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The Cost of Financing Insurancewith
Emphasis on Reinsurance
Glenn Meyers
ISO
CAS Ratemaking Seminar
March 10, 2005
Fifth Time at CAS Ratemaking Seminar
• 2001 – Proof of concepthttp://www.casact.org/pubs/forum/00sforum/meyers/index.htm
• 2002 – Applied to DFA Insurance Companyhttp://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
• Costs money to diversify
Diversification
$$
$ CostBenefit
At some point, it doesn’t pay to diversify.
Strategy – Reinsurance• Examples – Excess of Loss
– Coinsurance provisions– Treatment of ALAE– Stacked contracts with various inuring provisions
• Reinsurance costs money
Reinsurance
$$$ Cost
Benefit
•You can buy too much reinsurance.
•There are often a lot of messy details to be worked out.
Pretty Good
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.
Chart 3.1
Siz
e o
f L
os
s
Random Loss
Needed Assets
Expected Loss
Volatility Determines Capital NeedsLow Volatility
Volatility Determines Capital NeedsHigh Volatility
Chart 3.1
Siz
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f L
os
s
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:
2 and VarE K K c
Multiple Line Parameter Uncertainty
• Select from a distribution with E[] = 1 and Var[] = b.
• For each line h, multiply each loss by .
Multiple Line Parameter Uncertainty
A simple, but nontrivial example
1 2 31 3 , 1, 1 3b b
1 3 2Pr Pr 1/ 6 Pr 2 / 3and
E[] = 1 and Var[] = b
Low Volatility b = 0.01 r= 0.50
Chart 3.3
0
500
1,000
1,500
2,000
2,500
3,000
3,500
4,000
0 1,000 2,000 3,000 4,000
Y 1 = X 1
Y2=
X2
Low Volatility b = 0.03 r= 0.75
Chart 3.3
0
500
1,000
1,500
2,000
2,500
3,000
3,500
4,000
0 1,000 2,000 3,000 4,000
Y 1 = X 1
Y2=
X2
High Volatility b = 0.01 r= 0.25
Chart 3.3
0
500
1,000
1,500
2,000
2,500
3,000
3,500
4,000
0 1,000 2,000 3,000 4,000
Y 1 = X 1
Y2=
X2
High Volatility b = 0.03 r= 0.45
Chart 3.3
0
500
1,000
1,500
2,000
2,500
3,000
3,500
4,000
0 1,000 2,000 3,000 4,000
Y 1 = X 1
Y2=
X2
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 3.4Correlated Losses
0
1,000
2,000
3,000
4,000
5,000
6,000
7,000
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 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0
Random Multiplier
Su
m o
f R
an
do
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os
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s
Correlation and Capital b = 0.03
0
1,000
2,000
3,000
4,000
5,000
6,000
7,000
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 1.0 1.3 0.7 1.0 1.3 1.0 0.7 0.7 1.0
Random Multiplier
Su
m o
f R
an
do
m L
os
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s
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 = (X) E[X]
Step 2Allocate Capital
• How are you going to use allocated capital?
– Use it to set profitability targets.
• 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.
Expected Profit for Line Total Expected ProfitAllocated Capital for Line Total Capital
=
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
If then:k
k
P Pe
C C
D= º
D !k k
k k
PP P C P
C= D = D <å å
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 3Reinsurance
• Skip this for now
Step 4The 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 4The Cost of Financing InsuranceNet 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 4The Cost of Financing Insurance
Capital invested in year y+t C(t)
Capital needed in year y+t if division k is removed
Ck(t)
Marginal capital for division k Ck(t)=C(t)-Ck(t)
Sum of marginal capital SM(t)
Allocated capital for division k 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 4The Cost of Financing Insurance
Time Financial Support Allocated at time t
Amount Released at time t
0 Ak(0) 0
1 Ak(1) Relk(1) = Ak(0)(1+i) – Ak(1)
--- --- ---
t Ak(t) Relk(t) = Ak(t –1)(1+i) – Ak(t)
--- --- ---
1
Then 0 01
kk k t
t
Rel tP A
e
Back to Step 3Reinsurance 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 TransferThe Cost of Financing InsuranceTime Financial Support
Allocated at time t Amount Released
at time t 0 Ak(0)+Rk(0) 0
1 Ak(1) Relk(1) = Ak(0)(1+i) – Ak(1)
--- --- ---
t Ak(t) Relk(t) = Ak(t –1)(1+i) – Ak(t)
--- --- ---
1
Then 0 0 01
kk k k t
t
Rel tP R
eA
The Allocated $$ should be reduced with risk transfer.
Step 4 Without Risk TransferThe Cost of Financing Insurance
Time Financial Support Allocated at time t
Amount Released at time t
0 Ak(0) 0
1 Ak(1) Relk(1) = Ak(0)(1+i) – Ak(1)
--- --- ---
t Ak(t) Relk(t) = Ak(t –1)(1+i) – Ak(t)
--- --- ---
1
Then 0 01
kk k t
t
Rel tP A
e
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 Lossfor small insurer is
10 times less,
Various RiskMeasures
Various RiskMeasures
Different measures of risk imply different amounts of capital
Implied Capital
2xStd. Dev. VaR@99% TVaR@99%
Am
ou
nt
CapitalLiabilities
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
CMP-M CMP-S HO-M HO-S Auto-M Auto-S Cat-M Cat-S Total-M Total-S
Am
ou
nt
Diversification Benefit
Note capital isallocated 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 ReinsuranceStrategies
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.
1
Cost of Financing 01
kk t
t
Rel tA
e
Big InsurerCost of Financing with
No Reinsurance
Small InsurerCost of Financing with
No Reinsurance
Big InsurerCost of Financing with
Cat Reinsurance
Small InsurerCost of Financing with
Cat Reinsurance
Big InsurerCost of Financing withCat Reinsurance and
XS of Loss Reinsurance
Small InsurerCost of Financing withCat Reinsurance and
XS of Loss Reinsurance
Optimize reinsurance by minimizing the cost of financing
No Re CatRe
All Re No Re CatRe
All Re
Net ReinsCapital
Big Insurer Small Insurer
Note: Small insurer costs multiplied by 10.
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.
