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The Cost of Financing Insurance with Emphasis on Reinsurance Glenn Meyers ISO CAS Ratemaking Seminar March 10, 2005 Fifth 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 • 2004 – No new papers • 2005 – Emphasis on Reinsurance 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. 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 $$$ Cost Benefit At some point, it doesn’t pay to diversify. Diversification Strategy – Reinsurance • Examples – Excess of Loss – Coinsurance provisions – Treatment of ALAE – Stacked contracts with various inuring provisions • Reinsurance costs money •You can buy too much Pretty Good reinsurance. $$$ Cost •There are often a lot of Benefit messy details to be worked out. Reinsurance 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. Volatility Determines Capital Needs Low Volatility Chart 3.1 Size of Loss Random Loss Needed Assets Expected Loss Volatility Determines Capital Needs High Volatility Chart 3.1 Size of Loss 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 3,000 Y 2 = X 2 2,500 2,000 1,500 1,000 500 0 0 1,000 2,000 3,000 4,000 Y 1 = X 1 Low Volatility b = 0.03 r = 0.75 Chart 3.3 4,000 3,500 3,000 Y 2 = X 2 2,500 2,000 1,500 1,000 500 0 0 1,000 2,000 3,000 4,000 Y 1 = X 1 High Volatility b = 0.01 r = 0.25 Chart 3.3 4,000 3,500 3,000 Y 2 = X 2 2,500 2,000 1,500 1,000 500 0 0 1,000 2,000 3,000 4,000 Y 1 = X 1 High Volatility b = 0.03 r = 0.45 Chart 3.3 4,000 3,500 3,000 Y 2 = X 2 2,500 2,000 1,500 1,000 500 0 0 1,000 2,000 3,000 4,000 Y 1 = X 1 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 7,000 Sum of Random Losses 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 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 Random Multiplier Correlation and Capital b = 0.03 Chart 3.4 Correlated Losses 7,000 Sum of Random Losses 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 1.0 1.3 0.7 1.0 1.3 1.0 0.7 0.7 1.0 Random Multiplier Calculating an Insurer’s 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. 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 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. • Calculate the cost of financing for each reinsurance strategy. • Which reinsurance strategy is the most 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 DPk P P If = º e then: P = å DPk = å DCk < P ! DCk C k 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. 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. 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 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 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 Ck(t) is removed 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 4 The Cost of Financing Insurance Time Financial Support Amount Released Allocated at time t 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) --- --- --- Rel k t Then Pk 0 Ak 0 1 e t t 1 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 Financial Support Amount Released Allocated at time t 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) --- --- --- Rel k t Then Pk 0 Ak 0 Rk 0 1 e t t 1 The Allocated $$ should be reduced with risk transfer. Step 4 Without Risk Transfer The Cost of Financing Insurance Time Financial Support Amount Released Allocated at time t 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) --- --- --- Rel k t Then Pk 0 Ak 0 1 e t t 1 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 Expected Loss for small insurer is 10 times less, Various Risk Measures Various Risk Measures Different measures of risk imply different amounts of capital Implied Capital Amount Capital Liabilities 2xStd. Dev. VaR@99% TVaR@99% 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. Capital for Multiline vs Standalone Insurer Diversification Benefit Amount CMP-M CMP-S HO-M HO-S Auto-M Auto-S Cat-M Cat-S Total-M Total-S Note capital is allocated to loss reserves 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 List of Reinsurance Strategies 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. Rel k t Cost of Financing Ak 0 1 e t t 1 Big Insurer Cost of Financing with No Reinsurance Small Insurer Cost of Financing with No Reinsurance Big Insurer Cost of Financing with Cat Reinsurance Small Insurer Cost of Financing with Cat Reinsurance Big Insurer Cost of Financing with Cat Reinsurance and XS of Loss Reinsurance Small Insurer Cost of Financing with Cat Reinsurance and XS of Loss Reinsurance Optimize reinsurance by minimizing the cost of financing Big Insurer Small Insurer Net Reins Capital Note: Small insurer costs multiplied by 10. No Re Cat All Re No Re Cat All Re Re Re 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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posted: | 8/29/2010 |

language: | English |

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