Back in 1628, Ambrose Acroyd published a book called Tables of Leasses and Interest. Acroyd was an administrator at Trinity College from 1615-1625, including a role as senior bursar–the modern equivalent might be “chief financial officer”–for several years. The first table of his book focused on a specific situation:
Acroyd’s unusual Table i envisions a very specific situation: one party owns an annuity with a full term of twenty-one years, several of those years have already expired, and the annuity owner wishes to pay to extend that annuity’s term back to a full twenty-one-years. Imagine someone possessing an annuity paying £1 annually for twenty-one years and fourteen years had already expired, leaving seven remaining. Acroyd’s first table states how much should be paid to “fill up” that annuity back to twenty-one years: in this case, £3 5s.11d., found on row “14” in the table.
This is an example of a “net present value” or “present discounted value” calculation: that is, how much does one have to pay in the present to receive a stream of income for some years into the future? Acroyd’s book is just one of several printed in England in the first few decades of the 17th century that included these kinds of formulas. These authors did not invent the formulas for translating a stream of future payments into a present value: the mathematics goes back at least to the Leonardo of Pisa (called “Fibonacci”) several centuries before. But why does this calculation become important at this place and time? Indeed, authors were still using his book and asking “who was Acroyd, anyway?” a century and more later.
William Deringer tells the story in “Mr. Aecroid’s Tables: Economic Calculations and Social Customs in the Early Modern Countryside,” Journal of Modern History, March 2024, 96:1). Peter Dizikes offers a readable short overview in MIT News (June 6, 2024).
The social problem came up as a result of a surge of price inflation. As Deringer writes:
Compared to prices in the decade 1501–10, average prices for foodstuffs in England were 3.0 times greater in 1551–60, 5.0 times greater in 1601–10, and 6.5 times greater in 1651–60. This constituted a radical break from prior experience, when prices had generally been stable, even falling slightly in the period from 1400 to 1500.
Thus, rents paid by tenant farmers had been essentially stable for a century or more. Raising those rents would have been viewed as an act of social aggression, and tenants could and did push back against it with protests and through the courts. But as price inflation arrived, the nominal rental payments became smaller and smaller. The landowners, including the Church of England and church institutions like Trinity College, were squeezed. Again, raising the nominal rents seemed socially and politically impossible. So the landowners reacted by raising the fee for entering or renewing a lease, called a “fine.” The size of these one-time “fines” was linked to the profitability of the land over the number of years of the lease, which in turn was determined by some combination of past experience and land surveys.
As Deringer emphasizes, 17th century England is a time and place when the logic of supply and demand and market outcome was not even in the social discussion. Instead, this was a time when payments were judged in terms of fairness, given the social roles and obligations of the parties. In this setting, the books of net present value formulas became the solution to a social bargaining problem faced by landowners and tenants after inflation had dramatically disrupted their earlier fixed-nominal-rent annual payments. Acroyd’s 1628 book was almost all tables, with very little text, but apparently, a number of the surviving copies had a Latin epigraph written into the beginning of many copies. them. Deringer explains:
The poem, comprising four elegiac couplets, does not appear to stem from any earlier source. A fairly literal translation would be:
Deviant fraud often afflicts us, but the forthright rule of the law of arithmetic teaches [us] what is useful and just. Nature taught mortals to cheat; perhaps it will be for art to prevent the treachery of frauds. Let the crowd of tricksters, the quarrelsome people cry out in protest [against this book]. Buyers, sellers, consult [it]! You will be prudent [to do so]. Those who promote fair exchange among people, if they wish to avoid praise for doing so, should be able to avoid ill-will for it as well.
This epigraph frames Acroyd’s book as an instrument for combating fraud and promoting just exchange. This tool is presented as accommodating buyers and sellers, landlords and tenants—whoever sought fair, prudent, and honest commerce. In leasing conflicts, both landlords and tenants might be guilty of certain frauds and deceits (fraus): landlords, by demanding exorbitant fines or deploying coercive tactics like selling reversionary leases; tenants, by concealing the value of their property or denying landlords their fair share by unreasonable appeals to custom. Arithmetical calculation could advance justice and harmony by enabling reasonable economic practices and curtailing fraudulent ones.
To put it differently, those of us in the modern age tend to think of formulas for net present value as part of financial decision-making, which it is. When an investor thinks about buying a stock, the investor needs to think about what present payment would be equal to the returns expected from owning that stock over time. When a bank lends money, it is calculating how much of a present payment (to you) would be equal to the amount you will repay over time. When you buy a home, you are thinking about whether the price you are paying can be justified by the stream of benefits you expect to receive while owning the home, along with an expected resale price in the figure. When government considers a program for building infrastructure or improving child nutrition or reducing pollution, part of the analysis is to compare the amount spent in the present to the value received over time. Deringer quotes a 2016 book by financial historian William N. Goetzmann to the effect that “the method of ‘net present value’ is the most important tool in modern finance.”
But back in the 17th century, the net present value formula served a different social function: most people didn’t understand the details of the math, but they understood enough of the basic idea to believe that the math represented a rule that placed limits on opportunistic behavior by both parties.
But as Deringer points out, this interpretation is both true and incomplete. The specific interest rates used by Acroyd to determine the present value payments made by “fines” were pretty high–in the range of 11-13%. When relatively high interest rates are used to discount future benefits, the present value of those payments will be relatively small. Thus, it turned out that well-to-do people and political favorites were often able to lease land from the Church of England at these preferential rates, while the less powerful who were trying to lease land in the marketplace were less protected. Deringer draws a trenchant parallel between the use of “net present value” in the 1600s and the use of algorithms to make decisions today. He writes:
Yet there is also something strikingly modern about how “Mr. Aecroid’s Tables” encoded economic equity in abstruse calculations. Particularly remarkable is how those recondite pages of figures were empowered to make judgments about what was fair and equitable on behalf of people with no understanding of how the mathematics worked. … We can assume that the vast majority of church tenants found those cryptic tables largely inscrutable. Based on evidence … so too did some of the church officials tasked with using them. If, from one direction, we might see Acroyd’s Tables as the evolution of medieval conceptions of just price, from another perspective we might see their adoption as an important early chapter in what Rodrigo Ochigame recently called “the long history of algorithmic fairness.”
Today, the use of opaque computational procedures, “black box algorithms,” to make socially contentious decisions is a familiar phenomenon. Across many fields, complicated questions about what is fair—a fair interest rate, a fair use of public resources or distribution of public benefits, a fair punishment—are delegated to algorithms, bolstered by a belief that they are less susceptible to human bias and error. …
From one perspective, those calculative techniques did succeed in creating a mutually acceptable, sustainable solution to the problem of dividing agricultural revenues on church lands. They provided landlords a way to increase their revenues over time to accommodate inflation, while protecting tenants against arbitrariness and exploitation. Ossified in institutional routines, Acroyd’s Tables came to function like a synthetic custom.
At the same time, the adoption of discounting on church lands was a component of other profound transformations in landlord-tenant relations, most notably the shift from a system in which tenants paid based on fixed “ancient” rents to one based on the surveyed profitability of the land. This seismic transformation was unquestionably to tenants’ detriment. The institutionalization of discounting tables on church lands shielded tenants on church lands from the worst consequences of this change. At the same time, though, we can reasonably speculate that the church’s policies served to legitimate this shift and exacerbated the erosion of customary protections for tenants on nonchurch estates. Insofar as many of those who benefited from favorable church leases were comparatively wealthy and well connected, the institutionalization of Acroyd’s algorithm might actually have served to harden preexisting disparities—a recurrent pattern in the subsequent history of algorithmic judgment. To put it another way: calculated fairness was a form of collusion, a bargain struck between a subset of interested parties without the input—and often to the detriment—of many others who never got to be part of the equation.
