The Cost of Financing Insurance
with
Emphasis on Reinsurance
Glenn Meyers
ISO
CAS Ratemaking Seminar
March 10, 2005
Fifth 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
• 2004 – No new papers
• 2005 – Emphasis on Reinsurance
Underlying Themes
• The insurer's risk, as measured by its
stochastic distribution of outcomes,
provides a meaningful yardstick that can
be used to set capital requirements.
• Risk  Capital  Costs money.
• Develop strategy to make most efficient
use of capital.
Strategy – Diversification
• Examples
– Increase volume / Law of large numbers
– Manage concentrations in property insurance
– Decide where to grow and/or shrink
At some point,
it doesn’t pay
to diversify.
$$$
• Costs money to diversify
Cost
Benefit
Diversification
Strategy – Reinsurance
• Examples – Excess of Loss
– Coinsurance provisions
– Treatment of ALAE
– Stacked contracts with various inuring provisions
• Reinsurance costs money
•There are often a lot of
messy details to be
worked out.
Pretty Good
$$$
•You can buy too much
reinsurance.
Cost
Benefit
Reinsurance
Outline of Insurance Strategy
• Grow in lines of business where risk is
adequate rewarded.
• Shrink in lines of business where risk is
not adequately rewarded.
• Diversify when cost effective.
• Buy reinsurance when cost effective.
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
Calculating an Insurer’s
Underwriting Risk
• Use the collective risk model.
– Separate claim frequency and severity analysis
• For each line of insurance:
– Select a random claim count.
– Select random claim size for each claim.
• The aggregate loss for all lines = sum of all the
random claim amounts for all lines.
– Reflect the correlation between lines of insurance.
Consider the Time Dimension
• How long must insurer hold capital?
– 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.
• 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.
• Calculate the cost of financing for each
reinsurance strategy.
• Which reinsurance strategy is the most
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.
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.
Ways to Allocate Capital #2
• Average marginal capital, where
average is taken over all entry orders.
– Shapley Value
– Economic justification - Game theory
• Additive co-measures – Kreps
• Capital consumption – Mango
Remember the time dimension.
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
Examples
• Use ISO Underwriting Risk Model
• Parameterization based on analysis of
industry data.
• Big and small insurer
– Big Insurer is 10 x Small Insurer
• Three reinsurance strategies
Expected Loss
for small insurer is
10 times less,
Various Risk
Measures
Various Risk
Measures
Different measures of risk imply
different amounts of capital
Amount
Implied Capital
Capital
Liabilities
2xStd. Dev.
VaR@99%
TVaR@99%
Allocating (Cost of) Capital
• Calculate marginal capital for each profit
center.
• Calculate the sum of the marginal capitals for
all capital centers.
• Diversification multiplier equals the total
capital divided by the sum of the marginal
capitals.
• Allocated capital for each profit center equals
the product of the diversification multiplier
and the marginal capital for the profit
center.
Capital for Multiline vs Standalone Insurer
Amount
Diversification Benefit
CMP-M CMP-S HO-M
HO-S
Auto-M Auto-S Cat-M
Cat-S Total-M Total-S
Note capital is
allocated to
loss reserves
Optimizing Reinsurance
• User input
– Target return on capital
– Return on investments (sensitivity analysis
on investment income)
– Corporate income tax rate
– Cost of reinsurance
– Insurer expense provisions
List of Reinsurance
Strategies
Cost of Financing Insurance =
Cost of Capital + Net Cost of Reinsurance
• Cost of capital = target return x capital
• Net cost of reinsurance
= Premium – Expected Recovery
• Minimize the cost of financing.

Cost of Financing  Ak  0   
t 1
Rel k  t 
1  e 
t
Big Insurer
Cost of Financing with
No Reinsurance
Small Insurer
Cost of Financing with
No Reinsurance
Big Insurer
Cost of Financing with
Cat Reinsurance
Small Insurer
Cost of Financing with
Cat Reinsurance
Big Insurer
Cost of Financing with
Cat Reinsurance and
XS of Loss Reinsurance
Small Insurer
Cost of Financing with
Cat Reinsurance and
XS of Loss Reinsurance
Optimize reinsurance by
minimizing the cost of financing
Big Insurer
Small Insurer
Net Reins
Capital
Note: Small insurer
costs multiplied by 10.
No Re
Cat
Re
All Re
No Re
Cat
Re
All Re
Discussion of Behavioral Issues
• Smooth out earnings – Wall Street punishes
shock losses.
• Question – Cat limit to capital ratio?
– Answer – 10 to 15%.
• Impairment issues – Can you raise additional
capital if you lose 1/3 of capital?
• Silos – Divisional incentives work against
corporate objectives.