RISK BASED CAPITAL

MODELLING FOR P&C INSURERS

AND FINANCIAL SENSITIVITY

Laura Ballotta Nino Savelli

Cass Business School Catholic University of Milan
L.Ballotta@city.ac.uk nino.savelli@unicatt.it

XXXVI ASTIN Colloquium - Zurich, 4-7 September 2005

Motivation: Insurance Risk Management
and Solvency (1)
MAIN PILLARS OF THE INSURANCE MANAGEMENT:
the strategic triangle of competing forces:
market share
return for stockholders’ capital
financial strength/stability
RISK-BASED CAPITAL REQUIREMENTS:
to assess the risk capital of the company according to its own real risk profile.
Simulation models may be used for defining New Rules for Capital Adequacy
(see e.g. IAA Solvency Working Party)
INTERNAL RISK MODELS (IRM):
to be used
for solvency purposes (e.g. Pillar 1&2 of Solvency II)
to define the most appropriate management’s strategies
IRM could allow for a more comprehensive representation of the business of an
individual firm than the standard formula, with capital requirements more
significantly aligned to the effective risk of the company

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Motivation: Insurance Risk Management
and Solvency (2)
A NEW APPROACH FOR THE SUPERVISORY AUTHORITIES:
Stress testing in order to assess the solvency profile of the Insurer
Validation and approval of the Insurer IRM on the basis of
Prudential requirements
Comparability & consistency requirements (with respect to the
supervisor’s view of key minimum performance criteria)
Indication of the appropriate course of action to follow in case of
an excessive risk of insolvency over the short term

THE AIM OF THIS PAPER:

to propose a possible simple risk model for a P&C insurer
incorporating
the underwriting risk
the financial risk
to analyse specifically the impact of different asset allocation
strategies on the risk profile of the insurer

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 Agenda
General framework of the model
The Insurance sub-model
The Investment sub-model
The asset allocation rule
The numerical experiment
The impact of various asset allocations on
the Insurer’s Risk Based Capital
Final Comments

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 General framework of the model

Company: Investment Portfolio:

General Insurer with only 1 line of 1 category of assets for Equities and
casualty insurance other 5 categories for Gov.Bonds,
Time Horizon: 3 years differentiated according to time to
maturity (1, 2, 3, 5 and 10 years)
Aggregate Claim amount:
Investment Return:
Compound Mixed Poisson Process
- Geometric Brownian motion for
Number of Claims: equities
Negative Binomial distribution - CIR process for interest rates
Claim Size: Asset Allocation rule:
LogNormal distribution constant proportion
Dynamic Ins. Portfolio: Volume of Monte Carlo approach:
premiums increases every year 100,000 simulations
according to real growth and claim
inflation Risks not included:
Reinsurance: Claim Reserve risk
Credit and Operational risk
reinsurance cover is ignored
ALM risk

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 Risk Reserve process (Ut)

U% = (1+ %j )⋅U% +(π − X% − E ) + %j ⋅ L% −TX% − D

%
t t t −1 t t t t t −1 t t

Ut = Risk Reserve at time t jt = Investment return rate of

πt = Gross premiums at year t year t
Xt = Aggregate claims amount Lt-1 = Loss Reserve at time t-1
year t
TXt = Taxation amount year t
Et = general and acquisition
expenses year t Dt = Dividends year t

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 Gross Premiums, Safety Loading
and Loss Reserve
Gross Premiums: Loss Reserve:
Lt = δ* πt
πt = (1+i)*(1+g)* πt-1
with coefficient δ constant
i = claim inflation rate (constant)
g = real growth rate (constant) – The safety loading coefficient φ is
assumed not related to the market computed according to the standard
level of the premiums deviation principle:

Furthermore: (1−tx)⋅ ⎡⎣ϕ⋅ E(X%1) + L0 ⋅ E( %j1)⎤⎦ = b⋅ Var(X%1) + L20 ⋅Var( %j1)

πt = (1+ φ)*Ε(Xt) + c* πt in other words, the insurer is
asking for an expected profit
where: from the insurance business
- φ = safety loading coefficient equal to b (0.35) for each unit of
- c = expenses loading coefficient insurance risk (measured in terms
of standard deviation)
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 Aggregate Claims Amount (Xt)
k% t only 1 LoB is considered

X% t = ∑i=1
Z% i , t
CAT claims
not considered

kt = Claim Number of year t

here assumed to be Negative Binomial distributed, i.e.
k follows a Poisson distribution with a stochastic parameter n*q,
q is a multiplicative random structure variable with mean 1 and distributed as a Gamma(h,h),
which captures short-term fluctuations (we ignore systematic changes)
n is the expected number of claims increases with the real growth rate, i.e. nt=n0*(1+g)t

Zi,t = Claim Size for the i-th claim of year t (independent of k)

here assumed to be LogNormal distributed, with values increasing every year
according to the deterministic claim inflation (i) only.
The claim sizes Zt are assumed to be i.i.d. random variables

Xt are time independent variables. In the real world, though, long-term cycles are
present and then significant auto-correlation might be observed (especially for the case
of medium/long-term analyses).
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 The Investment Model (1)
The insurer invests
α% of the available resources in an equity index, S, and
(1- α)% in a portfolio of zero coupon bonds, P, with different redemption dates
β(τ) % is invested in the bond with time to maturity τ
the asset allocation and the asset mix are constant over time

dSt = µ St dt + σ St dWt ; drt = κ (θ − rt ) dt + v rt dZ t

dPt ( t + i ) = a ( t ; t + i ) Pt ( t + i ) dt + b ( t ; t + i ) Pt ( t + i ) dZ t
i ∈ N := {1,2,3,5,10}
dWt dZ t = ρ dt ⇒ Z t = ρWt + 1 − ρ 2 Yt
dAt
= ⎡⎣αµ + (1 − α ) a ( t ; t + i ) ⎤⎦ dt + ⎡⎣ασ + (1 − α ) ρΣ ( t ; t + i ) ⎤⎦ dWt + (1 − α ) 1 − ρ 2 Σ ( t; t + i ) dYt
At
a ( t; t + i ) = rt + λ ( t , r ) ; Σ ( t; t + i ) = ∑ β ( )b ( t; t + i )
i

i∈N

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 The Investment Model (2)
We also assume that every year the insurer invests the cashflows, F,
originated by the “pure” insurance business in the financial portfolio A
The cashflows arise from consideration of
the overall annual rate of growth of the premium (which depends on g and i)
the amount of claims deferred from the previous year and paid in the current year,
Ctd
the amount of claims occurred in the current year and settled during the same
period, Ctc

⎛ ~c d ⎞
⎡ ⎛ 1 ⎞⎤
Ft = (1 − c ) π t − ⎜ C t + C t ⎟ = π t ⎢(1 − c ) + δ ⎜ 1 − ⎟⎟ ⎥ − X% t
⎝ ⎠ ⎢⎣ ⎝ (1 + i )(1 + g ) ⎠ ⎥⎦
St ( i ) Pt ( t + i )
At = α [ At −1 + Ft −1 ] + (1 − α ) [ At −1 + Ft −1 ] ∑ β
St −1 i∈N Pt −1 ( t + i )
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 The Asset Allocation Rule
Asset allocation
Std. Insurer Insurer A Insurer B Insurer C
Equity α 15% 30% 50% 100%
Bond port. 1-α 85% 70% 50% 0%

Asset mix (bond portfolio)

Maturity (years) τ 1 2 3 5 10
Weight β(τ) 40% 25% 15% 10% 10%

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Parameters of the Insurance Model
Initial expected number of claims 20000
Variance structure variable 0.02
Skewness structure variable + 0.28
Initial expected claim size ($) 6000
Variability coefficient of claim size 7
NOTE:
Parameters as Var(q), E(Z) and
Claim Inflation 5% CV(Z)
Real Growth rate 5% are derived from the IAA Solvency
Expenses Loading coefficient 25% Working Party Report (2004)
Safety Loading coefficient (φ) Depending on asset allocation
Loss Reserve ratio 120%
Taxation rate 0%
Dividends rate 0%

Initial Risk Premium (mill $) 120.0

Initial Gross Premiums (mill $) π0 Depending on asset allocation

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 Parameters of the Investment Model
Equity Index Interest rate (CIR)
Expected rate of return µ 10% Long run mean θ 4%
Volatility σ 20% Speed κ 0.1
Diffusion υ 4.7%
Market price of interest rate risk λ -0.005
Correlation ρ -0.2
Current short rate r0 4.5%

Zero Yield Curve

1 year r(0,1) 4.47%
2 years r(0,2) 4.38%
3 years r(0,3) 4.39%
5 years r(0,5) 4.43%
Source: Bank of England (31/12/2004) 10 years r(0,10) 4.49%

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 The percentiles of the capital ratio Ũt/πt
(0.1%; 1%; 25%; 50%; 75%; 99%;99.9%)

Standard Insurer
Standard Insurer Insurer A
Insurer A
2 2

1.5 1.5

1 1

0.5 0.5

0 0

-0.5 -0.5

-1 -1
0 0.5 1 1.5 2 2.5 3 0 0.5 1 1.5 2 2.5 3
Year t Year t

2
Insurer B
Insurer B
2
Insurer C
Insurer C

1.5 1.5

1 1

0.5 0.5

0 0

-0.5 -0.5

-1 -1
0 0.5 1 1.5 2 2.5 3 0 0.5 1 1.5 2 2.5 3
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100,000 simulations
 Moments of capital ratio Ũt/πt
Standard Insurer Insurer A
(15% Equities - 85% Bonds ) (30% Equities - 70% Bonds )
T=1 T=2 T=3 T=1 T=2 T=3
Mean (%) 4.99 9.62 14.00 5.51 10.58 15.37
Std (%) 12.06 16.70 20.04 13.74 19.12 23.14
Skew - 0.29 -0.17 -0.11 -0.12 -0.01 0.07
Kurt 3.39 3.15 3.08 3.36 3.17 3.21

Insurer B Insurer C
(50% Equities - 50% Bonds ) (100% Equities - 0% Bonds )
Mean (%) 6.37 12.35 18.21 9.31 18.63 27.88
Std (%) 16.84 23.80 29.28 26.97 39.76 50.65
Skew 0.13 0.273 0.36 0.42 0.703 0.89
Kurt 3.26 3.35 3.43 3.41 4.03 4.53
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 Prob. Distrib. of the capital ratio ut
Standard Insurer (15% Equity - 85% Bonds) Standard Insurer (15% Equity - 85% Bonds)

Standard Insurer
(t =1) (t = 3)
3500 3500

Mean = 4.99% Mean = 14.00 %

3000

Std = 12.06%
15% equities 3000
Std = 20.04%
2500
Skew = -0.29 2500 Skew = -0.11
Kurt = 3.38 Kurt = 3.08
2000 2000
Frequency

Frequency
1500 1500

1000 1000

500
t=1 500
t=3
0 0
-1.5 -1 -0.5 0 0.5 1 1.5 2 -1.5 -1 -0.5 0 0.5 1 1.5 2
U/B U/ B

Insurer C (100% Equity - 0% Bonds) Insurer C (100% Equity - 0% B onds)

((t = 1)

Insurer C
(t = 3)
3500 3500

Mean = 9.31% Mean = 27.88%

3000

Std = 26.97%
100% equities 3000

Std = 50.65%
2500
Skew = +0.42 2500
Skew = +0.89
2000
Kurt = 3.41 2000
Kurt = 4.53
Frequency
Frequency

1500 1500

1000 1000

500
t=1 500
t=3
0 0
-1.5 -1 -0.5 0 0.5 1 1.5 2 -1.5 -1 -0.5 0 0.5 1 1.5 2
U/B U/ B

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 The Risk Based Capital
Standard Insurer Insurer A
(15% Equities - 85% Bonds ) (30% Equities - 70% Bonds )
Safety loading
coefficient and
initial gross
φ = -1.42% Î π0 = 157.73 φ = -2.08% Î π0 = 156.67
premium
RBC ratio
(per unit of initial T=1 T=2 T=3 T=1 T=2 T=3
Gross Premiums π0)

CL: 99.0 % 26.75 34.71 40.14 28.93 37.27 42.89

CL: 99.9 % 40.13 52.28 60.28 42.34 53.19 63.30

Insurer B Insurer C
(50% Equities - 50% Bonds ) (100% Equities - 0% Bonds )
φ = -2.59% Î π0 = 155.86 φ = -2.92% Î π0 = 155.33
CL: 99.0 % 32.95 42.17 48.24 46.69 57.00 62.41
CL: 99.9 % 46.59 60.39 69.24 62.43 75.58 82.77
100,000 simulations
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 The claim distribution X1
Simulated Distribution of X at time t=1 - Gross of reins.
600

Mean = 132,30 mill $

Std = 19.80 mill $
500 Î Variability coefficient = 15%

400
Frequency

300

200

100

0
-0.5 0 0.5 1 1.5 2 2.5 3
XXXVI ASTIN Colloquium
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Claim Amount X 8
x 10
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 Investment returns distribution j (0,1)
4 Standard Insurer (15% Equity - 85% Bonds) 4 Insurer A (30% Equity - 70% Bonds)
x 10 x 10
3 3

µ1 (j) = 5.66% µ1 (j) = 6.51%

2.5 2.5
σ (j) = 3.77% σ (j) = 6.95%
skew(j) = 0.4736 skew(j) = 0.5753
2 kurt(j) = 3.4813 2 kurt(j) = 3.6179

1.5 1.5

1 1

0.5 0.5

0 0
-0.5 -0.2 0.1 0.4 0.7 1 -0.5 -0.2 0.1 0.4 0.7 1
j j

4 4
x 10 Insurer B (50% Equity - 50% Bonds) x 10 Insurer C (100% Equity - 0% Bonds)
3 3

2.5
µ1(j) = 7.65% 2.5
µ1(j) = 10.50%
σ (j) = 11.30% σ (j) = 22.28%
skew(j) = 0.6021 skew(j) = 0.6145
2 2
kurt(j) = 3.6585 kurt(j) = 3.6790

1.5 1.5

1 1

0.5 0.5

0 0
-0.5 -0.2 0.1 0.4 0.7 1 -0.5 -0.2 0.1 0.4 0.7 1
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500,000 simulations
 Investment returns distribution j (0,3)
4 4
x 10 Standard Insurer (15% Equity - 85% Bonds) x 10 Insurer A (30% Equity - 70% Bonds)
2 2

µ1(j) = 17.38% µ1(j) = 20.38%

1.5 σ (j) = 6.77% 1.5 σ (j) = 13.35%
skew(j) = 0.4316 skew(j) = 0.5762
kurt(j) = 3.3824 kurt(j) = 3.6440

1 1

0.5 0.5

0 0
-0.5 -0.1 0.3 0.7 1.1 1.5 1.7 -0.5 -0.1 0.3 0.7 1.1 1.5 1.7
j j

4 4
x 10 Insurer B (50% Equity - 50% Bonds) x 10 Insurer C (100% Equity - 0% Bonds)
2 2

µ1(j) = 24.45% µ1(j) = 34.96%

1.5 σ (j) = 22.63% 1.5 σ (j) = 48.21%
skew(j) = 0.7356 skew(j) = 1.1130
kurt(j) = 4.0334 kurt(j) = 5.3720
1 1

0.5 0.5

0 0
-0.5 -0.1 0.3 0.7 1.1 1.5 1.7 -0.5 -0.1 0.3 0.7 1.1 1.5 1.7
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500,000simulations
500,000 simulations
 Final Comments
The introduction in the general framework of the financial risk
originates
Higher instability over time of the capital ratio
Stronger capital requirement for solvency purposes
Different asset allocation affects the safety loading coefficient
A higher percentage in equity, α, means a lower safety loading, φ
However, the corresponding increase in the risk based capital
means that a higher contribution has to come from shareholders
Work in progress
Multiline insurer with underwriting cycle
Correlation between LoBs
Reinsurance
Link between safety loading and claim growth rate
Alternative rules for the calculation of the safety loading, φ

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