The concepts of “actuarial fairness”
The concepts of “actuarial fairness”
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Abstract
The concept of actuarial fairness stems from an Aristotelian tradition in which
fairness requires equality between the goods exchanged. When dealing with aleatory
contracts, this principle evolved, among medieval scholars, into equality in risk:
benefits and losses should be proportional to the risks undertaken. The formalization
of this principle gave rise to the concept of mathematical expectation, first
implemented in the calculation of the fair price of gambles. The concept of an
actuarial fair price was first theoretically articulated in the 17th century as an
implementation of this same Aristotelian principle in the field of life insurance. For a
practical estimation of fair actuarial prices it was necessary to build mortality tables,
assuming that the major risk factor was age. Yet, in the 18th and 19th centuries, we find
no agreement among proto-actuaries about the proper construction of these tables.
Among the obstacles they found, we want to highlight their early awareness of the
possibility of adverse selection: buyers and sellers could manipulate the risk
assessment for their own private interests, in a way that would either make fair
companies collapse or fair customers be cheated. The paradox in the concept of
actuarial fairness is that as soon as it was formally articulated, markets made clear it
could never be implemented in actual pricing. The concepts of “actuarial fairness”
1. Actuarial fairness revisited
The advent of big data raises concern about insurability, as the minimization of
classes of risk may ultimately dissolve mutuality and cause market failure –see e. g.
(Charpentier, Denuit, & Elie, 2015). In this respect, fair pricing seems to contradict
insurability: this opposition is especially challenging for actuaries, since they draw on
an old intellectual tradition about actuarial fairness. In the last few years, we have
witnessed a growing debate on what is a fair actuarial price (Johnson, 2015; Landes,
2015; Lehtonen & Liukko, 2011). The discussion is motivated partly by the impact of
big data on the assessment of individual risks, and partly by the changing regulation
of insurance in the wake of the latest financial crisis. Unlike other fields in finance,
actuaries draw on an old intellectual tradition within their own discipline regarding
fairness: a fair premium is understood, by default, as the expected value of the
insured quantity. We want to examine this historical tradition in order to articulate a
precise response to the question in our title: what was fair in the concept of actuarial
fairness? Then we want to propose a conjecture about why, despite its clear
conceptual articulation, actuarial fairness was never implemented in the calculation
of actuarial prices in the almost two centuries elapsed between 1671 (Jan De Witt’s
foundational text) and 1829 (the British issue of life annuities after the Napoleonic
wars)The concepts of “actuarial fairness” .
From a historical standpoint, the concept of actuarial fairness articulates two
different principles about the fair pricing of an insurance contract (section 2): equality
in exchange and equality in risk. Equality in exchange is an Aristotelian principle about
fairness in trade: in an exchange, both parties should leave even. Equality in risk is a
variation of the former, stemming from a Scholastic debate on contracts with
uncertain outcomes (e.g., insurance): the distribution of costs and benefits would
only be fair if it was proportional to the risks each of the contracting parties took. If
the parties were taking the same risks, equality in exchange applied: they should have
equal costs and benefits.
Despite their successful posterity, both principles are inevitably vague: what should
count as equal or proportional? How to measure the risks? We will follow a
mathematical thread: a significant part of the debate on what these principles
entailed was developed through formal analogies. Aristotle exemplified equality in
exchange in terms of arithmetical means. More crucially for our analysis, early in the
17th century, equality in risk was spelled out in terms of expected values (a probability
weighted mean). This is where our current debates on actuarial fairness originate.
The expected value of an insurance contract is usually considered its fair actuarial
price (its expected value). The fairness of this price, we contend, is grounded on the
equality in risk principle. In sections 3-4, we will study how expected values captured
this principle, without an explicit definition of mathematical probabilities. Risks were
subjectively estimated by the contracting parties, under the assumption that they all
knew the same about the uncertain outcome on which their contract hinged The concepts of “actuarial fairness” . I
