Life insurance contracts, life annuities, and an array of transactions relating to estates and inheritances are contingent on lifespan. It would make sense that some method of valuing these life-contingent transactions would have been formulated long ago, even in the absence of good demographic data. As it happens, life annuities were common in ancient Rome and there seems to have been accepted ways of calculating their value. We know of this today largely because of surviving Roman legal texts; life annuities and life interests in property were commonly encountered by Roman courts.
The value of a life annuity depends on expected lifespan. In a life annuity, payments are received at some regular interval for so long as the beneficiary, or annuitant, is alive. An annuity is valued by discounting each of these expected future payments at some discount rate, even if zero, and the extent of this expected chain of future payments is, of course, dependent on the expected remaining lifespan of the life referenced in the annuity, often that of the annuitant himself.
So, accurately valuing life annuities requires a good understanding of lifespans. Today, demographic data abounds but this was not always the case. Not all societies have kept records on births and deaths and where they existed, these records were not always accurate, free from bias, or accessible by statisticians. Thus, historians often presume that computing fair valuations of life annuities was a problem only solved in the 17th century when some of the first actuarial scientists began compiling data on lifespans.
Whether fairly valued or not, life annuities and life interests in property were common in ancient Rome and certain means of valuing them were commonly accepted. Roman estate law included a rule known as the lex Falcidia. The lex Falcidia was a Roman law from 40 B.C. establishing that at least one-fourth of an estate must be left to certain protected heirs, the ‘Falcidian portion’, with the remaining three-fourths able to be bequeathed however the testator chose.
If a will left more than three-fourths of the estate to persons other than legally protected heirs, those bequests would be scaled down pro rata. The proportions were based on the entire capitalized value of the estate in question, including annuities and life interests in any property. This raised the question of how to value life annuities and similar arrangements for purposes of ensuring that a will was compliant with this rule.
This was not an issue only rarely encountered. Life annuities and life interests in property were common in Roman estates and inheritances. To keep an estate intact, a testator’s will might stipulate that one person receive the estate in its entirety, but on the condition that they pay a life annuity to an heir, perhaps covering the required Falcidian portion. Because much of Roman law was primarily concerned with issues of inheritances and legacies, courts had to employ an efficient method to ensure the lex Falcidia was being followed and thus a way to value annuities.
Macer and Ulpian
One of the legal experts opining on the question of how to value a life annuity was Aemilius Macer. A Roman jurist of the 3rd century A.D., Macer wrote a treatise on the Roman inheritance tax. This treatise covered how to value a life interest. A portion of Macer’s text survived in the Corpus Juris Civilis, a compilation of Roman law from the 6th century, and specifically in the Digesta, one of the four component parts of the Corpus Juris Civilis, and essentially an encyclopedia of legal opinions from centuries of Roman legal history.
Macer conveys an approach used by an earlier Roman jurist, Ulpian, to value an annuity. According to Macer, Ulpian estimated the value of a life annuity using the age of the referenced life and the following approach:
“The amount bequeathed to anyone for this purpose from the first to the twentieth year is computed to have lasted for thirty years, and the Falcidian portion of that sum shall be reserved. From twenty to twenty-five years, the amount is calculated for twenty-eight years, from twenty-five to thirty years, the amount is calculated for twenty-five years; from thirty to thirty-five years, the amount is calculated for twenty-two years, from thirty-five to forty years, it is computed for twenty years; from forty to fifty years, the computation is made for as many years as the party lacks of the sixtieth year after having omitted one year; from the fiftieth to the fifty-fifth, the amount is calculated for nine years; from the fifty-fifth to the sixtieth year, it is calculated for seven years; and for any age above sixty, no matter what it may be, the computation is made for five years.”Corpus Juris Civilis, Digesta, Book 35, 2.68
The more complex valuation for lives between the ages of forty and fifty was likely to avoid a large abrupt step down in value when sticking to the otherwise consistent use of five-year buckets with fixed values. The resulting table was still simple, giving no explicit role for an interest rate different than zero percent; thus, it would seem a payment received twenty years in the future was not valued any differently than one paid next year.
Ulpian’s approach replaced an even simpler rule of thumb which called for the valuation of any life annuity given to someone under the age of thirty at thirty years’ worth of payments and the difference of sixty and that person’s age for persons older than thirty. It would appear this approach would assign no value at all to a life annuity on someone older than sixty.
It isn’t easy to verify the accuracy of Ulpian’s rule of thumb. First of all, the fact it employs broad age buckets clearly illustrates its limits; after all, a life annuity on a man of twenty was valued the same as one on the child of twelve. Scholars today have even less relevant Roman data to work with than Ulpian likely had access to. Good data on Roman lifespans, fit for statistical evaluation, has not survived to the present.
It is possible that Ulpian may have simply consulted Roman graveyards, as others have done, to determine the distribution of years of death and accepted the biases inherent with such an approach. After all, not all Roman dead, especially deceased infants, would have had a tomb with an epitaph allowing for a determination of lifespan. Some argue that Ulpian, a jurist, was concerned less with actuarial accuracy than ensuring efficient justice. On the other hand, so little is known about the origin of the rule attributed to Ulpian that it is entirely possible that the method was not even his own creation.
Inheritance Tax and Estate Sales
Despite what is plainly stated in the excerpt from the Corpus Juris Civilis above, Macer’s work was originally aimed at addressing a tax question and not the lex Falcidia. A Roman inheritance tax, the lex Julia de vicesima hereditatium, was implemented by Augustus in A.D. 6 to support the military. It was levied at a rate of 5% on the value of estates.
This tax was repealed by the time that the Corpus Juris Civilis was compiled and so the section of Macer’s text included in that compilation of the law was edited to refer to the lex Falcidia instead, which had remained in place. Regardless, the issue of valuing an annuity was just as relevant to calculating the amount of tax owed as it was to ensuring compliance with the lex Falcidia.
Valuing life annuities or life interests in property was also crucial to the private sale of estates. It was common in ancient Rome for heirs to sell an inherited estate rather than take possession of it. However, any buyer of the estate also had to honor the payment of life annuities set out by the will. These annuities were secured by the estate. Thus, valuing these annuities was essential to coming to a price.
Because there was a private market in estates including life annuities, another Roman jurist, Marcellus, simply suggested consulting the market price of estates with life annuities and life interests to solve these valuation problems in judicial proceedings. The opinion of Marcellus, recorded in the Digesta as well, says that “to establish the proportion payable by the heir and other legatees, [the judge] should estimate the value of the legacy at whatever it could have brought … it being uncertain how long [the annuitant] might live”. It is possible Ulpian’s valuation approach may have simply been a restating of a common market convention, or his suggested improvements to such convention.
Accurately valuing life contingent arrangements like annuities requires a good understanding of human lifespans. Prior to the advent of well-constructed life tables, how could someone know what a life annuity was worth? While not an insignificant question, uncertainty here did not stop life annuities from being created. Financial contracts, if useful enough, will be entered into even without optimal information.
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1. De Vries, T., and W.J. Zwalve. “Roman Actuarial Science and Ulpian’s Life Expectancy Table.” Roman Rule and Civic Life: Local and Regional Perspectives, 2004, pp. 275–297.
2. Frier, Bruce. “Roman Life Expectancy: Ulpian’s Evidence.” Harvard Studies in Classical Philology, vol. 86, 1982, pp. 213–251.
3. Greenwood, M. “A Statistical Mare’s Nest?” Journal of the Royal Statistical Society, vol. 103, no. 2, 1940, pp. 246–248.
4. Scott, Samuel P., translator. The Digest or Pandects of Justinian. 1932.