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

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Random Loss

Needed Assets

Expected Loss

Volatility Determines Capital NeedsLow Volatility

Volatility Determines Capital NeedsHigh Volatility

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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

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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

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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