The Policy Rate and a Bond Yield: What Separates Them

A policy rate is decided and announced by an authority for one very short lending period, so exactly one of it exists. A bond yield is solved out of a price paid for a dated set of payments, so one exists for every maturity being dealt in. On the invented SPOT curve here, the one year and thirty year rates sit 1.70 percentage points apart at one instant.

Underneath every other difference between these two numbers sits a single one: where each of them comes from. One is chosen by somebody and published. The other is not chosen by anybody at all. A bond yield is the answer to a sum, and the sum starts from a price that a buyer and a seller already agreed on. Because a price exists separately at every maturity that anyone is dealing in, the answer to that sum exists separately at every maturity too, and so there is a whole line of them at any instant rather than a single figure.

Where each number comes from produces every other difference between them. One authority announces one figure, so there is one policy rate; prices exist at every maturity, so there is a whole curve of bond yields. A policy rate changes on a decision. A bond yield changes the moment a price changes. A policy rate can be looked up. A bond yield has to be worked out. Asking, of any rate that arrives, whether it was announced or solved dissolves most of the confusion in this subject before any arithmetic has been done.

Who decides a policy rate, and what exactly does it attach to?

One of these two objects was written down by somebody, so start there. A policy rate is an announced rate. An authority meets, decides on a number, and publishes it, and the number then applies to lending or borrowing between that authority and the banking system for a stated and very short period. Nothing about it is derived. There is no sum behind it whose answer happens to be that figure; there is a decision, and then there is an announcement of the decision.

Two properties of that announced rate do all the work: it is SET rather than solved, and it attaches to ONE very short period rather than to a range of maturities. Hold both of them. Almost every mistake made with policy rates comes from forgetting the second one while remembering the first.

Think about what "very short" means here. The lending this rate attaches to is measured in days. The lending is money placed and returned inside the money marketThe place where money is lent and borrowed for very short periods, counted in days rather than in years.. In that part of the borrowing world the clock runs in days rather than in years. A ten year bond, a thirty year bond, a five year corporate borrowing: none of them are that. Longer borrowings sit far out on the same axis, and the announced rate does not sit on top of them.

Here is the everyday version. A shopkeeper who lends a neighbour money until Friday and a shopkeeper who lends a nephew money for a decade are not doing the same thing, even at the same shop. If the first shopkeeper announces a new rate for lending until Friday, they have changed one number, for one length of time, on one kind of arrangement. Whether the ten year arrangement changes at all is a separate matter and a separate conversation.

The level the announced rate is set at, what the corridorA band of announced rates sitting around a central one. What it contains and how each rate in it is used is published by the Reserve Bank of India. of announced rates around it contains, by what process the decision is taken, and through which operations it is carried into the money market, are all settled by the Reserve Bank of India and published by the Reserve Bank of India at rbi.org.in. A level written down for any of them would be stale within a day and simply wrong within a week. The shape of the distinction being learned does not move at all.

A policy rate is one announced number, attached to a period of days, decided by an authority whose process and whose level are published at rbi.org.in. Every property that matters for the comparison below survives without the level.

Where does a bond yield come from, if nobody sets it?

A bond is a promise to hand over stated amounts on stated dates. Somebody pays a price today to receive that stream. The yield is the single rate at which those dated amounts, each pulled back to today, add up to exactly the price that was paid. Nothing else. The yield is the answer to a question of the form: at what rate does this pile of future money shrink to that price?

The direction is the whole point: the price comes first and the rate is solved out of it afterwards. Nobody sets a bond yield, and there is no announcement anywhere to go and look up. A yield is a description of a bargain that has already been struck.

Take Bond A, an invented ten year bond carrying an 8.50 per cent annual coupon on Rs 1,000.00/- of face. Every year for ten years it hands over Rs 85.00/-, and in the tenth year it also hands back the Rs 1,000.00/- redemption amountThe amount handed back when a borrowing ends, separate from the last interest payment even though the two arrive together.. Suppose the price agreed is Rs 1,000.00/-. Try 8.50 per cent as the rate and pull each of the eleven amounts back to today at that rate. The total lands on Rs 1,000.000000/-, exactly the price. So the yield is 8.50 per cent a year. Notice what happened and what did not: nobody decided on 8.50; the price decided it, and the arithmetic found it.

The same direction explains why a yield can change without anybody announcing anything. Somebody sells Bond A tomorrow at a different price. The dated amounts do not change. Each one was fixed when the bond was issued, and nothing in the world can alter it. The price is different. So the rate that makes those unchanged amounts add to the new price must be different too. The yield moved because a price moved, and no meeting was held.

Bond A and every other price shown here are struck on annual compounding: the discounting takes one step a year, so 8.50 per cent a year means dividing an amount by 1.085 as many times as there are years to wait for it. Annual compounding is not bookkeeping detail and it is not a preference. Discounted on a different pace, the identical set of payments comes out at a different price. The convention therefore travels with the number rather than sitting in a footnote.

Why is one of them a single number and the other a whole curve?

Now put the two objects at the same instant and count them. One authority announced one number, so there is one announced rate. How many bond yields are there? As many as there are maturities being dealt in. Each maturity is a different set of dates, has its own price, and therefore has its own solved rate. There is nothing to stop the one year answer and the ten year answer from being far apart, and on any real day they are.

At one instant the market has already produced a different answer at each maturity, and those answers are not close together, so a single announced number cannot be the rate for every maturity at once.

A SPOT rate covers money handed over now and handed back on one named future date, with nothing at all happening in between. The invented curve here reads: one year 5.90 per cent, two years 6.25 per cent, three years 6.55 per cent, five years 6.90 per cent, ten years 7.35 per cent, thirty years 7.60 per cent, every one of them on annual compounding. The six figures are made up for teaching, and the curve they draw is a made up curve.

Why must every rate carry the word SPOT?

Because rates of two completely different kinds can read almost the same, and a rate written down without its label can no longer be identified as either. A SPOT rate covers a loan starting now and ending on one named date. A FORWARD rate covers a loan that starts later and ends later still. The two are not rival opinions about anything: a FORWARD rate is already buried inside a line of SPOT rates, and arithmetic digs it out. Pulling one out is covered under forward rates.

The invented curve used throughout contains a near miss: a FORWARD rate pulled out of it lands close enough to one of the SPOT rates on it that an unlabelled figure could honestly be either. The near miss is not an accident: on any smooth curve forwards do land near spots. The cure is to label every single rate rather than to move any number. The word SPOT therefore appears beside every rate, and a rate that arrives without its label should not be used.

How far apart do the rates on one curve sit at a single instant?

Rather than assert that the spread of rates is wide, the subtraction settles it. At the two ends of the invented SPOT curve, the one year SPOT rate is 5.90 per cent a year and the thirty year SPOT rate is 7.60 per cent a year, so 7.60 less 5.90 is 1.70 percentage points. One percentage point contains a hundred basis points, so the same distance written the other way reads as 170 basis points. Same quantity, two units, and they are never used interchangeably here because doing so is the fastest way to be wrong by a factor of a hundred.

The same subtraction on a shorter span shows this is not a trick played with the extremes. The two year SPOT rate is 6.25 per cent a year and the ten year SPOT rate is 7.35 per cent a year, so 7.35 less 6.25 is 1.10 percentage points, or 110 basis points. Even that inner stretch, ignoring both ends of the curve, is wider than most announced rate changes ever reported.

No single announced number can be 5.90 and 6.25 and 6.55 and 6.90 and 7.35 and 7.60 at the same instant, and that is not a limitation of the authority, it is the arithmetic refusing. Six different sets of dates were priced by six different bargains, and six different answers came out.

The five steps in that table add to 1.70 percentage points, the figure the end to end row reads. All five are positive, and that is what makes the drawn line rise from left to right. Notice that the steps are not equal. The largest single step in this invented curve is the 0.45 points between five years and ten years, and the smallest is the 0.25 points between ten years and thirty years. The line rises and flattens. A rising, flattening line is a shape, and a shape has more in it than a level does. One announced number could not carry that information even if somebody wanted it to.

What does an announced rate actually reach directly?

Having accepted that a policy rate is one number attached to a period of days, the natural next thought is that everything else follows along behind it. Something does happen, and the word for the route is transmission, but the route is worth looking at carefully rather than assuming.

An announced rate lands directly on the thing it is attached to: very short lending and borrowing between the authority and the banking system. Very short lending is the only place it lands by definition rather than by anybody's choice. Everything beyond that period is reached only because people who deal in longer paper decide to do something differently, and a decision is not an equation.

The route from an announced very short rate to a thirty year price runs through other people's judgement, not through arithmetic, so nothing in the mathematics forces a long dated price to move at all when a very short rate is changed.

The everyday version has exactly the same shape. Take it first. A landlord who changes the rent on next month's tenancy has changed one number, directly, for one month, with nobody's agreement needed but their own. The rent being asked on a ten year lease in the same building is a completely different question. The ten year rent depends on what other people take that one month change to mean about the next ten years: whether it is the start of something, whether it will be reversed next month, whether it says anything at all about a decade. Some of them will conclude one thing and some another, and the ten year figure moves, or does not, on the balance of those conclusions. The landlord did not set it.

The same shape in bonds. A ten year bond's price is a bargain about ten years of dated payments. Somebody buying it at a price is making a statement about what those ten years are worth to them today. When an announced very short rate changes, that buyer has learned something, and what they do with it depends entirely on what they read into it. If they take the change as a one off with nothing behind it, the ten year bargain may barely shift. If they take it as the first step in a long sequence, they may reprice the whole decade. Both are judgements, and no arithmetic settles which one people make.

There is a second, quieter reason the route is indirect, and it is arithmetic rather than psychology. Even if a very short rate were somehow to reach directly into a long bond, it would only be entitled to touch the part of that bond that arrives soon. Almost nothing about a ten year bond arrives soon. Pulling Bond A's price apart year by year shows how little of it arrives soon.

Why does a ten year yield not follow a very short rate step for step?

Here is where the arithmetic settles the argument. Bond A at a price of Rs 1,000.000000/- is not one payment; it is ten of them, arriving one a year for ten years, and the price is the sum of what each of them is worth today. So ask a simple question of that price: how much of it is money arriving soon, and how much of it is money arriving late?

Pull each payment back to today at 8.50 per cent, one step a year, and look at where the price actually sits. The first year's Rs 85.00/- comes back to Rs 78.341014/-, or 7.8341 per cent of the Rs 1,000.000000/- price, with the price as the base. The tenth year is different in kind. Year ten carries the last coupon and the redemption amount together: Rs 1,085.00/- arriving, coming back to Rs 479.879675/-, or 47.9880 per cent of the same price on the same base.

Nearly half of everything paid for Bond A is a single amount arriving ten years away, and a rate attached to a period of days has no direct hold on it whatsoever. The weighting is the whole answer to why a long dated yield does not follow an announced very short rate step for step. The lag is not stubbornness on the part of the market. The two numbers are about different money.

Two things are worth noting about that table. First, each of the ten printed shares was rounded to four places on its own, and nine of the roundings happened to go the same way, so the last column adds to 100.0001 per cent rather than to 100.0000. The unrounded shares add to exactly one hundred; the printed ones do not, and the check runs on the unrounded figures. Second, the split that does close cleanly is the two part one: the nine coupon dates come back to Rs 520.120325/-, or 52.0120 per cent of the price, and year ten comes back to Rs 479.879675/-, or 47.9880 per cent. Those two shares do print to 100.0000. Carried to enough places a split closes; carried to too few it stops closing. The rule is worth more than the example that taught it.

A reader meeting year ten for the first time often assumes it is huge for some mysterious reason, so split it open. Nothing about it is mysterious. Of the Rs 479.879675/- sitting in year ten, Rs 37.594260/- is the last coupon pulled back ten years and Rs 442.285415/- is the redemption amount pulled back ten years. The two add to Rs 479.879675/- exactly. The tenth year is enormous because it is the only year in which the borrower hands back the principal, and that is a property of how the bond was written, not of the discounting.

How do the two objects compare when they are set on one row each?

Descriptions in turn are easy to nod along to and hard to use. Both objects on the same five criteria, side by side, are harder to confuse. The criteria are the ones that actually separate them: who produces it, what period it attaches to, how many exist at one instant, what it takes to change it, and how it is read.

Down the middle column the grid gives one number with one job; down the right hand column it gives a whole line of numbers, none of which anybody chose.

Bond A makes the last row concrete, so close the grid on it. Bond A is priced at parA price equal to the amount handed back at the end. Nothing more and nothing less passes over the life of the borrowing than the stated interest., at Rs 1,000.00/-, and it yields 8.50 per cent a year for exactly one reason: Rs 1,000.00/- is what its dated payments come back to when they are discounted at that rate. Nobody announced 8.50. If tomorrow somebody pays a different price for the same unchanged payments, the yield is a different number tomorrow, and there will be no announcement of that either.

Why does the compounding convention have to sit inside the arithmetic rather than under it?

Because the same figures on a different pace give a different answer, and the difference is large enough to matter while leaving no trace in the printed number.

The mismatch shows on Bond A. Every single thing about the bond stays the same: the same Rs 85.00/- arriving in each of ten years, the same Rs 1,000.00/- redemption amount in year ten, the same 8.50 per cent written on the front. Only the pace of the discounting changes: instead of one step of 8.50 per cent a year there are two steps of 4.25 per cent a half year. The half yearly pace is really 1.0425 multiplied by itself, giving 8.680625 per cent a year, or 8.68 per cent to two places. Discounted that way, the identical schedule comes out at Rs 988.243355/-, or Rs 988.2434/- to four places, against Rs 1,000.000000/- on the annual pace.

Identical payments, an identical rate written on the front, and a price Rs 11.7566/- apart on a Rs 1,000.00/- face, purely from the pace at which the discounting was taken. Round the two prices and it reads as Rs 11.76/-, but the gap is a difference of two full precision prices rather than a subtraction of two rounded ones. The four place figure is the one to carry.

Two things must be said clearly about that Rs 988.2434/-. The second figure demonstrates a mismatch; it is not a second convention for this instrument. Bond A is an annual compounding instrument throughout this material, and the second figure exists only to show what happens when somebody discounts it on a pace it was not written for. And the demonstration belongs in a treatment of policy rates and bond yields for this reason: a yield is a number solved out of a price, so a yield quoted without its pace is not a fact anybody can check. With the convention written beside the number a reader can reproduce the sum. Without it they cannot, no matter how many decimal places accompany the figure.

What may and may not be concluded when a policy rate changes?

Suppose a reader learns that an announced rate has been changed. Which conclusions follow?

One thing follows: an announced rate has been changed. The change is a fact about one number attached to a period of days, and it may be repeated.

The direction of any bond price, the size of any move, and the idea that the whole curve has shifted by the same amount do not follow. The reason is not caution. A single curve read at a single instant cannot show how one rate responds to another.

A curve read at one instant is a photograph, not a film. A photograph cannot show whether a curve shifts in parallel, twists, or barely moves when an announced rate changes. All three would look identical in the single frame. So any account of what a ten year price does when an announced rate changes would have to make up the evidence for its own conclusion, dress the invention as an illustration, and leave the reader with a habit that will cost them the first time the made up pattern and the real one disagree.

There is an honest reading available and it is worth having. A change in an announced rate is information. People who deal in longer paper receive that information and act on it or do not, according to what they take it to say about the years in between. The response of a price to that information is a separate question, and answering it takes evidence about moves that a single instant cannot supply.

The practical version, for the next headline. A reader may write down that a number was changed. A reader may write down which period that number attaches to. A reader may not write down what a ten year bond is now worth. A rule of thumb that converts one into the other is a made up relationship, and no arithmetic on the curve produces it.

Who actually uses this distinction, and for what?

The separation is not academic. Three different people meet it in three different shapes, and the same confusion costs each of them something different.

A treasury desk inside a lender lives on the difference. Its very short borrowing and lending genuinely does sit close to the announced rate: days are the period the announced rate attaches to. Its longer lending does not. So the desk cannot manage both sides of its own book with one number, and the first discipline anybody learns there is to say which maturity a rate belongs to before quoting it to a colleague. The same discipline that a headline calls for is, in that room, a rule about how sentences are spoken.

An analyst valuing a set of dated payments needs a rate for each date. A curve of SPOT rates is precisely that, and an announced rate is precisely not. Handed only an announced rate, the analyst has one number for the shortest money and nothing at all for years two through thirty. Handed a curve, they have a rate for every date and can do the work. Notice that this is a statement about arithmetic requirements, not about opinions: the sum literally cannot be completed with a single announced figure unless every payment arrives within days.

A household meets it in the plainest form of all. Somebody with a running home loan and somebody holding a ten year deposit have both read the same headline, and it is doing quite different things to them. The home loan is often tied by contract to something that moves with announced rates, so the connection there can be close and mechanical. The ten year deposit was struck at a rate that was fixed when it was struck, and no announcement rewrites it. Two people, one headline, two entirely unrelated consequences, and what separates them is exactly the question at issue throughout: what period does the rate attach to?

In all three shapes the useful habit is identical: before doing anything with a rate, establish who produced it and which stretch of time it covers. A number that fails either test is not yet usable.

Several rules, levels and timetables that this subject brushes against are set elsewhere and change without notice. The date on which a purchase of government securitiesBorrowings issued by a government. Who may hold them, and how a purchase is paid for and delivered, is decided by the Reserve Bank of India. settles on, the settlement conventionThe agreed timetable on which a purchase is paid for and the security handed over. The timetable itself is covered separately. that decides when money and paper actually change hands, the day count conventionThe agreed rule for counting the days between two dates when interest is worked out. The rule itself is covered separately. that governs how the days between two dates are counted, and what a company issuing corporate debtBorrowings issued by a company rather than by a government. The Securities and Exchange Board of India decides what must be disclosed about them. has to disclose: all of them are set by somebody, all of them change, and none of them is written here. The arithmetic above needs only one convention to be reproducible: the annual compounding stated inside every sum. A second market or a changed rule adds a row below rather than rewriting the argument.

Bond A and the SPOT curve it is set beside are invented.
Educational material. Not advice on any investment, tax, budget or market position.