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A Day’s Work for New Dimensions an International Consulting Firm Glenn Meyers Insurance Services Office, Inc. CAS/ARIA Financial Risk Management Seminar DFA - Dynamic Financial Analysis • Coined by the CAS in 1994. • Best defined in terms of the problems it seeks to solve. – How much capital does an insurer need? – For how much time is the capital needed? – What decisions does an insurer make to provide the greatest return on its capital? • Underwriting • Asset management (Include hedges) Outline of Talk • Multi-dimensional aspects of insurer capital management • Provide simple (perhaps artificial) examples focusing on particular dimensions. – Short and long tailed lines – Catastrophe options and reinsurance • Describe (but not solve) a multi-dimensional insurer problem in capital management. • Compare approach with efficient frontier methods. Assignment #1 Lineland Life Insurance Company • • • • Writes one life insurance policy Face value $1 t is the term of the policy Mortality assumptions – Probability of death in [0,t] = q – Uniform distribution of deaths within [0,t] Assignment #1 Lineland Life Insurance Company • Investors provide $1 of capital. • Capital is invested at rate I compounded continuously. • In return for exposing the capital to loss they demand a return of R compounded continuously. R > I • Find minimum premium, P, it must get. Assignment #1 Lineland Life Insurance Company Case 1 - Claim occurs at time T The return is a continuous annuity of I R T 1 e PV withclaim I R E PV with claim R t 1 e 0 I R z F G H q d t I 1 e R t q 1 R t R I J K Assignment #1 Lineland Life Insurance Company Case 2 - Claim does not occur Return = PV[Annuity] + PV of Capital R t 1 e PV without claim I R e R t F 1 e E PV without claim a 1 qf G H R t I R e R t I J K Assignment #1 Lineland Life Insurance Company • Receives P immediately. • Receives annuity until claim occurs or the term ends. 1 = P + E[PV with Claim] + E[PV without Claim] Assignment #1 Lineland Life Insurance Company I R q 6% 10% 0.100 t P P-q 1 2 3 4 5 6 7 0.131 0.160 0.185 0.208 0.229 0.248 0.264 0.031 0.060 0.085 0.108 0.129 0.148 0.164 P increases when capital must be held longer. Background - Capital Requirements Define Terms X = Random Insurer Loss F( x) = Pr{ X x} f ( x) = F ( x) af LEV L z a1 F(L)f L x f ( x ) dx L 0 = Standard Deviation of X C = Required Insurer Capital Background - Capital Requirements Three Formulas #1 Probabililty of Ruin F(C E[ X ]) 1 is determined by judgment of insurer management. Insurer management always knows what the rating agencies - NAIC, Best, S&P think they should have. Value at Risk -- VaR = C+E[X] Background - Capital Requirements Three Formulas #2 Expected Policyholder Deficit (EPD) a f LEV C E[ x] 1 E[ X] is determined by judgment of insurer management. Sensitive to amount of insolvency Background - Capital Requirements Three Formulas #3 Standard Deviation Formula C T T is determined by judgment of insurer management. Normal approximation to ruin formula, but you can use this formula as is. Easiest to work with Assignment #2 Lineland Property Insurance Company • Losses have a Gamma(100,100) distribution. • Claims settle quickly – Time value of money is not an issue. • Investors expect 10% ROE. • Find the Cost of Capital. Gamma Distribution Mathematics Cumulative Distribution Function a f F( x) ; x / a f GammaDist x, , , TRUE Expected Value a f af 1 EX Excel Formula Gamma Distribution Mathematics Limited Expected Value (LEV) Function a f a af 1 LEV L 1;L / L 1 1;L / f b a fg a f xa 1 GammaDist( x, , , TRUE)f exp GammLn( 1) GammaLn() GammaDist( x, 1, , TRUE) Variance E X2 a f af 2 2 2 1 2 Var X E X 2 E X a f 2 2 Excel Formula Assignment #2 Lineland Property Insurance Company Probability of Ruin • E[X] = 10,000 • F(12,472) = 0.99 Capital = 2,472 @ 1.0% Level • Cost of capital = 247 Assignment #2 Lineland Property Insurance Company Expected Policyholder Deficit • E[X] = 10,000 • LEV[12,091] = 9,990 LEV[12,091] EPD 1 0.0010 E[ X] Capital = 2,091 @ 0.10% Level • Cost of Capital = 209 Assignment #2 Lineland Property Insurance Company Standard Deviation • E[X] = 10,000 • Std[X] = 1000 • Select T = 2.33 Capital = 2,330 • Cost of Capital = 233 Cost of Capital Depends Upon: Economic Environment e.g. interest rates How long Capital is held Volatility of Net Worth Parameter Uncertainty for Gamma(,) • Let be a random variable – E[] = 1 – Var[] = b • Select at random • Conditional distribution given Gamma(,) Parameter Uncertainty for Gamma(,) A simple, but nontrivial example 1 1 3b, 2 1, 3 1 3b k p k p k p Pr 1 Pr 3 1 / 6 and Pr 2 2 / 3 E[] = 1 and Var[] = b Assignment # 2´ Capital Requirements with Parameter Uncertainty b 0.02 1 100 1 75.51 2 100 2 100.00 3 100 2 124.49 a af f a f a , x / 1 2 , x / 2 , x / 3 FU x 6 3 6 f Assignment # 2´ Capital Requirements with Parameter Uncertainty Probability of Ruin • E[X] = 10,000 • FU(14,443) = 0.99 Capital = 14,443 @ 1.0% Level • Cost of capital = 444 Assignment # 2´ Capital Requirements with Parameter Uncertainty Probability of Ruin Capital Capital Threshold w/o PU with PU 1.0% 2,472 4,443 Expected Policyholder Deficit Capital Capital Threshold w/o PU with PU 0.10% 2,091 4,129 Standard Deviation Capital Capital Threshold w/o PU with PU 2.33 2,330 4,049 Assignment #3 Lineland Property Insurance Company Considers Renewing a Policy • The renewal business has a Gamma(100,1) loss distribution. • Lineland has a Gamma(100,99) loss distribution without the renewal. Property of the Gamma Distribution • Lineland has a Gamma(100,100) loss distribution with the renewal. This Property Assumes Independence Assignment #3 Lineland Property Insurance Company Considers Renewing a Policy • What is the marginal capital needed for the renewal business? • Calculate capital needed without the business. • Calculate capital needed with the business. • Marginal capital is the difference. Assignment #3´ Find Marginal Capital Assuming Parameter Uncertainty • The random variable affects all business (including renewal) simultaneously. • The renewal’s parameter changes at the same time as the for the remaining business. • The renewal’s losses are correlated with the rest of the losses. In case you are interested -- = 0.195 Assignment #3 and #3´ Results Probability of Ruin @ 1.0% C b C-R C 0.00 2,460.59 2,472.26 11.67 0.02 4,409.12 4,443.25 34.13 Expected Policyholder Deficit @0.1% C b C-R C 0.00 2,083.58 2,091.11 7.53 0.02 4,100.04 4,129.19 29.15 Standard Deviation @ 2.33 C b C-R C 0.00 2,318.32 2,330.00 11.68 0.02 4,015.75 4,049.11 33.66 With Parameter Uncertainty Total Capital Double Marginal Capital Triple + How do you use the marginal cost of capital? • Allocate the total cost of capital in proportion to the marginal cost of capital. – No consensus among actuaries yet. • Add the allocated cost of capital to the expected loss and expense to see if you can make money at the “going market premium.” • Can be done at individual insured level, or the line of business level. Assignment #4 Flatland Casualty Insurance Company • Claim count distribution is negative binomial - by settlement lag. • Claim severity distribution is mixed exponential - by settlement lag. Name Lag 0 Lag 1 Lag 2 Summary Statistics by Settlement Lag E[Count] Std[Count] E[Severity] Std[Severity] 1,200 244 40,349 160,219 600 123 59,798 194,452 300 63 79,248 221,804 Assignment #4 Flatland Casualty Insurance Company Outstanding Aggregate Loss Statistics Lags 0-2 Lags 1-2 Lag 2 Aggregate Loss Statistics for OS Losses E[Loss] 99th Pct EPD = 0.1% Std Dev 108,071,943 158,505,938 155,520,667 19,835,337 59,653,299 91,387,990 90,579,282 12,265,291 23,774,319 40,533,916 41,250,295 6,283,149 The aggregate loss model included parameter uncertainty affecting all claim count distributions simultaneously. (g =.02 - analogous to b =.02 above.) Assignment #4 Flatland Casualty Insurance Company Capital is released over time as losses are paid. Required Capital for OS Losses Pr{Ruin}@1.0% [email protected]% Std Dev x 2.33 Lags 0-2 50,433,995 47,448,724 46,216,335 Lags 1-2 31,734,691 30,925,983 28,578,127 Lag 2 16,759,597 17,475,976 14,639,737 Assignment #4 Flatland Casualty Insurance Company What is the cost of providing the capital? i = Interest rate on invested capital r = Rate of return needed to attract capital. C0 = Capital needed at beginning of year 0. Re lease t C t 1 (1 i) C t The cost of capital, R, satisfies: 3 Re lease t C0 R t r t 1 1 a f Assignment #4 Given i = 6% and r = 10% What is the cost of providing the capital? Required Capital for OS Losses Pr{Ruin}@1.0% [email protected]% Std Dev x 2.33 Lags 0-2 50,433,995 47,448,724 46,216,335 Lags 1-2 31,734,691 30,925,983 28,578,127 Lag 2 16,759,597 17,475,976 14,639,737 Time t 1 2 3 Expected Return at Time t 21,725,344 19,369,665 20,411,187 16,879,175 15,305,566 15,653,078 17,765,172 18,524,534 15,518,122 Cost of Capital 3,386,713 3,272,953 3,065,288 Asset Management Reinsurance and Catastrophe Options • “Value will be determined not by the ability of an [insurance] enterprise to accumulate capital and sit on it. • Rather it will be determined by a company’s franchise with its customers and its ability to originate risk. • In this scenario the capital markets become the more efficient warehouse of [insurance] risk.” Asset Management Reinsurance and Catastrophe Options • Reduce the cost of financing insurance – Expected insurer costs – Cost of Capital – Cost of Capital Substitutes • Reinsurance • Contracts on a catastrophe index • Find the right mix of capital and capital substitutes Quantifying the Cost of Capital • We use the “easy” formula Cost of Capital = K T Where: = Standard deviation of total loss T = Factor reflecting risk aversion K = Rate of return needed to attract capital Quantifying Basis Risk Ran RMS cat model through insurers and index. Event 1 2 3 4 5 6 7 8 9 10 Index Value 100.0 89.04 87.56 83.48 83.20 82.15 80.95 80.55 79.19 77.48 Event Probability 0.00000121 0.00000121 0.00000181 0.00000702 0.00000702 0.00000466 0.00000791 0.00005060 0.00000702 0.00000181 Max Event Contract Direct Reinsurance Event Loss Probability Value Insurer Loss Recovery Given Max 0.00000121 1,125,200,000 1,212,550,269 16,000,000 71,350,269 0.00000121 1,021,700,000 1,509,161,589 16,000,000 471,461,589 0.00000181 1,021,700,000 1,303,694,653 16,000,000 265,994,653 0.00000702 939,300,000 761,956,629 16,000,000 (193,343,371) 0.00000702 939,300,000 734,137,782 16,000,000 (221,162,218) 0.00000466 939,300,000 735,660,852 16,000,000 (219,639,148) 0.00000791 939,300,000 1,004,861,128 16,000,000 49,561,128 0.00005060 939,300,000 1,071,076,934 16,000,000 115,776,934 0.00000702 856,900,000 688,269,904 16,000,000 (184,630,096) 0.00000181 856,900,000 1,652,933,116 16,000,000 780,033,116 + about 9000 more • Compare variability before and after • Is the risk reduction worth the cost? Minimize Sum of Cost Elements • Insurer Capital Cost of Capital = K T (Net Losses) • Reinsurance Transaction Cost + Expected Cost • Cat index contracts Transaction Cost + Expected Cost Use cat model results to back out transaction costs. References Missing transaction costs are in the first paper. • “The Cost of Financing Catastrophe Insurance” by Glenn Meyers and John Kollar 1998 DFA Call Paper Program • Catastrophe Risk Securitization: Insurer and Investor Perspectives” by Glenn Meyers and John Kollar - 1999 CAS Spring Meeting Call Paper Program Assignment #5 Analyze Three Insurers • Insurer #1 - A medium national insurer Highly correlated with the index • Insurer #2 - A large national insurer Moderately correlated with the index • Insurer #3 - A small regional insurer Slightly correlated with the index Search for Best Strategy to Minimize Cost of Financing Insurance • Search for the combination of index and reinsurance purchases that minimizes total cost of providing insurance. Questions • How many index contracts at each strike price? • What layer of reinsurance? Results of Search Contract Range 5-20 25-40 45-55 60-70 75-85 90-100 Number of Index Contracts Insurer #1 Insurer #2 Insurer #3 47,400 93,100 0 74,400 118,100 6,300 59,500 67,900 0 47,600 28,600 0 81,400 545,100 0 37,200 634,800 0 Reinsurance Retention 73,000,000 457,000,000 54,000,000 Limit 13,000,000 36,000,000 105,000,000 Financing With Reinsurance and Catastrophe Options Expected Net Loss Cost of Capital Cost of Reinsurance Cost of Catastrophe Options Cost of Financing Insurance Insurer #1 Insurer #2 Insurer #3 16,315,629 62,086,995 1,464,410 53,470,927 143,662,761 12,914,922 2,088,287 1,848,530 1,726,342 22,252,015 42,409,101 249,427 94,126,858 250,007,387 16,355,100 Financing Without Reinsurance and Catastrophe Options Expected Net Loss Cost of Capital Cost of Reinsurance Cost of Catastrophe Options Cost of Financing Insurance Insurer #1 Insurer #2 Insurer #3 34,839,348 95,417,229 2,385,629 68,768,384 166,962,499 15,356,683 0 0 0 0 0 0 103,607,732 262,379,728 17,742,312 Differences in Costs Without Reins & Options With Reins & Options Difference Pct Difference Insurer #1 Insurer #2 Insurer #3 103,607,732 262,379,728 17,742,312 94,126,858 250,007,387 16,355,100 9,480,874 12,372,341 1,387,212 9.2% 4.7% 7.8% Assignment #6 Spaceland Property and Casualty • Short tailed property exposure – Include catastrophe exposure • Long tailed casualty exposure – Include unsettled claims from prior years • Capital Management Questions – Catastrophe options/reinsurance? – Casualty reinsurance? Assignment #6 Spaceland Property and Casualty Underwriting Management Decisions • Allocate the cost of capital to the lines of insurance - in proportion to the marginal cost of capital. • Allocate the cost of reinsurance and/or catastrophe options to the lines of insurance - in proportion to the marginal costs. Assignment #6 Information and Technology Requirements • An Aggregate Loss Model • Size of loss distributions by settlement lag • Correlation structure between lines of insurance • A catastrophe model • Exposure underlying catastrophe index References • “Underwriting Risk” by Glenn Meyers – 1999 CARe Call Paper Program • “Estimating Between Line Correlations Generated by Parameter Uncertainty” by Glenn Meyers – 1999 DFA Call Paper Program • These papers should be eventually available at CAS web site. • Currently available on my personal web site http://www.crimcalc.com/glenn.htm Relationship Between this Capital Cost Allocation Method and the Efficient Frontier Methods • They are equivalent – (loosely speaking) • I say “loosely speaking” because: – There is a lot of loose speaking about the meaning of “risk.” – There is a lot of loose speaking about the meaning of “allocated cost of capital.” Relationship Between the Capital Cost Allocation Methods and the Efficient Frontier Methods • The intuition • Allocated cost of capital depends upon marginal risk. • Making decisions that yield a higher return on marginal capital moves you closer to the efficient frontier. Relationship Between the Capital Cost Allocation Methods and the Efficient Frontier Methods • Some History from PCAS – Kreps: Risk loads from marginal capital requirements, 1990 – Meyers: Risk loads from efficient frontiers (mimic CAPM), 1991 – Heckman: Kreps and Meyers are equivalent, 1993 (CAS Forum) – Meyers: Cat risk loads from marginal capital requirements, 1997 Relationship Between the Capital Cost Allocation Methods and the Efficient Frontier Methods If two are equivalent, why did I switch? • Easier to explain • Easier to extend – To different measures of risk – To different capital holding times