What was fair in actuarial fairness?

Abstract

In actuarial parlance, the price of an insurance policy is considered fair if customers bearing the same risk are charged the same price. The estimate of this fair amount hinges on the expected value obtained by weighting the different claims by their probability. We argue that, historically, this concept of actuarial fairness originates in an Aristotelian principle of justice in exchange (equality in risk). We will examine how this principle was formalized in the 16th century and shaped in life insurance during the following two hundred years, in two different interpretations. The Domatian account of actuarial fairness relied on subjective uncertainty: An agreement on risk was fair if both parties were equally ignorant about the chances of an uncertain event. The objectivist version grounded any agreement on an objective risk estimate drawn from a mortality table. We will show how the objectivist approach collapsed in the market for life annuities during the 18th century, leaving open the question of why we still speak of actuarial fairness as if it were an objective expected value.

Keywords

actuarial fairness annuity history life insurance mathematical expectation

Actuarial fairness revisited

Unlike in other fields of economics or finance, we find a normative concept at the core of the actuarial sciences: We speak of the actuarial fair price of an insurance policy when customers bearing the same risk pay the same amount. There is no separate theory of actuarial fairness, however. For at least three centuries, practitioners and theoreticians alike have taken for granted the principle that equal risks should be charged the same price, using this as a benchmark for internal accountancy. Today, however, the pricing policies of insurance companies are putting this principle to the test. On the one hand, new insurance regulations (such as the EU Solvency II Directive (2009/138/EC)) allow companies to manage their customers’ risks as they see fit, provided they meet strict controls of solvency. On the other hand, these risks can now be analysed with powerful statistical tools from the emerging world of Big Data. Whereas old actuaries classified customers according to broad risk profiles (for which there would be a single fair price), insurance companies are now refining their ability to predict the sum payable at the maturity of each of their policies, and have more freedom to set their prices. Yet rather than disappearing, the concept of actuarial fairness seems to be in transition. In March 2011, for example, the European Court of Justice gave a ruling (C-236/09) prohibiting the use of a person’s gender as a rating factor; the industry deemed this an ‘unfair’ decision (Landes, 2015).

We want to contribute to the ongoing redefinition of this concept of actuarial fairness with a historical outlook on its sources.¹ In actuarial parlance, the expected value of an insurance policy provides its fair premium. For instance, the claim amounts of an insurance policy X can be 0 (with probability 0.81), 50 (with probability 0.18), and 100 (with probability 0.01). Its expected value is 10: E(X) = 0 x 0.81 + 50 x 0.18 + 100 x 0.01 = 10. Why is this expected value the best rendition of the same risk, same price principle? In the first half of this article, we will defend that the identification of actuarial fair prices with the expected value of insurance policies is merely a historical contingency, not a matter of conceptual necessity. As we shall see in the following sections, in the 17th century, early probability theorists used expected values to formalize an Aristotelian principle about the justice of contracts involving random events. We will dub this principle equality in risk: the distribution of costs and benefits in such contracts would be fair only if it was proportional to the risk each of the contracting parties took. Our modern concept of actuarial justice (same risk, same price) was first articulated in the 17th century, when expected values were used to calculate the fair price of a particular insurance contract: life annuities.²

We will present two different interpretations of this formalization of equality in risk that coexisted in the 17th and 18th centuries. On the one hand, we find a legal interpretation of the fairness of an aleatory contract we will call Domatian (after Jean Domat), in which the contracting parties calculate the expected value of the insurance policy on the basis of their agreed estimates of the chances of each outcome. This agreement on risk is fair to the extent that both parties are equally uncertain about the risk they are betting on. On the other hand, we find an objectivist account in which the agreement on an actuarial price was fair if the expected value of the contract was grounded on a mortality table capturing the objective risk of death.

We will then illustrate how both versions of actuarial fairness were implemented to calculate the theoretically fair price of life annuities. For Jan De Witt (1625–72), Domatian fair prices emerged from the agreement of the contracting parties without any separate estimation of the probabilities involved. For Edmund Halley or Nicholas Bernoulli, objectivist fair prices were based on an empirical estimate of the actual chances of death. In our concluding section, we will contribute an analysis of how the proliferation of mortality tables led to the collapse of the objectivist account in the market for life annuities. We will show that, rather than leading to a converging set of fair actuarial prices, the different mortality estimates instead yielded conflicting prices that proto-actuaries rarely trusted for selling their policies.

Our inquiry is therefore a study in the history of the theoretical concept of actuarial fairness. We rely on already well-studied episodes in the development of modern probability and statistics, such as the emergence of expected values and mortality tables. Our original contribution is to make explicit the different conceptions of justice underlying the discussion of fair actuarial prices over the course of two centuries, together with an empirical test of sorts for the objectivist account of fairness: a comparison between actuarial prices drawn from the available mortality tables until the early 19th century. We thus focus on an abstract actuarial concept that reached our present evolving in parallel with a variety of pricing practices in insurance markets, without much influence on the latter. We hope our analysis will help in dispelling the illusion of objectivity that still pervades many contemporary debates on insurance prices.

The justice behind arithmetical means

The first step in our argument is to show what concept of justice was formalized in the expected value version of actuarial justice. Historically, the same risk, same price principle stems from an Aristotelian tradition in which justice in exchange was appraised in terms of means. In the fifth book of the Nichomachean Ethics (EN 1131b25–1132b20), Aristotle addressed the problem of the justice of contracts through a mathematical analogy. Suppose that we have two parties with equal claims on a given good, but they have received unequal shares of it (a, b): the fair division of this good is the arithmetical mean of those unequal shares (a + b) / 2. Although Aristotle did not discuss in detail what would count as equal in actual exchanges, his intuition was enormously influential, and reached almost verbatim his medieval commentators (Fleischacker, 2004). For Aquinas, for instance, a fair exchange was one in which the quantities traded did not deviate from the arithmetical mean of the total amount exchanged:

If, at the start, both persons have 5, and one of them receives 1 out of the other’s belongings, the one that is the receiver, will have 6, and the other will be left with 4: and so there will be justice if both be brought back to the mean, 1 being taken from him that has 6, and given to him that has 4, for then both will have 5 which is the mean. (Aquinas, 1947[1225-1274]).

Within this Aristotelian background, the Schoolmen extensively discussed the fairness of the so-called aleatory contracts, in which the benefits and losses depended on an uncertain event.³ Here emerged the principle of equality in risk, of which we will present a particular version by Domingo de Soto (1494–1560).⁴ De Soto was a Dominican theologian who systematized centuries of legal controversies among the Schoolmen in his monumental De Iustitia et Iure (De Soto, 1967[1556]). In the sixth and seventh questions of the sixth book in this series, de Soto discussed the fair distribution of benefits and losses in partnership formed through an aleatory contract. His major claim was that an Aristotelian division (an arithmetical mean) would be fair only if the partners were taking equal risk in their contribution (be it capital or labour). If the risk each undertook were different, they should divide the total amount proportionally to those risks.⁵ This is what we call the equality in risk principle, which incorporates uncertainty into commutative justice.

Equality in risk allowed de Soto to distinguish between insurance contracts and loans. In the latter, the owner of the money did not bear any risk in lending it: the recipient was obliged to return it, independently of the success of his venture, plus interest – hence the shadow of usury and, therefore, unfairness. In an insurance contract, both parties bore risk instead. The insured party would lose the insurance fee if no adversity occurred; the insuring party would cover the insured capital if it did. The premium was therefore compensation for covering this risk.

In the 17th century, equality in risk was formalized in early probability theory, when Pascal and Huygens articulated the concept of mathematical expectation in order to analyse distribution problems in a particular kind of aleatory contract: gambles (Daston, 1988: 49–110; Franklin, 2001: 306–16; Teira, 2006). In his Treatise on the Arithmetical Triangle (1665), Pascal addressed the so-called Problem of Points: how to distribute the bets in an interrupted gamble. According to Pascal, the distribution ‘should be strictly proportional to what they [the players] might rightfully expect from chance’ (Pascal, 1963: 57). How could anyone quantify this fair expectation? In De ratiociniis in ludo aleae (1657), Huygens provided an algorithm. To illustrate, let us assume a gamble in which two players may either earn a if they win or b if they lose. If I may expect either a or b, and am equally likely to receive either, then my expectation might be said to be worth (a + b) / 2 (Bernoulli, 2006: 133). Here we have an arithmetical mean, according to the Aristotelian principle of commutative justice. But the risk according to which values are assigned was now implicitly quantified. Today, we would read Huygens’ formula as a mathematical expectation: a probability-weighted average, in which both outcomes (a, b) are equiprobable.

This formalized equality in risk solved the Problem of Points: if the game was interrupted, each gambler should receive an amount equal to the expectation of the game. And this would be the fair price to pay for betting in such a gamble: for those who took the same risk, the price should be the same.

Measuring risk through contracts, in theory and practice

The expected value version of actuarial fairness thus originates in the Aristotelian equality in risk principle. Had Aristotle chosen a different analogy, we might never have appraised the fairness of a price in arithmetical terms. Indeed, historians have shown that Huygens did not quantify risk directly through mathematical probabilities. Following a standard procedure in commercial mathematics (Sylla, 2003), Huygens studied gambling contracts and sought equivalences between them. To the extent that various contracts entailed the same risk, they would have the same expected value. An implicit quantification of probabilities arose therein.

Here is the argument Huygens used to articulate the concept of mathematical expectation. Imagine a simple game α (a series of coin tosses, for example) with equal chances for the players to get outcomes a or b (where a < b). The two players start to play α, but they are interrupted before the series of coin tosses reaches its end. Two new players want to replace them, each one of them paying an amount x to gamble. The Problem of Points is about finding what x should amount to, given a and b, at the stage at which the game is interrupted. The two new players agree that the winner of this second gamble β will earn 2x, while the loser will still get a. Following the usual procedure in commercial arithmetic, these two gambles (α and β) will be equivalent if the winner in either of them gets the same prize, b. Hence, 2x − a should be equal to b. The amount x that the two new players of gamble β should pay to replace the original players of gamble α is equal to (a + b) / 2. This is the expected value of gamble β, and is the fair price of the (interrupted) gamble α. The one-half weight in the formula arises from 2 being the number of players betting in the gamble, not from a separate quantification of probability.⁶

Huygens thus provided the theoretical foundations of our current concept of actuarial fairness, formalizing the equality in risk principle. This is what we will call, from now on, the objectivist version of actuarial fair prices, whereby the calculation of the premium hinges on the quantification of the different variables involved in the algorithm, assuming this quantification is unique.

Yet almost nobody could estimate the actual equality in risk in an insurance contract at a time in which there was no separate quantification of probabilities. This was true at the time of de Soto, for whom there was no universal valuation of risk, and who advised that the contracting parties should reach an agreement on their own (De Soto, 1967[1556]: 580).⁷ This was still the case in the 17th and 18th centuries. Our best guess as to how the fairness of an insurance contract was actually established is again legal theory. Among Pascal’s closest friends, we find the jurist Jean Domat (1625–96), author of a systematic treatise on The Civil Law in Its Natural Order (1850[1689]), which is usually considered the first attempt at a rational systematization of French law.⁸

At various points in this book, Domat discussed the role of uncertainty in the fairness of an agreement. Consider, for instance, those covenants concerning an uncertain event, in which one of the contracting parties may, for example, renounce all profit, and free himself from all loss. Domat claimed that their justice was founded upon this:

One party prefers a certainty, whether of profit or loss, to an uncertain expectation of events; and the other party, on the contrary, finds it his advantage to hope for a better condition. Thus, there is made up between them a sort of equality in their bargains, which renders their agreement just. (Domat, 1850[1689], Vol. 1: 186)

If the contracting parties had complementary expectations about the outcome, the agreement was fair. Their expectation depended, of course, on their subjective estimation of their chances of suffering from a certain adversity. For Domat, this subjective estimate provided good enough grounds for a fair agreement inasmuch as the contracting parties were equally uncertain about the outcome. Imagine, for example, a ‘universal partnership’ in which all the partners contribute a given amount of money so that if any of them has a daughter, they will be able to fund the dowry from the ‘joint stock’ (ibid., Vol. 1: 354–5). According to Domat, this arrangement is fair because all the partners were ‘under the same uncertainty of the event [having a daughter], and with the same right, having rendered their condition equal, it made also their agreement just’.⁹

Here is a second version of actuarial fairness in which the equality of risk is assessed in terms of equal ignorance: no party can exploit the ignorance of the others for his own benefit. Let us call this the Domatian interpretation of equality in risk. Unlike the objectivist version, it does not presuppose an independent procedure for risk quantification (arising from the symmetries in the contract, as in Huygens, or whatever other source). In the Domatian approach, the contracting parties may use any risk figures they are willing to agree on, and the agreement will be fair (as will the subsequent price), provided none of them is hiding information that should be reflected in such figures. As we will see in the next two sections, both accounts in the interpretation of actuarial fairness coexisted in the transition to the 18th century.

Actuarial fairness, the Domatian version

Let us now see the Domatian version of actuarial fairness at work in Jan de Witt’s piece on the ‘Value of Life Annuities in Proportion to Redeemable Annuities’ (De Witt, 1995[1671]). De Witt presided over the United Provinces of Holland, and the paper was written as a report addressed to the States General of Holland. The country was considering the issuance of life annuities at a fixed price (that is, regardless of age), which De Witt wanted to compare to the price of perpetual annuities. In exchange for this price, a life annuity would provide a series of equal payments for the remainder of the buyer’s lifetime. Throughout the centuries, different institutions had been financing themselves with the sale of annuities, but De Witt is considered the first author to have computed the value of a life annuity as the sum of expected discounted future payments (Daston, 1988: 27–8; Hald, 1990: 123–31; Turnbull, 2016: 11–13).

In this calculation, De Witt operated within the normative framework presented above, exploiting the analogy between life annuities and gambling contracts. On the one hand, the prize of the gamble corresponded to the income the annuity buyer might obtain depending on the duration of his life. On the other hand, the chance of each outcome in the gamble corresponded to the chance of the buyer dying at any particular point in time. Hence, using Huygens’ approach, De Witt was able to calculate the expected value (a_x ) of a life annuity for a person of age x as follows. In our current notation (Hald, 1990: 128), the formula for his algorithm would be:

a_{x} = \frac{1}{l_{x}} \sum_{t = 1}^{w - x - 1} a_{t} d_{x + t}

The outcomes of the gamble are $a_{t}$ , where $a_{t}$ is the current value of an annuity paid every six months (at a 4% annual interest rate) over t half-years. The number of deaths in each period x + t is d _x+t. The sum of the number of deaths from age x (in half-years) onwards is l_x = d_x + d_x+1 +…+ d_w-1 , where w is the maximum number of half-years in which the annuity will be paid. As Hald (1990: 128) observes, d_x+t / l_x is a probability distribution. But this is not how De Witt estimated the chances of dying. He first divided a person’s life into four intervals: (3–53), (53–63), (63–73), and (73–80).¹⁰ The chance of anyone dying in any of these four intervals was estimated as follows:

Taking for example two persons of equal constitution, one aged 40 years, and the other 58 years, if these two persons made such a contract, that in case the person of 58 years should happen to die in less than 6 months, the one aged 40 were to inherit a sum of 2000 florins from the property of the defunct; but that if, on the other hand, the person aged 40 years should die in less than 6 months, the other aged 58 years were to have 3000 florins from the property of the deceased; such a contract cannot be considered disadvantageous for the person who would have the 3000 florins, if the event were favourable to him, and who, in the contrary event, would only lose 2000 florins. (De Witt, 1995[1671]: 2)

The chance of anyone dying in these two intervals was inferred from the fairness of the contract: for De Witt, the proportion between the chance of dying of a person whose age was in the range (53–63) and the chance of one in the (3–53) range was 3 to 2, because if the older person died, the amount to be paid would be only two-thirds of the amount due if the younger one died. This was the Domatian, rather than the objectivist, version of actuarial fairness: there was no independent quantification of risk, and the contracting parties implicitly agreed on their respective chances if they did not consider the payout ‘disadvantageous’.

In buying a life annuity at a price calculated according to De Witt’s formula, the contracting parties implicitly agreed on a given proportionality of chances for each age range: taking (3–53) as the baseline, the proportion would be 2/3 for (53–63), 1/2 for (63–73) and 1/3 for (73–80). De Witt’s formula would then comply with the equality in risk principle if two buyers falling within a given age interval considered themselves to have the same chance of dying, and the same proportion relative to the buyers in the other intervals. In accepting this distribution of chances, they acknowledged that they were all ‘under the same uncertainty of the event’; otherwise, they would have been exploiting someone else’s ignorance.¹¹

Actuarial fairness, the objectivist version

Let us now see how the objectivist account of actuarial fairness emerged at the turn of the 18th century thanks to mortality tables. These would become an independent source for the quantification of the risk of death, setting apart the two accounts we will examine here (those of Halley and Bernoulli) from De Witt’s Domatian approach.

In 1693, Edmund Halley published in the Philosophical Transactions of the Royal Society his ‘Estimate of the Degrees of Mortality of Mankind’, based on the records of the city of Breslaw (today’s Wrocław) (Daston, 1988: 125–38; Hald, 1990: 131–41; Turnbull, 2016: 13–16). For Halley, ‘the price of insurance upon lives ought to be regulated’ (Halley, 1693: 602) based on these mortality tables, as they provided an empirical estimate of the chance of people dying at a certain age. Whereas De Witt had estimated these chances on the basis of a contract, Halley now argued based on death frequencies, as follows. If we want to insure the lives of two men aged 20 and 50: ‘It being 100 to 1 that a Man of 20 dies not in a year, and but 38 to 1 for a Man of 50 Years of Age’ (ibid.). For Halley, ‘it is plain that the Purchaser ought to pay for only such a part of the value of the Annuity, as he has Chances that he is living’ (ibid.). In other words, what was plain to Halley was the equality in risk principle: same risk, same price. The only difference was that now the risk was inferred from statistical records. Nonetheless, adopting Breslaw’s mortality table as the source of every risk estimate was objectionable, as Halley himself acknowledged: ‘it may be objected, that the different Salubrity of places does hinder this proposal from being universal; nor can it be denied’ (ibid.: 619). For Halley, this was an empirical matter, and a matter for further investigation.

Nicolas Bernoulli concurred on this point with Halley. In 1711, he published a summary of his doctoral dissertation on the use of the ars conjectandi in law (Bernoulli, 1992[1711]; Daston, 1988: 136–7; Hald, 1990: 110–15). A couple of years later, Bernoulli published his uncle’s unfinished Ars Conjectandi, a landmark work establishing the foundations of modern probability theory. The framework of Bernoulli’s dissertation is still normative, however. In Chapter 4, Bernoulli discussed the legal foundations of the pricing of life annuities. In a Domatian spirit, he claimed that the only foundation for these prices was the reason of the contracting parties (ratione contrahentium), as the price should be fixed at the time of their agreement (and not when the outcome to which the contract relates happens). Yet Bernoulli shifted to the objectivist account of fairness:

It is clear that the price cannot be established without taking the buyer’s age and health into consideration, of which we should have the best knowledge in order to set the price of a life annuity. The same annuity cannot be sold indifferently to men of all ages. (Bernoulli, 1992[1711]: 62; our translation)

Drawing on a summary of John Graunt’s 1662 mortality table, Bernoulli proceeded to estimate the length of a human life. However, he observed that some additional data from an unidentified Swiss city disagreed with his estimates. These should therefore remain hypothetical, he concluded, inviting, like Halley, further research on the topic.

Thus, unlike in De Witt’s Domatian approach, the mortality tables set an objective assessment of risk upon which the fair price of an annuity could be calculated. Indeed, historians of probability have discussed Halley’s and Bernoulli’s contributions mostly in terms of their assumptions about the regularity of death: in particular, whether there was an underlying stability in mortality statistics that they were able to grasp with their data (Daston, 1988: 125–38; Hacking, 1975: 119–22). This was a powerful ideal that, as we will see next, inspired the construction of many other mortality tables during the following centuries.¹²

Yet historians of probability have so far neglected a second problem that any objectivist account of actuarial fairness was bound to encounter. This is what philosophers of probability today call the reference class problem (Hájek, 2007): Assigning a probability to the risk of death of a particular individual depends on the relevant risk factors used to classify that individual. The probability will change depending on whether the actuary takes into account tables that consider only ‘age and health’, for example, or also take into account ‘the different Salubrity of places’, or whatever other factors impinge on mortality. If the actuary could grasp in full the laws of mortality covering all these factors, they might be able to construct a single mortality table, and the calculation of actuarial fair prices would be as objective as it could possibly be. Yet, as we are going to see in the next section, actuaries have access only to partial mortality tables, often with diverging probability estimates. Under these circumstances, could they estimate anything like an actuarial fair price on an objective basis?

Actuarial justice in the actual markets

The best way to grasp the differences between the objectivist and the Domatian versions of actuarial justice is to see them in conflict. Eve Rosenhaft provides a splendid illustration in her analysis of the collapse of a German widows’ pensions fund in the late 18th century (Rosenhaft, 2010). Established in Hannover in 1767, the Calenbergische Witwenversorgungs-Gesellschaft (the Calenberg, hereafter) recruited in about a decade more than 5000 married couples from all over Europe. The Calenberg had been designed using the Süssmilch 1741 mortality table (Süssmilch, 1761) and contributions took age into account. It was therefore one of the first actuarially based funds, comparable to the British Equitable Society (Ogborn, 1962). Yet, early after its establishment, qualified experts like George Christian von Oeder and Johann Nicolaus Tetens criticized the actuarial foundations of the Calenberg, questioning its long-term solvency. In the early 1780s, a large group of subscribers refused to pay their contributions and recruited, among others, Tetens to advise them in their negotiation with the Calenberg administrators, who recruited a committee of lawyers from the University of Leipzig to adjudicate the case.

The protesters argued that there was a ‘fundamental error’ in the design of the fund (Rosenhaft, 2010: 29), because, in Tetens’ words, the ‘general calculation of the business’ had been made not by ‘algebrists’, but ‘mere adders-up’ (ibid.: 32). Therefore, the subscribers had been ‘cheated’. The widows already receiving pensions from the Calenberg replied that the fund was based on an aleatory contract, in which the subscribers had ‘knowingly [entered into] an insecure transaction in which he could win and [i.e. or] lose’ (ibid.: 34). The Leipzig jurist adjudicated that the Calenberg indeed depended ‘on the length of human life, and so on a completely uncertain outcome’ (ibid.).

In our terms, the protesters were invoking the objectivist version of actuarial justice. Given a mortality table, they argued, there was only one fair amount to set for the fund premium. The Leipzig jurists countered along Domatian lines: Ultimately, the fairness of the contract depended on the shared uncertainty about the husbands’ deaths, and mortality tables did not make it less uncertain for any of the fund stakeholders.¹³ The purported objectivity of actuarial prices might have been more credible in a world with a single mortality table, that is, a single life expectancy estimate. As Struyck declared in 1740, ‘I think if unbiased people were taking data on annuities from other accounts in other countries, considering all the people who bought insurance around the same time, dividing them into classes and noting the number of years during which they drew their pensions, the same way I did above, they would arrive at a nearly identical result’ (Struyck, 1912; our translation).

Yet, as Table 1 below shows, between 1662 and 1769, a growing number of mortality tables for different European populations were published, exhibiting striking differences in their mortality estimates (Figure 1), not only due to the increasing longevity of Europeans.

Table 1.

Mortality tables in Europe (1662–1830).

Author	Date	Data type	Hypothetical	Fact-based	Actuarial notations	Notes \| Sources
Ulpianus	ca. 225	Legal value of (life) annuity in years’ purchase	—	No	a_n	Can be interpreted as a_n
John Graunt	1662	Survival table for 10-year intervals	Yes	Allegedly	l ₇, l ₁₇, l ₂₇,…l ₇₇	(Birch, 1759)
John Graunt	1662	Survival table for 10-year intervals	Yes	Allegedly	l ₇, l ₁₇, l ₂₇,…l ₇₇	Disputed authorship: Petty (Le Bras, 1998)
Hôtel-Dieu, Paris	1668	Mean life, median life, life annuity price table	—	No	a_n	(Pradier, 2016)
Hôtel-Dieu, Paris	1668	Mean life, median life, life annuity price table	—	No	a_n	Life annuity price table
C. & L. Huygens	1669	Life annuity price table			e_n , a_n	(Rohrbasser & Véron, 1999)
C. & L. Huygens	1669	Life annuity price table			e_n , a_n	Data from Graunt’s 1662 table (Birch, 1759)
Jacob van Dael	1670	Unclear: survival table or income stream	Yes	No	Doubtful	(Hald, 1990: 121)
Johan de Witt	1671	Mortality rates, survival table	Yes	No	$a_{\bar{x} 4 %}$ $\frac{q_{x + n}}{q_{x}}$ a_x for some x, n.	(De Witt, 1995[1671])
Johan Hudde	1672	Compilation of mortality data, life annuity price table	Likely	Yes	a_n	(Van Ham, 2005)
Johan Hudde	1672	Compilation of mortality data, life annuity price table	Likely	Yes	a_n	Data from annuitants’ lives, with new annuitants aged between 1 and 50 years
Edmund Halley	1693	Survival table	Likely	Yes	l_x , v^x , a_x	(Bellhouse, 2011; Halley, 1693)
Abraham de Moivre	1725	Analytical simplification	Yes	No	$l_{x + h} = l_{x}$ $- \frac{h}{n} (l_{x} - l_{x + n})$	(De Moivre, 1725)
Abraham de Moivre	1725	Analytical simplification	Yes	No	$l_{x + h} = l_{x}$ $- \frac{h}{n} (l_{x} - l_{x + n})$	‘De Moivre’s law’ = linear survival function to fill missing values in Graunt’s 1662 table (Birch, 1759)
Isaac de Graaf	1729	Analytical simplification	Yes	No	$p_{x} = 1$ $- {(\frac{x}{92})}^{n}$ $w i t h n = 5$	(Struyck, 1912: 203)
John Smart	1738	Survival table	Yes	Unclear	l_x , d_x	(Smart, 1738)
Nicolaas Struyck	1740	Survival tables	Yes	Yes	l_x , a_x	(Struyck, 1912) Data from annuitants’ lives; separate tables for male and female
Kersseboom	1738 1742	Survival table		Yes	l_x , a_x	(Kersseboom, 1742)
Kersseboom	1738 1742	Survival table		Yes	l_x , a_x	Data from annuitants’ lives
Johann Süssmilch	1740	Survival table		Yes	l_x , a_x	(Süssmilch, 1761)
	1761					Compilation of data from previous works

Thomas Simpson	1742	Survival table	Yes	Yes	l_x , v^x , a_x	Computed from Smart (1738), correcting for migration using Halley (1693).
Antoine Deparcieux	1746	Survival table	Yes	Yes	l_x , a_x	(Deparcieux, 2003[1746])
Antoine Deparcieux	1746	Survival table	Yes	Yes	l_x , a_x	Data from annuitants’ lives to control adverse selection; de-clustering of 5-year clusters
Georges-Louis Leclerc de Buffon	1749	Median life		Yes	N/A	(Buffon, 1749)
James Dodson	1756	Survival table		Yes	l_x , d_x , a_x	(Dodson, 1756)¹⁴
Corbyn Morris	1759	Survival table		Yes	l_x	(Morris, 1759)
Daniel Bernoulli	1765	Hypothetical survival table	Yes	Yes	l_x , e_x	(Bernoulli, 1982[1765])
Daniel Bernoulli	1765	Hypothetical survival table	Yes	Yes	l_x , e_x	Table deduced from Halley (1693), with assumptions about mortality from smallpox
Pehr Wargentin	1765	Survival table		Yes	l_x , d_x	(Wargentin, 1995)
Pehr Wargentin	1765	Survival table		Yes	l_x , d_x	From census data
Richard Price	1769	Survival table		Yes	l_x , d_x	(Price, 1773)
Richard Price	1769	Survival table		Yes	l_x , d_x	Usually known as ‘Northampton table’
Louis Messance	1766	Survival table		Yes	l_x for 0, 5, 10, 15…	(Messance, 1766)
Jean-Henri Lambert	1772	Hypothetical survival table	Yes	Yes	l_x , d_x , e_x	(Lambert, 1772)
Jean-Henri Lambert	1772	Hypothetical survival table	Yes	Yes	l_x , d_x , e_x	Table deduced from Süssmilch (1761), with assumptions about mortality from smallpox
Jean-Baptiste Moheau	1778	Mean life			l_x , d_x , e_x	(Moheau, 1778)
Johann Augustin Kritter	1781	Survival tables		Yes	l_x , d_x , e_x	(Kritter, 1780)
Adrien Duvillard de Durand	1806	Survival table	Yes	Yes		(Duvillard, 1806)
Adrien Duvillard de Durand	1806	Survival table	Yes	Yes		Compilation of data from Moheau (1778) and from Geneva; adjustment for non-stationarity
Joshua Milne	1815	Survival table		Yes	l_x , d_x , e_x	(Milne, 1815)
Joshua Milne	1815	Survival table		Yes	l_x , d_x , e_x	Usually known as ‘Carlisle table’
James Finlaison	1829	Survival and life annuity price tables		Yes	l_x , q_x , a_x	(Finlaison, 1829)

Figure 1.

Survival curves (l_x ) from 1662 to 1781.

Although Halley’s approach set the standard for most of the subsequent tables, the methodology remained in constant development for two centuries. In the 1720s, de Moivre and De Graaf provided some analytical approximations for Graunt’s and Halley’s tables. Simpson (1742) addressed the (so far implicit) problem of assuming a constant population in constructing a life table from mortality data. Subsequent tables developed over the following century used larger samples and more sophisticated methodologies (Murray, 2016).

However, the discrepancies between all these mortality tables were not due only to sampling and methodology. The tables were also constructed with different goals, exhibiting a clear awareness that not all of them were fit for insurance valuation. For instance, Graunt, Buffon, and Moheau were addressing issues in demography: In Graunt’s own words, ‘it may now be asked, to what purpose tends all this laborious bustling and groping to know, 1. The number of people? 2. The number of male and female? 3. How many married and single?’ (Birch, 1759: 35). Other tables focused instead on proto-epidemiological questions, namely the increase in life expectancy resulting from inoculation of infants (Bernoulli, 1982[1765]; Lambert, 1772). Others were constructed instead for actuarial purposes, as was the case with De Witt’s.

According to Hup (2011), between 1662 and 1713 in the Low Countries, only 20% of the life annuities actually sold provided insurance for adults against future poverty; the remaining 80% were placed on healthy children with a view to maximizing the expected return of the annuity.¹⁵ With this gamble in mind, De Witt had argued in 1671 that an 8% price for life annuities was too generous, suggesting instead an amount closer to 6.67% (16 years’ purchase), computed to match the highest life expectancy among the annuitants. This was reflected on his mortality table: If we take Halley’s table as a purely descriptive benchmark, we see that De Witt’s underestimation of the mortality of the first age rank (3–18) raised the value of annuities placed on young lives only.

The consequences for the objectivist account of actuarial fair prices are straightforward. As Table 2 shows, in our array of mortality tables, life expectancies at age 6 ranged from 19 to 48 years; at 20, from 19 to 40; and at 50, from 10 to 20. Table 3 displays the corresponding price of life annuities: Although the variance is less, we still find an 80% difference between the lower and higher prices at 6, 60% at 20, 50% at 50, and so on. We concur with Clark (1999: 114–54) about the lack of solid reasons for proto-actuaries to use mortality tables. They were reasonably sceptical about the accuracy of the data for capturing the risk factors of their actual pool of customers and, trusting their own experience, they often succeeded at organizing viable financial schemes. A powerful rationale behind their scepticism was the reference class problem: When there was more than one probability estimate for the life expectancy of a given individual, which one would be the fair choice?

Table 2.

Life expectancy in European mortality tables (1662–1830).*

Age	Hudde 1672	Halley 1693	De Moivre 1725	de Graaf 1729	Smart 1738	Struyck 1740 F	Struyck 1740M	Süssmilch 1741	Simpson 1742	Deparcieux 1746	Dodson 1756	Wargentin 1766 F	Wargentin 1766 M	Price 1769	Kritter 1781 F	Kritter 1781M
6.0	40.0	41.0	19.7	48.9	38.6	43.7	39.7	42.7	35.6	47.8	38.5	47.4	45.0	40.6	N/A	N/A
20.0	30.1	32.9	19.1	34.9	28.7	34.3	30.6	34.5	28.2	39.8	28.6	38.7	36.5	32.9	36.3	35.0
50.0	15.5	16.0	13.3	9.9	15.6	17.0	14.7	16.4	14.8	20.0	15.6	18.8	17.2	17.5	18.2	16.5

* Sources in Table 1 above | F: female; M: male.

Table 3.

Fair price of life annuities in years’ purchase for a 5% interest rate.

Years’ purchase	Hudde 1672	Halley 1693	De Moivre 1725	de Graaf 1729	Smart 1738	Struyck 1740 F	Struyck 1740 M	Süssmilch 1741	Simpson 1742	Deparcieux 1746	Dodson 1756	Wargentin 1766 F	Wargentin 1766 M	Price 1769	Kritter 1781 F	Kritter 1781 M
6	16.5	16.3	10.8	18.9	16.2	16.9	16.3	16.5	15.3	17.0	16.2	17.2	16.9	16.0	N/A	N/A
20	14.3	15.3	10.9	16.8	14.2	15.5	14.7	15.6	14.1	16.5	14.1	16.5	16.1	15.0	15.9	15.8
50	10.6	10.8	9.4	8.1	10.3	11.0	10.0	10.9	10.2	12.4	10.3	12.0	11.3	11.3	11.7	10.9

* Sources in Table 1 above | F: female; M: Male.

Concluding remarks: The aftermath

In our view, the 18th century saw the collapse of the objectivist version of actuarial fairness: The proliferation of mortality tables dispelled any illusion of having an algorithm to calculate one single fair price for every life annuity. This conclusion raises at least two further questions. On the one hand, why did the concept of actuarial fairness survive, apparently untouched, until today? On the other hand, why did the Domatian interpretation not come to prevail over the objectivist account? We offer two conjectures by way of conclusion.

As to the survival of the objectivist account, our simple conjecture is that this concept of actuarial fairness was revived as a marketing tool in the United Kingdom and the United States at a point at which companies were converging on unified mortality tables.¹⁶ This convergence made the idea of the single fair price credible. In the market for annuities, this happened relatively early. In 1829, the British Treasury issued a large number of life annuities in order to repay government debt, choosing John Finlaison’s mortality tables to price them. Finlaison had earned himself an appointment as Actuary of the National Debt Office, showing the money his estimates could save the Treasury. The crucial difference was not in Finlaison’s method, but in the situation: In times of peace, there was no urgent need to squeeze money out of the public by offering high returns, in competition with private insurance companies.¹⁷ The British Treasury could now offer lower returns, grounded on Finlaison’s tables, but with fewer prospects of bankruptcy than any private seller. Even if there were competing mortality tables, the state set a risk benchmark in terms of solvency: Finlaison’s tables signalled the level of risk a state was willing to cover for.

Yet Finlaison’s table never became the benchmark for other life insurance markets in the UK, since the British government never sold anything other than annuities. Throughout the 19th century, British and American insurance companies (Murphy, 2010) chose one or another life table depending on whether they wanted to reduce premiums or pay dividends to their shareholders. According to Alborn (2009: 103), it took until the 1880s for British companies to adopt tables elaborated by the Institute of Actuaries and the Faculty of Actuaries, even if these tables represented the life expectancy of only one particular type of customer (certain types of healthy white males). Everybody else was simply charged in excess for their risk (ibid.: 116–21). In other words, even if there was a single mortality table, this table did not properly represent the risk of the entire customer pool, merely a majority (in the best possible scenario). The same risk, same price principle did not apply here. Following Bouk (2015: 4), our best conjecture is that, in a packed marketplace, equity became a marketing tool: ‘It allowed life insurers to attract sound lives with insurance offered at a lower price’.

If it had some marketing value, then, why could the Domatian interpretation of actuarial fairness not compete any longer with the objectivist account, despite its flaws? Again, this is an open research question and we can only speculate. The Domatian view of aleatory contracts was incorporated into the Napoleonic Code (arts 1964–1983) and exerted through it a significant influence on Continental law. In life annuities, for instance, for the uncertainty to be genuine, death was not allowed to be predictable for at least 20 days in advance.¹⁸ Otherwise, there would not be enough of an alea for such a contract to qualify as aleatory (Aubry and Rau, 1871: 584). Yet life insurance thrived under common law in Anglo-Saxon countries, where, according to Alborn (2009: 221), companies rarely tried to penalize events after the policy was issued. After the Gambling Act of 1774, the major legal issue in the Anglo-Saxon world seems to have been about the concept of insurable interest (Merkin, 1980). The Domatian view of justice may have had a life of its own in Continental law, outside life insurance, that remains to be explored.

The history of how the concept of actuarial fairness evolved through the 19th and 20th centuries to the present is yet to be written. Yet we may conclude that when actuaries today speak of fair insurance prices in terms of expected values, they are simply following a centuries-old tradition without real consideration of what was fair about them. On the one hand, as we have just shown, the formalized version of the equality in risk principle for life insurance went hand in hand with an objectivist account of risks that never fully materialized. On the other hand, the same risk, same price principle originates in an Aristotelian intuition about justice in exchange. This intuition hinges on a mathematical analogy, by which arithmetical means (with or without risk weights) capture the equality between the exchanging parties. After more than a century of neoclassical theorizing about markets, it is difficult to sustain the idea that prices reflect anything objective about the goods exchanged, other than an agreement between buyers and sellers. In this regard, the equality in risk principle has lost its original normative force, precisely because no sort of equality is a prerequisite for a fair exchange. Hence, it would not hurt to consider alternative principles of justice if we are to rethink actuarial fairness today (Meyers and Van Hoyweghen, 2018).

Footnotes

Declaration of conflicting interests

The authors declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

Funding

The authors disclosed receipt of the following financial support for the research, authorship, and/or publication of this article: David Teira received a research grant from the Spanish Ministerio de Economía y Ciencia (FFI2014-57258-P).

ORCID iD

David Teira

Notes

References

Alborn

(2009) Regulated Lives: Life Insurance and British Society, 1800–1914. Toronto: University of Toronto Press.

Aubry

Rau

(1871) Cours de droit civil français: Tome 4 [Course in French Civil Law: Vol. 4]. Paris: Imprimerie et librairie générale de jurisprudence.

Aquinas

(1947[1225-1274]) The Summa Theologica: Complete Edition, trans. Fathers of the English Dominican Province. New York: Benzinger Bros.

Bellhouse

D. R.

(2011) ‘A New Look at Halley’s Life Table’, Journal of the Royal Statistical Society: Series A (Statistics in Society) 174(3): 823–32.

Bernoulli

(1982[1765]) ‘Essai d’une nouvelle analyse de la mortalité causée par la petite vérole, et des avantages de l’inoculation pour la prévenir’ [An Attempt at a New Analysis of the Mortality Caused by Smallpox and of the Advantages of Inoculation to Prevent It], in Bouckaert

L. P.

Speiser

van der Waerden

B. L.

(eds) Die Werke von Daniel Bernoulli: Band 2. Analysis, Wahrscheinlichkeitsrechung [The Works of Daniel Bernoulli: Vol. 2. Analysis, Probability Calculation]. Basel: Birkhäuser, pp. 1–45.

Bernoulli

(2006) The Art of Conjecturing, Together With Letter to a Friend on Sets in Court Tennis, ed. and trans. Sylla

E. D.

Baltimore: Johns Hopkins University Press.

Bernoulli

(1992[1711]) L’usage de l’art de conjecturer en droit [The Use of the Art of Conjecturing in Law], ed. Meusnier

. Paris: Seminaire Histoire du calcul des probabilités et de la statistique.

Birch

, ed. (1759) A Collection of the Yearly Bills of Mortality, From 1657 to 1758 Inclusive. London: A. Millar.

Bouk

(2015) How Our Days Became Numbered: Risk and the Rise of the Statistical Individual. Chicago: University of Chicago Press.

10.

Buffon

G.-L.

(1749) Histoire naturelle, générale et particulière: Tome second [Natural History, General and Particular: Vol. 2]. Paris: Imprimerie royale.

11.

Ceccarelli

(2001) ‘Risky Business: Theological and Canonical Thought on Insurance From the Thirteenth to the Seventeenth Century’, Journal of Medieval and Early Modern Studies 31(3): 607–58.

12.

Clark

(1999) Betting on Lives: The Culture of Life Insurance in England, 1695–1775. Manchester: Manchester University Press.

13.

Coumet

(1970) ‘La théorie du hasard est-elle née par hasard?’ [Was the Theory of Chance Born by Chance?], Annales 25(3): 574–98.

14.

Daston

(1988) Classical Probability in the Enlightenment. Princeton: Princeton University Press.

15.

De Moivre

(1725) Annuities Upon Lives. London: W. P.

16.

Demonferrand

M. F.

(1835) Essai sur les lois de la population et de la mortalité en France [Essay on the Laws of Population and Mortality in France]. (n.p.).

17.

Deparcieux

(2003[1746]) Essai sur les probabilites de la duree de la vie humaine [Essay on the Probabilities of the Human Lifespan], ed. Behar

Gallais-Hamonno

Tietsch

Berthon

. Paris: Institut national d’etudes demographiques.

18.

De Santarém

(1971[1552]) Tractatus de assecurationibus et sponsionibus [The Treatise on Insurance] (2nd ed.). Lisbon: Grémio dos Seguradores.

19.

De Soto

(1967[1556]) De iustitia et iure libri decem: De la justicia y del derecho en diez libros [On Justice and Law in Ten Books], trans. Carro

. Madrid: Instituto de Estudios Políticos.

20.

De Witt

(1995[1671]) ‘Value of Life Annuities in Proportion to Redeemable Annuities’, in Haberman

Sibbett

T. A.

(eds) History of Actuarial Science (Vol. 1). London: William Pickering, pp. 144–64.

21.

Domat

(1850[1689]) The Civil Law in Its Natural Order (Vols 1–2), ed. Cushing

L. S.

, trans. Strahan

. Boston: Charles C. Little and James Brown.

22.

Duvillard

E. E.

(1806) Analyse et Tableaux de l’influence de la petite vérole sur la mortalité à chaque age [Analysis and Tables of the Influence of Smallpox on Mortality in Each Age]. Paris: Imprimerie Impériale.

23.

Finlaison

(1829) Report of John Finlaison, Actuary of the National Debt, on the Evidence and Elementary Facts on Which the Tables of Life Annuities Are Founded. London: H. M. Stationery Office.

24.

Fleischacker

(2004) A Short History of Distributive Justice. Cambridge, MA: Harvard University Press.

25.

Franklin

(2001) The Science of Conjecture: Evidence and Probability Before Pascal. Baltimore: Johns Hopkins University Press.

26.

Gordley

(1993) The Philosophical Origins of Modern Contract Doctrine. Oxford: Clarendon Press.

27.

Hacking

(1975) The Emergence of Probability: A Philosophical Study of Early Ideas About Probability, Induction and Statistical Inference. London: Cambridge University Press.

28.

Hájek

(2007) ‘The Reference Class Problem Is Your Problem Too’, Synthese 156(3): 563–85.

29.

Hald

(1990) A History of Probability and Statistics and Their Applications Before 1750. New York: Wiley.

30.

Halley

(1693) ‘An Estimate of the Degrees of the Mortality of Mankind’, Philosophical Transactions of the Royal Society 17(196): 596–610.

31.

Hup

(2011) ‘Life Annuities as a Resource of Public Finance in Holland, 1648-1713: Demand- or supply-driven?’, BSc thesis, Utrecht University.

32.

Iglesias Garzón

(2009) ‘Filosofía y derecho en Jean Domat’ [Philosophy and Law in Jean Domat], PhD thesis, Carlos III University of Madrid.

33.

Kersseboom

(1738) Verhandeling tot een proeve om te weeten de probable menigte des volks in de provintie van Hollandt en Westvrieslandt [Treatise on an Experiment to Determine the Probable Size of the Populations of the Provinces of Holland and West Friesland]. (n.p.).

34.

Kersseboom

(1742) Eerste verhandeling tot een proeve om te weeten de probable menigte des volks in de provintie van Hollandt en Westvrieslandt [Initial Treatise on an Experiment to Determine the Probable Size of the Populations of the Provinces of Holland and West Friesland]. (n.p.): Jan vanden Bergh.

35.

Kritter

J. A.

(1780) ‘Untersuchung des Unterscheides der Sterblichkeit der Männer und der Frauen von gleichen Alter’ [Investigation of the Difference in Mortality of Men and Women of Equal Age], Göttingisches Magazin der Wissenschaften und Litteratur 2: 229–58.

36.

Lambert

J. H.

(1772) Beyträge zum Gebrauche der Mathematik und deren Anwendung: Dritter Theil [Contribution to the Usage of Mathematics and Its Application: Vol. 3] . Berlin: Verlag des Buchladens der Realschule.

37.

Landes

(2015) ‘How Fair Is Actuarial Fairness?’, Journal of Business Ethics 128(3): 519–33.

38.

Le Bras

(1998) Le démon des origines: démographie et extrême droite [The Devil of Origins: Demography and Extreme Right]. La Tour-d’Aigues: L’Aube.

39.

Merkin

(1980) ‘Gambling by Insurance – A Study of the Life Assurance Act 1774’, Anglo-American Law Review 9(3): 331–63.

40.

Messance

(1766) Nouvelles recherches sur la population de la France, avec des remarques importantes sur divers objets d’administration [New Research on the Population of France, With Important Remarks on Various Objects of Administration]. Paris: Durand.

41.

Meyers

Van Hoyweghen

(2018) ‘Enacting Actuarial Fairness in Insurance: From Fair Discrimination to Behaviour-Based Fairness’, Science as Culture 27(4): 413–38.

42.

Milne

(1815) A Treatise on the Valuation of Annuities and Assurances on Lives and Survivorships (Vols 1–2). London: Longman, Hurst, Rees, Orme, and Brown.

43.

Moheau

J.-B.

(1778) Recherches et considérations sur la population de la France [Research and Considerations on the Population of France]. Paris: Moutard.

44.

Monsalve

(2014) ‘Scholastic Just Price Versus Current Market Price: Is It Merely a Matter of Labelling?’, The European Journal of the History of Economic Thought 21(1): 4–20.

45.

Morris

(1759) Observations on the Past Growth and Present State of the City of London. London: A. Millar.

46.

Murphy

S. A.

(2010) Investing in Life: Insurance in Antebellum America. Baltimore: Johns Hopkins University Press.

47.

Murray

L. L.

(2016) ‘Data Smoothing Techniques: Historical and Modern’, PhD thesis, University of Western Ontario.

48.

Ogborn

M. E.

(1962) Equitable Assurances: The Story of Life Assurance in the Experience of the Equitable Life Assurance Society, 1762–1962. London: George Allen & Unwin.

49.

Pascal

(1963) Œuvres complètes [Complete Works], ed. Lafuma

Paris: Seuil.

50.

Pradier

P.-C.

(2016) The Debt of the Hôtel-Dieu de Paris from 1660 to 1690: A Testbed for Sovereign Default. Paris: Centre d’économie de la Sorbonne.

51.

Price

(1773) Observations on Reversionary Payments (3rd ed.). London: T. Cadell.

52.

Rohrbasser

J.-M.

Véron

(1999) ‘Les frères Huygens et le “calcul des aages”: l’argument du pari équitable’ [The Huygens Brothers and the ‘Calcul des Ages’: The Argument of the ‘Pari Equitable’], Population 54(6): 993–1011.

53.

Rosenhaft

(2010) ‘How to Tame Chance: Evolving Languages of Risk, Trust, and Expertise in Eighteenth-Century German Proto-Insurances’, in Clark

Anderson

Thomann

von der Schulenburg

J.-M. G.

(eds) The Appeal of Insurance. Toronto: University of Toronto Press, pp. 16–42.

54.

Simpson

(1742) The Doctrine of Annuities and Reversions. London: J. Nourse.

55.

Smart

(1738) ‘A Table Showing the Probabilities of Life’, in The Bills of Mortality for the City of London. London: Guildhall Library.

56.

Stone

D. A.

(1993) ‘The Struggle for the Soul of Health Insurance’, Journal of Health Politics, Policy and Law 18(2): 287–317.

57.

Struyck

(1912) Les œuvres de Nicolas Struyck (1687-1769) qui se rapportent au calcul des chances, à la statistique générale, à la statistique des décès et aux rentes viagères [The Works of Nicolas Struyck (1687–1769) Regarding the Calculation of Chances, General Statistics, Death Statistics, and Life Annuities], ed. and trans. Vollgraff

J. A.

. Amsterdam: Martinus Nijhoff.

58.

Süssmilch

J. P.

(1761) Die göttliche Ordnung in den Veränderungen des menschlichen Geschlechts [The Divine Order in the Changes in the Human Race]. Berlin: Verlag des Buchladens der Realschule.

59.

Sylla

E. D.

(2003) ‘Business Ethics, Commercial Mathematics, and the Origins of Mathematical Probability’, History of Political Economy 35(Suppl.): 309–37.

60.

Teira

(2006) ‘On the Normative Dimension of the St. Petersburg Paradox’, Studies in History and Philosophy of Science Part A 37(2): 210–23.

61.

Turnbull

(2016) A History of British Actuarial Thought. New York: Palgrave Macmillan.

62.

Van Ham

(2005) ‘De tafel van afsterving van Johannes Hudde’ [Johannes Hudde’s Life Table], De Actuaris 12(6): 31–3.

63.

Wargentin

(1995) ‘Mortality in Sweden According to the “Tabell-Verket”‘, in Haberman

Sibbett

T. A.

(eds) History of Actuarial Science: Vol. 2. Life Tables and Survival Model, Part 2. London: William Pickering, pp. 11–38.