The Cost of Financing Insurance
Version 2.0
Glenn Meyers
Insurance Services Office Inc.
CAS Ratemaking Seminar
March 13, 2001
The Cost of Financing Insurance
Version 2.0 - Web Site
• Use DFA to set profitability targets by
line on insurance
• “The Cost of Financing Insurance”
– Sets forth the underlying theory
• “An Analysis of the Underwriting Risk of
DFA Insurance Company”
– Applies “Cost” paper to a very realistic
situation.
• Downloadable spreadsheets
Set Profitability Targets for
an Insurance Company
• The targets must reflect the cost of
capital needed to support each
division's contribution to the overall
underwriting risk.
• The insurer's risk, as measured by its
statistical distribution of outcomes,
provides a meaningful yardstick that can
be used to set capital requirements.
Volatility Determines Capital Needs
Low Volatility
Size of Loss
Chart 3.1
Random Loss
Needed Assets
Expected Loss
Volatility Determines Capital Needs
High Volatility
Size of Loss
Chart 3.1
Random Loss
Needed Assets
Expected Loss
Additional Considerations
• Correlation
– If bad things can happen at the same time,
you need more capital.
The Negative Binomial Distribution
• Select  at random from a gamma
distribution with mean 1 and variance c.
• Select the claim count K at random from
a Poisson distribution with mean .
• K has a negative binomial distribution
with:
E K   and Var K    c  
2
Multiple Line Parameter Uncertainty
• Select b from a distribution with E[b] =
1 and Var[b] = b.
• For each line h, multiply each loss by
b.
Multiple Line Parameter Uncertainty
A simple, but nontrivial example
1  1  3b, 2  1, 3  1  3b
Pr   1  Pr   3   1/ 6 and Pr   2   2 / 3
E[b] = 1 and Var[b] = b
Low Volatility
b = 0.01 r = 0.50
Chart 3.3
4,000
3,500
Y 2 = X 2
3,000
2,500
2,000
1,500
1,000
500
0
0
1,000
2,000
Y 1 = X 1
3,000
4,000
Low Volatility
b = 0.03 r = 0.75
Chart 3.3
4,000
3,500
Y 2 = X 2
3,000
2,500
2,000
1,500
1,000
500
0
0
1,000
2,000
Y 1 = X 1
3,000
4,000
High Volatility
b = 0.01 r = 0.25
Chart 3.3
4,000
3,500
Y 2 = X 2
3,000
2,500
2,000
1,500
1,000
500
0
0
1,000
2,000
Y 1 = X 1
3,000
4,000
High Volatility
b = 0.03 r = 0.45
Chart 3.3
4,000
3,500
Y 2 = X 2
3,000
2,500
2,000
1,500
1,000
500
0
0
1,000
2,000
Y 1 = X 1
3,000
4,000
About Correlation
• There is no direct connection between r
and b.
• Small insurers have large process risk
• Larger insurers will have larger
correlations.
• Pay attention to the process that
generates correlations.
Correlation and Capital
b = 0.00
Chart 3.4
Correlated Losses
Sum of Random Losses
7,000
6,000
5,000
4,000
3,000
2,000
1,000
0
1.0
1.0
1.0
1.0
1.0
1.0
1.0
1.0
1.0
1.0
1.0
1.0
1.0
1.0
1.0
1.0
Random Multiplier
1.0
1.0
1.0
1.0
1.0
1.0
1.0
1.0
1.0
Correlation and Capital
b = 0.03
Chart 3.4
Correlated Losses
Sum of Random Losses
7,000
6,000
5,000
4,000
3,000
2,000
1,000
0
0.7
1.3
1.3
1.0
1.0
0.7
1.0
0.7
1.3
1.3
0.7
1.3
1.3
1.0
0.7
0.7
Random Multiplier
1.0
1.3
0.7
1.0
1.3
1.0
0.7
0.7
1.0
Covariance Generators
• Can be estimated from data
• “Estimating Between Line Correlations
Generated by Parameter Uncertainty”
http://www.casact.org/pubs/forum/99sforum/99sf197.pdf
• Need to combine the data from several
insurers to get reliable estimates.
Additional Considerations
• Reinsurance
– Reduces the need for capital
– Is the cost of reinsurance less than the
cost of capital it releases?
• How long the capital is to be held
– The longer one holds capital to support a
line of insurance, the greater the cost of
writing the insurance.
– Capital can be released over time as risk is
reduced.
Additional Considerations
• Investment income generated by the
insurance operation
– Investment income on loss reserves
– Investment income on capital
The Cost of Financing Insurance
• Includes
– Cost of capital
– Transaction cost of reinsurance
• Transaction Cost of Reinsurance =
Total Cost - Expected Recovery
The To Do List
• Allocate the Cost of Financing back
each underwriting division.
• Express the result in terms of a “Target
Combined Ratio”
• Is reinsurance cost effective?
Doing it - The Steps
• Determine the amount of capital
• Allocate the capital
– To support losses in this accident year
– To support outstanding losses from prior
accident years
• Include reinsurance
• Calculate the cost of financing.
Step 1
Determine the Amount of Capital
• Generate the insurer’s aggregate loss
distribution
– Use ISO size of loss distributions
– Covariance generators estimated from
insurer data reported to ISO
– Include unsettled claims from prior years.
Step 1
Determine the Amount of Capital
• Decide on a measure of risk
• “Coherent Measures of Risk”
– Philippe Artzner, Freddy Delbaen, Jean-Marc
Eber and David Heath, Math. Finance 9 (1999),
no. 3, 203-228
http://www.math.ethz.ch/~delbaen/ftp/preprints/CoherentMF.pdf
– http://www.casact.org/pubs/forum/00sforum/meyers/Coherent
Measures of Risk.pdf
A List of Loss Scenarios
Scenario
1
2
3
4
5
6
7
8
9
10
Maximum Loss
X1
1.00
2.00
3.00
4.00
3.00
2.00
1.00
0.00
0.00
0.00
4.00
X2
0.00
0.00
0.00
1.00
2.00
3.00
4.00
3.00
2.00
1.00
4.00
X1+X2 X3 = 2*X1 X4 = X1+1
1.00
2.00
2.00
2.00
4.00
3.00
3.00
6.00
4.00
5.00
8.00
5.00
5.00
6.00
4.00
5.00
4.00
3.00
5.00
2.00
2.00
3.00
0.00
1.00
2.00
0.00
1.00
1.00
0.00
1.00
5.00
8.00
5.00
Define a measure of risk r(X) = Maximum{Xi}
Subadditivity
Scenario
1
2
3
4
5
6
7
8
9
10
Maximum Loss
X1
1.00
2.00
3.00
4.00
3.00
2.00
1.00
0.00
0.00
0.00
4.00
X2
0.00
0.00
0.00
1.00
2.00
3.00
4.00
3.00
2.00
1.00
4.00
X1+X2 X3 = 2*X1 X4 = X1+1
1.00
2.00
2.00
2.00
4.00
3.00
3.00
6.00
4.00
5.00
8.00
5.00
5.00
6.00
4.00
5.00
4.00
3.00
5.00
2.00
2.00
3.00
0.00
1.00
2.00
0.00
1.00
1.00
0.00
1.00
5.00
8.00
5.00
r(X+Y)  r(X)+r(Y)
Monotonicity
Scenario
1
2
3
4
5
6
7
8
9
10
Maximum Loss
X1
1.00
2.00
3.00
4.00
3.00
2.00
1.00
0.00
0.00
0.00
4.00
X2
0.00
0.00
0.00
1.00
2.00
3.00
4.00
3.00
2.00
1.00
4.00
X1+X2 X3 = 2*X1 X4 = X1+1
1.00
2.00
2.00
2.00
4.00
3.00
3.00
6.00
4.00
5.00
8.00
5.00
5.00
6.00
4.00
5.00
4.00
3.00
5.00
2.00
2.00
3.00
0.00
1.00
2.00
0.00
1.00
1.00
0.00
1.00
5.00
8.00
5.00
If X  Y for each scenario, then r(X)  r(Y)
Positive Homogeneity
Scenario
1
2
3
4
5
6
7
8
9
10
Maximum Loss
X1
1.00
2.00
3.00
4.00
3.00
2.00
1.00
0.00
0.00
0.00
4.00
X2
0.00
0.00
0.00
1.00
2.00
3.00
4.00
3.00
2.00
1.00
4.00
X1+X2 X3 = 2*X1 X4 = X1+1
1.00
2.00
2.00
2.00
4.00
3.00
3.00
6.00
4.00
5.00
8.00
5.00
5.00
6.00
4.00
5.00
4.00
3.00
5.00
2.00
2.00
3.00
0.00
1.00
2.00
0.00
1.00
1.00
0.00
1.00
5.00
8.00
5.00
For all   0 and random loss X, r(X) = r(Y)
Translation Invariance
Scenario
1
2
3
4
5
6
7
8
9
10
Maximum Loss
X1
1.00
2.00
3.00
4.00
3.00
2.00
1.00
0.00
0.00
0.00
4.00
X2
0.00
0.00
0.00
1.00
2.00
3.00
4.00
3.00
2.00
1.00
4.00
X1+X2 X3 = 2*X1 X4 = X1+1
1.00
2.00
2.00
2.00
4.00
3.00
3.00
6.00
4.00
5.00
8.00
5.00
5.00
6.00
4.00
5.00
4.00
3.00
5.00
2.00
2.00
3.00
0.00
1.00
2.00
0.00
1.00
1.00
0.00
1.00
5.00
8.00
5.00
For all random losses X and constants a
r(X+a) = r(X) + a
Axioms for Coherent Measures of Risk
Satisfied by our example
• Subadditivity – For all random losses X and Y,
r(X+Y)  r(X)+r(Y)
• Monotonicity – If X  Y for each scenario, then
r(X)  r(Y)
• Positive Homogeneity – For all   0 and random
loss X
r(X) = r(Y)
• Translation Invariance – For all random losses X
and constants a
r(X+a) = r(X) + a
Value at Risk/Probability of Ruin
is not coherent - violates subadditivity
Scenario
1
2
3
4
5
6
7
8
9
10
VaR@85%
X1
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.00
1.00
0.00
X2
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.00
1.00
0.00
0.00
X1+X2
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.00
1.00
1.00
1.00
0  r  X1   r  X 2   r  X1  X 2   1
Standard Deviation Principle
is not coherent - violates monotonicity
Scenario
1
2
3
4
5
6
7
8
9
10
E[Loss]
StDev[Loss]
E[Loss]+2*StDev[Loss]
X1
1.00
2.00
3.00
4.00
5.00
5.00
4.00
3.00
2.00
1.00
3.00
1.41
5.83
X2
5.00
5.00
5.00
5.00
5.00
5.00
5.00
5.00
5.00
5.00
5.00
0.00
5.00
The Representation Theorem
• Let  denote a finite set of scenarios.
• Let X be a loss associated with each
scenario.
• A risk measure, , is coherent if and
only if there exists a family, , of
probability measures defined on  such
that
r  X   supEP  X  P  P 
i.e. the maximum of a bunch of generalized scenarios
Probability Measures?
The Easiest Example
• Let A {Ai} be the set of one element
subsets of W. Let Xi be the loss for ai.
0 if   Ai 

Pi    
1 if   Ai 
• Then
r  X   sup EP  X  P  P   max  X i 
Probability Measures?
The Next Easiest Example
• Let A {Ai} be the set of n element
subsets of W. Let X be the loss for W
 0 if   Ai 

Pi     n
 if   Ai 
1

• Then
1

r  X   sup EP  X  P  P   max   X 
i
 n Ai

Proposed Measure of Risk
Tail Value at Risk
Value at Risk
VaRa  X   inf x Pr  X  x  a 
Tail Conditional Expectation
Tail Value at Risk
TCEa  X   TailVaRa  X   E  X X  VaRa  X 
Tail Value at Risk - (TVaR)
1.00
Cumulative Probability
0.90
0.80
0.70
Tail Value at Risk is
the average of all
losses above the
Value at Risk
0.60
0.50
0.40
0.30
0.20
0.10
0.00
Value At Risk
Subject Loss
Tail Value at Risk - (TVaR)
1.00
Cumulative Probability
0.90
0.80
0.70
0.60
0.50
0.40
VaR
EPD
0.30
0.20
Area TVaR
0.10
0.00
Value At Risk
Subject Loss
TVaR and
Expected Policyholder Deficit
TailVaRa  X   VaRa  X  
EPD VaRa  X  
1 a
The appeal of TVaR and EPD is that they
both address the question -- How bad is bad?
Step 1
Determine the Amount of Capital
• Decide on a measure of risk
– Tail Value at Risk
• Average of the top 1% of aggregate losses
– Standard Deviation of Aggregate Losses
• Note that the measure of risk is applied to
the insurer’s entire portfolio of losses.
• Capital determined by the risk measure.
C = (X) - E[X]
Step 2
Allocate Capital
• How are you going to use allocated capital?
– Use it to set profitability targets.
Expected Profit for Line
Total Expected Profit
=
Allocated Capital for Line
Total Capital
• How do you allocate capital?
– Any way that leads to correct economic
decisions, i.e. the insurer is better off if
you get your expected profit.
Better Off?
• Let P = Profit and C = Capital. Then the
insurer is better off by adding a line/policy if:
P  P P

C  C C
 P  C  C  P  C  P  P   C
P P


C C
 Marginal return on new business 
return on existing business.
OK - Set targets so that marginal return on
capital equal to insurer return on Capital?
• If risk measure is subadditive then:
Sum of Marginal Capitals is  Capital
• Will be strictly subadditive without perfect
correlation.
• If insurer is doing a good job, strict
subadditivity should be the rule.
OK - Set targets so that marginal return on
capital equal to insurer return on Capital?
If the insurer expects to make a return,
e = P/C
then at least some of its operating divisions
must have a return on its marginal capital
that is greater than e.
Proof by contradiction
DPk P
P
= º e then: P = å DPk = å DCk < P
If
DCk C
C k
k
!
Ways to Allocate Capital #1
• Gross up marginal capital by a factor to
force allocations to add up.
• Economic justification - Long run result
of insurers favoring lines with greatest
return on marginal capital in their
underwriting.
• Appropriate for stock insurers.
• I use it because it is easy.
Ways to Allocate Capital #2
• Average marginal capital, where
average is taken over all entry orders.
• Shapley Value
• Economic justification - Game theory
• Appropriate for mutual insurers
Ways to Allocate Capital #3
• Line headed by CEO’s kid brother gets
the marginal capital. Gross up all other
lines.
• Economic justification - ???
Allocate Capital to
Prior Years’ Reserves
•
•
•
•
•
Target Year 2001 - prospective
Reserve for 2000 - one year settled
Reserve for 1999 - two years settled
Reserve for 1998 - three years settled
etc
Step 3
Reinsurance
• Skip this for now
Step 4
The Cost of Financing Insurance
The cash flow for underwriting insurance
• Investors provide capital - In return they:
• Receive premium income
• Pay losses and other expenses
• Receive investment income
– Invested at interest rate i%
• Receive capital as liabilities become
certain.
Step 4
The Cost of Financing Insurance
Net out the loss and expense payments
• Investors provide capital - In return they:
• Receive profit provision in the premium
• Receive investment income from capital
as it is being held.
• Receive capital as liabilities become
certain.
• We want the present value of the income
to be equal to the capital invested at the
rate of return for equivalent risk
Step 4
The Cost of Financing Insurance
Capital invested in year y+t
C(t)
Capital needed in year y+t if division k
is removed
Marginal capital for division k
Ck(t)
Sum of marginal capital
Allocated capital for division k
Ck(t)=C(t)-Ck(t)
SM(t)
Ak(t)=Ck(t)*C(t)/SM(t)
Profit provision for division k
Pk(t)
Insurer’s return in investment
i
Insurer’s target return on capital
e
Step 4
The Cost of Financing Insurance
Time
0
Financial Support
Allocated at time t
Ak(0)
Amount Released
at time t
0
1
Ak(1)
Relk(1) = Ak(0)(1+i) – Ak(1)
---
---
---
t
Ak(t)
Relk(t) = Ak(t –1)(1+i) – Ak(t)
---
---
---
Then DPk (0) = A k (0) -
¥
å
t =1
Relk (t )
t
(1 + e)
Back to Step 3
Reinsurance and Other
Risk Transfer Costs
• Reinsurance can reduce the amount of,
and hence the cost of capital.
• When buying reinsurance, the
transaction cost (i.e. the reinsurance
premium less the provision for expected
loss) is substituted for capital.
Step 4 with Risk Transfer
The Cost of Financing Insurance
Time
0
Financial Support
Allocated at time t
Ak(0)+Rk(0)
Amount Released
at time t
0
1
Ak(1)
Relk(1) = Ak(0)(1+i) – Ak(1)
---
---
---
t
Ak(t)
Relk(t) = Ak(t –1)(1+i) – Ak(t)
---
---
---
Then DPk (0) = Ak (0) + Rk (0) -
¥
å
Relk (t )
t =1
t
(1 + e)
The Allocated $$ should be reduced with risk transfer.
Example
ABC Insurance Company
• Five Lines
– GL 5 lags
– PL 5 lags, slower payout than GL
– AL 3 lags
– Prop 1 lag
– Cat 1 lag 2% chance of big loss
Example
ABC Insurance Company
• The first four lines move together with
user input of covariance generator, b.
• Cat line is independent of other lines.
• All parameters can be changed.
• Spreadsheet is downloadable.
• Look at spreadsheet
Example
DFA Insurance Company
•
•
•
•
Diversified multi-line insurance company
Northeast/Midwest exposure
Some cat exposure
Details on CAS web site for DFA Call
Paper Program
Example
DFA Insurance Company
• Generated aggregate loss distributions using:
– ISO claim severity distributions by lag
– WC distributions by lag from an independent state
rating bureau
– Covariance generators from ISO study that varied
by line and lag
– Reinsurance information
• Calculated marginal TVaR and Standard
Deviations and then allocated capital.
Example
DFA Insurance Company
• Downloadable spreadsheet
• Aggregate loss distributions calculated
outside the spreadsheet
• All other parameters can be changed
• Multiple reinsurance strategies placed on
spreadsheet
• Look at spreadsheet