Longevidad y Mortalidad: ¿dos caras de la misma … · Concepto SCR Se trata de un colch on de...

Longevidad y Mortalidad: ¿dos caras de lamisma moneda en la formula estandar de

Solvencia II?Seminario de Investigacion

Master de Ciencias Actuariales y Financieras UB

Barcelona

Noviembre de 2017

Jaume Belles [email protected], [email protected]

Objetivo principal

Explicaros un artículo en el que hablamos de cómo se tratan las

dependencias entre los riesgos de longevidad y mortalidad en el cálculo

del SCR por FE

3 carterasde segurosde Vidainventadas

Calculamos SCR de longevidad, de mortalidad y la contribución conjunta según expresión cerrada fórmula estándar

Hacemos una propuestasobre como calcular losmismos SCR de longevidad, de mortalidad y la contribución conjunta según fórmula estándar

pero en base a simulaciones

Hacemos varias pruebas sobre el efecto que cambiar la estructura de dependencia de la fórmula estándar tiene en el SCR conjunto de longevidad y mortalidad

Concluimos

Objetivo principal



del SCR por FE

Objetivo principal



del SCR por FE

Objetivo secundario

Plantearos dudas, preguntas, cuestiones

que os puedan ser útiles o que os puedan ayudar

a plantear posibles trabajos de investigación

Objetivo principal



del SCR por FE

Objetivo secundario




¿Qué es un BEL?

Objetivo principal



del SCR por FE

Objetivo secundario




¿Qué es un BEL?

¿Qué es un SCR?

Objetivo principal



del SCR por FE

Objetivo secundario




¿Qué es un BEL?

¿Qué es un SCR?

¿Qué quiere decir

Fórmula Estándar?

Objetivo principal



del SCR por FE

Objetivo secundario




¿Qué es un BEL?

¿Qué es un SCR?

¿Qué quiere decir

Fórmula Estándar?

¿Qué son las tablas de

mortalidad?

Objetivo principal



del SCR por FE

Objetivo secundario




¿Qué es un BEL?

¿Qué es un SCR?

¿Qué quiere decir

Fórmula Estándar?


mortalidad?

¿Qué son los riesgos de

longevidad y mortalidad?

¿Qué entendemos por dependencias entre

riesgos?

Objetivo principal



del SCR por FE

Objetivo secundario




¿Qué es un BEL?

¿Qué es un SCR?

¿Qué quiere decir

Fórmula Estándar?


mortalidad?




riesgos?

Objetivo principal



del SCR por FE

Objetivo secundario




¿Qué es un BEL?

¿Qué es un SCR?

¿Qué quiere decir

Fórmula Estándar?


mortalidad?



¿Por qué las correlaciones lineales

no siempre son adecuadas?


riesgos?

Objetivo principal



del SCR por FE

Objetivo secundario




¿Qué es un BEL?

¿Qué es un SCR?

¿Qué quiere decir

Fórmula Estándar?


mortalidad?





¿En qué consisten las

cópulas?

¿Qué es la Función Actuarial?


riesgos?

Objetivo principal



del SCR por FE

Objetivo secundario




¿Qué es un BEL?

¿Qué es un SCR?

¿Qué quiere decir

Fórmula Estándar?


mortalidad?






cópulas?

¿Qué son las funciones fundamentales?



riesgos?

Objetivo principal



del SCR por FE

Objetivo secundario




¿Qué es un BEL?

¿Qué es un SCR?

¿Qué quiere decir

Fórmula Estándar?


mortalidad?






cópulas?

Preliminares Sobre el calculo del SCR en la FE

Principales mensajes a recordar sobre el SCR

Concepto SCR

Se trata de un colchon de capital para intentarevitar insolvencias en las companıa de seguroseuropeas.La carga de capital de solvencia (SCR) deberıacorresponderse con el VaR99.5%(Z), siendo VaRla medida de riesgo ‘Value-at-Risk’ y Z lavariable aleatoria ‘fondos propios basicos de lacompanıa a un ano’

Principales caracterısticas del SCR por SF

- Estructura modular: Modulo ' Tipologıa deriesgo

- Dependencias entre los modulos: a traves decorrelaciones lineales

- En cada modulo: 1 de 4 opciones de calculo(ver opciones)

- En cada modulo: hipotesis sobre v.a.subyacentes Normales o Lognormales

- Admite ciertas simplificaciones

Opciones de calculo del SCR

La cuantificacion de todo o parte de este SCRpuede hacerse mediante: formula estandar (SF),parametros especıficos de la empresa (USP) omodelos internos totales o parciales (IM).

Parametros especıficos de la empresa USP

Parametros especıficos de la empresa: soloalgunos modulos. SF pero con parametros quetienen en cuenta datos historicos de la companıa.

Modelos internos totales o parciales IM

- Principal ventaja: libertad para cuantificar elSCR como mejor considere la companıa

- Principal inconveniente: coste en tiempo yrecursos

Jaume Belles Sampera 2 / 29

Preliminares Sobre el calculo del SCR en la FE

SCR Basico

D2 =

Intan RestoAgrIntan 1 1

RestoAgr 1 1

D1 =

Merc Contr Salud Vida No Vida

Merc 1 0.25 0.25 0.25 0.25Contr 0.25 1 0.25 0.25 0.50Salud 0.25 0.25 1 0.25 0Vida 0.25 0.25 0.25 1 0

No Vida 0.25 0.5 0 0 1


Preliminares Tipologıas de calculo del VaR en la FE

Tipos de contribuciones al SCR

Ej. A1 - Riesgo de Prima y Reserva en No Vida

SCRnlpremres = 3σnl Vnl

Ej. A2 - Riesgo de Contraparte

SCRdef ,1 = 3

√√√√√Vinter +∑j

1, 5PDj (1 − PDj )

2, 5 − PDj

∑PDj

LGD2i

Ej. B1 - Riesgo de Mortalidad en Vida

Perdida en FFPP basicos resultante de ↑ 15% tasas de mortalidadpara el calculo de las provisiones tecnicas

Ej. B2 - Riesgo Catastrofico RC Autos

Perdida en FFPP basicos resultante de una perdida instantanea igualaLmotor = max (6.000.000,

50.000√

Na + 0, 05Nb + 0, 95 min(Nb, 20.000

))


Preliminares Tipologıas de calculo del VaR en la FE

Mapa aproximado de tipos de contribucion al SCR


Preliminares Dependencias entre riesgos en la FE

Estructuras de dependencia en la FE


Preliminares Dependencias entre riesgos en la FE

Dependencias para los riesgos de suscripcion Vida

- Como ya hemos comentado, lasdependencias entre (sub)riesgos en laFormula Estandar se expresan a travesde matrices de correlacion lineal

- Por ejemplo, en el caso de los riesgos desuscripcion Vida, la matriz propuestapor la FE es la siguiente:

Mort Long Disab Lapse Exp Revis CatMort 1 −0.25 0.25 0 0.25 0 0.25Long −0.25 1 0 0.25 0.25 0.25 0Disab 0.25 0 1 0 0.5 0 0.25Lapse 0 0.25 0 1 0.5 0 0.25Exp 0.25 0.25 0.5 0.5 1 0.5 25

Revis 0 0.25 0 0 0.5 1 0Cat 0.25 0 0.25 0.25 0.25 0 1


Preliminares Conceptos generales sobre dependencias

What do we know about dependencies between risks?

- It is not easy to determine dependencies between risks: type?formalization? quantification?

- Lack of enough amount of data ⇒ main source of troubles whentrying to calibrate dependencies

- It is not easy to appropriately understand and communicatedependencies between risks

- Dependencies between a pair of risks are not easy, but increasingdimensionality (i.e. playing with more than two risks) seriouslyincreases difficulties



Correlation measures 6= indicators of real dependency

- Linear correlation (Pearson’s correlation) is only one of the measuresused to simplify the description of dependency between risks

- Other dependency measures (which also simplify what a dependencyis) are the following

- Kendall’s tau- Spearman’s correlation coefficient- Tail dependency coefficients (upper and lower tail dependence)

- Causal relationships, multivariate joint distributions and copulas aretools to better depict dependencies between risks



Copulas and Sklar’s Theorem

- A 2-dimensional copula C is adistribution function on [0, 1]2 withstandard uniform distribution functions

- Sklar’s representation theorem Let FX

be a bivariate distribution function withunivariate distribution functions F1 andF2. Then there exists a copula C suchthat for all x ∈ R2,

FX (x1, x2) = C (F1(x1),F2(x2))

- If the marginal distribution functions arecontinuous, then the copula C is unique

- There is a huge amount of copulas



Copulas and Sklar’s Theorem

- A 2-dimensional copula C is adistribution function on [0, 1]2 withstandard uniform distribution functions

- Sklar’s representation theorem Let FX

be a bivariate distribution function withunivariate distribution functions F1 andF2. Then there exists a copula C suchthat for all x ∈ R2,

FX (x1, x2) = C (F1(x1),F2(x2))

- If the marginal distribution functions arecontinuous, then the copula C is unique

- There is a huge amount of copulas

- Copulas may be classified by differentcriteria. An usual classification is theone provided, for instance, in Mc Neil etal (2005). Examples:

? Fundamental copulas: representingspecial dependence structures(counter-comonotonicity,comonotonicity,...)

? Implicit copulas: its functional formmatches one known multivariatedistribution function (Normal, Studentt,...)

? Explicit copulas: they may berepresented by simple closed formformulas (Frank, Clayton, Gumbel,...)



Some transformations on copulas

Several functional transformations may be done in a given copula C .Two of these transformations are the following:

- Survival copula of C , denoted by C . It is the copula obtained by

C (u1, u2) = C (1− u1, 1− u2) + u1 + u2 − 1

- Co-copula of C , denoted by C∗. It is not a copula. It is obtained by

C∗(u1, u2) = 1− C (1− u1, 1− u2)


BEL BEL y ejemplos

Concepto

- BEL son las siglas de Best Estimate Liabilities. Son importes que tratan derepresentar las mejores estimaciones de los compromisos economicos que hemosasumido con los tomadores de las polizas.

- Por norma general, se calculan como valores actuales de las diferencias entre lasprestaciones futuras a las que estimamos que tendremos que hacer frente y losingresos (habitualmente, primas) que estimamos que percibiremos en funcion delcontrato en vigor.

- Destacar que cualquier prestacion o ingreso futuro hay que descontarloeconomicamente (de algun modo) al momento actual.

- Y que, a diferencia de lo que sucede cuando se tarifica, hay que tener en cuentatantos aspectos del comportamiento del tomador como seamos capaces (porejemplo: rescates).


BEL BEL y ejemplos

Un ejemplo

BEL(X1; rt ;Q; st ;YYYY ) =

N1∑i=1

∞∑t=max(65−x1i ,0)

(1 · tp(Q; st)x1i · f (rt ;YYYY ))−

N1∑i=1

max(65−x1i ,0)∑t=0

(p1(x1i ) · tp(Q; st)x1i ∆f (rt ;YYYY ))

donde

tp(Q; st)x1i = Πtl=0 (1 − (qx1i+l + st − qx1i+l · sl)) es la probabilidad de

que un asegurado de edad x1i a DD/MM/YYYY haya sobrevivido y nohaya rescatado despues de t anos segun la tabla de mortalidad Q y lastasas de rescate st .

f (rt ;YYYY ) es el factor de actualizacion de 1 unidad monetaria tras tanos desde la fecha DD/MM/YYYY y en base a los tipos de interes rt .

N1 y p1(x1i ) son el numero total de asegurados de la cartera delproducto 1 (45000) y la prima a pagar (calculada en 31/12/2012) porel asegurado de edad x1i , respectivamente.


Sobre el artıculo

Motivation

What if analysis...

Guideline 51 - The actuarial function of anundertaking using an internal model(BoS-14/253)

1.100. The undertaking should require theactuarial function to contribute to specify whichrisks within their domain of expertise are coveredby the internal model. The actuarial functionshould also contribute to how dependenciesbetween these risks and dependencies betweenthese risks and other risks are derived. Thiscontribution should be based on a technicalanalysis and should reflect the experience andexpertise of the function.


Sobre el artıculo

Motivation

What if analysis...

Guideline 51 - The actuarial function of anundertaking using an internal model(BoS-14/253)

1.100. The undertaking should require theactuarial function to contribute to specify whichrisks within their domain of expertise are coveredby the internal model. The actuarial functionshould also contribute to how dependenciesbetween these risks and dependencies betweenthese risks and other risks are derived. Thiscontribution should be based on a technicalanalysis and should reflect the experience andexpertise of the function.

...with a practical approach

- What do we know about dependencies (ingeneral)?

- What do we know about dependencies (inthe SCR Standard Formula)?

- More precisely, what about dependenciesbetween longevity and other risks?

- Are we able to put some of this knowledgedown into practice?


3 carterasde segurosde Vidainventadas

Calculamos SCR de longevidad, de mortalidad y la contribución conjunta según expresión cerrada fórmula estándar

Hacemos una propuestasobre como calcular losmismos SCR de longevidad, de mortalidad y la contribución conjunta según fórmula estándar

pero en base a simulaciones

Hacemos varias pruebas sobre el efecto que cambiar la estructura de dependencia de la fórmula estándar tiene en el SCR conjunto de longevidad y mortalidad

Concluimos

Sobre el artıculo Resumen grafico

How dependencies may be put down into practice? IBy means of underlying causal relationships. For instance

- Extremely opposite dependence

(u1, u2) s.t. u2 = 1− u1 (1)

- Extremely direct dependence(u1, u2) s.t. u2 = u1 (2)

- Linear dependence

(u1, u2) s.t. u2 = F2

(F1−1

(u1)a + b + e)

(3)

a = ρσ2

σ1, b = µ2 − aµ1

e ∼ N(

0, σ2

√1− ρ2

)


Sobre el artıculo Resumen grafico

How dependencies may be put down into practice? II

By means of an inversion algorithm for conditional distribution of copulas - when feasible (see,for instance, Genest and Rivest (1993), McNeil et al (2005) or Nelsen (2006))

Gaussian (ρ) : u2 = Φ(

Φ−1(t2)√

1− ρ2 + ρΦ−1(u1))

with t2 ∼ U(0, 1) (4)

Student t (ρ,m dgf ) : u2 = tm

t−1m+1(t2)

√1− ρ2√√√√ m + 1

m +(t−1m (u1

)2

+ ρt−1(u1)

(5)

Frank surv. copula (α) : u2 = 1−[−

1

αln

[exp(−α)− 1

11−t2

exp(−α(1− u1))− (exp(−α(1− u1))− 1)+ 1

]](6)

Clayton co-copula(α) : u2 = 1−[

(1− t2)α

1+α − 1

(1− u1)α+ 1

]− 1

α(7)


Sobre el artıculo Ejercicio realizado

What have we done? I- A portfolio of life insurance products subject to longevity and mortality risks has

been built, inspired by the portfolio proposed in van Gulick et al (2012)

- Some assumptions has been made in order to calculate Best Estimate Liabilities ofthis portfolio as of December 31st 2016

- Contributions to the Standard Formula SCR due to longevity and mortality risks ofthis portfolio have been calculated (SCRLong and SCRDeath), as well as theirmerged contribution based on the −25% correlation coefficient stated by the SF:

SCRLong&Death =√

SCR2Long + SCR2

Death − 0.25SCRLongSCRDeath

- Apart from the previous direct calculations, simulation based calculations of thesame amounts (SCRLong , SCRDeath and SCRLong&Death) have been done, based onthe assumptions described in EIOPA-14-322



What have we done? II

- It has been checked that direct and simulation based calculations are close enough

- Taking advantage of the simulation framework in place, different dependencestructures have been tested, in order to stress the underlying dependence structureof the SF. The tested structures are the following

1. Extremely opposite dependence

2. Extremely direct dependence

3. Linear dependence (as described inExpression 3)

4. Gaussian (ρ = −25%)

5. Student t - Basic (ρ = −25%,

m = 3)

6. Frank survival copula - Basic(α = −0, 3125)

7. Clayton co-copula - Basic (α = 1.5)

8. Student t - Alternative (ρ = −85%,m = 3)

9. Frank survival copula - Alternative(α = −15)

10. Clayton co-copula - Alternative

(α = 12)

- Obtained results have been analyzed, as it is discussed in the conclusions.

- Everything has been done in MS Excel spreadsheets.



Portfolio features

Three life products with three differentportfolios of policyholders have beenconsidered in the exercise:

- A deferred single life annuity that yields ayearly payment of 1 monetary unit every yearthat the insured in alive and older than 65

- A survivor annuity that yields a yearlypayment of 0.75 monetary units in every yearthat the spouse outlives the insured, if theinsured dies before age 65

- A death benefit insurance that yields asimple payment of 7.5 monetary units in theyear the insured dies, if the insured diesbefore age 65.

Three different portfolios but with the same agedistribution. N1 = 45, 000 andN2 = N3 = 15, 000 policyholders.



Assumptions for pricing and BELThe following simplifications have been made:

- It has been assumed that all the policyholders in the three portfolios are male and theirpartners are women two years younger than each one of them

- Premiums have been calculated 4 years before our simulated solvency calculations (i.e., pricinghas been done as of December 31st 2012). Mortality table used is PASEM. A flat annualdiscount rate of 3% has been assumed when pricing policies

- Pure pricing has been assumed: no room for commissions (neither external nor internal) hasbeen considered

- When calculating BEL, a flat surrender rate of 2% (independent of the age of the policyholderand of the aging of the policyholder in the portfolio) has been assumed for all products

- For pricing and BEL purposes, it has been assumed that claims (and premiums) are paid(acquired) at the end of the year

- A policyholder of age x is representative of all of the policyholders of the same age in theportfolios

- For BEL purposes, volatility adjustment has been added to the interest rate term structurepublished by EIOPA as of December 31st 2016.



Un ejemplo (revisitado)

BEL(X1; rt ;Q; st ;YYYY ) =

N1∑i=1

∞∑t=max(65−x1i ,0)

(1 · tp(Q; st)x1i · f (rt ;YYYY ))−

N1∑i=1

max(65−x1i ,0)∑t=0

(p1(x1i ) · tp(Q; st)x1i ∆f (rt ;YYYY ))

donde

tp(Q; st)x1i = Πtl=0 (1 − (qx1i+l + st − qx1i+l · sl)) es la probabilidad de

que un asegurado de edad x1i a 31/12/YYYY haya sobrevivido y nohaya rescatado despues de t anos segun la tabla de mortalidad Q y lastasas de rescate st .

f (rt ;YYYY ) es el factor de actualizacion de 1 unidad monetaria tras tanos desde la fecha 31/12/YYYY y en base a los tipos de interes rt .

N1 y p1(x1i ) son el numero total de asegurados de la cartera delproducto 1 (45000) y la prima a pagar (calculada en 31/12/2012) porel asegurado de edad x1i , respectivamente.



Simulation of SF underlying assumptions

As stated in EIOPA-14-322 , calibration of the life underwriting risk stress factors are considered to

be in line with the 99.5% VaR and a one-year time horizon. The calibration regarded of great

importance a study published in 2004

Longevity risk - Summary of quantitativeassumptions

- The annual mortality improvements follow a normal distribution

- The stress factor over the mortality table is of 80%

- It is assumed that the average mortality rate of the respectiveinsured persons does not increase by more than 10% each year

Mortality risk - Summary of quantitativeassumptions

- The probability distribution for mortality is skewed, with acurrent trend towards improving mortality















Taking also into account the document UNESPA - LRI (2009), the

following improving factors mL(u) and mD (u) have been used to

simulate (in a really simple way) the SF assumptions:

mL(u) = 1− (0% + σu)

mD (u) = 1− (0.0001% +15%

20%σu)

with σ =20%

Φ−1(99.5%)

when u take values of a standard uniform distribution.















- The annual mortality improvements mL follow a normaldistribution (mL ∼ N (1, σ))

- VaR0.5%(mL) = 1 − 20% = 80%

- The average mortality rate of the respective insured personsdoes not increase by more than 0% each year (in particular< 10%)


- The probability distribution for mortality is skewed, with acurrent trend towards improving mortality(E(mDqx ) = 0.9999%E(qx ) < E(qx ))

- VaR99.5%(mD ) = 99.9999% + 15% = 114.9999%



Mortality multipliers we have used



Sketch - Actual dependencies taken into account

1. Extremely opposite dependence




2. Extremely direct dependence




3. Linear dependence (as described in Expression 3)




4. Gaussian (ρ = −25%)




5. Student t - Basic (ρ = −25%, m = 3)




6. Frank survival copula - Basic (α = −0, 3125)




7. Clayton co-copula - Basic (α = 1.5)




8. Student t - Alternative (ρ = −85%, m = 3)




9. Frank survival copula - Alternative (α = −15)




10. Clayton co-copula - Alternative (α = 12)


Sobre el artıculo Resultados

Our findings I

Dependence hypothesis VaR99.5% BEL SCRLong&Death Rel Diff vs SF

Standard Formula (SF) 16,279.24 0.00%Extremely opposite 89,139.99 74,423.07 14,716.92 -9.69%Extremely direct 93,012.37 74,423.07 18,589.30 14.07%Lineal 90,711.24 74,423.62 16,287.62 0.05%Gaussian copula 90,752.50 74,423.63 16,328.87 0.30%Student t copula - Basic 90,854.94 74,423.53 16,431.41 0.93%Frank survival copula - Basic 91,167.69 74,423.65 16,744.04 2.86%Clayton co-copula - Basic 90,683.42 74,421.27 16,262.15 -0.10%Student t copula - Alternative 89,425.98 74,423.15 15,002.83 -7.84%Frank survival copula - Alternative 89,902.74 74,422.83 15,479.91 -4.91%Clayton co-copula - Alternative 89,950.16 74,421.97 15,528.19 -4.61%

Table: Results obtained with 200 simulations of each dependence scenario, exceptfor extremely opposite and direct relationship in which only 1 scenario has beenconsidered. For each scenario, 4,000 random values u2 have been taken intoaccount.


Sobre el artıculo Resultados

Our findings II

- Linear dependence and the Gaussian copula fit the SCR SFquite well (relative differences ±1%)

- ‘Basic’ copulas generate capital requirements close to the SCRSF. Parameters of these copulas where calibrated in order toachieve a Kendall’s correlation (rank correlation) similar to the-25% lineal correlation proposed by the SF. In that sense,obtained results do not allow to pick which dependencestructure could be underlying the SF

- Extremely opposite dependence is the one that generates ahigher capital saving with respect to the SF (' 10%), as it wasexpected. Intuitively, this kind of dependency should reflectappropriately the relationship between longevity and mortalityrisks

- ‘Alternative’ copulas generate capital requirements lessconservative than the SCR SF. This is an expected result, aslong as these copulas have been calibrated to depict morenegative dependence between longevity and mortality risks thanthe −25% linear correlation linked to the SF


Sobre el artıculo Conclusiones

Take aways

- Dependency structures between different risks are crucial whenquantifying aggregated losses

- Simple but indicative analysis can be done in MS Excel spreadsheets inthe bivariate case

- In our exercise, real savings on capital requirements (at pair-levelbetween longevity and mortality risks) can be achieved by modifyingthe dependence structure among these risks. Despite that, it has to benoted that these savings could be diluted when aggregating withlosses from other sources of risk

- As analyzing dependencies between risks is not an easy task, there isroom for proposing criteria that help actuarial functions of all overEurope to deal with this issue


¿Qué son las funciones fundamentales?



riesgos?

Objetivo principal



del SCR por FE

Objetivo secundario




¿Qué es un BEL?

¿Qué es un SCR?

¿Qué quiere decir

Fórmula Estándar?


mortalidad?






cópulas?

Referencias

References I

UNESPA Longevity Risk Investigation.

Technical report, UNESPA, 2009.

The underlying assumptions in the standard formula for the Solvency Capital Requirement

calculation.Technical Report EIOPA-14-322, EIOPA, 2014.

Guidelines on System of Governance.

Technical Report EIOPA-14-322, EIOPA, 2015.

Michel Denuit, Jan Dhaene, Marc Goovaerts, and Rob Kaas.

Actuarial Theory for Dependent Risks. Measures, Orders and Models.John Wiley & Sons Ltd, Chichester, 2005.

Antoni Ferri Vidal.

Estructuras de Dependencia aplicadas a la Gestion de Riesgos en Solvencia II.PhD thesis, University of Barcelona, 2010.

Christian Genest and Louis-Paul Rivest.

Statistical Inference Procedures for Bivariate Archimedean Copulas.Journal of the American Statistical Association, 88(423):1034–1043, 1993.

Luis Latorre Llorens.

Teorıa de Copulas. Introduccion y aplicaciones a Solvencia II.Fundacion Mapfre, 2017.


Referencias

References II

Alexander J. McNeil, Rudiger Frey, and Paul Embrechts.

Quantitative Risk Management.Princeton Series in Finance. Princeton University Press, New York, 2005.

Roger Nelsen.

An Introduction to Copulas.Springer, 2006.

Gerwald van Gulick, Anja De Waegenaere, and Henk Norde.

Excess based allocation of risk capital.Insurance: Mathematics and Economics, 50(1):26–42, 2012.


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