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The Cost of Financing Insurance Version 2.0 Glenn Meyers Insurance Services Office Inc. CAS Ratemaking Seminar March 13, 2001 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 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. Volatility Determines Capital Needs Low Volatility Size of Loss Chart 3.1 Random Loss Needed Assets Expected Loss Volatility Determines Capital Needs High Volatility Size of Loss Chart 3.1 Random Loss Needed Assets Expected Loss Additional Considerations • Correlation – If bad things can happen at the same time, you need more capital. 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: E K and Var K c 2 Multiple Line Parameter Uncertainty • Select b from a distribution with E[b] = 1 and Var[b] = b. • For each line h, multiply each loss by b. Multiple Line Parameter Uncertainty A simple, but nontrivial example 1 1 3b, 2 1, 3 1 3b Pr 1 Pr 3 1/ 6 and Pr 2 2 / 3 E[b] = 1 and Var[b] = b Low Volatility b = 0.01 r = 0.50 Chart 3.3 4,000 3,500 Y 2 = X 2 3,000 2,500 2,000 1,500 1,000 500 0 0 1,000 2,000 Y 1 = X 1 3,000 4,000 Low Volatility b = 0.03 r = 0.75 Chart 3.3 4,000 3,500 Y 2 = X 2 3,000 2,500 2,000 1,500 1,000 500 0 0 1,000 2,000 Y 1 = X 1 3,000 4,000 High Volatility b = 0.01 r = 0.25 Chart 3.3 4,000 3,500 Y 2 = X 2 3,000 2,500 2,000 1,500 1,000 500 0 0 1,000 2,000 Y 1 = X 1 3,000 4,000 High Volatility b = 0.03 r = 0.45 Chart 3.3 4,000 3,500 Y 2 = X 2 3,000 2,500 2,000 1,500 1,000 500 0 0 1,000 2,000 Y 1 = X 1 3,000 4,000 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 3.4 Correlated Losses Sum of Random Losses 7,000 6,000 5,000 4,000 3,000 2,000 1,000 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 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 Correlation and Capital b = 0.03 Chart 3.4 Correlated Losses Sum of Random Losses 7,000 6,000 5,000 4,000 3,000 2,000 1,000 0 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 Random Multiplier 1.0 1.3 0.7 1.0 1.3 1.0 0.7 0.7 1.0 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. 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 – Transaction cost of reinsurance • Transaction 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 • Generate the insurer’s aggregate loss distribution – Use ISO size of loss distributions – Covariance generators estimated from insurer data reported to ISO – Include unsettled claims from prior years. 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/preprints/CoherentMF.pdf – http://www.casact.org/pubs/forum/00sforum/meyers/Coherent Measures of Risk.pdf A List of Loss Scenarios Scenario 1 2 3 4 5 6 7 8 9 10 Maximum Loss X1 1.00 2.00 3.00 4.00 3.00 2.00 1.00 0.00 0.00 0.00 4.00 X2 0.00 0.00 0.00 1.00 2.00 3.00 4.00 3.00 2.00 1.00 4.00 X1+X2 X3 = 2*X1 X4 = X1+1 1.00 2.00 2.00 2.00 4.00 3.00 3.00 6.00 4.00 5.00 8.00 5.00 5.00 6.00 4.00 5.00 4.00 3.00 5.00 2.00 2.00 3.00 0.00 1.00 2.00 0.00 1.00 1.00 0.00 1.00 5.00 8.00 5.00 Define a measure of risk r(X) = Maximum{Xi} Subadditivity Scenario 1 2 3 4 5 6 7 8 9 10 Maximum Loss X1 1.00 2.00 3.00 4.00 3.00 2.00 1.00 0.00 0.00 0.00 4.00 X2 0.00 0.00 0.00 1.00 2.00 3.00 4.00 3.00 2.00 1.00 4.00 X1+X2 X3 = 2*X1 X4 = X1+1 1.00 2.00 2.00 2.00 4.00 3.00 3.00 6.00 4.00 5.00 8.00 5.00 5.00 6.00 4.00 5.00 4.00 3.00 5.00 2.00 2.00 3.00 0.00 1.00 2.00 0.00 1.00 1.00 0.00 1.00 5.00 8.00 5.00 r(X+Y) r(X)+r(Y) Monotonicity Scenario 1 2 3 4 5 6 7 8 9 10 Maximum Loss X1 1.00 2.00 3.00 4.00 3.00 2.00 1.00 0.00 0.00 0.00 4.00 X2 0.00 0.00 0.00 1.00 2.00 3.00 4.00 3.00 2.00 1.00 4.00 X1+X2 X3 = 2*X1 X4 = X1+1 1.00 2.00 2.00 2.00 4.00 3.00 3.00 6.00 4.00 5.00 8.00 5.00 5.00 6.00 4.00 5.00 4.00 3.00 5.00 2.00 2.00 3.00 0.00 1.00 2.00 0.00 1.00 1.00 0.00 1.00 5.00 8.00 5.00 If X Y for each scenario, then r(X) r(Y) Positive Homogeneity Scenario 1 2 3 4 5 6 7 8 9 10 Maximum Loss X1 1.00 2.00 3.00 4.00 3.00 2.00 1.00 0.00 0.00 0.00 4.00 X2 0.00 0.00 0.00 1.00 2.00 3.00 4.00 3.00 2.00 1.00 4.00 X1+X2 X3 = 2*X1 X4 = X1+1 1.00 2.00 2.00 2.00 4.00 3.00 3.00 6.00 4.00 5.00 8.00 5.00 5.00 6.00 4.00 5.00 4.00 3.00 5.00 2.00 2.00 3.00 0.00 1.00 2.00 0.00 1.00 1.00 0.00 1.00 5.00 8.00 5.00 For all 0 and random loss X, r(X) = r(Y) Translation Invariance Scenario 1 2 3 4 5 6 7 8 9 10 Maximum Loss X1 1.00 2.00 3.00 4.00 3.00 2.00 1.00 0.00 0.00 0.00 4.00 X2 0.00 0.00 0.00 1.00 2.00 3.00 4.00 3.00 2.00 1.00 4.00 X1+X2 X3 = 2*X1 X4 = X1+1 1.00 2.00 2.00 2.00 4.00 3.00 3.00 6.00 4.00 5.00 8.00 5.00 5.00 6.00 4.00 5.00 4.00 3.00 5.00 2.00 2.00 3.00 0.00 1.00 2.00 0.00 1.00 1.00 0.00 1.00 5.00 8.00 5.00 For all random losses X and constants a r(X+a) = r(X) + a Axioms for Coherent Measures of Risk Satisfied by our example • Subadditivity – For all random losses X and Y, r(X+Y) r(X)+r(Y) • Monotonicity – If X Y for each scenario, then r(X) r(Y) • Positive Homogeneity – For all 0 and random loss X r(X) = r(Y) • Translation Invariance – For all random losses X and constants a r(X+a) = r(X) + a Value at Risk/Probability of Ruin is not coherent - violates subadditivity Scenario 1 2 3 4 5 6 7 8 9 10 VaR@85% X1 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 1.00 0.00 X2 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 1.00 0.00 0.00 X1+X2 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 1.00 1.00 1.00 0 r X1 r X 2 r X1 X 2 1 Standard Deviation Principle is not coherent - violates monotonicity Scenario 1 2 3 4 5 6 7 8 9 10 E[Loss] StDev[Loss] E[Loss]+2*StDev[Loss] X1 1.00 2.00 3.00 4.00 5.00 5.00 4.00 3.00 2.00 1.00 3.00 1.41 5.83 X2 5.00 5.00 5.00 5.00 5.00 5.00 5.00 5.00 5.00 5.00 5.00 0.00 5.00 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 r X supEP X P P i.e. the maximum of a bunch of generalized scenarios Probability Measures? The Easiest Example • Let A {Ai} be the set of one element subsets of W. Let Xi be the loss for ai. 0 if Ai Pi 1 if Ai • Then r X sup EP X P P max X i Probability Measures? The Next Easiest Example • Let A {Ai} be the set of n element subsets of W. Let X be the loss for W 0 if Ai Pi n if Ai 1 • Then 1 r X sup EP X P P max X i n Ai Proposed Measure of Risk Tail Value at Risk Value at Risk VaRa X inf x Pr X x a Tail Conditional Expectation Tail Value at Risk TCEa X TailVaRa X E X X VaRa X Tail Value at Risk - (TVaR) 1.00 Cumulative Probability 0.90 0.80 0.70 Tail Value at Risk is the average of all losses above the Value at Risk 0.60 0.50 0.40 0.30 0.20 0.10 0.00 Value At Risk Subject Loss Tail Value at Risk - (TVaR) 1.00 Cumulative Probability 0.90 0.80 0.70 0.60 0.50 0.40 VaR EPD 0.30 0.20 Area TVaR 0.10 0.00 Value At Risk Subject Loss TVaR and Expected Policyholder Deficit TailVaRa X VaRa X EPD VaRa X 1 a The appeal of TVaR and EPD is that they both address the question -- How bad is bad? 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 insurer’s entire portfolio of losses. • Capital determined by the risk measure. C = (X) - E[X] Step 2 Allocate Capital • How are you going to use allocated capital? – Use it to set profitability targets. Expected Profit for Line Total Expected Profit = Allocated Capital for Line Total Capital • 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. 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 DPk P P = º e then: P = å DPk = å DCk < P If DCk C C k k ! 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. 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 - ??? 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 Step 3 Reinsurance • Skip this for now 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. 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 Step 4 The Cost of Financing Insurance Capital invested in year y+t C(t) Capital needed in year y+t if division k is removed Marginal capital for division k Ck(t) Sum of marginal capital Allocated capital for division k Ck(t)=C(t)-Ck(t) SM(t) 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 4 The Cost of Financing Insurance Time 0 Financial Support Allocated at time t Ak(0) Amount Released at time t 0 1 Ak(1) Relk(1) = Ak(0)(1+i) – Ak(1) --- --- --- t Ak(t) Relk(t) = Ak(t –1)(1+i) – Ak(t) --- --- --- Then DPk (0) = A k (0) - ¥ å t =1 Relk (t ) t (1 + e) 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. Step 4 with Risk Transfer The Cost of Financing Insurance Time 0 Financial Support Allocated at time t Ak(0)+Rk(0) Amount Released at time t 0 1 Ak(1) Relk(1) = Ak(0)(1+i) – Ak(1) --- --- --- t Ak(t) Relk(t) = Ak(t –1)(1+i) – Ak(t) --- --- --- Then DPk (0) = Ak (0) + Rk (0) - ¥ å Relk (t ) t =1 t (1 + e) The Allocated $$ should be reduced with risk transfer. 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 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 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 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. 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