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The Cost of Financing Insurance with Emphasis on Reinsurance

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Risk Capital Costs money. Develop strategy to make most efficient use of capital. ... If the insurer expects to make a return, e = P/C ... Ways to Allocate Capital #1 ... – PowerPoint PPT presentation

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Title: The Cost of Financing Insurance with Emphasis on Reinsurance


1
The Cost of Financing Insurance with Emphasis on
Reinsurance
  • Glenn Meyers
  • ISO
  • CAS Ratemaking Seminar
  • March 10, 2005

2
Fifth Time at CAS Ratemaking Seminar
  • 2001 Proof of concept
  • http//www.casact.org/pubs/forum/00sforum/meyers/i
    ndex.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/0
    3sforum/03sf015.pdf
  • Reinsurer http//www.casact.org/pubs/forum/03spfor
    um/03spf069.pdf
  • 2004 No new papers
  • 2005 Emphasis on Reinsurance

3
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.

4
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

At some point, it doesnt pay to diversify.
5
Strategy Reinsurance
  • Examples Excess of Loss
  • Coinsurance provisions
  • Treatment of ALAE
  • Stacked contracts with various inuring provisions
  • Reinsurance costs money

Pretty Good
  • You can buy too much reinsurance.
  • There are often a lot of messy details to be
    worked out.

6
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.

7
Volatility Determines Capital Needs Low Volatility
8
Volatility Determines Capital Needs High
Volatility
9
Additional Considerations
  • Correlation
  • If bad things can happen at the same time, you
    need more capital.
  • We will come back to this shortly.

10
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

11
Multiple Line Parameter Uncertainty
  • Select b from a distribution with Eb 1 and
    Varb b.
  • For each line h, multiply each loss by b.

12
Multiple Line Parameter Uncertainty A simple,
but nontrivial example
Eb 1 and Varb b
13
Low Volatility b 0.01 r 0.50
14
Low Volatility b 0.03 r 0.75
15
High Volatility b 0.01 r 0.25
16
High Volatility b 0.03 r 0.45
17
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.

18
Correlation and Capital b 0.00
19
Correlation and Capital b 0.03
20
Calculating an Insurers 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.

21
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

22
The Cost of Financing Insurance
  • Includes
  • Cost of capital
  • Net cost of reinsurance
  • Net Cost of Reinsurance
  • Total Cost Expected Recovery

23
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?

24
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.

25
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
  • ?(XY) ?(X) ?(Y)
  • i.e. diversification reduces total risk.

26
Step 1 Determine the Amount of Capital
  • Note that the measure of risk is applied to the
    insurers entire portfolio of losses.
  • r(X) Total Required Assets
  • Capital determined by the risk measure.
  • C r(X) - EX

27
Step 2 Allocate 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.

28
Better Off?
  • Let P Profit and C Capital. Then the insurer
    is better off by adding a line/policy if

? Marginal return on new business ? return
on existing business.
29
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.

30
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
31
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.

32
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.

33
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

34
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

35
Step 3 Reinsurance
  • Skip this for now

36
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.

37
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

38
Step 4 The Cost of Financing Insurance
39
Step 4 The Cost of Financing Insurance
40
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.

41
Step 4 with Risk Transfer The Cost of Financing
Insurance
The Allocated should be reduced with risk
transfer.
42
Step 4 Without Risk Transfer The Cost of
Financing Insurance
43
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

44
Expected Loss for small insurer is 10 times less,
45
Various Risk Measures
46
Various Risk Measures
47
Different measures of risk imply different
amounts of capital
48
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.

49
Diversification Benefit
50
Note capital is allocated to loss reserves
51
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

52
List of Reinsurance Strategies
53
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.

54
(No Transcript)
55
Big Insurer Cost of Financing with No Reinsurance
56
Small Insurer Cost of Financing with No
Reinsurance
57
Big Insurer Cost of Financing with Cat
Reinsurance
58
Small Insurer Cost of Financing with Cat
Reinsurance
59
Big Insurer Cost of Financing with Cat
Reinsurance and XS of Loss Reinsurance
60
Small Insurer Cost of Financing with Cat
Reinsurance and XS of Loss Reinsurance
61
Optimize reinsurance by minimizing the cost of
financing
Big Insurer
Small Insurer
Note Small insurer costs multiplied by 10.
62
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.
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