Overview

In the simplest case, insurance is a protection from negative shocks consisting in a payment to the victim.

Insurance covers events, uncertain in terms of actual occurrence, timing and amount of damage, by spreading the risk to a pool of mutually connected people or through business transactions with an institution, which assumes the risk in exchange for a payment, called "premium".

More or less regulated by government, an entire industry provides an extensive range of differentiated insurance products and contracts, for general and special purposes. Areas commonly covered by insurance are life and non-life events, such as road accidents, physical damages to property (e.g. because of hurricanes, floods, fires), work injuries, healthcare, life duration (including pension schemes), business operations (e.g. harvest losses because of weather conditions), and also macro-risks, e.g. unemployment and climate change impacts (including the so-called "Losses and damages" topics of UNFCCC negotiations).

There are six independent roles in the market: who decides on the insurance, who is the agent of the damage covered, who pays for the premium, who is the victim of the damage, who reintegrates the damage, who further shares the risk. Depending on the scheme, some of these roles are played by the same person or institution.

In this paper, we shall discuss the main issues of insurance theory by taking advantage of a computable formal model of insurance expressed as a standard Excel file, allowing you to follow, explore and modify our arguments, mainly but not only related to non-life insurance schemes.

Insurance does not recover the physical or immaterial loss: it compensates with money the evaluated monetary loss (depending on the scheme, the evaluation is about the price of past purchase, the price of repurchasing a substitute now or a mix, possibly taking into account inflation). For instance, biodiversity losses due to climate change cannot recovered by insurance; however the cost of protecting biodiversity might be funded by a scheme that includes an insurance element.

The advantage of being insured is apparent when the damage actually occurs and it is (partially or totally) recovered thanks to the scheme. In turn, this allows people and firms to undertake risky activities to which the potential damage is connected, thus the social advantage of insurance is enriched by this second element: that certain activities do take place (and with a wider range of actors than without insurance).

Mutual insurance

The basic case of insurance is a group of people-at-risk who decide to mutually guarantee that, in case of damage to any of them, they all will share the cost of it [1]. They sum up all claims (requests for recompensation) and divide them in equal parts.

In model terms, each person faces a probability of a negative shock of given amount (severity) [2], where the probability can be influenced by what happens to others and what happened before. Negative shocks can thus be correlated to each other and auto-correlated over time. The influence can be positive (adding to the final probability) or negative (subtracting to the final probability) or zero (no influence: events are independent). This is much more realistic and flexible than most neoclassical models (that are based on the constraint of independence).

At the end of each period, the total damage for all negative shocks is computed and divided equally for all the participants to the mutual insurance scheme. In this way, everybody will have a larger probability of having to pay something but a lower probability to pay a lot. Since the payment is ex-post, people don't need to know the probability of the events, even less the complex relations among the events. This is realistic because the probability of negative events to a specific person are largely unknown, whereas only attribution to a class of events allow for some predictability, as we shall see later on.

By experimenting with the software, you shall discover that the advantage of being insured is higher:

1. the larger the amount of damage;

2. the lower the probability of occurrence;

3. the larger the pool of people mutually insured, depending however on the actual composition;

4. the more negative is the correlation between events;

5. the more negative is the auto-correlation over time of damages.

For high-risk people, the advantage to be insured is larger if low-risk people take part to the scheme.

In other words, if there is a very little probability of a very high damage, by pooling with others you get a relatively small and constant premium to pay, smoothing the negative shock if and when it occurs. If the probability is high, the premium will approach the level of the damage. If the probability is one, the premium will be equal to the damage, less the taming effect of the presence of the others.

The first mitigating effect is due to the presence in the pool of people to which no negative shock occur in the period, but pay for the premium, reducing the effect on the others. On average, low-risk people subsidize high-risk people. This is the inter-personal mitigating effect.

The second mitigating effect is due to negative correlations across shocks: if a negative occurrence reduces the probability of another one, the presence of both in the pool keeps the premium relatively low and constant. Adding a few participants whose damages have a negative correlation with the existing pool is a major leverage to improve the advantage of insurance for everybody.

The third factor is the inter-temporal mitigating effect: over time, the "periods in which the shock does not occur" pay for "the periods where the damage gets real". This happens even if a person is alone in the pool and just cumulate over time payments with itself [3].

The relative imporance of the three effects is shown in the sheets "Advantage of insurance" and "Inter-temporal insurance".

A further advantage of insurance is the reduction of the likelyhood of going bankrupt (a damage larger than income or even wealth), which in turn is particularly important if the damage is provoked to somebody else and has to be paid by you. Your bankruptcy deprives the victim of the payment (or part of it).

All this to say that people may have a widely different advantage in being insured, with some unwilling to participate if they think they will be subsidizing others all the time. By including them, however, the total advantage of insurance can rise, which would explain a system-level rational for mandating participation [4].

Please note that this scheme of mutual insurance does not fix in advance the amount to be paid; if nothing happens, nobody will pay anything. The payment varies depending on negative shocks really occurring.

This scheme of mutual insurance, albeit simple, faces in reality a number of troubles. People may be unwilling to pay at the end of the period, trying to escape their commitment; they may disagree (and they have an interest in doing that!) about the amount of damage or whether the damage actually took place. Furthermore, their pocket may be unsufficient as for the actual payment due [5].

Enforcement costs and legal costs can be high, with the effect that the ex-ante credibility of the scheme may be low. These - and other - reasons explain the presence of institutions providing insurance.

Insuring institutions in competition

A stable and well known institution can take part as a contracting side to transactions aimed at insuring people and properties. By exhibiting a large asset and an history of actual payments to victims, the institution has the trustworthiness and reputation to cover losses and damages raising from new commitments, even beyond what standard estimation could indicate for negative shocks: assets are used as buffering tools just as inventories are used in manufacturing to cope with erratic shocks in demand[6].

Cumulated profits over time and across different insurance pools serves to the same purpuse. Moreover, it can enter into re-insurance agreements with other institutions, sharing and transfering the risks.

All this allows for premiums paid in advance instead of the ex-post agreements we analysed beforehand: clients can believe to the promise of covering losses and let the insurer propose a premium, based on its own computation of probabilities and damage amounts, with a shift in computational burden. The institution has a much wider capability to compute and categorize events than individuals. It has a large array of lawyers and field experts to analyse the risks, propose arrangements mitigating and preventing the damages, and enforce the contract across the pool of people insured.

If this is allowed by regulation, the insurer can adjust the premium to specific circumstances and commitments taken by the insured, providing some flexibility with respect to the equal-premium-for-all case we explored before. If people are not mandated to buy an insurance, this flexibility can attract people that otherwise would not participate to the scheme. Conversely, the inter-personal insurance advantage is weakened.

The presence of an institutional partner is particularly important for cognitive reasons. The probability of the negative shock is largely an interpretation of an external eye attributing the event to a category, possibly using pieces of information and historical time series to compute likelihood without entering into the actual chains of events, responsibilities and coincidences that generates the occurrence.

For instance a car accident will depend on a very precise number of mistakes, difficulties, ineptitudes, risk-taking behaviour and misfortunes; an external observer will see it as an occurrence out of a percentage equal to "yearly car accidents rate of the country taking the last 10 years as reference" but another could (equally well!) consider it as an occurrence out of "January car-with-truck male-driven accidents in our sub-national region taking the last 5 years as reference". In the first case, it might have a 5% chance of occurring, in the second, say, 1.5%. Also the value of the damage may differ (e.g. 4000$ or 12000$).

External observers disagree over categorization, differ in information and parameters of the data to be considered (e.g. the extension of the time series), ways of elaborating information, internal guidelines, to the effect that they have different subjective probabilities.

Which was the "likelihood" of the accident serving as the basis for the computation of the premium to be requested by the insurance company is much more dependent on conventions, referred data and interpretation than the neoclassical theory would like to admit.

This heterogeneity in points of views is reflected in competitive markets where several institutions operate: competition across insurance companies, if significant, should provide alternatives to the customer, free to choose the best deal of coverage and options.

In the opposite direction, in the insurance industry, there are often methods to standardize the key ratios and elements on which premiums are calculated and proposed to customers. Actuarial tables covering events for which insurance offer products are common tools.

This practice has, in part, the aim of broadly agreeing ranges of the premium level (even, in certain situation, are used in order to develop an informal cartel fixing the same price across different companies).

When the clients are all mandated to sign a policy and there is just one insuring company (or all tacitly agree to ask the same price), the higher the premium, the higher the profits. Precise calculation about the likelihood of the event become just floors for the premium, with much higher requests being comfortably presented. You can explore this issue in the sheet "Companies" in the spreadsheet.

Competition across insurers should bring the price down to a level where one insurer, having computed a lowered-probability assessment, lower-than-usual administrative costs (and profits) is ready to propose the lowest possible premium for the policy. If does so aggressively on every occasion, it is possible that this insurer will operate with an inadequate organization and will face large losses in case of actual claims. If the market is unregulated, it may even go bankrupt, with all insured parties being deprived of the protection they paid for.

If the regulation, explicitly or implicitly, avoids bankruptcy ("too big to fail"), risk-prone companies will ask extremely low prices of their premium (with respect to the likelihood and damage of the event), with clients buying them while ignoring the possibility the the company will fail or relying on a failure that happens later than their claims or relying on the regulation that will shift the losses to the state or to other insuring companies (leading to bigger firms).

You can experiment several price scenarios, reflecting some of these situations, in the "Companies" sheet of the software.

However, usually the problem is the opposite: insurance companies share and generate a not-too-competitive market, with some geographical or customary partition of the market, premium requests not too dissimilar - and significantly higher than the "fair premium" resulting from the expected value of the losses (the probability times the severity). Customer might value the "name" and reputation of the insurer, without extensively searching for the cheapest possible policy (possibly because of high search costs but also because of cognitive, emotional, attention and time limitations).

Moreover, in many countries, there are monitoring authorities asking, influencing or mandating prices, capital requirements, re-insurance agreements, and other methods to share and transfer the risks. In particular, the customer may be free to remain uninsured, in principle providing incentives for companies to make more attractive conditions, in practice leaving a stock of people effectively not covered and with negative effects on the people that deal with them (e.g. people having an accident with uninsured car owner).

By exploring the "Company" sheet, you will see how a given flows of events over time impact on the companies that attracted customers. In the basic option, companies need to propose the same (lowest) price in order to have clients, who randomly choose which one of the institutions offering it will actually serve them. By repeatedly computing the results, you shall see that attracting customer is not the only issue for insurers: the key to profitability is to attract "good customers" - those who pay premiums without filing claims (because they were lucky enough not to have damages).

Needless to say, the large array of lawyers and field experts can be directed by the institution to devise "fine prints" clauses difficult to understand and refuse, having the goal of limiting claims and discouraging "bad clients" - which indeed are those who need insurance the most.

The profitability goal of insurers may stand against the maximization of total advantage of insurance, as you can experiment with the Excel file.

Income - related issues

Income distribution and income differences among poor, middle class and the rich are very important when risks of damages and insurance coverage are considered. Given a certain poverty lines, the poor have a larger probability of going below it for any given negative shock, while having a larger probability of going broken if the shock is large enough. Moreover, the poor might not afford the same premium as the middle class or the rich, with the consequence of larger share of people taking the risky and short-sighted choice of remaing uninsured, if only voluntary schemes are in place, or great pain in participating if mandated.

In absence of correlation between income and damage risk (and amount), it may happen that a "lucky" poor who did not suffer from negative shock does subsidy the "unlucky" rich that was in the opposite case, with ex-post cash tranfer from the poor to the rich.

The more so, if the damage is proportional to wealth (e.g. because the damage occurs to assets such as houses and the comulative bundle): the poor, having a smaller wealth, could quite systematically subsidize the rich over time.

Income-adjusted premiums in mutual insurance schemes are effective in reducing this injustice. A similar outcome could be achieved by tax-funded subsidies dependent on income, both in the mutual insurance scheme as in the case with institutions.

These arguments would be strengthened if you would consider the differences between rich and poor in wider terms: the rich can better absorb negative shocks, can afford higher premiums, has more connections to (rich) relatives and friends to cope with negative shocks through solidarity networks.

Please experiment with the software, changing the values of income, wealth and the fraction of income that constrains the affordability of insurance policy - in the "Basic area data". You will see the reasons of being uninsured and of pain under a mandate, after running the simulation in the "Companies" sheet.

Voluntary participation

Let's now consider the case where the people can participate to the scheme or refuse to participate. To model this choice, we use the simplest tool: a reserve price for the premium, which we already used in a general consumer purchase analysis. If the premium asked by the insurer is higher than the reserve price, then the people refuse to buy the insurance policy and remain uninsured.

The determination of the reserve price "reserveP" can be flexibly expressed by the following formula

reserveP =aE(X) + bEpsilon

where you find the following elements: a multiplier coefficient "a" expressing risk-aversion and, inextricably, a cognitive bias for people that ignore the computation of the expected value of damage "E(X)" and ignore the complex correlations and interrelations with the other events over time as well. Epsilon is a random variable uniformly distributed between - 1 and 1. In this way you can, by selecting different sets of parametres "a" and "b", cover the cases of risk aversion, risk-neutrality, risk-taking and a dilution of knowledge and computational requirements down to the case where the reserveP is simply a random extraction.

To include a known phenomenon of bounded rationality, a second condition you can impose is a minimum probabilty for the event to be relevant: if the probabilty is lower than the minimum threshold, the people will ignore the risk and refuse to participate.

A third condition is affordability of the premium with respect to income: you can fix a percentage of income over which the people cannot afford to buy (e.g. 10% of income, which is a relatively realistic cost for health insurance; 100% is for the full income, set more than 100% if you want to consider a case of loans and external subsidies).

Experiment with different parameters (set in the "Basic area data") checking the "V" (for "Voluntary") cell in the "Companies" sheet: you shall see that under many parameters, some people remain uninsured, take the risk, and are ruined by their own risky and short-sighted choice. For the reasons of lack of coverage, see the results shown in the "Basic area data".

Insurers' refusal to sell

Conversely, some systems allow the insurer to refuse to sell, basically because it anticipates too high claim costs over time. Instead of making money by pooling risk, the insurer want to make profit on each client, which runs against the very principle of insurance (the accepted client would be better off by saving scheme alone, without participate to the scheme).

To model this choice, and the consequences in terms of involuntary uninsured people, we shall use the same formula for the insurer to compute an expectation of claims for each perspective customer.

reserveP =aE(X) + bEpsilon +cE(Xall)

We added a third parameter "c" to include the possibility that the insurer "knows something" of the average damage across all people. You shall soon discover that if the insurer is not risk-taking (i.e. a and c significantly lower than 1) it would refuse all customer (if the premium is set at its fair level, equal to expected losses). In order the market to exist, the insurer might raise the premium to compensate. In other words, high premiums might be the result of risk-aversion of the insurer, transmitting to the customer its inefficiency or, better, "inability" (indeed a risk averted insurer seems an oxymoron).

Refusal can derive from all the other factors in the equation, so by private judgement of the firm, as well as by the sensibility to a large average probability of claim.

Reversed adverse selection

If the customer are mandated to buy and there is a cartel of insurers, allowed to refuse customers, risk-averting insurers would raise premiums, leaving the poor and the middle class largely uninsured, while making large profits (which can be masked as "administrative costs" including high salary, perks and benefits for the management) and providing just a minimal mitigation of damage: clients were carefully chosen not to be risky and most people-at-risk remain uninsured. We define this as "reversed adverse selection": it's the insurer that select only low-risk customers.

By contrast, the standard "adverse selection" is about risky customers flocking to insurers that cannot differentiate premiums, to the effect that the portfolio of insurers is disproportionally rich in high-risk customers (and any increase in premiums would only worsen the situation).

Without further changes, introducing a mandate when there is "reversed adverse selection" to get insurance would ruin many uninsured, because they cannot afford the premiums (although the mandate at least improves the damage mitigation).

As extreme case, a customer might not fulfil its mandate if refused by all insurers (possibly because it has a relatively high risk or because of classification and perception biases and prejudices) or because it cannot afford the proposals by the only (or the few) insurers that are not refusing to sell.

In terms of market organization, price differentiation across firms (each proposing the same price to every potential customer) can be sustained if the lowest-price insurers do reject potential customers, leaving them the only choice to accept higher prices by more accomodating insurers. Both categories of firms enjoy the situation (the former because they have a lower-risk portfolio, the second because of the higher prices they can charge).

Premium differentiation

In principle, the participating to the scheme without the institution could agree any division of the ex-post pooled damage. Similarly, income-adjusted premiums can be legally established.

When an insuring institution comes in, it could offer personalized premium proposals; the interaction between the two formula we introduced earlier on will produce a range of acceptable deals. If you want a simple model of negotiation, one might propose a random variable within that range. However, the institution could be smart enough to try to compute the reserve price of the prospective customer and ask the largest acceptable deal. Conversely, a pro-active client could ask for several estimated offers by different insurers and choose the cheapest. All these activities might likely produce a reduction in inter-personal mitigating effect of the scheme (as the most risky people will be paying the most).

In actual institutions, premium differentiation is obtained by applying company-wide underwriting guidelines, implementing the view of the institution, in terms of knowledge base, historical data, categories emerging as relevant (e.g. in GLM - Generalized Linear Models), down to automatic routines that generate the special price proposed to any applicant or instead with a person or group of people taking the responsibility to adapting the pricing to the case at hand. Industry-wide often exist official documents expressing which kind of variables should be taken into consideration.

Moral hazard in customer behaviour and in the supply chain

Until now, we considered cases where the people-at-risk cannot influence the probability and the amount of losses and claims. However, in certain cases, a number of precautions and routines could reduce both variables. If these activities are costly, insured people might, according to some, renounce to them, to the effect that the provision of insurance increases expected losses. Uninsured people would be more cautious

Conversely, somebody would argue that it's precisely because it allows people to undertake risky activities (which produce positive results) that insurances are so advantageous to society. For instance, medical insurance in trips abroad allows people to travel in places where they would not go if not covered.

At the same time, there is another - more systemic - process of moral hazard which we would like to underline: the interconnected changes in pricing and routines along the supply chain due to the presence of insurance. When the suppliers of goods and services needed to repair the damage know that the customer is covered by insurance, they can significantly raise the prices without resistance from the customer. This routinely happens with car repair after accidents covered by insurers, with soaring prices of spare parts and machine shop services. These sections of the supply chain profit from accidents covered by insurance. This is the result of a process over time moving the average claim up (and premiums with it).

Partial coverage, co-payments and life-time limits

In certain insurance contracts, the victims of damage still have to cover part of the losses, either as a fixed percentage or until a threshold of minimum losses is achieved. This significantly reduces the expenditure of the insurer, which can price these contracts at a rabate rate. In principle, co-payments reduce moral hazard, as the victim has no interest at all in suffering the loss.

A contract can also cap the total amount covered, leaving the upper part back to the insured, which can turn out to be quite "un-insured".

Re-insurance

The inclusion of re-insurance reduces the fluctuations in income flows of insurers, providing a stop-losses mechanism, for which losses beyond a certain threshold are transferred to other (very large and with a lot of assets) businesses.

Reinsurance gives signals to all insurers operating on the market about some general features of it (in particular probabilities and severity of actual events and their trends, which might be unobserved or observed with imperfection by the single insurer) [7].

Policies for the case of reversed adverse selection

Which changes should be introduced in order the insurance industry to provide affordable coverage and significant damage mitigation when insurer companies charge "too much" and exclude from coverage the very people that would need it?

1. Stronger competition to avoid oligopolistic (implicit or explicit) collusion in premium prices and refusal to sell. If this turns out to be impossible when insurers are too similar in institutional settings and goals, "institution building" could generate new organization types with divergent business models, practices, and risk-aversion (possibly including "public options" by non-profit and government-backed institutions).

2. As protection for insurers, an extensive use of re-insurance instead of cumulated profits and large assets. Instead of having huge companies, which necessarily are just few and with a strong incentive to collude, it would be good to have a wider number of companies, operating across non-correlated or even negatively correlated geographical markets and all having significant re-insurance.

3. A regulatory authority checking actual damage events and making other tools to establish risk-aversion of insurers, including excessive administrative costs, attentive to consumer denounciations and charges, with a legal capability to cap premiums. The authority might also suspend the mandate for people that can prove they cannot afford the premiums ("hardship clause") and give them an alternative.

Policies for the active involvement of insurance institutions in public protection

In principle, insurance company should have a material interest in reducing actual occurrences of negative shocks, for any given level of premium (making money out of non-occurrence). This might, under certain institutional settings, lead them to make extensive data collection, careful evaluation of risks, accurate research to establish key determinants of occurrence, to devise and explore various options on mitigating activities, and even direct funding for their implementation. All this just out of their self interest.

Such companies would be great partners to public authorities and other industries and stakeholders in increasing the protection of the public. If properly working, these partnerships might inherit certain commitments and goals of the Welfare state, with a potential increase of efficiency and effectiveness due to superior knowledge and capability of taking action.

Within the system there would be a new funding source of activities reducing the likelihood and severity of negative shocks, together with full coverage of risks and decreasing premiums over time.

In practice, however, many real insurance system are at odds with this description, making easy money out of each single customer, with low competition, high premiums, reverse adverse selection, refusing to help those that are at risk, and high lobbying expenditure to influence regulators and law-makers.

A thorough analysis of the conditions in your country is needed to establish the actual dynamics and sources of profits, if any.

Policies for the active involvement of insurance institutions in public protection might begin with "moral suasion" and public announcement of the desirability of such changes in front of the general public, the consumers' associations and the industry. If nothing happens, the law could mandate that a certain percentage of premiums collected has to be directed to data collection, research, policy proposals and their implementation, with transparency being the key throghout the process.

In a more discretionary process, a tender might be made for additional insurance coverage of all events of a certain kind in the country or a sub-national region (e.g. all car accidents or work injury cases). A policy table with all stakeholders, including the winner of the tender, would generate actions to avoid and mitigate the negative shocks, partially funded by the winner, leading a lower-than-expected rate of occurrences. Any reduction in premiums should be smaller than the fall in the claim rate, increasing both the short-term and the long-term profitability of the winner (which can however be substituted with a new tender if cooperation proves ineffective).

Technical appendix on risk correlation and path-dependency

The Excel spreadsheet emboding the formal model used in this paper introduces a new two-stage procedure to generate and manage non-independent stochastic events. This computational procedure opens to modellers a feasible and flexible tool for new models.

In the first stage, prodromal symptoms are generated, according to a certain probability for each event, independently from each other. In the second stage, actual occurrence is obtained by taking into account the presence or absence of prodromal symtoms of all events.

More formally, in the first stage a random variable is generated and compared to a threshold reflecting the independent probability. In the second stage, the threshold is increased or lowered depending on the precursors of all other events and the sensibility of the threshold to their occurrence.

The change in the threshold, compared with the random variable generated in the first stage, confirms or reverses the direction to which the premonitory symptoms were pointing.

For instance, if in the first stage the precursors to the onset of the occurrence of event A do appear and in the second stage the computation of the effects on the threshold by all other potential events (B, C, D,...) leads to an increase of the threshold (thus reinforcing the event), event A does take place.

As you shall see in the spreadsheet, the ingredients of the model are the following

1. a list of N possible events (called "negative shocks" to "area points" - e.g. a city, a coastal area, etc.);

2. a probability for prodromic symptoms for all N events;

3. a NxN matrix of "correlation" in which every prodromic symptom exerts a positive, negative or nihil influence on every actual occurrence, depending on the value of the cell;

4. the random variable (uniformely distributed between 0 and 1), extracted each time the model is run.

On the diagonal of the NxN matrix (sub. 3), we include the effect of previous period occurrence of the same event (a sort of auto-correlation over time). In this way, the whole system of probabilities can become non-stationary and path-dependent. The occurrence at time t of a certain number of events influences the probability of occurrence at time t+1 and the following.

The effects of symptoms in the matrix (sub. 3) is symmetric: in absence of the symptom, the influence has the opposite sign as in the presence of the symptom.

In the "Basic area data" you find the list of events (sub. 1) in row 2, the probabilities of prodromic symptoms (sub. 2) in row 3. The matrix (sub. 3) is in the sheet "Matrix of influences". You can change any of those values (including the number active area points) and run the model to see what occurs each run.

During the run, the sheet "Prodromic symptoms" will be updated, with row 4 showing what is going to occur in period 1 and the sheet "Actual occurrence" taking into account the changes in thresholds due to the Matrix of influences.

The random variables (sub. 4) are recorded in sheet "Triggering value". If the value is lower than the sub. 2 threshold, then prodromic symptom will occur (value: 1). If total influences are positive, the symptoms will be confirmed in the "Actual occurrence" sheet. If they are negative, depending on how deeply and which was the triggering value, the symptom are confirmed or not. For the formal code, see later on in the Appendix.

Commenting on this two-stages procedure, we emphasise that it is extremely flexible and powerful in generating many different relations across variables. It does not cycle back and forth between the two stages, in search for a "fixed point", because the symptoms and the events take place in real time, not in a logical time: it takes time for the symptoms to reveal and for the events to coalesce. For events like "car crashes", symptoms are like "high speed" and "low-skilled driver": there is no loop between the two.

Back to nitty-gritty details of the procedure: to read the formal computer code (in Visual Basic for Application language), just look in the routine labelled CommandButton1_Click(), i.e. activated by a click on the button.

If you don't have access to the code, we report it in what follows, under the convention that

"Basic area data" sheet is called in the code as "Foglio2"

"Triggering values" = Foglio6

"Prodromic symptoms" = Foglio1

"Actual occurrence" Foglio5.

For t = 1 To nperiods
For i = 1 To nareas
triggering = Rnd 'uniformely distributed random extraction of a value between 0 and 1
If triggering < Foglio2.Cells.Value(3, i + 1) Then
occurrence = 1
Else
occurrence = 0
End If
Foglio6.Cells(t + 3, i + 1) = triggering
Foglio1.Cells(t + 3, i + 1) = occurrence

Next i

For i = 1 To nareas 'influenced event
additionalthreshold = Foglio2.Cells.Value(3, i + 1)
For j = 1 To nareas 'influencing event

If i <> j Then
If Foglio1.Cells(t + 3, j + 1) = 1 Then
additionalthreshold = additionalthreshold + Foglio3.Cells.Value(i + 2, j + 2)
Else
additionalthreshold = additionalthreshold - Foglio3.Cells.Value(i + 2, j + 2)
End If
Else
If t > 1 Then
If Foglio1.Cells(t + 2, j + 1) = 1 Then
additionalthreshold = additionalthreshold + Foglio3.Cells.Value(i + 2, j + 2)
Else
additionalthreshold = additionalthreshold - Foglio3.Cells.Value(i + 2, j + 2)
End If
Else
End If
End If

Next j

If Foglio6.Cells(t + 3, i + 1) < additionalthreshold Then
occurrence = 1
Else
occurrence = 0
End If

Foglio5.Cells(t + 3, i + 1) = occurrence
Next i

Next t

NOTES

[1] In the software, people are called "area points" because it was devised for discussing the case of climate change damage and losses, which is gaining prominence in international negotiations. Furthermore, the expression "mutual insurance" is used here to represent a group of people without any institution except their contracts, whereas in US this expression refers to an institution owned by insurance policyholders.

[2] You can modify the software to have a range of amounts with different probabilities. For the moment, the amount of damage is just one.

[3] Since payments are ex-post, the calculation is about the average payment per period, but it would have been impossible to know in advance its amount. This is a weakness for the self-insurance (by saving) providing a further argument for not renouncing to insuring with a pool.

[4] The advantage is computed as the ratio between the dis-utility without insurance and the dis-utility with insurance, dis-utility being a quadratic function of damage. The quadratic form expresses the idea that a big loss is much more negative than a small loss (more than proportionally). You can experiment with other (dis)utility functions (e.g. cubic, etc.). What's important is the ranking of dis-utility values, as utility is in this case an ordinal and not a cardinal unit.

[5] To explore this point, look at income and wealth rows in the "Basic area data" sheet and the percentages of premiums on them.

[6] For our model of a manufacturer using inventories in such a way, see here.

[7] You can modify the file to include a re-insurance sheet and explore how one re-insurer or more re-insurers in competition would change the profitability of insurers, their pricing policies and the coverage of people-at-risk.