The cascade effect in insurance pricing.
The recent financial condition of the insurance industry and new risk-based capital requirements have led insurers to innovative financing techniques. One method of raising capital, the use of initial public offerings (IPOs), has been used successfully by a number of insurers.(1) Prior to the 1990s, the use of IPOs by insurers was relatively uncommon. However, as capital needs increase for the industry, IPOs are likely to become a significant source of financing.
A number of studies demonstrate that IPOs of common stocks are typically underpriced. A number of models have been proposed to explain this phenomenon. Welch (1992) developed a "cascade" pricing model to explain this tendency. According to this model, when a homogeneous common value good (many similar items that are sold at the same price to different buyers) - such as an IPO or high-yield debt - is sold to a large number of investors sequentially and there is some noise in each investor's signal about the true value of the good, the purchase decisions of later investors depend on what earlier investors have decided. If this is the case, later investors who can observe previous demand will sometimes base the decision to purchase exclusively on earlier sales and ignore their own signals about the true value, thus leading to a cascade. As a consequence of this cascade phenomenon, the danger of starting a negative cascade, in which no one will buy regardless of the signal received, may be so high that sellers always decide to underprice so that all goods are sold.
Whereas Welch applies the cascade model to pricing IPOs, this article applies the model to insurance. The assumptions of the cascade model are met in the marketing of certain insurance policies that require cooperation among a large number of insurers, or other risk bearers, in order to establish a price and other policy terms.(2) These policies, termed special risks in this article, are generally very large risks, with a maximum probable loss that significantly exceeds the capacity of an individual insurer, even with reinsurance, to bear. Thus, the insurance market provides an additional opportunity to test the cascade model.
The following section provides a brief overview of the securities market and IPOs with a review of the relevant literature, including a description of the cascade model. The next section provides a classification of insurance marketing systems into competitive and cooperative, to set the stage for a test of the cascade model for insurance pricing. The subsequent section presents three pricing models: competitive pricing, cooperative pricing without cascades, and cooperative pricing with cascades. Later sections compare these models and incorporate empirical results.
THE SECURITIES MARKET AND IPOs
Securities markets, such as the New York Stock Exchange (an organized exchange) or the NASDAQ (an electronic network), provide a method for determining security prices. These institutions are termed secondary markets, as they allow investors to trade securities that have previously been sold by the issuing firm. On organized exchanges prices are established by an open outcry auction market. For the NASDAQ markets, investors trade at the most favorable price quoted through the network. Thus, the prices of secondary market securities tend to reflect the consensus of their fair market value. Most trades are small enough that there are many potential participants willing to accept the entire transaction. Thus, the trade usually takes place at a single price with a single entity.
In contrast with the open bidding approach for secondary market securities, the primary market for a firm's initial public offering of stock involves negotiation with multiple potential buyers. Two methods of distributing IPOs exist. One is a best efforts offering, in which the underwriter tries to raise all of the desired capital at the negotiated price. If demand at this price is not sufficient, the offering fails. The alternative method is a firm commitment offering, in which the underwriter buys the entire issue at an agreed-upon price and is then responsible for reselling it. Under this method, the price is set after indications of interest have been received.
Determining the appropriate price for an IPO is difficult, since there is no observable market price prior to the offering and many issuing firms have little or no operating history. The number of shares offered is so large that no single buyer could accept the entire issue. The price of an IPO is set by the underwriter rather than by bidding among investors. If the underwriter sets the price too low, the issuer does not raise as much capital as it could. Conversely, if the price is too high, the underwriter has a financial loss or the issue is withdrawn and the issuer raises no capital. Thus, the price selected must satisfy two clienteles, the issuer and investors.
Ibbotson (1975) was among the first to document the underpricing of initial public offerings. Ibbotson, Sindelar, and Ritter (1988, 1994) and Smith (1986) provide convincing evidence that initial public offerings are, on average, significantly underpriced.(3)
Several models have been proposed to explain the underpricing phenomenon. Rock (1986) classifies investors as informed or uninformed about the correct price of an offering, with the participation of both needed for a successful offering. Without underpricing, uninformed investors would purchase a disproportionately large share of overpriced issues, a situation termed the "winner's curse." In order to keep uninformed investors in the IPO market, offering firms must issue the securities at a discount from their expected aftermarket prices.
In Welch's (1992) "cascade" model, investors use a Bayesian updating process to adjust their expected value of the security by observing earlier investors' behavior. In the new issue market, investors cannot observe other investors' signals; they only observe whether these investors purchased. Each potential investor receives a signal about the true value of the issue. However, since this signal contains a degree of error, the investor does not completely trust this signal. Investors also consider prior decisions made by earlier investors, which reveal information about prior signals. An individual investor interprets a successful initial sale to imply that earlier investors had favorable signals, giving the later investor an additional incentive to invest. Conversely, slow initial sales discourage subsequent investing. Beyond some point, investors base their decisions exclusively on earlier sales, ignoring their own signals. At this point, a cascade occurs. If a few early investors believe that the offering price is too high, they will refuse to buy and induce all future investors, even those with positive private signals, also to refuse to purchase. This is termed a negative cascade and results in the failure of the offering. Alternatively, if a few early investors indicate that the offering is a good value by deciding to purchase it, they can create an almost unlimited demand for this issue by generating a positive cascade. To avoid the possibility of a negative cascade, IPOs are underpriced.
Bikhchandani, Hirshleifer, and Welch (1992) generalize Welch's cascade effects and explain social phenomena such as fashion, custom, and cultural change as informational cascades. They demonstrate that an informational cascade occurs when it is optimal for an individual, having observed the actions of others, to follow the behavior of the preceding individuals regardless of his or her own information.
INSURANCE PRICING MECHANISMS
The cascade model assumes that the investor knows how many investors have been approached previously and how many have purchased. This is not exactly true in IPOs, because investors cannot be sure how many other investors have been approached or learn accurately how much of the offer has been subscribed, but those conditions may be the case in special risk insurance policies. Thus, insurance may be a better test of cascades than IPOs.(4)
Two distinct pricing mechanisms exist for establishing insurance prices, akin to the different pricing methods in equity markets for seasoned offerings and IPOs. In the conventional insurance market, insurers compete against one another. Each insurer is able to write the entire risk, or can make individual arrangements in the reinsurance market to cope with excess risk. Each insurer quotes a price, either based on the past experience of the risk or other factors the insurer considers relevant. The insurance buyer can shop the competing insurers to obtain the best combination of price, service, and quality. If each insurer is assumed to receive a signal about the appropriate price for a policy and to develop a rate based on that signal, the insurer with the lowest error term will quote the lowest price and tend to write the business. Since the error term can be negative, the insurer that is willing to underprice by the largest amount is likely to get the business. This is an example of the "winner's curse" for insurance.(5)
Since insurers know that this competitive pricing leads to underpricing, each insurer tends to consider the prices charged by other insurers when setting its own price. An insurer receiving a signal indicating that the appropriate price for a risk is below the market price would also be willing to write that risk at the market price. An insurer with a higher signal would forego writing the policy. This is a form of cooperative pricing without cascades, as decisions are not made sequentially.
The market for special risks functions differently. These risks are so large that no one insurer, even with reinsurance, can accept the entire risk. Each insurer writes only a small percentage of a special risk policy. For instance, property insurance on skyscrapers, hull coverage on jumbo jets, satellite coverage, and pollution liability coverage belong to this category. This market can function by sequential decisions, similar to IPOs, in which the policyholder and first insurer negotiate the policy terms. Once the price is established, any other insurer writing a portion of the policy applies the same price and other policy terms. However, if enough other participants cannot be found to cover the risk fully, then the policy is withdrawn and no insurer provides any coverage.
Lloyd's of London functions in this manner.(6) At Lloyd's, risks are placed by brokers who pass before the desks of the underwriters and present a "slip" which is the proposal of insurance. Each accepting underwriter signs the form and indicates the amount of insurance he or she is ready to assume. The broker goes first to a specialist who is a "lead" underwriter. After establishing the price and other policy conditions by placing some insurance with the lead underwriter, the broker then takes the slip around to as many following underwriters as are required to complete the slip. The reputation of the lead underwriter is an important factor in influencing other underwriters to accept the risk. By reviewing the slip, each underwriter can see which underwriters accepted the risk. By knowing the common pattern in which underwriters are approached, based on the Lloyd's floorplan, the underwriter can also infer which ones rejected the risk. This knowledge is a key element of the cascade model, and one that is not generally met by IPOs. If an underwriter refuses to assume a portion of the risk, then following underwriters may not accept the risk regardless of their information about the value of the risk, resulting in a negative cascade. Conversely, subsequent underwriters may decide also to accept the risk even if they receive an unfavorable signal (their own information suggests declining the risk), simply because the earlier underwriters accepted the risk. Thus, because of the high price elasticity of demand, other policy terms being the same, the broker must overprice the insurance premium to reduce the chance of a total failure to get insurance due to a negative cascade.(7)
In this section, several different approaches for determining an insurance price are applied: a competitive market (which was not examined by Welch), a cooperative market without cascades (similar to the benchmark IPO price in Welch), and a cooperative market with cascades (similar to Welch's model for IPOs). The models are placed in an insurance, as opposed to an IPO, framework and the results are compared to illustrate the impact of the cascade model on insurance pricing.
Several assumptions about the insurance market are made. First, the market consists of many insurers, each willing to provide coverage if the premium, P, exceeds the expected loss, L. Second, the policyowner is risk averse, willing to buy insurance even if the premium exceeds the expected loss. The exact expected loss is unknown to both the insurers and the policyowner. Each insurer knows that the expected losses lie between a and b. For simplicity, expected losses are assumed to follow a uniform distribution in this interval.(8) If x is the premium level, where x [0,1] is a linear transformation of [a,b], the probability of the insurer accepting the risk at premium P = a + x(b-a) is exactly x.(9) For example, when x is 0.3, P = a + 0.3(b-a), the probability of obtaining insurance is 0.3.
Competitive Market Pricing
In a competitive market, the customer obtains quotations from different insurers and buys insurance at the lowest price. Since each insurer has the same expected loss distribution, the probabilities of acceptance are the same among the insurers for a given premium level. For a given premium P, each insurer will provide insurance with probability x. If there are N insurers, the probability that the customer can obtain coverage at this premium level is 1-[(1-x).sup.N]. This is the same as one less the probability that all the insurers reject coverage at that price level. Since N insurers offer premium levels independently, the policyowner can buy insurance at the lowest price. The expected value of the lowest price is E(x) = 1/(N+1) (see the Appendix for details). The insurer with the lowest signal will write the policy. Given that the signals contain random error, this leads to underpricing unless insurers pay attention to competitors' prices. The underpricing can be avoided by some form of cooperative pricing as described next.
Cooperative Pricing with No Cascade Effects
If the expected loss is so large that no one insurer can assume the entire risk, the pricing decision must be made cooperatively, rather than competitively. The customer must get approval from n of N insurers to get insurance at a given premium level, with each insurer charging the same price. This is basically a "Dutch auction" situation, with the policyowner purchasing coverage at the lowest price full coverage is available. The expected value of the premium in this case is E(x) = n/(N+1) (see the Appendix for details).
In practice, competitive insurance markets may converge to this form of cooperative pricing without cascades. In the competitive pricing model, each insurer receives a signal for each customer and can provide full coverage for any given insured. However, for an entire line of business, such as private passenger automobile coverage in Chicago, no one insurer could write all the business. An insurer that is willing to write at a particular low premium level due to receipt of a favorable signal for that price level, is going to be looking at the rates charged by other insurers before establishing its own rate. Although willing to write the business at the level indicated by its signal, it would prefer to write at an even higher rate if potential insureds were willing to pay that level. In that case, the market price will converge to the nth order statistic level.
Depending on the number of insurers in the market, N, and the number of insurers needed to write the coverage, n, the expected value of the nth order statistic could be below, equal to, or above the expected cost of providing the coverage. If the market price is below the cost of providing coverage, the number of insurers willing to write a line is likely to decline, which would lead to a higher price. Conversely, if the market price exceeds the cost of providing coverage, then more insurers would be likely to enter the market, leading to a lower price. The equilibrium market price will occur when n = (N+1)/2, for at that point the expected value of the premium will equal the expected value of the coverage under the assumption that the signal each insurer receives is uniformly distributed.
Cooperative Pricing with Cascades
The third pricing model assumes insurance rates are again set cooperatively, but each insurer decides in sequence whether to accept or reject a risk at a particular price. Each insurer has one, and only one, chance to accept a risk at that price. When enough insurers have accepted a risk at the offered price, no other insurer has the chance to participate. If all insurers have been offered the opportunity to participate and not enough have accepted the risk to provide full coverage, the offer fails and no coverage is provided by any insurer.
Earlier insurers' actions can affect later insurers' decisions. Although later insurers cannot observe the earlier insurers' signals, they can observe their actions (acceptance or rejection). The first insurer approached for coverage will accept or reject the risk based on the signal it receives. Similarly, the second insurer will also decide to insure based on its signal if that signal agrees with what the first insurer did. If it receives a conflicting signal, it would be indifferent about insuring, knowing that of two signals, one indicated the risk should be accepted and one that it be rejected. Thus, its decision cannot be predicted. However, if both of the first two insurers either accept or reject the risk, the third insurer will do the same regardless of its signal. If the third signal contradicts the decision of the first two insurers, the third insurer will know that its signal is outnumbered, two to one, and that it would be best to follow the indication of the majority of the signals instead of relying on its own information. As shown in Welch (1992), the posterior expected value of x given n signals of which k are favorable is
E(x [where] k,n) = (k+1)/(n+2) Welch's Lemma 1. (1)
A cascade begins whenever an insurer ignores its own signal about the expected losses and relies solely on the decisions of earlier insurers. In this case, whenever two consecutive insurers make the same decision, all subsequent insurers will follow this lead, creating a cascade (see the Appendix for details). Knowing this fact, the price that is proposed for a given risk, as a result of the consultation between the lead underwriter and the insured, is determined considering the likelihood of starting a positive cascade and avoiding initiating a negative cascade. This decision is to select the x that maximizes the chance of successfully obtaining coverage.
ILLUSTRATION OF PRICING MODELS
In this section, the probability of obtaining insurance and the expected premium level for the different pricing mechanisms is presented by a numerical example. Assume there are nine insurers in the market and one customer that wants to buy insurance. Each insurer has the same information about the possible loss, and this information can be expressed as a uniform distribution over the range of 95 to 105. Further assume that, in the case of cooperative pricing, the customer needs to get approval from five insurers of the nine in the market. In this case, the probabilities and the expected values can be calculated with N = 9, n = 5, a = 95, b = 105. Given the uniform distribution, the expected value of this loss is (a+b)/2 = 100. This value does not depend on the pricing methodology.
Under competitive pricing, the expected value of x is 1/(N+1) = 0.1. Thus, the expected premium would be a + x(b-a) = 96. The insurer with the lowest signal is expected to quote a price of 96, and the insured will purchase coverage at this price from this insurer.
Under cooperative pricing without cascades, the insured needs to find a price at which five of the nine insurers would be willing to write the policy. The expected value of x is n/(N+1) = 5/10 = 0.5. The expected premium is 95 + 0.5(105-95) = 100. Thus, as indicated previously, the market clearing price of the expected losses occurs, since n was chosen to equal (N+1)/2.
With a cascade effect in the cooperative pricing model, when two consecutive insurers accept the risk, a positive cascade occurs, and when two consecutive insurers reject the risk, a negative cascade occurs. As long as either a positive cascade or no cascade occurs, the policyowner can get approval from at least five of the nine insurers. Based on Welch's approach, the expected value of x is 0.642, and the expected premium level is 101.42 (see the Appendix for this calculation).
Based on this analysis, the price of insurance coverage in competitive markets is below the cost of losses, the price of coverage in markets that are cooperative but without cascades is equal to the expected loss, and the price in markets with cascades is above the expected loss. Thus, insurers that write business in a cooperative manner should be more profitable than other forms of insurance pricing. This higher return is, perhaps, the reward for the additional risk cascades generate, as described next.
Table 1 and Figure 1 show the probabilities of getting insurance under each pricing mechanism at different premium levels. The probability that the customer can get insurance in a competitive market is much higher than in a cooperative pricing mechanism at all premium levels. However, the competitive pricing model applies when an individual insurer can accept the entire risk. For aggregate markets, the proper comparison is between cooperative markets with and without cascades. In this comparison, the probability of getting insurance at a low price is [TABULAR DATA FOR TABLE 1 OMITTED] higher under the cascade model than without cascades. The probability of an aggregate premium level of 96 being accepted in the cooperative pricing market without cascades is only 0.09 percent. Under the cascade model, this increases to 6.05 percent. This level represents considerable underpricing, since the expected loss is 100.
The probability of significant underpricing is higher under the cascade model than for cooperative pricing without cascades. This illustrates the much greater risk, for the insurer, with cascade pricing. Whenever an insurer, or Lloyd's underwriter, ignores its own signal about a risk and accepts it based on the decisions of others, then the possibility of a significant loss exists. This is the price of following the herd. Following the herd can lead to disaster, as Welch so aptly illustrates in his presentations of the cascade model by showing a copy of Breughel's painting, "The Parable of the Blind," showing a linked line of unsighted individuals sequentially tumbling into a ditch.
The relationships described in this section imply that to get adequate insurance in a special risk market such as Lloyd's of London, the customer should, on average, pay a higher premium compared with the conventional risk market. Therefore, the cascade model leads to the hypothesis that Lloyd's of London would be more profitable than conventional insurance markets. However, the chance of significant underpricing is also higher under a sequential decision-making market such as Lloyd's, compared with the conventional risk market.
THE DATA AND EMPIRICAL EVIDENCE
To test this hypothesis, the profitability of the conventional risk market is compared with the special risk market. Total operating results, as a percentage of premiums, are used as the basis of comparison. The experience of Lloyd's of London is used for the special risk market. Aggregate national experience in sixmajor countries is the basis for the conventional risk market. Total operating results are used instead of underwriting profit margins due to the different loss payout patterns involved in the two markets and different interest rates available over time and among the different countries. Lloyd's of London uses a three-year accounting system under which operating results are not finalized until three years after the policies are written, and any loss development after three years is rolled over into later account years. Since this practice is different from those of conventional markets that report calendar year experience, adjustments may be necessary to compare the performance of the two different markets.
Empirical data rarely conform exactly to the needs of the researcher, and this case is no exception. Lloyd's profitability may differ from that in other markets for a multitude of reasons other than cascades. It would be very useful to have more detailed profitability data to compare, such as risks in the different markets and a longer time period to smooth out shocks. Despite the data limitations, though, the findings suggest that the cascade model does explain some of the developments at Lloyd's and, as such, is a useful approach.
Table 2 shows the total operating results as a percentage of premiums for Lloyd's and six major countries during the period 1975 through 1992 for which [TABULAR DATA FOR TABLE 2 OMITTED] comparable data could be obtained. Over the entire period, Lloyd's was less profitable than the other six conventional risk markets. This result is heavily affected by the huge and unusual losses Lloyd's reported for 1988 through 1992.(10) During this period, Lloyd's lost almost [pounds]8 billion ($12.6 billion). The average operating results of the six countries changed by only between 0.1 and 2 percent when 1988 through 1992 data are excluded. However, in the case of Lloyd's, when 1988 through 1992 data are excluded, average operating results increase from -1.8 percent (1975 through 1992) to 10.9 percent.
The results for the entire period do not support the hypothesis that the cascade effect in insurance causes special risk policies to be overpriced compared with conventional risk coverage. However, this conclusion is caused by the post-1987 results. To examine the effects of the most recent years, the comparisons were run for the entire period and for 1975 through 1987. These results are displayed in Table 2.
For the period 1975 through 1987, Lloyd's is significantly more profitable than every country at the 0.5 percent level. Based on these data, there are economically significant differences in overall profitability between Lloyd's and the conventional market, supporting the cascade effect. But when the 1988 through 1992 results are included, the loss for Lloyd's is so large that it offsets the profitability of all prior years during the period 1975 through 1987, making Lloyd's less profitable than the conventional market. Figure 2 displays the operating results graphically.
Although data limitations prevent the inclusion in this article of results prior to 1975, research by Hitchings (1992) compares the profitability of Lloyd's with that of the U.K. and U.S. insurance industries during 1950 through 1989. Based on Hitchings' work, the average total insurance profits as a percentage of net written premium for Lloyd's is 6 percent compared to 3.9 percent for U.K. insurers and 2.9 percent for U.S. insurers. Lloyd's was more profitable than U.K. and U.S. insurers during each decade, although the profit advantage was greater during the 1950s and 1970s than it was during the 1960s and 1980s, two decades with significant natural disasters (Hurricanes Betsy and Hugo).
The marked divergence in results at Lloyd's prior to and after 1988 suggest that something may have drastically changed during this time. Evidence suggests that one change was the relationship between internal (those that work at Lloyd's) and external names. In the three years up to 1988, the average annual returns for external names was 2.3 percent compared to 2.5 percent for internal names; however, over the eight-year period up to 1990, while external names averaged a loss, internal names averaged significant profits (Raphael, 1995, pp. 169-171). During Lloyd's disastrous year of 1989, almost one-half of the syndicates produced profits and one-third of the total losses was generated by only five syndicates (Bray, 1992, and Shapiro, 1992). Thus, which syndicates an investor belonged to had a significant effect on actual financial results.
One result of the problems at Lloyd's is an increased availability of data. Lloyd's of London (1993), lists data by syndicate for 1987 through 1990, including profitability and the composition of the syndicate. Regressing individual syndicate results against the percent of a syndicate's capacity provided by external names, as shown in Table 3, shows that profitability was inversely related to externally supplied capital after 1987. Thus, there appears to be justification for examining a subset of the data, the pre-1988 period, in addition to the full data set.
The comparison of operating results shown on Figure 2 illustrates that Lloyd's tends to be more profitable than the conventional risk market, with values above the others for most years, but with some very unprofitable years that, in this [TABULAR DATA FOR TABLE 3 OMITTED] case, wipe out this profitability. This pattern. of both higher profitability and high loss years corresponds with the cascade model implications.
The pricing mechanism in insurance markets can play a key role in determining profitability. This article compares two different markets in terms of the pricing system and profitability. The pricing mechanism of the special risk market resembles the pricing of initial public offerings in that the price and other terms are determined by sequential decision-making of potential buyers. Thus, Welch's cascade model is potentially applicable to insurance markets. The hypothesis that the special risk market is more profitable than the conventional market is tested. Results suggest that the profitability of the special risk market was significantly higher than that of the conventional risk market until 1988, but after that point the reverse is true.
The variability of results suggests the need for additional study of this issue. Since many syndicates in Lloyd's of London also write conventional risks, such as automobile insurance, it would be useful to collect data excluding these lines. Also, many conventional insurance companies participate in the special risk market. Thus, in order to separate other effects on profitability, it would be helpful to compare profitability of each line of conventional insurance with the syndicates which deal in special risks exclusively. This would also deal with the econometric problem of comparing Lloyd's results with aggregate results by country.
Welch's model has proven useful in explaining returns for securities markets. Special risk insurance markets provide an even more appropriate test of this theory. The cascade model suggests that a market such as Lloyd's should be more profitable than the conventional insurance market and also have a greater likelihood of significant underpricing. Over the period 1975 through 1992, the profitability hypothesis is not supported by this study, although the indication that significant underpricing is more likely to occur is amply demonstrated.
Competitive Market Pricing
Since the lowest priced insurer sells the policy, an order statistic can be applied. Let [Y.sub.1] [less than] [Y.sub.2] [less than] ,..., [less than] [Y.sub.N] denote a random sample of size N from the distribution U [approximately] [0,1]. The probability density distribution of [Y.sub.1] is
[g.sub.1]([y.sub.1]) = N!/(N - 1)![(1 - [y.sub.1]).sup.N-1] = N[(1 - [y.sub.1]).sup.N-1], 0 [less than] [y.sub.1] [less than] 1,
= 0 elsewhere.
The expected value of the first-order statistic is
[Mathematical Expression Omitted].
Cooperative Pricing with No Cascade Effects
Similar to the competitive case, the probability that the policy-owner gets insurance at a price that n of N insurers will accept can be represented by the nth-order statistic:
[Mathematical Expression Omitted].
The expected value of the nth-order statistic is
E([y.sub.n]) = [integral of] [y.sub.n][g.sub.n] ([y.sub.n]) [dy.sub.n] between limits 1 and 0 = n/N + 1.
Cooperative Pricing with Cascades
With sequential decision-making, later insurers observe the earlier insurers' actions and revise their expectations. For premium level x, the posterior expected value of x if k insurers of the n approached accepted the risk is, from Welch (1992),
[Mathematical Expression Omitted].
According to this formula, once two consecutive insurers accept or reject a risk, all subsequent insurers will follow this lead, ignoring their own signals. As a result of this decision rule, the probability of a positive cascade, no cascade, or a negative cascade occurring after an even number of insurers n is
[Mathematical Expression Omitted].
For the illustrative case, the probability of getting full insurance is the sum of the probabilities of a positive cascade and of no cascade occurring during the first eight decisions (the highest even number less than N), which equals
F(x) = x(x + 1) [1 - [(x - [x.sup.2]).sup.4]]/2(1 - x + [x.sup.2]) + [(x - [x.sup.2]).sup.4]
[Mathematical Expression Omitted].
The authors thank David Coleridge and Heidi Hutter of Lloyd's of London for providing data, Jaekook Bae for his advice on the models, and three anonymous reviewers for their helpful suggestions.
1 Recent insurance IPOs include Horace Mann, Equitable, First Colony Life, Transamerica, American Re, Allstate, Prudential Reinsurance, and Risk Capital Holdings. For a more comprehensive listing, see Lee (1996).
2 Gogol (1993) discusses this process to illustrate the value of additional information on a risk.
3 However, despite this initial underpricing, a strategy of investing in IPO stocks produces below average long-term results (Loughran and Ritter, 1995).
4 Another insurance application of the cascade model may be the pricing of CAT (or disaster) bonds, as recently purchased by USAA (Scism, 1997).
5 Harrington and Danzon (1994) examine two possible causes of underpricing in commercial liability insurance in the mid-1980s: moral hazard and heterogeneous information. Moral hazard occurs when rewards and penalties for risk are asymmetric. Heterogeneous information is similar to the different signals of the Welch model. Their results indicate moral hazard was a more significant effect in explaining the winner's curse. A major difference in their analysis is that the commercial liability market is not a homogeneous, common value good priced sequentially.
6 For a more complete description of the Lloyd's market, see Carter and Diacon (1990), Paretzky (1988), and Raphael (1995).
7 The cascade model assumes that all investors have equal knowledge. In the case of Lloyd's of London, the lead and other early underwriters actually are expected to have superior knowledge. Thus, the signal provided by their participation, or lack thereof, is stronger than the cascade model would suggest, providing an ever greater likelihood of starting a cascade. Quantifying the effect of superior knowledge would be difficult, so the assumption of equal knowledge is retained. The direction of the difference in expected values of prices under the different models remains the same. The dispersion of prices under the cascade effect would be even larger if early decision-makers were assumed to have superior knowledge.
8 In practice, expected losses are likely to have a more complex distribution. The uniform distribution assumption follows Welch (1992) and indicates the direction of any differences among expected profits in the different types of markets, even if the exact amounts are different.
9 Another way of viewing x is to consider it to represent the level of water in a glass, with zero being empty and one being full. Select any point within the glass; x represents the likelihood that the selected point is in the water (insurance coverage would be obtained) or out of the water (the risk would be declined by the insurer).
10 These losses appear to have ended for account year 1993, for which preliminary results were recently reported.
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Stephen P. D'Arcy is Professor of Finance at the University of Illinois. Pyungsuk Oh is Assistant Professor in the Department of Finance and Insurance at Dongseo University, Korea.
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|Author:||D'Arcy, Stephen P.; Oh, Pyungsuk|
|Publication:||Journal of Risk and Insurance|
|Date:||Sep 1, 1997|
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