Carry: What You Earn Simply for Holding, at an Unchanged Yield

Carry is what a holding earns across a stated period when its yield finishes the period exactly where it started. Hold a bond bought at par for a year on those terms and the coupon is the whole of it. Hold a bond that pays nothing at all and no cash arrives, yet the price climbs by the same rate.

The mechanism underneath that is short enough to state in one breath. A bond is worth whatever its remaining payments discount to. Let a day pass and every one of those payments is a day nearer, so at an unchanged yield the discounting is applied across a shorter stretch and the value rises by itself. Carry is that rise, plus whatever cash landed in the holder's hands along the way. Nothing else is in it.

Everything below is built rather than quoted. Two invented instruments are worked through the same twelve months, every sum is set out with its terms visible so it can be rerun, and the yield is pinned in place to show what the calendar alone does. A total nobody can reproduce teaches nothing at all, so both instruments discount on a once a year clock, and each sum states it.

What is carry, and over what stretch of time is it measured?

Carry is the return a holding produces over a period on the assumption that nothing about its yield changes across that period. The assumption at the end of that sentence is doing all the work. Read it twice. A carry figure is a counterfactualA statement about what would have happened under a condition that did not actually hold. A counterfactual is measured rather than predicted, and it stops being true the moment the condition breaks., not a prediction: it measures what a holding would earn if the only thing that moved was the calendar.

The distinction between a measurement and a prediction matters more than it looks. Somebody who quotes a carry figure is not stating what a holding will earn. A carry figure hands over a decomposition. A period's return has two sources, and only two: the time that passed, and the yield that moved. Carry isolates the first and sets the second to nothing. Whatever a holding actually earns beyond its carry is attributable to the yield having moved, and whatever it falls short by is attributable to the same thing.

Notice the second thing that sentence contains. It is a period. Ask somebody what a bond's carry is and the honest reply begins with a question back: over what stretch of time? A carry figure with no period attached is not a quantity at all, in the same way that a speed with no unit of time in it is not a speed. Every figure in this guide is stated for twelve months, and every one of them says so.

Stripped of the vocabulary, this is something most households already act on without a second thought. Money goes into a deposit that pays nothing until the day it matures. Asked six months in whether the money earned anything, anybody in that household answers yes without a pause, even though not one rupee has arrived. Nobody finds a deposit confusing on this point. A bond earns in exactly the same way, and for some reason the point stops being obvious the moment the word bond is involved.

What does a bond bought at par earn over a year if its yield does not move?

Take the simplest case available and work it all the way through. Bond A is an invented ten year bond carrying an 8.50 per cent annual coupon on Rs 1,000.00/- of face amount, and it is bought at par. Buying at par means Rs 1,000.00/- is paid for it. Because the price paid equals the amount repaid, the yield on it is 8.50 per cent a year as well. The 8.50 per cent is now pinned there for the whole of this exercise. The yield is an input supplied to a sum, never a level read off any market.

Now move the calendar forward by exactly one year and change nothing else. Two things happened across those twelve months. A coupon of Rs 85/- was paid, being 8.50 per cent applied to the Rs 1,000.00/- of face amount. The bond also aged from a ten year bond into a nine year one. So the whole question is what the nine year version is worth at the same 8.50 per cent yield.

Evaluated separately, the two halves give an answer worth pausing on. The nine coupons still to come discount to Rs 520.1203/-. The Rs 1,000.00/- of face amount, nine years out, discounts to Rs 479.8797/-. The two halves add to Rs 1,000.0000/-, the price the bond started at. The price did not move at all, and that is not a coincidence: a bond whose yield equals its contracted rate is priced at its face amount at every remaining life, so shortening the life by a year changes nothing whatsoever.

So the whole of the year is the Rs 85/- of cash. Rs 85/- set against the Rs 1,000.00/- that left the buyer's hand is 8.500000 per cent for the year, and the baseThe amount a ratio is divided by. Change it and the very same rate turns into a different quantity of money. of that ratio is the price paid rather than the face amount, which on this one instrument happen to be the same figure. Carry for the year on Bond A is therefore 8.500000 per cent, made up of Rs 85/- of cash and a price change of exactly Rs 0.0000/-.

A trap sits inside those two component figures, and it is the sort of thing that ruins an otherwise sound calculation. Rs 520.1203/- and Rs 479.8797/- also split Bond A's price at the very start, before any year had passed, into the first nine dates and the tenth. Same two numbers, and they close on Rs 1,000.0000/- either way. But they mean two completely different things. One split separates coupons from the amount repaid. The other separates the dates before the end from the final date, and that final date carries a coupon and the face amount together. Two splits that both close on the same total are still two different splits, so the label on a split is the part that has to be checked.

Why does the period have to be named, and what clock is the counting done on?

Every price above was struck with the discounting applied once a year. Treating that as housekeeping, or relegating it to a footnote, breaks the sum for anybody trying to check it. The clock is an input to the sum in exactly the way the 8.50 per cent is, and leaving it out makes the arithmetic unreproducible. On this clock, 8.50 per cent a year means one single division by 1.085 for every year that goes by, with no second division hiding inside any of those years.

Now watch what that does to the period question. If carry for twelve months is 8.500000 per cent, what is carry for six months? The number a reader reaches for is half of 8.50, or 4.250000 per cent. Half is wrong, and the reason is the clock. Six months of growth on a once a year clock is the square root of 1.085, or 1.04163333. Six months of carry is therefore 4.163333 per cent. Two of those six month steps multiply back to 1.085 exactly and give 8.500000 per cent for the full year. The check proves the smaller figure is the right one.

The gap between the two readings is 0.086667 percentage points, or 8.6667 basis points. The two numbers are two units for one distance rather than two facts. A hundred basis points make a percentage point. Sliding between the two units is how a small difference gets written down as a large one, and that is why they are worth keeping apart.

Put it in rupees on an instrument and it stops being abstract. Bond B, worked properly further down, is worth Rs 559.4640/- today. Six months on at an unchanged yield it is worth Rs 582.7564/-, so those six months added Rs 23.2924/-. Half of the full year's gain of Rs 47.5544/- would have been Rs 23.7772/-. The second half of a year compounds on a larger amount, so at an unchanged yield the first half earns less than the second. Halving the rate quietly denies exactly that.

Why does a bond that pays nothing at all earn the same rate?

Bond B is an invented bond that pays no coupon on any date. Bond B repays Rs 1,000.00/- once, after 7.1191 years, and hands over nothing at all before then. The same 8.50 per cent yield as Bond A is read on it, with the discounting ticking once a year for it too. Its price today is Rs 1,000.00/- discounted across 7.1191 years, or Rs 559.4640/-.

A reader who checks that price needs one note about it first. Bond B's maturity appears in this record both as 7.1191 years and unrounded as 7.11906264 years. Priced on the shorter figure the answer is Rs 559.4640/-, printed as Rs 559.46/-. Priced on the longer one the answer is Rs 559.4657/-, printed as Rs 559.47/-. The whole of the Rs 0.0017/- between them comes from how many decimals were carried in the exponent and from nothing else. Both readings appear here together for a plain reason: where the entire claim is that the arithmetic can be reproduced, a reader who types the shorter maturity into a calculator and lands a paisa away from a printed figure must be able to see why, rather than assume the fault is theirs.

Now run the same twelve months. Nothing is paid, so the cash column is empty. The maturity is what changes. The bond that had 7.1191 years to run now has 6.1191 years to run, and Rs 1,000.00/- discounted across 6.1191 years at the same 8.50 per cent is Rs 607.0184/-. The holding is worth Rs 47.5544/- more than it was, and Rs 47.5544/- set against the Rs 559.4640/- that was paid for it is 8.500000 per cent for the year.

The same 8.500000 per cent came out of an instrument that paid not one rupee across the whole twelve months, and that is the single most useful fact about carry. It is not an approximation and it is not close to Bond A's figure. Bond B's rate is identical, reached by a route with no cash in it anywhere. And once the arithmetic is set out, the reason is almost embarrassing in its simplicity. Dividing by 1.085 one time fewer is the same as multiplying by 1.085, so the price a year on is the price today times 1.085. The gain is then 8.50 per cent of the starting price by construction rather than by luck.

The cancelling is worth naming as forced arithmetic rather than as a happy result. The face amount, the number of years and the size of the price all disappear from the ratio, so the answer cannot be anything except the yield. Which also means the answer never changes: the first year of the holding, the last year, and every year in between all read 8.500000 per cent. The number of rupees that rate stands for does change, and changes a great deal.

Is carry just another word for the coupon?

No, and the two worked cases already on the table settle it between them. On Bond A the coupon turned out to be the whole of the carry. On Bond B the carry was the whole of the price movement and the coupon was nothing at all, because there is no coupon. Two instruments, one rate, and the coupon accounted for everything in one case and nothing in the other.

The reason the two coincided on Bond A is not that carry means coupon; it is that the bond was bought at par, so its price had nowhere to travel as it aged. Take that condition away and the two figures separate immediately. Think about what has to be true on the last day of any bond's life: whatever was paid for it, it repays its face amount and no more. So a bond bought for more than its face amount has a price that must come down to the face amount by the end, and a bond bought for less has a price that must climb up to it. At an unchanged yield, some of that journey happens every single year.

Which means a bond bought above its face amount has a carry that is its coupon minus the slide back towards the face amount. The coupon still arrives in full, every rupee of it, on the date it was promised. But the holding is worth slightly less at the end of the year than at the start, and the year's carry is the two effects netted against each other. A bond bought below its face amount runs the other way: its carry is the coupon plus a climb.

The direction each price travels is one question; how far it travels is another. Putting a number on how far a price moves is a different measurement with its own machinery, and it is covered separately. The direction belongs here. Naming it stops anybody writing the coupon down as the carry and walking away.

Why is a rate not an amount, and what happens when the base changes?

Both bonds earned 8.500000 per cent across the same twelve months. Now look at what that rate stood for in money. On Bond A it was Rs 85/- of cash on a Rs 1,000.00/- base. On Bond B it was Rs 47.5544/- of price on a Rs 559.4640/- base. Same rate, and Rs 37.4456/- between them.

8.50 per cent of Rs 1,000.00/- and 8.50 per cent of Rs 559.4640/- are the identical rate and are nowhere near the same amount of money. Name the base of the ratio and the period of the rate in the same sentence, every single time. A carry figure is usually quoted as a bare percentage with no base attached to it at all, so carry is where that discipline gets broken most often. Handed such a figure, an analyst can work out precisely nothing in rupees. The rate is not yet a quantity.

There is a second difference between those two amounts, and it is not about size. Bond A's Rs 85/- is cash. The cash arrived, it can be spent, it can be put somewhere else. Bond B's Rs 47.5544/- is an unrealisedWorth something on paper and not yet turned into money. An unrealised gain is real in what a holding is worth and cannot be spent until the holding is sold or repaid. gain: it is entirely a price the holder has not received and cannot use for anything until the bond is sold or repaid. The rate does not know the difference. Carry is a rate of return, and a rate of return does not care whether the return arrives as cash or as a higher price. Whether the holder cares is a different question, and it belongs to whoever has bills to pay.

What do the two bonds look like set out side by side?

Here is the whole exercise in one place. Both columns cover the same twelve months, both are struck on a once a year discounting clock, and the yield is held at 8.50 per cent a year from the first row to the last. Holding the yield still is the assumption that makes any of it a carry calculation rather than a description of something that happened.

The whole argument sits across the bottom two rows of the table. One instrument paid Rs 85/- and did not move, the other paid nothing and moved Rs 47.5544/-, and the twelve months came to the same rate for both. Then read the very last row, the one most likely to be skipped. Neither column has a cost of funding in it. Both figures are gross, and what a gross figure leaves out is set out below, along with the reason the row is drawn empty rather than filled with a zero.

What is missing from every carry figure?

Every figure above is a gross figure. The word gross is doing real work and it is not a hedge. Carry is very often written as what a holding earns after what it costs to fund it, and the second half of that sentence needs a number the arithmetic above never supplies: no funding rateWhat somebody pays to borrow the money used to buy a holding. A funding rate is set in a market, it moves, and no level for it is assumed above., financing cost or borrowing rate of any kind. Not a low one, not a high one. None.

So the row where that cost would sit is drawn empty, with the reason written inside it, and it is not filled with a zero. The difference between those two choices is the whole of the honesty here. An empty row says nothing is known about what funding this holding costs. A zero would say funding it costs nothing. Costing nothing is a claim, and a false one. Anybody who has ever borrowed to buy anything knows which of the two is nearer the truth.

The absence of a funding rate means something in practice, and it is worth stating plainly. A holding funded at a rate above its own yield produces a negative net carry out of exactly the gross arithmetic shown here. The funding cost is subtracted afterwards and never enters the price, so the gross figure stays 8.500000 per cent for the year in both columns whatever the funding costs. Which side of that line any particular holding sits on cannot be settled from the gross arithmetic alone: it takes a funding rate, and no funding rate appears here.

Who actually reaches for a carry figure, and what do they do with it?

The household version is the same idea without the vocabulary. Start there. A household holds a deposit that pays nothing until it matures and asks whether it is worth breaking early to fund something else. The honest way to answer that is to work out what the deposit produces over the stretch of time in question if nothing about the terms changes, and set it against what breaking it costs. The comparison is a carry calculation. Nobody in the household calls it one.

The same shape shows up in three places where money is at stake. Somebody deciding whether to keep a holding for another month wants to know what the month produces if the market simply does nothing. That figure is the hurdle any alternative use of the money has to beat. Somebody explaining last quarter's result after the fact wants to split what was earned into the part the calendar produced and the part the market produced. Only the market's part tells anybody anything about a decision. And somebody comparing two holdings of different sizes needs the rate rather than the rupees. The base gets named every time for that reason.

In each of those three uses the carry figure is a benchmark rather than a forecast. Its assumption makes it fit for exactly that. Asked what a holding would earn if nothing happened, carry gives a number worth having, precisely because the answer is almost never zero. Asked what it will earn, carry is being put to a question it was never built to answer.

Two habits go with using it properly. The first is stating the period, and annualisedRestated as a rate for a full year so that stretches of different length can be set beside each other. The restating is arithmetic, not a promise about a year. is not a substitute for it: a month's carry annualised is a month's carry, restated, and it does not become a year's worth of anything by being written that way. The second is knowing at what value the holding is being carried in the books in the first place. A return computed on a price is a different object from one computed on a mark to marketRestating what a holding is worth at a price taken from the market rather than at what was paid for it. valuation or on a cost figure. Which of those applies is set by an authority rather than by arithmetic, and the block below says where to read it.

What can a carry figure never tell an analyst?

Carry holds the yield still, and a yield held still is the one thing that never actually happens. Holding the yield still is not a weakness in the measure. The stillness is the definition of carry, and it is where the measure stops.

So carry is not a forecast of a return and cannot be treated as one. Carry cannot be added to a view about where rates are going to produce an expected return. The two do not sit side by side. The moment the yield moves, the price effect swamps the arithmetic, as the Rs 20.6309/- against Rs 85/- above shows for a yield move of only 1.10 percentage points. Carry is a decomposition of a period that has already happened, or a measurement of a period under a stated condition. Nothing more.

There are three further things it does not contain, and each is missing for its own reason. Carry contains no cost of funding, and no rate for funding is assumed above. Carry contains no adjustment for a general rise in prices, so a carry figure is never a real yieldA yield with the effect of rising prices stripped out of it. Carry is measured before that stripping, so carry and a real yield are different objects.. And it contains no allowance for the borrower failing to pay, a separate subject with its own arithmetic.

One last discipline governs every rate above. The 8.50 per cent used throughout is a yield on an invented instrument, supplied as an input to a sum. The 8.50 per cent is not a level read off the invented SPOT curve used elsewhere, where a SPOT rate attaches to one future date and runs from today out to it. Nor is it a FORWARD rate. A FORWARD rate fixes a rate now for a period whose first day is still ahead, and no FORWARD rate is worked above, for want of the two SPOT rates that would have to stand behind it. Keeping those three objects apart by name is not pedantry: a reader who meets an unlabelled rate has no way of knowing which of the three they are holding.

Bond A and Bond B are invented.
Educational material. Not advice on any investment, tax, budget or market position.