Carry and Roll-Down: Splitting a Year on an Unmoved Curve

Carry is what a claim earns at the rate it was bought at. Roll-down is the separate price change that arrives when the claim reaches a shorter maturity and gets repriced at whatever rate the curve fixes there. Held together for a year on a curve that never moves, the two add to the FORWARD rate those SPOT rates already contained, not to the SPOT rate paid.

The panel opens on an invented holding built here: Rs 1,000.00/- of face, three years to run, a coupon of 7.00 per cent, bought at a yield of 6.55 per cent for Rs 1,011.907192/-, financed at a repoShort for a repurchase agreement: cash borrowed for a short spell against securities handed over and bought back later at an agreed price. rate of 6.00 per cent and held one year while the curve stands still. The coupon collects Rs 70.00/- and the funding costs Rs 60.71/-, so carry is Rs 9.29/-. The holding reaches the two year point, where the curve fixes a yield of 6.25 per cent, and the price rises Rs 5.52/- as it is repriced onto that lower yield: that part, and only that part, is roll-down. Pull to parThe drift of a price toward the face amount as the payment date comes nearer, which happens even when no yield anywhere has moved. takes away Rs 3.72/- and the yield move is set to nothing, so the year comes to Rs 11.09/-, or 1.095041 per cent of the price paid.

Carry reads two ways and the panel holds both. A holding financed in the repo market collects the coupon and pays the funding, and the difference is carry. A claim bought outright has neither. Its earnings at the rate it was bought at are its price accreting toward face, and that accretion is the pull to par line. With the coupon and the funding rate set to nothing, the panel reproduces the unfunded arithmetic the rest of this guide works through, to the paisa.

Both halves exist for one reason: a curve carries different rates at different maturities, and this calculation holds that curve still while a claim travels across it. With either condition taken away, one half vanishes. A curve with an identical rate at every maturity has no second rate to be repriced at, so it produces carry and nothing else. A curve that moves produces an outcome this arithmetic was never measuring, and the panel above prices that move once it is declared.

Everything here runs on one invented SPOT curve, built for teaching and matching no market anywhere. The curve fixes six maturities and nothing between them, all on annual compounding. The claims priced against it were made up too, along with the 7.00 per cent coupon and the 6.00 per cent funding rate the panel opens on, and the Rs 1,000.00/- face was built for this guide rather than carried in from any instrument.

What is carry, and what is roll-down, and why keep them apart?

Carry is the plain part. A claim bought to yield a given rate earns that rate for as long as it is held, less whatever the money to hold it costs. Buy an unfunded claim at the two year SPOT rate of 6.25 per cent and carry over one year is 6.25 per cent. Nothing about the rest of the curve changes it.

Roll-down is the part that comes from the shape underneath, not from the claim itself. A year passes and a two year claim has become a one year claim. The claim has not changed hands, its payment date has not moved and its face has not moved. The one thing that has changed is which point of the curve prices the claim: the two year SPOT rate on the day it was bought, the one year SPOT rate a year later. If those two rates differ the price reflects the difference, and that price change is roll-down.

Keeping them apart matters because they have different causes. Carry comes from the rate on the claim against the rate on the money funding it. Roll-down comes from the distance between two rates on the curve. Flatten the curve until every maturity carries an identical rate. Carry survives untouched and roll-down disappears completely, and the panel above does exactly that the moment the two yields are set equal. If a change to the curve's shape moves the number, the number was roll-down.

Here is the everyday version. A woman runs a tea stall on the pavement outside an office building. Some of what she earns comes from being open every morning with a working stove and a stock of milk, and that part barely varies. Some comes from the building's queue reaching her counter as a new floor of tenants moves in, and the queue depends entirely on something outside her stall. Both show up as one number at the end of the month. If she never separates them she cannot tell whether the stove or the building was responsible, so a good month teaches her nothing and a bad month cannot be explained.

Which two rates does a roll-down calculation actually need?

Two, and this is where the whole difficulty of the calculation sits. A roll-down calculation needs the rate the claim was bought at, and it needs the rate the curve fixes at the maturity the claim will have reached. Both have to be rates the curve actually states. Miss either and there is no calculation to run, only an estimate wearing the clothes of one.

Take an invented zero coupon claimA claim that pays one single amount on one single date, with nothing in between, so nothing else about it can move while the curve is being studied. of Rs 1,000.00/- face amountThe single sum a claim pays on its date. Only the price somebody pays today for the right to receive it changes., bought at the two year SPOT rate of 6.25 per cent on annual compoundingInterest reckoned once a year, so a rate of 6.25 per cent turns one rupee into 1.0625 rupees after a year. The same figures on a half yearly basis would give different prices.. DiscountingTurning a future amount into what it is worth today, by dividing it by the growth the rate would have produced over the waiting time. Rs 1,000.00/- for two years at that rate gives Rs 885.813149/-. One year later the claim has one year left, and this curve fixes the one year SPOT rate at 5.90 per cent, so the same Rs 1,000.00/- discounted for one year is worth Rs 944.287063/-.

Both of those are rates the curve states outright. Stating both is the only condition under which this arithmetic is honest. The word for a maturity where a curve actually fixes a rate is a nodeA maturity at which this record states a rate outright. Six of them exist here, and the spaces between them state nothing at all., and a one year roll needs a node at the start and a node at the finish. Six nodes sit on this curve and nothing at all in the gaps between them. The count of nodes decides a great deal about what can be worked on it.

What does the whole year add up to, in points and then in rupees?

In points the split closes exactly and nothing has to be rounded, so work it in percentage points first. The claim cost Rs 885.813149/- and is worth Rs 944.287063/- a year later, a return of 6.601157 per cent. Carry was the 6.25 per cent it was bought at, and subtracting that leaves roll-down of 0.351157 percentage points, or 35.1157 basis pointsOne hundredth of a percentage point. A hundred of them make one percentage point. Swap the two words and a figure moves by a factor of a hundred..

Now the same year in rupees. Carry of 6.25 per cent on a price of Rs 885.813149/- is Rs 55.363322/-, or Rs 55.36/- at the paisa, and roll-down is what is left of the gain, Rs 3.110593/-, or Rs 3.11/-. The two rounded figures come to Rs 58.47/-, and the gain worked straight from the two prices, Rs 58.473914/-, rounds to the same. Both routes agree at the paisa here. The agreement is a fact about these particular numbers rather than a guarantee about any others. Round two components separately and some pair will land a paisa away from the rounded total, which is why the points version leads.

Why does the year come to the FORWARD rate rather than the rate paid?

Because it was always going to. Take the two SPOT rates behind the claim and derive the FORWARD rate they imply. Two years of growth factorOne plus a rate, written as a multiplier. A rate of 6.25 per cent has a growth factor of 1.0625, and joining years means multiplying growth factors rather than adding rates. at the two year SPOT rate of 6.25 per cent is 1.0625 multiplied by itself, or 1.12890625. One year of growth at the one year SPOT rate of 5.90 per cent is 1.0590. The first divided by the second is 1.06601157, so the one year rate one year FORWARD is 6.601157 per cent.

The year returned exactly that figure, to the last decimal, and the match is neither coincidence nor approximation: the price a year later is the face discounted at the one year SPOT rate, the price today is the face discounted at the two year SPOT rate across two years, and dividing one by the other leaves precisely the ratio of growth factors that defines the FORWARD rate. A claim held for a year against a curve that does not move earns the FORWARD rate, not the SPOT rate it was bought at.

A FORWARD rate reads like a forecast to anybody meeting one for the first time. It is not. A FORWARD rate is arithmetic already sitting inside today's two SPOT rates, and it says nothing about what anybody expects. The FORWARD rate turns up for a narrower reason: it is what the year comes to when the curve is held still, and holding the curve still is a condition declared rather than something anybody observed.

There is a trap sitting beside this figure. The one year rate one year FORWARD is 6.601157 per cent and the three year SPOT rate on this same curve is 6.55 per cent. The two sit 0.051157 percentage points apart, or 5.1157 basis points. The two are completely different objects: one is derived from two other rates, the other is stated by the curve, and on any scale wide enough to show the whole curve they sit on top of each other. The risk of confusing them is why every rate in this guide carries the word SPOT or the word FORWARD, every time.

Why is the three year claim's roll-down larger on a smaller step?

Run the same arithmetic a second time, on the only other claim this curve supports. The invented three year claim of Rs 1,000.00/- face, bought at the three year SPOT rate of 6.55 per cent, costs Rs 826.684201/-, and a year later it is a two year claim worth Rs 885.813149/-, the same price the other claim was bought at. The year's total, worked below, is the one year rate two years FORWARD, off the three year SPOT rate compounded across three years divided by the two year SPOT rate compounded across two.

Put the two rolls beside each other now. What comes next looks like a mistake. The two year claim stepped 35 basis points, from the two year SPOT rate of 6.25 per cent onto the one year SPOT rate of 5.90 per cent, and earned roll-down of 0.351157 percentage points. The three year claim stepped only 30 basis points, from the three year SPOT rate of 6.55 per cent onto the two year SPOT rate of 6.25 per cent, and earned 0.602544 percentage points, nearly twice as much. The smaller step produced the larger roll-down, and both figures are correct.

The reason is that roll-down is not the step. Roll-down is what the step is worth once applied to a price, and a rate applies across every year of waiting still left in the claim. After its roll the two year claim has one year left, so its 35 basis point step gets one year of discounting to work on; the three year claim has two years left, so its 30 basis point step gets two. Two years of a slightly smaller step beats one year of a slightly larger one, comfortably. Roll-down depends on the size of the step and on how far the claim still has to run, and reading only the first produces a confident wrong answer.

How is a hold of more than one year worked out?

Hold the invented three year claim for two years instead of one, still against a curve declared to stand still. Year one takes it from the three year SPOT node at 6.55 per cent to the two year SPOT node at 6.25 per cent and returns 7.152544 per cent. Year two takes it from the two year SPOT node to the one year SPOT node at 5.90 per cent and returns 6.601157 per cent. Two years, two returns, both already worked above.

The obvious next move is to add them. Carried at full precision the two returns come to 13.753700 per cent, and that figure is wrong. Money left in place earns the second year's return on the first year's gain as well, so the two growth factors multiply rather than the two rates adding. Check it against the prices, which cannot argue. The claim cost Rs 826.684201/- and is worth Rs 944.287063/- two years later, a total of 14.225851 per cent, or 0.472151 percentage points and 47.2151 basis points more.

The gap is not the size of the error so much as the shape of it: 13.753700 per cent sits in exactly the range a reader expects, and it survives every check except recomputation. It is close to the right answer and it has the right number of digits. AnnualisingTurning a multi year total into a per year figure by taking the root rather than by dividing, so that the years compound the way money actually does. the correct total means taking the square root of the two year growth factor rather than halving the percentage. The root gives 6.876495 per cent a year, and 6.876495 per cent a year is the two year rate one year FORWARD, off the one year SPOT rate of 5.90 per cent and the three year SPOT rate of 6.55 per cent, the same identity one maturity along. The panel above prints both annualising routes side by side.

When does roll-down go below zero?

Roll-down is not a fee anybody collects and nothing in it has to be positive. Roll-down is positive when the curve fixes a lower rate at the shorter maturity. The claim arrives somewhere it is discounted less harshly than it was priced for. On this curve the one year SPOT rate of 5.90 per cent sits below the two year SPOT rate of 6.25 per cent, so the two year claim rolls into gentler discounting and roll-down is positive.

Turn that round. On a curve where the shorter maturity carries the higher rate, a claim rolls onto a rate above the one it was priced at, is discounted harder, and loses on the price. Roll-down takes its sign from the shape of the curve, so a treatment that only ever shows it positive has taught it as a property of bonds when it is a property of a shape. There is nothing pathological about the negative case: it is the same division on a curve arranged the other way.

The control below sweeps the one year SPOT rate from 4.90 per cent to 7.00 per cent, a declared range chosen so the crossing falls inside it and observed nowhere. Holding the two year SPOT rate at 6.25 per cent throughout freezes carry. The roll-down bar shrinks, reaches exactly nothing when the one year SPOT rate meets 6.25 per cent, then grows the other way below the line while the carry reading has not moved.

Two things are worth noticing while the control moves. Nothing on the control touches the rate the claim was bought at, so the carry reading never changes by a digit. At exactly 6.25 per cent the claim is repriced at precisely the rate it was priced for, so roll-down is not nearly nothing but nothing. Neither this control nor the split itself can say how much of the year a move in the curve would take away, and that silence is where a carry and roll figure is most often misread.

Which claims on this curve can be rolled at all?

A one year roll needs the SPOT rate at a maturity one year shorter than the claim. Needing a rate one year shorter disqualifies most of what could be priced here. A roll exists only where two of the six maturities sit exactly one year apart, and two such pairs do: two years with one year, and three years with two years. There are exactly two one year rolls on this curve and there is no third.

The problem is not the price, and saying so is worth a sentence. Every node has a price, and the table below prints five of them. The missing rate is the one at the far end of the roll: a four year SPOT rate, a nine year SPOT rate, a twenty nine year SPOT rate. None is fixed here, and none is manufactured. An account that quietly draws a four year SPOT rate off the line between the three year and five year points has invented the answer it then teaches.

How does anybody actually use this split?

Somebody reporting on a year of a bond position has one number to explain and several people asking where it came from. Splitting it into carry and roll-down turns one unexplainable result into two explainable ones. The coupon is fixed and the funding rate is agreed, so carry was largely settled on the day of purchase. Roll-down was not. Roll-down depended on where the curve's shorter end sat when the year ended, and nobody chose that.

The same split shows up in a household without anyone naming it. A person who buys a three year fixed deposit knows what it pays, and that is the carry half. Something held that might be sold before it matures pays instead whatever rate is applied at that shorter length of time on the day of sale, and no promise covers that rate. The lesson is not which to prefer, but that the second has two moving parts and the first has one.

An analyst reading somebody else's reported year uses it in reverse. A return well above the rate the position was bought at came from the curve's shape or from the curve moving, and the two are worth separating before anybody calls the result skill. A return below that rate has negative roll-down, or funding above the coupon, among its ordinary explanations rather than as a sign that something has gone wrong. In both directions the split asks a better question and settles nothing about what anybody should hold.

What does the whole tool assume, and what cannot be known from it?

Every figure in this guide rests on one supposition, declared here rather than observed anywhere: the curve does not move for the length of the hold, unless a move is declared on the panel and priced there. Nobody can say how likely a still curve is, and the supposition makes no claim that it is likely. A still curve is the condition under which carry and roll-down mean anything at all, and the condition belongs beside every figure they produce rather than in a note underneath.

Two questions follow, and neither can be answered here. How often a curve does stand still needs a history of curves, and there is none. Which yield move ought to be tested needs a view about what is coming, and the panel prices any declared move while saying nothing about which to declare. A blank cell that says why it is blank is an answer and a filled one would be a fabrication, so both are drawn below as empty cells with the reason written inside them.

The six node SPOT curve and the claims priced against it are invented.
Educational material. Not advice on any investment, tax, budget or market position.