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liability firms
109
Insurance: Mathematics and Economics 10 (1991) 109-123 North-Holland
Reinsurance retention levels for property/ liability firms A managerial portfolio selection framework * Yoram Kroll Hebrew University, Jerusalem, Israel Ruppia Institute, Emek Hefer, Israel University of Florida, Gainesville, FL 3261 I, USA
David Nye University of Florida, Gainesville, FL 32611, USA Received June 1988 Revised March 1990
Abstract: This paper presents a managerial portfolio selection model which analyzes the reinsurance decision of the ceding insurer. Alternative goal functions for management are assumed and then comparative statics, as well as illustrative numerical examples are used to develop testable implications concerning the optimal proportional reinsurance retention level for property/liability insurance firms.
Keyword:
Goal functions, Optima1 retention, Portfolio selection, Property-liability,
Proportional reinsurance.
1. Introduction In perfect capital markets, reinsurance appears to be redundant since stockholders of insurance firms can use portfolio diversification as a substitute for the risk reduction achieved by the insurer’s purchase of reinsurance [Doherty and Schlesinger (1983)]. However, this conclusion is contradicted by the virtual universal use of reinsurance by insurance companies. * This is due to incomplete markets as well as limited diversification possibilities for the original buyers of insurance contracts. For example, Doherty and Tinic (1980) noted that policy holders, who have limited diversification possibilities, would exhibit higher demand for insurance policies issued by firms with lower ruin risk. Mayers and Smith (1981) argue that industry structure and alternative insurance instruments can be explained by the various agency problems and conflicting interests in the insurance industry. They also note that reinsurance is a device to reduce the probability of bankruptcy and thereby attract potential insurance buyers. Accordingly, Mayers and Smith (1981, p. 430) state that ‘the probability of use of reinsurance increases (1) the smaller the insurance company, (2) the larger the maximum claim under the policy, (3) the higher the covariance of policy payoffs with the existing set of company policies and (4) if the insurance company is mutual’. * The authors wish to thank Phelim Boyle for his thorough review of our model, its assumptions and comparative statics analysis. His suggestions resulted in a substantially improved paper. ’ For example, in 1988, U.S. direct premiums written in property insurance were $211 billion. Reinsurance ceded totalled $142 billion and reinsurance assumed was $134 billion. Best’s Aggregates and Averages, A.M. Best Company, Oldwich, New Jersey, 1989. 0167-6687/91/$03.50
0 1991 - Elsevier Science Publishers B.V. (North-Holland)
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firms
This paper argues that in addition to policyholders, managers (agents) who make the reinsurance decision are also sensitive to ruin possibilities and are more exposed to firm risk than the stockholders of the firm. This follows from agency theory and the extensive empirical findings concerning the managers (agents) motive to reduce risk below what is preferred by owners (principals) [Amihud and Lev (1981), Blume et al. (1980), Fama (1980), Jensen and Meckling (1976), and Roy (1952)]. We assume that firm managers are risk averse and are compensated by shares in the firm ’ resulting in an undiversified portfolio for the managers. It can be argued this is reasonable since many firms (especially small to medium-sized companies) have owner-managers who wish to retain control of the firm and therefore will hold all or a large percentage of the firms shares in their portfolio. Often this results in a relatively undiversified portfolio for those individuals. This paper analyzes the reinsurance decision within the context of the overall portfolio of financial opportunities available to the firm. The contribution of this paper is to place the reinsurance decision within the context o’f both the insurance market and the capital markets. The bulk of the literature on optimal reinsurance has developed theories of the insurance firm with reference only to insurance markets. This study defines in broad terms the kinds of impacts that might be expected in an agency context from risk-shifting contracts, such as proportional reinsurance, when capital market transactions are permitted. No attempt is made to formulate a general equilibrium model of the insurance process that can accommodate all possible goals - indeed, such a paper is probably not attainable with out current state of knowledge. The objective in this paper is to gain insights into an extremely complex phenomenon and thereby stimulate further research - both empirical and theoretical. The manager’s objective function is formulated in two different ways: first, to minimize ruin for a target return goal and second, to maximize return for a given threshold of risk. 3 The paper confirms the reinsurance determinants hypothesized by Mayers and Smith, provides an indication of their relative importance and identifies other important variables. In Section 2 the model is developed and an expression for the optimal reinsurance proportion is derived. Section 3 presents the comparative statics analysis. In Section 4, the model is demonstrated using illustrative numerical examples to suggest which of the determinants for reinsurance may be found to be significant in future empirical studies. Section 5 contains our conclusions. 2. The model In this partial equilibrium model, it is assumed that the ceding insurance firm 4 has sold a predetermined amount of insurance and that the random claims on these policies will be paid on a known future date. Unlimited proportional reinsurance is available at a given price and the ceding firm can invest its funds in two types of financial assets: riskless and risky. The risky asset is not necessarily the CAPM ‘market portfolio’ since various institutional constraints may limit management in its asset selection. In any case, it is assumed that, in addition to the purchase of reinsurance to reduce risk, management can exchange risk for return by modifying the proportion of funds invested in the risky and riskless asset. Unlimited borrowing is also assumed. The opportunity to exchange risk for return in the financial market has an important role in determining the optimal reinsurance retention and hence the general market equilibrium price of reinsurance (although our partial equilibrium model does not derive an equilibrium insurance price). This is a one-period model and given these assumptions, the terminal wealth, IV, of the ceding firm is W=BR+{(l-Xq)P+S-B}X-(l-q&
(1)
’ If necessary, one can also assume that the managers receive a fixed amount of money as compensation in addition to their shares and this cost is embedded in the transactions cost of the policy. This additional assumption does not change the model or the conclusions presented in this paper. 3 The same or similar goals have been used by other researchers in this field. For example, Centeno (1985) utilized a goal of minimizing the coefficient of variation of net claims and minimizing the probability of ruin (1986). Zecchin (1987) and Gerber (1984a) assumed that the insurer sought to maximize expected utility. 4 A ceding insurance firm refers to a firm which is the buyer of insurance.
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where W B R h
= = = =
q P S X L
= = = = =
terminal equity one period hence, amount invested in the risk-free asset, one plus the risk-free rate of interest, the loading of the reinsurance firm (this is negotiable and determined in the market; normally it is greater than one), the amount of proportional reinsurance purchased by the firm on all policies, aggregate premiums = CP,, i = 1, 2,. l., n (net of expenses), initial insurance company equity (policyholder surplus), one plus the rate of return on the risky portfolio, total claims amount for the portfolio = Eli where 1; is equal to the claims amount of each insured. L is proportional to P.
Equation (1) shows that terminal wealth consists of investment returns from the riskless and risky assets, initial equity, S, and policyholder premiums, P, minus the reinsurance premium (XqP), and net aggregate claims (1 - q)L payable by the insurer. Management has two choice variables: the amount of reinsurance, q, and the mix between the amount invested in the riskless and risky assets, B and ((1 - Xq)P + S - B), respectively. 2.1. Alternative
managerial
goal functions
Two formulations of the objective function are examined. First, it is assumed that managers determine target (or threshold) expected return on equity and minimize the probability of ruin for this target return. If the return on the portfolios of risky assets and insurance policies can be approximated by a normal distribution, minimizing the probability of ruin is equivalent to minimizing the variance of terminal wealth. By scanning a wide range of target mean returns, one can cover the optimal decisions of many different managers with this type of goal function. The second formulation of the objective function is to maximize expected return on equity for a given threshold of risk (variance) of return on equity. Once again, scanning various risk thresholds permits many different managers to be included in this category. In the case of a mean-variance world, these two formulations lead to equivalent strategies. These goals are consistent with lexicographic utility functions under which decision makers first reach threshold levels of one important variable and then maximize the goal function which is defined over the remaining variable. 5 The wide range of admissible alternative goals in each of the cases suggests that if certain reinsurance policy determinants have similar effects for each goal, then it is more likely their effect can be detected empirically. 2.2. Optimal reinsurance
retention
Denote q* and q * * as the optimal reinsurance retentions which maximize each of the goal functions. The expression for each retention level is presented below in equations (2) and (3) respectively. 7 4
’
*_V(L)y2-[WO-R(S+P)+E(L)][XPR-E(L)]-P[W,-XPR-R(S+P)+2E(L)][E(X)-R] V(L)y’+[APR-E(L)]*+2p[hPR-E(L)][E(X)-R]
The ‘Safety First’ rule proposed by Roy (1952) is an example of a lexicographic rule. The derivation of the optimal retention is given in the appentix. Since reinsurance prices are given, our analysis represents only partial equilibrium conditions. Raviv (1979) and Doherty and Schlesinger (1983) analyse equilibrium prices of insurance contracts using the expected utility maximization principle but do not incorporate the other financial activities of the firm in their analysis.
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where Y = [E(X) - RI/4 n p= cov(X, L)/V(X).
and E(a) and V(e) and cov(., l) are the expectations, variances and covariances of the random variables in equation (1). Gamma (y) is the price of risk in the financial market and p measures the systematic risk of the insurance portfolio with respect to the risky portfolio X. Empirical findings indicate that claims on property/liability insurance policies are not significantly correlated with returns in financial markets [Cummins and Nye (1980)]. Therefore, in equations (3) and (4) below it is assumed for simplicity of presentation that cov( X, L) = 0.' Under the second goal formulation, management maximizes expected return on equity for a given risk (variance) and the optimal reinsurance retention, q * *, is: 4
%sW)
**=I-
(3)
[l +m(L)y2'
where u,,(W) = the threshold risk level as measured by the standard deviation of terminal wealth, and
XPR-E(L) 44
6=
I
(E(X)-@ u(x)
(4)
.
6 can be interpreted as the price of risk in the reinsurance market relative to the price of risk in the financial market. Note the ‘market’ price of risk, [E(X) - RI/u(X), affects the optimal reinsurance retention ratio even when independence between X and L is assumed. Additional impacts of various parameters on q* and 4 * * are presented in the next section. The numerical and graphical presentation in Section 4 demonstrates the potential magnitude of these effects.
3. Comparative
statics
The results of our comparative statics analysis conform with the relationships identified in earlier work [Mayers and Smith (1981)]. The effects reported below for q* also hold for q* * since these coincide for each particular risk-return point (a,,( IV), w,,) on the efficient frontier. Since q* and q* * have the same value at the same point they respond in the same way to a given change in one of the variables. 9~10 Continuing with the assumption that cov( X, L) = 0, q * can be rewritten as q*=
V(L)y2-[W-R(S+P)+E(L)][XPR-E(L)] V(L)y’+
(5)
[hPR-E(L)]’
Recall that y = (E(X) - R)/u( X) can be interpreted as the price of risk in the financial market. R is greater than one as is X under most circumstances. P - E(L) is positive or only slightly negative so it seems reasonable to assume that the ‘actuarial spread’, [ XPR - Fi( L)] is greater than or equal to zero. * In the numerical examples in Section 4 we note the impact of assuming cov( X, L) = 0. 9 The authors are grateful to Professor Boyle for noting this point and deriving the proof. Copies of the proof are available from either of the authors upon request. lo The efficient frontier can also be derived in terms of expected return and variance and is equal to
a;(w)
=
(w,-RS+PR(X-I)*)v(x)~(L) (E(X)-
R)‘v(L)+(XPR-E(L))*V(X)
’
Derivation of this equation is available upon request from either of the authors. I’ Note that one can replace W, in equations (2) or (5) by SK where K represents one plus the required rate of return on equity and K could be calculated from a capital asset pricing type model.
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Since W, represents the ‘target’ terminal equity goal of management, it is also reasonable to assume that W, > RS (otherwise, stockholders would earn below the riskless rate on their equity). ‘i Given these assumptions, q* decreases with: (i) (ii) (iii) (iv) (v)
the the the the the
reinsurance loading factor, X, of the reinsurer, reinsurance ‘actuarial spread’, [ XPR - E(L)], price of risk, y, amount of equity of the ceding firm, S, l2 target return on equity, K, where K = W/S.
Conversely, q*
increases with:
(i) the variance of aggregate loss, V(L), (ii) the actuarial spread, RP - E(L), of the ceding firm. These results conform with the predictions of Mayers and Smith. Namely, the reinsurance retention ratio is higher when the firm is smaller, and when the risk associated with a claim is larger, i.e., the variance of each claim is larger or there is a positive covariance between individual claims such that V(L) is larger.
4. Numerical illustrations This section provides numerical examples to illustrate the model. Hypothetical values were selected for each variable. In order to avoid using large numbers, firm size was kept small while preserving realistic relationships between the variables. The desirability of seeking positively correlated assets and liabilities for the insurance firm is noted first. The impact on terminal wealth of different reinsurance strategies is then examined followed by a sensitivity analysis of the impact of several important variables on the firm’s optimal reinsurance proportion. Input data for a hypothetical insurer is given in Table 1. Actual expenses are assumed equal to expected I2 This can be seen when we replace W, in equation (5) by SK and K > R (see footnote 10).
Table 1 Input data for a hypothetical insurer. Variable
Value
Expected loss per insured Underwriting profit Gross premium Number of policies sold Total premiums Initial equity Variance of loss per insured Reinsurer loading factor One plus the risk-free rate One plus the expected return on the market Variance of return on the market
$10 5% B $10.53 b 250 $2,632 $1,000 $400 1.1 1.06 1.12 0.015
a A number of financial theories have been developed in finance and later applied to insurance which attempt to determine an appropriate profit margin for insurance contracts. Examples of such theories are the capital asset pricing model and the option pricing model. A comprehensive and in-depth discussion of each of these models is provided in D’Arcy and Doherty (1988). b The insurer’s expenses are assumed paid at the start of the policy and therefore are unavailable for investment.
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expenses and to have been paid at time 0. Thus, current premiums reflect only expected losses and an expected underwriting profit of 5 percent. One year policies are issued at the beginning of the time period and the investable funds are assumed to be available, on average, for six months. Assume initially that the firm purchases no reinsurance (i.e., q = 0) which yields the following expression for the mean and variance of terminal equity: E(W,)=RB+E(X)A-E(L), V(W,)=A*V(X)+V(L)-2Acov(X,
L),
where A = P + S - B. If all funds are placed in the riskless asset, A = 0, B = P.-t S and expected terminal equity is R( P + S) - E(L) while it will be E( X)( P + S) - E(L) if all funds are put into the risky asset (B = 0). The standard deviation of terminal equity will depend upon the correlation between claims and the return on the risky portfolio. If all funds are invested in the riskless asset, then a(W) = u(L) regardless of the covariance between financial returns and insurance claims. If all funds are placed in the risky asset, then: a,=(P+S)a,-a,
if
(X,
L)=l,
uW= [(P+S)*U,+U,]~‘~
if
p(X,
L)=O,
u,=(P+S)u,+u,
if
p(X,
L) = -1,
where p( X, L) = cov( X, L)/u,u,. 4.1. Effect of covariance Figure 1 depicts the risk-return relation for different values of p( X, L) assuming no reinsurance. The horizontal axis represents expected terminal equity (in $1,000) and the vertical axis represents the standard deviation of terminal wealth. Line AB is the locus of points obtained by changing the mix of risky and riskless investments when p = 1. Point A occurs when all funds are placed in the riskless asset and the insurers risk is due only to
Slgmo
(W)
Rho = 1
800
600
R$l
/-‘-“iI
Fig. 1. Variability of terminal equity (4 = 0)
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115
random claims. Point B arises when all funds are placed in the risky asset. Lines AC and AD are achieved in a similar fashion for the cases p = 0 and p = - 1, respectively. Points on line AB dominate those on lines AC and AD, When only risky assets are considered for a portfolio, those with negative correlation are used to minimize risk for a given expected return. In the current case, positive correlation between risky assets and risky liabilities yields a lower risk level for a given expected return. Thus, if management of the insurer decided to invest in common stocks, it is more efficient to seek insurance lines with claims which are positively related to the market return. However, empirical evidence indicates that the correlation between the market and claims is close to zero for most lines [Cummins and Nye (1980)] so that assumption is made throughout the rest of the numerical examples. 4.2. Terminal
wealth under different reinsurance
When reinsurance
is purchased,
q >
strategies
0 and the risk-return
equations
E(W,)=BR+[(l-hq)P+S-B]E(X)-(l-q)E(L),
for terminal
equity
become (6)
V( W,) = A2V( X) + (1 - q)*V( L) - 2A(l
- q) cov( X, L),
(7)
where A = (1 - Aq)P + S - B and other variables are as before. Figure 2 graphs the risk-expected return trade off under three circumstances: when no reinsurance is purchased, a fixed amount (30 percent) of reinsurance is purchased and when reinsurance is optimized. The locus of terminal wealth points under the optimal reinsurance strategy was achieved by altering both the amount of reinsurance purchased and the mix between risky and riskless assets. The other two lines in Figure 2 were produced by changing only the asset mix. Figure 2 demonstrates the magnitude by which the optimal reinsurance strategy dominates other strategies at various wealth levels. Note, however, that tangencies also occur between the other graphs and the optimal graph, i.e., there is always one risk-return possibility from a non-optimal strategy which is as good as the optimal strategy. In the fixed reinsurances case, this occurs at a wealth point of about $1,100 while in the non-reinsurance case it occurs at about $1,250. Not all points on the graph are feasible. Points to the right of A on the optimal reinsurance graph
Standard
Devlotlon
(B)
1200 -
Optimal Rein. II
1000 -
W/O Rein. A " Fixed
Rein.
600 -
Termlnol
Weolth
($, 000)
Fig. 2. Hypothetical insurer risk-return space.
116
Y. Kroll, D. Nye / Retention levels for property/liability
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Table 2 Terminal equity for a hypothetical insurer under different reinsurance strategies. Terminal equity
Standard deviation of terminal equity
$1,080 1,100 1,120 1,140 1,160 1,180 1,200 1,220 1,240 1,260 1,280 1,300 1,320 1,340 1,360
Optimal reinsurance
Fixed reinsurance
Without reinsurance
$208 221 234 247 260 273 286 299 312 325 338 351 364 377 390
$231 222 242 285 342 408 479 552 628 705 783 862 941 1,020 1,100
$728 655 585 518 456 401 357 327 316 326 355 399 453 515 581
Source: Calculated using data from Table 1
involve negative values of q *, i.e., the firm must sell reinsurance and achieve the reinsurer’s profit spread in order to reach its wealth goal. When 30 percent reinsurance is purchased, only the points between B and C are feasible. To the left of point B, the firm borrows at the risky rate to cut its profit to the wealth goal while points to the right of C are attained by borrowing at the riskless rate. A similar situation occurs for points D and E on the no-reinsurance graph. Points to the left of D are inefficient and attained only by borrowing at the risky rate and buying the riskless asset. Points to the right of E are achieved by reversing this procedure. In summary, risk-return combinations between points B and A are feasible under the reinsurance optimizing strategy. These points offer rates of return of initial wealth between 10 percent and 24 percent (i.e. terminal wealth ranges from $1,100 to $1,240). This range of returns is inefficient if the firm’s managers decide not to purchase any reinsurance, although the degree of inefficiency becomes negligible at the 24 percent return figure. Similarly, the efficiency difference between the optimal reinsurance strategy and purchasing an arbitrarily selected proportion of reinsurance is minimal at the lower end of the feasible return range. 4.3. Sensitivity
analysis
In the following numerical examples the relative impact on q* of several variables is examined. Specifically, the relationship between the reinsurance level and the market price of risk, loading, variance of loss per insured, premium written to surplus ratio and underwriting profit is illustrated. I. Market price of risk. Equation (5) contains y which is the price of risk in the capital market. Differentiating (5) with respect to y yields
w
av=
2V(L)y[(XPR-E(L))][W,-RS+PR(X-l)] [Vet
+ [XPR - E(L)]~]’
(8) ’
Since each term is positive, the sign of equation (8) is positive but the impact of y on q * is relatively small. If the expected return on the risky asset is.8 percent, y equals 0.0816 and q* is 0.2311. An increase in the expected return to 16 percent causes q * to increase to 0.2806 or an increase of 5 percentage points. Thus,
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a very large increase in the market price of risk (rising from 0.08 to 0.41) induced a relatively modest increase in q *. If equation (5) is differentiated with respect to X, the result is 2. The reinsurance loading factor. g
= { -PR{V(L)y2[W,-RS+PR(h-l)] +(hRR-E(L))[V(L)+(W,-R(S+R)+E(L)(ARR-E(L))]}
V(L)y’[W,-RRS+RP(X-l)]
= -PR i
([Vet+
[APR-E(L)]~]~
+
(XPR - E(L)q*
(9)
[v(L)Y~+(APR-E(L)~]
Now the first term within the brackets is positive. The second term is positive as long as q* is positive. Hence, a necessary condition for the total expression to be negative is for q* to be positive. When equation (9) is evaluated, the relationship is as expected for values of A greater than 1, i.e., as reinsurance becomes more expensive, the optimal reinsurance proportion declines. For example, when the reinsurer charges 5 percent (X = 1.05), q* equals 0.351 and this declines to 0.155 when X is increased to 1.20. Under extremely competitive conditions, it is possible for A to be less than one, e.g., at times of high interest rates and excess capacity in the market. Under these conditions, the results are ambiguous. 3. Variance of loss per insured. This variable is important to insurer management because of its potential to control risk. First party insurance coverages are generally recognized as having less variance than liability lines so management can limit the variability of terminal equity by emphasizing first party lines in its book of business. If high variance lines must be written, then possibly reinsurance can be used to help control overall risk as management attempts to reach its profit goals. To derivative of q * with respect to V(L) is
-=a4* WL)
y’[XPR-E(L)][W,-RS+PR(X-l)] [V(L)+
[xPR-E(L)]‘]’
(10) ’
Equation (10) demonstrates there is a positive relationship between q* and the loss variance. An increase in variance per risk from $50 to $1,000 (on an expected loss of $10) causes an increase in the reinsurance proportion of 4.5 percentage points. This result suggests that, ceterus paribus, liability insurers should select reinsurance proportions which are not too different from those of property insurers. Assume that curve EF is the initial efficient frontier and that management has located at point A. As one moves from point E to point F along the curve, the amount of reinsurance purchased decreases. Assume an increase in loss variance takes place (with no change in the other variables) which causes the efficient frontier to shift to the right and the new curve becomes E’F’. If management had a return threshold to maintain, then, depending upon the actual value of the variables, reinsurance would either be unchanged or increase as management restructured the firm to locate at point C. Conversely, if any increase in risk was unacceptable, management would move to B by purchasing more reinsurance (and possibly rebalancing its asset portfolio) thereby leaving risk unchanged but reducing expected return. An empirical investigation concerning the impact of V(L) on the optimal reinsurance retention ratio may provide an indication about which goal function is more plausible. 4. Premiums written-to-surplus ratio. Another important question is the degree to which the optimal reinsurance proportion is affected by an increase in premiums written relative to initial surplus. This effect was measured by finding the derivative of q * with respect to the number of policies. The result shows that increased premium volume leads to increased demand for reinsurance. At a premium written to surplus
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Y. Kroll, D. Nye / Retention IeveIsfor property/liability firms
E(w)
E
E’
Sigma
(w)
Fig. 3. Change in reinsurance proportion when claim variance increases.
ratio of 1.6 to 1, the optimal reinsurance proportion is 12.9 percent. This increases to 33.4 percent at a premium to surplus ratio of 4.2 to 1. 5. Underwriting profit. The final variable analyzed is underwriting profit. As noted earlier, expenses are paid at the beginning of the time period and are assumed not to deviate from expected expenses. Therefore, the gross premium is equal to the expected loss divided by one minus the profit rate. One minus underwriting profit is the claims ratio, (r, given by a: = E( L)/P. The positive relationship between underwriting profit and reinsurance demand conforms with expectations and may be interpreted as relaxing a budget constraint. Increased underwriting profit increases the
Table 3 Asset allocations for a hypothetical insurer under different reinsurance strategies. Wealth goal
$1,080 1,110 1,120 1,140 1,160 1,180 1,200 1,220 1,240 1,260 1,280 1,300 1,320 1,340 1,360
Fixed (30%) reinsurance
No reinsurance
4*
Risky asset
Riskless asset
Risky asset
Riskless asset
Risky asset
Riskless asset
0.35 0.31 0.27 0.23 0.19 0.15 0.11 0.07 0.03 - 0.01 - 0.05 -0.10 -0.14 -0.18 - 0.22
$270 287 304 320 337 354 371 388 404 421 438 455 472 488 505
$2,348 2,448 2,549 2,649 2,750 2,850 2,950 3,051 3,151 3,251 3,352 3,452 3,553 3,653 3,753
$-535 132 798 1,465 2,132 2,798 3.465 4,132 4,798 5,465 6,132 6,798 7,465 8,132 8,798
$3,298 2,632 1,965 1,298 632 -35 -702 - 1,368 - 2,035 - 2,702 - 3,368 - 4,035 - 4,702 - 5,368 - 6,035
$-5,351 - 4,684 - 4,018 - 3,351 - 2,684 - 2,018 - 1,352 - 684 -18 649 1,316 1,982 2,649 3,316 3,982
$8,982 8,316 7,649 6,982 6,316 5.649 4,982 4,316 3,649 2,982 2,316 1,649 982 316 - 351
Optimal reinsurance
Source: Calculated using data from Table 1.
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‘affordability’ of reinsurance and management is able to purchase additional coverage without sacrificing its profit goal. Underwriting losses cause the firm to fall short of its return goal and this is remedied by purchasing less reinsurance and eventually becoming a seller of reinsurance.
5. Conclusion This paper has developed a model for finding the optimal amount of proportional reinsurance. The model does not attempt to represent all goal structures but simply two reasonable goals. The firm’s managers are assumed to either minimize risk (variance) for a given expected return on equity, or to maximize terminal equity for a given risk level. The firm has two ways to control risk; alter the mix between the risky and riskless assets and purchase proportional reinsurance. Expressions for the optimal amount of reinsurance for the two formulations of the goal function were derived and included the significant exogenous and endogenous variables. Exogenous variables were the risk-free rate, expected return and standard deviation of return for the risky asset and transaction costs of the reinsurance market. Endogenous variables were underwriting profit, premiums written-to-surplus ratio and the variance of loss per insured. Numerical values for each variable were selected and an analysis of the model was performed. It was noted that insurance lines which are positively correlated with the asset market are most desirable because the model incorporates a risky asset (the market) and a risky liability (the portfolio of policies) rather than two risky assets. Empirical evidence suggests that the correlation between insurance lines and the market is not significantly different from zero so that assumption was adopted. The optimal reinsurance proportion was analyzed to determine its sensitivity to a number of different factors. The exogenous variables had a relatively mild effect upon the optimal reinsurance proportion except for reinsurer loadings. Significant endogenous variables were the premiums written-to-surplus ratio and underwriting profit. Variance of loss per insured had approximately the same impact as the market price of risk. These results suggest some interesting avenues for future research. Empirical work needs to be undertaken to measure insurers actual goal specifications and to provide a test of the validity of the alternative models. If an operational model can be developed, it may assist insurer management by identifying important variables in the decision making process and exploiting their interrelationships to achieve the firm’s goals. Finally, research is needed on incorporating the excess of loss reinsurance decision into the framework developed in this paper. This is difficult because we do not have a good theory of how the price of reinsurance depends upon the retention limit and this is needed in order to work out the first-order conditions associated with the choice of the optimal retention limit. Moreover, even if we have the functional form of the excess of loss premium, it is very unlikely that a closed form solution could be obtained since the excess of loss retention limit enters the analysis in a non-linear fashion as the limit of an integral.
Appendix A.I. The development
of q* and q * *
Under the first goal function, management minimizes the probability of ruin for a given expected return. Namely, the manager is assumed to choose a target value of E(W) and minimize the risk, V(W), from seeking this target return. In this model, disk depends upon two factors: the amount invested in the risky asset and the amount of reinsurance. Thus, risk is minimized with respect to B and q for a given target equity and the problem is written as YiYw) subject to
(A-1) E(W)
= W,,
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120
firms
where 4 = the proportion B = the amount of W, = target terminal E(W) and V(W) are in the text:
of reinsurance purchased, the riskless asset purchased or borrowed, equity. obtained by taking the expected value and variance of W as expressed in equation (1)
E(W)=BR+{(l-Xq)P+S-B}E(X)-(l-q)E(L),
(A-2)
V(W)=A2V(X)+(1-q)2V(L)-2A(1-q)cov(X,
L),
(A-3) (A-4)
A=(l-Xq)P+S-B.
For each value of q, the optimal investment in the riskless asset (B *) which satisfies E(W) = W, can be calculated from (A.2) to.find B* = w, - E(X)[S
+ (I- Xq)Rl R-E(X)
+ (I-
q)E(L)
the constraint
(A-5)
This change permits the minimization problem to be rewritten as MinV( W)
(A4
4
subject to
B = B*
The first-order conditions for a maximum in (A.6) are
Ww)
aA* =2A*V(X)T-2(1-q*)V(L)+2A*cov(X,
a4 B=
B*,
L),
(A-7)
q=q*,
-2(1-q)
cov(x,
L)F
=o,
where A* is as given in equation (A.4) when B * = B. Substituting the expression for B* (A.4) yields W,-E(X)[S+(l-Xq)P]
A*=(l-Xq)P+S-
+(l-q)E(L)
R-E(X)
from (A.5) into
(‘4.8)
Eq. (A.8) can be rewritten as + W,-R(S+P)+E(L) E(X)-R
.
(A-9)
For simplicity, denote D=
Af’R-E(L) E(X)-R
F=
w,-R(S+P)
(A.lO)
’
+ E(L)
E(X)-R
’
(A.ll)
Thus, (A.8) can be rewritten as (A.12)
A*=qD+F
and aA*/aq B=B*,
= D,
(A.13)
Y. Kroll, D. Nye / Retention levels for property/liability
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121
Thus, the first-order conditions in equation (A.7) can be written as F=[,*D+F]V(X)D-(I-q*)V(L)+[q*D+F]cov(X,
L)-(1-q*)cov(X,
= 0,
q* =
(A.14)
B= B*.
q=q*,
Thus, q*
L)D
which solves the first-order condition is V(L)-V(X)FD+cov(X,
L)(D-F)
V(X)D2+V(L)+2Dcov(X,
L)
(A.15)
’
By using the definitions of D and F and rearranging terms we obtain q*=
{V(L)(E(X)-R)2-V(X)[W,-R(S+P)+E(L)][XPR-E(L)] +cov(X,
L)[APR-2E(L)-
w,+R(S+P)](E(X)-R)}
x{V(L)[(E(X)-R]2+V(X)[hPR-E(L)]2 +2cov(X,
L)[hPR-E(L)](E(X)-R)}-l.
(~.16)
Denote y as the price of risk; Y = (E(X)
(A.17)
-@/e(X),
and /3 as the regression coefficient of L on X: P=cov(X,
(A.18)
L)/V(X),
to obtain q* =
V(L)y2-[[W,-R(S+P)+E(L)][XPR-E(L)]-8[W,-hPR-R(S+P)+2E(L)][E(X)-R]
(A.19)
V(L)y*+[hPR-E(L)]*+2/3[hPR-E(L)](E(X)-R)
Equation (A.19) is equivalent to equation (2) in the text. A.2. The development
of q * *
According to the second goal function, the maximization problem is (A.20)
YyE(W) subject to
V(W)
= V,,(W).
Assume for simplicity that cov( X, L) = 0 which reduces equation (A.3) to V(W)
+ (1 -q)2V(L).
=A’(V(X)
(A.21)
According to (A.21) the A,, that maintains V(W) = V,,(W) is A = + -
V,(W) [
-
(1 -
4)2V(L)
V(X)
1. 1’2
(A.22)
According to equation (A.4) and equation (A.22) the corresponding B,=(l-Aq)P+Sf
V,(W)
- (1 - q)2v( L) V(X)
1 1’2
B,, is: (A.23)
122
Y. Kroll, D. Nye / Retention levels
for property/liability
firms
In order to obtain the maximum and not the minimum value of A, in (A.22), the positive square root should be taken (this is because zero variance can be obtained only for A, = 0 and E(X) > R).Thus,the maximization problem in (A~20) can be restated as M$RB,+E(X)A,-(l-q)E(L)]. The first-order
condition
(A.24)
is
dE(W) =R%+E(X)%+E(L)=O.
(A.25)
a4
By taking
the derivative
E(X)-R o(X) Define
of B, and A, and rearranging V(L)(I
terms we find that equation
(A.25) is equivalent
- 4)
to
(~.26)
[v,(w)-(I-q)2~(Q]1’z
=XPR-E(L).
6 as 8 = ARR- E(L) E(X)-R
and denote
o(X)
(A.27)
I V(L)
p = 1 - q to express
the first-order
condition
as (~.28)
s = [V,(W) Squaring (denoted
-pplv(L)y~
both sides of (A.28), rearranging by q* *) which is
4 **=1*
which is identical
terms and replacing
1
p with q, yields the optimum
value of
kJw> [l + SW( L)y2
q
(A.29) ’
to (3) in the text.
References Amihud, Y. and B. Lev (1981). Risk reduction as a managerial motive for conglomerate mergers. The Bell Journal of Economics 12, 605-617. Andreadakis, M. and H.R. Waters (1980). The effect of reinsurance on the degree of risk associated with an insurer’s portfolio. Astin Bulletin 11, 119-135. AyIing, D.E. (1984). Underwriting Decisions under Uncertainty: The Catastrophe Market. Gower, Famborough. Blume, M.E., I. Friend and R.W. Westerfield (1980). Impediment to capital formation. Rodney L. White Center for Financial Research, The Wharton School, University of Pennsylvania, Philadelphia, PA. Borch, K. (1976). The Mathematical Theory of Insurance. Heath, Lexington, MA. Centeno, L. (1985). On combining quota-share and excess of loss, Astin Bulletin 1.5, 49-63. Centeno, L. (1986). Measuring the effects of reinsurance by the adjustment coefficient. Insurance: Mathematics and Economics 5, 169-182. Cummins, J.D. and David J. Nye (1980). The stochastic characteristics of property-liability insurance company underwriting profits. Journal of Risk and Insurance 47, 61. D’Arcy, Stephen P. and Neil A. Doherty (1988). The Financial Theory of Pricing Property-Liability Insurance Contracts. Monograph no. 15. S.S. Huebner Foundation for Insurance Education, Wharton School, University of Pennsylvania, Philadelphia, PA. Doherty, N.A. and H. Schlesinger (1983). Optimal insurance in incomplete markets. Journal of Political Economy 91. Doherty, N.A. and S.M. Tinic (1980). Reinsurance under conditions of capital market equilibrium: A note. Journal of Finance 36, no. 4, 949.
Fama, E.F. (1980). Agency problems and the theory of the firm. Journal of Political Economy 88, 288-307. Gerber, Hans U. (1984a). Chains of reinsurance. Insurance: Mathematics and Economics 3, 43-48. Gerber, H.U. (1984b). Equilibria in a porportional reinsurance market, Insurance: Mathematics and Economics
3, 97-100.
Y. Kroli, D. Nye / Retention levels for property/liability
firms
123
Jensen, M.C. and W.H. Meckling (1976). Theory of the firm: Managerial behavior, agency costs and ownership structure, Journal of Financial Economics 3, 305-398. Mayers, D. and C.W. Smith (1981). Contractual provisions, organizational structure, and conflict control in insurance markets. Journal of Business 54, 407-433. Roy, A.D. (1952). Safety first and the holding of assets. Econometrica 20, 431-443. Roy, Yves and J. David Cummins (1985). A stochastic simulation model for reinsurance decision making by ceding companies. In J. David Cummins, ed., Strategic Planning and Modeling in Property-Liability Insurance. Kluwer-NiJhoff F’ublishing, Boston, MA. Raviv, A. (1979). The design of an optimal insurance policy. American Economic Review 69, 223-239. Waters, H.R. (1983). Some mathematical aspects of reinsurance. Insurance; Mathematics and Economics 2, 17-26. Zecchin, Marco (1987). Calculation of the maximum in XL reinsurance. Insurance: Mathematics and Economics 6, 169-178.
Insurance: Mathematics and Economics 10 (1991) 109-123 North-Holland
Reinsurance retention levels for property/ liability firms A managerial portfolio selection framework * Yoram Kroll Hebrew University, Jerusalem, Israel Ruppia Institute, Emek Hefer, Israel University of Florida, Gainesville, FL 3261 I, USA
David Nye University of Florida, Gainesville, FL 32611, USA Received June 1988 Revised March 1990
Abstract: This paper presents a managerial portfolio selection model which analyzes the reinsurance decision of the ceding insurer. Alternative goal functions for management are assumed and then comparative statics, as well as illustrative numerical examples are used to develop testable implications concerning the optimal proportional reinsurance retention level for property/liability insurance firms.
Keyword:
Goal functions, Optima1 retention, Portfolio selection, Property-liability,
Proportional reinsurance.
1. Introduction In perfect capital markets, reinsurance appears to be redundant since stockholders of insurance firms can use portfolio diversification as a substitute for the risk reduction achieved by the insurer’s purchase of reinsurance [Doherty and Schlesinger (1983)]. However, this conclusion is contradicted by the virtual universal use of reinsurance by insurance companies. * This is due to incomplete markets as well as limited diversification possibilities for the original buyers of insurance contracts. For example, Doherty and Tinic (1980) noted that policy holders, who have limited diversification possibilities, would exhibit higher demand for insurance policies issued by firms with lower ruin risk. Mayers and Smith (1981) argue that industry structure and alternative insurance instruments can be explained by the various agency problems and conflicting interests in the insurance industry. They also note that reinsurance is a device to reduce the probability of bankruptcy and thereby attract potential insurance buyers. Accordingly, Mayers and Smith (1981, p. 430) state that ‘the probability of use of reinsurance increases (1) the smaller the insurance company, (2) the larger the maximum claim under the policy, (3) the higher the covariance of policy payoffs with the existing set of company policies and (4) if the insurance company is mutual’. * The authors wish to thank Phelim Boyle for his thorough review of our model, its assumptions and comparative statics analysis. His suggestions resulted in a substantially improved paper. ’ For example, in 1988, U.S. direct premiums written in property insurance were $211 billion. Reinsurance ceded totalled $142 billion and reinsurance assumed was $134 billion. Best’s Aggregates and Averages, A.M. Best Company, Oldwich, New Jersey, 1989. 0167-6687/91/$03.50
0 1991 - Elsevier Science Publishers B.V. (North-Holland)
Y. Kroll, D. Nye / Retention levels for property/liability
110
firms
This paper argues that in addition to policyholders, managers (agents) who make the reinsurance decision are also sensitive to ruin possibilities and are more exposed to firm risk than the stockholders of the firm. This follows from agency theory and the extensive empirical findings concerning the managers (agents) motive to reduce risk below what is preferred by owners (principals) [Amihud and Lev (1981), Blume et al. (1980), Fama (1980), Jensen and Meckling (1976), and Roy (1952)]. We assume that firm managers are risk averse and are compensated by shares in the firm ’ resulting in an undiversified portfolio for the managers. It can be argued this is reasonable since many firms (especially small to medium-sized companies) have owner-managers who wish to retain control of the firm and therefore will hold all or a large percentage of the firms shares in their portfolio. Often this results in a relatively undiversified portfolio for those individuals. This paper analyzes the reinsurance decision within the context of the overall portfolio of financial opportunities available to the firm. The contribution of this paper is to place the reinsurance decision within the context o’f both the insurance market and the capital markets. The bulk of the literature on optimal reinsurance has developed theories of the insurance firm with reference only to insurance markets. This study defines in broad terms the kinds of impacts that might be expected in an agency context from risk-shifting contracts, such as proportional reinsurance, when capital market transactions are permitted. No attempt is made to formulate a general equilibrium model of the insurance process that can accommodate all possible goals - indeed, such a paper is probably not attainable with out current state of knowledge. The objective in this paper is to gain insights into an extremely complex phenomenon and thereby stimulate further research - both empirical and theoretical. The manager’s objective function is formulated in two different ways: first, to minimize ruin for a target return goal and second, to maximize return for a given threshold of risk. 3 The paper confirms the reinsurance determinants hypothesized by Mayers and Smith, provides an indication of their relative importance and identifies other important variables. In Section 2 the model is developed and an expression for the optimal reinsurance proportion is derived. Section 3 presents the comparative statics analysis. In Section 4, the model is demonstrated using illustrative numerical examples to suggest which of the determinants for reinsurance may be found to be significant in future empirical studies. Section 5 contains our conclusions. 2. The model In this partial equilibrium model, it is assumed that the ceding insurance firm 4 has sold a predetermined amount of insurance and that the random claims on these policies will be paid on a known future date. Unlimited proportional reinsurance is available at a given price and the ceding firm can invest its funds in two types of financial assets: riskless and risky. The risky asset is not necessarily the CAPM ‘market portfolio’ since various institutional constraints may limit management in its asset selection. In any case, it is assumed that, in addition to the purchase of reinsurance to reduce risk, management can exchange risk for return by modifying the proportion of funds invested in the risky and riskless asset. Unlimited borrowing is also assumed. The opportunity to exchange risk for return in the financial market has an important role in determining the optimal reinsurance retention and hence the general market equilibrium price of reinsurance (although our partial equilibrium model does not derive an equilibrium insurance price). This is a one-period model and given these assumptions, the terminal wealth, IV, of the ceding firm is W=BR+{(l-Xq)P+S-B}X-(l-q&
(1)
’ If necessary, one can also assume that the managers receive a fixed amount of money as compensation in addition to their shares and this cost is embedded in the transactions cost of the policy. This additional assumption does not change the model or the conclusions presented in this paper. 3 The same or similar goals have been used by other researchers in this field. For example, Centeno (1985) utilized a goal of minimizing the coefficient of variation of net claims and minimizing the probability of ruin (1986). Zecchin (1987) and Gerber (1984a) assumed that the insurer sought to maximize expected utility. 4 A ceding insurance firm refers to a firm which is the buyer of insurance.
Y. Kroll, D. Nye / Reienlion levelsfor property/liability firm
111
where W B R h
= = = =
q P S X L
= = = = =
terminal equity one period hence, amount invested in the risk-free asset, one plus the risk-free rate of interest, the loading of the reinsurance firm (this is negotiable and determined in the market; normally it is greater than one), the amount of proportional reinsurance purchased by the firm on all policies, aggregate premiums = CP,, i = 1, 2,. l., n (net of expenses), initial insurance company equity (policyholder surplus), one plus the rate of return on the risky portfolio, total claims amount for the portfolio = Eli where 1; is equal to the claims amount of each insured. L is proportional to P.
Equation (1) shows that terminal wealth consists of investment returns from the riskless and risky assets, initial equity, S, and policyholder premiums, P, minus the reinsurance premium (XqP), and net aggregate claims (1 - q)L payable by the insurer. Management has two choice variables: the amount of reinsurance, q, and the mix between the amount invested in the riskless and risky assets, B and ((1 - Xq)P + S - B), respectively. 2.1. Alternative
managerial
goal functions
Two formulations of the objective function are examined. First, it is assumed that managers determine target (or threshold) expected return on equity and minimize the probability of ruin for this target return. If the return on the portfolios of risky assets and insurance policies can be approximated by a normal distribution, minimizing the probability of ruin is equivalent to minimizing the variance of terminal wealth. By scanning a wide range of target mean returns, one can cover the optimal decisions of many different managers with this type of goal function. The second formulation of the objective function is to maximize expected return on equity for a given threshold of risk (variance) of return on equity. Once again, scanning various risk thresholds permits many different managers to be included in this category. In the case of a mean-variance world, these two formulations lead to equivalent strategies. These goals are consistent with lexicographic utility functions under which decision makers first reach threshold levels of one important variable and then maximize the goal function which is defined over the remaining variable. 5 The wide range of admissible alternative goals in each of the cases suggests that if certain reinsurance policy determinants have similar effects for each goal, then it is more likely their effect can be detected empirically. 2.2. Optimal reinsurance
retention
Denote q* and q * * as the optimal reinsurance retentions which maximize each of the goal functions. The expression for each retention level is presented below in equations (2) and (3) respectively. 7 4
’
*_V(L)y2-[WO-R(S+P)+E(L)][XPR-E(L)]-P[W,-XPR-R(S+P)+2E(L)][E(X)-R] V(L)y’+[APR-E(L)]*+2p[hPR-E(L)][E(X)-R]
The ‘Safety First’ rule proposed by Roy (1952) is an example of a lexicographic rule. The derivation of the optimal retention is given in the appentix. Since reinsurance prices are given, our analysis represents only partial equilibrium conditions. Raviv (1979) and Doherty and Schlesinger (1983) analyse equilibrium prices of insurance contracts using the expected utility maximization principle but do not incorporate the other financial activities of the firm in their analysis.
Y. Kroll, D. Nye / Retention levels for property/liability firms
112
where Y = [E(X) - RI/4 n p= cov(X, L)/V(X).
and E(a) and V(e) and cov(., l) are the expectations, variances and covariances of the random variables in equation (1). Gamma (y) is the price of risk in the financial market and p measures the systematic risk of the insurance portfolio with respect to the risky portfolio X. Empirical findings indicate that claims on property/liability insurance policies are not significantly correlated with returns in financial markets [Cummins and Nye (1980)]. Therefore, in equations (3) and (4) below it is assumed for simplicity of presentation that cov( X, L) = 0.' Under the second goal formulation, management maximizes expected return on equity for a given risk (variance) and the optimal reinsurance retention, q * *, is: 4
%sW)
**=I-
(3)
[l +m(L)y2'
where u,,(W) = the threshold risk level as measured by the standard deviation of terminal wealth, and
XPR-E(L) 44
6=
I
(E(X)-@ u(x)
(4)
.
6 can be interpreted as the price of risk in the reinsurance market relative to the price of risk in the financial market. Note the ‘market’ price of risk, [E(X) - RI/u(X), affects the optimal reinsurance retention ratio even when independence between X and L is assumed. Additional impacts of various parameters on q* and 4 * * are presented in the next section. The numerical and graphical presentation in Section 4 demonstrates the potential magnitude of these effects.
3. Comparative
statics
The results of our comparative statics analysis conform with the relationships identified in earlier work [Mayers and Smith (1981)]. The effects reported below for q* also hold for q* * since these coincide for each particular risk-return point (a,,( IV), w,,) on the efficient frontier. Since q* and q* * have the same value at the same point they respond in the same way to a given change in one of the variables. 9~10 Continuing with the assumption that cov( X, L) = 0, q * can be rewritten as q*=
V(L)y2-[W-R(S+P)+E(L)][XPR-E(L)] V(L)y’+
(5)
[hPR-E(L)]’
Recall that y = (E(X) - R)/u( X) can be interpreted as the price of risk in the financial market. R is greater than one as is X under most circumstances. P - E(L) is positive or only slightly negative so it seems reasonable to assume that the ‘actuarial spread’, [ XPR - Fi( L)] is greater than or equal to zero. * In the numerical examples in Section 4 we note the impact of assuming cov( X, L) = 0. 9 The authors are grateful to Professor Boyle for noting this point and deriving the proof. Copies of the proof are available from either of the authors upon request. lo The efficient frontier can also be derived in terms of expected return and variance and is equal to
a;(w)
=
(w,-RS+PR(X-I)*)v(x)~(L) (E(X)-
R)‘v(L)+(XPR-E(L))*V(X)
’
Derivation of this equation is available upon request from either of the authors. I’ Note that one can replace W, in equations (2) or (5) by SK where K represents one plus the required rate of return on equity and K could be calculated from a capital asset pricing type model.
Y. Kroil, D. Nye / Retention levels for property/liability
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113
Since W, represents the ‘target’ terminal equity goal of management, it is also reasonable to assume that W, > RS (otherwise, stockholders would earn below the riskless rate on their equity). ‘i Given these assumptions, q* decreases with: (i) (ii) (iii) (iv) (v)
the the the the the
reinsurance loading factor, X, of the reinsurer, reinsurance ‘actuarial spread’, [ XPR - E(L)], price of risk, y, amount of equity of the ceding firm, S, l2 target return on equity, K, where K = W/S.
Conversely, q*
increases with:
(i) the variance of aggregate loss, V(L), (ii) the actuarial spread, RP - E(L), of the ceding firm. These results conform with the predictions of Mayers and Smith. Namely, the reinsurance retention ratio is higher when the firm is smaller, and when the risk associated with a claim is larger, i.e., the variance of each claim is larger or there is a positive covariance between individual claims such that V(L) is larger.
4. Numerical illustrations This section provides numerical examples to illustrate the model. Hypothetical values were selected for each variable. In order to avoid using large numbers, firm size was kept small while preserving realistic relationships between the variables. The desirability of seeking positively correlated assets and liabilities for the insurance firm is noted first. The impact on terminal wealth of different reinsurance strategies is then examined followed by a sensitivity analysis of the impact of several important variables on the firm’s optimal reinsurance proportion. Input data for a hypothetical insurer is given in Table 1. Actual expenses are assumed equal to expected I2 This can be seen when we replace W, in equation (5) by SK and K > R (see footnote 10).
Table 1 Input data for a hypothetical insurer. Variable
Value
Expected loss per insured Underwriting profit Gross premium Number of policies sold Total premiums Initial equity Variance of loss per insured Reinsurer loading factor One plus the risk-free rate One plus the expected return on the market Variance of return on the market
$10 5% B $10.53 b 250 $2,632 $1,000 $400 1.1 1.06 1.12 0.015
a A number of financial theories have been developed in finance and later applied to insurance which attempt to determine an appropriate profit margin for insurance contracts. Examples of such theories are the capital asset pricing model and the option pricing model. A comprehensive and in-depth discussion of each of these models is provided in D’Arcy and Doherty (1988). b The insurer’s expenses are assumed paid at the start of the policy and therefore are unavailable for investment.
Y. Kroll, D. Nye / Retention levels for property/liability firms
114
expenses and to have been paid at time 0. Thus, current premiums reflect only expected losses and an expected underwriting profit of 5 percent. One year policies are issued at the beginning of the time period and the investable funds are assumed to be available, on average, for six months. Assume initially that the firm purchases no reinsurance (i.e., q = 0) which yields the following expression for the mean and variance of terminal equity: E(W,)=RB+E(X)A-E(L), V(W,)=A*V(X)+V(L)-2Acov(X,
L),
where A = P + S - B. If all funds are placed in the riskless asset, A = 0, B = P.-t S and expected terminal equity is R( P + S) - E(L) while it will be E( X)( P + S) - E(L) if all funds are put into the risky asset (B = 0). The standard deviation of terminal equity will depend upon the correlation between claims and the return on the risky portfolio. If all funds are invested in the riskless asset, then a(W) = u(L) regardless of the covariance between financial returns and insurance claims. If all funds are placed in the risky asset, then: a,=(P+S)a,-a,
if
(X,
L)=l,
uW= [(P+S)*U,+U,]~‘~
if
p(X,
L)=O,
u,=(P+S)u,+u,
if
p(X,
L) = -1,
where p( X, L) = cov( X, L)/u,u,. 4.1. Effect of covariance Figure 1 depicts the risk-return relation for different values of p( X, L) assuming no reinsurance. The horizontal axis represents expected terminal equity (in $1,000) and the vertical axis represents the standard deviation of terminal wealth. Line AB is the locus of points obtained by changing the mix of risky and riskless investments when p = 1. Point A occurs when all funds are placed in the riskless asset and the insurers risk is due only to
Slgmo
(W)
Rho = 1
800
600
R$l
/-‘-“iI
Fig. 1. Variability of terminal equity (4 = 0)
Y. Kroll, D. Nye / Reteniion Ieuels for property / liabilityfirms
115
random claims. Point B arises when all funds are placed in the risky asset. Lines AC and AD are achieved in a similar fashion for the cases p = 0 and p = - 1, respectively. Points on line AB dominate those on lines AC and AD, When only risky assets are considered for a portfolio, those with negative correlation are used to minimize risk for a given expected return. In the current case, positive correlation between risky assets and risky liabilities yields a lower risk level for a given expected return. Thus, if management of the insurer decided to invest in common stocks, it is more efficient to seek insurance lines with claims which are positively related to the market return. However, empirical evidence indicates that the correlation between the market and claims is close to zero for most lines [Cummins and Nye (1980)] so that assumption is made throughout the rest of the numerical examples. 4.2. Terminal
wealth under different reinsurance
When reinsurance
is purchased,
q >
strategies
0 and the risk-return
equations
E(W,)=BR+[(l-hq)P+S-B]E(X)-(l-q)E(L),
for terminal
equity
become (6)
V( W,) = A2V( X) + (1 - q)*V( L) - 2A(l
- q) cov( X, L),
(7)
where A = (1 - Aq)P + S - B and other variables are as before. Figure 2 graphs the risk-expected return trade off under three circumstances: when no reinsurance is purchased, a fixed amount (30 percent) of reinsurance is purchased and when reinsurance is optimized. The locus of terminal wealth points under the optimal reinsurance strategy was achieved by altering both the amount of reinsurance purchased and the mix between risky and riskless assets. The other two lines in Figure 2 were produced by changing only the asset mix. Figure 2 demonstrates the magnitude by which the optimal reinsurance strategy dominates other strategies at various wealth levels. Note, however, that tangencies also occur between the other graphs and the optimal graph, i.e., there is always one risk-return possibility from a non-optimal strategy which is as good as the optimal strategy. In the fixed reinsurances case, this occurs at a wealth point of about $1,100 while in the non-reinsurance case it occurs at about $1,250. Not all points on the graph are feasible. Points to the right of A on the optimal reinsurance graph
Standard
Devlotlon
(B)
1200 -
Optimal Rein. II
1000 -
W/O Rein. A " Fixed
Rein.
600 -
Termlnol
Weolth
($, 000)
Fig. 2. Hypothetical insurer risk-return space.
116
Y. Kroll, D. Nye / Retention levels for property/liability
firms
Table 2 Terminal equity for a hypothetical insurer under different reinsurance strategies. Terminal equity
Standard deviation of terminal equity
$1,080 1,100 1,120 1,140 1,160 1,180 1,200 1,220 1,240 1,260 1,280 1,300 1,320 1,340 1,360
Optimal reinsurance
Fixed reinsurance
Without reinsurance
$208 221 234 247 260 273 286 299 312 325 338 351 364 377 390
$231 222 242 285 342 408 479 552 628 705 783 862 941 1,020 1,100
$728 655 585 518 456 401 357 327 316 326 355 399 453 515 581
Source: Calculated using data from Table 1
involve negative values of q *, i.e., the firm must sell reinsurance and achieve the reinsurer’s profit spread in order to reach its wealth goal. When 30 percent reinsurance is purchased, only the points between B and C are feasible. To the left of point B, the firm borrows at the risky rate to cut its profit to the wealth goal while points to the right of C are attained by borrowing at the riskless rate. A similar situation occurs for points D and E on the no-reinsurance graph. Points to the left of D are inefficient and attained only by borrowing at the risky rate and buying the riskless asset. Points to the right of E are achieved by reversing this procedure. In summary, risk-return combinations between points B and A are feasible under the reinsurance optimizing strategy. These points offer rates of return of initial wealth between 10 percent and 24 percent (i.e. terminal wealth ranges from $1,100 to $1,240). This range of returns is inefficient if the firm’s managers decide not to purchase any reinsurance, although the degree of inefficiency becomes negligible at the 24 percent return figure. Similarly, the efficiency difference between the optimal reinsurance strategy and purchasing an arbitrarily selected proportion of reinsurance is minimal at the lower end of the feasible return range. 4.3. Sensitivity
analysis
In the following numerical examples the relative impact on q* of several variables is examined. Specifically, the relationship between the reinsurance level and the market price of risk, loading, variance of loss per insured, premium written to surplus ratio and underwriting profit is illustrated. I. Market price of risk. Equation (5) contains y which is the price of risk in the capital market. Differentiating (5) with respect to y yields
w
av=
2V(L)y[(XPR-E(L))][W,-RS+PR(X-l)] [Vet
+ [XPR - E(L)]~]’
(8) ’
Since each term is positive, the sign of equation (8) is positive but the impact of y on q * is relatively small. If the expected return on the risky asset is.8 percent, y equals 0.0816 and q* is 0.2311. An increase in the expected return to 16 percent causes q * to increase to 0.2806 or an increase of 5 percentage points. Thus,
Y. Kroll, D. Nye / Retention levels for property/liability firm
117
a very large increase in the market price of risk (rising from 0.08 to 0.41) induced a relatively modest increase in q *. If equation (5) is differentiated with respect to X, the result is 2. The reinsurance loading factor. g
= { -PR{V(L)y2[W,-RS+PR(h-l)] +(hRR-E(L))[V(L)+(W,-R(S+R)+E(L)(ARR-E(L))]}
V(L)y’[W,-RRS+RP(X-l)]
= -PR i
([Vet+
[APR-E(L)]~]~
+
(XPR - E(L)q*
(9)
[v(L)Y~+(APR-E(L)~]
Now the first term within the brackets is positive. The second term is positive as long as q* is positive. Hence, a necessary condition for the total expression to be negative is for q* to be positive. When equation (9) is evaluated, the relationship is as expected for values of A greater than 1, i.e., as reinsurance becomes more expensive, the optimal reinsurance proportion declines. For example, when the reinsurer charges 5 percent (X = 1.05), q* equals 0.351 and this declines to 0.155 when X is increased to 1.20. Under extremely competitive conditions, it is possible for A to be less than one, e.g., at times of high interest rates and excess capacity in the market. Under these conditions, the results are ambiguous. 3. Variance of loss per insured. This variable is important to insurer management because of its potential to control risk. First party insurance coverages are generally recognized as having less variance than liability lines so management can limit the variability of terminal equity by emphasizing first party lines in its book of business. If high variance lines must be written, then possibly reinsurance can be used to help control overall risk as management attempts to reach its profit goals. To derivative of q * with respect to V(L) is
-=a4* WL)
y’[XPR-E(L)][W,-RS+PR(X-l)] [V(L)+
[xPR-E(L)]‘]’
(10) ’
Equation (10) demonstrates there is a positive relationship between q* and the loss variance. An increase in variance per risk from $50 to $1,000 (on an expected loss of $10) causes an increase in the reinsurance proportion of 4.5 percentage points. This result suggests that, ceterus paribus, liability insurers should select reinsurance proportions which are not too different from those of property insurers. Assume that curve EF is the initial efficient frontier and that management has located at point A. As one moves from point E to point F along the curve, the amount of reinsurance purchased decreases. Assume an increase in loss variance takes place (with no change in the other variables) which causes the efficient frontier to shift to the right and the new curve becomes E’F’. If management had a return threshold to maintain, then, depending upon the actual value of the variables, reinsurance would either be unchanged or increase as management restructured the firm to locate at point C. Conversely, if any increase in risk was unacceptable, management would move to B by purchasing more reinsurance (and possibly rebalancing its asset portfolio) thereby leaving risk unchanged but reducing expected return. An empirical investigation concerning the impact of V(L) on the optimal reinsurance retention ratio may provide an indication about which goal function is more plausible. 4. Premiums written-to-surplus ratio. Another important question is the degree to which the optimal reinsurance proportion is affected by an increase in premiums written relative to initial surplus. This effect was measured by finding the derivative of q * with respect to the number of policies. The result shows that increased premium volume leads to increased demand for reinsurance. At a premium written to surplus
118
Y. Kroll, D. Nye / Retention IeveIsfor property/liability firms
E(w)
E
E’
Sigma
(w)
Fig. 3. Change in reinsurance proportion when claim variance increases.
ratio of 1.6 to 1, the optimal reinsurance proportion is 12.9 percent. This increases to 33.4 percent at a premium to surplus ratio of 4.2 to 1. 5. Underwriting profit. The final variable analyzed is underwriting profit. As noted earlier, expenses are paid at the beginning of the time period and are assumed not to deviate from expected expenses. Therefore, the gross premium is equal to the expected loss divided by one minus the profit rate. One minus underwriting profit is the claims ratio, (r, given by a: = E( L)/P. The positive relationship between underwriting profit and reinsurance demand conforms with expectations and may be interpreted as relaxing a budget constraint. Increased underwriting profit increases the
Table 3 Asset allocations for a hypothetical insurer under different reinsurance strategies. Wealth goal
$1,080 1,110 1,120 1,140 1,160 1,180 1,200 1,220 1,240 1,260 1,280 1,300 1,320 1,340 1,360
Fixed (30%) reinsurance
No reinsurance
4*
Risky asset
Riskless asset
Risky asset
Riskless asset
Risky asset
Riskless asset
0.35 0.31 0.27 0.23 0.19 0.15 0.11 0.07 0.03 - 0.01 - 0.05 -0.10 -0.14 -0.18 - 0.22
$270 287 304 320 337 354 371 388 404 421 438 455 472 488 505
$2,348 2,448 2,549 2,649 2,750 2,850 2,950 3,051 3,151 3,251 3,352 3,452 3,553 3,653 3,753
$-535 132 798 1,465 2,132 2,798 3.465 4,132 4,798 5,465 6,132 6,798 7,465 8,132 8,798
$3,298 2,632 1,965 1,298 632 -35 -702 - 1,368 - 2,035 - 2,702 - 3,368 - 4,035 - 4,702 - 5,368 - 6,035
$-5,351 - 4,684 - 4,018 - 3,351 - 2,684 - 2,018 - 1,352 - 684 -18 649 1,316 1,982 2,649 3,316 3,982
$8,982 8,316 7,649 6,982 6,316 5.649 4,982 4,316 3,649 2,982 2,316 1,649 982 316 - 351
Optimal reinsurance
Source: Calculated using data from Table 1.
Y. Kroll, D. Nye / Retention leuels for property / liabilityfirms
119
‘affordability’ of reinsurance and management is able to purchase additional coverage without sacrificing its profit goal. Underwriting losses cause the firm to fall short of its return goal and this is remedied by purchasing less reinsurance and eventually becoming a seller of reinsurance.
5. Conclusion This paper has developed a model for finding the optimal amount of proportional reinsurance. The model does not attempt to represent all goal structures but simply two reasonable goals. The firm’s managers are assumed to either minimize risk (variance) for a given expected return on equity, or to maximize terminal equity for a given risk level. The firm has two ways to control risk; alter the mix between the risky and riskless assets and purchase proportional reinsurance. Expressions for the optimal amount of reinsurance for the two formulations of the goal function were derived and included the significant exogenous and endogenous variables. Exogenous variables were the risk-free rate, expected return and standard deviation of return for the risky asset and transaction costs of the reinsurance market. Endogenous variables were underwriting profit, premiums written-to-surplus ratio and the variance of loss per insured. Numerical values for each variable were selected and an analysis of the model was performed. It was noted that insurance lines which are positively correlated with the asset market are most desirable because the model incorporates a risky asset (the market) and a risky liability (the portfolio of policies) rather than two risky assets. Empirical evidence suggests that the correlation between insurance lines and the market is not significantly different from zero so that assumption was adopted. The optimal reinsurance proportion was analyzed to determine its sensitivity to a number of different factors. The exogenous variables had a relatively mild effect upon the optimal reinsurance proportion except for reinsurer loadings. Significant endogenous variables were the premiums written-to-surplus ratio and underwriting profit. Variance of loss per insured had approximately the same impact as the market price of risk. These results suggest some interesting avenues for future research. Empirical work needs to be undertaken to measure insurers actual goal specifications and to provide a test of the validity of the alternative models. If an operational model can be developed, it may assist insurer management by identifying important variables in the decision making process and exploiting their interrelationships to achieve the firm’s goals. Finally, research is needed on incorporating the excess of loss reinsurance decision into the framework developed in this paper. This is difficult because we do not have a good theory of how the price of reinsurance depends upon the retention limit and this is needed in order to work out the first-order conditions associated with the choice of the optimal retention limit. Moreover, even if we have the functional form of the excess of loss premium, it is very unlikely that a closed form solution could be obtained since the excess of loss retention limit enters the analysis in a non-linear fashion as the limit of an integral.
Appendix A.I. The development
of q* and q * *
Under the first goal function, management minimizes the probability of ruin for a given expected return. Namely, the manager is assumed to choose a target value of E(W) and minimize the risk, V(W), from seeking this target return. In this model, disk depends upon two factors: the amount invested in the risky asset and the amount of reinsurance. Thus, risk is minimized with respect to B and q for a given target equity and the problem is written as YiYw) subject to
(A-1) E(W)
= W,,
Y. Kroll, D. Nye / Retention levels for property/liability
120
firms
where 4 = the proportion B = the amount of W, = target terminal E(W) and V(W) are in the text:
of reinsurance purchased, the riskless asset purchased or borrowed, equity. obtained by taking the expected value and variance of W as expressed in equation (1)
E(W)=BR+{(l-Xq)P+S-B}E(X)-(l-q)E(L),
(A-2)
V(W)=A2V(X)+(1-q)2V(L)-2A(1-q)cov(X,
L),
(A-3) (A-4)
A=(l-Xq)P+S-B.
For each value of q, the optimal investment in the riskless asset (B *) which satisfies E(W) = W, can be calculated from (A.2) to.find B* = w, - E(X)[S
+ (I- Xq)Rl R-E(X)
+ (I-
q)E(L)
the constraint
(A-5)
This change permits the minimization problem to be rewritten as MinV( W)
(A4
4
subject to
B = B*
The first-order conditions for a maximum in (A.6) are
Ww)
aA* =2A*V(X)T-2(1-q*)V(L)+2A*cov(X,
a4 B=
B*,
L),
(A-7)
q=q*,
-2(1-q)
cov(x,
L)F
=o,
where A* is as given in equation (A.4) when B * = B. Substituting the expression for B* (A.4) yields W,-E(X)[S+(l-Xq)P]
A*=(l-Xq)P+S-
+(l-q)E(L)
R-E(X)
from (A.5) into
(‘4.8)
Eq. (A.8) can be rewritten as + W,-R(S+P)+E(L) E(X)-R
.
(A-9)
For simplicity, denote D=
Af’R-E(L) E(X)-R
F=
w,-R(S+P)
(A.lO)
’
+ E(L)
E(X)-R
’
(A.ll)
Thus, (A.8) can be rewritten as (A.12)
A*=qD+F
and aA*/aq B=B*,
= D,
(A.13)
Y. Kroll, D. Nye / Retention levels for property/liability
firm
121
Thus, the first-order conditions in equation (A.7) can be written as F=[,*D+F]V(X)D-(I-q*)V(L)+[q*D+F]cov(X,
L)-(1-q*)cov(X,
= 0,
q* =
(A.14)
B= B*.
q=q*,
Thus, q*
L)D
which solves the first-order condition is V(L)-V(X)FD+cov(X,
L)(D-F)
V(X)D2+V(L)+2Dcov(X,
L)
(A.15)
’
By using the definitions of D and F and rearranging terms we obtain q*=
{V(L)(E(X)-R)2-V(X)[W,-R(S+P)+E(L)][XPR-E(L)] +cov(X,
L)[APR-2E(L)-
w,+R(S+P)](E(X)-R)}
x{V(L)[(E(X)-R]2+V(X)[hPR-E(L)]2 +2cov(X,
L)[hPR-E(L)](E(X)-R)}-l.
(~.16)
Denote y as the price of risk; Y = (E(X)
(A.17)
-@/e(X),
and /3 as the regression coefficient of L on X: P=cov(X,
(A.18)
L)/V(X),
to obtain q* =
V(L)y2-[[W,-R(S+P)+E(L)][XPR-E(L)]-8[W,-hPR-R(S+P)+2E(L)][E(X)-R]
(A.19)
V(L)y*+[hPR-E(L)]*+2/3[hPR-E(L)](E(X)-R)
Equation (A.19) is equivalent to equation (2) in the text. A.2. The development
of q * *
According to the second goal function, the maximization problem is (A.20)
YyE(W) subject to
V(W)
= V,,(W).
Assume for simplicity that cov( X, L) = 0 which reduces equation (A.3) to V(W)
+ (1 -q)2V(L).
=A’(V(X)
(A.21)
According to (A.21) the A,, that maintains V(W) = V,,(W) is A = + -
V,(W) [
-
(1 -
4)2V(L)
V(X)
1. 1’2
(A.22)
According to equation (A.4) and equation (A.22) the corresponding B,=(l-Aq)P+Sf
V,(W)
- (1 - q)2v( L) V(X)
1 1’2
B,, is: (A.23)
122
Y. Kroll, D. Nye / Retention levels
for property/liability
firms
In order to obtain the maximum and not the minimum value of A, in (A.22), the positive square root should be taken (this is because zero variance can be obtained only for A, = 0 and E(X) > R).Thus,the maximization problem in (A~20) can be restated as M$RB,+E(X)A,-(l-q)E(L)]. The first-order
condition
(A.24)
is
dE(W) =R%+E(X)%+E(L)=O.
(A.25)
a4
By taking
the derivative
E(X)-R o(X) Define
of B, and A, and rearranging V(L)(I
terms we find that equation
(A.25) is equivalent
- 4)
to
(~.26)
[v,(w)-(I-q)2~(Q]1’z
=XPR-E(L).
6 as 8 = ARR- E(L) E(X)-R
and denote
o(X)
(A.27)
I V(L)
p = 1 - q to express
the first-order
condition
as (~.28)
s = [V,(W) Squaring (denoted
-pplv(L)y~
both sides of (A.28), rearranging by q* *) which is
4 **=1*
which is identical
terms and replacing
1
p with q, yields the optimum
value of
kJw> [l + SW( L)y2
q
(A.29) ’
to (3) in the text.
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