1998 CAS DFA Seminar Managing Risk in a Portfolio Context Reinsurance and Reserves Gary G. Venter Sedgwick Re Insurance Strategy, Inc. (INSTRAT) Investment and Reinsurance Risk Goal Manage investment and reinsurance risk simultaneously Test strategies by their impact on bottom line income probability distributions Create risk/return efficient frontiers from strategies tested Risk/Return Efficient Frontier @ probability = p Any strategy above or to the right of the efficient frontier provides less benefit than points on or below the efficient frontier. Inefficient options can be quickly recognized. By establishing an efficient frontier of options, one can discover and create new or hybrid solutions that provide greater benefit Measuring Risk Problems with standard deviation Inconsistent meaning among distributions Treats upside and downside risk the same Look at key individual percentiles Preference might be to increase mean return, reduce downside risk by giving up chance of big gain Risks to Income Investment performance Market results Need to liquidate to pay losses Underwriting result Current year Reserve development Reinsurance Modeling Requirements Model the risk elements Investment market, current u/w, develop-ment, reinsurance, cash flow, taxes, GAAP, statutory Get probability distribution of income for each strategy to find efficient frontier of strategies at each probability level Need to generate scenarios by probability, not just wide variety Modeling Investment Risk Yield curve Diffusion model Other assets Regression models Why a Diffusion Model? Rates are moving continuously Process generating movement of short-term rates also generates yield curve Can guarantee arbitrage-free yield curve movements in model Can calibrate to market - e.g., to bond options The CIR Model dr = a(b - r)dt + sr1/2dz. r is short-term rate b is reverting mean a is speed of reversion s is volatility measure What is z? z is standard Brownian motion Starts at zero After time t, z is normally distributed with mean 0 and variance t Notation sdz means that after time t, variance is s2t Thus for CIR sr1/2dz means that after time t, variance is s2rt Use this to simulate short intervals t Yield Curve under CIR Y(T) = A(T) + rB(T) for term T A(T) = -2(ab/s2T)lnC(T) – 2aby/s2 B(T) = [1 – C(T)]/yT C(T) = (1 + xyeT/x – xy)-1 x = [(a - )2 + 2s2]-1/2 y = (a - x)/2 This has a free parameter called the market price of risk Yield is a linear function of r for fixed T Testing - Yield Curve Model Should: Closely approximate current yield curve Produce patterns of changes in the short-term rate that match history Over longer simulations, produce distribu-tions of yield curves that, for any given short-term rate, match historical conditional distribution of yield curves Historical Yield Curve Distribution Measure shape by 1st and 2nd differences E.g., 1 year rate minus 3 month rate Look at historical shape measures as a function of r Yield Spread 3-Month to 1-Year 1-Year to 3-Year Yield Spread CIR Compared to Historical Andersen & Lund Model Comparison Andersen & Lund Comparison A&L + Variable Market Price of Risk Long spread matches historical Modeling Loss Development Need paid development for cash flow Need reserve development for liabilities One strategy: simulate paid diagonal and develop triangle to get new incurred Simulating whole diagonal - as opposed to just simulating sum - requires model of development process Diagonal Development Issues Is development proportional to emerged? Is development proportional to ultimate? Is it inflation sensitive? What is variance proportional to? Can you tell from data? Classifying Development Models Six yes-no questions Can test the questions using triangles Provides 64 classes of development models May be several models in each class Other class schemes possible Six Questions Development depends on emerged? Purely multiplicative development? Independent of diagonal effects? Stable parameters - e.g. factors? Normally distributed disturbances? Constant variance of disturbances? For each age - not proportional to anything E.g., All Yes new incrmnt = f*[prev emerged] + e e normal in 0 and s2 MLE of f is chain ladder (i.e., link ratio) estimate Alternatives Development proportional to ultimate - BF new incrmnt = f*ultimate + e Additive not multiplicative - Cape Cod new incrmnt = a + e Diagonals inflation sensitive - separation new incrmnt = f*[prev emerged]*(1+i) + e More Alternatives Factors change over time - smoothing Last 3 diagonals, exponential smoothing . . . Lognormal disturbances - take logs Disturbance proportional - weighted MLE Inverse weights to variance Proportional to emerged to date: weighted least squares takes ratios of column means Still More Combined models - e.g.: new incrmnt = a + f*ultimate*(1+i) + e Reduced parameters - e.g.: new incrmnt = (1+i)*ultimate/(1+j)k + e Which model does the data like? Fit models to the triangle Measure goodness of fit Statistical significance of parameters Sum of squared errors Penalize for extra parameters Divide sse by (obs - params)2 Testing a Triangle 0 to 1 is a constant Lag 1 vs. Lag 0 Losses 9000 8000 7000 6000 5000 4000 3000 2000 1000 0 0 1000 2000 3000 4000 5000 6000 Adjusted SSE’s for Incrementals SSE Model Prms Simulation Formula 157,902 CL 9 qw,d = fdcw,d + e 81,167 BF 18 qw,d = fdhw + e 75,409 CC 9 qw,d = fdh + e 52,360 BF-CC 9 qw,d = fdhw + e h5 - h9 same, h3=(h2+h4)/2, f1=f2, f6=f7, f8=f9 44,701 BF-CC+ 7 qw,d = fdhwgw+d + e 4 unique age factors, 2 diagonal, 1 acc. yr. Stability of Factors 2 nd t o 3 r d 5 - t er m m oving aver age 2 .4 2 .2 2 1 .8 1 .6 1 .4 1 .2 1 0 .8 Normal Residuals Is Disturbance Proportional? Errors as Function of Independent Var Check for Linearity of Fit 3000 2000 1000 0 -1000 -2000 -3000 -4000 0 1000 2000 3000 4000 5000 6000 Reserve Model Choice Affects Risk No Inflation on Reserves Short Medium Mean 3422 3443 1% 3085 2600 10% 3220 3024 90% 3626 3786 99% 3821 4096 Long Stocks+ 3470 3540 2182 1762 2801 2287 4117 4661 4784 6120 With Post-Event Inflation Short Medium Mean 3429 3438 1% 3021 2589 10% 3205 2972 90% 3635 3816 99% 3859 4289 Long Stocks+ 3538 3569 1899 1358 2848 2294 4242 4613 4879 6197 Reinsurance & Investment Scenarios Higher Retention Short Medium Mean 3422 3443 1% 3085 2600 10% 3220 3024 90% 3626 3786 99% 3821 4096 Long Stocks+ 3470 3540 2182 1762 2801 2287 4117 4661 4784 6120 Lower Retention Short Medium Mean 3227 3255 1% 3271 2884 10% 3351 3156 90% 3500 3749 99% 3618 3951 Long Stocks+ 3345 3473 2197 1773 2865 2564 4202 4909 4630 5642 Conclusions DFA can be used to jointly manage reinsurance and investment strategy Requires probabilistic generation of asset and underwriting scenarios Modeling issues Getting reserve model right necessary for relevant analysis Interest rates follow complex processes that require careful attention