The Cost of Financing Insurance Glenn Meyers Insurance Services Office Inc. CAS Ratemaking Seminar...
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Transcript of The Cost of Financing Insurance Glenn Meyers Insurance Services Office Inc. CAS Ratemaking Seminar...
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 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
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.
Siz
e o
f L
os
s
Random Loss
Needed Assets
Expected Loss
Volatility Determines Capital NeedsLow Volatility
Volatility Determines Capital NeedsHigh Volatility
Siz
e o
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
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
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
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
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
m L
os
se
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
se
s
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 = (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.
• 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 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
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 SuppliesExpected Loss
We need to knowhow long allocatedcapital 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
Aggregate Mean and
Standard Deviation
Capital and TVaRat 99% Level
No Reinsurance
Aggregate Mean and
Standard Deviation
Capital and TVaRat 99% Level
Cat ReinsuranceWith $50MRetention
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
Note capital isallocated to loss reserves
Cat ReinsuranceWith $50MRetention
Constant ofProportionality
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
List of ReinsuranceStrategies
User Input inBlue Fonts
Allocated capital in current andfuture accident years
No Reinsurance
Cat Reinsurance XS $50 M
Best ReinsuranceStrategy
Cat Reinsurance XS $50 M+
XS of Loss Reinsurance over 1Mfor non–cat lines
Standard RatemakingExhibit
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 - $32,560,481Best strategy for big
Big - $35,554,037Best 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 Expected Capacity Cap Chg as
Contract Retention Limit Loss Charge % Exp Loss
Comm Auto Liab A 500,000 500,000 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%
General Liability A 500,000 500,000 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%
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
Scatter Plot for Contract A
Reference Portfolio Earthquake
Ear
thq
uak
e C
ontr
act
Capacity Charge =
102% of Expected Loss
Scatter Plot for Contract C
Reference Portfolio Earthquake
Ear
thq
uak
e C
ontr
act
Capacity Charge =
33% of Expected Loss
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 partyhttp://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