The Term Premium: Compensation for Holding Longer, and What Is Left

A term premium is the part of a long SPOT rate that is not the expected average of future short rates. The premium is a leftover after a subtraction, and the quantity being subtracted is observable nowhere. On this invented curve one supposition about expectations leaves 1.45 percentage points at ten years and a second leaves nothing at all, from the identical six recorded rates.

Two quite different things could explain a long rate standing above a short one. People might expect short rates to be higher in later years, so a long rate is only the average of a rising sequence. Or people might want paying for tying money up across a long stretch, whatever they expect. A curve reports one figure per maturity, so one observed rate is being asked to give away two quantities at once, and it cannot.

Every working below runs on one invented six node SPOT curve: one year 5.90 per cent, two years 6.25 per cent, three years 6.55 per cent, five years 6.90 per cent, ten years 7.35 per cent, thirty years 7.60 per cent. A SPOT rateWhat a placement made today earns if it is left alone until one stated date and handed back in a single lump at that date. Nothing is paid out along the way, so there is only one date to attach it to. is what a placement made today earns if it is left alone until one stated date and handed back in a single lump then. ANNUAL compounding means a rate is applied once a year and whatever it earns is left in to earn again the next year. Every sum below runs on that basis. The basis has to be said wherever a sum is worked. These same six figures on a twice yearly basis would price the same promise differently. Rates get compared here in two units and the units are not interchangeable: a percentage point is a whole unit of a rate, a basis point is a hundredth of one, so 1.45 percentage points and 145 basis points are one distance written twice.

What is a term premium, and what exactly gets taken away to leave it?

Think of a shopkeeper closing the till on a Saturday evening. Total takings, counted, are Rs 41,200/-. She wants to know how much of that came from the new counter she opened last month, so she subtracts what the old counters would ordinarily have brought in on a Saturday. Whatever is left, she calls the new counter's contribution. Notice the structure. One of those two figures she counted. The other she estimated. The thing she ends up reporting is a residualA quantity arrived at by taking one thing away from another, rather than by observing it directly. Whatever is left after the subtraction is the whole of it, including any error in either figure., and it inherits every bit of slack in the estimate she subtracted.

A term premium has exactly that shape: take a long SPOT rate, take away what short rates are expected to average across the same span, and whatever is left is the term premium. It is the compensation for holding across the whole period in one commitment rather than placing money for a year at a time and doing it again. The term premium is therefore defined by subtraction rather than by observation. Nothing about it is controversial. The structure of that subtraction implies something, and the implication arrives immediately.

The first of the two quantities is sitting in plain view. The invented curve records 7.35 per cent at the ten year node, and anybody can read it. The second quantity, what short rates are expected to average across the coming ten years, appears on no curve, in no price and on no screen. The expected average lives in the heads of the people placing money. So the term premium is a residual computed from one number that exists and one number that has to be brought in from somewhere else entirely.

Read that term table once more and notice which row has no source attached to it. The first row points at the curve. The last row points at the calendar. The middle row points at a survey, a forecast or a model, all of them somebody's work rather than a market. Every disagreement about term premiums in the whole subject is a disagreement about that middle row, and none of it is a disagreement about the arithmetic.

Why can a term premium not be read off a yield curve, however carefully it is examined?

The invented curve records 7.35 per cent at its ten year node. The recorded figure is the sum of two quantities, and the curve never separates them. The information required to do the splitting is not held in the curve, so no amount of care with the curve will split it either. The limit is easy to mistake for a complaint about this particular curve, or about a curve invented for teaching. Sitting with it for a moment is worth the trouble.

The limit is neither. Any curve anywhere reports one rate per maturity, so every split of that rate arrives from an assumption, a survey or a model brought in from outside the curve. A curve with six nodes has this problem. Each extra node reports one more rate carrying its own pair of quantities inside it, so a curve with sixty nodes has exactly the same problem sixty times over. Adding maturities gives a better picture of the curve. Adding maturities gives nothing at all towards the subtraction.

Here is the same point drawn. Three readers each take the recorded 7.35 per cent and each draws a line across it, and the three lines sit in three different places. Every one of the three splits adds back to 7.35 per cent exactly. The only thing the curve ever said was 7.35, so it is content with all three and has no way to prefer any of them.

So what follows in the next two sections is not a calculation of the term premium on this curve. The two sections instead set out two suppositionSomething assumed for the sake of the working and stated openly rather than slipped in. Change the supposition and every figure downstream of it changes with it.s, each one labelled as a supposition every time it produces a figure, applied to the same six recorded rates. Read them as a demonstration of how far apart two honest answers can sit, not as two attempts at one right answer.

What does this curve give if expectations are supposed never to change?

Suppose, and the word suppose will keep recurring, that the one year SPOT rate is expected to sit at today's 5.90 per cent in every future year. Nobody has stated that. Nobody has measured it. The flat path is simply a choice made in the open. Nothing is simpler to suppose about a rate than that it stays where it is.

The supposition does something convenient to the arithmetic. If the one year rate is expected to be 5.90 per cent next year and the year after and every year after that, then the expected average of short rates over any span whatsoever is also 5.90 per cent. An average of a constant is the constant. So the subtraction in the definition collapses to something that can be done with two figures off the curve and nothing else.

Run it across the six recorded nodes and the leftovers come out like this. At two years the distance from 5.90 is 0.35 percentage points. At three years, 0.65 percentage points. At five years, 1.00 percentage points. At ten years, 1.45 percentage points. At thirty years, 1.70 percentage points. At one year the leftover is necessarily nought. The front node is the thing being subtracted from itself. Every one of those five figures belongs to the supposition and not to the curve, and that sentence stands beside the numbers rather than sitting in a note underneath.

The picture actually shows the curve, redrawn with its first node pulled to the floor. Nothing has been measured, nothing has been surveyed, and no new information entered between the curve and the chart. A reader who came in cold, saw a rising set of bars headed term premium, and did not read the word supposition would walk away believing something had been established. Nothing was.

What does the same curve give if expectations are supposed to be the rates it already implies?

Here is a second supposition, and it is every bit as fair as the first. Suppose people expect the one year rate covering each future year to be exactly the rate this curve already implies for that year. A FORWARD rateA rate attached to a window of time that opens later. The rate carries both the length of that window and the day the window opens, and it is squeezed out of two SPOT rates rather than quoted by anybody. is precisely that: a rate attached to a window of time that opens later, carrying the length of that window and its opening day, and squeezed out of two SPOT rates rather than quoted by anybody. Supposing expectations equal the forward rates is the pure form of the expectations reading of a curve, and plenty of careful people hold it.

The effect of the supposition on the arithmetic is worked out under forward rates. The chain of FORWARD segments running along this curve multiplies out to the recorded growth at every node: money placed for a single year at the one year SPOT rate, then for the next year at the one year rate one year FORWARD, then again for each year after that until the end, arrives at exactly what a single placement at the long SPOT rate would have given. The five year rate five years FORWARD is one link in that same chain, and fixing it takes the five year and the ten year SPOT rates together.

So under supposition two the expected average of short rates equals the long SPOT rate at every maturity, and the leftover is 0.00 percentage points everywhere. Not small. Not roughly nothing. Exactly nothing, by construction, at one year and at two, at three and at five, at ten and at thirty alike. The chain of forward segments had already accounted for the entire long rate, so the subtraction takes the whole of it away and leaves no remainder.

The second supposition is entirely ordinary, and worth being blunt about. The supposition is not a trick position adopted to make a teaching point. A person saying a curve shows where rates are headed is asserting precisely that supposition. If the curve genuinely shows where rates are headed, then people expect the forwards, and if people expect the forwards there is no compensation for length inside the curve at all. The two beliefs cannot both be held comfortably, and most readers meeting a term premium for the first time are holding them both.

What happens when both suppositions are set side by side?

The same curve. The same six recorded rates. The same arithmetic, run correctly both times, by two people neither of whom has made a mistake. At ten years the first supposition leaves 1.45 percentage points of term premium and the second leaves 0.00 percentage points. The gap between two honest answers is the whole of the thing being reported.

Set the two out as a table and the disagreement stops being a claim about ten years and becomes a claim about the whole curve.

The heading on the second column says where it came from, and so do the headings on the third and the fourth. Naming the source is the only honest way to head a column of this kind. Two of those columns are arithmetic on a stated assumption. One is a set of recorded figures. A reader glancing at the table and taking away a term premium of 1.45 percentage points has read the third column as though it were the second.

The useful thing about a term premium is not its size but this: anybody quoting one has brought an assumption about expectations along with them, and most of what they report is that assumption. Neither of the two people in this section made an error. Neither ought to withdraw their figure. Each of them owes a reader the sentence naming what they subtracted, and once that sentence is in hand the number can be judged. Without it the number is unreadable, however many decimal places it carries.

The disagreement is not a peculiarity of an invented curve either. Two people working from one real market, on the same afternoon, with identical screens in front of them, can report term premiums that disagree, and the reason is never that one of them is unable to subtract. The reason is that one filled the unobservable row from a surveyA count of what people say they expect, gathered by asking them. A survey supplies a quantity that cannot be read off any price, and no price will ever supply it. of what forecasters say they expect, while the other filled it from a statistical model fitted to years of curve history. Different fillings, different leftovers, same subtraction.

Past 7.35 per cent something worth seeing happens. The leftover crosses nought and keeps going, and the right hand panel drops the bar below its zero line to show it. A negative term premium is not a broken calculation. A negative leftover says that on that supposition people expect short rates to average more over the next ten years than the ten year rate pays. Holders are accepting less to commit for the long stretch than they think they could get by rolling short. Nothing in the recorded curve moved throughout, and the whole of the movement was in the supposition.

What does a FORWARD rate actually state, if not a premium?

Everything so far has been about a quantity the curve refuses to give. Here is something the curve does give, completely and without any assumption, and it is worth having because it is what most people are reaching for when they ask about a term premium.

Suppose a household has Rs 1,000.00/- to place and two years before it is needed. Two routes are open. Place it once for two whole years, at the two year SPOT rate, entered at 6.25 per cent, and it returns Rs 1,128.906250/-. Or place it for a single year first, at the one year SPOT rate, listed at 5.90 per cent, collect Rs 1,059.000000/- twelve months from now, and place that for a second year at whatever the one year SPOT rate happens to be by then. Nobody has that second figure yet. The second route is rolling shortPlacing money for a short stretch, then placing it again when that stretch ends, over and over, rather than committing it once for the whole span., and the household has no idea today what the second year will pay.

There is one rate for that second year which makes the two routes finish in exactly the same place, and it can be worked out today from the two SPOT rates already on the curve. Rs 1,128.906250/- divided by Rs 1,059.000000/- is 1.06601157, so the second year would have to pay 6.601157 per cent. The figure of 6.601157 per cent is a break-evenThe level at which two courses of action finish exactly level, with neither one ahead. Above the break-even one course wins and below it the other does., which is to say the level the second year has to beat for the rolling route to have paid more, and it is a fact about today's two SPOT rates rather than a premium or a forecast. Its proper name, worked out under forward rates, is the one year rate one year FORWARD.

The break-even claims very little. Nothing in it says the second year will pay 6.601157 per cent, or that anybody expects it to, or that the rolling route is riskier and the household should be paid for taking it. All it says is where the dividing line falls, given two rates that are already recorded today. A break-even is a smaller statement than a term premium and a far more solid one. The two objects are worth keeping apart for precisely that reason.

What happens when the future one year SPOT rate lands away from that break-even?

Two suppositions again, and both of them are declared inputs rather than anything observed, surveyed or expected by anybody. The two are taken one at a time.

Suppose the one year SPOT rate twelve months from now turns out to be 5.60 per cent. The rolled route is holding Rs 1,059.000000/- at that point, and a second year at 5.60 per cent brings it to Rs 1,118.304000/-. The single two year placement finished at Rs 1,128.906250/-, so the rolled route came up short by Rs 10.602250/- on every Rs 1,000.00/- placed. The shortfall is the arithmetic of landing below the break-even.

Now suppose instead the one year SPOT rate then is 7.65 per cent. The same Rs 1,059.000000/- grows to Rs 1,140.013500/-. The rolled route ends Rs 11.107250/- above what the two year route delivered. Landing above the break-even and landing below it change which route paid more, and the break-even is the only thing that separates the two cases.

Now notice carefully what this section did not do. Neither outcome was called likely, and no rate on this curve makes one more likely than the other. The household was not told to choose one route over the other. Whether a longer commitment is worth making is covered separately. Neither difference was turned into a premium. A difference measured after the fact between two routes is a result, and a premium is a compensation demanded in advance. The two get confused constantly and they are not the same object at all.

A caterer quoting for a wedding eighteen months away meets the same structure without any rates in it. A price fixed now is a known number. A price agreed at whatever vegetables cost that week might come out better or worse. There is some level of future vegetable price at which the two arrangements come out level, and that level is a break-even, computable from today's quoted fixed price. The break-even is not a statement about what vegetables will cost, and no caterer would mistake it for one.

What would it take to measure a term premium, and what is missing here?

A measurement needs the quantity being subtracted. The definition is a subtraction, and one of its two terms has to come from somewhere, so there is no way round it and no clever route to it. In practice it arrives one of two ways. In a survey, somebody asks a large number of people what they expect short rates to be over the coming years and averages the replies. In the second, somebody fits a model to a long run of curve history, and the model produces expectations from data rather than from asking. Both are real bodies of work, both are done carefully, and both supply a number for the row the curve leaves blank.

No survey, no forecastA statement about what somebody thinks will actually happen. Arithmetic worked out of today's figures can look like a prediction and is not one. and no measured expectation of any future rate stands behind these six recorded rates, so the missing term in the subtraction cannot be supplied from them. A figure produced by picking an expectation and presenting the leftover as a finding wears the clothes of a measurement without being one, which is a worse outcome than a visible absence.

Two routes lead further than this guide goes. The estimation methods, and the names attached to them, live in the published research and can be found through the repositoryA catalogue where published academic work is deposited and indexed, so that a name attached to an idea can be checked before anybody writes it down. at ideas.repec.org. Any measured or surveyed Indian series would come from the Reserve Bank of India's data site at dbie.rbi.org.in. Attaching the wrong name to an account of the term structure of interest rates hands a reader a mistake they keep for years. Open both before writing anything down.

One thing the arithmetic above does not need from that list. Every sum above is written free of any convention except the compounding basis, and the compounding basis is stated inside the arithmetic itself, as ANNUAL. Not a single sum could be reproduced without it. A different market with a different convention would therefore be an addition to the block above rather than a rewrite of the working.

Where does any of this land on somebody's desk?

Three places, and none of them involves producing a term premium at all.

The first is reading. An analyst opens somebody else's note and finds a term premium quoted at some maturity, sometimes to two decimal places, sometimes with a chart of how it has moved over the years. The single most useful thing that analyst can do is turn straight to the sentence naming the expectation that was subtracted, and if there is no such sentence, treat the figure as unreadable rather than as approximately right. That is not scepticism for its own sake. A premium computed against a survey and a premium computed against a fitted model are two different quantities wearing one name, and stacking them into the same chart produces a series that measures the change of method as much as anything else.

The second is reporting. Somebody on a desk is asked what the curve is paying for length and has to write an answer that will be read by people who were not in the room. The disciplined form of that answer carries the assumption inside it: on the supposition that short rates hold near today's level, the ten year node stands 1.45 percentage points above the front of the curve, and on the supposition that short rates follow the FORWARD rates the curve already implies, nothing stands there at all. The disciplined sentence takes twice as long to write and cannot be misquoted. Being impossible to misquote is the whole of its value. A bare figure travels further and arrives meaning something the writer never claimed.

The third is the household, and it is the one nearly everybody meets. A person with money to place for two years is choosing between a two year deposit and a one year deposit that will be renewed once. Such a person is not computing a term premium and never will. The break-even is what they can compute from two rates already quoted to them: the rate the second year would have to pay for the renewing route to catch up. Whether the second year clears that level is unknown to everybody. But knowing where the line falls turns a vague sense that longer pays better into a specific question with a specific number attached, and that is a better place to stand.

Notice that the third use needs no assumption whatever. The first two are entirely about handling one. The split runs through the whole subject. The parts of a curve that can be computed cleanly are computed cleanly and never need an argument. The moment the word premium appears, somebody has supplied a quantity nobody can see, and the only remaining question is who supplied it and how.

One more thing about that second step, the division by ten. The division fails for a separate reason worth naming on its own. The slip catches people who have understood everything else. The leftover of 1.45 percentage points is already a rate a year. The leftover is not a total piled up over ten years waiting to be spread out. Dividing it by ten produces a tenth of a yearly figure rather than a yearly figure, and a tenth of a yearly figure corresponds to nothing at all. The same slip turns up wherever a rate gets mistaken for an amount, and it is the sort of error that survives review precisely because the division itself is correct.

What is a term premium not?

Four things get mistaken for it, and each mistake is worth naming separately because each one comes from a different direction.

A term premium is not the slope of the curve. The slope is a difference between two observed rates and it needs no assumption of any kind: the ten year node standing at 7.35 per cent sits 1.45 percentage points above the front node, and that is true whoever reads it and whatever they believe about expectations. The slope is the thing the failure block reported under the wrong name. The slope is a perfectly good quantity with a perfectly good name of its own.

A term premium is not a FORWARD rate. A forward rate is derived arithmetic, fixed by two SPOT rates that are both already recorded, and what it states is a break-even. The two objects do not even have the same shape. A forward rate belongs to one dated future period. A term premium belongs to a whole span running from today. The two land near each other on paper often enough that stripping the labels off runs them together in a reader's head. Every rate above carries either SPOT or FORWARD written out for that reason.

A term premium is not a payment. No cash flow anywhere carries that label. Nothing is credited on any date, nothing appears on any statement, and no receipt exists. A term premium is an accounting of a rate, performed after the rate has been split by an assumption. Some readers hold a picture of a premium being handed over the way an insurance premium is, and there is no such handing over anywhere in this subject.

And a term premium is not a reason to hold anything. Whether a longer commitment is worth making depends on a great deal that is not in a curve at all, is covered separately, and is not settled by the size of a leftover, particularly a leftover whose size is mostly the assumption behind it.

The shape matters more than the figures. A term premium is a subtraction with one term that can be seen and one that cannot, so the answer is whatever the invisible term was chosen to be. On this invented curve two reasonable choices produced 1.45 percentage points and nothing at all, from the identical six recorded rates, and no third figure would be any better founded than those two. The dependence is not a weakness in the idea but the idea, stated accurately.