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 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

Assignment #1 Lineland Life Insurance Company Case 2 - Claim does not occur Return = PV[Annuity] + PV of Capital

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 P increases when capital must be held longer.

Background - Capital Requirements Define Terms

Background - Capital Requirements Three Formulas #1 Probabililty of Ruin  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)  is determined by judgment of insurer management. Sensitive to amount of insolvency

Background - Capital Requirements Three Formulas #3 Standard Deviation Formula 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 C  T 

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 Expected Value Excel Formula

Gamma Distribution Mathematics Limited Expected Value (LEV) Function Variance Excel Formula

Assignment #2 Lineland Property Insurance Company Probability of Ruin E[X] = 10,000 F(12,472) = 0.99  Capital = 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  Capital = 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 Volatility of Net Worth How long Capital is held

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 E[  ] = 1 and Var[  ] = b

Assignment # 2´ Capital Requirements with Parameter Uncertainty

Probability of Ruin E[X] = 10,000 F U (14,443) = 0.99  Capital = 1.0% Level Cost of capital = 444

Assignment # 2´ Capital Requirements with Parameter Uncertainty

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 Total Capital  Double Marginal Capital  Triple + With Parameter Uncertainty

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.

Assignment #4 Flatland Casualty Insurance Company Outstanding Aggregate Loss Statistics 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.

Assignment #4 Flatland Casualty Insurance Company What is the cost of providing the capital? r = Rate of return needed to attract capital. i = Interest rate on invested capital C 0 = Capital needed at beginning of year 0. The cost of capital, R, satisfies:

Assignment #4 Given i = 6% and r = 10% What is the cost of providing the capital?

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. Compare variability before and after Is the risk reduction worth the cost? + about 9000 more

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 U se 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 DFA Call Paper Program Catastrophe Risk Securitization: Insurer and Investor Perspectives” by Glenn Meyers and John Kollar 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

Financing With Reinsurance and Catastrophe Options

Financing Without Reinsurance and Catastrophe Options

Differences in Costs

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

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