How to Read a Yield Curve, Node by Node and Gap by Gap
Read a yield curve in seven passes. Name which borrowing it is drawn from, whether its rates are SPOT or FORWARD, and on what compounding. List the maturities it actually fixes. Take two subtractions for the shape. Divide each step between neighbouring maturities by the years it spans. Then derive the forward rates already sitting inside it. Everything else is an opinion.
A curve is a bundle of separate rates that happen to have been drawn near one another, and nearly every misreading starts with treating the picture as the thing. The passes below pull the picture apart and return the rates. Every statement made about the curve then traces to a subtraction somebody else could repeat and land on the same figure.
Where these numbers came from
All six rates on the curve below were written for teaching. Every other figure in this guide is arithmetic run on those six, worked in full rather than carried in. Nothing was copied off a screen, a vendor feed or a published curve, and no date attaches to any of it.
Two of the derived rates also sit in the working record rounded to two places, at 6.60 per cent and 7.80 per cent. The two rates are carried further below, to 6.601157 per cent and 7.801894 per cent, with the whole chain of working printed beside each one. Both figures can then be arrived at rather than accepted on trust. Where a reading of this kind gets carried through to a price, the rupee amount involved is an invented zero coupon claim rather than anything the record publishes, and pricing is covered separately.
What has to be labelled before a curve can be read at all?
Three labels, and a curve missing any one of them cannot be read. Not read badly. Not read. The numbers sit there and refuse to mean anything.
The first label says whether each rate is a SPOT rate or a FORWARD rate. The two cover completely different stretches of time. A SPOT rate is priced from now: the clock starts today and stops at the maturity written into its name. A FORWARD rate covers a stretch that has not started yet, pinned between two future dates, and it is squeezed out of two SPOT rates rather than seen anywhere. Put a 6.55 and a 6.60 side by side without those words attached and a reader will merge them into one idea. Merging them is the single most expensive thing that happens on a curve.
The second label is the compounding convention. Every rate on the invented curve read here is stated on ANNUAL compounding: the interest is credited once a year and from then on earns beside the amount that threw it off. Running the identical six numbers on a convention that credits twice a year describes a different set of prices, from figures that look the same on the screen. The convention is therefore stated inside every sum below rather than in a footnote under them.
The third label names the borrowing the curve is drawn from. A curve assembled from one kind of borrower is not a curve assembled from another, and the choice of which security counts as the reference securityThe particular borrowing that gets treated as the standard at a given length of time, so that everything else can be measured against it. Somebody decides which one it is. at each maturity is somebody's decision rather than a fact about the market. So is the method that fills a published curve out from the handful of securities that actually trade, which goes under names like bootstrappingWorking a set of pure single payment rates out of the prices of borrowings that pay more than once, one length of time at a time, each answer feeding the next. and curve fittingFitting a smooth mathematical shape through a scatter of observed prices so that a rate can be quoted at any length of time, including ones nothing actually traded at.. None of that should be taken from memory: how a benchmark government curve is put together and published, and which security is treated as the reference at a given maturity, are both set by the Reserve Bank of India at rbi.org.in, and both move.
Which of these does NOT have to be established before a curve can be read?
Which of these rows must be filled in before touching a real curve?
Seven rules sit behind any real curve, and every one is decided by a body that can revise it. A figure remembered from last year would be a hostage to the next revision. The current position is read at the site beside the row before a real curve is taken anywhere.
| The rule behind a published curve | Who settles it |
|---|---|
| Assembling and publishing a benchmark government yield curve | Reserve Bank of India, rbi.org.in |
| Picking which security stands as the reference at a given maturity | Reserve Bank of India, rbi.org.in |
| The compounding basis a published yield is stated on | Reserve Bank of India, rbi.org.in |
| The day count a yield calculation has to run on | Reserve Bank of India, rbi.org.in |
| Quoting a government security's price, and against what | Reserve Bank of India, rbi.org.in |
| The tenorsThe lengths of time a borrowing is offered for. A menu of stretches, not a continuous range. government borrowing is offered at | Reserve Bank of India, rbi.org.in |
| The valuation normThe rule fixing the price at which a holding is written into the books, which need not be the price anyone paid. behind a carrying price | Reserve Bank of India, rbi.org.in |
Above this block, the only convention anything leans on is the compounding basis. The basis sits inside each sum rather than in a note, and not one figure above reproduces without it. Meeting a market that runs on different conventions adds one row to this table; it does not force the arithmetic above to be written again. Where corporate borrowing enters a reading, the disclosures required of the borrower when it offers terms are settled by the Securities and Exchange Board of India (SEBI) at sebi.gov.in.
Which rates does this curve actually fix, and what sits in the gaps?
The invented SPOT curve read in this guide fixes six rates and six only. At the one year node the SPOT rate reads 5.90 per cent. At the two year node, 6.25 per cent. The three year node carries 6.55 per cent, the five year node 6.90 per cent, the ten year node 7.35 per cent, and the thirty year node 7.60 per cent. Every one of those six is a SPOT rate, and every one is stated on ANNUAL compounding.
Between any two of those maturities the curve fixes nothing whatsoever. Not a small amount. Nothing. The line joining two points on a chart was put there by whoever drew the chart, and a reader who slides a finger along it and reads off a four year SPOT rate has just quoted a number that never existed. Sliding a finger along the line is worth watching for. The move is quick, it feels like reading, and it produces a figure that looks exactly like the six that are real.
So the nodes are counted before anything else, and the count written down. Six here. Every statement made afterwards is capped by that count: six rates give fifteen possible pairs to subtract, and not one rate at any other maturity. A reader who has written down the number six will notice when a colleague's note quotes a seven year rate. A reader who skipped that pass will not.
The gaps are part of what is being read, not empty space between the interesting bits. A bus timetable with six departures on it works the same way. The timetable does not say there is no bus at eleven o'clock; it says the timetable has nothing at eleven o'clock, and those are different statements. The honest reading of a gap is that the record is silent there, and silence is a finding.
How many rates does the invented SPOT curve fix?
What are the two subtractions that describe the shape?
Shape is described by subtraction, and by nothing else here. Two readings do the work, and both of them carry their maturities around with them like a surname.
The slope is one rate less another, with both maturities named every time it is quoted. Put the ten year node's 7.35 per cent against what the two year node carries, 6.25 per cent, and 1.10 percentage points is what lies between them. One basis point is a hundredth of a single percentage point, so a hundred of them stack into one percentage point, and 1.10 percentage points is therefore 110 basis points. One distance, two units, and putting the wrong word after the number multiplies it a hundredfold.
The qualifier is the part that gets dropped. Six nodes give fifteen pairs, and a slope exists for every one of them. Setting the thirty year node's 7.60 per cent against the SPOT level at five years gives a gap of 0.70 percentage points. Setting the same thirty year figure against the ten year node instead gives 0.25. Three genuine slopes on one unchanged curve, reading 1.10, 0.70 and 0.25, and a note reporting the slope without naming which pair produced it has told nobody anything. Naming the maturities each time sounds pedantic for about a week and then it saves the reader.
The butterfly reading works a middle maturity against the two outside it: double whatever stands at the five year node, then take off the two year node and the ten year node in turn. Double 6.90 per cent is 13.80. The two outer SPOT levels together, 6.25 plus 7.35, come to 13.60. The reading is 0.20 percentage points, or 20 basis points. The reading says the middle of this stretch stands clear of a chord run between the two year node and the ten year node, and 0.20 percentage points measures exactly how far clear.
Why does the distance between two neighbouring rates mislead on its own?
The distance between two neighbouring rates is the sharpest trap on any curve, and this curve was built so that the trap bites. Take the five steps between neighbouring recorded nodes and lay them out raw.
One year to two years is 0.35 percentage points, or 35 basis points. Two years to three years is 0.30 percentage points, or 30 basis points. Three years to five years is 0.35 percentage points, or 35 basis points. Five years to ten years is 0.45 percentage points, or 45 basis points. Ten years to thirty years is 0.25 percentage points, or 25 basis points.
Read that list and the curve looks like it climbs at a broadly even pace, with one bigger stretch in the middle and a quiet long end. Now divide each step by the number of years it spans, and the whole picture turns over.
| Between | Years it spans | Raw step, basis points | Basis points a year |
|---|---|---|---|
| one year and two years | 1 | 35 | 35.00 |
| two years and three years | 1 | 30 | 30.00 |
| three years and five years | 2 | 35 | 17.50 |
| five years and ten years | 5 | 45 | 9.00 |
| ten years and thirty years | 20 | 25 | 1.25 |
The raw column is not even in order. The raw figures go 35, then 30, then back to 35, then 45, then 25. No reading can hold on to a pattern like that. The per year column falls at every single one of the four handovers: 35.00, then 30.00, then 17.50, then 9.00, then 1.25. Five figures, strictly falling, no exceptions. And the front of the curve is charging exactly twenty eight times what the back of it charges for one more year of waiting.
Look hardest at rows one and three. Both raw figures are 35. Both rows carry the same number. Per year, one is 35.00 basis points and the other is 17.50, precisely half. The two steps are not comparable and never were; they were only ever printed in the same column.
Part of why this survives is that curves get drawn on axes that squash the long end, either by spacing the points evenly whatever the years between them, or by putting the time axis on a logarithmic scaleAn axis where equal distances stand for equal multiples rather than equal amounts, so the far end is compressed into a small strip.. On a picture like that the last stretch looks like a modest step. Twenty years of curve carrying 25 basis points between its ends is not a modest step. The last stretch is almost nothing a year, and the drawing was telling the opposite of the arithmetic.
Before anything moves: the step from the one year SPOT rate to the two year SPOT rate is 35 basis points and the step from the three year SPOT rate to the five year SPOT rate is also 35 basis points. Does the curve climb at the same pace in both places?
Move the thirty year node and watch the raw distance and the per year distance come apart
Every rate apart from the thirty year one stays at the level this record wrote for it, and that includes whatever the ten year node holds, 7.35 per cent. Only the thirty year point travels. The two bars underneath are the same distance measured twice: once raw, and once shared out across the twenty years it is spread over. The two bars share one scale, so the difference between them can be seen rather than merely asserted. The horizontal axis switches between spacing by years and spacing by node. The switch changes the drawing and changes not one number.
The thirty year SPOT rate is now 7.60 per cent, which stands 0.25 percentage points above the 7.35 per cent held at the ten year node, and that works out at 1.25 basis points of climb for each of the twenty years between the two.
The step from the ten year SPOT rate to the thirty year SPOT rate is 25 basis points, the smallest raw step on this curve. What is that step per year?
The one year node holds a SPOT rate of 5.90 per cent, and 6.25 per cent sits at the two year node. Will the one year rate one year FORWARD land above or below that two year level?
What do the forward rates already inside the curve report?
Every gap between two SPOT rates already holds a rate for the stretch between them, and it falls out of arithmetic rather than out of anybody's view about what happens next. Nothing has to be added to the curve to get it. The forward rate was there from the outset.
Here is the chain for the nearer one. Money placed for two years at the two year node's SPOT level of 6.25 per cent grows by 1.0625 twice over, for a factor of 1.12890625. Money placed for one year at 5.90 per cent, the SPOT level at the one year node, grows by 1.0590. Divide the longer factor by the shorter one and the growth belonging to the second year alone is left behind: 1.06601157. Take one away and that second year prices at 6.601157 per cent, running from the end of year one to the end of year two. Its proper name is the one year rate one year FORWARD. Rounded to two places, the working record carries that identical figure at 6.60 per cent.
| sa | the SPOT rate at the nearer maturity, as a decimal, taken straight off the recorded curve |
| sb | the SPOT rate at the further maturity, as a decimal, also taken straight off the curve |
| fa,b | the FORWARD rate covering the stretch that begins at the nearer maturity and ends at the further one |
| b less a | how many years that stretch runs for, which is what the left hand side is raised to |
The longer one runs the same way with one extra move. Compounding 6.90 per cent, the SPOT level at the five year node, across its five years gives a factor of 1.396010. Compounding 7.35 per cent at the ten year node across its ten years gives 2.032453. Dividing the longer by the shorter leaves 1.455901 as the total that the five years lying between those two nodes have to deliver in total. Because that total is spread over five years rather than one, the fifth root of it is taken. The fifth root is the geometric meanThe average obtained by multiplying the terms together and taking the matching root, rather than by adding them and dividing. The geometric mean is the right average whenever a quantity compounds. of those five years: 1.078019. Take one away and 7.801894 per cent is the five year rate five years FORWARD. The working record carries the same figure to two places as 7.80 per cent.
A FORWARD rate reports the implication already carried by the SPOT levels standing on the curve today, and it refuses to report what any rate will turn out to be. Nothing in the arithmetic measured an expectation. Nobody was surveyed. A reader who treats 6.601157 per cent as a prediction of next year's one year rate has added a claim the division never made, and has quietly turned bookkeeping into a forecast.
Now here is where the labels earn their keep. Worked out a moment ago, the one year rate one year FORWARD came to 6.601157 per cent. Sitting on the very same curve, the three year node carries a SPOT rate of 6.55 per cent. Between the two lie 0.051157 percentage points, or 5.1157 basis points, and at two decimal places they print as 6.60 and 6.55. Two completely different objects, five basis points from each other, and the only thing keeping them apart in the text is the word SPOT or the word FORWARD attached to each. On the sort of axis that has to fit thirty years of curve onto it, the two land on top of one another and no eye can pull them apart. The gap is real, it matters, and it is invisible at the scale a curve is normally drawn on.
A FORWARD rate derived from two SPOT rates reports which of these?
What can one curve never report, however carefully it is read?
Three things, and each of them is worth writing down beside the reading rather than left to be discovered by whoever picks the note up next.
Unusual is a comparison, and a comparison needs something on the other side of it. One curve cannot say whether its own shape is unusual. Calling this curve steep needs a run of other curves to set it against, and this record holds exactly one. Calling this curve steep is the same as calling a person tall after meeting one person. The sentence has a shape that sounds like a finding and no content behind it.
One curve cannot say what anybody expects. An expectation is a state of mind and a price is a number, and no arithmetic run on prices turns the second into the first. The term premiumThe extra return a lender is sometimes said to want for tying money up longer, over and above what short rates are thought likely to average. Separating it out needs a model and a run of history, neither of which sits here. would come in here. Separating a term premium out of a curve needs a model and a measured history. The result is a reading of the curve plus a large number of assumptions, and the assumptions are doing most of the work.
One curve cannot supply a rate at any maturity it does not fix. On this curve that leaves every maturity apart from six. Asked for the four year SPOT rate, the honest answer is that there is not one. Asked for the eleven year, the answer is the same. The gaps are not a rounding problem to be tidied away; they are the extent of what the record holds.
So a note calling the curve steep has smuggled a comparison in through the side door. A note recording that the SPOT level at ten years stands 1.10 percentage points clear of the SPOT level at two years has not, and it will still be true and still be checkable when somebody picks it up in six months.
A note says the curve is steep. What is missing from that reading?
How does anybody actually use a reading like this?
Picture a treasury desk at a mid sized manufacturer that has to raise money and can choose how long to borrow for. Nobody there is going to forecast rates for a living. The desk needs a defensible statement about what an extra year of borrowing costs at different points along the curve, and the per year column is exactly that statement. Moving from three years to five costs 17.50 basis points for each of the two extra years. Moving from ten years to thirty costs 1.25 for each of the twenty. Those two figures make a comparison a treasurer can put in a board note and a board member can check with a calculator.
A lender reads the same column from the other side and reaches the opposite conclusion about where the effort is worth spending. The front of this curve is where the price of an extra year moves fastest, so a small change in how long a facility runs for changes the price meaningfully. Out at the long end an extra year barely registers. The flatness is a fact about this curve rather than a rule about curves, and the treasurer and the lender are looking at the identical five numbers.
An analyst valuing a stream of dated payments uses the reading differently again. Each payment gets discounted at the SPOT rate for its own date, so the analyst needs to know immediately which dates the curve actually covers and which ones it does not. The count of nodes decides what can be valued honestly and what needs an assumption declared on the face of the workings. Where a payment falls into one of the five gaps, the analyst either says so and states what was assumed, or produces a figure nobody can reproduce.
And a household making the same choice in miniature runs into it too. Take a term deposit offered at several lengths, or a home loan offered fixed for two, three or five years. The sensible question is never which rate is highest. The question is what each extra year is costing or paying, and answering it takes a subtraction and then a division. A one year old daughter's education fund and a thirty year retirement pot sit at opposite ends of exactly the pattern in the table above.
In what order do the seven passes run?
The order matters because each pass uses only what the passes before it settled. Run out of sequence, they produce a slope quoted before the number of nodes is known, or a forward rate derived before the curve has been established as stated on annual compounding.
Pass one: write down which borrowing the curve is drawn from, whether its rates are SPOT or FORWARD, and the compounding convention, confirming each of those at its own source rather than assuming it. Pass two: list the maturities the curve fixes, and count them. Pass three: take the slope between two named maturities, and put the pair into the name of the reading. Pass four: take the butterfly reading across three named maturities. Pass five: take each step between neighbouring maturities and divide it by the years it spans before comparing any two of them. Pass six: derive the forward rates required from the two SPOT rates behind each one, and quote none. Pass seven: write down the questions this curve cannot answer. A reading that does not record its own limits gets used later as though it had none.
Of the seven passes, which one comes last?
What does one whole reading look like, run end to end?
The full reading of the invented SPOT curve follows, run in the order of the seven passes. Every line can be checked against the ones above it.
| Pass | What it settles | The reading on this curve |
|---|---|---|
| One | The three labels | The rates are SPOT rather than FORWARD, the compounding is ANNUAL, and the curve was written for teaching rather than taken from anywhere. |
| Two | The nodes, and the count | Six maturities are fixed: one year at 5.90 per cent, two years at 6.25, three years at 6.55, five years at 6.90, ten years at 7.35 and thirty years at 7.60 per cent. Nothing at all is fixed between them. |
| Three | The slope, with its pair named | Ten years less two years is 1.10 percentage points, which is 110 basis points. Thirty years less five years is 0.70 percentage points. Thirty years less ten years is 0.25 percentage points. Three slopes, three different numbers, one unchanged curve. |
| Four | The butterfly reading | Twice 6.90 is 13.80; 6.25 plus 7.35 is 13.60; the reading is 0.20 percentage points, which is 20 basis points. |
| Five | Each step, per year | Raw: 35, 30, 35, 45 and 25 basis points. Per year: 35.00, 30.00, 17.50, 9.00 and 1.25 basis points. The raw column is not in order; the per year column falls at every handover. |
| Six | The forward rates, derived | Dividing 1.12890625 by 1.0590 gives the one year rate one year FORWARD, 6.601157 per cent. A fifth root on 2.032453 over 1.396010 yields the five year rate five years FORWARD, 7.801894 per cent. |
| Seven | What the curve cannot answer | Whether this shape is unusual, what anybody expects, and the rate at any of the maturities it does not fix. |
Every figure in that table is arithmetic on the six recorded nodes and on nothing else. There is no seventh input hiding anywhere. A reader who disagrees with a line can take it apart and find out which subtraction the disagreement rests on, and that is the whole reason for reading a curve this way.
The reading that gets made, and what it costs
The failure is comparing two steps of different lengths as though they were the same measurement, and it happens in the second the two figures land in the same column.
A reader lists the five steps and sees 35 basis points from one year to two years and 35 basis points from three years to five years. The figures are identical, so the note says the curve rises at the same pace in both places and the reader moves on. It does not. The first is spread over one year and the second over two, so per year they are 35.00 basis points and 17.50 basis points, one of them exactly half the other. The same reader then sees the 25 basis points from ten years to thirty years, the smallest raw step on the curve, and reports the long end as nearly flat. The conclusion happens to be true, but it is not what the figure shows. Across twenty years the step is 1.25 basis points a year against 35.00 at the front, a difference of twenty eight times rather than the third the raw numbers suggest.
Who makes it: anybody reading a curve off a chart, where the horizontal axis is nearly always squashed at the long end and the picture is telling the eye the opposite of what the arithmetic says. The cost: two statements about the shape of the same curve that cannot both stand, carried into a note nobody rechecks, and quoted six months on by a reader who never saw the workings. The repair takes one division, and it has to be done before any two steps are compared rather than after somebody queries the note.
Which authority settles each rule, and where it is published
Each row names the body that decides a rule this reading leans on. Any figure for one of those rules dates from the day the body next revises it.
| Who keeps it | What it decides | Site | Confirmed |
|---|---|---|---|
| Reserve Bank of India | How a benchmark government yield curve gets assembled and made public, which security counts as the reference at a maturity, the compounding basis a published yield is stated on, the day count a yield calculation must run, how a price is quoted, the tenors borrowing is offered at, and the valuation norm behind a carrying price. | rbi.org.in | 28 August 2026 |
| Reserve Bank of India, database route | The route to a measured series of rates, which is needed before any shape can be called unusual. | dbie.rbi.org.in | 28 August 2026 |
| SEBI | The disclosures a corporate borrower must make about the terms it offers, named because a reader arriving from the credit material will look for it. | sebi.gov.in | 28 August 2026 |
| Repository of named academic work | Named academic readings of the term structure, to be opened and checked before a name and a year get typed anywhere. | ideas.repec.org | 28 August 2026 |
The six node yield curve read here is invented.
Educational material. Not advice on any investment, tax, budget or market position.