The Forward Rate: The Rate Implied for a Future Period

A forward rate is the rate attaching to a stretch of time that has not started yet, and nobody quotes one: two SPOT rates already contain it. On the invented curve, the one year SPOT rate, standing at 5.90 per cent, and the two year SPOT rate, standing at 6.25 per cent, between them fix the one year rate one year FORWARD at 6.601157 per cent, on annual compounding.

Underneath that sits one very ordinary observation. Two ways of putting money away can finish on the same calendar date: one placement running the whole distance, or a shorter placement followed by a second one. If the rate on the whole distance and the rate on the first leg are both settled today, then the rate on the second leg is settled today as well. Nobody gets to choose it. The rest is arithmetic on that one observation.

What is a forward rate, and which two dates does one always carry?

Start with the thing a forward rate is measured against. A SPOT rateMoney handed over today and returned in a single lump on one named date later, with the whole of the interest settled by that one rate. is what a lender is paid for handing money over today and getting it back, all at once, on one named date in the future. A SPOT rate has one end fixed to today, so a single number of years describes it completely. Everybody already knows when it starts.

A FORWARD rate is the rate attaching to a stretch of time that has not begun yet, so it needs two pieces of information rather than one: how long it runs, and how far ahead it starts. Drop either one and what is left is not a rate at all. Somebody who quotes "the one year forward rate" has stated a length and nothing else, and a one year period starting twelve months from now is a different object from a one year period starting three years from now, carrying a different rate.

Every forward rate below is written in one fixed word order: the length first, then how far ahead the period begins. The one worked hardest here is the one year rate one year FORWARD. The period runs for one year, starting twelve months from today and finishing twenty four months from today. The two year rate three years FORWARD appears later, starting thirty six months from today and running until sixty months from today. Once those names are read slowly, the word order stops feeling awkward.

Think of it the way a landlord thinks about a room. Saying "the rent is Rs 18,000/- a month" is complete. Saying "the rent for a year" is not. A year beginning this April and a year beginning three Aprils from now are two different lettings, and nobody would quote them at the same figure without saying so. A forward rate is the same shape of statement: the length alone never identifies the thing being priced.

Every sum below depends on one convention, so it is settled first. Every rate in this guide is stated on annual compounding: interest is added once a year and then earns alongside the money that produced it. The convention sits inside the arithmetic rather than beside it. None of the amounts below can be reproduced without knowing it. And where distances between rates are quoted, one basis point is one hundredth of a percentage point, so a gap of 5 basis points is a gap of 0.05 percentage points. The two units are not interchangeable and neither is ever swapped for the other here.

If nobody quotes a forward rate, where does one come from?

One question trips people up. If the one year rate one year FORWARD appears on no screen anywhere, quoted by nobody and published by nobody, how can a six decimal figure be put on it? The answer is that the figure was never separate from the curve. The forward rate is impliedPinned down by working on figures already available, so nobody had to quote it and nobody had to guess it. by two rates that are already there, in the way that knowing the total of a bill and knowing what one item cost fixes what the other item cost. Nobody had to publish the second item.

Set two journeys side by side. Both start with Rs 1,000.00/- today and both finish exactly twenty four months from today.

The first journey is one placement running the whole distance: Rs 1,000.00/- placed today for two years at the two year SPOT rate, recorded on this curve at 6.25 per cent. The two year rate covers the whole span, so where the money lands is settled the moment the placement is made.

The second journey is two placements in a row: Rs 1,000.00/- placed today for twelve months at the one year SPOT rate, recorded on this curve at 5.90 per cent, and then whatever that produces placed again for the second year. The first leg is settled today too. The second leg is the only unknown in either journey.

Now put the two journeys next to each other. Both begin on the same day with the same amount. Both end on the same day. Three of the four ingredients are fixed by rates the curve already records. So there is exactly one rate the second leg can carry that makes the two journeys finish level, and that rate is the one year rate one year FORWARD. The rate did not come from a view about the future. The requirement behind it is that two things which start together and end together should not end in two different places.

The word implied earns its keep right there. A rate that is implied has been pinned down by arithmetic on figures already visible. Nobody quoted it, nobody forecast it, and nobody was asked their opinion. The forward rate sits inside the term structure of interest ratesThe whole set of rates for different lengths of time, standing side by side as one object rather than as separate quotations. that these rates make up, and it is settled entirely by what that structure already says.

How do two SPOT rates fix the one year rate one year FORWARD?

People believe money, so the money version comes first. Rs 1,000.00/- placed at the one year SPOT rate, standing at 5.90 per cent, on annual compounding, stands at Rs 1,059.000000/- after twelve months. The same Rs 1,000.00/- placed at the two year SPOT rate, standing at 6.25 per cent, stands at Rs 1,128.906250/- after twenty four months. Both of those come straight off the invented curve, with no forward rate anywhere in sight yet.

Now ask the only question left. Which rate carries Rs 1,059.000000/- across the second year and lands it on Rs 1,128.906250/-? The second year added Rs 69.906250/-, and it added that on a base of Rs 1,059.000000/-, not on the original Rs 1,000.00/-. Setting the addition against the base it grew from gives 6.601157 per cent, the one year rate one year FORWARD. One division is the whole derivation.

Notice what just happened. Most readers skip it. The forward rate was found by a division, not a subtraction. Rs 69.906250/- on its own is a gain in rupees and it is not a rate until it is set against what produced it. Taken as a ratio instead, the two amounts give 1.06601157, and stripping the 1 off the front leaves the rate itself. Same operation, written two ways.

Now the same thing in rates. The rate route carries to any pair of nodes on the curve. A growth factorWhat one rupee multiplies by across a stated stretch of time. One plus the rate, raised to the number of years. is what one rupee turns into over a stated stretch of time, and it is simply one plus the rate raised to the number of years. The two year growth factor here is 1.0625 squared, or 1.12890625. The one year growth factor is 1.0590. Dividing the first by the second gives 1.06601157, exactly the ratio the money route produced. The Rs 1,000.00/- was common to both journeys and cancelled itself out.

The equation has one unknown in it, so it can be rearranged straight away. Divide both sides by the one year growth factor and the forward rate is alone on the left.

Both routes give 6.601157 per cent and neither is the shortcut. The agreement is not a coincidence and it is not corroboration either. The arithmetic forces it: the Rs 1,000.00/- appears in the numerator and the denominator of the money route and cancels out entirely, leaving nothing but the ratio of the two growth factors. The money route run with Rs 40,000/- or with Rs 4/- lands on 6.601157 per cent again.

The general form matters more than the particular one. The curve has segments of one, two, five and twenty years on it, and none of the longer ones can be handled by the two year version above. The only change is that after the division the growth factor covers several years, so the root matching the length of the period is what converts it back to a rate.

Why does the forward rate land above both SPOT rates it was built from?

Put the three figures in a column and something looks odd at first. The invented curve puts the one year SPOT rate at 5.90 per cent and the two year SPOT rate at 6.25 per cent. The one year rate one year FORWARD is 6.601157 per cent, higher than either of the two rates it was derived from. Nothing was added to produce that. No premium was charged, nobody demanded compensation, and no view about the future was inserted.

The two year SPOT rate is what the whole two year stretch earns on average, so if the first year earns less than that average the second year has to earn more, and the second year has to carry the entire shortfall on its own. The explanation is complete, and it also settles the direction in every other case.

Work the sizes. The first year sits 0.35 percentage pointsThe unit a rate itself is measured in. A move from 6.25 to 7.25 is one percentage point. It is a different unit from a per cent change in a quantity. below the two year SPOT rate: 6.25 less 5.90 is 0.35. The second year sits 0.351157 percentage points above it, and 6.601157 less 6.25 is 0.351157. The two distances are nearly the same size and only nearly, and the small difference between them is 0.001157 percentage points. The near-equality is not decoration. It is the reason a bad shortcut survives for years, and that failure is examined further on.

Generalised, the direction gives a rule worth keeping. Where the curve rises from one node to the next, the later stretch carries the whole of the rise, so every forward segment reads above both SPOT rates that produced it. Where the curve is flat between two nodes, no rise is left for anything to carry, and the forward reads exactly the same as both of them. And where the curve falls from one node to the next, the forward drops below both, by the same logic run in reverse. All three appear in the simulation below. Its control runs across a range wide enough to include the flat case.

How close does the forward sit to the three year SPOT rate, and does that closeness mean anything?

The sharpest trap on the invented curve was left in deliberately rather than engineered away. Take the forward just derived, 6.601157 per cent. Take the three year SPOT rate recorded on the same curve, 6.55 per cent. Subtract one from the other and they sit 0.051157 percentage points apart, a distance of 5.1157 basis points. A screen carrying two decimal places would print 6.60 beside 6.55, close enough that a tired reader treats the pair as one thing.

They are not the same thing, they are not comparable, and the closeness of the two numbers means precisely nothing. Look at what each one covers. The forward covers a single year, and that year begins twelve months from today and ends twenty four months from today. The three year SPOT rate covers three years, and those three years begin today. One of them describes twelve months of the future that has not started; the other describes thirty six months starting this morning. They share neither a start date nor a length.

Why does this happen at all? Because a smooth curve makes it almost unavoidable. Forward rates are pulled out of the same set of nodes that the SPOT rates sit on, so a forward for some segment will nearly always land somewhere near a SPOT rate for some maturity. The near-landing is a property of smoothness rather than a discovery about anything. Treated as evidence, it is a coincidence read as a finding.

The defence is not cleverness. The defence is labelling. Every rate in this guide carries either the word SPOT or the word FORWARD, in the drawings, in the captions and inside the quiz feedback, and a rate that carries neither is not a figure anybody can use. Think of a plumber's van holding two spools of pipe, both looking identical from a metre away, one rated for drinking water and one not. Nobody solves that by learning to tell them apart by eye. Plumbers solve it by writing on the spool.

Is a forward rate a forecast of what a rate will be?

No, and this is the misreading that does the most damage. A forward rate looks exactly like a prediction. The rate carries a future date, a specific figure to six decimal places, and a place on a curve that people talk about as though it knew something. Everything about its appearance suggests somebody looked ahead and formed a view.

Nobody formed a view: the one year rate one year FORWARD of 6.601157 per cent is what makes two journeys already available today finish level, and that is arithmetic on two numbers a reader can see rather than an opinion about the second year. The two SPOT rates it came from were not forecasts either. Both are rates at which money can be placed today for one year and for two years.

The proof is that nothing needs to move for the check to work. Rs 1,000.00/- placed today for two years at the two year SPOT rate, standing at 6.25 per cent, returns Rs 1,128.906250/- on the stated date, and that amount was fixed the day the placement was made. Whatever the one year SPOT rate turns out to be twelve months from now, lower or higher or unchanged, that placement returns exactly what it always returned. The forward rate was never a promise about the second year. The forward rate described what today's two SPOT rates already contain, and a description does not stop being accurate when the future disagrees with it.

There is a second question hiding behind the first. Even granting that a forward rate is arithmetic rather than a forecast, a reader might still ask whether part of the gap between a short rate and a long one is payment for the discomfort of committing money for longer. The quantity has a name, the term premiumWhatever portion of a longer rate is payment for committing money for longer, rather than a statement about where rates go. It is covered separately., and separating it from anything else inside a forward rate is covered separately. Where the whole of a forward rate comes from is settled above. How it splits into parts is a separate question.

How does the whole SPOT curve break into a chain of forward periods?

Everything so far used two nodes. The invented curve has six, and the payoff of the derivation is what happens when all six are run. Six recorded SPOT rates give five forward segments, one for each gap between adjacent nodes, and the chain that results is not an approximation of the curve. The chain is the curve, written a different way.

Work it as one table, starting from Rs 1,000.00/- and using nothing but the recorded SPOT rates. Each row grows the money to a node using that node's SPOT rate, then reads off the forward segment implied between the previous node and this one.

Every rate in the right hand column was produced by dividing the growth figure on its own row by the growth figure on the row above it, then taking the root that matches the number of years the segment spans. The two year rate three years FORWARD, for instance, is Rs 1,396.009990/- divided by Rs 1,209.651761/-, square rooted for a segment two years long, less one, and it reads 7.427157 per cent. Nothing was quoted from anywhere and nothing was assumed.

Running the chain in the other direction is the check that matters. Rs 1,000.00/- grown one year at 5.90 per cent reaches Rs 1,059.000000/-. Grown a further year at 6.601157 per cent it reaches Rs 1,128.906250/-, exactly what the two year SPOT rate gives on its own. A further year at 7.152544 per cent reaches Rs 1,209.651761/-, matching the three year SPOT rate. Applying 7.427157 per cent twice reaches Rs 1,396.009990/-, matching the five year SPOT rate. Applying 7.801894 per cent five times reaches Rs 2,032.452889/-, matching the ten year SPOT rate. Applying 7.725218 per cent twenty times lands on Rs 9,002.603850/-, precisely what a single placement at the thirty year SPOT rate, standing at 7.60 per cent, gives.

A recorded SPOT curve and a chain of forward segments hold the same information written two ways, so the chain closes at every node with no residual anywhere. Careful rounding did not produce that. Each forward segment was defined as whatever makes those two routes agree, so agreeing is the only thing it could possibly do.

One more thing this table quietly proves. Forward periods do not have to start where the previous one ended, and the curve fixes overlapping ones too. The three year node and the one year node are both recorded, so the two year rate one year FORWARD is fixed as well: divide Rs 1,209.651761/- by Rs 1,059.000000/-, take the square root because the span is two years, subtract one, and it reads 6.876495 per cent. The two year rate one year FORWARD overlaps the one year rate one year FORWARD entirely, and neither contradicts the other. The curve is simply being asked two different questions.

Which forward rates does this curve fix, and which does it not contain at all?

Six recorded nodes give five forward segments, and here is the part that gets glossed over: the segments are whatever lengths the gaps between the nodes happen to be. They are not a tidy set of one year steps. On this yield curveThe picture drawn when a set of rates is plotted against the length of time each one runs for. It is covered separately. the gaps run one year, one year, one year, two years, five years and twenty years, so the forward segments come out at exactly those lengths and no others.

The awkward case follows from that. A SPOT rate is recorded at three years and another at five years, so the curve fixes the two year rate three years FORWARD at 7.427157 per cent. Now suppose the one year rate three years FORWARD, or the one year rate four years FORWARD, is wanted separately. A SPOT rate at four years was never recorded, so the curve fixes neither of them: without that node there is nothing to divide by.

The tempting move is to reach for the two year figure and use it for one of those single years. It does not work. The 7.427157 per cent was solved for a two year span, and it says nothing whatever about how the return is distributed inside the span. It is consistent with a fourth year at one level and a fifth at another, and it is equally consistent with both years being identical, and the curve has no opinion between those pictures. Splitting the segment requires an assumption about where inside the two years the return sits, and that assumption belongs to whoever makes it rather than to the curve.

Think of it as a taxi fare for a two stop trip. The driver quotes Rs 340/- for the whole journey. The quoted fare says nothing reliable about what the first leg alone would have cost, and dividing by two produces a number that feels like an answer and is not one. A price for the first leg has to come from somebody who priced it.

The absence is better stated than filled. A reference that hands over a rate for a period the curve never fixed has invented a number, and an invented number that arrives dressed as a derivation is worse than an honest blank. Nobody downstream can tell the difference.

Two things are worth watching as the control moves. The first is the pace. The second year alone absorbs a rise spread over two years, so a shift of 5 basis points in the control moves the forward by roughly twice that. The second is the landmark at the very bottom of the range. At a control setting of 5.90 per cent the first two years are flat, so the marker sits exactly on the line where the forward and the control read alike, and the two bars stand at the same height. The flat setting is the only place on the whole control range where a forward rate and the SPOT rate behind it read the same, and every step away from it opens a gap that grows about twice as fast as the input.

How is a forward rate computed, step by step?

Here is the whole procedure in the order it is actually performed. The procedure is short, and only one step in it is regularly skipped.

Step one carries the weight. A forward period with only one end sitting on a recorded node cannot be computed at all, and that is where a reader who cannot find a recorded node reaches for something nearby instead. Steps two, six and seven are all easy to get wrong as well, but each of them produces a figure that looks visibly odd: the wrong compounding basis or the wrong root gives an answer that is obviously too big or too small, and a missing label gets caught the first time somebody asks what period the rate covers. A missing node produces a number that looks perfectly reasonable and is not in the curve at all.

Where does this arithmetic actually get used by somebody at work?

Take a treasury officer at a mid sized manufacturer who knows a payment of Rs 8 crore falls due in three years and has cash arriving in one. The question in front of them is not what rates will do. The useful question is what the market already implies about the stretch between year one and year three. The comparison shows whether committing now for three years or committing for one and again for two leaves them in the same place. The two year rate one year FORWARD of 6.876495 per cent, derived from the one year and three year nodes, is exactly that comparison, and it is available without anybody forecasting anything.

An analyst covering a lender uses the chain in a different way. When a lender books a five year loan funded by one year deposits rolled over, the arithmetic of the roll is the forward chain: the segments between one year and five years say what the funding has to cost, on today's curve, for the loan to keep the margin it was written at. The forward chain turns a vague worry about refinancing into a specific set of rates that can be written down and checked, and that is most of what makes it worth learning.

A household version exists too, and it is closer than it looks. Somebody choosing between a five year deposit and a one year deposit they intend to renew four times is choosing between exactly the two journeys this guide opened with. The forward chain is what the second choice needs to earn on each renewal for the two to finish level. The forward chain does not pick between them. The arithmetic turns a preference into a comparison.

All three have the direction of the question in common. None of them asks the curve what will happen. Each asks what today's benchmark government curveThe reference set of government rates a market uses as its yardstick. How one is assembled and made public is settled by an authority rather than by arithmetic. already implies, then compares that against something they have to decide anyway. When somebody says a forward rate looks high or low, the useful reply is to ask what it is being compared against. The forward on its own is not a judgement about anything.