working paper series...c61: optimization techniques; programming models keywords goal programming,...

College of Business Administration

University of Rhode Island

2004/2005 No. 16

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exchange of ideas. Inclusion here does notpreclude publication elsewhere.

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Office of the DeanCollege of Business AdministrationBallentine Hall7 Lippitt RoadKingston, RI 02881401-874-2337www.cba.uri.edu

William A. Orme

Gordon H. Dash, Jr. and Nina Kajiji

A Nonlinear Goal Programming Model for Efficient Asset-Liability Managementof Property-Liability Insurers

A Nonlinear Goal Programming Model for Efficient Asset-Liability Management

of Property-Liability Insurers *

By

Gordon H. Dash, Jr.**

Associate Professor of Finance and Insurance

Nina Kajiji ***

Assistant Professor of Research

JEL CLASSIFICATION

G22: Insurance; Insurance Companies C61: Optimization Techniques; Programming Models

KEYWORDS

Goal programming, Nonlinear programming, Asset-liability models, Insurance, Risk management, Property-liability insurers, Portfolio optimization

2004 By Gordon H. Dash, Jr. and Nina Kajiji. All rights reserved. Explicit permission to quote is required for text quotes beyond two paragraphs. In all cases, full credit, including notice is required.

The research presented in this paper was completed under grants from: The Office of the Provost, University of Rhode Island; and, The NKD Group, Inc. (www.nkd-group.com), the producers of the WinORSfx financial engineering software used for the application presented in this manuscript.

http://www.nkd-group.com

Nonlinear Heirachical Modeling for Insurers.

Page -2-

ABSTRACT

Optimization of the firm-level asset-liability model (ALM) is an important part of enterprise risk management. In the context of the property-liability insurer we increase the credibility of the ALM by explicitly unifying the efficient management of financial risk factors across both sides of the economic balance sheet. The ALM presented in this research produces a simultaneous solution to the Markowitz mean-variance (MV) allocation of asset- and liability-side resources within a complex hierarchical goal environment. The nonlinear optimization method applied to the dual MV problem that is defined within the overall ALM is a separable program that encapsulates a vector optimized goal-program (NLGP). In addition to the identification of efficient combinations of traded assets and not-traded liabilities within a complex goal environment, the NLGP ALM also proves suitable for the extant characterization of credit, liquidity, and profit margin objectives.

1. Introduction

The liquidity strains of the late 1970s and early 1980s caused property-liability insurers

to follow the example set by the U.S. banking industry and its commitment to asset-liability

management (ALM) models as means by which to mitigate the ill effects of enterprise risk

management. For example, by the late 1980s banks had already advanced the state of ALM

frontier methods in their quest for efficient economic asset strategies given a particular liability

profile and overall business plan. In a recent review on insurer valuation, Babble (2001) noted

that while ALM tools are well developed what are lacking are useful stochastic valuation models

that unify both sides of the economic balance sheet. Asset-side stochastic model formulations

were already prevalent in the literature. In the late 1980s Booth and Bessler (1989) provided a

substantial review of relevant literature and a model extension that featured a futures-based

hedge to efficiently mitigate asset-side interest-rate risk. Almost a decade later, Boender, et. al.

(1998) produced a summary of contemporary ALM models and conveniently identified the

stylized relationships needed to simulate consistent decision-making scenarios. As researchers

entered the new Millennium model building efforts focused on two new areas: complex models

characterized by multi-criteria and hierarchical firm objectives and applications in the global

financial services industry. In the former case Babbel and Hogan (1992) and Korhonen (2001)

provide a time-line of evidence for complex hierarchical and multi-criteria decision-making. In

the latter case, Kosmidou and Zopounidis (2002) provide international evidence of prevailing

asset-side risk mitigating multi-criteria ALM in an application to a Greek bank. Dash and Kajiji

(2002) further advance the globally oriented stochastic multi-criteria ALM synthesis to include

domestic demand curve nonlinearities of an Indian bank.


Page -4-

It is widely accepted that insurers, particularly property-liability insurers, also rely upon

ALM for the determination of efficient firm planning. In a recent review of the empirical issues

on insurer consolidations and economies of scope, Cummins et. al. (2003) synthesized the long

standing realization that property-liability insurance firms face ALM complexities that are not

dissimilar to those faced by depository financial institutions. In citing the two distinct activities

carried out by property-liability insurers -- the selling of future claims against themselves (the

issuance of insurance contracts, or underwriting) and the diversification of asset side investments

-- Babbel (2001) argues persuasively that measurable improvements to insurer value will come

from the use of improved ALMs and not necessarily from improved security valuation methods.

He asserts that, there is a deep need for a unified valuation model one that efficiently takes

into account the important financial risk factors across all major asset categories and over both

sides of the economic balance sheet. From the early ALM formulations that attempted to mesh

evolving financial valuation theory with efficient firm-level risk management, Hofflander and

Drandell (1969), Drandell (1977), and Cummins and Nye (1981) and others have continually

sought to improve asset-side property-liability ALM specificity.1

1 See Lamm-Tennant, J. (1993). "Stock Versus Mutual Ownership Structures: The Risk Implications." Journal of Business

66(March): 29-46. for a discussion on organizational form and the management of firm-specific risk. BarNiv, R. and J. B. McDonald (1992). "Identifying Financial Distress in the Insurance Industry: A Synthesis of Methodological and Empirical Issues." Journal of Risk and Insurance

49(4): 543-574. provide a focused review of specific ALM methodologies that targets insolvency problems within the insurance industry. Similarly, Petroni, K. R. (1992). "Optimistic Reporting in the Property-Casualty Insurance Industry." Journal of Accounting and Economics

15(4): 485-508. presents evidence that supports the adoption of ALM approaches by documenting how weak insurers are more prone to forego the use of ALM methodologies as way to avoid the inherent objectivity the method brings to the description of adverse news (eg., required estimates of claim losses). Cummins, D. J., M. A. Weiss, et al. (1999). "Organizational Form and Efficiency: An Analysis of Stock and Mutual Property-Liability Insurers." Management Science

45: 1254-1269. focus on organizational form and efficiency for both stock and mutual property-liability insurers.


Page -5-

Although it is not strictly an insurer-based model, Sharpe and Tint (1990) extended the

analysis of the firm s across-sheet balance sheet MV problem. In an effort to capture liability

side effects, the Sharpe and Tint model focused on the firm s net surplus. The surplus model

was a notable extension but it has limited practical significance. The model s introduction

preceded contemporary economic realities such as the passage of the Financial Institutions

Modernization Act of 1999 (FIMA), emerging international opportunities, and new technological

innovation. The rapid change to the insurer s economic landscape have fueled model builders to

renew their focus on how to best estimate the cost of equity for insurers with increasingly diverse

underwriting portfolios. Cummins and Phillips (2003) address the issue through the

introduction of the full-information industry beta (FIB). The major contribution of the overall

firm FIB rests on its ability to employ the full-information betas of the underlying underwriting

lines. Of interest here is the fact that these results form a foundation by which to examine the

efficient trade-offs in alternative return generating profiles that define the composition of the

underwriting portfolio.

Given the context of the property-liability ALM process as recently summarized by

Babble (2001) on the one hand, and Cummins and Phillips (2003) on the other, we establish an

objective for this research to extend the ALM of property-liability insurers to include the

simultaneous solution of a dual Markowitz (1952) mean-variance framework one on each side

of the firm s economic balance sheet. In this research we extend the contemporary application

of efficient asset-side risk mitigation to explicitly include the simultaneous efficient mitigation of

liability-side risk. The paper proceeds as follows. Section 2 presents the foundations of a

separable program that explicitly solves the nonlinear hierarchical goal program. Section 3

develops a nonlinear hierarchical goal ALM of a hypothetical, or generalized, property-liability


Page -6-

firm. Because the purpose of this paper is illustrative, the model presented in this section does

not focus on a specific firm but is purposely generalized to meet the needs of any typical

property-liability insurer. Following the presentation of the model, solution results are discussed

in section 4. Section 5 is devoted to a summary of the findings.

2. A Separable Program for Nonlinear Hierarchical ALM

In goal programming it not always possible to satisfy all objectives simultaneously. Convex

goal programming is an appropriate and recognized optimization method by which to solve a

decision-model that is characterized by a defined organizational structure, heterogeneous

decision-making attributes, and hierarchical objectives. The nonlinear goal program applied to

the property-liability ALM presented in this paper is summarized as follows:

1

k

pp

Min Z h h

. .S T Ax Ih Ih b

, , 0x h h

where Z quantifies the k objectives (goals) so that an optimal solution to the linear program

associated with priority-one objectives is achieved prior to determining the optimal solution to

the linear program that defines priority-two objectives and so on. That is, in managerial

governance of the generalized property-liability insurer it is assumed that resource allocations

must be achieved in a prioritized sequence where Z1 > Z2 > > Zk. + and

- are constant terms

that indicate relative preference within each p-th goal; A is an m x n matrix of technological

coefficients created from the separable programming grid approximation; b is an m-component

vector of goal targets; h+ and h- are m-component column vectors that capture goal over- and


Page -7-

under-achievement, respectively. Lastly, x is a n-component column vector of decision

variables.

The method proceeds by solving a sequence of modified linear programs (MLP) that are

consistent with separable programming constraints. The use of finite bounds on the decision

variables permits the development of a piecewise linear approximation for each nonlinear

function in the problem statement. To form a piecewise linear approximation using r line

segments, it is necessary to select 1r values of the scalar x over its range of 0 x u . By the

use of a grid of 1jr points for each variable jx over its range, the separable programming

problem in x becomes the following modified linear program in where an adjacency criterion

is imposed on the new decision variable kj by use of a restricted basis entry rule:

1 0: ( ) ( )j

kj

n r

kj kjj kMin f f x

Subject to:

1 0( ) ( ) , 1...,jn r

i kj kij ij k kjg g x b i m

01, 1,...,jr

kjkj n

0, 0,..., , 1,..., .kj jk r j n 2

2 Several researchers discuss the intricacies of separable methods. See Stefanov, S. M. (2001). "Convex Separable Minimization Subject to Bounded Variables." Computational Optimization and Applications: An International Journal

18(1): 27-48. for both theory and methods. For additional discussion, also see Drandell, M. (1977). "A Resource Association Model for Insurance Management Utilizing Goal Programming." The Journal of Risk and Insurance

44(June): 331-315., Feijoo, B. and R. R. Meyer (1988). "Piecewise-Linear Approximation Methods for Nonseparable Convex Optimization." Management Science

34(3): 411-419., and Fare, R., S. Grosskopf, et al. (1988). "On Piecewise Reference Technologies." Management Science

38(12): 1507-1511..


Page -8-

Within the hierarchical goal program we define pC as the cost vector for goals at the p-th priority

level. The goal program proceeds to solve for the lexicographically smallest vector (c1x1, c2x2,

, cnxn). Given 1 | , 0x Ax b b the goal program proceeds by solving the first linear

program 1 1 1|LP Min C x x with optimal solution at x*. Then for each k-1 remaining x*, the

goal program iterates through all specified priority levels solving an LP at each step:

[ | ]p p pLP Min C x x , which implies that *1 1 1| , 1,2,..., 1p p p px x C x C x j p . For

parsimony, we note that alternative methods exist by which to formalize multi-criteria decision-

making in the treatment of resource allocation problems. For example, see Novikova and

Pospelova (2002) for an application to multicommodity networks.

3. The Hierarchical NLGP ALM for Property-Liability Insurers

The model presented here is forged from the generalized property-liability firm financial

statement as presented in table 1. The model is a one-period deterministic characterization of the

decision-making scenario. The notation is as follows. Beginning period values for asset

accounts are indicated by An , n = 1..N. Similarly, the beginning period value for each m-th

liability account is indicated by Lm for m =1..M. A buy or sell decision on n-th asset is

indicated by Bn and Sn , respectively. The rate used to make the buy and sell decision on n-th

asset is captured by rn .

<<< Insert Table 1>>>


Page -9-

Insurance lines are defined in table 2. The model specification includes eight insurance

lines that are divided into property lines (P-lines) and non-property lines (non P-lines). The

distinction is necessary to capture the traditional regulatory structure. The premiums collected

on all insurance lines, both P-lines and non P-lines is captured by Pp , for p = 1..P lines. The

parameters needed to complete the specification of the model are detailed in table 3. These

parameters include both endogenously and exogenously determined characteristics of asset-

liability behavior.



3.1 Asset-Liability Efficiency Decisions

The property-liability company s firm manager faces two efficient diversification

problems. One is the efficient asset allocation of N investment portfolio securities. The other is

found on the liability side. It is the efficient balancing of operational risks incurred by the

underwriting function across M insurance lines. Equations (1) through (10) define the two

portfolio problems. Equations (1) through (5) state the MV asset allocation problem for

investment securities, while equations (6) through (10) define the MV diversification problem

for insurer underwriting activities. The portfolio optimization sub-models are formulated

following the specification of the approximating quadratic Sharpe (1963) diagonal model. The

Sharpe diagonal model simplifies the accounting for asset covariance by the inclusion of an N+1

asset. We invoke the Sharpe diagonal model to exploit a reduction in modeling characteristics

under the separable approximation created to represent the dual MV optimizations. Inputs to the


Page -10-

Sharpe diagonal model are obtained by application of the single-index market model to the

individual traded assets: iS i i M iR a R , where ia is the component of security i's return

that is independent of the stock market's performance, MR

is the rate of return on the market

index, and i is a constant that measures the expected change in iSR given by a change in MR ,

and i is a random element with an expected value of zero and variance 2

i. In addition to 2

i,

which is the measure of unique (unsystematic) risk for the i-th security, the Sharpe diagonal

model also requires the variance of the market index, 2

MR , which is specified as the N+1 asset.3

The theoretical framework for estimating the systematic risk of non-traded assets held by

insurance firms was first presented by Fairley (1979). Not only did the Fairley results become

the foundation for rate application problems at the state level, but the theoretical underpinnings

of the model help to formalize the underwriting line diversification problem. Hill and

Modigliani (1987) presented a detailed case in the state of Massachusetts. More recent

contributions to the theory of underwriting line risk measurement are summarized in a review of

extensions by Lee and Cummins (1998) and Cummins and Phillips (2003).

3 A performance comparison of the diagonal and full-covariance model is reported by Cohen, K. J. and J. A. Pogue (1967). "An Empirical Evaluation of Alternative Selection Models." Journal of Business

40(2): 166-193. Representative extensions can be found in Frankfurter, G. M. (1976). "The Effect of Market Indexes on the Ex-post Performance of the Sharpe Portfolio Selection Model." The Journal of Finance

31: 949-955. For a summary of alternative computational methods, see Elton, E. J. and M. J. Gruber (1997). "Modern Portfolio Theory, 1950 to Date." Journal of Banking and Finance

1997: 1743-1759. More contemporary extensions that demonstrate alternative approaches is expemplified by Jobst, N. J., M. D. Horniman, et al. (2001). "Computational Aspects of Alternative Portfolio Selection Models in the Presence of Discrete Asset Choice Constraints." Quantitative Finace

1: 1-13. Finally, see Krokhmal, P., J. Palmquist, et al. (2001). "Portfolio Optimization with Conditional Value-At-Risk Objective and Constraints." Department of Industrial and Systems Engineering, University of Florida, Working Paper. for recent extensions that include value-at-risk specifications.


Page -11-

3.1.1 Optimal Investment Account Management

Equation (1) and (2) state the unsystematic and systematic risk, respectively for N

investment securities. In (1) the N+1 security is the market index and 1

2

NS is its variability.

The systematic risk level of individual asset relative to the market index pS is captured in (2).

Equation (3) forces the portfolio to be fully invested (short-sales are not permitted). Equation (4)

is the goal constraint for the managerially determined portfolio return, pSR . Equation (5) is an

accounting restatement of return.

12

11

0i

N

S ii

x d

(1)

1i p

N

S i Si

x

(2)

1

1N

ii

x

(3)

4 41

i p

N

S i Si

r x d d R

(4)

1s i

N

r i Si

C x r

(5)

3.1.2 Optimal Insurance Line Diversification

Equations (6) through (10) state the MV diversification problem for the insurers M

underwriting lines. In this specification jc is the fraction issued in underwriting line j, and pIR is

the managerially determined desired return from the insurance underwriting portfolio.

12

61

0j

M

I ij

c d

(6)


Page -12-

1

j p

M

I j Ij

c

(7)

1

1M

jj

c

(8)

9 91

j p

M

I j Ij

r c d d R

(9)

I jr j Ij

C c r

(10)

3.2 Premium Flows

Premium flows for j-th underwriting line are captured by jP . Equations (11) through

(18) are formulated to capture managerial goal setting. Total premiums written are defined

by 9P , in equation (19).

1 11 11 1 9P d d C P

(11)

2 12 12 2 9P d d C P

(12)

3 13 13 3 9P d d C P

(13)

4 14 14 4 9P d d C P

(14)

5 15 15 5 9P d d C P

(15)

6 16 16 6 9P d d C P

(16)

7 17 17 7 9P d d C P

(17)

8 18 18 8 9P d d C P

(18)

91

M

jj

P P

(19)


Page -13-

3.3 NAIC Policy Effects

The National Association of Insurance Commissioners (NAIC) implemented the

Insurance Regulatory Information System (IRIS) to monitor the financial condition of property-

liability firms. The IRIS system is defined by eleven ratios categorized into four financial ratio

domains: a) overall, b) profitability, c) liquidity and d) reserves. The NAIC also identified

ranges that are considered indicative of a financially sound insurer. For the model developed

here, NAIC policy is partially interrogated by the specification of equations (20) and (21).

Equation (20) defines variable P10 to report total P-line premiums while equation (21) defines

P11 to capture total non P-line premium flows. The premium sub-totals calculated here are used

to facilitate an examination of the insurer s general financial health by the IRIS overall domain.

To sample this domain, the ALM model calculates the net premiums written to surplus ratio as

well as the change in writings ratio.

10 1 2 3 4 8P P P P P P

(20)

11 5 6 7P P P P

(21)

3.4 The ALM Balance Sheet

The accounting relationships that define the firm s balance sheet are specified in

equations (22) to (37). The ending period account values for the asset side of the balance sheet

are defined in equations (22) to (26). Ending period values represent the amount of beginning

period values that did not mature plus purchases less any sales of the asset. Account purchases

are limited to forecasted values as stated in equations (27) to (31). Equations (32) to (36) limit

the insurer s ability to engage in short-selling. The accounting identity is stated in equation (37).


Page -14-

2 2 2 2(1 )A A a B S

(22)

3 3 3 3(1 )A A b B S

(23)

4 4 4 4(1 )A A c B S

(24)

5 5 5 5(1 )A A d B S

(25)

6 6 6 6(1 )A A e B S

(26)

2B f

(27)

3B g

(28)

4B h

(29)

5B k

(30)

6B l

(31)

2 2 (1 )S A a

(32)

3 3(1 )S A b

(33)

4 4 (1 )S A c

(34)

5 5 (1 )S A d

(35)

6 6 (1 )S A e

(36)

i ji j

A L

(37)

3.5 ALM Profitability Accounting

Equation (38) defines the policyholders' surplus account. For the model developed in this

research, all investment securities accounts are treated as perpetuities. Although this


Page -15-

assumption is somewhat unrealistic, it does not interfere with the purpose of the model and its

exposition. The managerial goal associated with this account is to minimize under-achieving

accumulations to the capital account. Policyholders surplus is defined to be a function of

beginning period values plus the accrued effect owing to changes in premiums written. Added to

this are the planning period changes in balance sheet values. For example, the equation includes

revenue for the fraction of existing government securities that did not mature, r2

(1-a)A2

, plus

revenue associated with the purchase of new securities, r2B

2, plus (less) capital gains associated

with security sales, [r2

- (r2

/r2) + 1]S

2, where, again, investment securities are all treated as

perpetuities. Except for the investment asset account, the revenue structure presented above is

similarly applied to all remaining asset side accounts. Income from investment assets is

calculated by adjusting the desired return PSR by under (over) achievement 4d ( 4d ) as

applied to the end of period investment assets values, 6A . Finally, the account definition

concludes by recording accumulated losses from the fraction of the change in loss-reserves (1-

i ) that is charged to the policyholders' surplus account for each of the four loss-reserve

accounts. Stated differently, the parameter i is the fraction of the change in loss-reserve that is

carried forward.

10 37 10 10 11 2 2 2 20.62 0.69 (1 )L d L E P P r a A r A

22 2 3 3 3 3

21 1rr S r b A r Br

33 3 4 4 4 4

31 1rr S r c A r Br

44 4 5 5 5 5

41 1rr S r d A r Br


Page -16-

55 5 4 4 6 1 5 5

51 1

PSrr S R d d A L Lr

2 6 6 3 7 7 4 8 81 1 1L L L L L L

(38)

3.6 Regulatory and Managerial Policy Effects

3.6.1 Liquidity

Equation (39) implements a basic regulatory requirement. This equation requires the

dollar investment in federal, state, municipal, and corporate bonds to equal total loss reserve for

all insurance lines with active premium flows.

2 3 4 39 39 2 3 4 5A A A d d L L L L

(39)

Equations (40) and (41) express the bounds on cash holdings as managerial goals. Taken

together, these two equations seek to produce an end-of-period cash account that falls between a

high and low dollar range. Both the upper- and lower-bound on cash holding are defined by

taking a percent of the total loss reserve. The upper bound is computed against loss reserves by

applying the upper bound percent, H . Similarly, the lower bound is computed by applying the

lower bound percent L .

1 40 40 2 3 4 5HA d d L L L L

(40)

1 41 41 2 3 4 5LA d d L L L L

(41)


Page -17-

3.6.2 Loss Reserve Accounting

The property-liability insurer must account for the relationship between current and prior

year loss reserves for each insurance line against which premiums are written. Because the loss

settlement history differs across the j lines, the value assigned to the premium roll-off factor, j ,

is set to reflect this history. A small (large) value is more likely for a long (short) settlement

period. Additionally, end-of-period loss reserves also include the portion of premium income

earmarked for loss payments throughout the planning period, j . For some lines very little

premium income is ever used to pay for losses; hence, only a small fraction of premium income

in that line is needed to be set-aside in a reserve. Equations (42) through (45) define this

relationship for the loss reserve for underwriting activity in non P-lines, miscellaneous liability,

workers compensation liability and automobile liability insurance. We note that the economic

relationship defined here is the one-year change in writings to surplus ratio as defined by IRIS

policy.

2 42 1 2 1 10L d L P

(42)

3 43 2 3 2 6L d L P

(43)

4 44 3 4 3 7L d L P

(44)

5 45 4 5 4 5L d L P

(45)

3.6.3 Unearned Premium Reserve Accounting

Equations (46) - (49) detail the behavior of unearned premium reserve (UPR). The UPR

is the liability item that reflects premium revenues received for contractual coverage not yet fully

provided. The ending period UPR account is directly related to a fractional amount of the


Page -18-

change in premium flows across two consecutive planning periods. Within the context of a one-

period model where premium flows are assumed to accrue evenly throughout the planning

period, a predetermined fraction of 0.5 with equal premium flows from one year to the next

would result in a constant UPR account as each dollar paid in losses would be replaced by

current premium writings.

6 46 6 10 100.5L d L P P

(46)

7 47 7 6 60.5L d L P P

(47)

8 48 8 5 50.5L d L P P

(48)

9 49 9 5 50.5L d L P P

(49)

3.6.4 Kenny Ratio

. During periods of remarkable profits the property-casualty industry could become

overly capitalized. The Kenny ratio is a well known industry specific approach that has been

favored by regulators to control an excessive accumulation of capital reserves. Generally, the

Kenny ratio is implemented to scale the premiums to surplus ratio at level of 2-1. Regulatory

review may be warranted when the ratio exceeds a level of 3-1. The Kenny ratio is implemented

across two equations. Equation (50) implements the upper bound ( U) of the Kenny dimension

as a constraint while equation (51) states the managerial goal ( L) for the relationship between

premiums and surplus.

9 50 10UP d L

(50)

9 51 51 10LP d d L

(51)


Page -19-

3.7 Maximization of Firm Value

Maximization of firm value is controlled through equations (52) and (53). Maximization

of firm value is goal-directed by use of the policyholders' surplus account. In equation (52) the

goal value is set to an arbitrarily high value; a value so high that it will cause the NLGP model to

produce a solution that reflects the most efficient allocation of firm resources within the context

of all defined financial relationships.

3710 52 10L d

(52)

Similarly, the ALM confronts potential agent-theoretic conflicts that may arise in the

process of seeking efficient solutions that maximize overall firm value. The Jensen and

Meckling (1976) view of agency theory suggests that firm managers may act in their own best

interest when not monitored. This agency problem arises for property-liability insurers as they

consider the implications of generating underwriting production by either the independent

agency system or through the employment of firm supported exclusive dealing insurers tied

agents (see Regan (1999) for a comprehensive retrospective on the hypothesis). The use of

independent agents is known to result in higher expense ratios for insurers. But, because the tied

agent's revenue is produced by a single insurer, when a significant adverse experience with a

particular underwriting line occurs, the rational behavior for the tied agent is to demand either

higher compensation for writing riskier policies, or the right to direct new underwritings to less

complex lines. This dilemma may cause agents to assert managerial goals that are in conflict

with the constrained efficiency standards inherent in value maximization principle.

Equation (53) is implemented to control for the potential of agency conflict actions. In

its most basic expression, agency conflict would reflect the agent's desire to increase income


Page -20-

through aggressive asset-side portfolio management while simultaneously limiting underwriting

combinations to less complex lines. The purpose of equation (53) is to utilize asset- and

liability-side MV sub-models to insulate the firm from agency conflict in the determination of

efficient investment asset and underwriting portfolio combinations.

53 53I sr r AC C d d R

(53)

3.8 The Hierarchical Objective Function

In the hierarchical goal programming model, it is the objective function that ties together

all goal and constraint equations. While the specification of the insurer ALM may be useful

across multiple individual firms within the industry, the model s goal hierarchy will certainly

differ at the firm level. The goal hierarchy modeled here is, like the ALM itself, a generalization

that represents the expression required to achieve efficient solutions to the embedded portfolio

problems while guided by the need to optimize firm value. Of course, more detailed and explicit

characterizations of firm goal hierarchy are possible in actual implementation of the model.

Equation (54) states the hierarchical objective function. The first priority (Z1), which

must be achieved before consideration of any lower level priorities, minimizes under-

achievement of two sub-objectives. Both sub-objectives have equal weight within this priority

level. Specifically, it is within this objective that we set the rate of return objectives for the two

MV portfolio diversification problems. The second priority (Z2) seeks to minimize over-

achievement in the portfolio risk dimension. When taken together, these two objectives operate

to fix the specified rate of return and then find the optimal allocation of investment-assets and


Page -21-

underwriting line premium flows that will best achieve the respective rate of return goal. The

third objective (Z3) captures the required shareholder maximization principle.

1 4 9 2 1 6 3 52:Min Z d d Z d d Z d

(54)

4. Application of the ALM to a Property-Liability Insurer

Alternative methods for solving the insurer ALM, including relevant forecasting

issues, are robustly discussed across Cummins and Nye (1981), Consiglio, et. al. (2002), Wang

and Yang (2002), and Yu, et. al. (2003). In this paper we introduce a scaled hypothetical

financial statement of a typical U.S. property-liability insurer as a means by which to explore the

contribution made by solving the dual MV across balance sheet ALM. To create the financial

statement of the hypothetical insurer we average reported balance sheet values across a sample

of U.S. headquartered property-liability insurers to obtain indicative individual account values.

Next, a common size balance sheet is derived to observe percent of asset/liability contribution

for each individual account. The hypothetical balance sheet is subsequently scaled to a $10,000

base based on the computed common size ratios. Beginning period values for the hypothetical

insurer financial statement are presented in table 4.


4.1 Data


Page -22-

The insurer ALM does not explicitly consider investment grade rankings for

individual investment account securities. The absence of this effect is managed by introducing

an upper bound constraint on the purchase of new assets (indicated by the maximum percent

increase). In the absence of such a constraint the ALM would simply take all of the liability

flows available without regard to the availability of interest earning assets. Finally, each asset

account specification includes a maturation parameter to capture the percentage of the asset that

matures (converts to cash) over the planning horizon.4 The upper bounds, forecast levels and

maturation parameters are presented in table 5.


Table 6 displays a sample of the forecasted dollar premium levels over growth (decay)

rates that vary from -25% to +100%. Associated interest rate parameters are presented in table

7. We note that variables with primes represent beginning period values. Conversely, variables

without primes capture the insurer s forecast of ending period rates that are expected to occur by

the end of the planning horizon.

<<< Insert Table 6, 7 >>>

Model data for policy parameters are presented in table 8. These values would be

observed directly from the firm s historical records or applicable regulatory statues. All

notations are as previously defined in section 3.

4 Under existing market definitions, some asset accounts may not mature (e.g., the investment assets account). In such cases, the maturation parameter is set to zero.


Page -23-


The data required to solve the dual Sharpe diagonal portfolio optimization problems is

presented in table 9. The table is divided into two sections. Data that specifies the

characteristics of each investment asset may be found in the left-most section of the table. This

data includes each securities expected rate of return, its systematic risk coefficient (beta), and a

measure of unsystematic risk ( 2). Parallel structure follows in the presentation of the risk and

return measures for the underwriting lines.


4.2 Representative ALM Solutions

Comparative ending- and beginning-period balance sheets produced by solving the model

under different economic scenarios are presented in tables 10 and 11, respectively. To obtain

these results, we varied two policy parameters one on each side of the balance sheet. On the

asset side of the balance sheet we vary the agency theoretic required rate of return, equation (53).

On the liability side of the balance sheet we explore alternative growth rate scenarios for

insurance underwriting premiums. For premium growth (decay), we started the simulation

process at extreme values. For example, for an insurer operating under a forecast for a 98

percent decline in premium flows, the model s solution finds that seeking a rate of return of one-

percent from the across balance sheet portfolio activities produces minimal total asset growth to

$10,491. While it is possible to generate any number of economic scenarios, we focused

substantial effort on solutions characterized by an eight percent growth rate in premiums.

Holding this policy dimension constant, we then allowed the agency rate of return that binds the


Page -24-

dual MV portfolio problems to vary between 8 and 50 percent (8-, 12-, 15-, 18-, and 50-percent

are shown in the tables).

<<< Insert Table 10, 11>>>

It is immediately obvious that the property-liability firm s total asset growth is impacted

by hierarchical goal resolution within the simultaneous risk mitigation strategies defined on both

sides of the balance sheet. Stated differently, the model s solution confirms the conventional

wisdom that an insurer s scale of operation is dependent upon the change in premium

underwritings. However, this wisdom is challenged in the face of excessive risk taking on the

asset side of the balance sheet as such activity can retard the growth in total assets. The model

exemplifies this fact as the decay in premium writings abates from a negative 98 percent to a

negative 25 percent. Total assets show growth to $11,697. This positive trend in total asset

growth continues as premium flows are subjected to higher and positive growth rates. However,

when we halt the growth rate of premiums at 8 percent and hold it constant, we note that total

asset growth stalls when management sets required asset returns between 12- and 18-percent. At

a 50 percent required rate of return the insurer is forced into ill-logical financial statement

combinations. That is, at this unrealistically high rate of return, the insurer simply accumulates

earnings in the cash account as forecasted purchase levels for securities and investment assets are

already at their maximum investment grade level. Upon first review, this may seem to be an

idiosyncrasy of the model. However, this characterization could be indicative of agency conflict

where tied agents attempt to increase overall firm returns by assuming excessive asset side

investment risk as a means by which to offset their desire to reduce new underwriting flows in

risky (but high yielding) insurance lines.


Page -25-

Returning to the plausible specification of 8 percent as a growth rate for premium

underwritings and as a required rate of return expected from the across balance sheet MV

portfolios, we present the optimal allocation of the investment asset and underwriting portfolio in

table 12. To accommodate the coarseness of the separable programming grid, for expository

purposes only a two percent minimum allocation constraint was added to the investment account

specification. By permitting the agent theoretic required rate of return to vary the efficient set is

produced and displayed in figure 1. As with the findings presented for the 8 percent level, any

point on the efficient set represents the efficient combination to the simultaneously obtained

across balance sheet portfolio problems. Stated differently, as the insurer interrogates alternative

agent theoretic expected returns, the ALM produces the optimal combination of investment

assets to hold as well as the optimal underwriting combinations to seek over the planning

horizon. The second efficient set shown in figure 1 is dominated by the one obtained from

solving the insurer ALM. The dominated efficient set, shown for visual reference only, provides

ancillary evidence that the simultaneous solution of the dual MV problem increases overall

insurer financial solvency.

<<< Insert Table 12 >>>

<<< Insert Figure 1 >>>

A closer examination of the 8 percent solution reveals an ending period value of $602.80

for the investment asset account (see table 10). This amount is optimally allocated at 13.47- and

72.53- percent between asset 2 and 7, respectively. All other securities enter at the 2-percent

structural constraint level. The return generated by the asset-side diversification strategy is 6.59

percent with a corresponding risk (standard deviation) of 4.19 percent. By contrast, the rate of


Page -26-

return from the insurance underwriting function is 1.30 percent. The ranked diversification path

for underwriting calls for the insurer to focus on: automobile liability, fire, homeowners

liability, commercial multiperil, and allied fire liability. All remaining underwriting lines enter

at a contribution level under 5 percent of the underwriting portfolio. When added together, the

solution produced an across balance sheet return of 7.90 percent. The difference between the

target 8 percent return and the actual results is attributed to rounding error attributable to the

coarse two-grid point approximation. Figure 1 also shows three points labeled B, K, and M.

These points represent the risk-return relationships of the insurance portfolios alone. Stated

differently, without the ability to invest in asset-side equity investments, property-liability firms

would generate risk-return profiles at these three interior points. The dominance principle

demonstrated by solving the across balance sheet efficient diversification problems provides

normative support for the full balance sheet hypothesis as advanced by Babbel (2001)

4.3 Premium Volume

At the 8% growth level, the model does not take all of the premiums that were forecasted.

This is directly related to the structural link between premium income and the optimal

underwriting combination. The goal over (under) achievement for the insurer underwriting lines

is reported in table 13 (a and b). Negative values suggest a failure to take premiums in an

amount equal to the optimal contribution. By contrast, positive values in this table report an

over-achievement of the optimal flow goal. When the forecasted change in premium flows is

negative, the insurer always takes the entire available forecast. However, as flow forecast

changes to positive values we can again observe the conservative nature of the ALM at work.

There is an increasing divergence between the ability of the insurer to employ new premium


Page -27-

volume effectively at growth rates that range from 4 percent up to 12 percent. Despite a small

narrowing (or elimination) of the divergence above 12 percent, at extremely high premium

growth rates (50- and 100-percent, respectively), the insurer accepts all forecasted premium

volume. However, as previously noted, this action is irrational insurer behavior as it leads to

distorted balance sheet allocations.

<<< Insert Table 13a and 13b >>>

4.4 Financial Policy and Regulatory Objectives

Under the 8 percent solution, cash holdings meet the bounds established by equations

(39) through (41). All loss-reserve and unearned premium reserve accounting are also met.

Importantly, the ALM specification lends itself to important ratio calculations. The insurer

solvency ratios produced under this scenario are presented in table 14. The impact of the

optimizing process is evident. For example, the Kenny ratio stays at a relative conservative level

reaching a value of 1.35. The ratio 1-year change in premium writings to policyholders surplus

shows a modest decline. This too is a reflection of the conservative solution generated by the

ALM when premium volume grows at the moderate rate of 8 percent per period. While the

conservative trend is reflected across the entire ratio analysis it is clear that the ALM produced

desired profitability while simultaneously increasing the overall efficiency of the generalized

property-liability insurer.

<<< Insert Table 14 >>>


Page -28-

4.5 Other Considerations

A basic premise uncovered by the model solution process suggests that the property-

liability insurer should avoid the sale of assets to raise liquidity. Instead, management should

focus on generating cash flow through its underwriting activity. Further, in the case where

underwriting activity is excessive and it is no longer possible to invest cash flow in traditional

investment grade assets, then management should search out "other assets" as well as hold cash

rather than reject new insurance business. However, as shown from alternative solutions of the

model, this latter scenario should be avoided due to the unacceptable balance sheet ending period

allocations. In summary, the model strategically and conservatively models efficient responses

to realistic economic scenarios while it sheds new light on the consequences from potentially

irrational insurer behavior that could arise owing to agency theoretic conflicts.

5. Summary and Conclusion

This research paper examined the performance of a one-period ALM for a hypothetical

property-liability insurer. The model extended prior model building efforts by explicitly

incorporating the extensions suggested by Babbel (2001) on the one hand, and Cummins and

Phillips (2003) on the other. The ALM model introduced in this research incorporated a dual

MV optimization problem that encompassed both sides of the insurer balance sheet. The asset

side MV diversification problem is well-known. Recent extensions to the liability side risk

measurement in the property-liability industry led to the model extension that permitted the

simultaneous incorporation of the second MV specification. Relying upon hypothetical

underwriting risk coefficients, coefficients that are easily generated by methods discussed in the

extant literature, the nonlinear hierarchical ALM produced both a simultaneous optimization of


Page -29-

the firm s across balance sheet management portfolio problems as well as a goal satisfying

solution that balanced the potential for agency conflict. Despite the need to create a linear

approximation of the nonlinear problem, the results produced from solving the model proved to

be consistent with normative applications in MV theory while clearly demonstrating the

usefulness of incorporating the MV problem with the specification of the firm s goal hierarchy.

The robustness of the current ALM solution argues well for a model extension that would

incorporate more complex economic environments that are best modeled by stochastic

relationships. Additionally, the model as currently configured ignores tax effects, derivative

instruments, timing decisions, and a more detailed profit-loss functional definition. The latter

point is of keen interest as a detailed profit-loss statement would permit a greater focus on

important ratios that are not fully enumerated in the current model (e.g., the combined ratio).

Finally, one additional aspect that was not explored in this paper is a model enhancement that

would permit the generation of an efficiency set of alternate insurer balance sheets. Such an

extension would allow insurers to explore the price of risk associated with the trade-off between

investment and underwriting opportunities. Regulators, by contrast, would be able to observe

useful information about the firm s ability to mediate risky managerial decisions and risky

economic environments.


Page -30-

Table 1. Generalized Property-Liability Firm Financial Statement

Assets Beginning Period Values

Purchase Variable

Sale Variable

Rate

Parameter

Liabilities Beginning

Period Values

Cash A1

Liabilities & Other L1

Federal Govt. Sec. A2

B2 S2 r2 Loss Reserves: P Lines L2

State & Local Sec. A3

B3 S3 r3 Loss Reserves: Misc. L3

Corp. Bonds A4

B4 S4 r4 Loss Reserves: Workers Comp. L4

Preferred Stock A5

B5 S5 r5 Loss Reserves: Auto-Bodily Injury L5

Investment Assets A6

B6 S6 r6 Unfunded Premiums: P Lines L6

Other Assets A7

B7 S7 r7 Unfunded Premiums: Misc. L7

Unfunded Prem. Res: Workers Comp. L8

Unfunded Prem. Res: Auto-Bodily Injury L9

Policyholders Surplus L10


Page -31-

Table 2. Insurance Lines and their Premium

Insurance Line Property Premium

Non-Property Premium

Fire P1

Allied Fire Liability P2

Homeowners Liability P3

Commercial-Multi Peril P4

Automobile Liability P5

Miscellaneous Liability P6

Workers Compensation P7

Automobile Physical Damage P8


Page -32-

Table 3. Modeling Parameters and Variables

Parameter Variable Description

N Number of investment securities

M Number of insurance lines

iS

Unsystematic risk for the i-th investment security (S)

jI

Unsystematic risk for the j-th insurance line (I)

iSr

Rate of return on the i-th investment security

jIr

Rate of return on the j-th insurance line

iS

Beta for the I-th investment security

jI

Beta for the j-th insurance line

ix

Fraction invested in the i-th security

jc

Fraction invested in the j-th insurance line

a Maturation fraction of federal government securities

b Maturation fraction of state and local obligations

c Maturation fraction of corporate bonds

d Maturation fraction of preferred stock

e Maturation fraction of investment assets

f Purchase forecast of federal government securities

g Purchase forecast of state and local obligations

h Purchase forecast of corporate bonds

k Purchase forecast of preferred stock

l Purchase forecast of investment assets

i

Change in policyholders surplus

j

The fraction of loss reserve of the j-th insurance line carried over to the next year

j

The fraction of premiums kept as reserve for the j-th insurance line

Kenny ratio parameter (see definition in sec 3.9.3)

Bounded Liquidity parameter (see definition in sec 3.9.1)

p Subscript referring to the portfolio


Page -33-

Table 4. Beginning Period Values

Assets Value

Liabilities Value

Cash 947

Liabilities & Other 958

Federal Govt. Sec. 1,043

Loss Reserves: P Lines 2,299

State & Local Sec. 4,187

Loss Reserves: Misc. 584

Corp. Bonds 1,261

Loss Reserves: Workers Comp. 638

Preferred Stock 378

Loss Reserves: Auto-Bodily Injury 1,132

Investment Assets 1,507

Unfunded Premiums: P Lines 1,015

Other Assets 677

Unfunded Premiums: Misc. 169

Unfunded Prem. Res: Workers Comp. 285

Unfunded Prem. Res: Auto-Bodily Injury 479

Policyholders Surplus 2,441

TOTAL $10,000

TOTAL $10,000

Table 5. Asset / Liability Forecast with Corresponding Maturation

Asset Category Upper Bound of Percent Increase

Forecast Parameter Maturation Parameter

Federal Govt. Sec. 30% f = 1,355.90 a = 0.80

State & Local Sec. 15% g = 4,815.05 b = 0.05

Corp. Bonds 50% h = 630.50 c = 0.00

Preferred Stock 30% k = 113.40 d = 0.00

Investment Assets 40% l = 602.80 e = 0.00


Page -34-

Table 6. Premium Dollar Values: Beginning Period and Various Forecast Levels

Insurance Lines Beginning Period

-25% 25% 30% 50% 100%

Fire 461.30 345.98

576.63 599.69 691.95 922.60

Allied Fire Liability 117.50 88.13

146.88 152.75 176.25 235.00

Homeowners Liability 277.70 208.27

347.12 361.01 416.55 555.40

Commercial-Multi Peril 243.00 182.25

303.75 315.90 364.50 486.00

Automobile Liability 433.20 324.90

541.50 563.16 649.80 866.40

Miscellaneous Liability 37.90 28.43

47.38 49.27 56.85 75.80

Workers Compensation 41.40 31.05

51.75 53.82 62.10 82.80

Auto. Physical Damage 77.00 57.75

96.25 100.10 115.50 154.00

Table 7. Interest Rate Parameters

Beginning Period

Rate Planning Period

Rate

r2

0.060 r2 0.070

r3

0.050 r3 0.054

r4

0.070 r4 0.073

r5

0.065 r5 0.069

Table 8. Policy Parameters

Parameter Value

i

1.47

1

0.15

2

1.54

3

1.54

4

1.47

i

0.20

H

4.00

L

2.00

H

0.15

L

0.10


Page -35-

Table 9. Portfolio Model Data

Security E(r) Beta

2 Insurance Line E(r) Beta

2

Sec_001 0.064

0.896

0.0266

Fire 0.010

0.850

0.0013

Sec_002 0.062

0.468

0.0072

Allied Fire Liability 0.050

1.895

0.0380

Sec_003 0.146

1.110

0.0441

Homeowners Liability -0.001

0.350

0.0270

Sec_004 0.173

1.361

0.0292

Commercial-Multi Peril 0.068

1.361

0.0901

Sec_005 0.198

1.273

0.0756

Automobile Liability -0.010

0.857

0.0035

Sec_006 0.056

0.590

0.0310

Miscellaneous Liability -0.014

1.313

0.1500

Sec_007 0.128

0.527

0.0207

Workers Compensation 0.028

0.981

0.0091

Sec_008 0.190

1.733

0.0384

Auto. Physical Damage 0.004

0.568

0.0280

Sec_009 0.116

1.041

0.0437

Market Index

0.0397

Insurance Industry Index

0.0800


Page -36-

Table 10. Projected Growth Rates and Optimal Asset Allocation Strategies

Projected Premium Growth Rates

E(r) Total Portfolio

Cash Federal Govt.

Securities

State & Local Securities

Corporate Bonds

Preferred Stock

Inv. Assets

Other Assets

Total Assets

-98.00%

1.00% 944.20 1355.90 4875.18 1891.50 113.40 602.80 708.15 10491.14

-25.00%

1.00% 1052.74 1355.90 6521.71 1261.00 113.40 602.80 789.55 11697.10

0.05%

18.00% 1313.62 1355.90 7955.30 1891.50 491.40 602.80 985.21 14595.73

4.00%

18.00% 1496.88 1564.50 7955.30 1891.50 491.40 3109.80 1122.66 16632.05

8.00%

8.00% 1093.06 1355.90 6268.70 1891.50 113.40 602.80 819.80 12145.17

8.00%

12.00% 1156.66 0.00 7955.30 1891.50 378.00 602.80 867.49 12851.75

8.00%

15.00% 1156.66 0.00 7955.30 1891.50 378.00 602.80 867.49 12851.75

8.00%

18.00% 1156.66 0.00 7955.30 1891.50 378.00 602.80 867.49 12851.49

8.00%

50.00% 1496.88 1564.50 7955.30 1891.50 491.40 2109.80 1122.66 16632.05

15.00%

18.00% 1140.83 208.60 7955.30 630.50 378.00 1507.00 855.62 12675.62

25.00%

18.00% 1496.88 1564.50 7955.30 1891.50 491.40 2109.80 1122.66 16632.05

30.00%

18.00% 1117.11 1355.90 6115.78 1891.50 491.40 602.80 837.83 12412.32

50.00%

18.00% 1156.96 1355.90 6866.78 1891.50 113.40 602.80 867.72 12855.06

100.0% 18.00% 1228.83 1355.90 7539.62 1891.50 113.40 602.80 921.62 13653.68


Page -37-

Table 11. Projected Growth Rates and Optimal Liability Allocation Strategies

Projected Premium Growth Rates

E(r) Total Portfolio

Liabilities and Other

Loss Reserve P Lines

Loss Reserve Misc.

Loss Reserve Workers Comp

Loss Reserve Auto B.I.

Un-earned Premium P

Lines

Un-earned Premium

Misc.

Un-earned Premium Workers Comp

Un-earned Premium Auto Bodily Injury

Policy-holders Surplus

-98.00% 1.00% 996.66 2302.53 585.17 639.28 1134.26 1615.02 188.33 306.11 521.83 2201.95

-25.00% 1.00% 1111.22 2431.36 627.77 685.82 1216.89 2044.44 202.16 321.23 679.95 2376.26

0.05% 18.00% 1386.59 2475.52 642.40 701.79 1245.25 2191.65 206.91 326.41 734.21 4685.01

4.00% 18.00% 1580.04 2479.00 644.70 704.31 1249.72 2203.27 207.66 327.23 742.76 6493.36

8.00% 8.00% 1153.79 2482.53 647.04 706.86 1254.25 2215.03 208.42 328.06 751.43 2397.77

8.00% 12.00% 1730.16 2482.53 647.04 706.86 1254.25 2215.03 208.42 328.06 751.43 2527.99

8.00% 15.00% 1730.16 2482.53 647.04 706.86 1254.25 2215.03 208.42 328.06 751.43 2527.99

8.00% 18.00% 1730.16 2482.53 647.04 706.86 1254.25 2215.03 208.42 328.06 751.43 2527.99

8.00% 50.00% 1580.04 2482.53 647.04 706.86 1254.25 2215.03 208.42 328.06 741.18 6468.65

15.00% 18.00% 1327.57 2501.95 651.12 711.32 1262.17 2279.74 209.74 329.51 766.59 2636.15

25.00% 18.00% 1580.04 2506.54 656.96 717.69 1273.49 2295.04 211.64 331.58 788.25 6270.82

30.00% 18.00% 1179.17 2514.84 659.88 720.88 1279.15 2322.71 212.59 332.61 760.64 2429.86

50.00% 18.00% 1221.23 2563.71 671.55 733.63 1301.79 2485.63 216.37 336.75 842.40 2481.99

100.0% 18.00% 1297.10 2651.95 700.73 765.51 1358.38 2779.75 225.85 347.10 950.70 2576.60


Page -38-

Table 12. Solution: Optimal Portfolio Allocations

Security xi Insurance Line Cj

Sec_001

2.00%

Fire 23.19%

Sec_002

13.47%

Allied Fire Liability 6.96%

Sec_003

2.00%

Homeowners Liability 16.44%

Sec_004

2.00%

Commercial-Multi Peril 14.39%

Sec_005

2.00%

Automobile Liability 29.77%

Sec_006

2.00%

Miscellaneous Liability 2.24%

Sec_007

72.53%

Workers Compensation 2.45%

Sec_008

2.00%

Auto. Physical Damage 4.56%

Sec_009

2.00%

Rp 6.59%

Rp 1.30%

p 4.19%

p 9.18%

Figure 1: Efficient Frontiers

Rat

e of

Ret

urn

Variance

R

ate

of R

etur

n


Page -39-

Table 13a. Premium Volume Goal Over (-under) Achievement

Growth -98% -25% 0.05% 4% 8%

Fire 3.67 137.70 183.99 167.41

151.23

Allied Fire -2.51 24.79 -125.56 -130.52

-135.54

Homeowners -3.11 -116.62 -155.87 -161.72

-167.94

Coml MP 4.10 153.83 205.20 213.30

221.51

Auto Liab 7.84 293.85 392.00 407.47

423.14

Misc. Liab -0.78 -29.32 -39.12 -40.66

-42.23

Worker Comp 0.67 30.90 41.32 42.95

44.61

Auto PD 1.54 57.75 77.04 80.08

83.16

Forecast 33.78 1266.75 33.78 1756.56

1824.12

Actual 33.78 1266.75 33.78 1733.03

1777.06

Table 13b. Premium Volume Goal Over (-under) Achievement (cont.)

Growth 15% 25% 30% 50% 100%

Fire 333.30 142.46 148.16 275.40 367.20

Allied Fire -144.32 -156.87 -163.15 -188.25 -251.00

Homeowners -178.82 -194.37 -125.27 -68.26 -311.00

Coml MP 235.87 256.38 266.63 307.65 410.20

Auto Liab 450.57 489.75 432.46 587.70 783.60

Misc. Liab -44.96 -48.87 -50.83 -58.65 -78.20

Worker Comp 47.51 51.64 53.71 61.99 82.70

Auto PD 88.55 96.25 100.10 115.50 154.00

Forecast 1942.35 2111.25 2195.70 2533.50 3378.00

Actual 1942.35 2024.21 2028.30 2533.50 3378.00


Page -40-

Table 14. Financial Ratios

Ratio Ending Period

Beginning Period

Total Premiums to Surplus 0.74 0.69

Liabilities to Liquid Assets 0.47 0.48

Loss Reserves to Surplus 1.60 1.44

Allied Fire to Fire Lines 0.28 0.25

1 Year Change in Writings to Policyholders Surplus 0.05 0.08

Kenny Ratio 1.35 1.31


Page -41-

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