The Price of Time

In 1648, a Dutch water authority responsible for maintaining a stretch of dike on the Lek river needed money for repairs. It did what borrowers did at the time. It issued a bond, written on goatskin, promising to pay interest to whoever held it.

Not for ten years. Not for a hundred. Forever.

That bond still pays. It was written out on the fifteenth of May, 1648, to a Mr Niclaes de Meijer, for a thousand Carolus guilders. One of the five known survivors is owned by Yale, which bought it for its finance archive in 2003 and periodically sends someone to the Netherlands to collect. In 2015 a curator came back with twelve years of arrears, amounting to about a hundred and thirty six euros.

The original rate was five percent. It was cut to three and a half, then to two and a half, in the seventeenth century, and two and a half is what it pays today. The institution that issued it has been reorganized more times than anyone has counted. The Dutch Republic that existed when the ink went onto the goatskin is gone. Three hundred and seventy odd years of European history have come and gone across that piece of skin, and the coupon has kept coming.

Britain did the same thing on a much larger scale. The consols, short for consolidated annuities, were government bonds with no maturity date. In February 2015 the Chancellor of the Exchequer redeemed 218 million pounds of four percent Consols, the first time in sixty-seven years that Britain had chosen to retire undated debt, and by that July every remaining undated gilt was gone from the portfolio.

Follow that particular debt backwards and it is worth the trip. The four percent Consols had been issued by Churchill in 1927, largely to refinance war bonds from the First World War. Those in turn had absorbed older obligations, including a bond Gladstone issued in 1853 to consolidate the capital stock of the South Sea Company, the enterprise whose collapse in 1720 gave us the phrase South Sea Bubble.

So a British taxpayer in 2015 made a final payment on a financial disaster that occurred under George the First. The debt had been rolled, renamed and refinanced across three centuries, and at no point in all that time did anyone have to redeem it, because it never came due.

I begin here because these objects make something visible that is otherwise hard to see. A perpetual bond has no maturity, which means the entire question of what it is worth reduces to a single number. Not the size of the payment. Not the creditworthiness of the issuer, at least not primarily.

The discount rate.

The most violent number in finance

Here is the arithmetic, and it is worth doing slowly, because almost everything in this chapter is a consequence of it.

Take an asset that pays you a thousand dollars a year, forever. What is it worth today?

The answer is the annual payment divided by the discount rate. That is the whole formula.

At a discount rate of ten percent, the asset is worth ten thousand dollars.

At five percent, twenty thousand.

At two percent, fifty thousand.

At one percent, a hundred thousand.

At a tenth of a percent, a million dollars.

At zero, there is no answer. Not a very large answer. No answer. The formula divides by zero and the value is not a number.

Look at the shape of that sequence. Moving from ten percent to five percent doubles the value. Moving from one percent to a tenth of a percent multiplies it by ten. The closer the rate gets to zero, the more violently the price responds to each further step down. The last few basis points matter more than everything that came before them.

Anyone who has traded long duration instruments knows this in their stomach rather than as a formula. It is why a thirty year bond moves several times as much as a two year bond on the same piece of news, and it is why pension funds with very long liabilities become almost hysterically sensitive to small changes in rates. Length amplifies. And a perpetuity is the limiting case of length.

So if you want to know what happens to the price of every permanent thing in an economy, you do not need to know very much. You need to know where the discount rate is going.

Where the discount rate comes from

Which raises the question of what the discount rate actually is, and why it sits where it does.

The standard framework goes back to Frank Ramsey, a Cambridge mathematician who wrote the founding paper on this in 1928 and then died two years later at the age of twenty six, which is either a bitter irony or an appropriate one for the subject of this book.

Ramsey’s rule says the rate has two components.

The first is growth. If the future is going to be richer than the present, then a dollar delivered to that richer future is worth less than a dollar now, because the recipient will already have more. This part is about circumstances.

The second is pure time preference. This is the part that says a given amount of happiness matters less simply because it happens later. Not because the recipient will be richer. Just because it is further away.

Economists have always been uncomfortable with the second term, because stated plainly it sounds like a confession of irrationality. Why exactly should something count for less merely by being in the future? Ramsey himself called the practice ethically indefensible and said we only do it out of weakness of imagination.

But there is a reading of pure time preference that is not irrational at all, and it is the one this book cares about.

You might not be there.

A promise of a thousand dollars in fifty years is not the same object as a thousand dollars today, even if you are certain the money will be paid and certain of what it will buy, because there is a real chance you will not be alive to receive it. That is not a psychological failing. It is an accurate assessment of a probability.

This is not a rhetorical point I am imposing on the literature. It is in the models explicitly. Olivier Blanchard’s widely used framework from 1985, sometimes called the perpetual youth model, handles mortality as a constant chance of dying in any given period, and that probability enters the effective discount rate directly. The higher your chance of death, the more heavily you discount. It is right there in the algebra, and it has been for forty years.

The most striking version of this appears in the climate debate, which I will come back to at the end of the chapter. When Nicholas Stern wrote his review of the economics of climate change for the British government, he had to choose a pure time preference rate, and his choice was famously low: one tenth of one percent. His justification was not psychological. He set the rate equal to the estimated probability that humanity ceases to exist.

He priced time as a mortality rate. That was not a metaphor. It was the actual construction.

So we have a chain, and every link in it is standard economics rather than anything I have invented. The value of every permanent asset depends on the discount rate. The discount rate contains a term for time preference. Time preference contains a term for the probability of not being there.

Extend life and you pull on the end of that chain.

The forty year decline nobody can fully explain

Now, the part of this chapter that is not theoretical.

Real interest rates across the developed world have been falling for roughly four decades. Not cyclically. Structurally. The measure economists use is r-star, the rate that would prevail with the economy at equilibrium, stripped of the business cycle and of whatever the central bank happens to be doing at the moment.

The standard estimates of it in the United States fluctuated between roughly two and two and a half percent from the 1990s through the mid 2000s, then fell to something near half a percent around 2009 and stayed there for years. Similar declines show up in Canada, the euro area and the United Kingdom, where recent estimates are the lowest in three decades.

This is one of the most consequential facts in modern finance and one of the least resolved. It has produced an enormous literature and a genuine puzzle, because the decline is too long and too broad to be explained by any central bank’s policy. It preceded the 2008 crisis, survived it, and continued through wildly different policy regimes in different countries.

The explanations offered are several, and most of them have merit. A global savings glut. Falling productivity growth. Rising inequality, since the rich save more. A shortage of safe assets. More recently, rates have risen again from their lows, which is a real development and which nobody should pretend away.

But one explanation appears in essentially every serious treatment, and it is demographics.

The mechanism is simple. People save for retirement. How much they need to save depends on how long they expect to be retired. As healthy old age extends, the required pile gets bigger, and the number of people in their peak saving years shifts. More desired saving chasing the same investment opportunities pushes the return on capital down. Work at the Bank of England attributed a substantial portion of the global decline since the 1980s to demographic factors alongside slower growth, and a body of academic work has since put numbers on the specific contribution of longer lifespans and shifting age structure.

I want to be careful about the strength of the claim, because this is the point in the book where a reader is entitled to be suspicious that I am reading my thesis into an ambiguous data series. So let me put it precisely. I am not claiming that longevity explains the fall in real rates. I am claiming something weaker and, I think, harder to argue with: that a mainstream, well documented body of research finds that longer lives push real rates down, that real rates have in fact gone down over exactly the period in which healthy old age has been extending, and that essentially nobody has asked what happens if the input keeps moving.

We have been treating this as a monetary policy story, because rates are what central banks do. Some of it is a monetary policy story. But underneath the cyclical noise there is a slow structural drift, and one of the things pushing on it is that people expect to be around longer.

Which means the transmission channel this book describes is not hypothetical. It is not something that switches on when a laboratory succeeds. It has been running, quietly, for forty years, and Japan is simply the country furthest along the curve.

What actually breaks

Push the rate further down and specific things stop working. Not metaphorically. Mechanically.

Pensions and insurance. A defined benefit pension is a promise to pay someone for as long as they live. The liability is long duration by construction, and a long duration liability is exactly the thing that explodes when rates fall. Every pension fund in the developed world already knows this, because it has spent the last two decades watching its deficit widen for reasons that had nothing to do with its investments. Now extend the payout period. The liability grows because the payments last longer, and the present value of each payment grows because the discount rate is lower, and both effects run the same way at the same time. Life insurance runs into the mirror image of the same problem, which is that a product priced on an actuarial table is a product priced on an assumption about when people die.

Anything permanent. Land, water rights, mineral rights, spectrum, orbital slots, the freehold on a building. These are perpetuities that happen to be made of dirt and paper. As the rate falls, their prices rise faster than the rate falls, and there is no ceiling in the formula. We have watched a version of this already: a substantial part of the rise in developed world house prices over recent decades is not a story about construction costs or about bricks. It is the same asset being capitalized at a lower rate. The building depreciates. The land underneath it is the perpetuity, and the perpetuity is what moved.

The idea of a fair price. This is the deepest of the three and the least discussed. Valuation is a technology for compressing an unbounded future into a single present number, and it works because the far future contributes almost nothing. Discount a payment eighty years out at eight percent and it is worth about two tenths of a cent on the dollar. It vanishes, which is convenient, because it means you do not have to know anything about the year 2106 in order to price something today. That convenience is a gift from the discount rate. Take the rate toward zero and the gift is withdrawn. The year 2400 starts to matter as much as next year, and no one has any idea how to price the year 2400.

The argument we are already having

There is one place where this fight has already broken into public view, and almost nobody noticed what it was really about.

In 2006 the Stern Review concluded that the world should spend heavily and immediately on climate change. William Nordhaus, who later won a Nobel for his work in the same area, concluded that the world should spend considerably less and more gradually.

The two men were not really disagreeing about climate science. They were disagreeing about the discount rate. Stern used a very low one. Nordhaus used a higher one. Because the costs of climate change land mostly on people in the distant future, and the costs of preventing it land on people now, the entire policy recommendation swings on that single parameter. Change the rate and the answer flips, with the physics held constant.

The debate that followed was fierce, and it was conducted in the language of ethics. What do we owe future generations? Is it defensible to weight a person less because they were born later?

Now put this book’s premise on top of that debate and watch what happens.

Stern’s low discount rate was an ethical assumption, contested precisely because it was an assumption. He was asking us to behave as though we cared about the far future roughly as much as the near one, and his critics said that was not how people actually behave.

If lifespans extend, that assumption stops being ethical and becomes descriptive. You do not have to be persuaded to care about the year 2200 out of moral obligation to strangers. You will care about it because you plan to be there, and so will everyone you are negotiating with.

I find this the single most interesting consequence in the chapter, and it cuts in an unexpected direction. Long lives may be very good for climate policy and for anything else that requires patience, because they align private incentives with the long run in a way that no amount of ethical argument has managed. The same force that breaks the pension system might build the sea walls.

That is worth holding onto, because this book is going to describe a great deal of machinery seizing up, and the honest accounting includes the things that get better. A civilization that discounts the future at nearly zero will build things we cannot currently justify building. Cathedrals, in effect.

It will also find that it has no idea what anything is worth.

The price of time

Everything in economics that involves the word “worth” runs through the discount rate. It is the exchange rate between now and later, and like any exchange rate, it is a price rather than a law.

We have been treating it as a law. We have built pension systems, insurance markets, valuation methods, regulatory cost benefit analysis and the entire apparatus of capital allocation on top of a number that we inherited rather than chose, and one of the things determining that number is how long the people using it expect to live.

That number has already started to move. The next four chapters are about the other four things it was holding up.