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The Cost of Financing Insurance Version 2.0

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Capital can be released over time as risk is reduced. Additional Considerations ... How are you going to use allocated capital? Use it to set profitability targets. ... – PowerPoint PPT presentation

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Title: The Cost of Financing Insurance Version 2.0


1
The Cost of Financing Insurance Version 2.0
  • Glenn Meyers
  • Insurance Services Office Inc.
  • CAS Ratemaking Seminar
  • March 13, 2001

2
The Cost of Financing Insurance Version 2.0 - Web
Site
  • Use DFA to set profitability targets by line on
    insurance
  • The Cost of Financing Insurance
  • Sets forth the underlying theory
  • An Analysis of the Underwriting Risk of DFA
    Insurance Company
  • Applies Cost paper to a very realistic
    situation.
  • Downloadable spreadsheets

3
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
    statistical distribution of outcomes, provides a
    meaningful yardstick that can be used to set
    capital requirements.

4
Volatility Determines Capital Needs Low Volatility
5
Volatility Determines Capital Needs High
Volatility
6
Additional Considerations
  • Correlation
  • If bad things can happen at the same time, you
    need more capital.

7
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

8
Multiple Line Parameter Uncertainty
  • Select b from a distribution with Eb 1 and
    Varb b.
  • For each line h, multiply each loss by b.

9
Multiple Line Parameter Uncertainty A simple,
but nontrivial example
Eb 1 and Varb b
10
Low Volatility b 0.01 r 0.50
11
Low Volatility b 0.03 r 0.75
12
High Volatility b 0.01 r 0.25
13
High Volatility b 0.03 r 0.45
14
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.

15
Correlation and Capital b 0.00
16
Correlation and Capital b 0.03
17
Covariance Generators
  • Can be estimated from data
  • Estimating Between Line Correlations Generated
    by Parameter Uncertainty
  • http//www.casact.org/pubs/forum/99sforum/99sf197.
    pdf
  • Need to combine the data from several insurers to
    get reliable estimates.

18
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.

19
Additional Considerations
  • Investment income generated by the insurance
    operation
  • Investment income on loss reserves
  • Investment income on capital

20
The Cost of Financing Insurance
  • Includes
  • Cost of capital
  • Transaction cost of reinsurance
  • Transaction Cost of Reinsurance
  • Total Cost - Expected Recovery

21
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?

22
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.

23
Step 1 Determine the Amount of Capital
  • Generate the insurers aggregate loss
    distribution
  • Use ISO size of loss distributions
  • Covariance generators estimated from insurer data
    reported to ISO
  • Include unsettled claims from prior years.

24
Step 1 Determine the Amount of Capital
  • Decide on a measure of risk
  • Coherent Measures of Risk
  • Philippe Artzner, Freddy Delbaen, Jean-Marc Eber
    and David Heath, Math. Finance 9 (1999), no. 3,
    203-228 http//www.math.ethz.ch/delbaen/ftp/prepr
    ints/CoherentMF.pdf
  • http//www.casact.org/pubs/forum/00sforum/meyers/C
    oherent Measures of Risk.pdf

25
A List of Loss Scenarios
Define a measure of risk r(X) MaximumXi
26
Subadditivity
r(XY) ? r(X)r(Y)
27
Monotonicity
If X ? Y for each scenario, then r(X) ? r(Y)
28
Positive Homogeneity
For all l ? 0 and random loss X, r(lX) lr(Y)
29
Translation Invariance
For all random losses X and constants a r(Xa)
r(X) a
30
Axioms for Coherent Measures of Risk Satisfied by
our example
  • Subadditivity For all random losses X and Y,
  • r(XY) ? r(X)r(Y)
  • Monotonicity If X ? Y for each scenario, then
  • r(X) ? r(Y)
  • Positive Homogeneity For all l ? 0 and random
    loss X
  • r(lX) lr(Y)
  • Translation Invariance For all random losses X
    and constants a
  • r(Xa) r(X) a

31
Value at Risk/Probability of Ruin is not coherent
- violates subadditivity
32
Standard Deviation Principle is not coherent -
violates monotonicity
33
The Representation Theorem
  • Let ? denote a finite set of scenarios.
  • Let X be a loss associated with each scenario.
  • A risk measure, ?, is coherent if and only if
    there exists a family, ?, of probability measures
    defined on ? such that

i.e. the maximum of a bunch of generalized
scenarios
34
Probability Measures? The Easiest Example
  • Let A Ai be the set of one element subsets of
    W. Let Xi be the loss for ai.
  • Then

35
Probability Measures? The Next Easiest Example
  • Let A Ai be the set of n element subsets of W.
    Let Xw be the loss for w?W
  • Then

36
Proposed Measure of Risk Tail Value at Risk
Value at Risk
Tail Conditional Expectation Tail Value at Risk
37
Tail Value at Risk is the average of all losses
above the Value at Risk
38
(No Transcript)
39
TVaR and Expected Policyholder Deficit
The appeal of TVaR and EPD is that they both
address the question -- How bad is bad?
40
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
  • Standard Deviation of Aggregate Losses
  • Note that the measure of risk is applied to the
    insurers entire portfolio of losses.
  • Capital determined by the risk measure.
  • C r(X) - EX

41
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.

42
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.
43
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.

44
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
45
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.
  • I use it because it is easy.

46
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

47
Ways to Allocate Capital 3
  • Line headed by CEOs kid brother gets the
    marginal capital. Gross up all other lines.
  • Economic justification -

???
48
Allocate Capital to Prior Years Reserves
  • Target Year 2001 - prospective
  • Reserve for 2000 - one year settled
  • Reserve for 1999 - two years settled
  • Reserve for 1998 - three years settled
  • etc

49
Step 3 Reinsurance
  • Skip this for now

50
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.

51
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

52
Step 4 The Cost of Financing Insurance
53
Step 4 The Cost of Financing Insurance
54
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.

55
Step 4 with Risk Transfer The Cost of Financing
Insurance
The Allocated should be reduced with risk
transfer.
56
Example ABC Insurance Company
  • Five Lines
  • GL 5 lags
  • PL 5 lags, slower payout than GL
  • AL 3 lags
  • Prop 1 lag
  • Cat 1 lag 2 chance of big loss

57
Example ABC Insurance Company
  • The first four lines move together with user
    input of covariance generator, b.
  • Cat line is independent of other lines.
  • All parameters can be changed.
  • Spreadsheet is downloadable.
  • Look at spreadsheet

58
Example DFA Insurance Company
  • Diversified multi-line insurance company
  • Northeast/Midwest exposure
  • Some cat exposure
  • Details on CAS web site for DFA Call Paper Program

59
Example DFA Insurance Company
  • Generated aggregate loss distributions using
  • ISO claim severity distributions by lag
  • WC distributions by lag from an independent state
    rating bureau
  • Covariance generators from ISO study that varied
    by line and lag
  • Reinsurance information
  • Calculated marginal TVaR and Standard Deviations
    and then allocated capital.

60
Example DFA Insurance Company
  • Downloadable spreadsheet
  • Aggregate loss distributions calculated outside
    the spreadsheet
  • All other parameters can be changed
  • Multiple reinsurance strategies placed on
    spreadsheet
  • Look at spreadsheet
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