The Actuary and Enterprise Risk Management www.guycarp.com
What is ERM? Involves a broad identification, assessment and control of risk Tries to incorporate all risks facing the company Allows for enterprise wide measures of risk Consistent risk quantification across the company Gives management perspective on the impact of strategic alternatives How do we grow lines profitability Acquisitions Reinsurance Etc. 1
Involves a Broad Identification of Risks Property-Casualty Insurance Company Enterprise Risk Insurance Hazard Financial (Asset) Operational & Strategic Underwriting Credit People / Processes / Systems Accumulation / Cat Market External Events Reserve Deterioration Liquidity Business Strategy 2
Works Best if Embedded in the Corporate Culture By articulating corporate strategy in terms of risk metrics, corporate risk culture is strengthened Disciplined Decision Making Design Successful Business Plans Corporate Alignment Transparency and Awareness with Boards 3
Organizational Benefits of ERM Risk Awareness The internal risk model is the hub of key decision processes It represents the Official Risk Record Banking term (Goldman Sachs book) On par with general ledger as official financial record of the firm Formally recording the corporation s risk opinion and position Transparency Documented support for risk opinions and decisions Increased buy in Objectivity Everyone can see the rationale Unearthing the intuitions so they can be discussed and vetted Capturing intellectual assets of the company Consistency Open communication on what the real risks are to the entity 4
Elements of ERM Quantitative model of company risks Big actuarial project to build this Risk measures that compare company capital to risk and can compare risks from different operations (investment, underwriting, ) and different businesses (marine, motor ) Also major actuarial role Establishing risk control and monitoring mechanisms, including limits written, reinsurance, investment controls Evaluation of risk and opportunity of every strategic and operational aspect of company management Will focus on first two 5
Good ERM Can Reduce Capital Needs ERM is a key driver for S&P rating upgrades (Rating agency needs rationale for rating change!) The ratings on Canadian based Manufacturers Life Insurance Co., (AAA/Stable/A 1+) were raised to 'AAA' from 'AA+' in November 2006. One of the factors behind the upgrade was the "Excellent" evaluation on the group's ERM. The ratings on Munich Reinsurance Co. (AA /Stable/ ) was raised to 'AA ' from 'A+' in December 2006, with the "Strong" evaluation on its ERM cited as a driver. The ratings on Zurich Financial Services (ZFS) was raised to 'AA ' from 'A+' in June 2007. One of the rationale was positive impetus from enterprise risk management program (ERM). ZFS' ERM program has improved to a strong level, providing management with an effective tool to maintain its risks at a level consistent with its risk tolerance. AM Best changed outlook to positive Dec 2007. Japan's Major Banks were upgraded in June 2007. The upgrades reflect the banks' "Adequate" ERM, stable improvement in asset quality and capitalization backed by the strong economy in Japan, and the diversification of revenue sources. 6
A.M. Best View of ERM Impact of ERM on BCAR Requirements Companies with strong risk management and low relative volatility will be allowed to maintain BCAR levels below the guideline Strong risk management: strong, traditional riskmanagement fundamentals, relative to the insurer s risk profile, in each of the five key risk types, and sound financial flexibility. Corollary is true for companies with weak risk management and/or high volatility Case by Case Basis 7
History of ERM Started with banking regulation Banks were not quantifying risk, going broke Basic risk quantification modeling developed Can t say this has been an unqualified success, as risks have become more complex Expanded to insurance regulation Attempts for application to strategic management Manufacturing looking at supply chains, etc. Financial institutions model financial risk Basically shifting from policyholder viewpoint to shareholder viewpoint 8
What ERM Modeling Does and What It Does Not Yet Do Does Quantify risk level of capital Quantify riskiness of business units Does not yet Show optimal level of capital Would like to be able to say what capital maximizes franchise value of the firm Show which business units are most profitable on a risk basis Comparing profit to allocated capital does not necessarily tell which units are adding most value to the firm 9
Model Building
Component Models Underwriting risk Generate scenarios using loss frequency, severity, parameter uncertainty, reserves, correlation by line of business Investment risk Investment outcome scenario generator by asset class Operational risk Losses from running a company personnel, facilities, breach of guidelines, etc. Some work on modeling done, but also need to identify and control operational risks Strategic risk like strategy for underwriting cycle 11
Some Actuarial Modeling Issues 1. Parameter risk and event risk Basic models do not capture enough risk sources 2. Correlation Not just how much correlation but which losses are correlated 3. Reserves Modeling risk of reserves more difficult than level of reserves 4. Assets Usually a major risk to returns but often not carefully modeled 5. The Cycle New area for modeling 12
1. Parameter Risk Main problem Basic risk models do not include enough risk sources Generally they say that the number of loss are random and the size of each loss is random so you can model an insurance company as a random sum of random losses But not all loss risk is from known frequency and severity fluctuations Only know distributions from data and data is always a sample Distributions change over time Including these elements gives a much more realistic risk model Parameter risk includes estimation risk, projection risk, and event risk It is a systematic risk it does not reduce by adding volume For large companies this could be the largest risk element, comparable to cat risk before reinsurance and greater than cat risk after reinsurance 13
Projection Risk Change in risk conditions from recent past Makes data unreliable for projecting changes In part due to uncertain trend Inflation, court cost trends, etc. can all change Includes change in exposures Change in work processes and driving patterns New types of fraud become more prevalent Etc. 14
Measuring Risk from Uncertain Trend Historical points could be known imprecisely Trend line does not fit perfectly All contribute to uncertainty around projected trend 15
Impact of Projection Risk J on Total Loss CV (CV is ratio of standard deviation to mean) (J is a projection risk factor with mean 1) CV(J) E(N): 2,000 20,000 200,000 0.05 16.6% 7.1% 5.2% 0.03 16.1% 5.8% 3.4% 0.01 15.8% 5.1% 1.9% 0.00 15.8% 5.0% 1.6% One risk measure is CV coefficient of variation = sd / mean This risk for total losses is less when loss frequency is higher more stability with greater frequency Adding projection risk to the model shows that large companies are still more stable, but not as much so as in simpler model 16
Translating CV Effect to Loss Ratio Risk E(LR)=65 CV(J)=0.05 E(N)=2,000 20,000 200,000 90th 79.2 71.0 69.4 95th 84.1 72.8 70.8 99th 94.1 76.4 73.3 CV(J)=0 90th 78.5 69.2 66.3 95th 83.1 70.5 66.7 99th 92.5 72.9 67.4 For fairly large company basic model shows little risk to loss ratio Adding projection risk shows a more typical risk level 17
Estimation Risk Data is never enough to know true probabilities for frequency and severity Statistical methods quantify how far off estimated parameters can be from true More data and better fits both reduce this risk but never gone With MLE this can be estimated by the Fisher information matrix, or by Bayesian methods 18
Estimation Risk Pareto Example typical 2 parameter severity distribution fit by maximum likelihood fit gives most likely distribution which has the highest probability possible distributions around it are nearly as likely and should be included in risk model to realistically capture spread of results 19
Other Parameter Risk Events One or several states decide to get tough on insurers Consumer groups decide company has been unfair and wins in court Court rules that repairs must use replacement parts from original car makers only Mold is suddenly a loss cause Biggest writer in market decides it needs to increase market share and reduce surplus so it lowers rates and others follow Rating downgrade These are major risks and can dwarf others. Hard to predict in future, but is an ongoing risk source and modeling needed. Might be considered part of operational risk or strategic risk 20
2. Correlation Issues Main problem It s not enough to recognize correlation also need to correlate right events Correlation is often stronger for large events Multi line losses in large events Modeled by copula methods: statistical models that include which events get correlated Quantifying correlation Degree of correlation Part of spectrum correlated Measure this, model it, or even use informed judgment, but don t ignore 21
Modeling via Copulas Association is put directly on the loss probabilities Inverse map probabilities to associate the losses Can specify where correlation takes place in the probability range 22
Formal Rules Bivariate Case F(x,y) = C(F X (x),f Y (y)) Joint distribution is copula evaluated at the marginal distributions Expresses joint distribution as inter dependency applied to the individual distributions C(u,v) = F(F X 1 (u),f Y 1 (v)) u and v are unit uniforms, F maps R 2 to [0,1] Shows that any bivariate distribution can be expressed via a copula F Y X (y) = C 1 (F X (x),f Y (y)) Derivative of the copula is the conditional distribution E.g., C(u,v) = uv, C 1 (u,v) = v = Pr(V<v U=u) So independence copula 23
Normal Copula Gives Weak Correlation on Large Losses Basic correlation method in many models 24
Gumbel Copula Correlates Large Losses More Strongly A more realistic alternative in many cases 25
Heavy Right Tail Copula Even More So Designed specifically for modeling insurance loss risk Fit data better in live test 26
Auto and Fire Claims in French Windstorms Les 736 Tempêtes ayant un coût supérieur à 1000 Francs dans les deux branches. 100 000 10 000 Auto en KF 1 000 100 10 1 1 10 100 1 000 10 000 100 000 1 000 MR en KF 000 10 000 000 27
MLE Estimates of Copulas Gumbel Normale HRT Frank Clayton Paramètre 1,323 0,378 1,445 2,318 3,378 Log Vraisemblance 77,223 55,428 84,070 50,330 16,447 τ de Kendall 0,244 0,247 0,257 0,245 0,129 28
Clues that Modeler is Using Normal Copula Uses terms like @Risk An Excel add in that uses the normal copula Cholesky decomposition A technical step in simulating from the normal copula 29
Extending to Multi Variate Case Single parameter not enough all variates would have same correlation You would like to have at least a parameter for each pair of variates to determine the strength of their dependency, and one overall for tail strength The t copula has this minimum set With this minimum you can control all correlations but all tail strengths are the same There are generalizations where you have a parameter for each variate and their interactions determine the tail strength for two variates Reduces to t case if two variates have the same parameter Grouped t and individualized t copulas work this way 30
Loss Scenario Generation for t Correlated Lines Generate a multi variate normal loss vector with the same correlation matrix. Divide each loss by (y/n) 0.5 where y is a number simulated from a chi squared distribution with n degrees of freedom. This gives a t distributed loss vector. Only difference in individuated t is you use a different degrees of freedom for each variate, but the same chi squared probability Apply the t distribution F n to each loss to get the probability vector generated for the t copula. The inverse severity distributions for each line can then be applied to get the by line losses for the scenario. 31
3. Reserve Issues Loss reserving runoff risk Quantifying risk of adverse loss development requires a model Many modelers assume just one model Better to test several models against actual data and see which explain it best E.g., see 1998 PCAS Testing the Assumptions of Age to Age Factors Correlation in runoff Changing conditions can affect several accident years at once Induces correlation in the runoff Even if reserving model does not measure this, it can be added to projected runoff scenarios 32
Zone Rated Development Accident Development Factors Year 1-2 2-3 3-4 4-5 5-6 6-7 7-8 8-9 9-10 10-11 11-12 12-13 1984 2.672 1.578 1.214 1.058 1.044 1.011 1.018 1.011 1.002 1.001 1.000 1.000 1985 2.581 1.505 1.224 1.070 1.024 1.015 1.004 1.022 1.005 1.000 1.013 1986 2.853 1.397 1.355 1.126 1.052 1.016 1.007 1.000 1.001 1.010 1987 2.514 1.595 1.215 1.126 1.063 1.030 1.006 1.017 1.000 1988 2.794 1.501 1.264 1.135 1.044 1.009 1.003 1.000 1989 2.613 1.464 1.230 1.072 1.019 1.017 1.022 1990 2.471 1.601 1.193 1.120 1.021 1.010 1991 2.690 1.469 1.304 1.115 1.023 1992 2.742 1.715 1.165 1.159 1993 2.679 1.440 1.195 1994 2.605 1.498 1995 2.580 33
Resulting Runoff Risk CV s of ultimate losses by accident year: 0.073 0.071 0.049 0.035 0.022 0.019 0.020 0.016 0.013 0.013 0.011 Assuming lognormal, the 99th percentile loss is above the mean by: 18.3% 17.6% 11.9% 8.3% 5.2% 4.5% 4.7% 3.8% 3.0% 3.1% 2.6% More developed years are more stable so need less capital But this is from one model only Can test fits of alternative models 34
Testing and Simulating Models Live Data Example SSE Model Params Simulation Formula 157,902 CL 9 q w,d = f d c w,d + e 81,167 BF 18 q w,d = f d h w + e 75,409 CC 9 q w,d = f d h + e 52,360 BF-CC 9 q w,d = f d h w + e 44,701 BF-CC + 7 q w,d = f d h w g w+d + e Some models fit better with fewer parameters Simulation and so development risk depends on model Two papers from 2007 Astin Colloquium look at reducing parameters without hurting fit, and using different distributions for the residuals 35
4. Asset Issues Asset risk can be significant for insurers so care in modeling is important Arbitrage free models No reward without some risk Probabilistic reality Modeled scenarios consistent with historical patterns Can use to combine asset and underwriting risk 36
Why No Arbitrage Is Important Key element of modern financial analysis Part of getting right distribution of scenarios Having arbitrage possibilities in scenario set distorts any optimization towards the arbitrage strategies 37
Testing Scenario Generators Bond models complex but basically understood Short term rate is highly auto regressive: correlated with previous values Also mean reverting: keeps coming back to current average level Current average level due to current conditions but it tends to gravitate over time to long term average level Volatility also changes over time but also reverts to a mean Volatility higher when rate is higher Longer term rates come from short term rate plus market price of risk, which also varies over time E.g., see 2004 ASTIN Bulletin Testing Distributions of Stochastically Generated Yield Curves Equity models still improving Basic models like Black Scholes do not have enough volatility Levi models with more short term volatility have too much long term volatility Several model approaches being tried to address these issues 38
5. Modeling the Cycle Not a fixed period Usually hard markets come from loss in industry capital Major event Asset drops Pricing gradually deteriorates Depends on capital in the industry Need models for price elasticity retention of business How fast can business grow in hard market if stay out in soft market? 39
Summary of Modeling Issues Parameter risk is a key issue for large companies Correlation should incorporate tail links to get true overall loss risk Reserve risk requires tests of alternative models to actual data Asset models should be arbitrage free and distributionally representative of history Modeling the cycle is an emerging discipline Getting the modeling right takes care and expertise, and is subject to many pitfalls 40
Applications to Capital Risk and Lines of Business
Capital Where We ve Been Capital = 1/3 rd of premium As reserves became more important, maybe Capital = ¼of liabilities, including reserves and UEPR But not all liabilities have equal risk, so RBC tried: Capital = sum of factors times income statement and balance sheet items Problems These are only very rough measures of risk and capital need Resulting capital not sensitive to many risk variables For instance, increasing adequacy of premium or reserves typically increases the RBC stated capital need 42
ERM Initial Attempt at Capital Analysis Capital = enough to make one year probability of ruin small enough Often called economic capital Advantages Consistent probabilistically across firms Low enough in information content that public disclosure will not help competitors Good criterion for regulatory measure 43
Disadvantages of Economic Capital Arbitrary choice of probability Only one way to measure risk Inaccuracy of models in extremes Not the capital a company needs in practice Want to optimize value Can be affected by perceptual issues, competitors, quality of risk management, reputation of management 44
What ERM Does Well: Quantifies Risk Level of Actual Capital Initial attempt: probability of default Problem: models not very accurate at extreme probabilities What can be done: Capital at multiples of various risk measures Capital = (perhaps) 4 standard deviations of earnings 3 times 1 in 100 TVaR 4 times 1 in 100 cat occurrence 1.5 times 2 1 in 100 aggregate loss years in a row Useful for comparisons of how different strategies that change risk can be imputed to change capital needs Does not tell you how much capital is optimal 45
Live Examples Capital = Multiple of probability of loss at quantifiable levels Swiss Re 12/31/2006: Times 99% VaR, 99.5% VaR and 99% TVaR Capital is: 3.6x 2.9x 2.4x Year over year comparisons in annual report Munich Re: Base capital = 2 aggregate 1 in 100 years in a row; actual capital is a multiple of base maybe 1.5x Endurance: Maximum capital loss of 25% at 1 in 100 event, or economic capital is 4 times the 1 in 100 loss ERM moves us from ¼of liabilities to 4 times 1 in 100 loss level A real improvement in probabilistic quantification 46
Optimal Capital Issues Basic You need enough to keep them buying, i.e., enough to support growth Can be affected by perceptual issues, competitors, reputation of management Reputation important Role of risk management, stability, etc. Competitive analysis What is capital level of competition? What sectors of business to compete for? How much does capital affect winning or losing quoted for business? Might have to talk to underwriters 47
What Might Be Done in Optimal Capital: Firm Value Models Find capital to optimize franchise value: market book Model value of firm as the expected present value of future cash flows to shareholders (at least since de Finetti in 1957) Look for capital level that maximizes value capital in this sense That is really finding optimum dividend policy Dividend paid earlier has higher present value But paying it could increase probability of ruin, ending future dividends Part of general theory of stochastic control You have a stochastic process (total capital) which you can control to some extent (paying dividends) and seek optimum strategy 48
Actuarial Progress in Stochastic Control of Insurer Capital Gerber and Shiu in NAAJ 2006 assume loss process is compound Poisson and severity is a mixture of exponentials, and find the optimum strategy Turns out to be a barrier strategy Pay out all capital over a certain level in dividends and none below it Another possible control is ceding reinsurance Can seek optimal strategy for reinsurance and capital Bather 1969 addressed this for proportional reinsurance Asmusson et al. 2000 try excess reinsurance Pricing rule for excess cover becomes an issue to address Costly reinsurance can be valuable to stave off ruin 49
Enter Finance De Finetti was in late 1950 s where finance world (MM) was saying risk transfer is not worthwhile One assumption of that is that external finance is always available and not costly Actuarial literature did not consider refinancing, so basically had infinite cost of external capital Peura 2003 Univ Helsinki thesis maybe first exception Froot and others introduced costly but possible external finance in the financial literature like new shares below latest price of existing Major adding costly external finance to actuarial approach 50
Traditional Risk Measurement for Business Lines Premium volume Exposures in force Limits in force Number of policies PML These don t compare well across businesses Not really probabilistically based 51
ERM Risk Measures Value at Risk (VaR) Pr(Y>VaR α ) = 1 α Tail Value at Risk (TVaR) E[Y Y>VaR α ] = TVaR α Both give probabilistically consistent ways of comparing units Both are tail measures Ignore risk that is probably important, in that you would charge for it Lower α s probably better, like 85% or even 60% 52
Issues with VaR and TVaR Sum of VaR s can be greater than VaR of sum Pool of risks each with loss probability of ½% each has VaR 0.99 = 0, so sum = 0, but VaR 0.99 of pool > 0 Actually not so critical in practice TVaR is linear in large losses, which is opposed to usual risk attitudes 53
Alternative Risk Measures Semi standard deviation (adverse deviations only) E[Ye cy/ey ] gives weight to tail but measures all risk Mean under transformed probabilities Gives more weight to tail but all risk included Financial value measures like CAPM and Black Scholes can be put in this form Useful for risk measure to give value of risk taken Can do TVaR with transformed probabilities No longer linear in large losses 54
Example of Transformed Mean Esscher transform for compound Poisson Severity density g(y) transformed to: g*(y) = g(y)e cy/ey /Ee cy/ey Frequency: λ* = λee cy/ey Constant c chosen to give overall load In one test, this fit well to reinsurance prices Is minimum entropy martingale transform Risk measure: transformed mean mean 55
Allocating Risk Allocating capital usually done by allocating risk Purpose is usually to get risk adjusted returns by business unit Typical approach: proportional allocation Measure risk for each unit and for firm, and allocate firm risk in proportion Problems with proportional allocation: Does not show the contribution of each unit to the risk of the firm Probably will not show marginal impact of change in unit on change in firm risk Growing units with higher return on risk will not necessarily increase return on risk for the whole firm 56
Contribution of Units to Firm Use co measures Prototype: sum of covariances of units with firm is the variance of the firm Co TVaR α for unit X j : E[X j Y>VaR α ] Shows contribution of X j to TVaR of firm Can be defined for any risk measure that can be expressed as a conditional expected value 57
Riskiness of Business Units Real issue is what is the contribution of the unit to the riskiness of the firm For homogeneous risk measures (i.e., r[ay] = a r[y]) this is done by Euler s Theorem Total risk measure for company is a multivariate function of the losses of the business units Euler: derivatives of risk measure with respect to units volumes sum up to entire risk measure Derivative is marginal impact of unit s volume of business on the entire company risk measure So can allocate by marginal impact and still sum to total Venter, Major, Kreps ASTIN 2006 do the derivatives for many risk measures 58
Details For risk measure ρ, company variable Y = sum of units X j r(x j ) = lim ε 0 [ρ(y) ρ(y εx j )]/ε. If ρ(y) = StD(Y), r(x j ) = Cov(X j,y)/std(y) Co TVaR is also marginal as is co VaR 59
Properties of Marginal Decomposition Better than separate individual unit measures For capital maintains pricing at the margin Growing units with higher profit/risk will increase profit/risk for the firm But only if growth is proportional, like reducing quota share or increasing shares of business written Approximation otherwise unless risk unit is transformed mean, then exact 60
Even so, Not Ideal Way to Risk Adjust Profits Arbitrary and artificial Not likely to reflect risk pricing principles Risk adjusting profit by allocating capital comes down to quantifying the value of the risk really increase in firm value Could work if risk pricing were the risk measure used in allocation, but allocation then not really needed A good risk pricing theory is needed to do it right A 1 st approximation is RTVaR risk adjust TVaR Conditional mean plus a percent of conditional standard deviation above the TVaR probability level Standard deviation pricing of tail risk Derivative for marginal calculation known Can use at probability of not meeting plan 61
Possible Next Step Model market value of firm Growth prospects, profitability prospects Risks to market value Allocate franchise value market value less capital to business units Stronger basis for strategic planning Need better theory of market value: Higher co moments, impact of jumps, customer attitudes to firm risk 62
What Might Be Done for Risk Pricing Two financial paradigms: CAPM and arbitrage theory Venter ASTIN 1991 showed that CAPM can be incorporated into arbitrage theory Transform the probabilities using conditional mean of market given the individual asset being priced to define new probabilities If returns are not normally distributed, it has been known since early 1970 s that maximizing investor utility requires pricing higher co moments than just covariance Some evidence of this in market prices Needed in insurance pricing Also jump risk may be an additional priced factor 63
Jump Risk In single period model not important But in continuous model, its existence can make the market incomplete without hedges for some risks You would think this would be of concern to investors So seems logical to price jump risk over and above moments This is done in many arbitrage free approaches Extending CAPM for jumps may be possible using co jumps That is jump for the company when there is a market jump Still some work needed to have a real pricing model for insurance risk 64
In Summary Economic capital better defined as multiple of loss at lower risk level, but still does not give the true capital need Tail risk measures have weaknesses If you re going to allocate, be marginal Even then not likely to give value of risk Strategic risk analysis through ERM has come a long way, but is not nearly done Modeling and optimizing franchise value of firm Measuring value contribution of business units, perhaps through a pricing theory 65