The Cost of Financing Insurance
Glenn Meyers
Insurance Services Office Inc.
CAS Ratemaking Seminar
March 11, 2004
Fourth Time at CAS
Ratemaking Seminar
• 2001 – Proof of concept
http://www.casact.org/pubs/forum/00sforum/meyers/index.htm
• 2002 – Applied to DFA Insurance Company
http://www.casact.org/pubs/forum/01spforum/meyers/index.htm
• 2003 – Additional realistic examples
– Primary insurer
http://www.casact.org/pubs/forum/03sforum/03sf015.pdf
– Reinsurer
http://www.casact.org/pubs/forum/03spforum/03spf069.pdf
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
stochastic 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.
• We will come back to this shortly.
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
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
– Net cost of reinsurance
• Net 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
• Decide on a measure of risk
– Tail Value at Risk
• Average of the top 1% of aggregate losses
• Example of a “Coherent Measure of Risk
– Standard Deviation of Aggregate Losses
• Expected Loss + K  Standard Deviation
– Both measures of risk are subadditive
• (X+Y) ≤ (X) + (Y)
• i.e. diversification reduces total risk.
Step 1
Determine the Amount of Capital
• Note that the measure of risk is applied to
the insurer’s entire portfolio of losses.
(X) = Total Required Assets
• Capital determined by the risk measure.
C = r(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
P
DPk P
= º e then: P = å DPk = å DCk < P
If
C k
DCk C
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.
• It is also 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 - ???
Reference
• The Economics of Capital Allocation
– By Glenn Meyers
– Presented at the 2003 Bowles Symposium
http://www.casact.org/pubs/forum/03fforum/03ff391.pdf
• The paper:
– Asks what insurer behavior makes
economic sense?
– Backs out the capital allocation method
that corresponds to this behavior.
Allocate Capital to
Prior Years’ Reserves
•
•
•
•
•
Target Year 2003 - prospective
Reserve for 2002 - one year settled
Reserve for 2001 - two years settled
Reserve for 2000 - 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)
---
---
--
Rel k  t 
t 1
1  e 
Then Pk  0   Ak  0   
t
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)
---
---
--
Rel k  t 
t 1
1  e 
Then Pk  0   Ak  0   Rk  0   
t
The Allocated $$ should be reduced with risk transfer.
Step 4 Without Risk Transfer
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)
---
---
--
Rel k  t 
t 1
1  e 
Then Pk  0   Ak  0   
t
Demonstration of Software
• OK – Now that we see that the “Cost of
Financing” can be quickly implemented,
lets look at screen shots.
Demo Will Cover
• Aggregate loss calculation
• Capital allocation
• Evaluating Reinsurance Programs
– Cat reinsurance
– Other reinsurance
– Show the effect of the size of insurer
Input
• Collective model input
– Claim severity distributions
– Claim count distribution parameters
– Covariance generators
• Per claim limit, retention and coinsurance
• Separate input by contract, line of
business, state, branch office etc.
ISO Severity
Distributions
User Severity
Distribution
User Supplies
Expected Loss
We need to know
how long allocated
capital will be held.
Up to 7 Years
Output
• Insurer aggregate loss distribution
– Calculates mean and standard deviation
– Calculates Value at Risk (VaR)
– Calculates Tail Value at Risk (TVaR)
– Used to derive needed capital
• Compare needed capital for different
reinsurance strategies
No Reinsurance
Aggregate
Mean and
Standard Deviation
Capital and TVaR
at 99% Level
Cat Reinsurance
With $50M
Retention
Aggregate
Mean and
Standard Deviation
Capital and TVaR
at 99% Level
Allocate Capital
• In this demo we allocate capital in
proportion to marginal TVaR99%
• Calculate TVaR99% with each
line/reserve removed
• Adjust by constant of proportionality
Cat Reinsurance
With $50M
Retention
Constant of
Proportionality
Note capital is
allocated to
loss reserves
Cost of Financing Insurance =
Cost of Capital + Net Cost of Reinsurance
• User input
– Target return on equity
– Cost of reinsurance
– Return on investments
– Insurer expense factors
• Objectives
– Evaluate reinsurance strategy
– Set underwriting targets
User Input in
Blue Fonts
List of Reinsurance
Strategies
Allocated capital in current and
future accident years
No Reinsurance
Best Reinsurance
Strategy
Cat Reinsurance XS $50 M
Cat Reinsurance XS $50 M
+
XS of Loss Reinsurance over 1M
for non–cat lines
Standard Ratemaking
Exhibit
Scroll to end –>
Cost of
Financing
Target
Combined
Ratio
The Effect of Insurer Size
• Divide all exposures by 10
– Non-cat lines → Divide all expected claim
counts by 10 and keep same limits
– Cat lines → Divide all claim amounts by
10, including the limit
• Examine reinsurance strategy
– No reinsurance
– Only cat reinsurance
– Cat + other reinsurance
Big - $32,763,664
Big - $32,560,481
Best strategy for big
Big - $35,554,037
Best strategy for small
Note Differences
• Cost of financing is not proportional ( x 10)
– No Re Small - $5,597,928 Big - $32,763,664
– Cat Re Small - $5,647,502 Big - $32,560,481
– All Re Small - $3,728,100 Big - $35,554,037
• Best reinsurance strategies are different.
Summary
• We have demonstrated
– How to calculate required capital
– How to evaluate reinsurance strategies
– How to calculate target combined ratios
that take capital management strategies
into account
Reinsurance Capacity Charges
• Generally speaking, the same principles apply
• Capacity charge is proportional to marginal cost
of capital over a reference portfolio.
• Reference – “The Aggregation and Correlation
of Reinsurance Exposure”
– By Glenn Meyers, Fred Klinker, and David Lalonde
Establish a Reference
Portfolio
• Represents current business
• Use as a base for calculating Marginal
Capital.
• Marginal Capital = Capital needed for
reference portfolio + new contract less
the capital needed for the reference
portfolio
Capacity Charge
• Proportional to marginal capital
• For long-tailed contracts, capital is
released over time.
• Earn reinsurer’s target return on capital
as long as capital is being held.
Casualty Insurance Examples
First Contract
Contract
Comm Auto Liab A
Retention
Loss
Charge % Exp Loss
500,000 1,000,000
14,525
1.45%
Comm Auto Liab B
1,000,000 1,000,000 1,000,000
14,942
1.49%
Comm Auto Liab C
1,000,000 5,000,000 1,000,000
21,174
2.12%
500,000 1,000,000
31,265
3.13%
General Liability B
1,000,000 1,000,000 1,000,000
32,484
3.25%
General Liability C
1,000,000 5,000,000 1,000,000
39,976
4.00%
General Liability A
500,000
Limit
Expected Capacity Cap Chg as
500,000
Explain Differences
• Capacity charges increase with
– Higher limits
– More volatility
– How long you have to hold capital
Capacity Charge for Cat
Covers
Reinsurance Expected Capacity Cap Chg as
Contract
Loss
Charge % Exp Loss
Earthquake A 303,947 310,554 102.17%
Earthquake B 593,735 529,436
89.17%
Earthquake C 2,760,151 919,608
33.32%
Earthquake D 371,200 350,656
94.47%
Hurricane A 123,008 348,911 283.65%
Hurricane B
75,723
15,894
20.99%
Hurricane C 640,824 589,125
91.93%
Hurricane D 462,064 306,564
66.35%
Explain Differences
• Contract that pays when rest of
contracts also pay are less desirable.
– Correlation
Earthquake Contract
Scatter Plot for Contract A
Capacity Charge =
102% of Expected
Loss
Reference Portfolio Earthquake
Earthquake Contract
Scatter Plot for Contract C
Capacity Charge =
33% of Expected Loss
Reference Portfolio Earthquake
Summary
• We have demonstrated
– How to calculate required capital
– How to evaluate reinsurance strategies
– How to calculate target combined ratios
that take capital management strategies
into account
– How to calculate capacity charges for
reinsurers.
Prediction (from RCM-2)
This how actuaries will include the cost
of capital in future insurance costing.
Obstacles to Overcome
• Fuzzy relationship between risk and capital
– See Recent work by IAA working party
http://www.actuaries.org/members/en/committees/WGRBC/documents.cfm
• Quantification of all risks
–
–
–
–
Underwriting risk – (Significant progress here)
Asset risk - Several commercial models
Operational risk
Other
• Consensus