Using the Public Access AFD Model: A Case Study, Essays (university) of Financial Management

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Using the Public Access DFA Model:

A Case Study

by Stephen E D'Arcy, FCAS, MAAA,

Richard W. Gorvett, FCAS, MAAA,

Thomas E. Hettinger, ACAS, MAAA,

and Robert J. Walling Ill, ACAS, MAAA

CAS Dynamic Financial Analysis Task Force on Variables

Dynamic Financial Analysis - Applications and Uses

1998 Call Paper Program

Using the Public Access DFA Model:

A Case Study

Submitted by

Stephen P. D'Arcy 1, Richard W. Gorvett 1, Thomas E. Hettinger z

and Robert J. Walling, IIIz

Department of Finance, University of Illinois at Urbana-Champaign

Miller, Rapp, Herbers & Terry, Inc., Consulting Actuaries

This paper describes the application of a DFA model to an actual insurance company. One goal of this w o r k is to help actuaries learn about DFA by observing the use of a working model in a realistic setting. The model described in this paper is publicly available and accessible over the Internet. The c o m p a n y that generously allowed its data to be used in this exercise has asked to remain anonymous. Thus, minor modifications have been made to the data to help preserve the a n o n y m i t y of this insurer. These changes do not affect the operation of the DFA model or obscure the data gathering process involved in running a DFA model.

Introduction

The DFA model used in this paper, termed Dynamo2, was developed by the actuarial consulting firm of Miller, Rapp, Herbers, & Terry, Inc. The model is accessible via their website ( w w w. m r h t. c o m ) and requires only Microsoft Excel and @Risk in order to run. For those w i t h o u t access to @Risk, a limited version of the model can also be run solely in Excel. The Excel version is also useful for running a small number of iterations quickly to check the reasonableness of input values. The general purpose of this model is to simulate a large number of possible outcomes from specific input data. By viewing the expected values and distributions of key variables, such as statutory surplus, premium-to-surplus ratios, and net income, the user can determine if these results are acceptable. If they are, then they validate the operating strategy of the company, subject to the general caveats of using DFA models. If not, then management can vary the input values to learn which changes would be effective in improving results to an acceptable level. The model, w h e n run using @Risk, allows the user to examine any of the stochastic parameters of interest determined as an @Risk function. Thus, users can v i e w the randomly generated values for all of the unacceptab{e outcomes to see if any factor tended to be responsible for a significant number of these cases. For example, if a large percentage of the cases in which surplus falls b e l o w a minimum standard involved a high level of catastrophe losses, then the c o m p a n y may be able to reduce catastrophe exposure by revising its reinsurance arrangements or shifting its geographic distribution. Management could use the DFA model to test the effects these changes would have on the results by re- running the model with the revised input before deciding w h e t h e r these approaches should be adopted. The basic operation of the model is to generate insurance c o m p a n y cash f l o w s and then evaluate the effect of these cash flows. The model integrates the cash flows from investments and underwriting, including catastrophes and taxes. The model consists of six different inter-related modules: underwriting, investments, catastrophes, taxation, an interest rate generator, and a p a y m e n t

pattern generator. Values generated in one module are shared with the other modules in subsequent calculations. This paper focuses on an application of DFA. In order to obtain a fuller understanding of DFA modeling, including the limitations of this approach, readers should refer to additional sources. Some useful sources are: D'Arcy, Gorvett, et. al. (1997}, D'Arcy, Gorvett, Herbers and Hettinger (1997), CAS Committee on Valuation and Financial Analysis (1995), CAS DFA Handbook (1996) and the multi-part Actuarial Review series " H o w DFA Can Help the Property-Casualty Industry" (1996-1998).

T h e T e s t C o m p a n y

The company used to test this model is a mid-sized property-liability insurer that operates nationwide. The major lines are private passenger and commercial automobile, commercial multi-peril, workers compensation and homeowners. The company has standard reinsurance contracts: excess of loss, quota share and catastrophe coverage. Since the company has been in operation for more than twenty years, enough historical information is available to generate loss payout triangles, frequency and severity trends, loss ratios by age of business, and the other input required for the DFA model. 1 Once the company's data were received, they were input into Dynamo2. Results from the model were generated, and incorporated in a report which was transmitted to the company. That report is included as an Appendix to this paper -

• in order to follow the progression of this project, the reader is advised to read the Appendix at this point. This initial report served as the basis for discussions on DFA at a meeting between the authors of this paper and representatives from the company; company personnel involved in these discussions included actuaries, investment personnel, and business planning staff. This report provides both an introduction to DFA and a starting point for a detailed dynamic financial analysis of the firm. The questions raised at that meeting will be covered later in this paper, after a detailed explanation of this DFA model.

T h e M o d e l

The DFA model used in this paper starts with detailed underwriting and financial data showing the historical and current positions of the company, randomly selects values for 4,387 (I) stochastic variables, calculates the effect on the company of each of these selected values, and then produces summary

I Generatingand gathering the data needed to run this model required the efforts of many people at the company, including the Chief FinancialOfficer, the Chief Investment Officer and the Chief Actuary, as well as members of their staff. We ere very grateful for their cooperation and willingness to supply us with their data; without their help, this paper could not have been written.

of freedom and non-centrality parameters being a function of the K, 8, and o parameters above. However, in this DFA model, w e approximate the discrete-time form of the CIR model using the following formula:

~r = a(b-r)~t + s~/

w h e r e ~ = the discrete-time (annual) change in the short-term interest rate, At = the discrete time interval (one year), and e = a random sampling from a standard normal distribution.

The CIR model separates interest rate changes into t w o components, one deterministic component, a ( b - r ) , and one stochastic component, s r °'66. The deterministic c o m p o n e n t moves the current interest rate part w a y (represented by a) back t o w a r d the long term m e a n b. The further the current interest rate is from this long term mean, the greater the deterministic c o m p o n e n t of the interest rate movement. The stochastic c o m p o n e n t causes the interest rate to jump around this otherwise level trend back t o w a r d the mean. Since the stochastic c o m p o n e n t is multiplied by the square root of the current interest rate, w h e n interest rates are low, the stochastic c o m p o n e n t is small. This reduces the likelihood that interest rates will become negative. (In the continuous time application of this model, interest rates cannot become negative because if the interest rate were ever to become zero, which a continuous line must cross before becoming negative, then the interest rate will have no stochastic c o m p o n e n t and will simply be pulled back t o w a r d the long term mean (it will actually become a ( b - r ) ). However, in the discrete approximation of this model, negative interest rates can occasionally occur.) In this interest rate model, the current interest rate is the actual short-term interest rate in the e c o n o m y at the time the model is run. As of mid-March, 1998, 3 m o n t h Treasury bills, a c o m m o n p r o x y for short t e r m rates, were yielding 4. 9 8 5 %. Thus, in this model, r(O) is set to 5%. The long-run mean, b, is also set at 5%. (Empirical tests of the CIR model on historical data indicate a value for the long-run mean of approximately 8%. These tests are based largely on data from the 1980s. When b is set at 8 % in this model, any investment strategy based on long-term bonds tends to under-perform a shorter-term portfolio, since interest rates would tend to move upward, depressing bond prices. To avoid introducing this bias, the long term mean was selected to be the same as the initial value of the short term interest rate. However, this is a variable that can, and should, be altered by the user to reflect individual v i e w s of interest rate movements, and to test the sensitivity of resutts to this variable.) Since, under the above parameter value selections, the value of b - r ( O ) is zero, the deterministic c o m p o n e n t of the interest rate change is zero in the first year. The stochastic component, then, determines the entire interest rate change.

In one run of the model, the value of e in the first year was randomly selected by the model to be - 1. 0 0 9 4 5. Thus, the calculation for the change in interest rates in that model run was:

~ r = sift e = (0.0854)(~/O.~)(-1.00945) = -0.

Since the interest rate started at 0.05, the change of - 0. 0 1 9 3 led to a n e w short- term interest rate of 0. 0 3 0 7 , or 3. 0 7 %. Once selected, the short term interest rate is used to generate the term structure of interest rates. Based on the interest rate model parameters selected, and upon the simulated short-term interest rate, rates on zero-coupon Treasury bonds are generated for each annual duration up to thirty years. This Treasury term structure is used to determine the market value of the c o m p a n y ' s bond holdings. The specific equations used to generate the term structure are taken from Cox, Ingersoll, and Ross (1985):

R(r,I,T) = rB(t,7) - blA(t,T~ T - t

where R is the yield-to-maturity at time t on a discount bond that matures at time T, and

A(t,T~ = [. 2yel(~'x'yxr°la ] z~°t° (K +~. +y)(e r~r-o_ I ) .2¥

B(t,7)) = 2(er(r-') _ 1)

(K +X +y)(e r(r-o_ 1) +2y

y ~ ((K+g)2+2o2) la

The short-term interest rate is also used to determine the general inflation rate, based on the following formula:

Line of Business Assumed Inflation in Payment Pattern

H o m e o w n e r s 0. 0 5 2

PP Auto - Liability 0. 0 6 7

PP Auto - Phys Dam 0. 0 4 3

Comm Auto - Phys Dam 0. 0 4 3

Comm A u t o - Liab 0. 0 6 7

CMP - Liab. 0. 0 4 5

CMP - Prop. 0. 0 4 5

Other Liab. 0. 0 7 3

Other Liab. - Umbrella 0. 0 7 3

WC 0. 0 6 8

b s Sample Line of Businesslnflation

The line of business inflation rates are used for t w o purposes. First, t h e y affect loss development. The initial loss reserves presume a specific inflation rate; the values selected for this run are listed on the above table. To the e x t e n t that the calculated line of business inflation rate differs from this value, loss payments will diverge from the initial loss reserves. The second effect of the line of business inflation rates is on loss severity, which drives the need for future rate increases. In the present application of this model for this specific company, frequency was assumed to be stable, so the only factor that affects the projected pure premium is the severity trend. Thus, the line of business inflation rate determines the indicated rate level change.

Jurisdictional Risk

Each state poses unique advantages and disadvantages t o the operation of an !nsurance company. Those advantages and disadvantages may take the form of judicial, legislative, or regulatory risk. For example, the likelihood of retroactive workers compensation benefit increases, mandated premium rebates, generous (for the policyholder) interpretations of contract provisions, and the ability to obtain rate increases all vary by state. In this model, jurisdiction risk is reflected in t w o ways. First, each state has a range of "acceptable" rate changes -- that is, there is associated with each state

a range of rate changes that can be implemented w i t h o u t extraordinary company cost (in terms of time or money) and/or additional insurance department scrutiny. GeneralJy, these ranges limit rate increases more than they do rate decreases, and the ranges are smaller in states with more restrictive regulation. The obvious effect of strict rate regulation is to prevent insurers from increasing rates to the degree they feel is necessary. However, a side effect of capping rate increases is to make companies more reluctant to lower rates as much as would be otherwise indicated if pure premiums are improving. The other effect of jurisdictional risk is to introduce a lag in implementing indicated rate changes. This lag, shown in the model in terms of years, is longer in states with restrictive rate regulation. The lags indicated on the jurisdictional risk exhibit included in the Appendix are estimated averages for rate increases and decreases; the average lags in the model are multiplied by 1.50 for rate increases and by 0.50 for rate decreases. The jurisdictional risk parameters are based on a Conning & Company study that ranks all states with respect to regulatory restrictiveness. States ranked as most restrictive were assigned the lowest acceptable rate ranges and the longest lags. The actual values were selected primarily based on the judgement of individuals with experience with rate filings in those states. As an example of jurisdictional risk in this DFA model, the range of Homeowners rate changes in Massachusetts is from .85 to 1.O6 (rates could be lowered by 15% or increased by 6% without significant additional company cost or regulatory scrutiny). Since the average lag is estimated to be ½ year, it woutd take 3 months to implement a decrease and 9 m~nths to implement an increase. The company's distribution of writings countrywide is used to determine the overall impact of jurisdictional risk.

Aging Phenomenon

The model reflects the aging phenomenon by separating writings for each line of business into new business, first renewals, and then second and subsequent renewals. Under the aging phenomenon, loss ratios gradually decline with the length of time the policies have been in force with the company. For more details on this experience, see Woll (1987), D'Arcy and Doherty (1989), D'Arcy and Doherty (1990) and Feldblum (1996). One requirement that this approach introduces is the need for the company to supply exposures end losses broken d o w n by age of the business. Although this allocation is not needed for

any statutory or accounting reports, many firms maintain this information for

internal reports, although not necessarily in the detail required for the DFA model. In this case, estimates of the loss frequency and severity by age of business can be tried and the resulting loss ratio indications checked for reasonableness, before finalizing these values. The overall result is that new business should have the highest loss ratio, first renewal business should have a slightly lower loss ratio,

Catastrophes

A catastrophe is defined as any natural disaster causing in excess of $ million in insured losses. The total number of catastrophes c o u n t r y w i d e is simulated based on a Poisson distribution, and then assigned to a "focal point" state based on historical catastrophe experience. The size of each catastrophe is then simulated based on a Iognornal distribution, the parameters of which vary according to the identity of the focal point state. For each simulated catastrophe, the contagion effect of the catastrophic losses from the focal point to other states, and by property line of business, is determined based on historical relationships. Finally, the effect of these catastrophes on the c o m p a n y is determined by the market share of the c o m p a n y in each state, by line of business. For example, in Florida the probability of any number of catastrophes occurring is determined based on a Poisson distribution with a mean of 0. 6 6 6 7. This value, relative to the parameters for all other states, determines the likelihood of a catastrophe being assigned to Florida. For each simulated catastrophe, the size is then determined based on the Iognormal distribution with a mean parameter of 2. 7 6 9 7 (in millions) and a variance parameter of 1.1563. For each catastrophe in which Florida is the focal point, 86 percent of the loss is assumed to be incurred in Florida, with the remaining 14 percent distributed to nearby states. All of these parameters were calculated based on data from Property Claim Services over the period 1 9 4 9 - 1 9 9 5. As an example, in one iteration of the model, no catastrophes occurred in Florida in 4 of the 5 years simulated; in the fifth year (2001), t w o catastrophes occurred, one causing $143 million in insured losses and the other $269 million in losses. It should be noted that the catastrophe module in this DFA model is meant to produce reasonable estimates, and is not intended to replace the more rigorous catastrophe models that are available. In fact, it is possible that the results from other commercially available catastrophe packages could be used in this DFA model.

Investment Results

Investment results for both fixed income securities and equities are determined in the investment module. For bonds, both the statutory value and the market values are calculated for each category of bond (Government, corporate, municipal) and for each maturity segment indicated in the Annual Statement (e.g., one year or less, one to five years, etc.). The market value is determined based on the term structure of interest rates obtained in. the interest rate generator module. The cash flows on bonds consider interest rates, coupon rates and default rates, generated stochastically based on historical patterns. The market value of equities is determined from a simulation based on the Capital Asset Pricing Model. The rate of return on equities is determined in a t w o

step approach. The initial expected market return is the risk free rate, as obtained in the interest rate generator, plus a market risk premium of 8.5% (historical average for 1926-1996). The adjusted market return is the initial expected return minus 4 times the simulated change in the short term interest rate. A random component based on a normal distribution with a mean of 0 and a standard deviation of 15 percent is generated and added to the adjusted market return to determine the overall market return for each year. The return for the company is then determined by applying the equity beta, which is an input value.

Collecting Data

One decision that needs to be made is h o w to deal with multiple companies operating under the same management. Many insurers have subsidiaries, but operations are coordinated within the group. In this case, the model should be run on the group as a whole, rather than for each individual company. However, if more detail is needed, then each company can be modeled separately. The primary source of input data for the model is the Annual Statement. However, additional information is also necessary, which requires the company to provide, or generate, some internal management reports. In addition, the company needs to provide information about exposure growth anticipated, by line for the next five 'years, and any shift in investment allocations that are contemplated. Examples of the specific data requirements are illustrated on the exhibits included in the Appendix. In a typical application of this model, some of the more problematic data areas might potentially include exposures and rates by renewal category, historic loss ratios by renewal category, and various aggregation issues (the trade-off between data volume and its homogeneity when examining lines and types of business). Also, in order to generate more credible cash flows, or to deal with homogeneous data, Annual Statement lines of business can be aggregated or split into separate components, as needed.

Running the Model

The first step in running the model (after the company-specific data has been input) is to determine where the industry stands in the underwriting cycle for each line of business. It is presumed that the insurance industry follows a time dependent cycle of competitiveness. In a soft market, premium increases tend to significantly reduce market share. Conversely in a hard market, policyholders find it difficult to obtain insurance, so it is easier for an insurer to increase market share. The next step is to determine the number of iterations to be run. The higher

Sheet Location

Cell Variable IDes~:ription Reference U/W Cycle Position Users viewpoint on current General Input C6 to C market conditions. Growth Rates Expected growth rates in Premium ~nput Row 22 exposures Renewal Ratio Premium Input Rows 30- Expense Provisions Commissions, General, Other Premium Input Rows 42, 46, 50, Acq,, Taxes, Dividends, and 54, 57, 59 Nonrecurring Expenses Q/S Ceding Premium Input Row 52 Commission Exposure Changes Use to Change Exposures Exposure Input and Market Shares by State Selected 1997 Loss Input Rows 167 to 169 Severities Selected 1997 Loss Input Rows 196 to 198 Frequencies Selected ULAE Loss Input Rows 227 to 233 Provisions O/S Arrangements Loss Input Rows 255- XOL Arrangements Includes Attachment Points Loss Input Rows 268 to 297 and Cost of Reinsurance Stop Loss Includes Attachment Points Loss Input Rows 349 to 353 Arrangements and Cost of Reinsurance Cat. Re Arrangements Includes Attachment Points Loss Input Rows 359 to 363 and Cost of Reinsurance Stock Betas Investment Input Rows 95 to 98 Capital Infusions Investment Input Rows 86 to 91 Reinvestment Investment Input Rows 109 to 125 Allocations

How Investment Income is Reinvested

Long-Run Interest Rate Interest Generator C

Current Interest Rate Interest Generator C General Inflation Interest Generator C35 to C Parameters LOB Inflation Interest Generator Rows 54 to 56 Parameters

U/W Cycle Parameters Includes Probability o'f I Changing Market Condition

and Supply/Demand Curves I

U/W Cycle Generator

C7 to H

Initial Reaction of the Company to the DFA Report

First Imnressions The c o m p a n y ' s first direct exposure to the DFA model occurred at a meeting b e t w e e n the authors and representatives of the c o m p a n y ' s actuarial, investment, and business planning departments. At this time the report included in the Appendix was delivered and a detailed explanation of the DFA model was presented. Many questions were raised at that point, a majority of which related to asking for an explanation of h o w the model w o r k e d. H o w e v e r , there were also a number of questions that will lead to model improvements and enhancements. Overall, c o m p a n y personnel were enthusiastic about the model and have hopes of using it in the future for strategic planning purposes. They also s a w it as a tool to help the different divisions of the c o m p a n y -- actuarial, financial, investment, and planning -- w o r k together. Finally, the c o m p a n y liked the s o f t w a r e platform on which the DFA model is based. The Excel spreadsheet format makes the model user-friendly and simple to change and enhance, and allows the user to examine the inner workings of the model in a non-black box environment.

Concerns The c o m p a n y expressed certain concerns regarding the model and the results that were initially supplied to them. It was evident that the Base Case indications were unacceptable (primarily due to the high g r o w t h goals of the company); however, the managers felt that constraining g r o w t h was not a viable alternative. Other options were explored, including increasing the n e w business renewal rate. For H o m e o w n e r s this value was 60 percent. Raising it to 8 0 - 9 0 percent caused some improvement, but not enough to turn results around completely. Another change was to modify the m a x i m u m ceded under the aggregate reinsurance contract. This also had a favorable effect on forecasted results. In order to gain a better understanding of w h a t was causing the results, t w o additional values, the short term interest rate and catastrophe losses, were added to the output page and the simulation re-run during the meeting. The ability to modify the model and quickly see the impact of the changes was v i e w e d very f a v o r a b l y. Some of the questions raised indicated the need for enhancements in future versions of the model. One question related to prepayments on bonds and CMOs as a function of interest rate.changes. Another w a n t e d to examine the effect of changing g r o w t h patterns by state, to examine the effect on the c o m p a n y of

Variable Adiustments During the presentation, several different computers loaded with the DFA model were available, allowing the managers to break into groups and test different DFA scenarios. For example, one group of managers adjusted the interest rate parameters. Specifically, they raised the long-run mean interest rate level to 10 percent and reduced the volatility parameter to 0, to observe the effect of increasing interest rates for a small sample of runs. Other groups ran the model after adjusting one or more of exposures, losses, the reinsurance program, catastrophe parameters, exposure g r o w t h assumptions, and investment variables. In still other cases, certain stochastic variables were "shut off" -- e.g., by setting the volatility parameter of the variable equal to zero. This allowed the user the opportunity to see the impact of certain stochastic variables w i t h o u t introducing additional "noise" from those variables that were turned off. In general, this exercise was seen as beneficial by all the groups, not just the actuaries. Having a viable DFA model will serve to help the different areas of the c o m p a n y w o r k more closely together, and facilitate coordinating the efforts of the various areas.

Presentation to UDDer Manaoement Members of the group raised several questions about h o w this model should be presented to the upper management of the company. In addition to needing to get comfortable with the model, they also wanted to be able to focus on h o w actual results differed from the projections. To do this, it was suggested that they might use the model to project results for last year (run the model w i t h o u t including data for the latest year and then compare the actual results with the o u t p u t from the model). In addition, they wanted to print out key financial exhibits for the situations that were unacceptable, so that they could focus on w h a t w e n t w r o n g in those cases. This feature is available in the @Risk version of the model, but currently not in the Excel version. Examining the effect of a c o m p a n y ' s use of a DFA model is a long term prospect. Modifications and enhancements to the model would be expected, as the c o m p a n y asks n e w questions after seeing initial indications. While it is too early to provide any information about the final effect of this process, the initial meeting and response suggest that the DFA model will provide a very useful management tool.

Future Enhancements

Enhancement of the public-access DFA model is an on-going process, input and suggestions from users and other interested parties are w e l c o m e d and encouraged. The following items represent some of the enhancements to the model which are currently being considered.

Determine the impact of callability provisions and other options embedded in insurer bond holdings. This will require identification of those bonds in the insurer's portfolio that have such options, information regarding w h e n during the life of the bond the option is exercisable, and the call premium or other parameters associated with the embedded option. The valuation f r a m e w o r k already incorporated within the DFA model -- i.e., market valuation of fixed-income securities based on the simulated term structure of interest rates -- will form the basis for the endogenous decision w h e t h e r or not to exercise .the option. Explicitly value mortgage-backed securities. These securities are comprising ever-larger proportions of insurer portfolios. In particular, for example, the prepayment risk associated with collateralized mortgage obligations will be simulated using the Public Securities Association (PSA) model of monthly prepayments on residential mortgages, with the parameters of the PSA model being impacted by simulated general economic conditions. Add state and/or regional detail in the underwriting module to facilitate measuring the effect of, for example, a change in the g r o w t h rate for a particular state. Continue to develop the underwriting cycle module and the associated demand curves, including their impact on business retention rates and jurisdictional risk. Implement correlations for the frequency and severity figures for business of different ages within a given line and b e t w e e n lines of business. Add tax-loss carry-forwards and carry-backs to the tax module. Add a module which produces risk-based capital results.

Conclusion

DFA is becoming an important concept for property-liability insurers, and it is likely that actuaries will be called upon to participate in, if not lead, this endeavor. This paper describes one DFA model. This model is publicly available and its use is encouraged, and comments on its effectiveness, limitations and potential improvements are actively solicited. While DFA for property-liability insurers is in a nascent stage, the intial reaction of c o m p a n y management to the application of this model to their operations was very favorable and provided evidence that DFA will prove valuable to the industry.