: The essential infimum . As an example, assume that a portfolio is currently worth 100, and the discount factor is 0.8 (corresponding to an interest rate of 25%): In this case the maximum loss is from 100 to 20 = 80, so the discounted maximum loss is simply 80 × 0.8 = 64 {\displaystyle 80\times 0.8=64} Quantile In statistics and probability , quantiles are cut points dividing

Expected shortfall - Misplaced Pages Continue

2688-459: The expected value of the portfolio. For a given portfolio, the expected shortfall ES q {\displaystyle \operatorname {ES} _{q}} is greater than or equal to the Value at Risk VaR q {\displaystyle \operatorname {VaR} _{q}} at the same q {\displaystyle q} level. Expected shortfall, in its standard form,

2772-434: The range of a probability distribution into continuous intervals with equal probabilities, or dividing the observations in a sample in the same way. There is one fewer quantile than the number of groups created. Common quantiles have special names, such as quartiles (four groups), deciles (ten groups), and percentiles (100 groups). The groups created are termed halves, thirds, quarters, etc., though sometimes

2856-416: The " p -quantile" is based on a real number p with 0 < p < 1 then p replaces k / q in the above formulas. This broader terminology is used when quantiles are used to parameterize continuous probability distributions . Moreover, some software programs (including Microsoft Excel ) regard the minimum and maximum as the 0th and 100th percentile, respectively. However, this broader terminology

2940-872: The VaR is equal to VaR α ⁡ ( X ) = { − μ − σ ξ [ ( − ln ⁡ α ) − ξ − 1 ] if  ξ ≠ 0 , − μ + σ ln ⁡ ( − ln ⁡ α ) if  ξ = 0. {\displaystyle \operatorname {VaR} _{\alpha }(X)={\begin{cases}-\mu -{\frac {\sigma }{\xi }}\left[(-\ln \alpha )^{-\xi }-1\right]&{\text{if }}\xi \neq 0,\\-\mu +\sigma \ln(-\ln \alpha )&{\text{if }}\xi =0.\end{cases}}} , where Γ ( s , x ) {\displaystyle \Gamma (s,x)}

3024-412: The appropriate index; the corresponding data value is the k -th q -quantile. On the other hand, if I p is an integer then any number from the data value at that index to the data value of the next index can be taken as the quantile, and it is conventional (though arbitrary) to take the average of those two values (see Estimating quantiles from a sample ). If, instead of using integers k and q ,

3108-670: The c.d.f. F ( x ) = ( 1 + e − x − μ s ) − 1 {\displaystyle F(x)=\left(1+e^{-{\frac {x-\mu }{s}}}\right)^{-1}} then the expected shortfall is equal to ES α ⁡ ( X ) = − μ + s ln ⁡ ( 1 − α ) 1 − 1 α α {\displaystyle \operatorname {ES} _{\alpha }(X)=-\mu +s\ln {\frac {(1-\alpha )^{1-{\frac {1}{\alpha }}}}{\alpha }}} . If

3192-689: The c.d.f. F ( x ) = { 1 − e − λ x if  x ≥ 0 , 0 if  x < 0. {\displaystyle F(x)={\begin{cases}1-e^{-\lambda x}&{\text{if }}x\geq 0,\\0&{\text{if }}x<0.\end{cases}}} then the expected shortfall is equal to ES α ⁡ ( L ) = − ln ⁡ ( 1 − α ) + 1 λ {\displaystyle \operatorname {ES} _{\alpha }(L)={\frac {-\ln(1-\alpha )+1}{\lambda }}} . If

3276-409: The c.d.f. F ( x ) = { 1 − e − ( x / λ ) k if  x ≥ 0 , 0 if  x < 0. {\displaystyle F(x)={\begin{cases}1-e^{-(x/\lambda )^{k}}&{\text{if }}x\geq 0,\\0&{\text{if }}x<0.\end{cases}}} then

3360-418: The c.d.f. F ( x ) = { 1 − ( x m / x ) a if  x ≥ x m , 0 if  x < x m . {\displaystyle F(x)={\begin{cases}1-(x_{m}/x)^{a}&{\text{if }}x\geq x_{m},\\0&{\text{if }}x<x_{m}.\end{cases}}} then

3444-448: The calculation of ES 0.05 {\displaystyle \operatorname {ES} _{0.05}} , the expectation in the worst 5% of cases. These cases belong to (are a subset of) row 1 in the profit table, which have a profit of −100 (total loss of the 100 invested). The expected profit for these cases is −100. Now consider the calculation of ES 0.20 {\displaystyle \operatorname {ES} _{0.20}} ,

Expected shortfall - Misplaced Pages Continue

3528-430: The case for the median (2-quantile) of a uniform probability distribution on a set of even size. Quantiles can also be applied to continuous distributions, providing a way to generalize rank statistics to continuous variables (see percentile rank ). When the cumulative distribution function of a random variable is known, the q -quantiles are the application of the quantile function (the inverse function of

3612-409: The cumulative distribution function crosses k / q . That is, x is a k -th q -quantile for a variable X if and For a finite population of N equally probable values indexed 1, …, N from lowest to highest, the k -th q -quantile of this population can equivalently be computed via the value of I p = N k / q . If I p is not an integer, then round up to the next integer to get

3696-770: The data, in essence the data are simply numbers or more generally, a set of items that can be ordered. These algorithms are computer science derived methods. Another class of algorithms exist which assume that the data are realizations of a random process. These are statistics derived methods, sequential nonparametric estimation algorithms in particular. There are a number of such algorithms such as those based on stochastic approximation or Hermite series estimators. These statistics based algorithms typically have constant update time and space complexity, but have different error bound guarantees compared to computer science type methods and make more assumptions. The statistics based algorithms do present certain advantages however, particularly in

3780-441: The dataset [3, 6, 7, 8, 8, 10, 13, 15, 16, 20] are [7, 9, 15]. If also required, the zeroth quartile is 3 and the fourth quartile is 20. Consider an ordered population of 11 data values [3, 6, 7, 8, 8, 9, 10, 13, 15, 16, 20]. What are the 4-quantiles (the "quartiles") of this dataset? So the first, second and third 4-quantiles (the "quartiles") of the dataset [3, 6, 7, 8, 8, 9, 10, 13, 15, 16, 20] are [7, 9, 15]. If also required,

3864-431: The default. The standard error of a quantile estimate can in general be estimated via the bootstrap . The Maritz–Jarrett method can also be used. The sample median is the most examined one amongst quantiles, being an alternative to estimate a location parameter, when the expected value of the distribution does not exist, and hence the sample mean is not a meaningful estimator of a population characteristic. Moreover,

3948-509: The discounted maximum loss can be written as − e s s . i n f ⁡ δ X = − sup δ { x ∈ R : P ( X ≥ x ) = 1 } {\displaystyle -\operatorname {ess.inf} \delta X=-\sup \delta \{x\in \mathbb {R} :\mathbb {P} (X\geq x)=1\}} where e s s . i n f {\displaystyle \operatorname {ess.inf} } denotes

4032-445: The distribution is discrete, then the distribution of the sample median and the other quantiles fails to be Normal (see examples in https://stats.stackexchange.com/a/86638/28746 ). A solution to this problem is to use an alternative definition of sample quantiles through the concept of the "mid-distribution" function, which is defined as The definition of sample quantiles through the concept of mid-distribution function can be seen as

4116-419: The end of the period: Now assume that we paid 100 at the beginning of the period for this portfolio. Then the profit in each case is ( ending value −100) or: From this table let us calculate the expected shortfall ES q {\displaystyle \operatorname {ES} _{q}} for a few values of q {\displaystyle q} : To see how these values were calculated, consider

4200-517: The estimate for the p -quantile (the k -th q -quantile, where p = k / q ) from a sample of size N by computing a real valued index h . When h is an integer, the h -th smallest of the N values, x h , is the quantile estimate. Otherwise a rounding or interpolation scheme is used to compute the quantile estimate from h , x ⌊ h ⌋ , and x ⌈ h ⌉ . (For notation, see floor and ceiling functions ). The first three are piecewise constant, changing abruptly at each data point, while

4284-429: The expectation in the worst 20 out of 100 cases. These cases are as follows: 10 cases from row one, and 10 cases from row two (note that 10+10 equals the desired 20 cases). For row 1 there is a profit of −100, while for row 2 a profit of −20. Using the expected value formula we get Similarly for any value of q {\displaystyle q} . We select as many rows starting from the top as are necessary to give

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4368-431: The expected shortfall is equal to ES α ⁡ ( L ) = x m a ( 1 − α ) 1 / a ( a − 1 ) {\displaystyle \operatorname {ES} _{\alpha }(L)={\frac {x_{m}a}{(1-\alpha )^{1/a}(a-1)}}} . If the loss of a portfolio L {\displaystyle L} follows

4452-530: The expected shortfall is equal to ES α ⁡ ( L ) = λ 1 − α Γ ( 1 + 1 k , − ln ⁡ ( 1 − α ) ) {\displaystyle \operatorname {ES} _{\alpha }(L)={\frac {\lambda }{1-\alpha }}\Gamma \left(1+{\frac {1}{k}},-\ln(1-\alpha )\right)} , where Γ ( s , x ) {\displaystyle \Gamma (s,x)}

4536-599: The expected shortfall is equal to ES α ⁡ ( L ) = μ + σ ν + ( T − 1 ( α ) ) 2 ν − 1 τ ( T − 1 ( α ) ) 1 − α {\displaystyle \operatorname {ES} _{\alpha }(L)=\mu +\sigma {\frac {\nu +(\mathrm {T} ^{-1}(\alpha ))^{2}}{\nu -1}}{\frac {\tau (\mathrm {T} ^{-1}(\alpha ))}{1-\alpha }}} . If

4620-412: The expected shortfall is equal to ES α ⁡ ( L ) = μ + σ φ ( Φ − 1 ( α ) ) 1 − α {\displaystyle \operatorname {ES} _{\alpha }(L)=\mu +\sigma {\frac {\varphi (\Phi ^{-1}(\alpha ))}{1-\alpha }}} . If the payoff of

4704-1047: The expected shortfall is equal to ES α ⁡ ( X ) = { − μ − σ α ξ [ Γ ( 1 − ξ , − ln ⁡ α ) − α ] if  ξ ≠ 0 , − μ − σ α [ li ( α ) − α ln ⁡ ( − ln ⁡ α ) ] if  ξ = 0. {\displaystyle \operatorname {ES} _{\alpha }(X)={\begin{cases}-\mu -{\frac {\sigma }{\alpha \xi }}{\big [}\Gamma (1-\xi ,-\ln \alpha )-\alpha {\big ]}&{\text{if }}\xi \neq 0,\\-\mu -{\frac {\sigma }{\alpha }}{\big [}{\text{li}}(\alpha )-\alpha \ln(-\ln \alpha ){\big ]}&{\text{if }}\xi =0.\end{cases}}} and

4788-1174: The expected shortfall is equal to ES α ⁡ ( X ) = − μ + σ ν + ( T − 1 ( α ) ) 2 ν − 1 τ ( T − 1 ( α ) ) α {\displaystyle \operatorname {ES} _{\alpha }(X)=-\mu +\sigma {\frac {\nu +(\mathrm {T} ^{-1}(\alpha ))^{2}}{\nu -1}}{\frac {\tau (\mathrm {T} ^{-1}(\alpha ))}{\alpha }}} , where τ ( x ) = Γ ( ν + 1 2 ) Γ ( ν 2 ) π ν ( 1 + x 2 ν ) − ν + 1 2 {\displaystyle \tau (x)={\frac {\Gamma {\bigl (}{\frac {\nu +1}{2}}{\bigr )}}{\Gamma {\bigl (}{\frac {\nu }{2}}{\bigr )}{\sqrt {\pi \nu }}}}{\Bigl (}1+{\frac {x^{2}}{\nu }}{\Bigr )}^{-{\frac {\nu +1}{2}}}}

4872-607: The expected shortfall is equal to ES α ⁡ ( X ) = − μ + σ φ ( Φ − 1 ( α ) ) α {\displaystyle \operatorname {ES} _{\alpha }(X)=-\mu +\sigma {\frac {\varphi (\Phi ^{-1}(\alpha ))}{\alpha }}} , where φ ( x ) = 1 2 π e − x 2 2 {\displaystyle \varphi (x)={\frac {1}{\sqrt {2\pi }}}e^{-{\frac {x^{2}}{2}}}}

4956-1044: The expected shortfall of a portfolio. The key contribution of Rockafellar and Uryasev in their 2000 paper is to introduce the auxiliary function F α ( w , γ ) {\displaystyle F_{\alpha }(w,\gamma )} for the expected shortfall: F α ( w , γ ) = γ + 1 1 − α ∫ ℓ ( w , x ) ≥ γ [ ℓ ( w , x ) − γ ] + p ( x ) d x {\displaystyle F_{\alpha }(w,\gamma )=\gamma +{1 \over {1-\alpha }}\int _{\ell (w,x)\geq \gamma }\left[\ell (w,x)-\gamma \right]_{+}p(x)\,dx} Where γ = VaR α ⁡ ( X ) {\displaystyle \gamma =\operatorname {VaR} _{\alpha }(X)} and ℓ ( w , x ) {\displaystyle \ell (w,x)}

5040-444: The family of data sketches that are subsets of Streaming Algorithms with useful properties: t-digest or KLL sketches can be combined. Computing the sketch for a very large vector of values can be split into trivially parallel processes where sketches are computed for partitions of the vector in parallel and merged later. The algorithms described so far directly approximate the empirical quantiles without any particular assumptions on

5124-418: The initial investment. How one does this comes down to personal preference; however, the worst possible alternative is generally considered to be the benchmark against which all other options are measured. The present value of this worst possible outcome is the discounted maximum loss. Given a finite state space S {\displaystyle S} , let X {\displaystyle X} be

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5208-413: The last six use linear interpolation between data points, and differ only in how the index h used to choose the point along the piecewise linear interpolation curve, is chosen. Mathematica , Matlab , R and GNU Octave programming languages support all nine sample quantile methods. SAS includes five sample quantile methods, SciPy and Maple both include eight, EViews and Julia include

5292-565: The left-tail case and some for the right-tail case, the following reconciliations can be useful: If the payoff of a portfolio X {\displaystyle X} follows the normal (Gaussian) distribution with p.d.f. f ( x ) = 1 2 π σ e − ( x − μ ) 2 2 σ 2 {\displaystyle f(x)={\frac {1}{{\sqrt {2\pi }}\sigma }}e^{-{\frac {(x-\mu )^{2}}{2\sigma ^{2}}}}} then

5376-481: The loss of a portfolio L {\displaystyle L} follows the exponential distribution with p.d.f. f ( x ) = { λ e − λ x if  x ≥ 0 , 0 if  x < 0. {\displaystyle f(x)={\begin{cases}\lambda e^{-\lambda x}&{\text{if }}x\geq 0,\\0&{\text{if }}x<0.\end{cases}}} and

5460-595: The loss of a portfolio L {\displaystyle L} follows the logistic distribution , the expected shortfall is equal to ES α ⁡ ( L ) = μ + s − α ln ⁡ α − ( 1 − α ) ln ⁡ ( 1 − α ) 1 − α {\displaystyle \operatorname {ES} _{\alpha }(L)=\mu +s{\frac {-\alpha \ln \alpha -(1-\alpha )\ln(1-\alpha )}{1-\alpha }}} . If

5544-587: The loss of a portfolio L {\displaystyle L} follows the Pareto distribution with p.d.f. f ( x ) = { a x m a x a + 1 if  x ≥ x m , 0 if  x < x m . {\displaystyle f(x)={\begin{cases}{\frac {ax_{m}^{a}}{x^{a+1}}}&{\text{if }}x\geq x_{m},\\0&{\text{if }}x<x_{m}.\end{cases}}} and

5628-695: The loss of a portfolio L {\displaystyle L} follows the Laplace distribution, the expected shortfall is equal to If the payoff of a portfolio X {\displaystyle X} follows the logistic distribution with p.d.f. f ( x ) = 1 s e − x − μ s ( 1 + e − x − μ s ) − 2 {\displaystyle f(x)={\frac {1}{s}}e^{-{\frac {x-\mu }{s}}}\left(1+e^{-{\frac {x-\mu }{s}}}\right)^{-2}} and

5712-416: The mean, then quantiles may be more useful descriptive statistics than means and other moment-related statistics. Closely related is the subject of least absolute deviations , a method of regression that is more robust to outliers than is least squares, in which the sum of the absolute value of the observed errors is used in place of the squared error. The connection is that the mean is the single estimate of

5796-430: The mean. This is because the exponential distribution has a long tail for positive values but is zero for negative numbers. Quantiles are useful measures because they are less susceptible than means to long-tailed distributions and outliers. Empirically, if the data being analyzed are not actually distributed according to an assumed distribution, or if there are other potential sources for outliers that are far removed from

5880-463: The non-stationary streaming setting i.e. time-varying data. The algorithms of both classes, along with some respective advantages and disadvantages have been recently surveyed. Standardized test results are commonly reported as a student scoring "in the 80th percentile", for example. This uses an alternative meaning of the word percentile as the interval between (in this case) the 80th and the 81st scalar percentile. This separate meaning of percentile

5964-590: The payoff of a portfolio X {\displaystyle X} follows the Laplace distribution with the p.d.f. and the c.d.f. then the expected shortfall is equal to ES α ⁡ ( X ) = − μ + b ( 1 − ln ⁡ 2 α ) {\displaystyle \operatorname {ES} _{\alpha }(X)=-\mu +b(1-\ln 2\alpha )} for α ≤ 0.5 {\displaystyle \alpha \leq 0.5} . If

6048-632: The payoff of a portfolio X {\displaystyle X} or a corresponding loss L = − X {\displaystyle L=-X} follows a specific continuous distribution. In the former case, the expected shortfall corresponds to the opposite number of the left-tail conditional expectation below − VaR α ⁡ ( X ) {\displaystyle -\operatorname {VaR} _{\alpha }(X)} : Typical values of α {\textstyle \alpha } in this case are 5% and 1%. For engineering or actuarial applications it

6132-545: The sample median is a more robust estimator than the sample mean. One peculiarity of the sample median is its asymptotic distribution: when the sample comes from a continuous distribution, then the sample median has the anticipated Normal asymptotic distribution, This extends to the other quantiles, where f ( x p ) is the value of the distribution density at the p -th population quantile ( x p = F − 1 ( p ) {\displaystyle x_{p}=F^{-1}(p)} ). But when

6216-422: The six piecewise linear functions, Stata includes two, Python includes two, and Microsoft Excel includes two. Mathematica, SciPy and Julia support arbitrary parameters for methods which allow for other, non-standard, methods. The estimate types and interpolation schemes used include: Notes: Of the techniques, Hyndman and Fan recommend R-8, but most statistical software packages have chosen R-6 or R-7 as

6300-420: The stream is made of a repetition of 100 times v1 and 100 times v2, there is no reason to keep a sorted list of 200 elements, it is enough to keep two elements and two counts to be able to recover the quantiles. With more values, these algorithms maintain a trade-off between the number of unique values stored and the precision of the resulting quantiles. Some values may be discarded from the stream and contribute to

6384-404: The terms for the quantile are used for the groups created, rather than for the cut points. q - quantiles are values that partition a finite set of values into q subsets of (nearly) equal sizes. There are q − 1 partitions of the q -quantiles, one for each integer k satisfying 0 < k < q . In some cases the value of a quantile may not be uniquely determined, as can be

6468-411: The time, what is our average loss". Expected shortfall can also be written as a distortion risk measure given by the distortion function Example 1. If we believe our average loss on the worst 5% of the possible outcomes for our portfolio is EUR 1000, then we could say our expected shortfall is EUR 1000 for the 5% tail. Example 2. Consider a portfolio that will have the following possible values at

6552-649: The underlying distribution for X {\displaystyle X} is a continuous distribution then the expected shortfall is equivalent to the tail conditional expectation defined by TCE α ⁡ ( X ) = E [ − X ∣ X ≤ − VaR α ⁡ ( X ) ] {\displaystyle \operatorname {TCE} _{\alpha }(X)=E[-X\mid X\leq -\operatorname {VaR} _{\alpha }(X)]} . Informally, and non-rigorously, this equation amounts to saying "in case of losses so severe that they occur only alpha percent of

6636-452: The value μ + zσ for z = −3 will never exceed Q ( p = 0.1) , the first decile. One problem which frequently arises is estimating a quantile of a (very large or infinite) population based on a finite sample of size N . Modern statistical packages rely on a number of techniques to estimate the quantiles. Hyndman and Fan compiled a taxonomy of nine algorithms used by various software packages. All methods compute Q p ,

6720-409: The value that is z standard deviations above the mean has a lower bound μ + z σ ≥ Q ( z 2 1 + z 2 ) ,   f o r   z ≥ 0. {\displaystyle \mu +z\sigma \geq Q\left({\frac {z^{2}}{1+z^{2}}}\right)\,,\mathrm {~for~} z\geq 0.} For example,

6804-631: The value that is z = 1 standard deviation above the mean is always greater than or equal to Q ( p = 0.5) , the median, and the value that is z = 2 standard deviations above the mean is always greater than or equal to Q ( p = 0.8) , the fourth quintile. When z ≤ 0 , there is instead an upper bound μ + z σ ≤ Q ( 1 1 + z 2 ) ,   f o r   z ≤ 0. {\displaystyle \mu +z\sigma \leq Q\left({\frac {1}{1+z^{2}}}\right)\,,\mathrm {~for~} z\leq 0.} For example,

6888-432: The weight of a nearby value without changing the quantile results too much. The t-digest maintains a data structure of bounded size using an approach motivated by k -means clustering to group similar values. The KLL algorithm uses a more sophisticated "compactor" method that leads to better control of the error bounds at the cost of requiring an unbounded size if errors must be bounded relative to p . Both methods belong to

6972-401: The worst losses. On the other hand, unlike the discounted maximum loss , even for lower values of q {\displaystyle q} the expected shortfall does not consider only the single most catastrophic outcome. A value of q {\displaystyle q} often used in practice is 5%. Expected shortfall is considered a more useful risk measure than VaR because it is

7056-608: The zeroth quartile is 3 and the fourth quartile is 20. For any population probability distribution on finitely many values, and generally for any probability distribution with a mean and variance, it is the case that μ − σ ⋅ 1 − p p ≤ Q ( p ) ≤ μ + σ ⋅ p 1 − p , {\displaystyle \mu -\sigma \cdot {\sqrt {\frac {1-p}{p}}}\leq Q(p)\leq \mu +\sigma \cdot {\sqrt {\frac {p}{1-p}}}\,,} where Q(p)

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