Ulpian’s Table and the Value of Life Annuities and Usufructs

Falcidian law, instituted in Rome in 40 BC, required that the legal heir receive at least one-fourth of a testator’s estate.1 This provision in law, known as the Falcidian fourth, allowed beneficiaries (other than the legal heir) to receive up to three-fourths of an estate in the form of money, property, usufructs,2 and cash annuities for a fixed term or for life. If the value of all bequests other than to the legal heir exceeded three-fourths of an estate, all such bequests were proportionally reduced in order to ensure that the legal heir received at least one-fourth of an estate. A valuation problem arose because the present value of annuities and usufructs had to be calculated in order to determine whether the legal heir received the mandated one-fourth. Life annuities and lifetime usufructs caused especially thorny valuation problems because there were no markets in which these products were traded. In about 220 AD, the jurist Domitius Ulpianus (circa 170 – 228) provided a table (Mays, 1971) for the purpose of valuing life annuities and lifetime usufructs. Ulpian’s table, the subject of this note, illustrates the inverse relation between age and the value of a lifetime stream of payments or benefits. A rigorous mathematical treatment of this basic idea did not occur until the seventeenth century (De Witt, 1671 and Halley, 1693). Like many things Roman, Ulpian’s table had staying power; commenting on the long history of the table, Haberman and Sibbett (1995) describe Ulpian’s table as “… by far the longest used table in history,”citing that the Tuscan Government used it to value life annuities as recently as 1814.