# Influence of Reinsurance on Solvency Capital Requirment

Document Sample

```					A. Lozano

Influence of Reinsurance on
Solvency Capital Requirement
Agenda

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

Coffee Break
Practical Case
– 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)
– 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)
– 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

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

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)
– VaRnet = (1-β)VaRgross
– TVaRnet = (1-β)TVaRgross

23
Quota Share
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

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
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

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
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

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

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%

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=

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

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

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

48
Retrospective Rating
Non Proportional
based on the current period actual loss experience.
Rate adjustment using the losses multiplied by a loss load and/or expense

Ingredients                                                1.80%

1.60%

– Minimum Rate                                            1.40%

1.20%

– Maximum Rate

Reinsurance Rate
1.00%

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 ))

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

+

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
– 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
– Double Prorrata or Prorrata Temporis
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 …

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 …

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
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

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

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

64

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

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