Sovereign Bonds: How a Government Borrows Across Time

A sovereign bond is a government borrowing on its own account: a dated promise by the state to pay stated amounts on stated dates. Because one government borrows over several horizons at once, its bonds record a schedule of rates rather than a single rate, and every other borrower in that market is priced as an amount stacked on top of the rate for the matching horizon.

Who is borrowing here, and over what horizons?

Start with what is not varying here. There is only one borrower. Not six borrowers, not six kinds of paper, not six risks. One state, borrowing money it will return at six separate future dates, and charged a different rate for each of those dates.

One schedule of rates, invented for teaching and belonging to no market anywhere, runs underneath everything that follows. The schedule records six SPOT ratesA SPOT rate is the rate for money placed today and returned at one stated future date. One rate, one start, one finish. and no others: the one year SPOT rate is 5.90 per cent a year, the two year SPOT rate is 6.25 per cent, the three year SPOT rate is 6.55 per cent, the five year SPOT rate is 6.90 per cent, the ten year SPOT rate is 7.35 per cent and the thirty year SPOT rate is 7.60 per cent. Nothing about the borrower changes between the one year point and the thirty year point; the only thing that changes is how long the lender waits to be repaid.

An everyday picture sits beside that before the arithmetic starts. A neighbourhood shopkeeper will sell the same steel almirah at a different price depending on how long the buyer takes to pay: cash today, six instalments, or twenty four. The almirah does not change. The shopkeeper does not change. The wait changes, and the wait has a price. The shopkeeper also quotes only three lengths, since those are the only three anyone has ever asked for. The schedule of government rates sits in exactly that position, with six horizons and nothing between them.

One convention has to be nailed down before a single sum is written, and it goes inside the arithmetic rather than in a note underneath it. Every rate and every price here is struck on annual compoundingOne discounting period a year. An amount due in three years is divided by one plus the rate, three separate times., meaning one discounting period a year, so an amount due in three years at the three year SPOT rate of 6.55 per cent is divided by 1.0655 three times over. The compounding convention is not housekeeping. The same six numbers on a semi-annual convention produce different prices and a different set of FORWARD rates, and nobody who is left guessing at the convention can reproduce a single figure below.

Spaced by real years rather than by even steps, the recorded schedule stops looking like a gentle ramp and starts looking like a wall followed by a plain: almost all of the climb is bought in the first three years of waiting, and the last twenty years add very little.

Try it out

One borrower quotes six SPOT rates for six horizons. Before any subtraction is done: is each extra year of waiting paid for at the same rate?

What does the shape say before any single level on it?

A schedule of rates is easiest to misread when the levels are taken one at a time. The shape comes first, and it is read by doing the subtraction rather than by describing it.

The ten year SPOT rate of 7.35 per cent less the two year SPOT rate of 6.25 per cent is 1.10 percentage points. The same quantity written in the other unit is 110 basis pointsOne hundredth of a percentage point. So 1.10 percentage points is 110 basis points, and 2.20 percentage points is 220 basis points.. Take the two outermost recorded horizons instead. The thirty year SPOT rate of 7.60 per cent less the one year SPOT rate of 5.90 per cent is 1.70 percentage points, or 170 basis points. A basis point and a percentage point are two units for one quantity, and writing 1.10 basis points where 110 basis points is meant is the commonest error in the whole of rate arithmetic.

Measured on one shared scale, the stretch from the two year to the ten year recorded horizon is worth 110 basis points and the stretch from the shortest recorded horizon to the longest is worth 170, so the outer pair really does buy the taller band.

Neither of those subtractions can see the middle. Both take a rate at one end and a rate at the other and ignore everything between, so a schedule that sagged in the centre and a schedule that bulged there would give the identical answer. Seeing the bend takes a third reading, and the one that does the job is the butterflyTwice a middle rate, less the two rates on either side of it. A positive answer says the middle sits above the straight line joining the wings.: twice a middle rate, less the two rates flanking it.

The reading that finds the bend

$$ B = 2 s_{5} - s_{2} - s_{10} $$

B	the butterfly, in percentage points, on this curve 0.20
s₅	the five year SPOT rate, 6.90 per cent a year, the middle of the three
s₂	the two year SPOT rate, 6.25 per cent a year, the near wing
s₁₀	the ten year SPOT rate, 7.35 per cent a year, the far wing

What it says in wordsDoubling the middle rate and taking away each of the two rates on either side of it leaves the amount by which the middle stands clear of the straight line joining its two neighbours, counted twice, and a positive answer means the middle stands above that line rather than below it.

Run it on the recorded rates. Twice 6.90 is 13.80. Take away 6.25 and 13.80 becomes 7.55. Take away 7.35 and 7.55 becomes 0.20. So the butterfly is 0.20 percentage points, or 20 basis points. The sign matters most. Positive says the five year SPOT rate sits above a straight line drawn between the two year and ten year SPOT rates when the three are weighted evenly. Halving the butterfly gives the plainer statement: the five year SPOT rate stands 0.10 percentage points clear of the simple average of its two neighbours, 6.80 per cent.

With the three horizons weighted evenly, the middle recorded rate is plotted a tenth of a percentage point clear of the straight line joining its neighbours, which is the bend that both of the two end-to-end subtractions above are blind to.

Try it out

The ten year SPOT rate is 7.35 per cent a year and the two year SPOT rate is 6.25 per cent a year. State the gap between them in both of its units.

What happens to those gaps once each one is divided by the years it buys?

Here is the step almost every reader skips, and skipping it is what makes a schedule of rates look tamer than it is. A gap bought with one extra year of waiting is not the same object as a gap bought with twenty extra years, even when the two gaps read the same size in print. Dividing each step by the number of years it covers puts like against like.

From the one year to the two year SPOT rate is 0.35 percentage points across one extra year, so 0.3500 points a year. From the two year to the three year SPOT rate is 0.30 across one year, so 0.3000. From the three year to the five year SPOT rate is 0.35 across two years, so 0.1750. From the five year to the ten year SPOT rate is 0.45 across five years, so 0.0900. And from the ten year to the thirty year SPOT rate is 0.25 across twenty years, so 0.0125. The first step charges 28 times what the last one charges for each additional year of waiting, and 0.3500 divided by 0.0125 gives that 28 exactly.

Once each recorded step is divided by the number of years it actually covers, the schedule collapses from five similar looking gaps into one very expensive first year and a twenty year stretch that is almost free.

What does a holder of one of these promises actually hold?

Strip a government bond back to its simplest possible form and it is a promise of one amount on one date. Everything else, every coupon and every schedule, is that single object repeated. So price the single object at each of the six recorded horizons and see what one borrower's schedule does to the same promise.

The price of one dated promise

$$ P = \frac{F}{(1 + s_{t})^{t}} $$

P	the price today, in rupees
F	the amount promised on the future date, here Rs 1,000.00/- of face
s_t	the SPOT rate recorded for that horizon, as a decimal
t	the number of years until the promised date, one of the six recorded

What it says in wordsThe price today of a single promised amount is that amount divided by one plus the SPOT rate for its own horizon, with the division repeated once for every year of waiting, which is what annual compounding means.

Rs 1,000.00/- promised in one year and discounted at the one year SPOT rate of 5.90 per cent costs Rs 944.287063/-. The same Rs 1,000.00/- promised in two years at the two year SPOT rate of 6.25 per cent costs Rs 885.813149/-, in three years at the three year SPOT rate of 6.55 per cent costs Rs 826.684201/-, in five years at the five year SPOT rate of 6.90 per cent costs Rs 716.327252/-, in ten years at the ten year SPOT rate of 7.35 per cent costs Rs 492.016324/-, and in thirty years at the thirty year SPOT rate of 7.60 per cent costs Rs 111.078974/-.

Count that last one twice before moving on. Rs 111.078974/- today for Rs 1,000.00/- in thirty years, from the very same borrower who charges Rs 944.287063/- for the very same Rs 1,000.00/- next year. Nothing about the borrower separates those two prices; the whole of the difference is the waiting. Each of those six numbers is a discount factorThe present value today of one rupee promised at a stated future date. Multiplied by the amount promised, it gives the price. multiplied by the face amount, and the discount factors themselves run 0.9442870633, 0.8858131488, 0.8266842011, 0.7163272523, 0.4920163244 and 0.1110789741.

The identical promise from the identical borrower is worth well over nine tenths of its face at the shortest recorded date and barely a ninth of it at the longest, so the whole of that collapse is the price of waiting rather than anything about the state.

Play with it

Move the horizon. Watch the rate climb while the price collapses.

One control, and it is the horizon. Everything else is nailed down and printed on the drawing: the same borrower, Rs 1,000.00/- of face, annual compounding, and no tax or dealing cost anywhere. Six horizons are all this record carries, so the control snaps to six positions and refuses every horizon in between. Slide it and watch the marker climb the rate axis while the bar underneath shrinks, and watch the shaded stretch that says nothing is recorded there.

Years to the promised date, snapping to the six recorded horizons

The control cannot be dragged to four years or to nine. There is no four year SPOT rate and no nine year SPOT rate in this record, and reading one off a drawn line would give an answer that depended on which line somebody drew.

Horizon

3 years

SPOT rate

6.55%

Cost today

826.684201

Share of face

82.67%

At the three year horizon the invented SPOT curve records a three year SPOT rate of 6.55 per cent a year on annual compounding, so Rs 1,000.00/- promised by that borrower in three years costs Rs 826.684201/- today, which is 82.67 per cent of what was promised.

The worked position, in plain text so it survives with the drawing stripped out. At three years the three year SPOT rate reads 6.55 per cent a year and Rs 1,000.00/- costs Rs 826.684201/- today. The two endpoints read Rs 944.287063/- at the one year SPOT rate of 5.90 per cent and Rs 111.078974/- at the thirty year SPOT rate of 7.60 per cent.

Try it out

Rs 1,000.00/- promised in one year costs Rs 944.287063/- at the one year SPOT rate of 5.90 per cent a year. What does the same Rs 1,000.00/- promised in thirty years cost at the thirty year SPOT rate of 7.60 per cent a year?

What is a par bond, and when does the coupon match the rate for that horizon exactly?

A par bondA bond whose price equals its face amount. Nothing is paid above or below the amount that will be returned at maturity. is a bond whose price is its face amount, no more and no less. The definition alone says nothing about which coupon delivers a par price, and that coupon is worked out below at two of the recorded horizons.

The easy case is where the rule hides, so take a one year government bond on Rs 1,000.00/- of face first. A one year bond has exactly one payment date, so exactly one SPOT rate can touch it. Rs 1,059.00/- arriving in one year, divided once by 1.0590, is Rs 1,000.000000/-. With a single payment date the par coupon has no choice at all. One payment discounted at one rate can only be pulled to face by that rate, so the par coupon is the one year SPOT rate of 5.90 per cent a year and nothing else.

Now walk three years out, where there are three payment dates and three different recorded SPOT rates. The first coupon is discounted at the one year SPOT rate of 5.90 per cent, the second at the two year SPOT rate of 6.25 per cent, and the third, together with the face amount, at the three year SPOT rate of 6.55 per cent. The level coupon that pulls the total to exactly Rs 1,000.00/- has to satisfy all three at once.

The coupon that puts the price at face

$$ c = \frac{1 - d_{n}}{d_{1} + d_{2} + \cdots + d_{n}} $$

c	the par coupon rate a year, as a decimal share of face
d_n	the discount factor for the final date, here 0.8266842011
d₁ ... d_n	the discount factors for every payment date, one per date
n	the number of payment dates, here three

What it says in wordsThe coupon that prices a bond at exactly its face amount is one less the discount factor for the final date, divided by the sum of the discount factors for all of its payment dates, so it is a weighted blend of every rate the bond touches rather than the rate for its last date alone.

Put the recorded numbers in. One less 0.8266842011 is 0.1733157989. Adding 0.9442870633, 0.8858131488 and 0.8266842011 gives 2.6567844132 for the three discount factors. The division gives 0.0652351760, so the three year par coupon rate is 6.523518 per cent a year, Rs 65.235176/- of coupon on Rs 1,000.00/- of face. Priced back on the same three recorded SPOT rates, the parts are Rs 61.600733/-, Rs 57.786177/- and Rs 880.613091/-. The three parts add to Rs 1,000.000000/-.

Hold on to this one: the three year par COUPON RATE of 6.523518 per cent a year is not the three year SPOT rate of 6.55 per cent a year. They sit 0.026482 percentage points apart, which is 2.6482 basis points, and the gap is a fact rather than a rounding error. The two earlier coupons are discounted at the lower one year and two year SPOT rates. Those lower rates make the early money worth more, and a slightly smaller level coupon can then do the job. On a schedule that rises with the horizon a par coupon must sit below the SPOT rate for its own final date, every time.

With one payment date the level coupon has to equal the rate for that date, while with three dates the two earlier payments meet lower recorded rates and drag the level coupon a couple of basis points below the rate for the final date.

Try it out

A three year government bond is to be priced at exactly its face amount on the recorded SPOT rates. Will its coupon equal the three year SPOT rate of 6.55 per cent a year?

What is a premium bond, and what exactly is the premium paying for?

A premium bondA bond whose price is above its face amount, so a buyer pays more today than the amount that will be returned at maturity. is a bond whose price stands above its face amount, and with no credit and no tax anywhere in the arithmetic there is exactly one thing that can put it there: a coupon fixed at an earlier issue that is higher than the coupon a bond issued today would carry.

Take an invented older government bond carrying 8.50 per cent a year on Rs 1,000.00/- of face, and price what is left of it at two of the recorded horizons. With one year left it pays Rs 1,085.00/- on its final date. Divided once by 1.0590 that is Rs 1,024.551464/-, so it stands Rs 24.551464/- above face while the one year SPOT rate reads 5.90 per cent a year. With three years left it pays Rs 85.00/-, Rs 85.00/- and Rs 1,085.00/-. On the three recorded SPOT rates those three payments are worth Rs 80.264400/-, Rs 75.294118/- and Rs 896.952358/-, adding to Rs 1,052.510876/-, a premium of Rs 52.510876/- over face.

What the premium is the present value of

$$ \text{Premium} = (c_{fixed} - c_{par}) \times F \times \sum_{t=1}^{n} d_{t} $$

c_fixed	the coupon rate fixed at the earlier issue, here 8.50 per cent a year
c_par	the coupon a bond issued today over the same dates would carry
F	the face amount, Rs 1,000.00/-
d_t	the discount factor for each remaining payment date

What it says in wordsThe premium is the extra coupon this bond pays over what a bond issued today at par would pay, multiplied by the face amount and then valued at today's recorded rates across every payment date that is left, so it is worth the saving on the remaining dates and not one rupee more.

A relationship shown once is a recipe; a relationship that closes on its own numbers has been tested. Run the check at both horizons. With one year left, a bond issued today at par would carry 5.90 per cent, so the extra coupon is 2.60 percentage points, Rs 26.00/- a year. Multiplied by the one year discount factor of 0.9442870633 that comes to Rs 24.551464/-, the premium exactly. With three years left, a bond issued today at par would carry 6.523518 per cent, so the extra coupon is 1.976482 percentage points, Rs 19.764824/- a year. Multiplied by the three discount factors added together, 2.6567844132, it comes to Rs 52.510876/-, again the premium exactly. The premium is the present value of the coupon above what a bond issued today at par would pay, and it is neither a quality mark nor a mistake in the price.

The everyday version makes the same point in one breath. A shop lease signed five years ago at a rent below today's rent changes hands for a payment, and everybody in the market knows roughly what that payment is worth: exactly the saving on the years still to run. Nobody pays for the years already used and nobody pays twice for the same saving. Notice too that the premium grew from Rs 24.551464/- to Rs 52.510876/- purely because there were more dates left to collect the extra coupon on.

The band above the face line more than doubles between the one year and three year cases without the coupon changing at all, because the extra coupon is collected on more remaining dates rather than because anything about the borrower moved.

Try it out

A government bond carrying an 8.50 per cent annual coupon prices at Rs 1,052.510876/- with three years left. What is the Rs 52.510876/- above face paying for?

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What is the curve saying about future rates, and what is it not saying?

Two kinds of rate now have to be held apart, and mixing them is the costliest mistake on the whole schedule. A SPOT rate covers money placed today. A FORWARD rateThe rate for money placed at one future date and returned at a later one. A FORWARD rate is already sitting inside the SPOT curve rather than being a separate opinion about the future. covers money placed at a future date and returned later still, and it is not a separate opinion about anything. The recorded schedule produces it by division.

Here is why. A lender with two years to spare has two roads. Road one: place money for two years at the two year SPOT rate of 6.25 per cent a year. Road two: place it for one year at the one year SPOT rate of 5.90 per cent a year, then place the proceeds for a second year at whatever rate the second year turns out to carry. If the schedule is to hold together, the rate that makes road two land exactly where road one lands is already determined by the two recorded rates, and it is the one year one year FORWARD rate.

Step one, the two roads must end together

$$ (1 + s_{1}) \times (1 + f_{1,1}) = (1 + s_{2})^{2} $$

s₁	the one year SPOT rate, 5.90 per cent a year, as a decimal
s₂	the two year SPOT rate, 6.25 per cent a year, as a decimal
f_1,1	the one year one year FORWARD rate, covering the second year alone

What it says in wordsOne rupee rolled for one year at the shortest recorded rate and then for a second year at the FORWARD rate has to finish exactly where one rupee placed for the whole two years at the two year recorded rate finishes, because both roads cover the same stretch of time.

The equation has one unknown in it, so divide both sides by one plus the one year SPOT rate and the unknown stands alone. Nothing has been assumed and nothing has been forecast; a factor has simply been moved across.

Step two, the FORWARD rate stands alone

$$ f_{1,1} = \frac{(1 + s_{2})^{2}}{1 + s_{1}} - 1 $$

f_1,1	the one year one year FORWARD rate, as a decimal, here 0.0660115675
s₂	the two year SPOT rate, 6.25 per cent a year, so 1.0625 squared is 1.1289062500
s₁	the one year SPOT rate, 5.90 per cent a year, so the divisor is 1.0590

What it says in wordsThe rate covering the second year alone is what one rupee grows to across two years at the two year recorded rate, divided by what it grows to across the first year at the one year recorded rate, less the one rupee that was there at the start.

Evaluated with every term visible: 1.0625 squared is 1.1289062500. Divided by 1.0590 that gives 1.0660115675. Taking away one and multiplying by a hundred, the one year one year FORWARD rate is 6.601157 per cent a year. The check runs the other way too. 1.0590 multiplied by 1.0660115675 returns 1.1289062500 exactly, and 1.1289062500 is 1.0625 squared. The FORWARD rate is arithmetic pulled out of today's recorded schedule, and nobody anywhere has promised that the one year SPOT rate a year from now will be anything at all.

Each rung of the ladder is a division anybody can repeat, and the two bars underneath show that the derived rate and the recorded three year rate cover stretches of time that barely overlap despite reading only five basis points apart.

Try it out

The one year SPOT rate is 5.90 per cent a year and the two year SPOT rate is 6.25 per cent a year. What rate for the second year alone is already implied by those two?

Which brings the labelling rule, and the arithmetic cannot be written without it. Every rate carries the word SPOT or the word FORWARD. A SPOT rate is the rate for money placed today and returned at one stated future date. A FORWARD rate is the rate for money placed at one future date and returned at a later one. The two are different objects, and on this invented schedule they sit close enough to be mistaken for one another. Nothing has been nudged apart to make them safer. The label does the work.

The error that gets made, and what it costs

A reader meets a table with the one year one year FORWARD rate of 6.601157 per cent a year on one line and the three year SPOT rate of 6.55 per cent a year on another. Two numbers, five basis points apart, no labels on the rows. The reader treats them as the same fact written twice.

The two rates are not the same fact, and they are not even about the same stretch of time. The three year SPOT rate of 6.55 per cent covers thirty six months starting now, money placed TODAY and returned in THREE years. The one year one year FORWARD rate, which came out of 1.0625 squared over 1.0590, covers months thirteen to twenty four, money placed A YEAR FROM NOW and returned a year after that. One of them includes the coming twelve months and the other one begins after they are over.

Who makes this error: everybody, the first time they read a schedule of rates whose rows are not labelled, and most often the reader who was handed a FORWARD rate quoted bare instead of derived. The cost: a comparison run between two rates that describe different stretches of time, so a conclusion reached about one gets acted on against the other. Nothing in the arithmetic is wrong, so the arithmetic never complains. The gap here is 0.051157 percentage points, 5.1157 basis points carried unrounded and five basis points at the rounding this record prints. A derived rate lands near a recorded one on any smooth schedule, and that is why the gap is small.

The repair is one line: write the word SPOT or the word FORWARD beside every rate, every time, and the trap cannot spring.

Try it out

The one year one year FORWARD rate works out at 6.601157 per cent a year from the two year SPOT rate of 6.25 per cent and the one year SPOT rate of 5.90 per cent, and the three year SPOT rate reads 6.55 per cent. What is each of the two measuring?

A forward rate already sits inside the sovereign curve. See what it never says.

Why is every other bond in this market priced against this borrower?

Palash Cements Limited, an invented cement maker, borrows for five years at 9.10 per cent a year. The five year SPOT rate on the invented government schedule is 6.90 per cent a year. Subtract, and 2.20 percentage points is what is left, 220 basis points in the other unit. The amount stacked on top of the government leg is a spreadThe amount a borrower pays above the government rate for the same horizon. A spread is always over something and always for a stated length of time., and what it implies about expected loss is covered separately.

A rate difference is easy to shrug at, so price it. Rs 1,000.00/- promised in five years costs Rs 716.327252/- discounted at the five year government SPOT rate. The identical promise from the company, discounted at its own 9.10 per cent a year, costs Rs 646.958238/-. The difference on every Rs 1,000.00/- of five year face is Rs 69.369015/-, and that number is a good deal harder to shrug at than 2.20.

The same Rs 1,000.00/- promised in five years	Discount rate a year	Cost today
Discounted at the five year government SPOT rate	6.90 per cent	Rs 716.327252/-
Discounted at Palash Cements Limited's own rate	9.10 per cent	Rs 646.958238/-
The gap, on Rs 1,000.00/- of five year face	2.20 points, 220 basis points	Rs 69.369015/-

The government leg is the reference rather than merely one quote among many because it is the only rate in the comparison that both borrowers share, so it is the only thing that can be held still while the other part moves. That is the whole of it. Not that the state is safe, but that the state borrows at every horizon anybody else borrows at, so there is always a matching leg to subtract. Consider two food stalls outside the same office building: the only way to say which is dearer is to compare the same plate on the same day, and the government schedule is the same plate on the same day for a borrowing market.

Try it out

Palash Cements Limited borrows for five years at 9.10 per cent a year while the five year government SPOT rate is 6.90 per cent a year. What is the gap worth on Rs 1,000.00/- of five year face?

Where does this schedule stop, and what may not be read off it?

Six horizons are recorded here and nothing between them: one, two, three, five, ten and thirty years. There is no four year SPOT rate in this record, no nine year SPOT rate and no twenty nine year SPOT rate, and there is nothing at all shorter than one year. Each recorded horizon is a nodeOne of the horizons at which a rate is actually recorded, as against a horizon read off a line somebody drew through the recorded points., and the stretches between the nodes are empty rather than merely undrawn.

No smooth line is drawn through the recorded points to read a value off. Interpolating produces figures that depend on the method somebody chose, so a straight line reading and a curved reading would disagree, and two treatments working from the same record would then print two different numbers for the same object. Where a reader expects a rate in between, the rate is simply not recorded. Drawing the empty stretches as empty is more honest than filling them, and it is the thing most pictures of a rate schedule get wrong.

The missing horizons have a cost right here. A three year government bond touches exactly three dates, and the one year, two year and three year SPOT rates are all recorded, so it can be priced. A five year bond touches the four year date as well, and no four year SPOT rate is recorded, so it cannot be priced. The par bond and the premium bond worked above therefore stop at three years rather than running out to five.

One writing rule travels with all of this. No rate goes up or down. Up and down mean the price in one sentence and the yield in the next, so a move is written as a rise in the yield or a fall in the yield, every single time. Prose that mixes the two ends up saying the opposite of its own arithmetic without anybody noticing.

Try it out

What is the four year SPOT rate on this schedule?

Who actually reads a schedule of government rates this way, and what for?

A lending desk reads it as the floor under its own quote. Before anybody argues about what a five year loan to a company should cost, somebody looks up what the state pays for the same five years. The government leg is common to both and can be held still, so the argument narrows to the amount stacked on top, a much smaller and much more answerable question than what should this loan cost.

An analyst reads the shape rather than the level, and reads it in the normalised form. Being told that the schedule rises 1.70 percentage points from the shortest recorded horizon to the longest carries very little; being told that the first extra year of waiting costs 0.3500 percentage points and the twentieth to thirtieth cost 0.0125 each carries a great deal. The second version says where in the schedule the money is actually being charged.

A household meets the same arithmetic without any of the vocabulary. Somebody offered a lump sum today instead of a fixed amount five years from now is doing exactly the discounting sum worked above, and the only question that matters is which rate is being used to shrink the future amount and whether that rate was disclosed. A price and a rate are the same statement written two ways, and being handed only one of them is how a household ends up unable to check the other.

And anybody who has to write about rates uses the labelling rule as a working discipline rather than as a style note. Writing SPOT or FORWARD beside every rate takes one extra word and removes an entire class of error that arithmetic alone cannot catch.

India

Where the rules on all of this actually live

No rule set governs any step above except the compounding convention, and that convention sits inside the arithmetic because a sum cannot be reproduced without it. Each row below is a real rule set the arithmetic brushes against.

How a government security is issued, and through what route. The Reserve Bank of India, rbi.org.in.
The mechanic by which a government security first reaches a holder. The Reserve Bank of India, rbi.org.in.
The schedule on which government borrowing is offered. The Reserve Bank of India, rbi.org.in.
How a benchmark government yield curve is constructed and published. The Clearing Corporation of India Limited, ccilindia.com, and the Reserve Bank of India, rbi.org.in.
Which security is treated as the reference at a given maturity, and how that is decided. The Clearing Corporation of India Limited, ccilindia.com.
Who may hold and deal in government securities, and under what conditions. The Reserve Bank of India, rbi.org.in.
The valuation norm that decides the price at which a holding is carried. The Reserve Bank of India, rbi.org.in.
The compounding convention a published yield is stated on. The Reserve Bank of India, rbi.org.in, with any measured series at its data site, dbie.rbi.org.in.
What an issuer must disclose in the terms of a bond it offers. The Securities and Exchange Board of India (SEBI), sebi.gov.in.

How a government security is issued, through what route it first reaches a holder, who may hold one and how the market for one is organised are all covered separately. Short-dated government borrowing is covered separately. How a policy rate and a bond yield are linked is covered separately. What a government borrows in a year, why it borrows and what a deficit is belong to the borrowing programme, covered separately; the subject here is the instrument. How far a price moves for a given move in yield is established earlier. What a spread implies about the loss an issuer might cause is covered separately. The par bond set against the premium bond in full detail is covered separately.

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References

Source	Named for	Where
The Reserve Bank of India	How a government security is issued and through what route, the mechanic by which one first reaches a holder, the schedule on which government borrowing is offered, who may hold and deal in government securities, the valuation norm that decides a carrying price, and the compounding convention a published yield is stated on	rbi.org.in
The Reserve Bank of India, data site	The route to any measured series of Indian government yields	dbie.rbi.org.in
The Clearing Corporation of India Limited	How a benchmark government yield curve is constructed and published, and which security is treated as the reference at a given maturity	ccilindia.com
SEBI	What an issuer must disclose in the terms of a bond it offers	sebi.gov.in

The schedule of SPOT rates, Palash Cements Limited and the older 8.50 per cent government bond are invented.
Educational material. Not advice on any investment, tax, budget or market position.