Docstoc

Influence of Reinsurance on Solvency Capital Requirment

Document Sample
Influence of Reinsurance on Solvency Capital Requirment Powered By Docstoc
					A. Lozano

Influence of Reinsurance on
Solvency Capital Requirement
Agenda


 Solvency Capital Requirement (SCR)
  – Risk Measures
 Risk Mitigation
  – Reinsurance
  – Standard Reinsurance Clauses
 Economic Value Added


 Coffee Break
 Practical Case
  – Need to buy reinsurance
  – What is the optimal reinsurance
  – EVA of reinsurance


                                      2
QIS 3

Solvency Capital Requirement
Solvency Capital
Main Target

 Ensure the fulfilments with policyholders in worse case scenarios
 The term ‘Worse case scenario’ is fuzzy and/or arbitrary
  – Possible maximum aggregate loss
  – Aggregate loss with small exceeding probability (Safety level)
  –…
 Time horizon
  – One year time horizon




                                                                     4
Solvency Capital


 Minimum Capital Requirement
 –Ultimate actions
 –Risks included in IAA definition      EC
 –‘Simple’ calculation
 Solvency Capital Requirement           SCR
  –Absorption of unexpected losses
  –Based upon standard or internal or
   both methods
  –Risks included in IAA definition
  –Actions to release SCR               MCR
 Excess Capital
  –Better approach to market
  –Rating agencies

                                              5
Risk Measure
Definition

  Let V be a set of real random variables of (Ω , Φ , P) such
   – If X1, X2 ∈V
          X1+X2 ∈V
         a·X1 ∈V, for all a ∈ ϒ
         b+X1 ∈V for all b ∈ ϒ
  Risk measure is a map ρ:Vαϒ ρ(X)=α ∈ϒ, X ∈V
  Not all function ρ would be appropriated for measuring Solvency
  Desirable properties
  Tolerance Level of Risk




                                                                    6
Coherent Risk Measures
Axiomatic definition of ρ

 Monotonicity
  – X,Y ∈ V,             if X   1   Y       then ρ(X)   ρ(Y)
 Subadditivity
  – For all X, Y ∈V       ρ(X+Y)        ρ(X)+ ρ(Y)
 Homogeneity
  – For all h ∈ ϒ\{0}    ρ(hX) = h ρ(X)
 Translation Invariant
  – X ∈V, a ∈ ϒ          ρ(a+X)= a + ρ(X)
 Perfect Dependency Additivity
  – X, Y ∈V co-monotonic        ρ(X+Y) = ρ(X)+ ρ(Y)
  – X, Y ∈V , Y counter-monotonic X
                               ρ(X+Y) = ρ(X) - ρ(Y)


                                                               7
Risk Measures
Common risks measures

 Standard deviation / semi-deviation
 Variance / semi variance
 PML (Probable Maximum Loss), ruin Probability or Value at Risk
  – Commonly used in Finance
  – Basel II standard for Bank and Financial Institutions
 TVaR
  – Mean of scenarios above a threshold
 Maximum Loss (if exist)
 PHT

                Solvency II tests VaR & TVaR



                                                                  8
                         Value at Risk
                         Continuous Distribution   Cumulative Distribution Function
                                                                 Property Lines
                                                                                                  1,677.55 VaR
                          1


                         0.9                                                                                         95.00% Level


                         0.8


                         0.7
Cumulative Probability




                         0.6


                         0.5
                                                                       VaRα ( X ) = FX−1 (α )
                         0.4


                         0.3
                                                           VaR (X) is the inverse of the CDF
                         0.2


                         0.1


                          0
                           600           800       1,000         1,200            1,400         1,600            1,800              2,000
                                                               One year aggregate Losses



                                                                                                                                        9
                         Value at Risk
                         Non continuous         Cumulative Distribution Function
                                                           Property Lines
                                                                                                       1,700. VaR
                          1


                         0.9                                                                                                95.58% Level


                         0.8


                         0.7
Cumulative Probability




                         0.6
                                                                    VaR ( X) inf {x| P[X<x]>   }
                         0.5


                         0.4


                         0.3


                         0.2


                         0.1


                          0
                           600            800   1,000     1,200             1,400                  1,600            1,800             2,000
                                                        One year aggregate Losses



                                                                                                                                           10
                         Tail Value at Risk
                                                Cumulative Distribution Function
                                                           Property Lines
                                                                       1,677.6 VaR
                          1


                         0.9                                                               95.0% Level


                         0.8


                         0.7
Cumulative Probability




                         0.6
                                                                                             TVaR ( X) E[X/X>VaRα( X)]

                         0.5

                                                                                                     TVaR0.995 = 1,849.6
                         0.4


                         0.3
                                                                                                      1 1                ⎡            ⎤
                                                                                           TVaRq =
                                                                                                     1− q ∫q VaRq dq = E ⎢ X X ≥ VaRq ⎥
                                                                                                                         ⎣            ⎦
                         0.2


                         0.1


                          0
                           600   800    1,000    1,200    1,400        1,600       1,800        2,000           2,200          2,400
                                                         One year aggregate Losses



                                                                                                                                          11
Solvency II Risk Measures
Calendar

QIS 2 Life SCR such P[SCR < 0] = 0.005 (VaR)
QIS 2 Non Life conditional mean E[X/X X0] withX0 P[X<X0]=0.01
 (TVaR)
QIS 3 Life / Non Life uses QIS 2 Life criteria of VaR0.005
QIS 5 2008 (?)
CEA Arguments (among others)
- VaR is easier to calculate where TVaR requires the knowledge of
  the entire loss distribution – difficulties in practice
- VaR became very popular in Financial
- VaR is a sub-additive measure in almost all insurance portfolios at
  the 0.995 level
- TVaR -> theoretical advantages


                                                                        12
Solvency II
Risk measure



Supervisor Targets under Solvency II
An insurance company should have a solvency
position that is sufficient to fulfil its obligations to
policyholders and other parties.

 – Minimize insured deficit                                    [
                                                            ϕ ( X − K )+        ]
      Losses exceeding the capital
                                                           is a non decreasing function


 – Weighting the cost of capital
     To a level of prudence
                                                            q ⋅ K q ∈(0,1)
    Insured Cost = Capital Cost + Insolvency Cost



                                                                                          13
Solvency II
Optimal Solvency Capital

 The expected value could be a reasonable no decreasing function
  – So the expected deficit cost of the policyholders

                     [
                   E (X − K )
                                +
                                    ]
 The Capital weight

                   q ⋅ K q ∈(0,1)
 The minimum cost
                                    ([      +
                                                ]
                           Cost = E ( X − K ) + K .q   )
 The optimal when
                             K = VaR1− q


                                                                   14
Optimal SCR

 Proof              [                 ]
               E ( X − K ) = E [X ] − E [X : K ] − K ⋅ (1 − P[X ≤ K ])
                                  +




                           ⎧
                               ∫0 xf (x )dx
                                 K
                           ⎪                             f continuous
 Where
               E[X : K ] = ⎨
                           ⎪∑ min≤inf {iX KX)P[X i ]
                                   (X ,                   Otherwise
                           ⎩    Xi        m , m ≥K }




   ([               ]         )
Δ K E ( X − K ) + q ⋅ K = −Δ K E [X : K ] + Δ K (K ) ⋅ P[X ≤ K ] + K ⋅ Δ K (P[X ≤ K ]) + Δ K (K ) ⋅ (q − 1)
                +



 Noting that


     Δ K E [ X : K ] = K ⋅ P[ X = K ]         and   K ⋅ Δ K (P[X ≤ K ]) = K ⋅ P[X = K ]


         ([                   ]           )
     Δ K E ( X − K ) + q ⋅ K = Δ K (K ) ⋅ P[X ≤ K ] + Δ K (K ) ⋅ (q − 1) = 0
                          +




    Then                P[ X ≤ K ] = 1 − q



                                                                                                              15
Optimal SCR



                                                                                                                            1.00
                                                                                                                            0.95
                    50000                                                                                                   0.90
                                                                                                                            0.85
                                                                                                                            0.80
                                                                                                                            0.75
                    40000
                                                                                                                            0.70
                                                                             t
                                                                           os                                               0.65
                                                                         tC
                                                                      rge
 Solvency Capital




                                                                                                                            0.60
                                                                   Ta




                                                                                                                                   F(x) = P[X<x]
                    30000                                                                   E[ (X-K)+]                      0.55
                                                                                            q·K                             0.50
                                                                                            C(X,q)                          0.45
                                                                                            SCR
                                                                                                                            0.40
                    20000                                                                   F(X)
                                                                      ht                                                    0.35
                                                                eig                         1-q
                                                             lw                                                             0.30
                                                         it a
                                                      Cap                                                                   0.25
                                                                                                                            0.20
                    10000
                                                                                                                            0.15
                                                                                    Expected Deficit                        0.10
                                                                                                                            0.05
                       0                                                                                                    0.00
                            0   50000   100000   150000   200000      250000   300000   350000   400000   450000   500000
                                                                           X




                                                                                                                                                   16
Solvency II
Optimal Cost Capital

 Cost Function


                       ([     +
                                  ]
             Cost = E ( X − K ) + K .q    )
 Optimal K

              K = VaR1− q
 Optimal cost


                       ([                 ]   )
          Cost = E (X − VaR1− q ) + VaR1− q .q ∝ TVaR1− q
                                      +




                                                            17
      Solvency Capital – Tails




          Solvency Capital


          0.5%
          Prob.




Capital

                                                     Expectation

                      Solvency Capital Requirement




                                                                   18
Reinsurance
Definition
Traditional Reinsurance Clause


 Let V be set of random variables of a portfolio
        If vi, vj™V are real random variables such
           - vi+ vj™V
           − α · vi™V
           − β + vi™V
        Where α,β are real numbers
 Reinsurance is a non decreasing map g: V        ϒ
      Gross: V
      Net Retention: N={v - g(v) | v™V }
      Ceded Portfolio C={ g(v) | vX™V }
 Main Types
  – Proportional Reinsurance: g is linear
  – Non Proportional: g is non-linear
                                                     20
Proportional Reinsurance
Types of Proportional
Prorrata treaties

  Obligatory
   – Quota Share Treaty
        Automatic reinsurance that requires the insurer to transfer (Cede),
        and the reinsurer to accept, a given percentage of every risk within a
        defined category of business written by the insurer
   – Surplus share Treaty
        Automatic Reinsurance that requires an insurer to transfer (Cede)
        and the reinsurer to accept the part of every risk that exceeds the
        insurer's predetermined retention limit (Sum Insured or EML)
          - Retained Line
          - Capacity of the Treaty
          - Number of Lines = Capacity/Retained line
  Facultative
   – Cession based on individual risks
  Fac/Oblig
   – Facultative for the insurer
   – Obligatory/Automatic for the reinsurer



                                                                                 22
Quota Share
Model

 Let V be the category of risk covered (set of real random variables)
  – For all v™V, g(v)= βv where β 1
      In particular Sm=X1 + X2 + … + Xm ™V then:
      Ceded portfolio is:
        - Y=g(X1 + X2 + … + Xm) =β(X1 + X2 + … + Xm)
        - X and Y are co-monotonic (perfect linear correlation)
 Quartile additive, thus
  – VaRnet = (1-β)VaRgross
  – TVaRnet = (1-β)TVaRgross




                                                                        23
  Quota Share
  Effect on Capital                                                                                Additional R/I
                                                                                                Commission 250,000


             Gross Results                           Net Results                                       Net Results
 Prob       VaR      TVaR                          VaR       TVaR                                    VaR       TVaR
  1.00%   -7956293 -1035191                      -3182517 -414076                                  -3432517 -664076
  5.00%   -7050082 -765012                       -2820033 -306005                                  -3070033 -556005
 10.00%   -6280911 -440302                       -2512364 -176121                                  -2762364 -426121
 15.00%   -5667968 -115120                       -2267187     -46048                               -2517187 -296048
 20.00%   -5148920    215643                     -2059568      86257                               -2309568 -163743
 25.00%   -4635135    555875                     -1854054    222350                                -2104054     -27650
 30.00%   -4141935    909734                     -1656774    363894                                -1906774    113894
 35.00%   -3642636 1279781      VaR & TVaR are   -1457054    511912        VaR & TVaR              -1707054    261912
 40.00%   -3178503 1670705       Homogeneous     -1271401    668282    Translation Invariance      -1521401    418282
 45.00%   -2651245 2088564                       -1060498    835426                                -1310498    585426
 50.00%   -2079371 2536269                        -831749 1014508                                  -1081749    764508
 55.00%   -1547696 3018407                        -619078 1207363                                   -869078    957363
 60.00%
 65.00%
           -987077 3556687
           -319703 4159595
                               ρ(hX) = h ρ(X)     -394831 1422675
                                                  -127881 1663838
                                                                       ρ(a+X)= a + ρ(X)             -644831 1172675
                                                                                                    -377881 1413838
 70.00%     407339 4844014                         162936 1937606                                    -87064 1687606
 75.00%    1199625 5653195                         479850 2261278                                    229850 2011278
 80.00%    2292637 6639074                         917055 2655630                                    667055 2405630
 85.00%    3642185 7893388                        1456874 3157355                                   1206874 2907355
 90.00%    5548621 9550385                        2219448 3820154                                   1969448 3570154
 95.00%    8045637 12423678                       3218255 4969471                                   2968255 4719471
 97.50%   10860450 15514324                       4344180 6205730                                   4094180 5955730
 99.00%   15396723 19549859                       6158689 7819943                                   5908689 7569943
 99.50%   18449761 22469135                       7379904 8987654                                   7129904 8737654
 99.75%   21513623 25011133                       8605449 10004453                                  8355449 9754453
100.00%   35840708 35840708                      14336283 14336283                                 14086283 14086283



             CDF                          Net Results = Net Losses – Net Premium + Net Expenses

Results = Losses – Premium

                                                                                                                     24
       Quota Share
       CDF Graphs                                 Results
                                                Quota Share


                           1


                          0.9
                                                                         Gross Loss    Net Result
                          0.8


                          0.7
Cumulative Probability




                          0.6


                          0.5


                          0.4


                          0.3


                          0.2


                          0.1


                            0
                         -20,000,000    0                                 20,000,000                40,000,000
                                       Losses - Premium + Expenses - R/I Comission



                                                                                                            25
Surplus Share
Model

 For all X™V, g(X)= βiX where βi                            ⎧
 depends on the Sum Insured and/or                          ⎪ 0           E≤R
 (EML – Estimate Maximum Loss)                              ⎪E − R
                                                            ⎪
                                            β (E , R, n ) = ⎨        R < E ≤ (n + 1)R
 and/or Limit (non random) of the ith                       ⎪ E
 policy                                                     ⎪ nR
                                                            ⎪ E
                                                                      (n + 1)R < E
                                                            ⎩
 Ingredients
  – E Sum Insured
  – R Retained Line
  – n is the number of lines
                                                 +
                                      ⎛ E − R nR ⎞      ⎛       ⎛ E − R nR ⎞ ⎞
                   β (E , L, n ) = min⎜      ,   ⎟ = max⎜ 0, min⎜
                                                        ⎜              ,   ⎟⎟
                                      ⎝ E      E ⎠      ⎝       ⎝ E      E ⎠⎟⎠
 Portfolios with large EML/SI/Limits
 uses to protect with several surplus
 shares
                                                      ⎛ R⎞
 If protection is ‘complete’      1 − β (E , R ) = min⎜1, ⎟
                                                      ⎝ E⎠

                                                                                        26
Surplus Share
Example


                                        Lines           Line
                   Retention              1           5,000,000
                   Number of lines        4          20,000,000
                                   Subtotal          25,000,000

                                    EML          Cession Retention
          Risk 1                    4,000,000          0%    100%
          Risk 2                   10,000,000         50%     50%
          Risk 3                   25,000,000         80%     20%
          Risk 4                   50,000,000         40%     60%

                                   =MAX(0,MIN(EML-Line,Capacity))/EML

                   Risk 3 Losses         Ceded          Retained
                                 50             40             10
                         1,000,000         800,000       200,000


                   Proportion of ceded Losses depends on EML
                               Not on the loss size!

                                                                        27
Surplus Share
Program

 Maximum EML €75M                              70
  – Surplus + Fac
                                               60
  – Retention €5M
 First Surplus                                 50




                            EML / SI / Limit
  – Number of Lines 8
  – Capacity placed €40M                       40

 Second Surplus                                30
  – Number of lines 5
  – Capacity Placed €20M                       20

 Surplus (FAC/Open Cover)                      10
  – €5M
                                               0
                                                    Pro-rata



                                                               28
Surplus Share
Program
                                                     5,000,000 45,000,000 65,000,000
 Portfolio Profile                                  40,000,000 20,000,000   5,000,000
                                                    1st Surplus 2nd Surplus   Fac
           EML             # Risk     Premium            β           β         β
         -       100,000    99,170      5,950,200         0.00%       0.00%     0.00%
     100,000     150,000    97,938     14,690,700         0.00%       0.00%     0.00%
     150,000     200,000    97,125     20,396,250         0.00%       0.00%     0.00%
     200,000     250,000    96,319     26,006,130         0.00%       0.00%     0.00%
     250,000     500,000    93,941     38,750,663         0.00%       0.00%     0.00%   Retention
     500,000     750,000    90,108     61,949,250         0.00%       0.00%     0.00%
     750,000   1,000,000    86,430     83,188,875         0.00%       0.00%     0.00%
   1,000,000   1,500,000    81,194    111,641,750         0.00%       0.00%     0.00%
   1,500,000   2,000,000    74,702    143,801,350         0.00%       0.00%     0.00%
   2,000,000   3,000,000    65,924    181,291,000         0.00%       0.00%     0.00%
   3,000,000   4,000,000    55,804    214,845,400         0.00%       0.00%     0.00%
   4,000,000   5,000,000    47,237    212,566,500         0.00%       0.00%     0.00%
   5,000,000   7,500,000    35,287    220,543,750        20.00%       0.00%     0.00%     80.0%
   7,500,000 10,000,000     23,262    203,542,500        42.86%       0.00%     0.00%     57.1%
  10,000,000 15,000,000     12,451    155,637,500        60.00%       0.00%     0.00%     40.0%
  15,000,000 20,000,000      5,411     85,223,250        71.43%       0.00%     0.00%     28.6%
  20,000,000 25,000,000      2,352     47,628,000        77.78%       0.00%     0.00%     22.2%
  25,000,000 30,000,000      1,022     25,294,500        81.82%       0.00%     0.00%     18.2%
  30,000,000 35,000,000        444     12,987,000        84.62%       0.00%     0.00%     15.4%
  35,000,000 40,000,000        193      5,066,250        86.67%       0.00%     0.00%     13.3%
  40,000,000 45,000,000          84     2,499,000        88.24%       0.00%     0.00%     11.8%
  45,000,000 50,000,000          36     1,197,000        84.21%       5.26%     0.00%     10.5%
  50,000,000 55,000,000          16       588,000        76.19%     14.29%      0.00%      9.5%
  55,000,000 60,000,000           7       241,500        69.57%     21.74%      0.00%      8.7%
  60,000,000 65,000,000           3       112,500        64.00%     28.00%      0.00%      8.0%
  65,000,000 70,000,000           1        40,500        59.26%     29.63%      3.70%      7.4%




                                                                                                    29
Surplus Treaty
Retention Share

                                                   Surplus Share
            1.0000


            0.9000


            0.8000


            0.7000
Retention




            0.6000


            0.5000


            0.4000


            0.3000


            0.2000


            0.1000


            0.0000
                     0   2000000   4000000   6000000   8000000   10000000   12000000   14000000   16000000   18000000

                                                            EML


                                                                                                                        30
Surplus Share
Discussion

 Surplus as linear combination of a set of Quota Share
  – Retention proportion decreases as EML/SI or Limit increases
 There Gross and Ceded are not co-monotones (perfect correlation)
  – Subadditivity property of TVaR and VaR (for certain percentiles)
  – Highly dependences on the right tail
       Assuming that large risks would causes large losses
 For heavy right tail distribution Surplus Share would be more
 efficient in terms of Capital than Quota Share, but
  – Sum insured, Limit and/or EML is required
        Not applicable for unlimited policies (Auto Liability in France)
  – Do not cap losses (i.e. Catastrophe losses, Conflagration, mass
    losses, etc.)



                                                                           31
Surplus Treaty
Dependency of Non Cat Losses                Surplus

                   1
                                                                      Worse scenarios
                  0.9                                                  Capital zone

                  0.8       τ= 0.92
                  0.7
       FX[g(x)]



                  0.6
       Ceded




                  0.5


                  0.4


                  0.3
                                                        ρ= 0.98
                  0.2


                  0.1


                   0
                        0       0.2   0.4         0.6     0.8     1

                                        Gross
                                        FX(X)

                                                                                    32
  Surplus Share                                                                                          Additional
  Effect on Capital                                                                                  Commission from
                                                                                                    Reinsurer of 250000
                                                                                                           Net Results
             Gross Results                               Net Results
                                                                                                         VaR       TVaR
 Prob       VaR      TVaR                              VaR       TVaR
                                                                                                       -3157950 -378970
  1.00%   -7956293 -1035191                          -2907950 -378970
                                                                                                       -2334172 -286478
  5.00%   -7050082 -765012                           -2334172 -286478
                                                                                                       -1982192 -183419
 10.00%   -6280911 -440302                           -1982192 -183419
                                                                                                       -1723553     -85521
 15.00%   -5667968 -115120                           -1723553     -85521
                                                                                                       -1516799      10168
 20.00%   -5148920    215643                         -1516799      10168
                                                                                                       -1309859    104643
 25.00%   -4635135    555875                         -1309859    104643
                                                                                                       -1141164    199758
 30.00%   -4141935    909734                         -1141164    199758
                                                                                   Recall               -962344    295965
 35.00%   -3642636 1279781                            -962344    295965
                                                                                                        -823218    394792
 40.00%   -3178503 1670705     Simulating Surplus     -823218    394792        VaR & TVaR
                                                                                                        -662162    498138
 45.00%   -2651245 2088564                            -662162    498138    Translation Invariance       -500738    606268
 50.00%   -2079371 2536269                            -500738    606268
                                                                                                        -352809    720910
 55.00%   -1547696 3018407                            -352809    720910
                                                                                                        -205756    846423
 60.00%    -987077 3556687                            -205756    846423
 65.00%
 70.00%
           -319703 4159595
            407339 4844014
                                                       -17887    983365
                                                       174269 1134874
                                                                           ρ(a+X)= a + ρ(X)              -17887    983365
                                                                                                         174269 1134874
                                                                                                         370838 1308268
 75.00%    1199625 5653195                             370838 1308268
                                                                                                         643689 1509778
 80.00%    2292637 6639074                             643689 1509778
                                                                                                         943835 1752219
 85.00%    3642185 7893388                             943835 1752219
                                                                                                        1327656 2070320
 90.00%    5548621 9550385                            1327656 2070320
                                                                                                        1885439 2564301
 95.00%    8045637 12423678                           1885439 2564301
                                                                                                        2417598 2999243
 97.50%   10860450 15514324                           2417598 2999243
                                                                                                        2947191 3522680
 99.00%   15396723 19549859                           2947191 3522680
                                                                                                        3396008 3886097
 99.50%   18449761 22469135                           3396008 3886097
                                                                                                        3677592 4215632
 99.75%   21513623 25011133                           3677592 4215632
                                                                                                        4825667 4825667
100.00%   35840708 35840708                           4825667 4825667



             CDF                                    Net Results = Net Losses – Net Premium

Results = Losses – Premium

                                                                                                                             33
       Surplus Share
       CDF Graphs                                               Results
                                                             Surplus Treaty


                           1


                          0.9


                          0.8                                       Gross Results    Net result

                          0.7
Cumulative Probability




                                                                  Break-even
                          0.6


                          0.5


                          0.4


                          0.3


                          0.2


                          0.1


                            0
                         -20,000,000   -10,000,000    0              10,000,000         20,000,000   30,000,000   40,000,000
                                                     Losses - Premium + Expenses - R/I Comission



                                                                                                                          34
Non Proportional Reinsurance
Stop Loss Treaty
Dependency                             Stop Loss

                  1


                 0.9


                 0.8


                 0.7
      FX[g(x)]



                 0.6
      Ceded




                 0.5


                 0.4


                 0.3


                 0.2


                 0.1


                  0
                       0   0.2   0.4          0.6   0.8   1

                                   Gross
                                   FX(X)

                                                              36
Type of Non Proportional Reinsurance

 Stop loss
  – Protects a cedant against an aggregate amount of claims over a
    period, in excess of a specified percentage of the earned
    premium income
 Excess of loss
  – The reinsurer pays all losses that exceed certain amount
    (priority) up to a limit (limit)
       Excess of loss per risk
         - Recoveries: losses occurred to one every risks (or policy)
       CAT Excess of loss
         - Recoveries: all aggregate losses that arises for one single
            event
 Other types
  – COSIMAX, Excès du Côte moyenne, etc


                                                                         37
Stop Loss
Model

 Let V be the category of risk covered (set of real random variables)
  – Let S=X1 + X2 + … + Xn + … ™V be aggregate losses (one reinsurance
    year) then:

                              ⎧ 0         S≤r        ⎫
                              ⎪                      ⎪
             SL(S ; r , l ) = ⎨S − r   r < S ≤ r + l ⎬ = max(0, min (S − r , l ))
                              ⎪ l        r +l < S ⎪
                              ⎩                      ⎭


 The gross and ceded are comonotones
  – SL is non decreasing that depends on S
  – Gross is a non decreasing (identity) that depends on S
  – However, dependency is non linear
 S and SL are quartile additive



                                                                                    38
  Stop Loss                                  Priority      6,000,000
  Effect on Capital                          Limit         12,000,000


                                                                                                      Net Results
          Gross Loss & ALAE                          Net Loss & ALAE
                                                                                                    VaR       TVaR
 Prob       VaR      TVaR                             VaR       TVaR
                                                                                                  -5119205 -602286
  1.00%    1043707 1043707                           1043707 1043707
                                                                                                  -3962995 -443838
  5.00%    1949918 1949918                           1949918 1949918
                                                                                                  -3193823 -272108
 10.00%    2719089 2719089                           2719089 2719089
                                                                                                  -2580880 -118544
 15.00%    3332032 3332032                           3332032 3332032
                                                                                                  -2061832      19159
 20.00%    3851080 3851080                           3851080 3851080
                                                                                                  -1548048    140602
 25.00%    4364865 4364865                           4364865 4364865
                                                                                 Recall           -1054847    243364
 30.00%    4858065 4858065                           4858065 4858065
                               Co-monotones                                  VaR & TVaR            -555549    324772
 35.00%    5357364 5357364                           5357364 5357364
                                                                         Translation Invariance     -91415    378967
 40.00%    5821497 5821497    percentile additive    5821497 5821497
                                                                                                     87087    408679
 45.00%    6348755 6348755                           6000000 6000000
                                                                                                     87087    440825
 50.00%    6920629 6920629                           6000000 6000000
                                                                                                     87087    480112
 55.00%    7452304 7452304                           6000000 6000000
                                                                                                     87087    529215
 60.00%
 65.00%
           8012923 8012923
           8680297 8680297
                                                     6000000 6000000
                                                     6000000 6000000    ρ(a+X)= a + ρ(X)             87087
                                                                                                     87087
                                                                                                              592340
                                                                                                              676493
 70.00%    9407339 9407339                           6000000 6000000
                                                                                                     87087    794280
 75.00%   10199625 10199625                          6000000 6000000
                                                                                                     87087    970901
 80.00%   11292637 11292637                          6000000 6000000
                                                                                                     87087 1265113
 85.00%   12642185 12642185                          6000000 6000000
                                                                                                     87087 1852950
 90.00%   14548621 14548621                          6000000 6000000
                                                                                                     87087 3611778
 95.00%   17045637 17045637                          6000000 6000000
                                                                                                   1947538 6601412
 97.50%   19860450 19860450                          7860450 7860450
                                                                                                   6483811 10636946
 99.00%   24396723 24396723                         12396723 12396723
                                                                                                   9536848 13556222
 99.50%   27449761 27449761                         15449761 15449761
                                                                                                  12600711 16098221
 99.75%   30513623 30513623                         18513623 18513623
                                                                                                  26927796 26927796
100.00%   44840708 44840708                         32840708 32840708



             CDF
Results = Losses – Premium

                                                                                                                        39
       Stop Loss
       CDF Graphs                                              Results
                                                           Stop Loss Treaty


                           1


                          0.9


                          0.8                                       Gross Results    Net result

                          0.7
Cumulative Probability




                          0.6


                          0.5


                          0.4


                          0.3


                          0.2


                          0.1


                            0
                         -20,000,000   -10,000,000    0              10,000,000         20,000,000   30,000,000   40,000,000
                                                     Losses - Premium + Expenses - R/I Comission



                                                                                                                          40
Excess of Loss
Model

 Let V be the category of risk covered (set of real random variables)
  – Let X1 , X2 , … , Xn … ™V be the size of single losses then:



                               ⎧ 0           X ≤r ⎫
                               ⎪                       ⎪
             XL( X ; r , l ) = ⎨ X − r   r < X ≤ r + l ⎬ = max(0, min ( X − r , l ))
                               ⎪ l
                               ⎩           r +l < X ⎪  ⎭

 The aggregate losses are not comonotones
  – XL is non decreasing that depends on X
  – Losses below the priority r that are not perfectly dependent
 S and XL(S) are right tail dependent




                                                                                       41
Excess of Loss Treaty
Dependency                           Excess of Loss Treaty

                  1


                 0.9


                 0.8       τ= 0.68
                 0.7
      FX[g(x)]



                 0.6
      Ceded




                 0.5


                 0.4


                 0.3


                 0.2
                                                               ρ= 0.95
                 0.1


                  0
                       0       0.2     0.4        0.6        0.8         1

                                         Gross
                                         FX(X)

                                                                             42
Excess of Loss
Effect on Capital

                          Gross Results                            Net Results
               Prob      VaR       TVaR                          VaR       TVaR
                1.00% -12150687 -1781611                       -8587070 -2737688
                5.00% -10162815 -1394559                       -7010245 -2529617
               10.00% -9104673     -940496                     -6182521 -2307039
               15.00% -8284498     -485351                     -5660143 -2094563
               20.00% -7486837      -23592                     -5209904 -1885741
               25.00% -6827090      450871                     -4793918 -1677981
               30.00% -6149721      948061                     -4370712 -1469262
               35.00% -5542003     1470156                     -3982126 -1260152
               40.00% -4931335     2030524   Simulating XL     -3602095 -1048399
               45.00% -4311504     2637563                     -3222595 -834651
               50.00% -3700217     3300053                     -2889046 -612655
               55.00% -3064497     4045752                     -2536138 -379186
               60.00% -2415376     4894861                     -2261527 -126832
               65.00% -1580979     5883479                     -1885010    151943
               70.00%   -679051    7055391                     -1499559    459685
               75.00%    362396    8500461                     -1042112    808093
               80.00%   1708337 10374893                        -508938 1202432
               85.00%   3510822 12970357                          72612 1680490
               90.00%   6154428 17123066                         818680 2311710
               95.00% 11315542 25730476                         1961476 3318962
               97.50% 19276153 36458853                         2947581 4223868
               99.00% 30944844 55977327                         4163075 5361935
               99.50% 42693546 74903649                         5060743 6042905
               99.75% 59600660 96949728                         5779416 6624821
              100.00% 219064671 219064671                       8542461 8542461



                          CDF                          Net Results = Net Losses – Net Premium
              Results = Losses – Premium

                                                                                                43
       Excess of Loss Treaty
       CDF Graphs                                                Results
                                                           Excess of Loss Treaty


                           1


                          0.9


                          0.8                                           Gross Results    Net result

                          0.7
Cumulative Probability




                          0.6


                          0.5


                          0.4


                          0.3


                          0.2


                          0.1


                            0
                         -20,000,000   -10,000,000   0           10,000,000      20,000,000           30,000,000   40,000,000   50,000,000
                                                         Losses - Premium + Expenses - R/I Comission



                                                                                                                                        44
Common clauses

Reinsurance Clauses
Sliding Scale Commission
Proportional Treaties

 Commission that has an                                                         Sliding Scale Commission


 inverse relationship to                     50.00%


 the aggregate loss
 experience on the                           45.00%



 business ceded                              40.00%
                                                                                                             C1 − C0
                                                                                                    α=                >0




                           % of Commission
 Ingredients:                                                                                                X1 − X 0
                                             35.00%

  – Minimum
    Commission                               30.00%


  – Maximum                                                        ⎧      C1              X ≤ X0
                                                                   ⎪
    Commission                               25.00%      C ( X ) = ⎨C0 + α ( X 1 − X ) X 0 < X ≤ X 1 C0 ≤ C1
                                                                   ⎪                      X1 < X
  – Scale                                                          ⎩      C0
                                             20.00%
                                                   30%        40%         50%         60%              70%        80%      90%        100%
                                                                                        Loss Ratio Treaty




                                                  C ( X ) = max(C0 , min (C1 , C0 + α ( X 1 − X )))
        Loss Ceded
 X=
      Earned Premium

        Commission commonly applied on Written Premium
       Sliding Scale Commission is a non increasing function
      Less capital effective than intermediate fixed commission
                                                                                                                                 46
Loss Corridor
Proportional / Non Proportional Treaties

 Cedant participates in
 aggregate loss in excess
                                                                                     Loss Corridor
 of some amount up to a                              8.00%

 limit
                                                     7.00%



 Ingredients:                                        6.00%



  – Participation Share                              5.00%




                                % of Loss Corridor
                                                                                                               Limit
  – Minimum Loss Ratio                               4.00%



  – Maximum Loss Ratio                               3.00%        Priority
                                                                                                              Slope = Participation
                                                     2.00%



                                                     1.00%



                                                     0.00%
                                                          55%   60%    65%   70%   75%         80%           85%   90%   95%    100%   105%
                                                                                         Loss Ratio Treaty

        Loss Ceded
 X=
      Earned Premium                                            LossCorridor = SL( X , X 0 , X 1 − X 0 )

  Loss Corridor can be defined as Counter- Stop Loss Reinsurance
                  Cedant reinsures the reinsurer

                                                                                                                                              47
Profit Commission
Proportional / Non Proportional

  Additional Commission
  based on the                                                                                            Profit Commission

  reinsurance profitability                        14.00%



  Ingredients:                                     12.00%



   – Share                                         10.00%


   – Margin
                              % of Loss Corridor
                                                   8.00%



                                                   6.00%



                                                   4.00%



                                                   2.00%


 X = Ceded Losses                                  0.00%
                                                            0%   5%   10%   15%   20%   25%   30%   35%   40%   45%   50%   55%     60%   65%   70%   75%   80%   85%   90%   95% 100%
                                                                                                                Loss Ratio Treaty

EP = Earned Premium

ProfitCommission = α (EP(1 − β ) − X − Commission )
                                                                                                                              +




                                                                                                                                                                                    48
Retrospective Rating
Non Proportional
 The reinsurance premium is adjusted (subject to minimum and maximum)
 based on the current period actual loss experience.
  – Loss Loaded Rating
    Rate adjustment using the losses multiplied by a loss load and/or expense
    load
                                                                                                          Loss Loaded Rate



 Ingredients                                                1.80%


                                                            1.60%


  – Minimum Rate                                            1.40%


                                                            1.20%

  – Maximum Rate



                                         Reinsurance Rate
                                                            1.00%



  – Loading Factor > 1                                      0.80%


                                                            0.60%




          ⎧π0         βX ≤ π 0
                                                            0.40%




          ⎪
                                                            0.20%




π ( X ) = ⎨β X     π 0 < βX ≤ π 1
                                                            0.00%
                                                                0.40%   0.50%   0.60%   0.70%    0.80%     0.90%      1.00%         1.10%   1.20%   1.30%   1.40%   1.50%   1.60%
                                                                                                Burning Cost of the Excess Treaty



          ⎪π          π 1 < βX
          ⎩ 1                                               π ( X ) = max(π o , min (βX , π 1 ))

         Reinsurance premium commonly adjusted to the subject premium (Rate On OGNPI)
                         Minimum & deposit premium is paid in advance
                         Final adjustment at the end of the run-off period


                                                                                                                                                                                    49
Aggregate Deductible


The total amount an reinsured is responsible to retain for the sum of all
losses during a reinsurance period.
Once the annual aggregate deductible has been reached by the
accumulation of payment by the reinsured, the reinsurance responds to the
remaining claims up to the occurrence limit




        XLAD (S ) = (S − AD ) = max(S − AD,0 )
                                  +




                                                                            50
Aggregate Limit


The maximum amount an reinsurer is responsible to pay for the sum of all
losses during a reinsurance period.
Once the annual aggregate limit has been reached by the accumulation of
payment by the reinsured, the reinsurance does not respond to the
remaining claims




              XLAL (S ) = min (S , AL )




                                                                           51
Reinstatements
Excess of loss

 When the amount of reinsurance coverage provided under a contract is
 reduced by the payment of loss as the result of one occurrence, the
 reinsurance cover is automatically reinstated
  – Sometimes subject to the payment of a specified reinstatement
    premium.
  – Reinsurance contracts may provide for an unlimited number of
    reinstatements or for a specific number of reinstatements
 Aggregate Limit
  – The Aggregate Limit with reinstatements
      (Number of reinstatement + 1) x occurrence limit
 Paid reinstatement
  – Single Prorrata or Prorrata Capita
       Additional Premium = Paid Loss / Occurrence Limit x
       x Upfront Premium
  – Double Prorrata or Prorrata Temporis
       Additional Premium =
       Paid Loss / Occurrence Limit x Upfront Premium x
       x (# days from date of loss to expiring date)/(# days R/I Contract)


                                                                             52
   Reinstatements
   Model .




  S               3,000,000                6,000,000                     9,000,000            12,000,000 …



2 reinstatements @ 100% Additional premium



             Additional Premium        Additional Premium


  S                                       Aggregate Limit
                   3,000,000                 6,000,000                   9,000,000



                                                                     +
                                     ⎛       ⎛ X −r ⎞            ⎞
               Recoveries = min⎜ ( X − r ) − ⎜      ⎟π , (l − π )⎟ + 2lmin·l ( X − r , l )
                                                                                          +
                                     ⎜                           ⎟
                            S ≤ 2 ·l
                                     ⎝       ⎝ l ⎠               ⎠     ≤ S <3




                                                     π
                                  Rate on line =
                                                       l
                                                                                                             53
   Reinstatements
   Model ..




  S               3,000,000                6,000,000                     9,000,000               12,000,000 …



1st free and 2nd @ 100% Additional premium



                                     Additional Premium


  S                                       Aggregate Limit
                   3,000,000                 6,000,000                    9,000,000




      Recoveries = min ( X − r , l ) + min                 (( X − r )(1 − ROL ), (l − π ))+ + 3lmin·l ( X − r , l )+
                                      +
                      S ≤l                   1·l < S ≤ 2·l                                      ≤S <2




                                                                                                                       54
Economic Value Added
    Reinsurance impact on Capital


                                                              Reinsurance Cost




Exceeding value as at Prob. 0.50%
(200 to 1 )




                 Marginal Capital      Net Solvency Capital

                                    Gross Solvency Capital




                                                                                 56
Economic Value Added


 Marginal Capital = Gross Capital – Net Capital
  – Cost of Capital = Return on Risk Capital (Net of risk free interest)
 CC = Cost Saved = Marginal Capital · R.O.R.A.C
 Cost of reinsurance (CoR)
  – CoR=Expected Net Present Value of Cash Flows ceded to the
    reinsurance (discounted or economic result)
  – CoR=Gross Economic Result – Net Economic Result
 EVA = CC - CoR




                                                                           57
Split of Capital (QIS 2)
Capital allocation




                             8         13
              Underwriting



              Market                        100
                                                    R.O.R.A.C. 18%



              Credit



              Operational
                                 100        R.O.E 12%




                                                                     58
EVA Calculation Process



      Figures in Thousand Euros                                   Company ABC

      Gross Capital*
         133,149          +
                                   Capital Saved
                                      92,172
                                                                 x             Cost Saved
                          -
      Net Capital *                                                              13,826     +
         40,977                   Cost of Capital **                                            EVA
                                       15.0%                                                    12,880


                                                                           -
      Gross Loss *
         10,616           -
                                  Reinsurance Cost
                                        946
                          +
         Net Loss *
          11,562
      * Economic Result                                ** Net of Riskfree Rate




                                                                                                         59
                        Optimal Reinsurance
                                                                  Cost - Benefit

                      100.0
                                                                               Total Cost
                       90.0                                                    Cost of Capital
                                                                               Efficient Frontier
                                                                               Alternative 1
                       80.0                                                    Alternativa 2
                                                                               Alternative 3
                                                                               Alternative 4
                       70.0                                                    Bare
Cost of Reinsurance




                       60.0


                       50.0


                       40.0
                                                                                         Risk tolerance is a “dynamic” tradeoff between the
                                                                                          return on the risk portfolio and the cost of capital
                       30.0
                                                                 Optimal = Minimum Cost
                       20.0


                       10.0
                                        Tan( α ) = R.O.R.AC - Risk Free Rate
                        0.0
                              0   100                200                300                    400             500              600              700
                                                                    Solvency Capital Requirement




                                                                                                                                                  60
But Don’t forget other targets
Decision Process




                                                                 Company ABC Reinsurance 2007
                                                   Relative Performance of Reinsurance structures by "Metric"

                                                        Profit for 95%ile Loss


                                                                                                                Option 2



                                                                                                                Option 3



                                                                                                                Option 4
      Net Profit Volatility (SDV)                                                               Net Profit

                                                                                                                Option 5



                                                                                                                Option 6



                                                                                                                Option 8



                                                                                                                Option 9



                                                                                                                Option 10
                            Largest 5%ile Profit                                 Cost / Benefit Ratio




                                                                                                                            61
Now You …
Auto Liability


 Parameters
  – Min Loss 500,000 €
  – Max Loss 50,000,000 €
  – Loss Severity: Simple Pareto
       alpha 1.62
       Standard deviation of alpha = 0.31
  – Frequency: Poisson Lambda = 7.2
       Effective Years = 1.2
 Reinsurance Program
  – 49,000,000 in excess of 1,000,000
  – Aggregate Deductible = 2,000,000
  – Premium = 2,500,000



                                            63
General Third Party Liability


 Parameters
  – Min Loss 200,000 €
  – Max Loss 7,000,000 €
  – Loss Severity: Simple Pareto
       alpha =1.22
       Standard Deviation = 0.2
  – Frequency: Poisson
       mean = 3.0
       Effective Years= 0.85
 Reinsurance Program GTPL
  – 5,500,000 in excess of 500,000
  – 1 reinstatement @ 100% Additional Premium
  – Premium = 1,000,000


                                                64
Tasks


 Graph of Gross / Net
 VaR 0.995
 TVaR 0.99
 Cost of reinsurance
 Stop Loss covering MTPL + GTPL
  – Priority 15,000,000
  – Limit 75,000,000
  – R/I Price 6,000,000
 Calculate the EVA
  – R.O.R.A.C. = 17.25%
  – Risk free rate = 4.23%



                                  65

				
DOCUMENT INFO
Shared By:
Categories:
Tags:
Stats:
views:6
posted:10/21/2012
language:English
pages:65