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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 concept http://www.casact.org/pubs/forum/00sforum/meyers/index.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/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. 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. • 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: 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 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 = r(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 P DPk P = º e then: P = å DPk = å DCk < P If C k DCk C 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. • 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 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) --- --- -- Rel k t t 1 1 e Then Pk 0 Ak 0 t 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) --- --- -- Rel k t t 1 1 e Then Pk 0 Ak 0 Rk 0 t The Allocated $$ should be reduced with risk transfer. Step 4 Without Risk Transfer 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) --- --- -- Rel k t t 1 1 e Then Pk 0 Ak 0 t 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 Supplies Expected Loss We need to know how long allocated capital 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 No Reinsurance Aggregate Mean and Standard Deviation Capital and TVaR at 99% Level Cat Reinsurance With $50M Retention Aggregate Mean and Standard Deviation Capital and TVaR at 99% Level 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 Cat Reinsurance With $50M Retention Constant of Proportionality Note capital is allocated to loss reserves 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 User Input in Blue Fonts List of Reinsurance Strategies Allocated capital in current and future accident years No Reinsurance Best Reinsurance Strategy Cat Reinsurance XS $50 M Cat Reinsurance XS $50 M + XS of Loss Reinsurance over 1M for non–cat lines Standard Ratemaking Exhibit 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 - $32,560,481 Best strategy for big Big - $35,554,037 Best 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 Contract Comm Auto Liab A Retention Loss Charge % Exp Loss 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% 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% General Liability A 500,000 Limit Expected Capacity Cap Chg as 500,000 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 Earthquake Contract Scatter Plot for Contract A Capacity Charge = 102% of Expected Loss Reference Portfolio Earthquake Earthquake Contract Scatter Plot for Contract C Capacity Charge = 33% of Expected Loss Reference Portfolio Earthquake 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 party http://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