Enterprise Risk Modeling

Getting the Risk Right – Problems and Pitfalls

Gary Venter, July 2002

Overview

Common problems and options for improving enterprise models

Capturing the risk– Key details needed to get risk right

Capital need and capital allocation– Critical to business managers– Alternative methods may improve

rationality of approach

Issues

1. Assets2. Reserves3. Parameter risk and event risk4. Correlation 5. Capital needed and allocation

1. Asset Issues

Arbitrage-free models– No reward without some risk

Probabilistic reality– Modeled scenarios consistent with

historical patterns Balancing asset and underwriting

risk

Arbitrage-Free Yield Curves

Long-term rates built from market expectations of short-rate changes plus a risk charge

Financial theory specifies required features of the risk charge– Called market price of risk– Adds a usually upward drift to the

short rate to get longer term rates

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

Balancing Asset and Underwriting Risk

Look at efficient investment frontier and how that changes with different reinsurance programs

Can review offsetting insurance risk with investment risk for optimal balance by adjusting reinsurance program to fit best to investment portfolio

Constrained Asset Efficient Frontier with Current Reinsurance Program

Frontier of Constrained After-Tax Operating Income / Assets1-year horizon: 2003

-0.03

-0.025

-0.02

-0.015

-0.01

-0.005

0

0.005

0.01

0.02 0.03 0.04 0.05 0.06 0.07

Std Deviation

Mea

n

Frontier

Company

Probability of Returns on Frontier

0.02 0.03 0.04 0.05 0.06 0.07

STD

-0.15

-0.12

-0.09

-0.06

-0.03

0.00

0.03

0.06

0.09

0.12

0.15

0.18

Variability of portfolios on after tax OI/Assets frontier

Ef. frQ01Q05Q25Q75Q95Q99

Vary

rein

sura

nce

an

d

investm

ents

2. Reserve Issues

Loss reserving models UEPR and current underwriting risk Time capital must be held

Loss Reserving Models

Actuaries start with development factors and Bornheutter method

Many more models are out there Key issue is measuring correlation

between inflation and development E.g., see 1998 PCAS Testing the

Assumptions of Age-to-Age Factors

Six Questions Give 64-Way Classification of Reserve

Models Do the losses that emerge in a period depend

on the losses already emerged? Is all loss emergence proportional? Is emergence independent of calendar year

events? Are the parameters stable? Are the disturbance terms generated from a

normal distribution? Do all the disturbance terms have the same

variance?

Testing and Simulating Models

Live Data ExampleSSE Model Params Simulation Formula157,902 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 44,701 BF-CC+ 7 qw,d = fdhwgw+d + e

Some models fit better with fewer parameters

Simulation and so development risk depends on model

Best fitting model has future paid responsive to future inflation

UEPR and Current Underwriting Risk

Different from loss reserve risk– Backward projection of reserve risk does not

model the risk situation Can be quantified through risk elements

– Frequency risk– Severity risk– Correlation among lines

Risk usually considered in terms of uncertainty about ultimate results, not just one year of stated values

Metarisk model designed to measure this risk gross and net of reinsurance

Time Period Reserve Capital is Needed

Capital needed to support an accident year until it runs off

Declining capital needed as losses settle Looking at capital needed for just one year

of runoff is generally felt to understate reserve capital need

Modelers sometimes understate this capital and thus allocate too little to long-tailed lines

Zone Rated Development Development Factors

Accident

19842.67

21.57

81.21

41.05

81.04

41.01

11.01

81.01

11.00

21.00

11.00

0

19852.58

11.50

51.22

41.07

01.02

41.01

51.00

41.02

21.00

51.00

0

19862.85

31.39

71.35

51.12

61.05

21.01

61.00

71.00

01.00

1

19872.51

41.59

51.21

51.12

61.06

31.03

01.00

61.01

7

19882.79

41.50

11.26

41.13

51.04

41.00

91.00

3

19892.61

31.46

41.23

01.07

21.01

91.01

7

19902.47

11.60

11.19

31.12

01.02

1

19912.69

01.46

91.30

41.11

5

19922.74

21.71

51.16

5

19932.67

91.44

0

19942.60

5

1995

Year 1-2 2-3 3-4 4-5 5-6 6-7 7-8 8-9 9-10 10-11 11-12 12-131.00

01.01

31.01

01.00

01.00

01.02

21.01

01.02

31.15

91.19

51.49

82.58

0

Murphy Method for Triangle Risk Residuals from fit give estimated sigma2

Also estimate variance of each lag’s factors Accident year n variance of ultimate losses =

process variance + parameter variance process var(n) = process var(n–1) *

factor(n)2 + est. sig(n)2 * cum dvlp(n–1). Start with process var(1) = last actual * est sig(1)2

param var(n) = var of factor(n) * cum dvlp(n)2 + mean sq factor(n) * param var(n–1). Start this with param var(1) = last actual2 * var of factor(1)

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 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%

Select Risk Measure Cost of capital for risk A = c*cov(A,

market) + d*cov(A, company) , or Cost of capital for risk A = a*corr(A,

market)* (std dev A) + b*corr(A, company)*(std dev A)

Assumed correlation structure: Correlations for Unit with:Market Company Loss 20% 35% Investment Income 80% 50%

Cost of capital for risk A = a*corr(A,market)*std. dev.(A) + b*corr(A,company)*std. dev.(A) a = 0.5 b= 0.5

Premium: 129,870,130 Loss Ratio 77% Exps Ratio 30% Invest Rate 7% cv invst inc 2% Target Return =Loss 100,000,000

End Year U/W End Yr U/W Avg. Cash U/W + Inv Cost Captl Cost Captl AnnualYear cv of ult % paid Std. Dev. Cash Flow Balance Avail in Yr Invst Inc. Cash Flow for Losses for Inv Inc Captl Cost

0 11.0000% 0.0000% 11,000,000 0 0 0 0 0 0 0 01 7.3409% 16.8144% 7,340,878 74,094,714 74,094,714 37,047,357 2,593,315 76,688,029 2,521,871 16,857 2,538,727 2 7.1005% 27.6792% 7,100,497 (27,679,230) 46,415,485 62,848,415 4,399,389 (23,279,841) 1,985,689 45,453 2,031,142 3 4.9039% 23.1066% 4,903,870 (23,106,572) 23,308,913 41,854,903 2,929,843 (20,176,729) 1,650,601 47,640 1,698,241 4 3.4743% 15.5151% 3,474,287 (15,515,060) 7,793,853 25,473,930 1,783,175 (13,731,885) 1,151,997 30,635 1,182,631 5 2.2092% 9.0859% 2,209,200 (9,085,922) (1,292,069) 14,956,614 1,046,963 (8,038,959) 781,479 18,396 799,875 6 1.8818% 3.3307% 1,881,826 (3,330,733) (4,622,802) 9,795,250 685,667 (2,645,066) 562,516 11,262 573,778 7 1.9953% 1.4485% 1,995,253 (1,448,452) (6,071,254) 8,091,325 566,393 (882,059) 533,098 8,138 541,237 8 1.6217% 0.9955% 1,621,701 (995,543) (7,066,798) 7,435,720 520,500 (475,043) 497,331 7,065 504,396 9 1.2905% 0.9460% 1,290,495 (946,049) (8,012,846) 6,985,424 488,980 (457,069) 400,427 6,562 406,989

10 1.3243% 0.1724% 1,324,330 (172,358) (8,185,204) 6,915,200 484,064 311,706 359,538 6,325 365,863 11 1.0892% 0.3605% 1,089,170 (360,462) (8,545,666) 7,132,854 499,300 138,838 331,856 6,392 338,248 12 0.8000% 0.5449% 800,000 (544,881) (9,090,548) 7,179,483 502,564 (42,318) 259,761 6,512 266,273 13 0.0000% 0.0004% - (362) (9,090,909) 7,409,425 518,660 518,298 110,000 6,638 116,638

Total 100.0000% (9,090,909) 17,018,813 7,927,904 11,364,038 Correlations for Unit with: Market CompanyLoss 20% 35%Investment Income 80% 50%

Assumed weighting coefficientsAssumed weighting coefficients

Investment Income CVInvestment Income CV

Total capital costs exceed total profitsTotal capital costs exceed total profits

Is This Line Profitable Enough?

Usual test is to compare costs on a discounted basis

Capital cost could be considered an outgoing cash flow each year

At any interest rate over 2%, present value of annual capital cost is less than present value of underwriting cash flow

A lot of work needed on risk measures and weights – probably fixed correlations wrong

3. Parameter Risk All loss risk not coming from known

frequency and severity fluctuations Includes estimation risk, projection

risk, and event risk Systematic risk – 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

Projection Risk

Change in risk conditions from recent past

In part due to uncertain trend Can include change in exposures

– More driving as gas prices change and other transportation looks risky

– New types of fraud become more prevalent

Measuring Risk from Uncertain Trend

Impact of Projection Risk J on Aggregate CV

(CV is ratio of standard deviation to mean)

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%

Translating CV Effect to Loss Ratio

Probabilities E(LR)=65, 3 E(N)’sCV(J)=0.05 E(N)=2,000 20,000 200,000

90th 79.2 71.0 69.495th 84.1 72.8 70.899th 94.1 76.4 73.3

CV(J)=090th 78.5 69.2 66.395th 83.1 70.5 66.799th 92.5 72.9 67.4

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

Estimation Risk – Pareto Example

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 replace-ment 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 big bucks risks and can dwarf others. Hard to predict in future, but must be considered an ongoing risk source and build into random effects.

4. Correlation Issues

Correlation is stronger for large events– Multi-line losses in large events– Modeled by copula methods

Quantifying correlation– Degree of correlation– Part of spectrum correlated– Measure, model, or guess

Modeling via Copulas

Correlate on probabilities Inverse map probabilities to

correlate losses Can specify where correlation

takes place in the probability range

Gumbel Copula Correlates Large Losses

Heavy Right Tail Copula Even More So

Normal Copula Doesn’t

Quantifying Dependency

Directly measure degree of and location of dependency– Fit to copulas by matching

measurement functions Model dependency through

generating process– For example losses and asset returns

could be fed by inflation

Concentration Measurement Functions for Right and Left Tail – Conditional Probability

of Both in Tail if One Is

Using Measurement Functions in Fitting

5. Capital Needed and Allocation

RAROC or RORAC? Economic Capital Target Coherent Measures of Risk Matching Capital and Return Allocation Methodologies

RAROC or RORAC?

Capital, not return, usually risk-adjusted

Sometimes return adjusted to replace cat losses by expected

Return targets often do not reflect value of favorable insurance pricing and availability provided to mutual company policyholders

Economic Capital Target Comparison to bond ratings

– E.g., 99.97% chance of not defaulting Measuring 1-year default probability accurately

for large company almost impossible– Strongly affected by risk guesses made– Projecting out to tails of distributions with no data to

tell if the tail is right– Single year default of A-rated insurer takes unusual

circumstances not even in models, like Enron-type accounting, management fraud, ratings downgrade below A-, not meeting debt service, substantial hidden reserve deficiencies, etc.

More realistic to set probability target for partial surplus loss, such as:– 99% chance of not losing more than 20% of surplus

Coherent Measures of Risk Mathematical consistency requirement

for risk measures VAR does not meet requirement

– For instance, combination of independent risks can increase VAR beyond the sum of the individual VARs

TVAR does meet requirement– Average loss above VAR threshold– More relevant to policyholders– Other coherent measures being researched

Matching Capital and Return

Each business unit generates investment returns on cash flow and on capital supporting the business

That income is part of return of unit That income and the capital needed

to support those investments both need to be charged to the business unit to properly evaluate the unit’s economic contribution

Alternatives for Capital Allocation and Performance Measurementa. Allocate by risk measure

– Coherently– Incoherently

b. Allocate by price of bearing riskc. Charge capital costs against profits

– Marginal capital costs of the business– Value of risk guarantee of parent

d. Compare value of float generated by the business to a leveraged investment fund with the same risk

a. Allocate by Risk Measure

Pick a risk measure– Coherent, such as TVAR– Not coherent, such as VAR

Pick an allocation method– Maybe spread in proportion to

marginal contribution to company risk– Or use the Kreps method of creating

additive co-measures, like co-TVAR, that give 100% additive allocation and consistent splits to subunits

Definition of Co-Measures Suppose a risk measure for risk X with

mean m can be defined as: R(X) = E[(X– am)g(x)|condition] for some

value a and function g, and X is the sum of n portfolios Xi each with mean mi

Then the co-measure for Xi is: CoR(Xi) = E[(Xi– ami)g(x)|condition] Note that CoR(X1)+CoR(X2) = CoR(X1+X2)

and so the sum of the CoR’s of the n Xi’s is R(X)

Example: EPD If X is losses and b total assets, the

expected policyholder deficit is EPD = E[(X – b)S(b)|X>b] where S(b)=1 – F(b)

Let a = 1 and g(x) = S(b)(X – b)/(X – m) Then with condition = X>b, R(X) = EPD CoEPD(Xi) = E[(Xi – mi)g(X)| X>b] =

E[S(b)(X – b)(Xi – mi)/(X – m)|X>b] Each portfolio gets a fraction of the overall

deficit given by the ratio of its adverse losses to the total annual adverse losses in each scenario

Allocating Capital by CoEPD Each portfolio charged in proportion to its

contribution to overall default Does not equalize portfolio expected

default costs across portfolios Additive across sub-portfolios and up to

total losses For instance, you could allocate capital for

each line to state, then add up all lines to get total state capital

Example: TVaR

TVARq = E[X|X>xq] where F(xq) = q. Note that if xq = assets, then:

EPD = default probability * (TVARq –assets)

Thus TVaR at default and EPD rank all risks identically

For a=0, CoTVaRq(Xi) = E[Xi |X>xq] Charges each portfolio for its part of

total losses in those cases where total losses exceed threshold value

Coherence of TVaR TVaR is a coherent measure, which means,

among other things, that for a fixed q the sum of the TVaR’s of any collection of loss portfolios will be the same or greater than the TVaR of the combined portfolio

Not true for EPD or for VaR with fixed q TVaR criticized for ignoring losses below

threshold and for not differentiating among risks that have the same mean above thresh-old – other coherent measures better there

Problems with Allocation by Risk Measure

Arbitrary choices of measure and method

Business units will favor choices that favor them, and there will be no underlying theory to fall back on

Pricing to equalize returns may not tie in to risk pricing standards

b. Allocate by Price of Bearing Risk

Financial theory gives market price guidelines for risk bearing

Can be calibrated to insurance market situation

Business units can be evaluated by profit vs. risk-pricing standards

Can allocate capital in proportion to target profitability

c. Charge Capital Cost against Profits Instead of return rate, subtract cost of

capital from unit profitability Use true marginal capital costs of

business being evaluated, instead of an allocation of entire firm capital– If evaluating growing business 10%, charge

the cost of the capital needed for that much growth

– If evaluating stopping writing in a line, use the capital that the company would save by eliminating that line

This maintains financial principle of comparing profits to marginal costs

Fixed and Marginal Capital Costs Company X buys a widget maker

and pays a big fee each year for mortgage costs

Running it and producing widgets is cheap

How does it decide whether or not to make more widgets?

Compare revenue with marginal cost of production

Fixed and Marginal Costs Similarly for insurance company

– Expanding or contracting a business unit evaluated based on revenue vs. marginal costs, especially marginal cost of new capital needed or capital released

– This includes capital needed for reserves and investment income on funds generated

– Separate analysis needed for strategies for fixed costs

Calculating Marginal Capital Costs

Could use change in overall risk measure of firm that results from the marginal business

Or set capital cost of a business segment as the value of the financial guarantee the firm provides to the clients of the business segment

Value of Financial Guarantee

Cost of capital for subsidiary is a difference between two put options:– 1. The cost of the guarantee provided by the

corporation to cover any losses of the subsidiary– 2. The cost to the clients of the subsidiary in the event

of the bankruptcy of the corporation Economic value added of the subsidiary is risk-

adjusted profit less cost of capital– Profit risk adjusted to account for long-term average

costs of highly unstable risks, like cats If EVA is positive, it is worth growing the

subsidiary

Allocation and Evaluation Summary

Allocating by risk measure straightforward but arbitrary

Using risk pricing appropriate for comparing profitability

Actual marginal surplus most useful for determining economic contributions of business units. This is not the same as allocation in proportion to marginal risk.

Leveraged mutual fund comparison is appropriate for evaluating return on total capital and the marginal contribution of each business unit to that

Conclusions Asset models should be arbitrage-free and

distributionally representative of history Reserve risk requires alternative models and is

easy to understate, both on time capital held and UEPR reserves

Parameter risk is a key issue for large companies and is difficult to quantify

Correlation should incorporate tail links to get true large loss risk

VAR is not the best overall capital standard, nor is allocation of total capital the best way to evaluate profitability

Getting the modeling right takes care and expertise, and is subject to many pitfalls