Reciprocity and the difference between usury and interest

At the IASH meeting on the Human-Business interface I attended last week, Michael Northcott discussed the relationship between capitalism and sustainability.  Michael is an authority on both subjects, having written on environmental ethics and debt.  At the end of Michael's presentation I made the point that theologians distinguished interest ad usury in the past, a distinction that was missing in Michael's discussion.

Usury derives from the Latin usus, meaning 'use',  and referred to the charging of a fee for the use of money.  Interest comes from the Latin interesse, meaning 'compensation for loss', and originated, in the Roman legal codes as the fee someone was paid  if they suffered a loss as a result of a contract being broken.  So a lender could charge interest to compensate for a loss, but they could not make a gain by lending.

It is easier to understand this distinction with a simple example.  A farmer lends a cow to their cousin for a year.  In the normal course of events, the cow would give birth to a calf  and the cousin would gain the benefit of the cow's milk.  At the end of the loan, the farmer could expect the cow and the calf to be returned.  The interest rate is 100%, but it is an interest since the farmer, if they had not lent the cow to their cousin, would have expected to end the year with a cow and a calf.  Similarly, if the farmer lent out grain, they could expect to get the loan plus a premium on the basis that their cousin planted the grain, he would reap a harvest far greater than the sum lent.

Because money is 'barren', unlike land or labour it could not 'produce' anything.  As a result, money can have no intrinsic value, other than its use to facilitate exchange, and so charging for the lending of money is essentially selling something that has no value.  Thomas Aquinas argued that

To take usury for money lent is unjust in itself, because this is to sell what does not exist, and this evidently leads to inequality which is contrary to justice.

So, usury contradicts 'natural law'.  Even if you could convince the canon lawyers that you were, in fact, selling something that did exist, the theologians might argue that usury was an affront to God because, since money was barren, the usurer was charging for time, and "time was God's exclusive property''.

In theory, this is all very clear, in practice there was still the question of where the dividing line between usury and  interest was and almost everyone who was handling money was looking to charge as much interest as was permissible.

Around 1236, an English professor of canon (church) law, Alanus Anglicus,  argued that usury did not exist if the future price of the good was uncertain in the mind of the merchant.  These theories became established in the medieval legal system between 1246 and 1253 by Pope Innocent IV, a former professor of law at Bologna.  Not only could a merchant adjust the 'just price' to cover their labour and expenses, but also they could also adjust the price to take into account the risk they bore,  called an aleatory contract, from the Latin word alea for chance.  In establishing this principle, a Catholic jurist initiated the scientific study of financial risk.

Today, financial economics models interest through a force of interest, r, and so the value of a loan of  X₀ at time t = 0 would be repaid by an amount X_T given by

This implies that the repayment amount X_t is the solution to the most basic differential equation,

that is, X_t grows at a constant rate r. This links to Piketty’s thesis that capitalism induces inequality because r, the return on money, is greater than g the growth rate of the economy. This is conventional economic theory.

I argue that at the heart of financial economics is not growth but reciprocity, so how do I account for interest?

 In 1837 Poisson wrote Recherches sur la probabilité des jugements en matiére criminelle et en matiére civile (‘Research on the Probability of Judgments in Criminal and Civil Matters’). Despite its title, most of Possion’s book was a development of ‘probability calculus’, and according to the historian of probability, Ivo Schneider, after its publication “there was hardly anything left that could justify a young mathematician from taking up probability theory”. The heart of Recherches was a single chapter on determining the probability of someone being convicted in a court, by a majority of twelve jurors, each of whom is “is subject to a given probability of not being wrong” and taking into account the police’s assessment of the accused’s guilt. In order to answer this problem, Poisson needed to understand what has become known as the ‘Law of Rare Events’, in contrast to the Law of Large Numbers. Poisson’s starting point was the Binomial Model, based on two possible outcomes such as the toss of a coin, or the establishment of innocence or guilt. De Moivre had considered what would happen as the number of steps in the ‘random walk’ of the Binomial Model became very large, with the probability of a success being about half. Poisson considered what would happen if, as the number of steps increased, the chance of a success decreased simultaneously, so that it became very small.

On this basis, Poisson worked out that if the rate of a rare events occurring, the number of wins per round, was r, then the chance of there being k wins in n rounds was given by  

the Poisson distribution. Apart from being one of the key models in probability, along with the Binomial and the Normal, the Poisson distribution has an important financial interpretation.

Consider a banker lending a sum of money, X₀.  The banker is concerned that the borrower does not default, which is hopefully a rare event, and will eventually pay back the loan. Say the banker assesses that the borrower will default at a rate of r defaults a day, and the loan will last T days. The banker might also assume that they will get all their money back, providing the borrower makes no defaults in the T days, and nothing if the borrower makes one or more defaults. On this basis the bankers mathematical expectation of the value of the loan is

E[loan] =   (Probability of no defaults × X₀) + (Probability at least one default × 0)

we can ignore the second expression, since it is zero, and for the first, using the Law of Rare Events, the probability of no defaults is given when k = 0 we have that (rT)⁰ = 1 and 0! = 1, so
E[loan] =  X₀e^-rT.

So the banker is handing over X₀ with the expectation of only getting X₀e^-rT < X ₀ back. To make the initial loan amount equal the expected repayment, the banker needs to inflate the expected repayment by e^rT ,

 That is, the repayment amount needs to be

.

We can interpret interest in two ways, as a means of "growing" ones wealth, which would be usurious in the Scholastic sense, or as a compensation.  If it is a compensation the wealth is not expected to grow, that is, Piketty's whole argument becomes somewhat meaningless.

Michael Northcott argued that the Christian prohibition on usury/interest was related to an intent to inhibit human bondage and he noted that contemporary Islamic finance still prohibited the charging of interest.  There in lies a counter argument to the interest equates to slavery claim, it was capitalist Britain that led the way in the emancipation of slavery, Islamic jurisdictions retained slavery into the second half of the twentieth century.

I agree with Michael that usury is a form of bondage, but I do not agree that the charging of interest is usurious, particularly if the interest is determined on the principle of balanced reciprocity.  The problem contemporary finance faces is that it emphasising economic growth over social cohesion, the distinction between usury and interest is obscured.