Abstract:
A reinsurance participation plan allows an insurance carrier to provide novel loss participation plans to insureds. The insurance carrier cedes a portion of the risk to a reinsurance company, such as a captive reinsurance company. The captive reinsurance carrier then enters into a contractual agreement with the insured for the participation plan, subject to the appropriate regulations for the primary line of insurance in question. The participation plan may be a non-linear participation plan, such as a curvilinear participation plan, where the non-linearities allow the plan to be offered to smaller companies than would otherwise qualify for more traditional retrospective participation plans, and to provide more advantageous plans to larger companies than they would otherwise be offered. The plans may be offered on a multi-year basis with particular diligence in informed consent for the prospective insured. Plans may be offered to small companies whose loss experience is aggregated and then divided according to relative premium amounts among the small companies such that the aggregate losses have a distribution with skewness comparable to that of a medium sized company.

Description:
CROSS-REFERENCE TO RELATED APPLICATION 
     This application is a continuation of U.S. nonprovisional patent application “Reinsurance Participation Plan”, Ser. No. 12/696,256, filed Jan. 29, 2010. Said application is incorporated in its entirety herein by reference. 
     Said nonprovisional patent application Ser. No. 12/696,256, in turn, claims the benefit of U.S. provisional patent application “Reinsurance Participation Plan”, Ser. No. 61/148,560, filed Jan. 30, 2009. Said application is incorporated in its entirety herein by reference. 
     Said continuation application hereby claims priority from said nonprovisional patent application Ser. No. 12/696,256 and said provisional patent application Ser. No. 61/148,560. 
    
    
     FIELD OF INVENTION 
     This disclosure is generally in the field of insurance. 
     BACKGROUND 
     There is long felt need for an insurance product that more closely matches an insured&#39;s perception of the risk of suffering various levels of aggregate loss and preferences regarding different final cost outcomes. 
     SUMMARY OF THE INVENTION 
     The Summary of the Invention is provided as a guide to understanding the invention. It does not necessarily describe the most generic embodiment of the invention or all species of the invention disclosed herein. 
     A small to medium sized company&#39;s perceived risk of incurring a given level of insurance loss can be more closely matched to an insurance carrier&#39;s needs to collect enough premium to cover all expected losses from all insureds and comply with state insurance regulations if the insurance carrier cedes a portion of the total risk to a reinsurance company and if the reinsurance company, in turn, provides a risk sharing participation program to the insured. 
     The risk sharing participation program is structured such that the insured&#39;s net premium payment will vary in a non-linear manner with respect to their actual losses. In particular, there will be accelerated savings in premiums for particularly low losses over a given period of time. 
     The risk sharing participation program is suitable for workers&#39; compensation insurance as well as insurance coverage for other risks, such as general liability and health risks. Coverage may be provided separately or in combination. 
    
    
     
       BRIEF DESCRIPTION OF DRAWINGS 
         FIG. 1  is an illustration of a workers&#39; compensation loss distribution for large companies. 
         FIG. 2  is an illustration of a conventional linear retrospective premium plan for workers&#39; compensation insurance for large companies. 
         FIG. 3  is an illustration of a workers&#39; compensation loss distribution for medium sized companies. 
         FIG. 4  illustrates the difficulties inherent in offering conventional linear retrospective premium plans to medium sized companies. 
         FIG. 5  illustrates the ability of an exemplary non-linear participation plan to overcome the limitations of a linear plan. 
         FIG. 6  illustrates an exemplary curvilinear participation plan. 
         FIG. 7  illustrates an exemplary system for providing a reinsurance participation plan that is in compliance with insurance regulations. 
         FIG. 8  is a Lee diagram which illustrates the relationship between loss ratio and cumulative distribution function for medium sized companies. 
         FIGS. 9A and 9B  compare the Lee diagrams for the loss distributions of medium sized companies and large sized companies. 
         FIG. 10  is a Smith diagram which illustrates the relationship between premium ratio and cumulative distribution function for a non-linear retrospective premium plan. 
         FIGS. 11A and 11B  compare the combined Smith diagrams and Lee diagrams for medium sized and large sized companies. 
         FIGS. 12A and 12B  compare the Smith diagrams for a fixed premium plan and a non-linear premium plan. 
         FIGS. 13A and 13B  compare Smith diagrams for non-linear premium plans with an adjustable maximum premium and minimum premium. 
         FIGS. 14A and 14B  compare Smith diagrams for non-linear premium plans that allow independent adjustment of maximum premium, Basic, Guaranteed Cost premium and minimum premium. 
         FIG. 15A  illustrates how the loss conversion factor varies with actual losses for an exemplary embodiment of the invention. 
         FIGS. 15B and 15C  illustrate a non-linear premium plan graphed on a Smith diagram and graphed relative to loss ratio. 
     
    
    
     DETAILED DESCRIPTION OF INVENTION 
     The following detailed description discloses various embodiments and features of the invention. These embodiments and features are meant to be exemplary and not limiting. 
     Loss Distributions and Linear Retrospective Premium Plans 
       FIG. 1  illustrates a distribution of actual workers&#39; compensation insurance losses (loss distribution) experienced by large companies. Curve  100  shows the relative number of companies that experience a loss of a given size over a given period of time, such as one year. This is known as a frequency distribution of losses. 
     By “loss” it is meant the amount that a given insurance carrier pays to settle claims by injured workers employed by a single company covered by the insurance carrier in a given year. This graph takes into account the fact that an injured worker may make claims, such as for medical care reimbursement or lost wages, over a period of many years after an accident occurs. 
     The curve is based on data collected by various agencies, such as the National Council of Compensation Insurers. These agencies report out loss experience data in table form. “Table M” produced by the National Council of Compensation Insurers is an example of such a table. The current Table M as of the filing date is incorporated herein by reference. 
     Table M categorizes companies by their average expected worker&#39;s compensation losses. The categories are defined as “Expected Ultimate Loss Groups” or EULGs. 
     Each group covers a range of losses. As used herein, when a group of companies is described as having expected losses of a particular value, it is meant that their values fall in the range of the corresponding EULG. A company that has expected losses of $160,000, for example would fall in EULG  55 . EULG  55  covers companies with expected losses in the range of $159,002.01 to $171,340.00. 
     Data points  110  illustrate underlying data which the frequency distribution is based on. Data points are only shown for the tail of the curve for clarity purposes. Full data sets would show points along the entire length of the curve. There is a certain amount of scatter in the data due to random fluctuations, as well as systematic differences between the types of workers represented. The circle data points  112  represent relatively low risk occupations, such as office workers. These data points tend not to extend out to the higher losses. The triangle data points  114  represent relatively high risk occupations, such as construction workers. These data points tend to extend out to the higher losses due in part to the higher probability of a worker suffering a long term disabling injury. 
     The frequency distribution curve illustrates that for companies seeking to purchase workers&#39; compensation insurance, there can be a difference in perception between what the company feels its “expected losses” are and what the insurance company feels its “expected losses” are. This difference can lead to a difference in opinion as to what the appropriate insurance premium should be and can make the sales process difficult. 
     The curve presented in  FIG. 1  is for companies that, on average, experience $4,000,000 in workers&#39; compensation losses. These are large companies with several thousand employees. Insurance for these companies is often bought by a professional risk manager who is very familiar with the nature of their losses. 
     An important part of the nature of workers&#39; compensation losses, is that the loss distribution has a long tail  102 . This means that the losses experienced by most companies are fairly low, but on relatively rare occasions, a catastrophic event can lead to very large losses. These large losses increase the high-end tail of the distribution and pull the overall average  106  up above the median  104 . 
     An insurance company considers the average losses to be the expected losses, since on average, this is what they expect to pay per insured. An insured company, however, may consider the median to be its expected losses, since that is what they normally expect to suffer. Hence there can be a mismatch in what the insurance company feels is a fair premium and what the company feels is a fair premium. 
     This dichotomy has lead to the development of retrospective premium plans.  FIG. 2  illustrates a comparison between a standard Guaranteed Cost insurance plan  210 , and a participating linear retrospective premium insurance plan  220 . Both of these types of plans are approved by the individual state insurance departments in the U.S. and therefore can be offered by admitted carriers to companies that meet certain criteria. The corresponding frequency distribution of loss  202  is also shown for reference purposes. 
     Guaranteed Cost plans are quite simple. The insured company pays a fixed premium no matter what its subsequent loss experience is for the term of its insurance coverage. This fixed premium is illustrated by the horizontal line  210 . 
     The fixed premium can be thought of as equaling a Basic  212  plus the average losses  214 . The Basic is the estimated cost of providing the insurance, not including claims. It includes sales, underwriting, profit and other fixed costs. The average losses is the expected average claims that will have to be paid.  FIG. 2  illustrates that a company with expected average losses of $4 million per year might be charged a premium of $6 million per year. $4 million is to cover payment of the losses. $2 million is to cover the other costs of providing the insurance. 
     A participating linear retrospective premium insurance plan  220  varies the premium that a company will pay based on its actual losses during a coverage period. In the illustrated example, the minimum is set at the Basic. The insurance premium then increases linearly along region  222  with respect to actual losses until it reaches a maximum at plateau region  226 . Thereafter, the premium is fixed. The maximum is set by the crossover point  224  and the shape of the underlying frequency distribution  202 . 
     The standard equation describing the relationship between premium and actual losses over the linear region  222  is:
 
Premium=Basic+C*Actual Losses
 
where C is known as the Loss Conversion Factor.
 
     An exemplary relationship between premiums and actual losses is illustrated in Table 1. 
     
       
         
               
             
               
               
               
               
               
               
             
           
               
                 TABLE 1 
               
             
             
               
                   
               
               
                 Conventional Retrospective Premium Plan with  
               
               
                 Constant Loss Conversion Factor 
               
             
          
           
               
                   
                   
                 Variable  
                   
                   
                 C Loss  
               
               
                 Actual 
                 Basic  
                 Loss  
                   
                   
                 Conversion  
               
               
                 Losses 
                 Expense 
                 Expense 
                 Taxes 
                 Premium 
                 Factor 
               
               
                   
               
               
                 $0 
                 $199,806 
                 $0 
                 $7,193 
                 $206,999 
                   
               
               
                 $256,868 
                 $199,806 
                 $64,217 
                 $18,752 
                 $539,643 
                 1.295000 
               
               
                 $513,736 
                 $199,806 
                 $128,434  
                 $30,311 
                 $872,287 
                 1.295000 
               
               
                 $770,603 
                 $199,806 
                 $192,651  
                 $41,870 
                 $1,204,930 
                 1.295000 
               
               
                 $1,027,471  
                 $199,806 
                 $256,868  
                 $53,429 
                 $1,537,574 
                 1.295000 
               
               
                 $1,284,339 
                 $199,806 
                 $321,085  
                 $64,988 
                 $1,870,218 
                 1.295000 
               
               
                 $1,541,207  
                 $199,806 
                 $385,302  
                 $75,547 
                 $2,202,862 
                 1.295000 
               
               
                 $1,798,075  
                 $199,806 
                 $449,519  
                 $88,106 
                 $2,535,506 
                 1.295000 
               
               
                 $2,054,943 
                 $199,806 
                 $513,736  
                 $99,665 
                 $2,868,150 
                 1.295000 
               
               
                 $2,311,811 
                 $199,806 
                 $577,953  
                 $111,225  
                 $3,200,795 
                 1.295000 
               
               
                 $2,568,327 
                 $199,806 
                 $642,082  
                 $122,768  
                 $3,532,983 
                 1.295000 
               
               
                 &gt;$2,568,327 
                 $199,806 
                   
                   
                 $3,532,983 
                 NA 
               
               
                   
               
             
          
         
       
     
     The Loss Conversion Factor is constant, or at least constant to within the numerical accuracy of the system calculating the premiums. This, in part, is due to the fact that there has been no motivation to modify a Loss Conversion Factor and it is therefore easiest to keep it the same over the linear portion of the retrospective rating plan. 
     For losses higher than $2,568,327 (last row of table 1), the premium is capped at the maximum, $3,532,983. The loss conversion factor is not applicable in this range (NA). 
     Only large companies, such as those with expected losses of at least $500,000 per year, can qualify for retrospective plans in the US. Small and medium sized companies are usually limited to Guaranteed Cost insurance. Also, the only retrospective plans that are available are linear ones. This is due in part to governmental and other regulatory requirements as well as the computational difficulties inherent in providing premium quotes for a broad range of companies that vary in a non-linear manner. The computational and practical challenges of providing non-linear plans and the reasons why they have not been available prior to the disclosures provided herein, are described in more detail in Crouse, Charles, “On Non-Linear Retrospective Rating”, Proceedings of the Casualty Actuarial Society, Nov. 18, 1949. Said publication is incorporated herein by reference. 
     Non-Linear Retrospective Premium Plans for Medium Sized Companies 
       FIG. 3  illustrates the frequency distribution  300  of actual losses for medium sized companies. It is dramatically different than the frequency distribution of actual losses for large companies shown in  FIG. 1 . The frequency distribution presented is for companies that have average losses of $160,000 per year. These companies might have several hundred employees. 
     The peak of the frequency distribution has shifted to zero. This means that it is fairly likely that some companies will experience no losses at all in a given year. In this case, the probability that a company will experience no loss is about 10%. On the other hand, the tail  302  has become much longer. This means that companies that do experience losses are much more likely to experience losses that are much higher than the average. The net effect is that average losses  306  are much higher than median losses  304 . The size of the difference between average losses  306  and median losses  304  dramatically reduces the viability of linear retrospective plans for these companies and hence tonly Guaranteed Cost plans are available to them. 
       FIG. 4  further illustrates why a linear plan  420  is not viable for medium and small sized companies. The corresponding Guaranteed Cost plan  410  and frequency distribution of losses  400  are shown for comparison. If the minimum is set to the Basic  412 , and the crossover  424  with the Guaranteed Cost plan  410  is pegged at the average losses, then the linear portion of the curve must extend to a relatively much higher level  426  than a large company in order for there to be enough premium collected to cover the overall cost of claims. The very high maximum means that the policy is no longer effectively insurance for companies that suffer large losses. This is because the cost of the premiums and the amount of the losses themselves are of the same magnitude. Also, there is little or no risk-sharing between the companies that suffer large losses with those companies that do not suffer any losses. 
       FIG. 5  illustrates a class of non-linear premium functions which address many of the above limitations and allow participating insurance plans to be effectively offered to medium and even small companies. 
     The non-linear premium function  530  illustrated in  FIG. 5  comprises an initial relatively steep portion  532 , a breakpoint  538 , a subsequent relatively shallow portion  534 , and a plateau portion  536 . A corresponding linear plan  520 , Guaranteed Cost plan  510  and frequency distribution of losses  500 , are shown for comparison. 
     The non-linear plan is set at the Basic  512  for zero actual losses. It is pegged  524  at the level of the Guaranteed Cost plan at the average. Because there is a breakpoint  538  in the function, the plateau portion  536  of the non-linear plan  530  can be much lower than the plateau  526  of the corresponding linear plan  520 . The reason is that more premium is collected at lower loss levels where most insured companies will wind up. This extra premium is available to compensate for the higher losses that the smaller fraction of insured companies will experience. 
     From a customer standpoint, this non-linear plan has an advantage over a linear plan of still providing meaningful savings in premiums for companies with losses somewhat below the average, the possibility of very large savings in premiums for companies with exceptionally low losses, and a much lower cap on maximum premiums for companies with large losses. 
     From an insurance carrier standpoint the non-linear approach provides an additional parameter (e.g. the breakpoint  538 ) which can be adjusted during the sales process to better meet the perceived needs of the customer. 
     Curvilinear Premium Function 
       FIG. 6  illustrates a non-linear premium function  610  with curvilinear properties. A corresponding linear plan  620  and frequency distribution of losses  600  are shown for comparison. 
     The curvilinear function  610  comprises an initial feathered portion  612 , a dimple  624 , and a subsequent feathered portion  614 . A plateau (not shown) may also be present at very high actual loss levels. 
     Similar to the corresponding linear plan  620 , the curvilinear function intersects the Y axis at the Basic  622  and has a premium equal to the corresponding guaranteed premium at the average of the actual losses  624 . From a company perspective, the curvilinear approach presents a smoother looking curve which shows increasing benefit for exemplary safety performance (lower actual losses). 
     From an insurance carrier perspective, the accelerated increase in premium shown in feathered portion  614  after the dimple  624  provides more premium dollars to help keep the upper plateau as low as possible. The curvilinear approach also allows small incremental increases  626  in premiums even if actual losses almost triple so that there is always some premium savings incentive for continued safety vigilance even in years when large losses have already occurred. 
     Reinsurance Participation Plan 
     One of the challenges of introducing a fundamentally new premium structure into the marketplace is that the structure must be approved by the respective insurance departments regulating the sale of insurance in the states in which the insureds operate. 
     In the United States, each state has its own insurance department and each insurance department must give its approval to sell insurance with a given premium plan in its respective jurisdiction. Getting approval can be extremely time consuming and expensive, particularly with novel approaches that a department hasn&#39;t had experience with before. Also, many states require insurance companies to only offer small sized and medium sized companies a Guaranteed Cost plan, without the option of a retrospective plan. In part, this is because of governmental rules and laws that regulate the insurance industry. 
     Disclosed herein is a reinsurance based approach to providing non-linear retrospective premium plans to insureds that may not have the option of such a plan directly. It also has the surprising ability to enable non-linear plans while at the same time complying with state regulations. 
       FIG. 7  illustrates an exemplary physical system and method tied to particular machines which allows for the provision of improved insurance premium plans in compliance with regulatory requirements that do not make specific provision for these plans. It is based on the fact that an insurance carrier can cede a certain portion of an insurance risk to a reinsurance company. Said reinsurance company can, in turn, enter into a separate Participation Agreement with the insured whereby a credit or debit is assessed on the insured as a function of the losses experienced by each insured. 
     An admitted insurance carrier  730  has a license from a state insurance department  760  to sell Guaranteed Cost workers&#39; compensation insurance in a given state. The insurance carrier obtains approval by using an industry standard Guaranteed Cost policy and filing premium rate requests with the insurance department  735 . The insurance department, already familiar with the policy, approves the rates  765 . 
     The insurance carrier then contractually arranges with a broker  750  to sell said standard policies to a targeted class of companies. These targeted classes include small sized  702  and medium sized  704  companies. As used herein, a small company has average losses of $60,000 per year or less. A medium sized company has average losses in the range of $60,000 to $500,000 per year. A large company has average losses of $500,000 per year or more. In this instance, the insurance carrier elects not to offer the policies to large companies  706  for competitive reasons. 
     As used herein “broker” is used in a broad sense to include independent brokers, captive agents, independent agents, the insurance carrier&#39;s own sales force, and other entities licensed to sell insurance. 
     Insured companies are shown in  FIG. 7  as stick figures  710 . The width  712  of a stick figure corresponds to the average loss rates for a given company. The height  714  of a stick figure corresponds to the actual losses experienced in a given year. The hat on a stick figure corresponds to the aggregate riskiness of the jobs in the company. A hard hat  718  (e.g. construction) corresponds to relatively high risk jobs. A mortar board  716  (e.g. office) corresponds to relatively low risk jobs. 
     In order to assist the broker in selling the insurance product, the insurance carrier  730  develops a computer implemented sales tool  732 . This is transferred  731  to the broker  750 . The broker makes additional modifications  752  to adapt it to its own needs (e.g. installed broker logos). The broker then uses the tool to illustrate  751  policies to prospective insureds  760 . Whereas the policies are Guaranteed Cost policies  754 , the broker nonetheless has a certain amount of freedom  756  to adjust the premiums to meet market demands. 
     If an insurance offering meets a prospective insured&#39;s needs, then it may apply  761  for coverage. A portion of the application information may then be transferred  733  to the insurance carrier  730  for underwriting purposes. If approved, the prospective insured then pays a premium and coverage is bound for the next year. 
     In order to provide a certain amount of loss participation to a prospective insured, the insurance carrier  730  may cede  737  a portion  734  of the insured risk to a reinsurance company  740  and pay a corresponding premium to said reinsurance company. The reinsurance company may be a captive reinsurer. In an exemplary embodiment, the insurance carrier  730  may retain the initial 40% of the risk, cede the next 20% of the risk to the reinsurance company  740  and retain the final 40%. If the insurance carrier  730  as a whole experiences total losses greater than 40% of the expected losses, then the reinsurance company  740  will pay  747  up to the next 20%. 
     The reinsurance company  740  can now provide funds to implement a non-linear retrospective rating plan as a “participation plan”. The reinsurance company does this by entering into a separate contractual arrangement directly with the insured. If the insured has lower than average losses in the next year, then the reinsurance company can provide a premium reduction  744  according to the participation plan. If the insured has higher than average losses in a given year, then the reinsurance company will assess additional premium  746  accordingly. The insured can now, in effect, have a retrospective rating plan because of the arrangement among the insurance carrier  730 , the reinsurance company  740  and the insured even though, in fact, the insured has Guaranteed Cost insurance coverage with the insurance carrier  730 . 
     The technology  742  to illustrate the participation plan can be transferred  741  to the broker  750  so that the broker will have the technology  758  to illustrate the plan to a prospective insured. One of the advantages of the participation plans described herein is that the broker  750  has greater freedom  759  to adjust the plan to meet the requirements of a prospective insured than it would have with either a Guaranteed Cost plan or a conventional linear retrospective plan. 
     Companies eligible for the participation plan might be medium sized companies. The broker would target these companies and would present the combined insurance  755  and participation plans  757  to a given prospective insured  780 . If the prospective insured applied  781  for both offerings, then the necessary information  733 ,  743  would be transferred to the insurance carrier  730  and the reinsurance company  740  such that each could enter into its respective agreements. 
     The non-linear plans described herein, such as a curvilinear plan, may even be offered  771 ,  773  to small companies if the loss experience of a multiplicity of small companies is aggregated into a cell  770 . The companies within a given aggregation cell may not be aware of who the other companies are. Each one must make separate application  775  for coverage. Enough small companies should be present in a cell so that the collective expected losses are comparable to a medium sized company. 
     Aggregation may also be over time. A relatively small company, for example, may be able to qualify by itself for a non-linear plan if it is willing to make a firm commitment for three year participation so that it&#39;s average losses over three years are comparable to the average annual losses of a medium sized company. The insurance policies themselves are one year policies, but the separate participation plan agreement is for a three year term. 
     It may be helpful in time-aggregation if prospective insureds are redundantly notified over the course of the sales process and thereafter with very explicit language that the participation agreement is for several years and not just one. To accomplish this, for, example, multiple signatures by responsible parties in the prospective insureds for said redundant notifications may be required. This reduces the exposure of the insurance carrier, broker and reinsurance company to complaints by the insured company once the participation agreement is in force due to the fact that the insured company “didn&#39;t realize” that the agreement was for a term of multiple years. Similar considerations are useful for aggregations over multiple companies. 
     Smith Premium Ratio Diagrams 
     Another barrier to providing non-linear retrospective plans has been the inordinate complexity of calculating the appropriate premiums for companies of various sizes and presenting said premiums to prospective insureds. It has been surprisingly discovered, however, that a new method of graphically representing these plans largely overcomes these difficulties. This new form of graphical representation is termed a “Smith Premium Ratio diagram” or simply a Smith diagram. Smith diagrams are used in combination with Lee Loss Ratio diagrams (or simply “Lee diagrams”) to substantially simplify the calculation of the appropriate relationship between actual losses and premium for non-linear retrospective plans. 
       FIG. 8  is a Lee Loss Ratio diagram which illustrates the relationship between loss ratio (r) and cumulative distribution function of actual losses (CDF) for medium sized companies. Lee diagrams are more fully described in Lee, Yoong-Sin, “ The Mathematics of Excess Loss Coverages and Retrospective Rating—A Graphical Approach”, Section  4, PCAS LXX, 1983. Said publication is incorporated herein by reference. 
     A loss ratio is the ratio of actual losses to expected average losses. If a company has expected average losses of $160,000 per year, and experiences an actual loss of $80,000 in a given year, it will have a loss ratio r of 0.5 for said given year. 
     The cumulative distribution function of actual losses (CDF) of a company is the rank of that company&#39;s actual losses in a given year relative to the actual losses of all of the other companies with the same expected average losses. If a company has a CDF of 0.5, then half of the other companies with the same expected losses had actual losses greater then said company had. 
     Curve  808  in  FIG. 8  shows the loss ratio versus CDF for companies with expected average losses of $160,000. This curve is calculated from the above referenced Table M. The scatter in the curve is due in part to rounding errors in the data presented in Table M. 
     The loss ratio curve intersects the x axis  804  at a CDF of 0.1. This means that 10% of the companies in this class will have zero losses in a given year. 
     The loss ratio has a value of 4 at a CDF of 0.95. This means that 5% of the companies  806  in this class will have losses that are at least four times the average in a given year. 
     The area under the curve  802  represents the total losses for all companies in this class. Since the y axis has been normalized by dividing the actual losses by the expected average losses, the area under the curve is unity. This is true for all classes of company size. The area under the curve includes the area not shown for values of loss ratio above 4. 
       FIGS. 9A and 9B  compare the Lee diagram for medium sized companies and the Lee diagram for large companies. Relative to the loss ratio curve for medium sized companies  904 , the loss ratio curve for large companies  914  is shorter and fatter. The probability that a large company has no losses  914  is very small. Likewise the probability that a large company has losses greater than 4 times the expected average loss  916  is also very small. The area under the curve for large companies  922 , however, is still unity. 
       FIG. 10  illustrates an exemplary Smith diagram for designing a non-linear retrospective premium plans. The Smith diagram is similar to the Lee diagram in the sense that the x axis is the CDF of the actual losses experienced by the companies in a given size class. The y axis, however, is a designed Premium ratio (p) instead of an experienced Loss ratio (r). A Premium ratio is defined as the ratio of the actual premium charged to a given company divided by the Guaranteed Cost premium. A curve on a Smith diagram is a Premium ratio curve. It indicates what premium an insured company will be charged as a function of the CDF of their actual losses, as opposed to the actual losses themselves. The area  1002  under the premium ratio curve  1010  will be equal to unity for plans which are designed to collect the same amount of premium as if all insureds paid the Guaranteed Cost premium. Premium curves with higher and lower areas may be used depending upon the market requirements. 
     A designer of a non-linear retrospective premium plan has tremendous freedom using a Smith diagram. In general, plans designed on a Smith diagram will be non-linear when premiums are graphed versus losses, due to the non-linear nature of the loss curves. This will be discussed in more detail below. 
       FIG. 10  illustrates an exemplary non-linear premium plan  1010  with a variety of design features. The plan may have horizontal portions  1004  when insureds in a certain range of losses should be paying the same premium. This would include insureds in relatively small size classes that are expected to have zero losses at low CDFs. 
     The plan may have regions that increase linearly with CDF  1006  where it is desired that premiums increase linearly with increasing losses. As indicated above, premiums may increase linearly with CDF, but they will not increase linearly with actual losses due to the non-linear relationship between loss ratios and CDF. 
     The plan may have dimple sections  1008  where the slope of the premium increases. There can also be curved portions  1012  and step changes  1014 . 
     The premium curve should generally increase or at least stay the same as CDF increases. 
     The premium curve should also be single valued at a given CDF so that only one premium will be charged for a given CDF. 
       FIG. 11A  shows how a Smith Premium Ratio diagram can be overlaid with a Lee Loss Ratio diagram so that the relationship between the loss ratio (r) and premium ratio (p) can be calculated. This, in turn, can be used to calculate the relationship between actual premium and actual losses. The premium ratio curve is shown as item  1100 . The loss ratio curve is shown as item  1108 . The CDF scales are matched and the loss ratio and premium ratio scales are also matched. 
     For a given loss ratio  1102 , the corresponding CDF  1104  is read off of the loss ratio curve  1108 . The premium ratio  1106  is then read off of the premium ratio curve at the same CDF. In this example, a loss ratio of 1.00 corresponds to a CDF of 0.75. A CDF of 0.75, in turn corresponds to a premium ratio of 1.35. The loss ratio curve  1108  is for medium sized companies with average losses of $160,000 per year. An approved Guaranteed Cost premium for said companies might be $208,000. A company participating in this plan that has a loss ratio of 1.00, therefore, will pay a premium of $280,800 (p=1.35). 
       FIG. 11   b  shows that the same premium ratio curve  1110  can be used for substantially larger companies, such as those with average losses of $4,000,000 per year. The premium ratio for a given loss ratio is calculated in the same manner, but using the loss ratio curve for large companies  1118  of this size. At  1112  in  FIG. 11   b , loss ratio of 1.00, for example, corresponds  1116  to a premium ratio of 1.2 and not the premium ratio of 1.35 as for smaller companies. 
     The steps for determining the relationship between premium and loss for non-linear plans can be automated using appropriate look up tables for loss ratio and premium ratio curves. Analytic forms can also be used. The method has the surprising ability to provide accurate and reproducible premiums despite scatter in the underlying data of the loss ratio curves. 
       FIGS. 12A through 14B  present the Smith diagrams for a range of premium plans. 
       FIG. 12A  presents the Smith diagram from a simple Guaranteed Cost plan where the premium that the insured pays is the same no matter what its actual losses (CDF) are. 
     The premium curve  1204  is a horizontal straight line. The area  1202  under the curve is one (i.e. 1.00×1.0). 
       FIG. 12B  presents a Smith diagram for a straight line plan  1214 . It is suitable for large companies where the probability of having zero losses is low. It has a premium that is linear with respect to CDF, but is non-linear with respect to actual losses. As discussed above, this is due to the curvature in the loss ratio curve of the corresponding Lee diagram. 
     The area  1212  under the premium ratio curve is one (i.e. ½×2.00×1.0). This plan charges no premium to an insured if their losses are zero (CDF=0.0). The maximum premium is capped at twice the Guaranteed Cost premium (p=2.00). All insureds, however, have an opportunity to get a discount relative to the maximum premium, no matter how high their losses are. Even if an insured has losses that are in the top 90% (CDF&gt;0.9), it can still have a discount of 10% relative to the maximum. 
       FIG. 13A  presents a Smith diagram for a straight line plan  1304  with a horizontal premium cap  1306 . The horizontal premium cap allows the insured to significantly lower the maximum premium  1308  should they experience high losses in exchange for a somewhat higher premium  1302  when they experience lower losses. 
       FIG. 13B  presents a Smith diagram for a straight line plan  1314  that has a positive y intercept  1316 . The intercept corresponds to the Basic. This allows insureds to tradeoff a reduction in maximum premium  1318  in exchange for an increase  1312  in minimum premium while at the same time being able to earn a discount relative to the maximum no matter how high their losses are. 
       FIG. 14A  presents a Smith diagram for a straight line plan  1406  that has a horizontal portion  1402  of constant premium ratio at low CDFs. Plans with this feature are suitable for medium sized and even small sized companies where a significant fraction of insureds will not have any losses. In the case illustrated, the plan is suitable for companies that have up to a 20% chance of being loss free in a given year. This design also incorporates a horizontal cap  1408  on the maximum premium. 
       FIG. 14B  presents a Smith diagram for a premium plan that comprises a dimple  1414 . The dimple allows the insurance company to anchor a reference premium, such as the Guaranteed Cost premium (p=1.00) at a particular CDF (e.g. 0.5). This plan also anchors the minimum premium  1412  at the Basic. A sales person presenting this plan to a potential insured has the option of independently lowering the maximum  1418  by increasing the intermediate values  1416  of the plan while at the same time keeping minimum pinned at the Basic and the median (CDF=0.5) pegged at the Guaranteed Cost premium. Thus, compared to conventional linear retrospective plans, this plan offers a lower maximum for the same Basic and Guaranteed Cost premium. 
     Example 1 
     Company A applies for workers compensation insurance coverage from insurance carrier B. Company A has expected annual losses of $162,513. This places them in Expected Ultimate Loss Group 55. They would normally qualify for a Guaranteed Cost Premium plan but would be too small for any available linear retrospective premium plan. 
     Reinsurer C, however, offers a reinsurance participation plan to Company A through Insurance Carrier B&#39;s sales force. The reinsurance participation plan stipulates that Company A will receive a discount if its losses are low but will be liable for a surcharge if its losses are high. The company must also commit to participating in the plan for 3 years. 
     In order to provide premium versus loss data to Company A (the prospective insured), an insurance agent for Insurance Carrier B inputs Company A&#39;s expected annual losses and other necessary information, such as number employees, work type, etc., into an input device, such as a laptop computer. The laptop computer then determines the EULG for Company A. 
     The laptop computer has been specifically modified to calculate the premium versus loss data by at least appropriate programming and downloading of necessary data, such a Table M. The laptop calculates the premium versus loss data such that the data is described by the equation:
 
premium=basic*VC(actual_losses)*actual_losses
 
Where:
         premium is a premium;   basic is a Basic;   actual_losses is an actual loss; and   VC(actual_losses) is a Variable Loss Conversion Factor Function that describes the variation of a Loss Conversion Factor with respect to actual losses.
 
VC(actual_losses) may be provided as a look-up table or analytic or other function. If VC(actual_losses) is provided as a look-up table, then interpolation or other smoothing may be used to estimate values between table values.
       

     Premium versus loss data is presented to the prospective insured using an output device (e.g. said laptop). The premium (“Collectible Premium”) versus loss data (“Actual Losses”) for this example is presented in Table 2 along with the corresponding Loss Conversion Factors. 
     
       
         
               
             
               
               
               
             
           
               
                 TABLE 2 
               
             
             
               
                   
               
               
                 Retrospective Premium Plan with Variable Loss Conversion Factor 
               
             
          
           
               
                   
                   
                 Loss 
               
               
                 Actual 
                 Collectible 
                 Conversion 
               
               
                 Losses 
                 Premium 
                 Factors 
               
               
                   
               
               
                     $0 
                 $124,179 
                   
               
               
                  $16,370 
                 $142,089 
                 1.09 
               
               
                  $24,278 
                 $146,608 
                 0.92 
               
               
                  $32,187 
                 $150,005 
                 0.80 
               
               
                  $40,095 
                 $153,386 
                 0.73 
               
               
                  $49,133 
                 $155,645 
                 0.64 
               
               
                  $57,041 
                 $157,904 
                 0.59 
               
               
                  $64,989 
                 $160,164 
                 0.55 
               
               
                  $72,858 
                 $161,301 
                 0.51 
               
               
                  $80,766 
                 $163,561 
                 0.49 
               
               
                  $88,674 
                 $165,820 
                 0.47 
               
               
                  $102,231 
                 $183,796 
                 0.58 
               
               
                  $121,437 
                 $203,107 
                 0.65 
               
               
                  $142,902 
                 $212,144 
                 0.62 
               
               
                  $167,757 
                 $213,265 
                 0.53 
               
               
                  $194,871 
                 $213,265 
                 0.46 
               
               
                  $223,114 
                 $214,387 
                 0.40 
               
               
                 &gt;$223,114 
                 $214,387 
                 NA 
               
               
                   
               
             
          
         
       
     
     The minimum premium for no actual loses is $124,179. The maximum premium for losses greater than $223,114 is $214,387. The corresponding Guaranteed Cost premium is $162,513. 
       FIG. 15A  illustrates the Variable Loss Conversion Factor Function  1500  versus actual losses for this example. The Loss Conversion Factors range from 0.4 to 1.09. This is a factor of more than 2.5 (1.09/0.4=2.7). The smallest change in Loss Conversion Factor is from 0.49 to 0.47. This corresponds to about a 4% change (0.49/0.47=1.038). Benefits can be achieved even if the change in loss conversion factor is as small as 1%. 
     The Variable Loss Conversion Factor Function actually increases  1501  with increasing actual losses for an intermediate range of actual losses. This is manifested as a dimple in a plot of premium versus actual losses (item  624 ,  FIG. 6  and item  1546 ,  FIG. 15C ). The increase in Variable Loss Conversion Factor over a range of actual losses allows for a lower maximum premium because more premium is collected from said intermediate range than if said Loss Conversion Factors were not allowed to increase. In this example, the intermediate range is set to begin at about the ½ of the expected annual losses ($88,674/162,513=0.55), and end at about 0.8 times the expected annual losses ($121,437/$162,513=0.75) that correspond to the EULG. This allows the prospective insured to more readily compare this offering with a standard Guaranteed Cost policy. The intermediate range can also be over different ranges depending upon the needs of the prospective insured. 
     The premium is constant for losses greater than $223,114. The Loss Conversion Factor is not applicable in this range. 
       FIG. 15B  presents a Smith diagram for a premium ratio function used to determine at least in part the premium versus loss data of Table 2. The Smith diagram shows the premium ratio versus Cumulative Distribution Function, said Cumulative Distribution Function being with respect to actual losses of insured with a given EULG. 
     The premium ratio function has a minimum  1502 , a first CDF range  1504 , a dimple  1506 , and a second CDF range  1508 , with a horizontal maximum  1512 . 
     The first CDF range is from 0 to about 0.6. A suitable equation for describing the premium ratio over said first range is:
 
 p=A*CDF+p   o  
 
Where:
         p is the premium ratio;   CDF is the Cumulative Distribution Function;   A is the increase in p for a unit increase in CDF; and   P o  is the premium ratio at CDF=0       

     In this example A is about 0.4 and P o  is about 0.8. 
     The second CDF range of from about 0.6 to about 0.8. A suitable equation for describing the premium ratio function over said second range is:
 
 p =( P   1   −P   2 )*(1−exp(−(CDF−CDF o )/CDF * ))+ P   1  
 
Where:
         p is the premium ratio;   CDF is the Cumulative Distribution Function;   P 1  is the premium ratio at the start of the second CDF range;   P 2  is the premium ratio at the end of the second CDF range;   CDF o  is the CDF at the start of the second CDF range; and   CDF* is a curvature parameter indicating how quickly p changes from P 1  to P 2          

     In this example, P 1  is about 1.0, P 2  is about 1.3, and CDF* is about 0.05. 
     The salesperson presenting the plan to the company can independently vary the parameters of the height of the dimple, the maximum and the total height  1516  of the premium plan in response the company&#39;s requirements even after the insurance carrier has underwritten the company. The software presenting the premium versus loss data to the company can alter the curvature of the premium ratio function over the second CDF range as the sales person adjusts the parameters so that the total area under the curve falls within a desired range. 
     The sales person presenting the plan also has the option of adjusting the area under the curve to be somewhat larger or smaller than unity by adjusting the total height of the curve  1516 . The sales person is subject to the constraint, however, that the sum of the premium weighted areas for his/her total book of business (i.e. the other companies that the sales person has signed up) average to unity plus or minus a small percentage, such as 5%. Thus a sales person can provide discounts in certain competitive situations provided those discounts are matched by more profitable pricing as other competitive situations allow. The insurance carrier will then collect enough premium overall to cover expenses and claims. 
     The insurance sales person presenting the reinsurance participation plan to the company uses a particular machine to calculate a set of losses versus premiums in real time. The particular machine comprises a work station located with the sales person and a pricing server which may be located remotely. The communications link between the work station and the pricing server has sufficiently high bandwidth so that the sales person can present data in real time (e.g. lags of less than 10 seconds). The pricing server has the pricing algorithms including the premium ratio curve as well as loss ratio curve. The workstation receives input data from the company and transmits said data to the pricing server. The pricing server selects the appropriate loss distribution  1522 , and determines a set of loss ratio  1526  and corresponding premium ratio  1528  pairs wherein the loss ratio and premium ratio in each pair have the same CDF to within a suitable accuracy, such as +/−5 percent. This information is then transmitted to the workstation. 
       FIG. 15C  presents the loss ratio/premium ratio pairs for the data presented in Table 2 above. The Basic  1542 , dimple  1546 , and maximum  1552  are positioned where expected. 
     Company A considers itself safer than its peers and is pleased with the opportunity to save up to 20% in its premium. It elects to purchase the policy and commit to the three year term of the reinsurance participation plan. 
     Detect and Correct for Adverse Selection 
     Allowing the insured to adjust the premium ratio curve to best meet its needs can help an insurance carrier detect and correct for adverse selection. Adverse selection means that the company has better information about its future losses than the insurance carrier has and can therefore select a form of coverage that may not leave the insurance company with enough premium to cover claims and expenses. For example, a company that anticipated higher than normal losses in a given year might select a premium plan that had a very low maximum cap but a high Basic. Companies that anticipated lower than normal losses might select plans with very low Basics but high maximum caps. 
     In principle, it should not matter what premium plan a company selected if the insurance carrier has done the proper job of underwriting. As a practical matter, however, the company has more information about its future plans and operations than the insurance carrier does, so the carrier&#39;s underwriting may have a systematic error. 
     Adverse selection may be compensated for at least in part by adjusting the overall area under the premium ratio curve in a Smith diagram in response to the choices a prospective insured makes. The area might be increased (i.e. more premium on average) if a company was a bit too concerned about the maximum premium. Conversely, it might be acceptable to decrease the area if the company exhibited very little concern about the maximum premium. 
     These adjustments can be made by the sales software. 
     Synergies with Bundled Employee Services 
     There are surprising synergies when employee services are bundled with the insurance coverages and participation plans described herein. This is particularly true if the employee services are payroll payment services. 
     It has been discovered that the data available from employer payroll services can be used to assess risk and to reduce fraud in workers&#39; compensation insurance. This fraud might be on the part of the insured company. There are very large differences between required insurance premiums for high risk occupations and low risk occupations. There is a motivation, therefore, for insured companies to incorrectly categorize the occupations of its employees in order to obtain a rate reduction. This, however, can result in an imbalance in payroll. The payroll company can detect this and the insurance carrier, in turn, can insist that the insured company have the correct job classification codes for its employees and thus collect the appropriate amount of premium. 
     Other Insurance Coverages 
     The non-linear retrospective premium plans and reinsurance participation plans can be applied to other insurance coverages, such as general liability, professional liability, auto, health and others as long as appropriate loss ratio data is available or calculable. 
     CONCLUSION 
     As used herein, the terms “about”, “approximately”, and their synonyms mean within plus or minus 10 percent of a given value, unless explicitly indicated otherwise or indicated otherwise by the context in which they are used. 
     While the disclosure has been described with reference to one or more different exemplary embodiments, it will be understood by those skilled in the art that various changes may be made and equivalents may be substituted for elements thereof without departing from the scope of the disclosure. In addition, many modifications may be made to adapt to a particular situation without departing from the essential scope or teachings thereof. Therefore, it is intended that the disclosure not be limited to the particular embodiment disclosed as the best mode contemplated for carrying out this invention.