Previously to Halley's time, and apparently for many years subsequently, all dealings with life annuities were based upon mere conjectural estimates. The earliest known reference to any estimate of the value of life annuities rose out of the requirements of the Falcidian law, which (40 B.C.) was adopted in the Roman empire, and which declared that a testator should not give more than three-fourths of his property in legacies, so that at least one-fourth must go to his legal representatives. It is easy to see how it would occasionally become necessary, while this law was in force, to value life annuities charged upon a testator's estate. Aemilius Macer (A.D. 230) states that the method which had been in common use at that time was as follows:--From the earliest age until 30 take 30 years' purchase, and for each age after 30 deduct 1 year. It is obvious that no consideration of compound interest can have entered into this estimate; and it is easy to see that it is equivalent to assuming that all persons who attain the age of 30 will certainly live to the age of 60, and then certainly die. Compared with this estimate, that which was propounded by the praetorian prefect Ulpian was a great improvement. His table is as follows:--
+-------------+-----------+------------+-----------+ | Age. | Years' | Age. | Years' | | | Purchase. | | Purchase | +-------------+-----------+------------+-----------+ | Birth to 20 | 30 | 45 to 46 | 14 | | 20 " 25 | 28 | 46 " 47 | 13 | | 25 " 30 | 25 | 47 " 48 | 12 | | 30 " 35 | 22 | 48 " 49 | 11 | | 35 " 40 | 20 | 49 " 50 | 10 | | 40 " 41 | 19 | 50 " 55 | 9 | | 41 " 42 | 18 | 55 " 60 | 7 | | 42 " 43 | 17 | 60 and | | | 43 " 44 | 16 | upwards | 5 | | 44 " 45 | 15 | | | +-------------+-----------+------------+-----------+
Here also we have no reason to suppose that the element of interest was taken into consideration; and the assumption, that between the ages of 40 and 50 each addition of a year to the nominee's age diminishes the value of the annuity by one year's purchase, is equivalent to assuming that there is no probability of the nominee dying between the ages of 40 and 50. Considered, however, simply as a table of the average duration of life, the values are fairly accurate. At all events, no more correct estimate appears to have been arrived at until the close of the 17th century.