design Georg Ch. Pug a,b,

### ∗

,Anna Timonina a , Stefan Hohrainer-Stigler a aInternationalInstitute forAppliedSystems Analysis, Laxenburg, Austria

b

InstituteofStatistis andOperationsResearh,University ofVienna, Oskar Morgenstern-Platz1, A-1090,

Vienna, Austria

Abstrat

In stohasti optimization models, the optimal solution heavily depends on the hosen model

for the senarios. However, the senariomodelsare hosenon the basis ofstatistial estimates

and are therefore subjet to model error. We demonstrate here how the model unertainty

an be inorporated into the deision making proess. We use a nonparametri approah for

quantifying the model unertainty and a minimax setup to nd model-robust solutions. The

methodisillustratedbyariskmanagementprobleminvolvingtheoptimaldesignofaninsurane

ontrat.

Keywords: insurane optimization,modelerror, minimax solution,distributionalrobustness

1. Introdution

Common approah in risk assessment and risk management is to base the risk estimates

on observed data and to use the statistially obtained estimates for nding the optimal risk

management strategies. However, the fat that statistialestimates an never give preise

val-ues of unknown parametersdue to anestimation error,is quiteoftennegleted. Moreover, the

hoie of the probability model, i.e. the lass of possible distributions, is typially hosen by

the statistiian and is not further questioned.

In general, statistialestimation proedures do not allow tosingle out one spei probability

model, but onlya whole set of models an be determined, inwhih the true modellies with a

prespeied probability. This ondene set an be taken as the set of models for a minimax

deision,wherethe best deisionunderthe worst modelinthe modelset is soughtfor. We all

suh sets of models ambiguity sets. The minimax solutioninthis ase isalled distributionally

robust.

Modeling unertainty. Eonomi deisions are made under some assumptions of the

dei-sion relevant parameters. In deterministi optimization, the parameters are onsidered to be

known and xed. Already intheearlydays ofoptimization,this assumption wasonsideredas

too narrow. Two possible setups have then been developed: (i) in robust optimization, a set

of possible parameters is determined, while (ii) in stohasti optimization the parameters are

onsidered tofollowa ertainprobability distribution. In robustoptimization, the parameters

are not weighted and one has to use minimax strategies (to nd the best strategy under the

worst ase). Probability models ome with a lot more of possible strategies: expeted utility

maximization or minimization of risk (shortfall risk, variane risk, tail risk, et.). However,

probability models depend heavily on the hosen probability model, whih is typially based

### ∗

Correspondingauthor

Email addresses: georg.pflugunivie.a.at(GeorgCh. Pug), timoninaiiasa.a.at(Anna

timation proedures, whih may ontain estimation error. This dependeny has been ignored

for quite a while, when most researh was put into deisionmaking under a given xed

prob-ability model. However, if the model is ompletely xed, only aleatori unertainty, i.e. the

unertainty about the realizationsis tobeonsidered.

Asalready notied,the modelhoie istypiallybasedonassumptionsandestimations,i.e. an

error in model hoie annot be exluded. The ambiguity in model seletionis often referred

to as epistemi unertainty. By inluding the epistemi unertainty into the deision making

proess, we an alsoto aertain extent reonile this approahto the pure senarioapproah.

While in the latter all senarios are in priniple possible, stohasti ambiguity models would

allowsenarioprobabilities tovary inorder toaommodate for modelunertainty. In the

ex-tremalasethat allprobabilitydistributionsoverthe senarios wouldbetheoretiallypossible,

the ambiguity modeling oinides with pure senario analysis. In most appliation areas we

know the importane or likelihood of senarios at least to a ertain extent, though we

typi-ally do not know the exat ourrene probabilities. This is why model ambiguity beomes

an important issue in deision making. In ambiguous modeling, the possible model error is

inorporated into the deisionproess, that allows to nd robust deisions.

We summarizethis asfollows:

If allparameters of anoptimizationproblem are known, weallit deterministi.

If some parametersare not exatlyknown, but known tolie insome set, then we have a

robust program

Iffortheunknown parametersarandomdistributionisspeied,we allthis astohasti

program.

Ifthedistributionoftherandomparametersisnotknown,butknowntolieinsomefamily

of distribution,then the problemisalled ambiguity problemand itsminimax solutionis

alled distributionally robust.

Bibliographi remarks. The idea of optimal deisions under several stohasti models

(i.e., min-max solutions) appears for the rst time in Sarf [15℄ in a linear inventory problem

seeking the stokage poliy, whih maximizesthe minimum prot onsidering all demand

dis-tributionswith given meanandgivenstandard deviation. More thoroughstudiesofambiguous

deision problems than a minimax problems were initiated by Dupaová [3, 4℄ for the lass of

stohasti linearproblems with reourse undergeneralassumptionsfor the ambiguity set. The

formulation is in a game theoreti setup, where the rst player hooses the deision and the

seondplayerhooses theprobabilitymodel. Thereare alternativenamesused inliterature for

the ambiguity problem, suh as minimax stohasti optimization, model unertainty problem

or distributionalrobustness problem. Many proposals for ambiguitysets inthe two-stage ase

have been made and analyzed. A list of popular lasses of ambiguous models is presented by

Dupaová [5℄.

Literature dealing with ambiguity either from theory or appliation point of view is growing

rapidly. The situation when the ambiguity set onsists of allprobabilities with given rst two

momentswasstudiedby Jagannathan[9℄ forthe linear ase. Shapiro andKleywegt [16℄ dene

anambiguitysetastheonvexhullofaniteolletionofmodels;AhmedandShapiro[17℄

on-sidersetsofmodelsgivenbymomentinequalities;asimilarapproahisadopted byEdirishinge.

Calaore [1℄ uses the Kullbak-Leiblerdivergene todene neighborhoodsof a baseline model

asambiguity sets. Thiele[18℄ onsiders the

### L

### 1

balls of densitiesasambiguitysets. Delage and Ye [2℄ onsider ambiguity set given by inequalities for the mean and the ovariane matrix.the multistagease. Hansen and Sargent onsider in their 2007 book [8℄ alternativemodels of

multistagestohastioptimizationproblemsgivenby maximaldeviationfromabaselinemodel

in Kullbak-Leibler divergene. Goh and Sim [7℄ study multistage ambiguity sets whih are

dened by a mean, whih must lie in some onial set, a given ovariane matrix and some

upperbounds on the exponentialmoments and extend this to multistage.

In this paper, we investigate the problem to determine an optimal insurane portfolio under

model ambiguity. To simplifythe approah, we onsider onlya single stage deision problem:

A government has todeide about investment in and insurane for infrastruture for the next

budget period. The infrastruture is subjet to natural hazard suh as earthquake, oods or

tropial storms and the problem is to nd the best mix between investment and insurane,

i.e. between inreased produtivity and higher protetion. The modelis similar to the IIASA

CATSIM model.

2. Model desription

To determine the optimal design of an insurane sheme is a typial problemof stohasti

optimization: Adoptionof a robust approahwould not makesense, assoon asit would mean

thatveryrare eventswould beonsideredasimportantasquitefrequentevents andthiswould

result in anoverly pessimistiresult. However, the probability modelmay be subjet to error

and this givesan argumentfor distributionally robustdeisions.

The total insurable infrastruture stok of the ountry underonsideration attime 0 is

### S

### 0

. We assume that the ountry is under hight risk of natural hazards and we denote by### L

theannual lossvariable. We further onsider a stop-loss insurane with exit level

### Y

, its paymentfuntion is

### min(

### L, Y

### )

. It is well known that stop-loss insurane ontrats are "optimal"(see Raviv [14℄). The stop-loss payment funtionis shown in Figure1.## damage

## payment

## exit

## level y

The infrastruture

### S

### 1

at the end of the period (typially one year) is given by the previous amount### S

### 1

minus the random damage and the amount obtained as ompensation from the insurane plus the investment### X

.### S

### 1

### =

### S

### 0

### −

### L

### + min(

### L, Y

### ) +

### X

The premiumfor the stop-lossinsuraneis denoted by

### π

### (

### L, Y

### )

. Thereis abudget### B

available, whihmay beused for investment and for infrastruture protetion by insurane.The deision problem onsidered here is to nd the optimal mix between investment

### X

and insurane with exit level

### Y

for the given budget### B

, the objetive to be maximised is thevariane-orreted expetation of

### S

### 1

. The omplete (yetunambiguous)modelis### max

### Y

### E

_{(}

_{S}

_{1}

_{)}

_{−}

_{λ}

### V

_{ar}

_{(}

_{S}

_{1}

_{)}

(2.1)
s.t. ### S

### 1

### =

### S

### 0

### −

### L

### + min(

### L, Y

### ) +

### X

### X

### +

### π

### (

### L, Y

### ) =

### B

### X

### ≥

### 0

### , Y

### ≥

### 0

### .

To summarize, wehave introdued the following symbols.

### S

### 0

the (deterministi)infrastruturevalue atthe beginningof the period### S

### 1

the (stohasti)infrastruture value atthe end ofthe period### L

the stohasti lossvariable### B

the budgetforeseen for investment and insurane### X

the investment ininfrastruture### Y

the exitlevelfor the insurane ontrat### λ

the penalty parameter forthe variane### π

### (

### L, Y

### )

the premiumof aninsurane with relative lossvariable### ξ

and exit level### Y

. For further use we introdue some relativevalues### ξ

the relativelossvariable,### ξ

### =

### L/S

### 0

### y

the relativeexit level,### y

### =

### Y /S

### 0

and### π

### y

### =

### π

### (min(

### L, yS

### 0

### )

### /S

### 0

### .

The insurane premium. Weassumethattheinsuranepremiumisalulatedaording

to a ombination the distortion and the utility priniple (see [10℄): Suppose that

### F

### L

is the

distributionfuntionofthelossvariable

### L

. Usingadistortionfuntion### g

on[0,1℄,whihsatises### g

### (

### u

### )

### ≥

### u

,### g

### (0) = 0

,### g

### (1) = 1

, the lossdistribution is distorted tothe new lossdistribution### F

### g

### L

### (

### u

### ) = 1

### −

### g

### (1

### −

### F

### L

### (

### u

### ))

### .

Under the monotoni and onvex (dis)utility funtion

### V

, the premiumfor the distorted loss### L

is

### π

### (

### L

### ) =

### V

### −

### 1

### Z

### ∞

### 0

### V

### (

### u

### )

### d

### [1

### −

### g

### (1

### −

### F

### L

### (

### u

### ))]

where### V

### −

### 1

isthe inverse (dis)utility.

If the overage is apped at

### Y

, then the distribution funtion of the damage variable### min(

### L, Y

### )

is### F

### min(

### L,Y

### )

### (

### u

### ) =

### F

### L

_{(}

_{u}

_{)}

if ### u

### ≤

### Y

### 1

otherwise.Assuming that

### F

### L

has adensity### f

### L

, the premiumfor the lossapped at

### Y

is### π

### (min(

### L, Y

### )) =

### V

### −

### 1

### Z

### V

### 0

### (

### u

### )

### g

### ′

### (1

### −

### F

### L

### (

### u

### ))

### f

### L

### (

### u

### )

### du

### +

### V

### (

### Y

### )

### ·

### g

### (1

### −

### F

### L

### (

### Y

### ))

with### g

### ′

_{(}

_{u}

_{) =}

### ∂

### ∂u

### g

### (

### u

### )

.The relative lossvariable

### ξ

### =

### L/S

### 0

has distributionfuntion### F

### ξ

_{(}

_{w}

_{) =}

_{F}

### L

_{(}

_{w/S}

### 0

### )

,### w

### ∈

### [0

### ,

### 1]

with density### f

### ξ

_{(}

_{w}

_{) = 1}

_{/S}

### 0

### f

### L

### (

### w/S

### 0

### )

. For the relative exit level### y

### =

### Y /S

### 0

, one gets for the premium### π

### y

of the ontrat with this exitlevel### π

### y

### (

### ξ

### ) =

### π

### (min(

### yS

### 0

### , ξS

### 0

### ) =

### V

### −

### 1

### Z

### y

### 0

### V

### (

### wS

### 0

### )

### g

### ′

### (1

### −

### F

### ξ

### (

### w

### ))

### f

### ξ

### (

### w

### )

### du

### +

### V

### (

### yS

### 0

### )

### g

### (1

### −

### F

### ξ

### (

### y

### ))]

### =

### S

### 0

### V

### 1

### −

### 1

### Z

### y

### 0

### V

### 1

### (

### w

### )

### g

### ′

### (1

### −

### F

### ξ

### (

### w

### ))

### f

### ξ

### (

### w

### )

### dw

with### V

### 1

### (

### w

### ) =

### V

### (

### S

### 0

### w

### )

.FortheExamplebelow,wehavehosenthepowerdistortionfuntion

### g

### (

### u

### ) =

### u

### r

for

### 0

### < r <

### 1

and the (dis)utility funtion

### V

### (

### u

### ) = (

### a

### −

### u/S

### 0

### )

### −

### q

_{−}

_{a}

### −

### q

, i.e.

### V

### 1

### (

### w

### ) = (

### a

### −

### w

### )

### −

### q

_{−}

_{a}

### −

### q

. With

this hoie,the premiumformulaan beonretized to

### π

### y

### (

### ξ

### ) =

### a

### −

### Z

### y

### 0

### [(

### a

### −

### w

### )

### −

### q

### −

### a

### −

### q

### ]

### r

### (1

### −

### F

### ξ

### (

### w

### ))

### r

### −

### 1

### f

### ξ

### (

### w

### )

### dw

### + [(

### a

### −

### y

### )

### −

### q

### −

### a

### −

### q

### ](1

### −

### F

### ξ

### (

### y

### ))

### r

### −

### 1

### /q

### .

(2.2)For the Madagasar Example (with apointmass at0 and a pieewise onstant density, see

below) and the parameterhoie

### a

### = 0

### .

### 2

,### q

### = 1

### .

### 1

,### r

### = 0

### .

### 95

,the funtion### y

### 7→

### π

### y

isdepited in Figure2.### 0

### 0.02

### 0.04

### 0.06

### 0.08

### 0.1

### 0.12

### 0.14

### 0.16

### 0.18

### 0.2

### 0

### 0.5

### 1

### 1.5

### 2

### 2.5

### 3

### 3.5

### 4

### 4.5

### x 10

### −3

Figure 2: Therelativepremium

### πy

as funtion ofthe relativeexit level### y

. Thedashed urveis the expeted relativelossto beoveredbythestop-lossontrat.Notie that

### S

### 1

### =

### S

### 0

### (1

### −

### ξ

### ) +

### S

### 0

### min(

### ξ, y

### )

### −

### S

### 0

### π

### y

### (

### ξ

### ) +

### B.

Sine

### E

### [

### S

### 0

### (1

### −

### ξ

### + min(

### ξ, y

### ))] =

### S

### 0

### E

### [1

### −

### ξ

### + min(

### ξ, y

### )]

and### V

### ar

### (

### S

### 1

### ) =

### S

### 2

### 0

### V

### ar

### [1

### −

### ξ

### + min(

### ξ, y

### )]

the followingprogram isequivalent to(2.1)

### max

### y

### E

_{[1}

_{−}

_{ξ}

_{+ min(}

_{ξ, y}

_{)]}

_{−}

_{π}

_{y}

_{(}

_{ξ}

_{)}

_{−}

_{γ}

### V

_{ar}

_{[1}

_{−}

_{ξ}

_{+ min(}

_{ξ, y}

_{)]}

(2.3)
s.t. ### π

### y

### (

### ξ

### )

### ≤

### B

### S

### 0

### y

### ≥

### 0

with

### γ

### =

### λ

### ·

### S

### 0

. Notie, that this isa one-dimensional optimizationproblem.For the Madagasar Example, the objetive funtion with the setting

### γ

### = 100

and### B/S

### 0

### =

### 0

### .

### 0182

isshown inthe Figure 3below.### 0.04

### 0.06

### 0.08

### 0.1

### 0.12

### 0.14

### 0.16

### 0.18

### 0.996

### 0.9965

### 0.997

### 0.9975

### 0.998

### 0.9985

### relative exit level

### objective function

Figure3: Theobjetiveasfuntion oftherelativeexitlevel

### y

For the formulation of the distributionally robust problem, we assume that

### ξ

has a pointmass

### F

### 1

at### z

### 1

### = 0

and for### z

### ∈

### (0

### ,

### 1]

a density whih is onstant in the intervals### (

### z

### i

### , z

### i

### +1

### )

for### i

### = 1

### , . . . , k

### −

### 1

. The distributionfuntion is alinear interpolationbetween the points### (

### z

### i

### , F

### i

### )

,### i

### = 1

### , . . . , k

,where### F

### i

are the umulativeprobabilities. Notie, that### F

### k

### = 1

.3. Alternative models

As we have introdued it, the baseline damage distribution is given by a disrete list of

breakpoints

### z

### 1

### , . . . , z

### k

togetherwithalistofumulativeprobabilities### F

### ˆ

### 1

### , . . . ,

### F

### ˆ

### k

. Thealternative modelswillhavethe sameintervalboundaries### z

### i

anddieronlyinthe umulativeprobabilities### F

### i

. A simple ambiguityset### P

isgiven by### P

### =

### {

### F

### :

### X

### i

### |

### F

### i

### −

### F

### i

### ∗

### | ≤

### ǫ

### }

or

### P

### =

### {

### F

### :

### |

### F

### i

### −

### F

### i

### ∗

### | ≤

### ǫ

### ∀

### i

### }

However, suh a neighborhoods do not take into aount the values

### z

### i

and are therefore notappropriate.

We use the Wasserstein distane as the basi metri for loss distributions. It has not only

the advantage of taking the values

### z

### i

into aount, but it is based on a distane on the realline, whihmay beadapted tothe needsof the problemathand. Forinstane, wemay use the

basi distane

### dist

### (

### z, w

### ) =

### |

### z

### s

### −

### w

### s

### |

(3.1)for some

### s >

### 1

, meaning that the higher relative damages get higher distanes, beause theyare more relevantfor the insurane pries. However, inthis paper we have set

### s

### = 1

.The Wasserstein distane between the two relative loss variables

### ξ

resp.### ξ

### ˆ

with distribution funtions### F

resp.### F

### ˆ

is dened as the optimalvalue of the (seeminglyinnite) linearprogram### min

### E

_{[}

_{dist}

_{(}

_{ξ,}

_{ξ}

### ˆ

_{)]}

(3.2)
s.t.

### ξ

### ∼

### F

(3.3)### ˆ

### −1

### −0.5

### 0

### 0.5

### 1

### −1

### −0.8

### −0.6

### −0.4

### −0.2

### 0

### 0.2

### 0.4

### 0.6

### 0.8

### 1

The levelsets of the funtion

### h

.The minimumis taken over alljointdistributions withgiven marginals

### F

resp.### F

### ˆ

. Denote the optimalvalue of this programby### d

### (

### F,

### F

### ˆ

### )

. Ifthe distane ishosenas in (3.1),then### d

### (

### F,

### F

### ˆ

### ) =

### Z

### |

### F

### (

### t

### )

### −

### F

### ˆ

### (

### t

### )|

### st

### s

### −

### 1

### dt

whihisaslightextensionofaresultbyVallander(iteVall74). Ifthetwodistributionfuntions

are pieewise linear with the same breakpoints

### z

### i

, then one nds after some alulation thatthe Wasserstein distane isgiven by

### d

### (

### F,

### F

### ˆ

### ) =

### 1

### 2

### k

### −

### 1

### X

### i

### =1

### (

### z

### (

### i

### + 1)

### −

### z

### (

### i

### ))

### ·

### h

### (

### F

### i

### −

### F

### ˆ

### i

### , F

### i

### +1

### −

### F

### ˆ

### i

### +1

### )

where### h

### (

### a, b

### ) =

###

###

###

###

###

###

###

### a

### +

### b,

if### a

### ≥

### 0

### , b

### ≥

### 0

### (

### a

### 2

_{+}

_{b}

### 2

_{)}

_{/}

_{(}

_{a}

_{−}

_{b}

_{)}

_{,}

if ### a

### ≥

### 0

### , b <

### 0

### (

### a

### 2

_{+}

_{b}

### 2

_{)}

_{/}

_{(}

_{b}

_{−}

_{a}

_{)}

_{,}

if ### a <

### 0

### , b

### ≥

### 0

### −

### a

### −

### b,

if### a <

### 0

### , b <

### 0

Notie that

### h

is a onvex funtion and that the sets### {

### F

### i

### :

### d

### (

### F,

### F

### ˆ

### )

### ≤

### ǫ

### }

are onvex in the parameters### F

### i

's.The distributionally robust solutionof our optimal insurane problemis the solutionof

### max

### y

_{{}

_{F}

_{:}

_{d}

### min

_{(}

_{F,}

_{F}

### ˆ

_{)}

_{≤}

_{ǫ}

_{}}

### E

_{F}

_{[1}

_{−}

_{ξ}

_{+ min(}

_{ξ, y}

_{)]}

_{−}

_{π}

_{y}

_{(}

_{ξ}

_{)}

_{−}

_{γ}

### V

_{ar}

_{F}

_{[1}

_{−}

_{ξ}

_{+ min(}

_{ξ, y}

_{)]}

(3.5)
s.t.

### ξ

### ∼

### F

(3.6)### F

has a density in### (0

### ,

### 1]

,whih ispieewise onstant in### [

### z

### i

### , z

### i

### +1

### ]

(3.7)### π

### (

### y

### )

### ≤

### B/S

### 0

### y

### ≥

### 0

### .

Here

### E

### F

resp.### V

### ar

### F

denotethe expetationresp. the variane, whenthe distribution funtion of### ξ

is### F

.4. A ase study: Madagasar

Madagasar has one ofthe highestylone risksworldwide, espeiallytheeast oast, whih

butusuallyfallsshortinprovidingadequateresoures. Consequently,thereisakeeninterestin

possibleinsurane mehanism(or aregionalinsurane pool)thatould helpnaningdisasters

in a proative manner. However, for suh kind of assessment an annual loss distribution for

yloneeventsonthe ountry levelhas tobeestimated rst. Toestimatethe damagepotential

of ylones, dierent tehniques an be used, e.g. stohasti or engineering approahes for

estimatingphysialvulnerabilityoftheassetsexposedandombiningthemwithhazardimpats

and orresponding probabilities of given events (see Woo 2012 for a detailed disussion on

atastrophe modeling approahes). However, as this kind of detailed information is not yet

available yet historial losses have to be used for the risk assessment instead. There are two

databases availablethat anbeused forsuhkindofanalysis. Oneisthe open-soure EMDAT

disasterdatabase(EM-DAT,2012)maintainedbytheCentreforResearhontheEpidemiology

of Disasters at the Université Catholique de Louvain. EMDAT urrently lists informationon

peoplekilled,madehomeless,andaetedaswellasoverallnaniallossesformorethan16,000

sudden-onset (suh asoods, storms, earthquakes) and slow-onset (drought) events from 1900

to present. Data are ompiledfrom various soures, inludingUN agenies, non-governmental

organizations, insurane ompanies, researh institutes and press agenies. The seond one

is the newly produed time-series loss data by Malagasy oials based on the "Damage and

losses assessment methodology" (from now onalled MLoss). It onsists of past publi setor

loss estimates for the Analanjiroforegion from 1980 to 2012 separated into dierent setorial

impats. Furthermore, the results for the Analanjirofo were upsaled to the national level by

assumingthe sameexposureandvulnerabilitylevelsinotherareas. Theestimatesarebasedon

the assumption that losses belong to the maximum domain of attration of an extreme value

distribution(andaslossesarealwaysadownsiderisk)the Frehettypedistributionwashosen

as the basi loss distribution. For estimating the shape as well as the loation parameter,

a non-linear optimization model was built, whih best ts the urve with the data at hand.

Furthermore, toinrease the robustness of the results, other models - suh as the Generalized

Paretomodel-weretestedandimprovedinastep-basedmannertosatisfatorylevels(basedon

graphialtestssuhasP-PplotsandQ-Qplots,seeEmbrehtsetal. 1997formoreinformation

onthesetehniques). Theparametersobtained withthismethodwereused toalulateannual

lossreturn periods.

The same approahwas used for the MLoss dataset. This dataset inludeslosses of the publi

setor, whih are separated into dierent dimensions inluding damages to shools, hospitals,

the teleommuniations system, the environment, and transportation from 1980 to 2012. In

total 4 dierent loss distributions were estimated all with dierent assumptions as well as

dierent estimation tehniques or datasets used, see Table 1.

For these loss distributions, extreme value distributions were tted and then the relative

losses were approximated by pieewise linear df'swith knots at

### z

### = [0

### ,

### 0

### .

### 0018

### ,

### 0

### .

### 0027

### ,

### 0

### .

### 0036

### ,

### 0

### .

### 0055

### ,

### 0

### .

### 0091

### ,

### 0

### .

### 0137

### ,

### 0

### .

### 0182

### ,

### 0

### .

### 0365

### ,

### 0

### .

### 0547

### ,

Noloss 0.406 0.607 0.406 0.406 20 2655 114 372 149.0 50 3991 409 510 204.2 80 4773 775 581 232.5 100 5172 1046 614 245.9 150 5943 1802 676 270.4 200 6530 2646 719 287.7 250 7010 3562 752 301.1 300 7419 4541 780 312.1 400 8097 6657 823 329.4 500 8651 8954 857 342.9

Table 1: Soure: Based on Hohrainer 2014. Estimated lossreturn periods for publi ylonerisk based on

dierentestimationmethods. Losses(onstant2000)inmillionUSD,publisetorlosses

5. Solving the Madagasar problem

If weonsider the EMDATmodelasthe one and onlymodel, weare faed witha standard

stohastioptimizationproblemwithjustonerealdeisionvariable

### y

. Thepertainingobjetivefuntion is shown in the Figure 4 below. The optimal exit level is

### y

### ∗

_{= 0}

_{.}

_{1188}

, i.e. to ap the

insurane at6510 mill. USD witha premiumof92.0 mill. Theexpeted insuredlossesare 51.6

mill.

### 0.04

### 0.06

### 0.08

### 0.1

### 0.12

### 0.14

### 0.16

### 0.18

### 0.996

### 0.9965

### 0.997

### 0.9975

### 0.998

### 0.9985

### relative exit level

### objective function

Figure4: Theobjetivefuntion.

If

### F

### (

### j

### )

_{, j}

_{= 1}

_{, . . . , ℓ}

is a nite set of dierent possible loss distributions,we solve

### max

### y

### j

### =1

### min

### ,...,ℓ

### E

### F

### (j)

### [1

### −

### ξ

### + min(

### ξ, y

### )]

### −

### π

### y

### (

### ξ

### )

### −

### γ

### V

### ar

### F

### (j)

### [1

### −

### ξ

### + min(

### ξ, y

### )]

s.t.### π

### (

### ξ

### 1

### , y

### )

### ≤

### B/S

### 0

### y

### ≥

### 0

### .

Notie that### F

### (1)

isthe basi model, onthe basis of whih allinsurane premia are alulated.

In the Madagasar ase, there are 3 possible loss distribution whih are plotted in Figure 5

below

Itturnsoutthattheinlusionofthealternativemodelsdoesnothangetheoptimalexitpoints

(

### y

### ∗

_{= 0}

_{.}

_{1188}

), sine the EMDAT loss distribution dominates the other models. The minimal

objetive funtionis the objetive funtion of the EMDAT ase.

However, we solved the full maximin problem for dierent radius of the ambiguity set, we

### 0

### 0.02

### 0.04

### 0.06

### 0.08

### 0.1

### 0.12

### 0.14

### 0.16

### 0.18

### 0.2

### 0.985

### 0.99

### 0.995

### 1

Figure5: Lossdistributions.

ambiguity radius

### ǫ

optimalexit level inmill.0 6,510 0.001 6,788 0.005 6,943 0.01 7,073 0.03 7,471 0.05 7,774 0.07 8,044 0.1 8,711 0.15 9,221 0.2 9,331

The relative values of optimal exit points are shown in Figure 6 and the pertainingworst

ase models are depited in Figure 7. Together with the optimal solutions, they desribe the

saddlepoints of our problem.

### 10

### −3

### 10

### −2

### 10

### −1

### 10

### 0

### 0.12

### 0.13

### 0.14

### 0.15

### 0.16

### 0.17

### 0.18

### Ambiguity radius (log−scale)

### optimal exit point

Figure 6: Dependenyoftheoptimalexitpointonthesizeoftheambiguityset.

### 0

### 0.02

### 0.04

### 0.06

### 0.08

### 0.1

### 0.12

### 0.14

### 0.16

### 0.18

### 0.2

### −0.2

### 0

### 0.2

### 0.4

### 0.6

### 0.8

### 1

### 1.2

Figure7: Theworstasedistributionfuntionsforradii

### ǫ

### = 0

### .

### 001

### ,

### 0

### .

### 005

### ,

### 0

### .

### 005

### ,

### 0

### .

### 01

Referenes

[1℄ G. Calaore. Ambiguous risk measures and optimal robust portfolios. SIAM Journal of

ControlOptimization,18 (3), pp. 853-877,2007.

[2℄ E.DelageandY.Ye.Distributionallyrobustoptimizationundermomentunertainty with

appliation todata-driven problems. OperationsResearh, 58,pp. 596-612,2010

[3℄ J. Dupaová.On minimaxdeision rule instohasti linear programing. Studiesin

Math-ematialprogramming, pp. 47-60,1980.

[4℄ J. Dupaová. Theminimax approahtostohasti programmingandanillustrative

appli-ation. Stohastis, 20,pp. 73-88, 1987.

[5℄ J.Dupaová.Unertaintiesinminimaxstohastiprograms. Optimization,1,pp.191-220,

2010.

[6℄ N. C. P. Edirishinge. Stohasti Programming Approximations using Limited Moment

InformationwithAppliationtoAsset Alloation.In: StohastiProgramming: The state

ofthe ArtinHonor ofGeorgeB.Dantzig. InternationalSeriesinOperationsResearhand

Management Siene. Springer, pp. 97-138, 2011.

[7℄ J.GohandM.Sim.Distributionallyrobustoptimizationanditstratableapproximations.

Operations Researh,58, pp. 902-917,2010.

[8℄ L. P. Hansen and T. J.Sargent. Robustness. Prineton University Press, 2007.

[9℄ R. Jagannathan. Minimax proedure for a lass of linear programs under unertainty.

Operations Researh,25, pp. 173-177,1977.

[10℄ Cunun Luan. Insurane Premium Calulations with Antiipated Utility Theory. Astin

Bulletin31 (1), pp. 23-35, 2001.

[11℄ G. C. Pug, A. Pihler and D. Wozabal. The 1/n investment strategy is optimal under

high modelambiguity.J. Bankingand Finane,36, pp. 410-417.

[12℄ G.Ch.PugandD.Wozabal.Ambiguityinportfolioseletion.QuantitativeFinane,7(4),

pp. 435-442, 2007.

[13℄ G. Ch. Pug and A. Pihler. MultistageStohasti Optimization. Springer Verlag, 2014.

[14℄ A.Raviv. TheDesign ofanOptimalInsurane Poliy.The AmerianEonomiReview69

al Theory of Inventory and Prodution, Arrow KJ, Karlin S, Sarf H, (Eds.). Stanford

University Press, 1958.

[16℄ A. Shapiro and A. J. Kleywegt. Minimax analysis of stohasti problems. Optimization

Methodsand Software, 17,pp. 523-542,2002.

[17℄ A. Shapiroand Sh. Ahmed. On a lass of minimaxstohasti programs. SIAM Journal of

Optimization,14, pp. 1237-1249, 2004.

[18℄ A. Thiele. Robust stohasti programming with unertain probabilities. IMA Journal of

Management Mathematis,19, pp. 289-321,2008.

[19℄ S.S. Vallander,Calulationof the Wasserstein distane between probability distributions

onthe line. Theory of Probability and itsAppliations, 1974, 18(4), pp. 784-786

Appendix

Convexity of h. If

### h

isa nonnegative1-homogeneous funtion on### R

### n

(i.e.

### h

### (

### λx

### ) =

### λh

### (

### x

### )

for

### λ >

### 0

suh that itslevelsets are losed onvex. Then### h

is onvex.Let

### C

betheepigraphof### h

.### C

isaone. Foreahelement### (

### x, h

### (

### x

### ))

### 6= (0

### ,

### 0)

onthe boundary of### C

, let### (

proj### (

### x

### )

### ,

### 1)

be the element### (

### 1

### h

### (

### x

### )

### x,

### 1)

on the 1-level set. Suppose that### (

### x, h

### (

### x

### ))

and### (

### y, h

### (

### y

### ))

are in the boundary of### C

, but an element### (

### z, α

### )

on the line segment joiningthem is not in### C

. Then also### (

proj### (

### z

### )

### ,

### 1)

is not in### C

, but it is on the line segment joining### (

proj### (

### x

### )

### ,

### 1)

and### (

proj### (

### y

### )

### ,

### 1)

. But sine these pointsare inthe same levelset, this is aontradition to the assumption.The solution algorithm.

Notie thatthe objetivefuntion

### E

_{P}

_{[1}

_{−}

_{ξ}

_{+ min(}

_{ξ, y}

_{)]}

_{−}

_{π}

_{y}

_{(}

_{ξ}

_{)}

_{−}

_{γ}

### V

_{ar}

_{P}

_{[1}

_{−}

_{ξ}

_{+ min(}

_{ξ, y}

_{)]}

is onvex in the probability measure

### P

and hene the inner problem### min

### E

_{F}

_{[1}

_{−}

_{ξ}

_{+ min(}

_{ξ, y}

_{)]}

_{−}

_{π}

_{y}

_{(}

_{ξ}

_{)}

_{−}

_{γ}

### V

_{ar}

_{F}

_{[1}

_{−}

_{ξ}

_{+ min(}

_{ξ, y}

_{)]}

### F

has a density in### (0

### ,

### 1]

, whih is pieewise onstant in### [

### z

### i

### , z

### i

### +1

### ]

### d

### (

### F,

### ˆ(

### F

### ))

### ≤

### ǫ

is aonvex optimizationproblem in

### R

### k

.

The minimax problem is solved in a stepwise manner, for the details of see Algorithm 7.2