Probability of Ruin

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    ACTL3162 General Insurance Techniques (2015) / 5. Solvency and Ruin Theory

    ACTL3162 General Insurance Techniques

    5. Solvency and Ruin Theory∗

    A/Prof Benjamin Avanzi

    School of Risk and Actuarial Studies

    UNSW Australia Business School

    [email protected]

    S2 2015

    ∗References:   MW 10, 5.0–5.2 / FV / (A 13) / (D) / (K)1/67

    mailto:[email protected]?subject=ACTL3162http://papers.ssrn.com/sol3/papers.cfm?abstract_id=2319328http://dx.doi.org/10.1080/10920277.1998.10595667http://dx.doi.org/10.1080/10920277.1998.10595667http://papers.ssrn.com/sol3/papers.cfm?abstract_id=2319328mailto:[email protected]?subject=ACTL3162

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    ACTL3162 General Insurance Techniques (2015) / 5. Solvency and Ruin Theory

    Plan

    1   Solvency considerationsBalance sheet and solvency

    Risk moduelsInsurance liability variables

    2   Ruin theory in discrete timeSurplus process and ruinLundberg bound

    3   Ruin theory in continuous timeSurplus processThe stability problemThe probability of ruinApplications to reinsurancede Finetti’s modification

    4   Dependence modelling and copulasIntroduction to DependenceWhat is a copula?Archimedean copulasSimulation of copulas

    Fitting copulas: case study2/67

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    ACTL3162 General Insurance Techniques (2015) / 5. Solvency and Ruin Theory

    Solvency considerations

    Balance sheet and solvency

    Balance sheet

    Balance sheet: we would like  At  ≥ Lt , withAt : assets at time  t 

    Lt : liabilities at time  t Let C t  = At  − Lt be the continuous time ‘surplus process’. At first sight it is

    reasonable to require

    Pr

    inf t 

    C t  ≥ 0 C 0  = c o  = Prc 0

    inf t 

    At  − Lt ≥ 1− p .

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    ACTL3162 General Insurance Techniques (2015) / 5. Solvency and Ruin Theory

    Solvency considerations

    Balance sheet and solvency

    Risk measures

    The requirement above with time horizon 1,

    Prc 0

    [A1 − L1] ≥ 1− p ,

    is that of a Value-at Risk VaR1−p (L1 − A1)  on security level 1 − p More generally, one requires

    ρ(L1 − A1) ≤ 0.

    Solvency II uses VaR99.5%. There are other, arguably better ones,such as the TVaR (used, e.g., in the Swiss Solvency Test). For

    given portfolio of insurance business, one can adjust  c 0  and

    investment strategy A

    0  accordingly.4/67

    ACTL3162 G l I T h i (2015) / 5 S l d R i Th

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    ACTL3162 General Insurance Techniques (2015) / 5. Solvency and Ruin Theory

    Solvency considerations

    Balance sheet and solvency

    Market consistent values

    The main difficulty is to model  L1 − A1, both of which arestochastic and  not   independant.

    A1   is typically well defined and corresponding to assets which

    have a market value.

    However,  L1  does not have a market value. For comparability,we need to determine market-consistent values in amarked-to-model approach.

    We splitL1  = X 1 + L

    +1 ,

    where  X 1  are payments done over the next year, and  L+1   what

    remains to be paid at the end of the year.

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    ACTL3162 G l I T h i (2015) / 5 S l d R i Th

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    ACTL3162 General Insurance Techniques (2015) / 5. Solvency and Ruin Theory

    Solvency considerations

    Balance sheet and solvency

    Market consistent values:   L+1

    L+1   is hard to determine:

    We need a market-consistent value for those outstandingclaims liabilities

    This is different from the IBNR R(1) (we will discuss those inModule 7):

    1   R(1) are calculated on a nominal basis (no discounting). Amarket-consistent value must include discounting.

    2

      R(1) are conditional   expectations , given the information

    available at time 1. A market-consistent value must include aloading to recognise the uncertaintly that is associated tothose future cash flows.

    Think of it as a  transfer value , or run-off value.

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    ACTL3162 General Insurance Techniques (2015) / 5. Solvency and Ruin Theory

    Solvency considerations

    Risk moduels

    1   Solvency considerationsBalance sheet and solvency

    Risk moduelsInsurance liability variables

    2   Ruin theory in discrete timeSurplus process and ruinLundberg bound

    3   Ruin theory in continuous timeSurplus processThe stability problemThe probability of ruinApplications to reinsurancede Finetti’s modification

    4   Dependence modelling and copulasIntroduction to DependenceWhat is a copula?Archimedean copulasSimulation of copulas

    Fitting copulas: case study7/67

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    ACTL3162 General Insurance Techniques (2015) / 5. Solvency and Ruin Theory

    Solvency considerations

    Risk moduels

    Asset deficit ADt 

    LetADt  = Lt  −At  = X t  + L+t  − At 

    be the asset deficit  as of time  t .

    Obviously, we want  ρ(ADt ) ≤ 0.In practice, its modelling is broken down into  modules , whichare then re-aggregated using correlation matrices.

    Most relevant to us:

    Market risk: volatility of market prices of financial instruments

    Insurance risk: typically split into branches. GI comprises: (i)reserve risk, (ii) premium risk.Credit risk: conterparty default riskOperational risk: “risk of loss arising from inadequate or failedinteral processes, or from personnel and systems,

    or from external events”7/67

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    ACTL3162 General Insurance Techniques (2015) / 5. Solvency and Ruin Theory

    Solvency considerations

    Risk moduels

    Asset deficit AD0  and its evolution over one time period

    We decomposeAD0  = L0 − A0,

    where

    L0   =   LPY0   + L

    CY0   = L

    +0

    A0   =   c 0 + APY0   + π

    CY

    At the end of the year, we have  A1  and

    L1  = X 1 + L+1   =

    X PY1   + X 

    CY1   + X 

    Op1

    +L+,PY1   + L

    +,CY1

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    ACTL3162 General Insurance Techniques (2015) / 5. Solvency and Ruin Theory

    Solvency considerations

    Risk moduels

    Asset deficit AD1

    AD1   =X PY1   + X 

    CY1   + X 

    Op1

    +L+,PY1   + L

    +,CY1

    − A1

    (split into payments and outstanding loss liabilities)

    =X PY1   + L

    +,PY1

    +X CY1   + L

    +,CY1

    + X Op1   − A1

    (split into PY risk and CY risk)

    Market risk: all

    Insurance risk: all but  X Op1   and  A1

    Credit risk: mainly  A0   (! reinsurance)

    Operational risk:  X Op19/67

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    q ( ) / y y

    Solvency considerations

    Insurance liability variables

    1   Solvency considerationsBalance sheet and solvency

    Risk moduelsInsurance liability variables

    2   Ruin theory in discrete timeSurplus process and ruinLundberg bound

    3   Ruin theory in continuous timeSurplus processThe stability problemThe probability of ruinApplications to reinsurancede Finetti’s modification

    4   Dependence modelling and copulasIntroduction to DependenceWhat is a copula?Archimedean copulasSimulation of copulas

    Fitting copulas: case study10/67

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    q ( ) / y y

    Solvency considerations

    Insurance liability variables

    Market-consistent values

    We focus here on insurance liabilities:

    LIns1   =X PY1   + L

    +,PY1

    +X CY1   + L

    +,CY1

     =  L1 − X Op1

    Introducing deflators  ϕ’s (random) leads to

    LIns1   =  1

    ϕ1

    s ≥1

    E  [ϕs X s  |F 1 ] = X 1 +   1ϕ1

    s ≥2

    E  [ϕs X s  |F 1 ]

    if  ϕs   and  X s  are uncorrelated :

    =   1ϕ1

    s ≥1

    E  [ϕs  |F 1 ] E  [X s  |F 1 ]

    =

    s ≥1P (1, s )E  [X s  |F 1 ]

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    Solvency considerations

    Insurance liability variables

    Simplification

    Denote:

    p (1, s ) =   E  [P (1, s ) |F 0 ]x s    =   E  [X s 

    |F 0 ]

    We then use the approximation

    P (1, s )E  [X s  |F 1 ]= [p (1, s ) + (P (1, s )

    −p (1, s ))] [x s  + (E  [X s 

    |F 1 ]−

    x s )]

    ≈   p (1, s )x s  (expected value as of time 0)+ (P (1, s )− p (1, s )) x s   (discounting uncertainty)+p (1, s ) (E  [X s  |F 1 ]− x s )  (insurance cash flows uncertainty)

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    Solvency considerations

    Insurance liability variables

    AD1  after simplification

    We have then

    AD1   =

    s ≥1p (1, s )x s  + (P (1, s )− p (1, s )) x s  −A1

    ( = market and credit risks  Z 1)

    +s ≥1

    p (1, s ) (E  [X s  |F 1 ]− x s )

    ( = insurance risk  Z 2)

    +X Op1( = operational risk  Z 3)

    Hereafter we focus on  Z 2.

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    Solvency considerations

    Insurance liability variables

    Insurance risk  Z 2

    We consider  Z 2  = Z PY2   + Z CY2   where

    Z PY2   =s ≥1

    p (1, s )E X PYs    |F 1

    − x PYs 

    CY

    2   = s ≥1 p (1, s )E X CYs    |F 1 − x CYs  Assume that the proportion of cash flows paid in year  s   γ s   isdeterministic. Then

    Z PY2   = s ≥1

    p (1, s )γ PYs  X PY1   +R(1) −R(0)Z CY2   =

    s ≥1 p (1, s )γ 

    CYs 

    [E  [S 1 |F 1 ]− E  [S 1 |F 0 ]]

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    Solvency considerations

    Insurance liability variables

    Insurance risk on PY

    LetCDR1  = −

    X PY1   +R(1) −R(0)

    ,

    which has expected value  E [CDR1] = 0. Process:1   Calculate MSEP for each LoB

    2   Specify a correlation matrix between the LoB

    3   Aggregate (1) with the help of (2) and obtain an overall

    variance4   Fit a translated gamma or lognormal to (3), assuming that the

    mean is R(0), to approximate the distribution of CDR1

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    Solvency considerations

    Insurance liability variables

    Insurance risk on CY

    Aggregate claimsE  [S 1 |F 1 ]

    result from the premium exposure  πCY . This is called the  premiumliability . It is split into two independent random variables:

    S lc: large claims (>  threshold  M ), modelled with compoundPoisson model (CRM) per LoB with Pareto claims severities.Then use Panjer or FFT.

    S sc: small claims, whose moments are aggregated using an

    appropriate correlation matrix, and then fit to with a gammaor lognormal distribution.

    Note an assumption is required to aggregate  Z CY2   and  Z PY2   into  Z 2.

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    Ruin theory in discrete time

    Surplus process and ruin

    1   Solvency considerationsBalance sheet and solvencyRisk moduelsInsurance liability variables

    2   Ruin theory in discrete timeSurplus process and ruinLundberg bound

    3   Ruin theory in continuous timeSurplus processThe stability problemThe probability of ruinApplications to reinsurancede Finetti’s modification

    4   Dependence modelling and copulasIntroduction to DependenceWhat is a copula?Archimedean copulasSimulation of copulas

    Fitting copulas: case study16/67

    ACTL3162 General Insurance Techniques (2015) / 5. Solvency and Ruin Theory

    R i h i di i

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    Ruin theory in discrete time

    Surplus process and ruin

    Surplus process

    We extend the definition of the surplus to more than one year, anddefine the surplus

    C t  = C (c 0)t    = c 0 +

    u =1(πu − S u ),with

    c 0   is the initial capital

    (πt , S t )t =1,2,3,...   is an iid sequence with  πt  > 0 and  S t  ≥ 0.Furthermore, we assume that

    X t  = πt  − S t 

    are independant and stationary increments.16/67

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    R i th i di t ti

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    Ruin theory in discrete time

    Surplus process and ruin

    Ruin

    We hopeC t  ≥ 0 for all  t  ≥ 0.

    If not, then the ruin time  τ   is defined such that

    τ  = inf {s  ∈ N0; C s  

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    Ruin theory in discrete time

    Surplus process and ruin

    Theorem 5.4

    Depending on the sign of  E [X 1], the process behaves differently:

    1   if  E [X 1] <  0, the process blows down to −∞;2   if  E [X 1] >  0, the process blows up to

     ∞;

    3   if  E [X 1] = 0, the process either blows up or down to ±∞.So what?

    1   if  E [X 1] <  0 then  ψ(c 0) = 1 for any  c 0 ≥ 0;2   if  E [X 1] >  0 (Net Profit Condition—NPC), then  ψ(0) <  1.

    Furthermore, it is obvious that under the NPC

    ψ(c 0) ≤ ψ(0) ≤ 1.

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    Ruin theory in discrete time

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    Ruin theory in discrete time

    Lundberg bound

    1   Solvency considerationsBalance sheet and solvencyRisk moduelsInsurance liability variables

    2   Ruin theory in discrete timeSurplus process and ruinLundberg bound

    3   Ruin theory in continuous timeSurplus processThe stability problemThe probability of ruinApplications to reinsurancede Finetti’s modification

    4   Dependence modelling and copulasIntroduction to DependenceWhat is a copula?Archimedean copulasSimulation of copulas

    Fitting copulas: case study19/67

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    Ruin theory in discrete time

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    Ruin theory in discrete time

    Lundberg bound

    Lundberg coefficient / Adjustment coefficient

    Assume there exists an  R  > 0 such that

    M −X 1 (R ) = M S 1−π1 (R ) = 1.Then, this  R  > 0 is called ‘Lundberg coefficient’. If it exists then itis unique.It can be shown that

    ψ(c 0) ≤ e −Rc 0

    .This is called  Lundberg’s exponential bound .

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    Ruin theory in continuous time

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    Ruin theory in continuous time

    Surplus process

    1   Solvency considerationsBalance sheet and solvencyRisk moduelsInsurance liability variables

    2   Ruin theory in discrete timeSurplus process and ruinLundberg bound

    3   Ruin theory in continuous timeSurplus processThe stability problemThe probability of ruinApplications to reinsurancede Finetti’s modification

    4   Dependence modelling and copulasIntroduction to DependenceWhat is a copula?Archimedean copulasSimulation of copulas

    Fitting copulas: case study20/67

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    Ruin theory in continuous time

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    y

    Surplus process

    Surplus process

    We define now the continuous time surplus process

    C (t ) = c 0 + πt − S (t ),

    where

    c 0   is the initial surplus;

    π   is the premium rate:   c  = (1 + θ)λE [Y 1]

    θ   is the relative security loading (θ > 0 under the NPC)

    S (t ) = N (t )i =1   Y i  are aggregate losses up to time  t If furthermore, the losses {Y i }  are iid and independent of  {N (t )},and {N (t )}  is a Poisson process, that is,

     N (t )i =1   Y i   is compound

    Poisson, then

    {C (t )}  is called the Cramér-Lundberg process.20/67

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    Ruin theory in continuous time

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    y

    Surplus process

    0   1   2   3   4

            0

            5

            1        0

            1        5

    Poisson process N(t)

    {N (t )}counting process

    step function

    Poisson process iff 

    incrementsN (t  + h)−N (t ) ∼Poisson(λh)

    ⇐⇒

    time between jumps  W i  ∼exponential(1/λ)

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    Ruin theory in continuous time

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    Surplus process

    0   1   2   3   4

            0

            5

            1        0

            1        5

            2        0

            2        5

            3        0

            3        5

    Compound Poisson process S(t)

    We define

    S (t ) =

    N (t )

    i =1 X i .{S (t )}   is {N (t )}  but

    step   i  has heightX i   instead of 1

    Increments:S (t  + h)− S (t ) ∼CPoisson(λh, P (x ))

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    Surplus process

    0   1   2   3   4

        -        5

            0

            5

            1        0

    Cramer-Lundberg process U(t)

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    The stability problem

    1   Solvency considerationsBalance sheet and solvencyRisk moduelsInsurance liability variables

    2   Ruin theory in discrete timeSurplus process and ruinLundberg bound

    3   Ruin theory in continuous timeSurplus processThe stability problemThe probability of ruinApplications to reinsurancede Finetti’s modification

    4   Dependence modelling and copulasIntroduction to DependenceWhat is a copula?Archimedean copulasSimulation of copulas

    Fitting copulas: case study24/67

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    The stability problem

    The stability problem

    The “survival” of the insurance company will depend on certain(decision) variables:

    initial surplus (c 0)

    loading of premiums (θ)

    reinsurance (e.g.   α  or d  — see later)

    What is the "best" way to choose/monitor these variables? Of course, this depends on what criterion the assessment is based:

    probability of ruin—goes back to Lundberg (1909) and Cramér(1930, 1955)

    utility—goes back to von Neumann and Morgenstern (1944)

    present value of dividends—goes back to de Finetti (1957)

    . . .

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    The probability of ruin

    1   Solvency considerationsBalance sheet and solvencyRisk moduelsInsurance liability variables

    2   Ruin theory in discrete timeSurplus process and ruinLundberg bound

    3   Ruin theory in continuous timeSurplus processThe stability problemThe probability of ruinApplications to reinsurancede Finetti’s modification

    4   Dependence modelling and copulasIntroduction to DependenceWhat is a copula?Archimedean copulasSimulation of copulas

    Fitting copulas: case study25/67

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    The probability of ruin

    How to calculate the probability of ruin

    There are different ways of calculating  ψ:

    analytically

    ψ(u )  is very hard to calculate (in closed form), but possible forexponential and mixtures of exponential lossesψ(u , t )   is even more difficult to determine

    using Panjer’s recursion via a special trick (assignment 2009,see reference D)

    Monte-Carlo methods (simulations)In what follows, we assume we are using the Cramér-Lundbergmodel.

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    Th b bili f i

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    The probability of ruin

    The adjustment coefficient

    Consider the excess of losses over premiums over the interval  [0, t ]:

    S (t )− πt .

    We define the adjustment coefficient  R  as the first positive solutionof 

    M S (t )−πt (R ) = E e R (S (t )−πt )

     =  e −R πt e λt [M Y 1(R )−1] = 1.

    Furthermore

    E e −RC (t )

     =  e −Rc 0 for all  t  ≥ 0

    and thus e −RC (t )

     is a martingale.26/67

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    Th b bilit f i

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    The probability of ruin

    0.0   0.1   0.2   0.3   0.4   0.5   0.6

            1  .

            0

            1  .

            5

            2  .        0

            2  .

            5

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    The probability of ruin

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    The probability of ruin

    A Theorem 13.4.1

    If  {C (t )}   is a Cramér-Lundberg process with  θ > 0, then for  c 0 ≥ 0

    ψ(c 0) =  e −Rc 0

    e −RC (τ )|T   < ∞

    .

    Since  C (τ ) <  0, we have then (Lundberg’s exponential upperbound)

    ψ(c 0)  −y max   =⇒   ψ(c 0) > e −R (c 0+y max)

    Finally,e −R (c 0+y max) < ψ(c 0) 

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    The probability of ruin

    Example

    Assume  Y 1 ∼ exp(β ). Find  R   and  ψ(c 0).

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    The probability of ruin

    Example

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    Applications to reinsurance

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    pp

    1   Solvency considerationsBalance sheet and solvencyRisk moduels

    Insurance liability variables2   Ruin theory in discrete time

    Surplus process and ruinLundberg bound

    3   Ruin theory in continuous time

    Surplus processThe stability problemThe probability of ruinApplications to reinsurancede Finetti’s modification

    4   Dependence modelling and copulasIntroduction to DependenceWhat is a copula?Archimedean copulasSimulation of copulas

    Fitting copulas: case study31/67

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    pp

    Applications to reinsurance

    We have now a model to study options about reinsurance

    Based on the probability of ruin criterion, we will adjust the

    adjustment coefficient (hence its name..) to meet a goal, suchas

    maximise  R  ⇔  minimise  ψ(c 0)find the cheapest reinsurance such that  ψ(c 0)   is inferior tosome level

    Note that even if  ψ(c 0)  can’t be calculated, you can still playwith  R  and have qualitative results about  ψ(c 0).

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    Numerical example (excess of loss)

    Let  X  ∼ exp(1),  θ = 1.25,  θrins  = 1.4. Consider nonproportionalreinsurance:

    transferred loss = (X  − d )+.We have then

    πreins = 1.4λ  ∞d 

    (x  − d )e −x dx  = 1.4λe −d 

    and

    M Y ret (r ) =    d 0 e rx e −x dx  +   ∞

    d  e rd 

    e −x 

    dx  =

     1

    −re −d (1−r )

    1− r    .Hence, the (nonlinear) equation for  R ret   is

    1 + (1.25

    −1.4e −d )r 

     −

     1− re −d (1−r )

    1− r   = 0.

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    0.7   0.8   0.9   1.0   1.1   1.2   1.3

            0  .

            3        1

            0  .

            3        2

            0  .

            3        3

            0  .

            3        4

            0  .

            3        5

    d

         R_

         h

    Using R, we have

    d ∗  = 0.9632226

    and

    R ∗ret = 0.3493290,

    which is much higher (better)

    than the best we could achievewith proportional reinsurance.

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    A Theorem 14.5.1

    Theorem 14.5.1 states that if 

    we are in a Cramér-Lundberg setting

    we are considering two reinsurance treaties, one of which is

    excess of loss

    both treaties have same expected payments and samepremium loadings

    then

    the adjustment coefficient with the excess of loss treaty willalways be at least as good (high) as with any other type of reinsurance treaty

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    de Finetti’s modification

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    1   Solvency considerationsBalance sheet and solvencyRisk moduels

    Insurance liability variables2   Ruin theory in discrete time

    Surplus process and ruinLundberg bound

    3   Ruin theory in continuous time

    Surplus processThe stability problemThe probability of ruinApplications to reinsurancede Finetti’s modification

    4   Dependence modelling and copulasIntroduction to DependenceWhat is a copula?Archimedean copulasSimulation of copulas

    Fitting copulas: case study35/67

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    Using the probability of ruin as a criterion presents some issues:

    minimising ψ(c 0)  supposes that companies should let theirsurplus grow without limit, which is not realistic

    why should an older company hold more capital than a youngone, just because it is older?

    furthermore, if some of the surplus is distributed from time totime, calculations of  ψ(c 0)  are wrong

    Bruno de Finetti’s (1957) goal:

    to propose an alternative formulation that would avoid 

    the misconceptions of the classical Cramér-Lundberg model and that would be sufficiently realistic and tractable to “study the practical problems regarding risk and reinsurance” (our translation)

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    Optimal dividend strategies

    One has to find a rational way to model distribution of the surplusthe distribution of some of the surplus (to the shareholders) isconsidered as a dividend

    the answer to  how much  and  when  dividends should bedistributed is called a dividend strategy

    Consider the expected present value of dividends paid until ruin

    This is the value of the company according to the  Gordonmodel 

    shareholders (the decision makers) are likely to want to

    maximise this value

    This leads to the question of  optimal dividend strategies.

    optimal with respect to the expected present value of dividends, rather than the probability of ruin

    (which is usually 1 in this context).36/67

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    Optimal dividends in the Cramér-Lundberg model

    The optimal dividend strategy is a barrier strategy (in some cases):

    0   1   2   3   4   5   6   7

            0

            5

            1        0

            1        5

    Cramer-Lundberg process U(t)

    0   1   2   3   4   5   6   7

            0

            5

            1        0

            1        5

    Surplus X(t) and dividends D(t)

    EPV of dividends:  V (u ; b ) =

     T 

    0  e −δt dD (t )

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    Introduction to Dependence

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    1   Solvency considerationsBalance sheet and solvencyRisk moduels

    Insurance liability variables2   Ruin theory in discrete time

    Surplus process and ruinLundberg bound

    3   Ruin theory in continuous time

    Surplus processThe stability problemThe probability of ruinApplications to reinsurancede Finetti’s modification

    4   Dependence modelling and copulasIntroduction to DependenceWhat is a copula?Archimedean copulasSimulation of copulas

    Fitting copulas: case study38/67

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    Introduction to Dependence

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    Motivation

    How does dependence arise?

    Events affecting more than one variable

    Underlying economic factors affecting more than one risk area

    Reasons for modelling dependence:

    Pricing:inflows and outflows of capital

    Solvency assessment:bottom up: risks given

     → capital requirements

    Capital allocation:top down: capital given → allocation per riskPortfolio structure: (or strategic asset allocation)how does the capital move compared to risks?

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    Examples

    World Trade Centre causing losses to Property, Life, Workers’Compensation, Aviation insurers

    Enron causing losses to the stock market and to Surety Bonds,

    Errors & Omissions and Directors & Officers underwriters

    Dot.com market collapse causing losses to the stock marketand to insurers of financial institutions and D&O writers

    WTC / Enron / stock market losses causing impairment to

    reinsurers solvency, so increasing credit risk on payments byreinsurers

    Asbestos affecting many past liability years at once

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    Introduction to Dependence

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    Example of real actuarial data(Avanzi, Cassar and Wong, 2011)

    Data were provided by the SUVA (Swiss workers compensationinsurer)

    Random sample of 5% of accident claims in constructionsector with accident year 1999 (developped as of 2003)

    Two types of claims: 2249 medical cost claims, et 1099 dailyallowance claims

    1089 of those are common (!)

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    Scatterplot of the log of those 1089 common claims (LHS) etemprical copula (RHS):

    There is obvious right tail dependence.

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    Correlation = dependance?

    Correlation consumption of cheese (US) and deaths by becomingtangled in bedsheets (Tyler Vigen, 2015):

    Correlation = 0.95!!

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    Common fallacies

    Fallacy 1: a small correlation  ρ(X 1, X 2)  implies that  X 1  and  X 2  are close to being independent 

    wrong!

    Independence implies zero correlation BUTA correlation of zero does not always mean independence.

    See example 1 below.

    Fallacy 2 : marginal distributions and their correlation matrix uniquely determine the joint distribution.

    This is true only for elliptical families (including multivariatenormal),  but wrong in general!

    See example 2 below.

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    Example 1

    Company’s two risks  X 1  and  X 2

    Let  Z  ∼ N (0, 1)  and Pr(U  = −1) = 1/2 = Pr(U  = 1)U  stands for an economic stress generator,   independent  of  Z 

    Consider:

    X 1  = Z  ∼ N (0, 1)and

    X 2  = UZ  ∼ N (0, 1).

    Now Cov(X 

    1,X 

    2) = E 

    (X 

    1X 

    2) = E 

    (UZ 2

    ) = E 

    (U 

    )E 

    (Z 2

    ) = 0hence  ρ(X 1, X 2) = 0. However,  X 1  and  X 2  are strongly dependent , with 50% probability co-monotone and 50%counter-monotone.

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    Example 2

    Marginals and correlations—not enough to completely determine joint distribution

    Marginals: Gamma(5, 1)

    Correlation:   ρ = 0.75

    Different dependence structures: Normal copula vsCook-Johnson copula

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    What is a copula?

    1 Solvency considerations

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    1   Solvency considerationsBalance sheet and solvencyRisk moduels

    Insurance liability variables2   Ruin theory in discrete time

    Surplus process and ruinLundberg bound

    3   Ruin theory in continuous time

    Surplus processThe stability problemThe probability of ruinApplications to reinsurancede Finetti’s modification

    4   Dependence modelling and copulasIntroduction to DependenceWhat is a copula?Archimedean copulasSimulation of copulasFitting copulas: case study

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    What is a copula?

    Skl ’ i h

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    Sklar’s representation theorem

    The copula “couples”, “links”, or “connects” the joint distribution toits marginals.Sklar (1959): There exists a copula function  C   such that

    F  (x 1, x 2, ..., x n) = C  (F 1 (x 1) , F 2 (x 2) , ..., F n (x n))

    where  F k   is the marginal for  X k ,  k  = 1, 2, ...,n. Equivalently,

    Pr (X 1 ≤ x 1, ..., X n ≤ x n) = C  (Pr (X 1 ≤ x 1) , ..., Pr (X n ≤ x n)) .

    Under certain conditions, the copula

    C  (u 1, ...,u n) = F F −11   (u 1) , ..., F 

    −1n   (u n)

    is unique, where  F −1k    denote the respective quantile

    functions. This is one way of constructing copulas.47/67 ACTL3162 General Insurance Techniques (2015) / 5. Solvency and Ruin Theory

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    What is a copula?

    E l

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    Example

    Let

    F (x , y ) =

    (x +1)(e y −1)x +2e y −1   (x , y ) ∈ [−1, 1]× [0,∞]1− e −y  (x , y ) ∈ (1,∞]× [0,∞]

    0 elsewhereHence

    F (x ) =  x  + 1

    2  ,   x  ∈ [−1, 1]

    F −1(u ) =   2u − 1 =  x G (u ) =   1 =  e −y ,   y  ≥ 0

    G −1(u ) =   − ln(1− u ) = y 

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    Example

    Finally,

    C (u , v ) =  (2u − 1 + 1)[(1− v )−1 − 1]

    2u − 1 + 2(1− v )−1

    − 1=

      2u (1− 1 + v )(2u − 2)(1− v ) + 2

    =  2uv 

    2u 

    −2uv 

     −2 + 2v  + 2

    =   uv u  + v  − uv 

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    What is a copula?

    D it i t d ith l

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    Density associated with a copula

    For continuous marginals with respective pdf  f  1,...f  n, the jointpdf of  X  can be written as

    f    (x 1, ...,x n) = f  1 (x 1)

    · · ·f  n (x n)

    ×c  (F 1 (x 1) , ...,F n (x n))

    where the copula density  c   is given by

    c  (u 1, ..., u n) = ∂ nC  (u 1, ..., u n)

    ∂ u 1∂ u 2

    · · ·∂ u n

    .

    Observe that the copula  c  distorts the independence to inducethe actual dependence structure.

    If independent,  c  = 1.

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    Archimedean copulas

    1   Solvency considerations

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    Balance sheet and solvencyRisk moduels

    Insurance liability variables2   Ruin theory in discrete time

    Surplus process and ruinLundberg bound

    3   Ruin theory in continuous time

    Surplus processThe stability problemThe probability of ruinApplications to reinsurancede Finetti’s modification

    4   Dependence modelling and copulasIntroduction to DependenceWhat is a copula?Archimedean copulasSimulation of copulasFitting copulas: case study

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    Archimedean copulas

    The family of Archimedean copulas

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    The family of Archimedean copulas

    C   is Archimedean if it has the form

    C  (u 1

    , ..., u n

    ) = ψ−

    1 (ψ (u 1

    ) +· · ·

    + ψ (u n

    ))

    for all 0 ≤ u 1, ...,u n ≤ 1 and for some function  ψ  (called thegenerator) satisfying:

    ψ (1) = 0;

    ψ   is decreasing; andψ   is convex.

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    Archimedean copulas

    The Clayton copula

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    The Clayton copula

    The Clayton copula is defined by

    C  (u 1, ..., u n) =   n

    k =1u −θk    − n + 1

    −1/θ.

    Archimedean type with:

    ψ (t ) = t −θ − 1, θ > 1ψ−1 (s ) = (1 + s )−1

    The case of  n = 2:

    C  (u 1, u 2) =u −θ1   + u 

    −θ2   − 1

    −1/θ.

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    Archimedean copulas

    The Frank copula

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    The Frank copula

    The Frank copula is defined by

    C  (u 1

    , ...,u n

    ) =  1

    log θ log1 +

    nk =1 (θ

    u k 

    −1)

    (θ − 1)n−1 .Archimedean type with:

    ψ (t ) = − log

    θt −1θ−1

    , θ ≥ 0

    ψ−1

    (s ) = −  1

    log θ log 1− e −s  1− e −θTry the case of  n = 2.

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    Simulation of copulas

    1   Solvency considerationsB l h d l

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    Balance sheet and solvencyRisk moduels

    Insurance liability variables2   Ruin theory in discrete time

    Surplus process and ruinLundberg bound

    3   Ruin theory in continuous time

    Surplus processThe stability problemThe probability of ruinApplications to reinsurancede Finetti’s modification

    4   Dependence modelling and copulasIntroduction to DependenceWhat is a copula?Archimedean copulasSimulation of copulasFitting copulas: case study

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    Simulation of copulas

    Simulating random variables with a dependent structure

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    Simulating random variables with a dependent structure

    We will introduce the general  conditional distribution method

    The overarching idea is (for the bivariate case)

    simulate two independent uniform random variable  u  and  t 

    ‘tweak’  t   into a  v  ∈ [0, 1]  so that it has the right dependencestructure (w.r.t.   u ) with the help of the copulamap  u  and  v   into marginal  x   and  y  using their distributionfunction

    However, there are some specific, more efficient algorithms

    that are available for certain types of copulas (see referencebooks)

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    Preliminary: the conditional distribution function

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    Preliminary: the conditional distribution function

    We will need the conditional distribution function for  V   givenU  = u , which is denoted by  c u (v ) :

    c u (v ) = Pr[V  ≤ v |U  = u ]=   lim

    ∆u →0C (u  + ∆u , v )− C (u , v )

    ∆u 

    = ∂ C (u , v )

    ∂ u 

      .

    In particular, we will need its inverse.

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    Example

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    Example

    For the copula

    C (u , v ) =  uv 

    u  + v  − uv we have

    c u (v ) =  v (u  + v  − uv )− uv (1− v )

    (u  + v  − uv )2   =

      v 

    u  + v  − uv 2≡ t 

    c −1

    u    (t ) =

    √ tu 

    1−√ t (1− u ) ≡ v 

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    Simulation of copulas

    The conditional distribution method

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    The conditional distribution method

    Goal: generate a pair of pseudo-random variables  (X , Y )  with d.f.’sF   and  G , respectively, with dependence structure described by thecopula  C .Algorithm

    1  Generate two independent uniform  (0, 1)  pseudo-randomvariable  u  and  t 

    2   Set  v  = c −1u    (t )

    3   Map  (u , v )   into  (x , v ):

    x    =   F −1(u )

    v    =   G −1(v )

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    Simulation of copulas

    Example

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    Example

    Let  X   and  Y  be exponential with mean 1 and standard Normal,respectively. Furthermore, the copula describing their dependence issuch as in the previous example:

    C (u , v ) =  uv 

    u  + v  − uv Furthermore, you are given the following pseudo-random(independent) uniforms:

    0.3726791, 0.6189313, 0.75949099, 0.01801882

    Simulate two pairs of outcomes for  (X , Y ).

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    Exercise

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    Use of the conditional distribution method yields1   We can use the uniforms given in the question such that

    (u 1, t 1) = (0.3726791, 0.6189313)

    (u 2, t 2) = (0.75949099, 0.01801882)

    2   Set  v i   =   u i √ t i 1−(1−u i )√ t i  for  i  = 1, 2:v 1   =   0.5788953

    v 2   =   0.1053509

    3   Mapping  (u i , v i )   into  (x i , y i )  using

    x i  = F −1(u i ) = − ln(1− u i )  and  y i   = Φ−1(v i )  we have

    (x 1, y 1) = (0.4662971, 0.8648739)

    (x 2, y 2) = (

    −0.3247659,

    −1.251638)

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    Fitting copulas: case study

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    Balance sheet and solvencyRisk moduels

    Insurance liability variables2   Ruin theory in discrete time

    Surplus process and ruinLundberg bound

    3   Ruin theory in continuous time

    Surplus processThe stability problemThe probability of ruinApplications to reinsurancede Finetti’s modification

    4   Dependence modelling and copulasIntroduction to DependenceWhat is a copula?Archimedean copulasSimulation of copulasFitting copulas: case study

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    ACTL3162 General Insurance Techniques (2015) / 5. Solvency and Ruin Theory

    Dependence modelling and copulas

    Fitting copulas: case study

    Insurance company losses and expenses

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    p y p

    Data consists of 1 500 general liability claims

    Provided by the Insurance Services Office, Inc.

    X 1   is the loss, or amount of claims paid.

    X 2   is the ALAE, or allocated loss adjustment expense.

    Policy contains policy limits, and hence, censoring.

    δ   is the indicator for censoring so that the observed data

    consists of 

    (x 1i , x 2i , δ i )   for  i  = 1, 2, ..., 1500.

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    ACTL3162 General Insurance Techniques (2015) / 5. Solvency and Ruin Theory

    Dependence modelling and copulas

    Fitting copulas: case study

    Summary statistics of data

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    y

    Policy Loss LossLoss ALAE Limit (Uncensored) (Censored)

    Number 1 500 1 500 1 352 1 466 34Mean 41 208 12 588 559 098 37 110 217 491Median 12 000 5 471 500 000 11 048 100 000Std Deviation 102 748 28 146 418 649 92 513 258 205Minimum 10 15 5 000 10 5 000Maximum 2 173 595 501 863 7 500 000 2 173 595 1 000 000

    25th quantile 4 000 2 333 300 000 3 750 50 00075th quantile 35 000 12 577 1 000 000 32 000 300 000

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    ACTL3162 General Insurance Techniques (2015) / 5. Solvency and Ruin Theory

    Dependence modelling and copulas

    Fitting copulas: case study

    Figure 2: loss vs ALAE

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    2 4 6 8 10 12 14

          4

          6

          8

          1      0

          1      2

    LOSS vs ALAE on a log scale

    log(LOSS)

          l     o     g      (      A      L      A      E      )

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    ACTL3162 General Insurance Techniques (2015) / 5. Solvency and Ruin Theory

    Dependence modelling and copulas

    Fitting copulas: case study

    Maximum likelihood estimation

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    Case 1: loss variable is not censored, i.e.   δ  = 0.

    f    (x 1, x 2) = f  1 (x 1) f  2 (x 2)  ∂ 2

    ∂ x 1∂ x 2C  (F 1 (x 1) , F 2 (x 2))

    Case 2: loss variable is censored, i.e.   δ  = 1.

    ∂ 

    ∂ x 2P  (X 1  > x 1, X 2 ≤ x 2) =   ∂ 

    ∂ x 2[F 2 (x 2)− F  (x 1, x 2)]

    =   f  2 (x 2)−  ∂ 

    ∂ x 2 F  (x 1, x 2)

    =   f  2 (x 2)

    1−   ∂ 

    ∂ x 2C  (F 1 (x 1) , F 2 (x 2))

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    ACTL3162 General Insurance Techniques (2015) / 5. Solvency and Ruin Theory

    Dependence modelling and copulas

    Fitting copulas: case study

    Choice of marginals and copulas

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    Pareto marginals:   F k  (x k ) = 1−

      λk λk +x k 

    θk for k  = 1, 2.

    For the copulas, several candidates were used:

    Copula   C  (u 1, u 2)   C 2 (u 1, u 2) =   ∂ C  (u 1, u 2)∂ u 2C 12 (u 1, u 2) =  ∂ 

    2

    C  (u 1, u 2)∂ u 1∂ u 2

    Independent   u 1 × u 2   u 1   1

    Claytonu −α1   + u 

    −α2   − 1

    −1/α(C /u 2)

    α+1 (α + 1) C α · (C /u 1u 2)α+1

    Gumbel-Hougaard exp− ((− log u 1)α + (− log u 2)α)1/α

      log u 2log C 

    α−1C 

    u 2

    1

    C  C 1C 2 [1 + (α− 1) / (− log C )]

    Frank  1

    α log

    1 +

     (e αu 1 − 1) (e αu 2 − 1)e α − 1

      e αu 1 (e αu 2 − 1)(e α − 1) + (e αu 1 − 1) (e αu 2 − 1)

    α (e α − 1) e α(u 1+u 2)[(e α − 1) + (e αu 1 − 1) (e αu 2 − 1)]2

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    ACTL3162 General Insurance Techniques (2015) / 5. Solvency and Ruin Theory

    Dependence modelling and copulas

    Fitting copulas: case study

    Parameter estimates

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    Independence Clayton Gumbel-Hougaard Frank

    Parameter Estimate s.e. Estimate s.e Estimate s.e. Estimate s.e.

    Loss (X 1)   λ1   14 552 1 404 14 000 2 033 14 001 1 292 14 323 1 359θ1   1.139 0.067 1.143 0.093 1.120 0.062 1.106 0.064

    ALAE (X 2)   λ2   15 210 1 661 16 059 2 603 14 122 1 409 16 306 1 762θ2   2.231 0.178 2.315 0.261 2.108 0.151 2.274 0.181

    Dependence   α   na na 1.563 0.047 1.454 0.034 -3.162 0.175

    Loglik -31 950.81 -32 777.89 -31 748.81 -31 778.45

    AIC 42.61 43.71 42.34 42.38

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    Dependence modelling and copulas

    Fitting copulas: case study

    AIC criterion

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    Akaike Information Criterion (AIC)

    In the absence of a better way to choosing/selecting a copula

    model, one may use the AIC criterion defined by

    AIC = (−2 + 2m) /n

    where     is the value of maximised log-likelihood,  m   is the

    number of parameters estimated, and  n  is the sample size.

    Lower AIC generally is preferred.

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    Dependence modelling and copulas

    Fitting copulas: case study

    Summary

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    To find the distribution of the sum of dependent random variableswith copulas (one approach):

    Fit marginals independently

    Describle/fit dependence with a copula (roughly)

    Get a sense of data (scatter plots, dependence measures)Choose candidate copulasFor each candidate, estimate parameters via MLEChoose a copula based on nll(highest) or AIC(lowest)

    Perform simulations to look at the distributions of aggregates

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