Spot Rate and Forward Rate: Two Rates on the Same Curve

A SPOT rate runs from today to one future date and discounts the single amount arriving then. A FORWARD rate runs between two future dates, touching today at neither end, and nobody quotes it: the two SPOT rates on either side of that stretch already fix it. Deriving one uses nothing except the set of SPOT rates itself.

Underneath that sits one requirement, and everything else here follows from it. Money lent for two years, and money lent for one year and then lent again for a second, have to finish in the same place. If they did not, anybody could take the cheap route and give away the dear one until the two matched. The requirement has a name, no arbitrageA pricing requirement with one idea in it. Somebody would buy the cheap way and sell the dear way until they met, so two ways of getting to the same place at the same time cannot cost different amounts., and it leaves nothing to choose. Given a one year SPOT rate and a two year SPOT rate, the rate for the second year is settled, not estimated. A FORWARD rate is therefore not pulled out of a market that quotes it; it is solved out of a market that quotes something else.

What does a SPOT rate actually cover?

One amount, one arrival date, one rate. Those three are the whole of it. A SPOT rate starts its clock now and stops it on the day the money turns up, and the only job it has is to bring that single amount back to what it is worth today. Nothing else is inside it. Neither the days before today nor the days after the arrival date sit inside its span, so a SPOT rate has nothing to say about either.

The property that makes a SPOT rate usable is that it needs no other rate to do its work. Hand somebody one amount and one date, and one SPOT rate values it. There is no assembly required and no second figure to look up. Most other rates cannot do that alone. A set of SPOT rates is therefore the object every discounting job reaches for first.

A full set of them across the maturities is what people are pointing at when they say the curve, and taken as one object rather than as a list it is the term structureThe whole pattern of rates across the maturities, taken as one object rather than as a list of separate levels. Where that pattern comes from, and the competing accounts of its shape, are settled elsewhere.. The set used from here on, an invented one of six levels, compounds once a year at every maturity and is written out in full below. No issuer is attached to it, and none is needed: a SPOT rate is a level, not a promise by anybody.

Two things about the shape of that set come back later, so both are worth computing rather than describing. Subtract the two year level from the ten year one and the distance is 1.10 percentage points, or 110 basis points. Rate differences here are counted in basis points, and a hundred of them make a percentage point, so the two units are never swapped for one another. The second reading is the curvature: twice the five year level is 13.80, the two year and ten year levels added come to 13.60, and the difference is 0.20 percentage points.

Where a set like this comes from in a real market is a separate question with a separate answer, involving bootstrappingWorking a set of single date rates out of the prices of instruments that each pay on several dates, one maturity at a time, using the shorter answers already found to unlock the next one. from traded prices and a stated rule for the maturities in between the ones that actually trade. Bootstrapping is covered separately. Only two facts matter from here on: six levels exist, and each attaches to exactly one arrival date.

What does a FORWARD rate cover, and what is it not?

A FORWARD rate never touches today at either end. The stretch it is written for begins later and finishes later still. The one derived first here covers the twelve months beginning on the first anniversary and ending on the second, and it is named for both facts at once: the one year rate, one year forward. The first phrase is how long the stretch runs. The second is how far ahead it starts.

Get the next sentence wrong and everything after it goes wrong too, so it is worth slowing down for. A FORWARD rate is not anybody's estimate of what the one year SPOT rate will be a year from now. It is a figure that is already sitting inside today's set of levels, and any two people holding the same two SPOT rates will compute exactly the same FORWARD rate without consulting each other, without a view and without a forecast between them. The arithmetic has only two inputs and both of them are today's, so there is no room in it for an opinion.

Notice also that a FORWARD rate is annualisedExpressed as a rate per year, whatever length of time the rate actually applies to. Annualising lets a twelve month stretch and a five year stretch be written on the same scale and compared without conversion. like everything else here. The one year rate one year forward is quoted per year even though it covers exactly one year, and the five year rate five years forward is quoted per year even though it covers five. Without that, none of these figures could be set beside each other at all.

How does a FORWARD rate fall out of two SPOT rates?

Put Rs 2,56,000/- somewhere for two years. There are two ways to cover that ground. Lend it once, for the whole two years, at the two year SPOT rate. Or lend it for a year at the one year SPOT rate, take it back on the first anniversary, and lend it again for the second year at whatever rate applies to that second year. The second act, taking money back at the end of a short arrangement and putting it straight into another one, is what rolling overTaking money back at the end of a short arrangement and putting it straight into the next one. Two short stretches then cover the same ground as one long stretch. means, and it is the only thing the derivation needs.

The two ways have to finish level. Set them equal and the rate for the second year is no longer free to be anything.

Now put the recorded levels in. Two years at 6.25 per cent a year, compounding once a year, is 1.0625 multiplied by itself. The product is 1.12890625 exactly. One year at 5.90 per cent a year is 1.0590. Divide the first by the second and the answer is 1.06601157. The FORWARD rate is therefore 6.601157 per cent a year, printed as 6.6012. Nothing in that sequence was assumed, estimated or looked up: the FORWARD rate fell out of two figures that were already recorded above.

The same argument works between any two maturities, not only between the first and second anniversary. Divide the growth over the longer stretch by the growth over the shorter one and take the root that matches the number of years in between.

There is a rule attached to all of this, and it is stricter than it looks. The whole idea is that a FORWARD rate already sits inside the recorded set rather than arriving as separate news about the future. A FORWARD rate stated without the two SPOT rates it came from has taught nothing at all. Print 6.6012 on its own and it reads like a quotation from somewhere. Print it beside 5.90 and 6.25 and it reads as the consequence it is.

Can the derivation be checked in rupees?

The rupee route rounds nothing anywhere inside it, and that makes it the better one to lead with. A starting amount of Rs 2,56,000/- serves. The number is not arbitrary: 6.25 per cent is exactly one sixteenth, so 1.0625 is seventeen sixteenths, and sixteen squared is 256. At that base both routes land on whole rupees at every step. The whole check runs without a single decimal place.

Route one grows Rs 2,56,000/- at 6.25 per cent for a year, reaching Rs 2,72,000/-, then does it again and reaches Rs 2,89,000/-. Route two grows the same Rs 2,56,000/- at 5.90 per cent for a year, reaching Rs 2,71,104/-, and then applies the derived FORWARD rate for the second year. Route two arrives at Rs 2,89,000/- as well. The two middle amounts differ by Rs 896/-, and the destination does not differ at all, exactly as the requirement said it would.

Run that second leg on the printed FORWARD rate instead of the unrounded one and something small and useful happens. Rs 2,71,104/- grown at 6.6012 per cent a year comes to Rs 2,89,000.1172/-, eleven paise and change above where it should be. The gap is the four decimal places doing what four decimal places do. A display figure is not an input, and this is the cheapest possible demonstration of it: the printed rate is right to look at and slightly wrong to multiply by, and the rupee route is where that shows.

The version worth remembering runs the check in the other direction. Once a FORWARD rate has been derived, rolling at the one year SPOT rate and then at that FORWARD rate has to land on the two year SPOT rate compounded; if it does not, something upstream is wrong. Nothing has gone subtly astray in the reasoning; an input is wrong, or two compounding clocks have been mixed.

Which tests actually separate the two rates?

A set of tests chosen after the answers are known is not a test set at all, so the questions get fixed before the answers arrive. Five questions do the work here.

First, what stretch of time is the rate written for. Second, does either end of that stretch sit in today. Third, can the rate discount an amount by itself. Fourth, is the rate read off the recorded set or solved out of it. Fifth, does the rate say anything about where rates are going.

Four of those five separate the two rates cleanly, and the fifth does not. The fifth question gets the same answer from both rates. By the ordinary rule that makes it a wasted line, and it stays on the list precisely because that is where the mistake gets made. Neither a SPOT rate nor a FORWARD rate says anything whatever about the direction of rates. The difference between them on that line is not in the answer, it is in how the question feels: one of the two reads as though it were answering, and the other never did. A test that both sides pass identically is normally dead weight; this one earns its place by being the line people get wrong rather than by sorting anything.

How does each rate score on those five tests?

Read down the two right hand columns and the shape of the difference comes out.

The third line is the one that changes practice at a desk. No path from today is built into a FORWARD rate, so it cannot discount anything on its own. Its stretch begins on a date that has not arrived, so applying it to an amount sitting on the desk today discounts across a period that has nothing to do with the money. Bringing one amount back from a future date requires a rate whose clock starts now, and only a SPOT rate has one.

Why do a FORWARD rate and a SPOT rate so often read almost the same?

Set the two figures side by side. The one year rate one year forward, derived above, is 6.601157 per cent a year. The three year SPOT rate, recorded above, is 6.55 per cent a year. The distance between them is 0.051157 percentage points, or 5.1157 basis points, printed as 5.12. The two figures are about five basis points apart and are completely different objects: one covers twelve months beginning a year from now, and one covers thirty-six months beginning today.

The proximity is not an accident of this particular set of levels. A FORWARD rate is built out of neighbouring levels, and neighbouring levels are close together, so any set that rises smoothly will put its derived FORWARD rates near its recorded SPOT rates. The smoother the set, the closer they land. So a reader who meets both figures without labels will merge them, and the merging is not carelessness: there is genuinely nothing in the digits to tell them apart.

Which settles what the defence has to be. Arithmetic produced the collision in the first place, so arithmetic cannot be the defence. The only defence is labelling: every rate here carries the word SPOT or the word FORWARD, and a rate written without one of those two words is unusable however many decimal places it was quoted to. The rule costs one word and prevents every version of the confusion.

Where does a second derivation land, and why above both?

Run the method once more on a different pair. Take the five year rate five years forward, covering the stretch from the fifth anniversary to the tenth. The two SPOT rates on either side of that stretch are 6.90 per cent a year at five years and 7.35 at ten.

Grow money for ten years at the ten year level and the factor is 2.03245289. Grow it for five years at the five year level and the factor is 1.39600999. Divide the first by the second and 1.45590139 is left, the growth belonging to the second five years alone. The stretch is five years long, so take the fifth root of that, and the factor per year is 1.07801894. Subtract one and the answer is 7.801894 per cent a year, printed as 7.8019.

Two habits are worth flagging inside that table. A rate that comes out of a fifth root is sensitive to what went into it, so every intermediate figure is carried to eight places. One circulating version of this chain carries 2.03278, 1.39679 and 1.45532, and those three figures do not even reproduce their own answer. Put through the same fifth root, they give 7.7933 and that is 86 hundredths of a basis point away from the 7.8019 that same version then printed. Three wrong intermediates and a right final answer is the most dangerous shape a worked example can have. The answer vouches for the working.

Now to the answer that looks wrong and is not. The answer, 7.8019 per cent a year, sits above the five year level of 6.90 and above the ten year level of 7.35 as well. The answer sits higher than both of the numbers it was built from. The shape reads like an arithmetic slip and is a necessity. The ten year level is a geometric averageThe average of a set of growth factors, found by multiplying them together and taking the matching root, rather than by adding them and dividing. Money grows on the multiplied version, and the multiplied version is the one that matters. across all ten of the years it covers. If the first five of those years run at 6.90 and the whole ten average out at 7.35, then the second five have to make up the difference, and making up a difference means sitting above the average rather than merely above the first half.

Check it back the way the earlier derivation was checked. Multiply the five year factor of 1.39600999 by 1.07801894 five times over and 2.03245289 comes back, exactly the ten year factor. The two halves rebuild the whole, and that is what makes the second half's figure a consequence rather than a guess. Put Rs 1,00,00,000/- through it and the ten year route reaches Rs 2,03,24,528.8908/-, while the two leg route parks Rs 1,39,60,099.8964/- at the five year mark and then, on the printed 7.8019, reaches Rs 2,03,24,534.2700/-, which is Rs 5.3792/- above it. The residual is the printed rate again, doing the same thing it did with Rs 2,71,104/-, and the fix is the same: multiply by the unrounded figure and print the rounded one.

What does the whole set look like read a second way?

Plot the six recorded levels and lay the two derived stretches on the same picture and the relationship stops being algebra. Each derived stretch sits above the recorded levels on either side of it, and each one covers a slice of time between two of the recorded maturities rather than a point on the axis. The derived stretches are drawn as segments and the recorded levels as marks for that reason: a SPOT rate belongs to a date, and a FORWARD rate belongs to a span.

The derived rates are not extra information sitting beside the recorded set; they are the same set read a second way. Nothing was added at any step except multiplication and division. Anybody handed the six recorded levels could produce every FORWARD rate here in a few minutes, and anybody handed every FORWARD rate could work back to the six levels. Two descriptions, one object.

Three settings on that control are worth stopping at. With the two year level all the way down at 5.90 per cent a year, level with the one year level, the derived FORWARD rate collapses onto 5.9000 as well, sitting 65.00 basis points below the three year SPOT rate. The reason is immediate: if both years together earn what the first year earns, the second year has nothing left to add. At the top of the range at 6.90 the FORWARD rate reads 7.9094 per cent a year, standing 135.94 basis points clear of the three year SPOT rate on the upper side. A move of 100 basis points in one recorded level has swung the derived rate by more than twice that.

At the setting in between, the derived rate crosses the fixed mark rather than merely approaching it, and that setting is the one that teaches. At a two year SPOT rate of 6.20 per cent a year the FORWARD rate reads 6.5008, which is 4.92 basis points below the three year SPOT rate; one step later, at 6.25, it reads 6.6012 and sits 5.12 basis points above. The two coincide at a two year SPOT rate of 6.2245 per cent a year, a level falling between two settings of the control and therefore never displayed. Nothing whatever happens at the crossing. Two figures that describe completely different stretches of time briefly print the same digits, and then they part again. Each one has to carry its own word for exactly that reason.

What does neither rate establish?

Three absences, and they are not small print. Neither rate says where rates go. The FORWARD rate is arithmetic performed on today's recorded levels. No slot in the division holds a belief, so it would read 6.6012 per cent a year whether every person in the market expected 4 per cent next year or 9 per cent. The simulation above makes the point physically: the derived figure only moves when a recorded level moves, and a recorded level is a price, not an opinion poll. Whether prices themselves carry opinions is a question about markets and is covered separately.

Neither rate says whether any borrower pays. Both figures here belong to instruments with no credit element in them at all, so there is nothing in either number compensating anybody for the possibility of not being repaid. Reading a spread into these levels would be reading something that was never put in.

And neither rate says how the recorded set was built. Six maturities are written down here and the maturities in between are simply absent. Filling them in requires interpolationFilling in a value that sits between two known ones by a stated rule. Every rule produces a different filling, so the rule has to be named before the filled-in number means anything at all. under some named rule, and different rules give different answers at the four year point, so the rule matters more than the arithmetic does. Which set a supervised holder values against, and how it is assembled and published, is decided by the Reserve Bank of India at rbi.org.in.

The honest summary is that a FORWARD rate is a fact about today's prices wearing the grammar of a statement about tomorrow. It uses the future tense. A FORWARD rate names a year that has not begun. The figure reads like news. Every one of those impressions comes from the naming rather than from the number, and reading the grammar instead of the fact is the single failure worth guarding against.

How does somebody actually use a FORWARD rate?

Two counters at the same branch. One offers a twenty-four month deposit at a fixed rate. The other offers twelve months, after which the money comes back and gets deposited again at whatever the twelve month rate happens to be then. A household choosing between them is not choosing between safety and risk, or between long and short. The household is asking one question: how good would the second year have to be for the shorter path to catch up?

The question has an exact answer, and the answer is the FORWARD rate. On the levels recorded here, taking the one year path at 5.90 per cent a year means the second year has to pay 6.6012 per cent a year for the two paths to finish level. Below that the two year deposit wins; above it the rolling path wins. A FORWARD rate is a break-even, and a break-even is a completely different object from a forecast: one states what would have to happen, and the other claims to know what will. Nobody at either counter has expressed a view about anything.

An analyst uses the same figure the same way, with a bigger number attached. Somebody who wants to hold a two year instrument but is being offered a one year one asks what the second year has to deliver, gets 6.6012 per cent a year as the threshold, and then has a real question to argue about instead of a vague one. The threshold came out of the recorded levels; the argument about whether the second year clears it is entirely the analyst's own, and it belongs in a separate paragraph of the note with the word view attached to it.

A lender reads it as a consistency check rather than a decision. If a five year arrangement and a ten year arrangement are both on the table, the five year rate five years forward, at 7.8019 per cent a year, is what the second half of the longer one is implicitly charging. Somebody who cannot say out loud that they are content with that figure has not finished reading the longer arrangement. None of these three uses requires anybody to believe the FORWARD rate will happen, and that is what makes them usable. Each one treats it as a threshold that was already sitting inside prices somebody else quoted.

The six rate levels used throughout are invented.
Educational material. Not advice on any investment, tax, budget or market position.