Zero Rates: The Rate With No Intermediate Cash Flow

A zero rate is the constant rate that, applied over the whole period with no payment in between, turns one unit today into the amount the bond pays at maturity. The rate is the bond price restated in different units, and it contains exactly what the bond price contained: no more, no less.

Restating a price as a rate changes the units and nothing else. Every property the collection of bond prices already had survives the restatement intact. The single thing gained is that a rate can be set beside another maturity's rate and read directly, and a price cannot. The restatement below runs on the four invented parameters of the reverting short-rate model, and then stops on the one number in it that surprises people.

What is a zero rate, exactly?

The starting point is what the bond already provides. A zero-coupon bond pays one unit at a single future date and pays nothing at all before then. Its price today is a single number: at one year, under the four invented parameters of the reverting short-rate model, that number is 0.949216. On a Rs 100/- face that is Rs 94.92/- today for Rs 100/- at the end of the year. There is no coupon, no reinvestment, no schedule. One payment, one date, one price.

The zero rate is that price read out as a growth rate instead of as a level. Ask what constant rate, applied without interruption from now until maturity, would grow one unit today into exactly the payment the bond makes. The constant that does it is the zero rate. The question has exactly one answer: there is only one payment to match and only one period over which to match it.

Read that in reverse and the two are visibly the same object. Multiply the rate by the number of years, change the sign, take the exponential, and the price comes back. Nothing is lost in either direction, and that is what a change of units means.

The phrase in the title matters more than it looks. No intermediate cash flow is what makes the answer unique. If something were paid out halfway, what happened to it would have to be stated, and any rate quoted would depend on that answer as much as on the bond. Because nothing is paid out, no such assumption is needed, and the rate is a clean function of the price. Consider the discount factorThe number that converts one unit of money at a future date into its value today. The bond price is already that number. as the distance and the zero rate as the average speed for a journey with no stops: with no stops, the average speed is fixed by the distance and the time, and there is nothing to argue about. One stop in the middle, and how long it lasted would have to be stated before anyone could quote a speed.

What does restating a price as a rate change?

The restatement changes the units and it changes nothing else. The restatement feels like progress, so readers routinely credit the rate with content the price did not have. The rate has none. The four invented parameters that produced the price produced the rate. The shape of the collection, the ordering of the maturities, the far end behaviour, every one of those was already sitting inside the prices and simply comes through the arithmetic unchanged.

The gain is real, though narrow. Prices at different maturities are not comparable by eye. The one year price is 0.949216 and the thirty year price is 0.169551. The periods are different lengths, so staring at those two numbers reveals almost nothing about which maturity carries the higher rate of return per year. Dividing the logarithm by the number of years takes the length of the period out of the comparison. 5.211896 per cent against 5.915333 per cent is a comparison anyone can make instantly. Comparability is the whole benefit. No information is added.

Why does this rate rise at every maturity?

Run the restatement across all nine maturities and something regular happens. Every step up in maturity raises the rate. Not most steps: every one of the eight, from six months out to thirty years, with no reversal anywhere. The regularity is worth understanding rather than noting. The reason is entirely mechanical and has nothing to do with anyone expecting anything.

The short rate starts at 5 per cent today. The short rate is pulled toward a long-run levelThe value a reverting quantity is drawn back toward whenever it sits away from it. Here it is 6 per cent, set as a parameter and never estimated from anything. of 6 per cent, and the pull has a half-lifeThe time it takes for half of whatever gap remains to close. Here it is 1.386294 years, being the natural logarithm of two divided by the speed of reversion. of 1.386294 years. So the expected level of the short rate is 5 per cent now, 5.393469 per cent in a year, 5.917915 per cent in five years and 5.993262 per cent in ten. The expected level climbs and settles.

A zero rate is an average over the whole period, so a longer period puts more weight on the later, higher expected rates and less on today's 5 per cent. At six months the average is dominated by rates barely off the starting level, and the reading comes out at 5.114856 per cent. At thirty years almost the entire period is spent up near 6 per cent and only the first year or two drags the average down, so the reading is 5.915333 per cent. Nothing else is happening. The rise is monotoneMoving in one direction only, with no step that goes the other way. Here every increase in maturity raises the rate. because every extra year added to the horizon has a higher expected rate than the running average it joins.

An everyday version: a walker takes a route that starts flat and turns steadily uphill, and reports the average gradient of however much has been walked so far. Ten minutes in, the average is nearly flat. Two hours in, the flat opening is a small share of the walk and the average has climbed close to the gradient of the hill. The gradient of the hill never changed. Only the share of the walk spent on it did.

Where does the curve end up, and why is that not the long-run level?

Every reader who has followed the rise this far makes the same guess, and it is wrong. The curve is pulled toward a long-run level of 6 per cent, the rate climbs at every maturity, so surely pushing the maturity far enough out delivers 6 per cent in the limit. It does not. The curve approaches 0.059800, or 5.980000 per cent. It stops exactly 0.000200 short of the long-run level, permanently.

The gap is not a rounding artefact and it does not shrink with maturity. The gap has a closed form, and the closed form contains no maturity at all.

Look at what the subtracted term is made of. Squared volatility on top, squared speed underneath, and nothing else. The shortfall is uncertainty about the rate showing up as a lower rate. Only a rate volatility of nought would make it exactly nought. Set the volatility to zero and the curve does reach 6 per cent, because with no uncertainty there is nothing for the subtraction to subtract. The shortfall scales with the square, so halving the volatility from 1 percentage point to half a point cuts it by a factor of four, from 0.000200 to 0.000050.

The exact form of the whole curve makes the arithmetic checkable at any maturity. The model is affineA structure in which the logarithm of the bond price is a straight-line function of today's short rate, with a slope and an intercept that depend only on the maturity., so the zero rate is the limit plus a single correction that shrinks as the maturity grows.

Past about twenty years the response term has effectively finished climbing, sitting at 1.999909 at twenty years and 1.999999 at thirty. From there the bracket is fixed at minus 0.019400 and the whole curve collapses to one line of arithmetic: the zero rate is 0.059800 minus 0.019400 divided by the maturity. Check it at a hundred years. 0.019400 divided by 100 is 0.000194, and 0.059800 less that is 0.059606. The model gives exactly that. At two hundred years the deduction halves to 0.000097 and the reading is 0.059703. At thirty years the same line gives 0.059800 less 0.019400 over 30, or 0.059800 less 0.000647, which comes to 0.059153.

One line of arithmetic is the whole shape of the far end. The remaining distance to the limit is 1.94 percentage points divided by the maturity in years, so it halves every time the maturity doubles and reaches nought at no finite maturity at all. Doubling from a hundred to two hundred years takes the remaining gap from 1.94 basis points to 0.97. Getting it under a tenth of a basis pointOne hundredth of a percentage point. A rate of 5.980000 per cent is 2 basis points below one of 6.000000 per cent. would need a maturity of 1,940 years. Meanwhile the 0.000200 that the limit itself falls short is not moving at all, in either direction, ever.

Why does the limit sit below the average of the expected rates?

The averaging story told a few blocks back was almost right and one term short. Take the expected short rate at every instant between now and maturity and average those expectations over the period. The average of those expectations does climb to 6 per cent: 5.933333 per cent at thirty years, 5.990000 per cent at two hundred, and 6.000000 per cent in the limit. The average gets there. The zero rate does not, and the difference between the two is a single deduction.

Why is there a deduction at all, rather than a bonus? Because the bond price depends on the rate through an exponential, and an exponential bends. Averaging a bent function of an uncertain quantity does not give the same answer as applying the function to the average. Here the bending works in one direction: uncertainty raises the average bond price, and a higher price restates as a lower rate. The name for that bending is convexityThe curvature of a function. Where a function bends upward, the average of its values at a spread of inputs sits above its value at the average input, and the gap grows with the spread..

The clean everyday version comes from measurement rather than money. With half a journey driven at 40 kilometres an hour and the other half at 60, the average speed for the trip is not 50. More time is spent on the slow half than on the fast half, so the average is 48. Narrowing the spread to 45 and 55 raises the average speed to 49.5: the shortfall drops from 2 to 0.5, a factor of four for a halving of the spread, exactly the square law the rate shortfall obeys. With both halves driven at 50 the shortfall is exactly nought. The shortfall is not caused by the speeds being low; it is caused by the speeds differing, and it disappears the instant they stop differing.

What is the compounding convention, and why does a rate need one?

Everything above has quietly used one convention, and it is time to say so out loud. Every rate in this guide is continuously compoundedA convention in which growth is applied without interruption rather than at set intervals, so a rate multiplied by a period and exponentiated gives the growth factor., and that convention is why the definition uses a logarithm and the inverse uses an exponential. Change the convention and the same bond price produces a different number, and the number is not wrong under either convention. The other convention is answering a different question.

Work the one year price through each. 0.949216 gives 5.211896 per cent continuously compounded, 5.223231 per cent compounded monthly, 5.245999 per cent quarterly, 5.280400 per cent half-yearly and 5.350106 per cent annually. The spread across those five figures is 0.138210 percentage points, close to fourteen basis points, produced entirely by a bookkeeping choice on an unchanged bond. A rate without a stated convention cannot be turned back into the price it came from, so it is not a number.

The spread also widens with maturity, and the widening is the part that catches people. Take the ten year price of 0.560610. Continuously compounded it is 5.787294 per cent. Compounded annually it is 5.958035 per cent. Quoted as a simple rate with no compounding at all it is 7.837705 per cent, more than two percentage points adrift of the continuously compounded figure for exactly the same bond. Nobody made an error anywhere in that. Three conventions, three answers, one price.

What does a zero rate not reveal?

A zero rate does not reveal the long-run level. The trap that follows from assuming otherwise is not an exotic one. The reasoning that walks into it is perfectly sensible at every step: the model reverts to a long-run level, the curve is built from the model, the curve rises toward something, so the thing it rises toward must be the long-run level. Every link holds except the last.

There is a second thing a zero rate does not reveal, and it follows directly. The zero rate does not reveal the path. Two very different sequences of future short rates can average to the same zero rate over the same period, and the rate cannot separate them, in the same way that an average speed of 48 kilometres an hour reveals nothing about whether the trip was steady or was half crawling and half sprinting. The rate is one number summarising a whole period, and summarising is lossy by construction.

Who actually reads a zero rate, and what for?

The obvious reader is anyone who has to set two maturities beside each other and say which one carries more return per year. Comparison across maturities is the reason the restatement exists, and the only thing it adds. A person converting prices to rates is buying comparability and nothing else. If they think they are also buying insight into where rates are headed, they have paid for one thing and taken delivery of another.

The subtler reader is anyone checking a model against its own parameters. For that reader the shortfall stops being a curiosity. Suppose a curve arrives with the statement that it came from a reverting short-rate model, and the task is to work out roughly what parameters produced it. The far end of the curve gives the limit, not the long-run level, so recovering the long-run level means adding back the squared volatility over twice the squared speed. So the far end alone cannot recover the level: the volatility and the speed are needed as well, and they have to come from somewhere else on the curve or from somewhere else entirely.

The general lesson is that a limit of a curve and a parameter of a model are two different objects, and only one of them is written on the parameter sheet. Fitting a model to a curve is a separate craft, set out under calibration. But the reason fitting is hard is visible right here: the far end can be seen and the long-run level cannot, and the difference between them is made of two parameters that cannot be seen either.

One last check for anyone using these figures. The convention has to travel with the number. If a rate is going to be handed to somebody else, moved into a different calculation, or compared against a rate that came from elsewhere, the convention is part of the number and not a footnote. Fourteen basis points at one year and two hundred at ten is the size of the confusion available to anyone who forgets that.