An insurance and investment portfolio model using chance constrained programming

Omega, Int. J. Mgmt Sci. Vol. 23, No. 5, pp. 577-585, 1995

Pergamon

030541483(95)00019-4

Copyright © 1995 Elsevier Science Ltd Printed in Great Britain. All rights reserved 0305-0483/95 $9.50 + 0.00

An Insurance and Investment Portfolio Model Using Chance Constrained Programming* SX LI Adelphi University, N.Y., USA (Received March 1994; accepted after revision April 1995) An insurance and investment portfolio model is here formulated in terms of the 'P-Modeis' of Chance Constrained Programming, which is then related to the 'satisficing concepts' of Simon. For a given insurers' aspiration level of return on equity and risk levels of violating minimum requirements on return and on cash and liquid assets, we propose a method to maximize the insurers' probability of achieving their aspiration level, subject to two chance constraints and other regulatory and institutional constraints. An empirical example is given, based on the indnstry's aggregated data for a twenty year period.

Key words--modeling, optimization, portfolio analysis, chance constrained programming

1. INTRODUCTION AFTER FERRARI [12] extended Markowitz's [22, 23] portfolio analysis to portfolio selection for property-liability insurance companies, more notable subsequent work can be found in literature. Some of this work, for example [2, 13-15, 17], can be used to generate meanvariance efficient frontiers for property-liability insurance companies. However, in the absence of specific knowledge of an investor's preference, none of these models can tell which of any two efficient portfolios is better. Agnew et al. [1] extended Markowitz's criterion to investment portfolio in a casualty insurance firm. By using the chance-constraint technique they at first introduced the incorporation of insurer's risk into the investment portfolio of a normative model. However, their model can be used only to select a Markowitz investment efficient portfolio according to the *The author is grateful to three a n o n y m o u s referees and the editor for their helpful comments and suggestions. OME 23/~-H

577

insurer's preference. Thompson et al. [27] generalized Agnew et al.'s work by considering the premium volume as the only decision variable of the insurance side and determining the portfolios of premium volume and investments through chance-constrained programming. They also assumed that all the returns on investments and insurance activities are statistically independent. Real situations are more complicated than the problems such as Agnew et al.'s problem of selection of an investment portfolio and Thompson et al.'s problem of selection of a portfolio of premium volume and investments. More recent papers employing chanceconstrained programming in a portfolio analysis can be found in [4, 5, 9, 18, 20]. All of these chance constrained programming formulations utilize expected value optimizations and hence fall in the class that Charnes and Cooper [8] referred to as 'E-Models'. Here we turn to the more general class that Charnes and Cooper referred to as 'P-Models' and develop them in a manner that enables us to make contact with theories of behavior such as are

578

Li--lnsurance and Investment Portfolio Model

described by the 'satisficing concepts' of Simon [25]. For a given aspiration level of return on equity, insurers, when confronted with an environment subject to risk, cannot be certain of achieving this given level when choosing from the available portfolio selections. Our proposed model is to maximize the probability of achieving the aspiration level under two chance constraints and other regulatory and institutional constraints. The first chance constraint specifies that the probability that the total return will fall below the insurers' acceptable minimum level will not exceed their return risk level, and the second chance constraint allows the insurers' minimum requirement on cash and liquid assets to be violated, but not more than their liquidity risk level. The approach developed in this paper is different from the existing models in the literature in the following ways. First, our model could be utilized to handle statistically dependent multiple insurances and multiple investments. In contrast, the models developed by Thompson et al. can be used only in a situation of statistically independent single investment, and the model in Agnew et al. can be used only in situations where no insurance line is permitted. Secondly, in traditional mean-variance portfolio models, the rates of returns on investments and insurance lines are assumed to follow normal distributions, while in our model, we assume that the rates of returns follow more general distributions. This assumption allows us to deal with a wider class of portfolio problems and it captures more realistic situations because the rate of return on a single insurance policy usually has a skewed distribution. Lastly, the objectives in the traditional portfolio literature are to maximize expected returns or to minimize the variance of returns subject to constraints such as budget requirements and/or minimum returns on equity. In contrast, the objective of our model is to maximize the insurer's probability of obtaining at least his/her aspiration level of return on equity subject to two chance constraints that are used to achieve and maintain the company's minimum return requirement and liquidity. Our portfolio model is examined by the use of aggregated data during a twenty year period, thereby demonstrating the potential uses of the model for the property-liability insurance industry.

2. T H E M O D E L

In this section we consider in detail a oneperiod portfolio diversification model of property-liability insurance companies. The model is formulated as follows: We suppose that there are m types of possible investments and n candidate insurance lines, and the dollar incomes from investment and insurance activities are random variables with known distributions having finite expected values and finite variances. For any given period, the return on equity, ~, is defined to be equal to underwriting profit plus investment income, i.e. m+n

~=

a,f~+ i=l

~

a~.

(l)

i=m+l

The average (expected) value of the return on equity is equal to: m+n

# = E ( Y ) = 2 a,E(?,),

(2)

i=1

and the variance of the return on equity is: m+nra+n

a 2 = Var(~) = ~, ~ aiaj Cov(ri, rj), i=l

(3)

j=l

where ~ denotes a random variable, ai = ratio of investment in asset i to equity (i = 1, 2, 3 . . . . . m) or ratio of insurance premium for line i to equity (i = m + 1.....

m +n),

~,.= rate of return on investment i (i = 1,2 . . . . . m) or rate of profit on the insurance activity i (i = m + 1. . . . . m +n). A basic assumption used in the literature is that there are no market saturation and institutional constraints. Many models suggest that insurers should specialize in only the few most attractive lines and can sell any amount of insurance and get the same expected return and variance. However, this is not realistic. In practice, most property-liability insurance companies offer a full range of coverage. The costs of becoming established in any given line of business prohibit withdrawing from the line except under the most drastic conditions. Abrupt increases in premium volume in any

Omega, Vol. 23, Number 5

given line would tax the company's underwriting expertise and could lead to the acceptance of substandard business. These factors suggest the need for both maximum and minimum limits on the premium allocations to each line of insurance. This may be expressed by

amin<~ai<~a max ( i = m + l

..... m+n).

(4)

In the US, state insurance laws place restrictions on the investments of property-liability insurance companies. These laws regulate the minimum amount of equity being invested in 'risk-free' securities such as government bonds, high grade corporate bonds, and certain other securities of comparable safety. In order to diversify investments more efficiently, unsystematic financial risk must be minimized. We may therefore wish to constrain the minimum and maximum amounts to be invested in each of the different kinds of investments [20]:

amin<~ai<~amax

(i = 1. . . . . m).

m+n

i=l

~.

aigi,

leverage, which reflects how much premium is written for each dollar of equity. In general, claim payments are paid from the premiums collected. However, if premium receipts are not enough, insurers need to liquidate some of their liquid assets to meet cash demands. Thus, insurers must have enough cash and liquid assets, )7, to meet cash requirements, or they will not be able to stay in business. The cash and liquid aspects of equity are represented by the return on insurance portfolio m+t/

E

i=rn+

I

liquid assets and return on liquid assets k

a,(1 + ?i), i=l

and interest or other cash income from liquid parts of the non-liquid assets

(5)

Since the insurance industry is required to maintain a balance between sources of investable funds and their uses, we have the following balance sheet constraint:

a, = 1 +

579

(6)

i=m+l

~=k+l

a,(l, + d~).

Here di is the rate of cash returns on investment i(k + 1 <<.i <~m), l~ is the liquid part of the ith asset (k + 1 ~< i ~
jT= ~ a , ( l + ? , ) +

where the term

i=l m+n

a,(/, + di) + i=k+l

i=1

represents the assets per one dollar of equity and the term m+n

~

aigi

i=m+l

represents the total liabilities which are generated by premiums ai's. The parameter gi is called the funds-generating factor which is the ratio of liabilities to premium for insurance line i. Since the equality is expressed per unit of equity, equity is equal to 1. Regulation, marketing constraints, and various other rules of thumb used in the insurance industry usually limit the amount of premium volume for a given amount of equity. This is expressed by the following equation: m+n

Z

ai = 6,

(7)

i=m+l

where the parameter 6 is known as insurance

~

a,?,.

(8)

i=m+l

Since returns on equity are stochastic, we should consider the chance that the insurance company's return on equity and cash and liquid assets are less than insurers' minimum requirements. According to [10, 28], the concept of risk is defined as the possibility of loss or injury; hazard; peril; danger. In our portfolio analysis, we define the insurers' return risk as the probability that their return on equity is less than or equal to their minimum level, 131, and insurers' liquidity risk as the probability that their cash and liquid assets are less than or equal to their minimum requirement, [/2. When the portfolio selection is determined, the ratios of all assets and insurance premiums to equity are to be constrained by requiring that the insurers' minimum requirement on return and on cash and liquid assets could be violated, but at most at and c¢2 proportions of the time, respectively. The lower the values of ~1 and c¢2 chosen by the

Li--lnsurance and Investment Portfolio Model

580

insurers, the more conservative the insurers are considered, since they must then be taking into account far less possibility of loss. Hence, the higher the values of ~ and ct2, the higher the insurers' risk, and vice-versa. These probabilistic (chance) constraints can be written as follows P { x ~>fll} >/1 - ~ l ,

(9)

P { y >~f12} >~ 1 -~t2.

(10)

Insurers usually have an aspiration level, x0, for their return on equity. This value can be either imposed by an outside authority, as in the budgeting model of Stedry [26], or adopted by an individual for some activity as in the satisficing concept of Simon [25]. Insurers will regard it as satisfactory whenever this level is achieved. Therefore, the insurers' objective is, for given risk levels and the aspiration level, to maximize the probability of achieving this aspiration level, subject to several chance and deterministic constraints. In more precise form, insurers would like to Max P{Y/> x0} ai

with all the random variables. In this case, the decision maker wants to specify values for all his/her decisions after data observations are made on the random variables. Since the purpose of this research is to determine an optimal composition of underwriting and investment portfolios at the beginning of a given period, the class of zero order rules should be employed in our development. We can also adapt our use of zero order rules by interpreting them as a series of one-period-at-a-time applications with appropriate models, to allow for changing aspiration levels and other parameters, and regard these (in many situations) as approximations to the more complex solution procedures involved in developing higher order 'conditional' decision rules to deal with fullscale treatment of the dynamics. Proceeding in this one-period-at-a-time manner also allows us to bypass additional problems such as the sample size considerations which are encountered in dealing with multiple observations [11]. Let/1 = t/t, and di = tai for i = 1. . . . . m + n. It is shown in the Appendix that we can then replace the above problem with one quadratic programming problem as follows:

subject to constraints (4)-(7), (9) and (10). The model described above is actually a special form of satisficing P-model chance constrained programming introduced by Charnes and Cooper [8]. Since the problem involves random variables, it might be generally formulated in terms of choosing a suitable decision rule (at . . . . . am+,) = t~(rl . . . . .

rm+n)

fi,~li, t

s.t.

E ( Y c ) - fl, t >~ - ~

'(~,),

E(y) + ~-'(~)~ >1~t, m+n

£1i : t "[i=1

(II)

E

~ligi'

i=m+l md-n

d~ = 6t,

with a prescribed class of vector functions, 4>. An important question is what class of functions, tp, should be used in finding the optimal stochastic ratios (al . . . . . am+,) = There are two main classes of rules [8]: one is the class of zero order rules; the other is the class of linear rules. The class of zero order rules has been applied to situations where a decision maker (insurer) wants to know all of his/her decisions in advance o f any observations being made on the r a n d o m variables involved in the problem. In applying zero order rules, we work with decision variables as deterministic directly. The class of linear decision rules supposes that the decision variables have a linear relationship

(12)

Max E(~) - Xot

i=m+l

d~ -- ami" t >~ O,

i = l .....

re+n,

a~

i = l .....

re+n,

t -- gt~ >~ O,

1/> Var(~), /i 2/> Var(p), /i~>0,

t~>0.

where m+n i=1

i=m+l m+n

Y = Z ,~,(l + g,) + i=[

a,(l, + 4) + i=k+l

E

i=m+l

dig

Omega, Vol. 23, Number 5

and fi is the sample mean of fi with the sample size N. Algorithms for solving a class of problems as in (12) can be found in, for example, [12, 16]. We can see, if we look at (A3) in the Appendix, that the model (12) is consistent with the Markowitz model, since both efficiency definitions are the same in the sense of Pareto, i.e. a portfolio is called efficient if it is impossible to obtain a greater average (expected) return without incurring greater variance, or it is impossible to obtain smaller variance without giving up return on the average. The parameters for this problem are the expected returns on insurance lines and investments, the covariance matrix of these returns, the upper and lower bounds on these activities, and the insurers' risk and aspiration levels. The estimation of these parameters can be obtained by using the companies' historical data through some statistical technique. The simplified model, (12), can easily be solved through the use of any standard nonlinear programming software. 3. AN EMPIRICAL EXAMPLE

In estimating expected rates of returns, variances, and covariances for lines of insurance and assets and utilizing these values to generate efficient portfolios, the implicit assumption is that the insurer's expectations should be based on the estimated values. Best's PropertyCasualty Aggregates and Averages [3] was used to obtain data on ten lines for the period 1971-1990. Rates of return on stocks were calculated from Moody's composite index for ~For simplicity, we t r e a t b o n d s as liquid assets a n d s t o c k s as n o n l i q u i d assets. It is easy to c o n s i d e r m o r e d e t a i l e d liquid a n d n o n l i q u i d assets in the e x a m p l e .

581

the years 1971-1990, and the returns on bonds were calculated from the Federal Reserve Bulletin during 1971-1990. ~ The analysis can be extended to include more insurance lines and investments. The model implicitly assumes that we are deriving our efficient portfolios for 1991 based on estimations determined by pre-1990 operating results. The funds-generating factors of insurance lines were determined in the following manner. For each line, the ratio of the sum of loss reserves and unearned premium reserves to premiums earned in each year was obtained first. Then the averages of the ratios during the period 1971-1990 were used in the estimation of all the funds-generating factors. The annual rate of return on each insurance line was estimated by the results of that line minus the sum of the annual ratios of losses and expenses incurred to premiums. Then the annual rates of insurance profit for each line for the 20 year period were used to calculate the average (arithmetic) rate of return. The annual rate of return on stocks was calculated by adding the dividend yield to the relative change in Moody's composite index. Then the average rate of return was found by using the annual rates of return for this period. Yields to maturity, rather than holding period returns, were used for bonds because insurance companies value bonds at amortized cost and generally do not engage in active trading. The proportion of premiums allocated to each line and the proportion of investments in each asset were permitted to vary between their actual maximum and minimum values during the past twenty years. The correlation coefficients could be estimated by the average annual rate of underwriting profit for each line and rate of return for each investment. Thus the covariance matrix could be obtained by the correlation

Table I. Estimations of parameters Ratios to prem. Insurance lines and asset mix

Max

Min

1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12.

0.420 0.520 0.326 0.230 0.310 0.400 0.195 0.450 0.420 0.325 2.000 1.700

0.087 0.200 0.150 0.040 0.100 0.130 0.120 0.045 0.080 0.090 0.655 0.538

Allied lines Auto liability Auto physical damage Burglary Commercial multiple peril Fire Homeowners Inland marine Ocean marine Worker's compensation Liquid asset Nonliquid asset

Generating factors 0.428 0.658 0.400 0.320 0.445 0.450 0.350 0.200 0.705 0.800

E(ri)

Var(ri)

-0.005 -0.042 0.053 -0.013 0.010 -0.004 -0.025 0.008 -0.007 0.045 0.062 0.090

0.0080 0.0008 0.0038 0.0040 0.0105 0.0020 0.0030 0.0035 0.0040 0.0010 0.0030 0.0150

1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 1 I. 12.

Allied lines Auto liability Auto physical damage Burglary Commercial multiple peril Fire Homeowners Inland marine Ocean marine Worker's compensation Liquid asset Nonliquid asset

P { x * >i x o }

a (x*)

E(x*)

1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 1I. 12.

Insurance lines and asset mix

Allied lines Auto liability Auto physical damage Burglary Commercial multiple peril Fire Homeowners Inland marine Ocean marine Worker's compensation Liquid asset Nonliquid asset

0.0080 0.0006 0.0015 0.0010 0.0008 0.0015 0.0040 0.0020 0.0023 -0.0003 0.0006 0.0018

1

0.092 0.310 0.300 0.104 0.100 0.400 0.100 0.450 0.321 0.317 1.580 0.725 0.165 0.075 0.75

fl = 0.55

0.0008 0.0015 0.0040 0.0025 0.00028 0.00008 0.00090 0.00070 -0.00025 0.0007 0.0080

2

4

0.0020 0.0015 0.0015 0.0010 0.00020 0.0006 0.00047

6

0.0030 0.00026 0.0015 --0.00038 -- 0.0002 0.00006

7

0.092 0.310 0.297 0.080 0.100 0.400 0.095 0.427 0.095 0.317 1.435 0.658 0.148 0.033 0.67

B = 0.65

~t = 0.10

0.094 0.310 0.289 0.070 0.100 0.390 0.095 0.437 0.095 0.317 1.18 0.625 0.141 0.067 0.59

fl = 0.75 0.110 0.303 0.316 0.087 0.105 0.380 0.095 0.428 0.325 0.318 1.435 0.976 0.176 0.088 0.80

fl = 0.55

0.098 0.308 0.310 0.048 0.107 0.390 0.085 0.387 0.125 0.321 1.300 0.845 0.165 0.043 0.72

fl = 0.65

at = 0 . 2 0

Table 3. Efficient portfolios for different a and fl

0.0105 0.0009 0.0023 0.0035 0.00058 0.0034 0.0020 0.0015

5

Table 2. Covarianeematfix of rates of return

0.0038 -0.0015 0.0040 0.0620 -0.0006 -0.00015 0.0015 -0.0004 0.0017 0.0007 0.0015 -0.0007 0.0023 -0.0004 -0.0004 0.0012 0.0015 0.0015 -0.00007

3

0.62

0.100 0.304 0.304 0.036 0.120 0.290 0.084 0.398 0.158 0.321 1.00 0.695 0.156 0.069

fl = 0.75

0.0035 0.0020 --0.0003 0.0020 0.00075

8

0.120 0.276 0.326 0.042 0.120 0.375 0.075 0.410 0.330 0.320 0.867 1.50 0.184 0.097 0.86

fl = 0.55

0.0040 --0.0005 0.0015 0.0005

9

0.120 0.240 0.324 0.024 0.112 0.376 0.080 0.370 0.130 0.324 0.925 1.230 0.179 0.055 0.78

fl = 0.65

at = 0.30

0.0010 -- 0.0010 -0.0015

10

0.114 0.294 0.325 0.022 0.135 0.275 0.075 0.302 0.214 0.325 0.800 1.25 0.166 0.075 0.67

12

0.0150

fl = 0.75

0.0030 0.0018

11

I

t~

Omega, Vol. 23, Number 5

matrix and the standard deviations for all the insurance lines and assets [20]. We assume that the rate of cash returns on stock is 1.5% and its liquid part is 35%. The maximum and minimum values, funds-generating factors of insurance lines, the average rates of underwriting profit and investment income, as well as their associated variances for 16 years are illustrated in Table 1. The covariance matrix among these insurance lines and investments is shown in Table 2. In this empirical example, the portfolio model was solved for different return and liquidity risk levels, 0~ I : 0~2 : 0~, and minimum requirements on return and on cash and liquid assets, fl~ = f12 =/~. For demonstration, we only consider the case where the aspiration level is 10%. For each ~, we obtain different groups of satisficing portfolios under the three different minimum requirements on return and on cash and liquid assets. Table 3 illustrates the numerical results for three different ~ s using GINO software [21]. Referring to Table 3, several important results are observed. First, for each fixed value of fl, if the risk level, ~, increases, the impact of this on investments is to decrease the proportion of bond while increasing the stock portfolio. For example, for fl = 0.75, as the risk level moves from 0.10 to 0.30, the bond portfolio would fall from 1.18 to 0.80 (32% decrease) of equity while the stock would rise from 0.625 to 1.25 (100% increase) of equity. This is because the bonds' standard deviation is smaller than the stocks'. The smaller the insurers' risk, the more conservative the insurers. Second, for each value of fl, five ratios of premium to equity are increasing and the rest of them are decreasing as the risk level increases. Worker's compensation insurance, for example, increases. This is due to a relatively high rate of return (4.5% of premiums) and funds-generating factor (80.0%). The optimal ratios of premiums to equity for auto physical damage and commercial multiple peril are increasing as the insurers' risk level increases. These types of insurance lines are characterized by their positive rates of return and middle size of fundsgenerating factors. The optimal ratios of premiums to equity for allied lines and ocean marine are also non-decreasing as the insurer's risk level increases. This is due to their relatively high funds-generating factors and low standard deviations. Their relatively low expected losses (0.5 and 0.7%) are not sufficient to overcome

583

these favorable characteristics. The optimal ratio of premium to equity for auto liability is non-increasing as the insurer's risk level increases.This is because of its relatively high expected loss (4.2% of premiums) and positive covariances with most other insurance lines (90% of all insurance lines). Since the rates of returns on burglary and fire insurances are negative and their funds-generating factors are relatively low, the optimal ratios of premiums to equity for these two insurance lines are nonincreasing as the insurers' threshold risk level increases. Even though the rate of returns of inland marine is positive, the optimal ratio of premium to equity for this line is nonincreasing as the insurer's risk level increases. This is due to its relatively small rate of returns (0.8% of premiums) and its relatively small funds-generating factor (20.0%). Homeowner's insurance has a relatively high expected loss (2.5% of premiums) and standard deviation (5.5%), a low funds-generating factor, and positive covariances with most other insurance lines. Hence, its optimal ratio of premiums to equity decreases. As the risk increases, the insurers are willing to obtain high return by sacrificing those insurance lines which have negative rates of return or have relatively small rates of return and small funds-generating factors, and by increasing the insurance ratios of lines that have relatively high returns or relatively small losses with high funds-generating factors. Third, as the risk level increases, the optimal expected return on equity and the maximum chance of achieving the satisfactory level also increase. These results are intuitive and consistent with our sensitivity analysis: the higher the risk, the higher the optimal expected return and the objective value. The percentage changes of these optimal parameters are very pronounced as the insurer changes his/her risk level. 4. CONCLUDING REMARKS

The model developed in this paper employed Simon's 'satisficing' concept in determining optimal portfolios of insurance lines and investments for property-liability insurance companies. The model incorporated more realistic assumptions, such as generally distributed rates of return on insurance lines and investments, regulatory and institutional constraints,

584

Li--lnsurance and Investment Portfolio Model

as compared with conventional models in the portfolio literature. Aggregate data for US property-liability stock insurance companies were used to estimate expected returns, variances, and covariances which were then utilized to calculate the efficient insurance-investment portfolios for various alternative degrees of insurer's risk levels and minimum requirements on return and on cash and liquid assets. Further research is needed on the estimation of rate of return parameters for insurance companies, including the role of subjective information. Finally, since the model developed in this paper is static, it has a similar drawback to the classic mean-variance models, i.e. it cannot synchronize the cash flows of assets with liabilities over time. Therefore, our next research should be conducted to develop a dynamic optimization model for property-liability insurers as in [6].

~

m-t-n

ai = 1 + i=1

~

aigi,

i=m+l

m+n

Z

a i = 6,

i=m+l

ai >~ a rain,

i = 1. . . . .

a~X>~a;,

i=1 ..... re+n,

m + n

where m+n i=1

k

~,

m+n

fi=~a;(l+6)+

a;(l;+4)+

i= 1

i=k + I

~

a;6.

i=m+ I

Let us consider the following problem Max ~

(A2)

at,

S.t. ( E ( ~ ) -- x o ) / ( V a r ( ~ ) ) ~/2 >i ~,

E(~) + @-~(~, )(Var(ff)) ~/2t> fl, E ( y ) + (I)-1(~2) (Var(fi)) I/2/> f12,

APPENDIX

rn+n

ai = 1 +

Proof of the Equivalence Between the Original Model and (12) In general, the ?;s do not follow normal distributions. More usually, their distributions will be skewed to the right. However, the model developed in the text deals only with non-negative linear combinations of the ~;s with n or k + n terms. Typically, in real applications, n and k are both > 10. In such circumstances, under assumptions likely to be valid in practice, we can appeal to the Central Limit Theorem and assume the linear combinations have distributions well-approximated by the normal distribution. Therefore, utilizing the chance constrained programming technique, the original model in the main paper can be transformed to the problem

i=1

ai, 7

s.t. ( E ( £ ) - Xo)/(Var(£))'/2 >I @ - ' ( 7 ) ,

E(£) + @-l(~q )(Var(ff))~/2 i> ill, E(f) +

@-I(~2) (Var()7))l/2/>/~2,

(A1)

aigi,

m+,v

Z

a i = 3,

i=m+l

ai >~a ~ '~,

i = l .....

amax>~ai,

i=1 ..... m+n.

re+n,

Since @-~(7) is a strictly increasing function of 7, (A1) and (A2) are equivalent in the sense of the optimal solution structure. It is easy to see that (A2) is equivalent to the following problem Max (E(~) - x0)/(Var(~)) ~/2

(A3)

ai

S.t. E ( £ ) + @-l(ot~ )(Var(~)) ~/2~>fl,

E()7) + @-'(~0(Var(y))'/2 I>/~2, ~,

Max 7

Z i=m+l

m+n

a;= l + i= 1

~

aigi,

i=m+ [ m+n

a ; = 6, i=m+l

ai >>.a re'n,

i = l .....

m + n,

amax >~ ai,

i = l .....

m + n

We introduce two new non-negative spacer variables, ~o and /z, and then rewrite (A3) as

Omega, Vol. 23, Number 5

Max (E(g) -

Xo)/Co

(A4)

m,#,a i

S.t.

E(~) + ~ - t ( ~ ) c o >~/~l,

E ( : ) + • '(~2)~ >//%, ai = 1 + i=1

aigi, i=m+l

m+n

i=m+

I

ai >~ a mi~,

i = l .....

re+n,

a~X>>.a~,

i=1 ..... m+n,

m2 1> Var()?), p2/> Var()7), p~>0,

~>~0.

Utilizing the Charnes-Cooper transformation [7], let t = 1/o~, /~ = t p , and &i=ta~ for i -- 1. . . . , m + n. We can then replace the fractional programming problem (A4) by the quadratic programming problem (12) in the main paper. REFERENCES

1. Agnew NH, Agnew RA, Rasmussen J and Smith KF (1969) An application of chance constrained programming to portfolio selection in a casualty insurance finn. Mgmt Sci. 15, 512-520. 2. Bachman J (1978) Capitalization Requirements for Multiple Line Property-Liability Insurance Companies. S. S. Huebner Foundation, Philadelphia. 3. Best's Property-Casualty Aggregates and Averages (1991) A. M. Best Co., New York. 4. Brockett PL, Charnes A and Li SX (1992) Portfolio and line of business selection for a casualty insurance company with stochastic assurance. Working Paper, The University of Texas at Austin. 5. Brockett PL, Charnes A, Cooper WW, Kwon KH and Ruefli TW (1992) Chance constrained programming approach to empirical analyses of mutual fund investment strategies. Decis. Sci. 23, 385-408. 6. Carino DR, Kent T, Myers DH, Stacy C, Sylvanus M, Turner AL, Watanabe K and Ziemba WT (1994) The Russell-Yasuda kasai model: an asset/liability model for a Japanese insurance company using multistage stochastic programming. Interface 24, 29-49. 7. Charnes A and Cooper WW (1962) Programming with linear fractional functionals. Naval Res. Logist. Q. 3, 181-185. 8. Charnes A and Cooper WW (1963) Deterministic

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SX Li, School of Management and Business, Adelphi University, Garden City, Long Island, N Y 11530, USA.