Transcript Presented - actlab.ru

The Price of Risk in Insurance
Presented by Michel M. Dacorogna
?, Moscow, Russia, April 23-24,2008
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include general economic conditions, including in particular economic conditions; the frequency, severity and development of insured loss events
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Company. Any securities to be offered as part of a capital raising will not be registered under the US securities laws and may not be offered or sold in
the United States absent registration or an applicable exemption from the registration requirements of the US securities laws.
Economy of Risk in Insurance
Michel M. Dacorogna
April 23-24, 2008
2
Outline of the Talk
A simple example
Risk and risk measures
Risk-based capital and economic capital
Valuation methods
Conclusions
Economy of Risk in Insurance
Michel M. Dacorogna
April 23-24, 2008
3
A simple example for pricing risk
Assume an insurance customer approaches a company to
insure the following risk:
 He must pay 10 USD if he gets a six on a die, and nothing
otherwise.
 He must throw the die 6 times.
We will answer two questions:
1. What is the price for such a risk, independently of any other
liability the insurer has?
2. How would the price change if the insurer assume many of the
same risk?
Economy of Risk in Insurance
Michel M. Dacorogna
April 23-24, 2008
4
Outcome of Throwing the Die
Second Throw
First Throw
Out.
Prob.
Claim
6
1/6
10
Out.
Prob.
Claim
6
1/6
10
Out.
Prob.
Claim
<6
5/6
0
Total
Probabilit
Claims
y1 1 1
 
6 6 36
5/36
20
10
And so on …
Out.
Prob.
Claim
<6
5/6
0
Out.
Prob.
Claim
6
1/6
10
Out.
Prob.
Claim
<6
5/6
0
5/36
10
25/36
0
Economy of Risk in Insurance
Michel M. Dacorogna
April 23-24, 2008
5
The Cumulative Distribution of Our Example
Amount Probability
of Loss
of Loss
0
33.48%
10
40.17%
20
20.09%
30
5.36%
40
0.80%
50
0.06%
60
0.02%
Expected Loss = 10 USD
Cumulative
Probability
33.48%
73.65%
93.75%
99.11%
99.91%
99.98%
100.00%
Value-at-Risk(1%) = 30 USD
Economy of Risk in Insurance
Michel M. Dacorogna
April 23-24, 2008
6
What is the Correct Price ?
Pricing the risk at the expected loss plus costs means running the
risk of losing more (here there is 26% chances to pay more than
the expected 10 USD).
The risk is to have a claim that far exceeds the expected loss.
We define the risk as the unexpected loss.
An insurer guarantees that he will pay the loss even if it is above
expectation.
Thus the need to put up capital for covering this risk (Risk-Based
Capital, RBC).
Economy of Risk in Insurance
Michel M. Dacorogna
April 23-24, 2008
7
Determining the Capital to Cover the Risk
In order to quantify the risk, we need to define up to which
probability the inusrer is willing to guarantee his payment
This is the confidence threshold at which the company wants to
operate (let us choose here the “1 over 100 year event”)
Such a confidence threshold corresponds to a claim of 30 USD
In our case, the capital would be 20 USD (the 1% claim minus
expected claim)
Providing capital has a cost – investors want a return on
investment.
Let us assume in this case a cost of 15% before tax.
Economy of Risk in Insurance
Michel M. Dacorogna
April 23-24, 2008
8
Computing the Premium
Premium
Company
Structure and Capital
Expenses
0.5
Risk loading
3.0
0.15 * (30-10)
Loss model
Expected loss
10
13.5
Economy of Risk in Insurance
Michel M. Dacorogna
April 23-24, 2008
9
The Concept of Risk in Insurance
Risk describes the uncertainty of the future outcome of a current
decision or situation.
The premium should reflect the risk assumed and the
diversification of the insurer’s portfolio.
Insurance is the transfer of risk from an individual to a company
(group).
We all have expectations about results – but the actual outcome
is uncertain.
In a model, the possible outcomes can be adequately described
by a probability distribution.
Economy of Risk in Insurance
Michel M. Dacorogna
April 23-24, 2008
10
Risk and Risk Measures
This can be measured in terms of probability distributions but it is
better to use one number to express it, called risk measure.
We want a measure that can give us a risk in form of a capital
amount.
The risk measure should have the following properties
(coherence):
1. Scalable (twice the risk should give a twice bigger measure),
2. Ranks risks correctly (bigger risks get bigger measure),
3. Allows for diversification (aggregated risks should have a lower
measure),
Economy of Risk in Insurance
Michel M. Dacorogna
April 23-24, 2008
11
Loss Model and Risk Measures
Mean
Probability / Bin of 250'000
VaR
Standard Deviation
Measures typical
size of fluctuations
+s
1.60%
Value-at-Risk (VaR)
Measures position of
99th percentile,
„happens once in a
hundred years“
1.20%
0.80%
0.40%
0.00%
450 498 547 595 643 691 739
Gross Losses Incurred ($ M)
Expected Shortfall (ES)
is the weighted
average VaR beyond
the 1% threshold.
Economy of Risk in Insurance
Michel M. Dacorogna
April 23-24, 2008
12
Appropriate Risk Measures
We want to measure the extreme risks so VaR and ES are more
appropriate.
We want to ensure that diversification is appropriately accounted for:
if two risks are added together the total risk should be at maximum
equal the sum of both (sub-additivity):
R( x1 + x2 )  R( x1 ) + R( x2 )
Among the measures presented, only the Expected Shortfall or t-VaR
has this property for the type of insurance risks we are facing. It is a
coherent measure of risk.
In general ES is more conservative than VaR but one can choose the
threshold (1% or 0.4%).
Economy of Risk in Insurance
Michel M. Dacorogna
April 23-24, 2008
13
Examples of Risk Evaluation
Typical gross natural catastrophe exposures VaR and ES (in MUSD).
Measure
Hurricane
Earthquake
Total
Expected
62
16
78
Std. Dev.
84
60
104
VaR(1%)
418
332
544
VaR(0.4%)
596
478
690
ES(1%)
575
500
678
ES(0.4%)
700
598
770
>
<
Economy of Risk in Insurance
Michel M. Dacorogna
April 23-24, 2008
14
Diversification:
Insuring Many Independent Risks together
Assume that the insurer takes on not only the risk of one
policyholder but many
Each policyholder insures the risk that he has to pay EUR 10
in each case a 6 appears on a die at 6 throws
Many risks will constitute now a portfolio of risks
How will the premium change due to diversification?
Remember: The expected loss per policy was EUR 10,
expense EUR 0.5 and the risk loading EUR 3 for one policy
seen in isolation
Economy of Risk in Insurance
Michel M. Dacorogna
April 23-24, 2008
15
Influence of Diversification on the Premium
Number of
Policies
1
5
10
50
100
1000
10000
Cost of
Capital
3.0000
1.5000
1.0500
0.4638
0.3257
0.1009
0.0257
As can be seen, if the risks diversify, the risk loading per policy
reduces the more, the more policies are in the insurer’s portfolio
Economy of Risk in Insurance
Michel M. Dacorogna
April 23-24, 2008
16
Limits to Diversification
Assume now that not all risk are diversifiable.
Assume that the policyholders play the die all in the same casino and
that with a given probability p, they will have a crooked croupier. In
that case, they will all lose EUR 60, i.e. each throw of a die will always
show a 6.
1
5
10
50
100
1000
10000
No Fraud
Risk
Premium
3.0000
1.5000
1.0500
0.4638
0.3257
0.1009
0.0257
10%
6.7517
6.7529
6.7442
6.7623
6.7818
6.7564
6.5269
5%
7.1249
7.1249
7.1253
7.1202
7.1286
7.1192
7.1266
1%
6.7103
4.9527
4.5834
3.9828
3.6925
3.5212
1.3321
0.10%
3.1163
1.4925
1.0441
0.4647
0.3239
0.1803
0.1251
Expected Loss
10.0000
15
12.5
10.5
10.05
Number of
Policies
Probability of Fraud
Diversification is significantly reduced if there are underlying risk factors
affecting all policies simultaneously (e.g. a crooked croupier)
Economy of Risk in Insurance
Michel M. Dacorogna
April 23-24, 2008
17
From Risk Loading to Cost of Capital
The traditional approach to pricing in insurance was to load the
expected loss by the uncertainty of the outcome through a
factor times the standard deviation or more generally:
P  E  L + k  r  L  + m
Where L are the losses, r is a risk measure (like s, s2 or
Value-at-Risk), k the risk loading factor and m the costs.
This approach is not compatible with premiums that depend on
the losses, which is very common in reinsurance
(reinstatements).
It also completely neglects portfolio effect, the cost of capital or
target profitability and the payout patterns of the losses.
Economy of Risk in Insurance
Michel M. Dacorogna
April 23-24, 2008
18
From Pricing the Losses to Pricing for Profit
Moreover, the traditional approach with certain risk measures
(standard deviation, VaR) is not always additive.
Introducing some basic finance idea we should price the profit to
be expected rather than the loss.
Let X be a reinsurance treaty with a profit P for the reinsurer:
P = Premium-Losses-Expenses
We can now simply introduce the time value of money by
computing the Net Present Value (NPV) of P discounting it to
today.
Economy of Risk in Insurance
Michel M. Dacorogna
April 23-24, 2008
19
Some Alternative Pricing Principles
We now use X=NPV(P) as the variable to compute the
premium.
Distorted Probability: Denneberg in 1988 and Wang
independently in 1995 proposed to find a distortion function
G:[0,1][0,1] increasing, surjective and concave such that:
E  X  :

+
 x d (G
!
FX )( x )  0

They then define the technical premium as the one for which X
satisfies the above equation.
Such a principle is additive.
Applying this methodology one can derive the risk neutral
probability that is used in finance for pricing derivatives.
Economy of Risk in Insurance
Michel M. Dacorogna
April 23-24, 2008
20
Coming up with a Quotation
Profitability
RoRBC
Treaty Features &
Profit Distribution
RBC
Performance
Excess
Risk
Loading
NPV
Expenses
Expenses
Loss Model
Conditions
Losses
Expected
Loss
Pure Losses
Economy of Risk in Insurance
Michel M. Dacorogna
April 23-24, 2008
21
Introducing Diversification and Discounting
The simple example before does not elaborate on two facts:
1.
The insurer should price against his portfolio,
2.
The payout patterns of the losses count: when does the insurer
pay the loss?
We need to introduce here more complicated notions of capital
allocation and discounting.
Allocating capital against the portfolio requires to know the
dependence between treaties and to use a risk measure that
accounts for diversification (sub-additive).
Economy of Risk in Insurance
Michel M. Dacorogna
April 23-24, 2008
22
Some Conventions
For the sake of simplicity, we always assume sufficient
differentiability, e.g.
 Each random variable is assumed to have a density.
 Empirical distributions can be approximated by smooth distributions
(for our purpose, as exactly as we wish).
For a random variable S, we denote by FS the cumulative distribution
function of S.
We use as our basic variable the NPV of the profit of a treaty X:
X = NPV(Premium-Losses-Expenses)
X represents the random variable while X is the full treaty.
Economy of Risk in Insurance
Michel M. Dacorogna
April 23-24, 2008
23
What Is an Appropriate Amount of Profit?
Clearly the expectation of X, E(X), should be positive.
It should also cover the cost of capital to be paid back to
the investors.
It should cover the expenses of the operation.
It should include a safety loading as seen before:
 The higher the risk, the higher the loading,
 the higher the dependence with the portfolio the higher the
loading,
 And the longer it takes to develop to ultimate the more capital is
needed.
Economy of Risk in Insurance
Michel M. Dacorogna
April 23-24, 2008
24
The Portfolio Viewpoint
Let us consider the following portfolio Z:
X i i 1,,m : Z
where Xi are the different risks (=treaties).
The portfolio Z is supported by a Risk Based Capital K.
An allocation of capital Ki to Xi requires a technical premium
such that
E [X i ]= ht i ×Ki
where ti is the duration of the risk Xi and h is the profit target.
Economy of Risk in Insurance
Michel M. Dacorogna
April 23-24, 2008
25
Capital Allocation: Euler Principle
We allocate capital to a sub-portfolio S (e.g., treaty, Line of
Business) in Z according to the Euler principle:
d
KS 
r (Z + tS )
dt t 0
Assume all ti=1. Then, roughly speaking, this is the only allocation
principle satisfying the following property (D. Tasche, 1999):
If the premium is higher (lower) than the technical one, then a
small increase (decrease) of the participation in X will improve
(lower) the return on RBC of the entire portfolio.
Steering the portfolio through pricing.
Economy of Risk in Insurance
Michel M. Dacorogna
April 23-24, 2008
26
Tasche’s result
Theorem (D. Tasche, 1999). Under the above assumptions and
some mild differentiability assumptions we have:
1
KS   E (S | Z  FZ ( ))
Thus we allocate capital to a line of business according to its
contribution to the bad performance of the whole portfolio.
In order to use this principle in practise, we need a sound
portfolio model!
To this end, we need a sound model to describe dependencies.
We use here copulae, CX,Y .
Economy of Risk in Insurance
Michel M. Dacorogna
April 23-24, 2008
27
Allocation of Capital to a Treaty
C. Hummel (2002) showed that if we are given a treaty S of Z
and the copula CS,Z between S and Z, then:
+
KS   E S | Z  FZ ( )     s d ( H S FS )( s)
1

with H S (u ) 
1

C S ,Z ( u ,  )
We call HS the Diversification Function of S in Z.
The distorted probability depends on the diversification of S
within Z.
Note that we do not need to know FZ to calculate KS..
Economy of Risk in Insurance
Michel M. Dacorogna
April 23-24, 2008
28
Interpretation
Consequently, the technical premium should insure a profit X for
treaty X that satisfies
+
 x dF
X
( x)  ht

+
 x d (H
X
FX )( x)

This is equivalent to (C. Hummel 2002):
+
 x d (G

X
FX )( x)  0
with
p + ht H X ( p)
GX ( p ) 
1 + ht
Compare this to Denneberg and Wang’s premium principles: In our
setup, the distorted probabilities differ from treaty to treaty and are
determined from the diversification effect of the treaty In the
reinsurer’s portfolio.
Economy of Risk in Insurance
Michel M. Dacorogna
April 23-24, 2008
29
Hierarchical Dependences
Z
Dependence between
Line of Business (LoB)’s
Y1
Dependence
between contracts
Y3
X 23
X 34
X 15
X 31
X 14
X11
X 12
Y2
X 22
X13
X 21 X
33
Economy of Risk in Insurance
Michel M. Dacorogna
April 23-24, 2008
X 32
30
A Central Assumption
Given this structure, the model is completely defined if we
also require that:
P( X  x | Y  y, Z  z )  P( X  x | Y  y )
for all LoB Y and all risks X in Y.
In other words, given that the result of Y influences the
information about the result in Z, the latter is not influenced by
the distribution of X in Y.
Economy of Risk in Insurance
Michel M. Dacorogna
April 23-24, 2008
31
Copula between the Risk X and the portfolio Z
From the model for the LoB Y we get CX,Y.
From the distribution of the LoB Y and its Copula to the
portfolio Z, we get CY,Z.
It is then possible, with relatively mild assumptions, to
compute the copula between X and Z :
1
C X , Z (u, w)  
0
C X ,Y
v
(u, v)
CY , Z
v
(v, w) dv
Economy of Risk in Insurance
Michel M. Dacorogna
April 23-24, 2008
32
The Diversification Function
Given the copula, CX,Y, between the risk X and the LoB Y, it is
possible to define a diversification function, HX(u), as follows:
H X (u)
  1C X ,Z (u,  )

1
C X ,Y
CY ,Z
(
u
,
v
)
( v,  ) dv
0 v
v
1
C X ,Y

(u, v ) dH Y (v )
v
0
1
C X ,Y (u, vm +1 )  C X ,Y (u, vm )

 HY (vm+1 )  HY (vm )  .
vm +1  vm
m 1
M
Assuming that the grid 0  v0  v1 
 vM  1 is fine enough.
Economy of Risk in Insurance
Michel M. Dacorogna
April 23-24, 2008
33
Only the Diversification Function within the LoB is
Relevant
We just saw that:
M
CX ,Y (u, vm+1 )  CX ,Y (u, vm )
m 1
vm+1  vm
H X (u)  
 HY (vm+1 )  HY (vm ) 
From this expression it follows:
 To be able to price a risk within a line of business, we do not
need to compute the copulae between the different LoB’s.
 We only need to implement the diversification function, HY,
with:
0  v0  v1   vM  1
Economy of Risk in Insurance
Michel M. Dacorogna
April 23-24, 2008
34
Coming up with a Quotation
Profitability
RoRBC
Treaty Features &
Profit Distribution
RBC
Performance
Excess
Risk
Loading
NPV
Expenses
Expenses
Loss Model
Conditions
Losses
Expected
Loss
Pure Losses
Economy of Risk in Insurance
Michel M. Dacorogna
April 23-24, 2008
35
Using the Traditional Method for Pricing
Risk Loading for Various
CAT Programs
Loading / StDev
0.3
A
B
C
D
E
0.2
Using the standard deviation
loading makes all these
programs lie on a straight line
since they present very similar
risk characteristics.
Risk Rate on Line
0.1
RRoL 
0.0
0
1
2
-Log(RRoL)
ExpectedLoss
Granted Limit
3
Economy of Risk in Insurance
Michel M. Dacorogna
April 23-24, 2008
36
Active Portfolio-Management: An Example
Risk Loading for Various
CAT-Programs
2nd Layer Prg. A
A
B
C
D
E
Loading / StDev
0.4
0.3
0.2
0.1
0
1
2
Example:
The distribution of the second
layer A and D are almost
identical.
A presents a stronger, D a
weaker dependence to the rest
of the Portfolio.
2nd Layer Prg. D
0.0
The capital allocation taking into
account the diversification effects
within the portfolio results in different
loading for similar risks.
3
-Log(RRoL)
Economy of Risk in Insurance
Michel M. Dacorogna
April 23-24, 2008
37
Active Portfolio-Management:
Example (II)
An
Risk Loading for Various
CAT-Programs
A
B
C
D
E
Loading / StDev
0.4
0.3
0.2
Diversification or risk accumulation
are favored respectively penalized in
the price.
As a result, the pricing mechanism
implicitly optimizes the portfolio.
0.1
0.0
0
1
2
3
-Log(RRoL)
Economy of Risk in Insurance
Michel M. Dacorogna
April 23-24, 2008
38
The Die Example Priced in our Portfolio
The price for the example, we presented at the beginning is of
course depending on the portfolio of the insurer.
We ran this example through our pricing tool MARS and
got:11.5 for this example taken in our credit & surety book
(dependence to the portfolio).
Let us modify the example by increasing the risk with the
same expected loss: we pay 60 USD for one draw of a six.
The price standalone in this case would be: 10 + 0.5 + 7.5 = 18
and MARS would give 12.5.
Economy of Risk in Insurance
Michel M. Dacorogna
April 23-24, 2008
39
Conclusion
The concept of Risk-Based Capital is central for understanding the
value creation of an insurance company.
The definition of RBC depends on the risk measure used and the
risk appetite.
Even if the measure and the threshold are defined: there are
different ways of defining the RBC, and each of them is valid in a
certain context.
A sound capital allocation methodology allows to price the risk of an
insurance contract to provide the appropriate return on equity.
Modeling the dependencies in a hierarchical way and using
expected shortfall as a risk measure allow to price deals against
the portfolio.
Economy of Risk in Insurance
Michel M. Dacorogna
April 23-24, 2008
40
References
H. Bühlmann, An Economic Premium Principle, Astin Bulletin 11 (1980), 52-60.
M. Denault, Coherent Allocation of Risk Capital, Ecole des H.E.C Montreal, Sept. 1999,
revised Jan. 2001, www.risklab.ch/Papers.html#Denault1999 .
D. Denneberg, Verzerrte Wahrscheinlichkeiten in der Versicherungsmathematik,
quantilsabhängige Prämienprinzipien, Universität Bremen, 1989.
C. Hummel, Capital Allocation in the Presence of Tail Dependencies, May 2002,
Presentation at the Eurandom Workshop on Reinsurance Eindhoven, The Netherlands.
D. Tasche, Risk contributions and performance measurement, Zentrum Mathematik
(SCA), TU München, Jun. 1999, revised Feb. 2000, www-m4.mathematik.tumuenchen.de/m4/pers/tasche/
D. Tasche, Conditional Expectation as Quantile Derivative, Nov. 2000, --- “ ---.
S. Wang, Premium Calculation by Transforming the Layer Premium Density, Astin Bulletin
26 (1996), 71-92.
Economy of Risk in Insurance
Michel M. Dacorogna
April 23-24, 2008
41