What a Bond Yield Measures, and How the Measures Differ

A bond yield is the single rate that makes every future payment, discounted back to today, add up to the price being paid. The measures differ in how much of that arithmetic each one looks at: the coupon rate reads the terms the issuer fixed, the current yield reads one year of cash against the price, and the yield to maturity reads every dated payment at once.

Underneath all of it is one plain fact. A bond is a set of dated payments and nothing else. Each payment pushed back to today, and the pushed-back amounts added up, is the price. Every measure in this guide is a way of squeezing that same set of dated payments into one number, and what separates the measures is simply how much of the set each one is allowed to see. So the price is built first, in full, and only then does the naming start.

Building the price first and naming the measures afterwards matters more than it sounds. A total handed over cannot be checked, and nobody tries. A total built line by line can be rebuilt from scratch the next day. So wherever the answer could be written down, the addition is written instead and the answer falls out of it.

Where does the Bond Price come from in the first place?

Bond A, the invented ten year bond above, carries an 8.50 per cent annual coupon on Rs 1,000.00/- of face amountThe sum the issuer hands back on the last day, and the base the coupon percentage is applied to. It does not move when the price moves.. No issuer stands behind Bond A because this record supplies none. Bond A is a bulletA shape of bond that hands back the whole face amount in one lump on the final day, with nothing repaid earlier., so the face amount comes back once, at the end. The terms just listed are the whole instrument.

Eight and a half per cent of Rs 1,000.00/- is Rs 85/-, so the payments are Rs 85/- at the end of each of ten years, and Rs 1,000.00/- more at the end of the tenth. Eleven amounts in total, and the tenth year carries two of them. Now discountWorking backwards from a future amount to what it is worth today, by dividing it by one plus the rate, once for every period of waiting. each one at 8.50 per cent a year, with a running total kept alongside.

The price is not looked up anywhere; it is built, one line at a time, and the running total in the last column is the whole of the argument. Read the table downwards and watch it climb.

Two things fall out of that table before any measure has been named. The eleven payments add to Rs 1,850.00/- piled up undiscounted, and to Rs 1,000.000000/- once each has been pushed back to today. The Rs 850.00/- between those two totals is not a fee and it is not lost; it is the whole of the discounting, and it is what waiting costs. And the tenth year alone is worth Rs 479.879675/- today, very nearly half the price. Rs 1,085/- falls due on that single date: the tenth coupon and the repayment of the face amount arrive together.

One small trap in that paragraph is worth defusing before it misleads anybody. The Rs 850.00/- of discounting happens to equal the ten coupons of Rs 85/- added up, and it would be easy to read some meaning into that. There is none to read. The match is forced arithmetic and nothing else: the undiscounted pile is ten coupons plus the face amount, the price here is the face amount because this bond is priced at par, and subtracting one from the other can only leave the ten coupons. Priced anywhere other than at par, the same bond has the two figures separate immediately.

Why is the compounding convention part of the arithmetic rather than a footnote under it?

Every price in this guide is struck on annual compoundingTwelve months to a step. An amount due in five years is divided five separate times and no oftener, so the calendar and the arithmetic keep the same pace.: one discounting period a year, so a rate of 8.50 per cent a year means the amount is divided by 1.085 once for each year it is away. The convention is not housekeeping, and here is why it is not.

Take Bond B, an invented zero coupon bondA bond that pays nothing at all until the final day, when it hands over the face amount by itself. with Rs 1,000.00/- of face and a stated maturity of 7.1191 years, priced at the same 8.50 per cent a year. On annual compounding it comes to Rs 559.4640/-. Now demonstrate what a mismatched convention does to it: halve the rate to 4.25 per cent, apply it twice as often, and the identical terms come to Rs 552.8781/-. The two answers are Rs 6.5859/- apart on the same Rs 1,000.00/- of face, out of figures that look identical written down. The second number exposes the mismatch rather than offering Bond B a second convention; this record prices everything annually and Rs 559.4640/- is the price. A price with no convention beside it is a number nobody can check, and a number nobody can check has to be taken on trust rather than understood.

Bond B also carries a rounding point worth stating out loud rather than hiding. Its maturity is stated to four decimal places as 7.1191 years, and at that stated maturity the price is Rs 559.4640/-. Rounded, that is Rs 559.46/-. The unrounded maturity behind it is 7.119062643353 years, and at that the price is Rs 559.4657/-. Rounded, that is Rs 559.47/-. Both are correct and this record carries Rs 559.47/-. The gap is seventeen ten-thousandths of a rupee and it exists only because a maturity was rounded before it was used. Both figures are printed. Silently choosing one of two defensible figures teaches a reader to trust a number they cannot reproduce.

Why does one rate have to serve every date?

The table above used a single 8.50 per cent for all ten years, and that is easy to let pass. A single rate for all ten years should not pass unexamined. Money placed for one year and money placed for three years are not the same thing and are not priced the same. A whole curve of rates, not one rate, is what this record carries.

The invented curve used throughout carries a one year SPOT rateA rate covering one unbroken stretch that begins now and finishes on a named day. The only part of it lying ahead is the day it finishes on. of 5.90 per cent a year, a two year SPOT rate of 6.25 per cent and a three year SPOT rate of 6.55 per cent, all on annual compounding. Take just Bond A's first three coupons of Rs 85/- and discount each one at its own SPOT rate rather than at a single rate.

Now the question this whole guide turns on. Which single rate, applied to those same three cash flows, would land on the same Rs 225.826675/-? The answer is 6.329812 per cent a year. 6.329812 per cent is not the 6.233333 per cent average of the three SPOT rates, and it is not any one of the three: a yield is a compression, and something is always lost in a compression.

Why the single rate lands above the simple average is worth a sentence. The reason is the same reason a shopkeeper's average margin is not the average of the labels on the shelf. The weights are not the count of the payments; they are the discounted amounts. Being divided three times over rather than once, the three year payment is discounted at the highest of the three rates and is also the payment a change in the single rate moves most. So the later, higher-rate dates pull the answer up.

Why does every rate here say SPOT?

A SPOT rate is the rate for money placed today and returned at one stated future date. A FORWARD rateA rate covering a stretch that has not begun yet, running from one named day to a later one. It is worked out of the SPOT rates rather than watched on its own. is the rate for money placed at one future date and returned at a later one, and it is not a separate opinion about what happens next: it is already sitting inside the SPOT curve and can be pulled out of it by arithmetic. The extraction is covered separately.

The reason for the discipline is that on any smooth curve a FORWARD rate lands close to some SPOT rate, purely as a matter of shape, and a reader who meets two such numbers without labels will quietly merge them into one idea. Every rate here is a SPOT rate, and every one of them is labelled. The fix for two things that look alike is a label on each, never a nudge to one of the numbers.

So what is a Bond Yield, exactly?

Now the definition can be stated. The arithmetic behind it already exists above. A bond yield is the single rate that makes the discounted payments equal the price. A bond yield is no more than that: not a promise, not a return, and not a forecast.

The direction is what people reverse, and reversing it is the commonest confusion in this whole subject: the price is the observable thing, and the yield is solved backwards out of it, so a yield does not cause a price, it reads one. Nobody sets a yield and then works out what to charge. Somebody agrees a price, and the yield is what falls out of that price once the question is put: what one rate would have produced it.

Here is the everyday version. A small shop sells ten different items at ten different margins. The whole shop can still be described with one average margin, and the figure is genuinely useful: it compresses ten numbers into one that can be set against the shop next door. But nobody sets the average first and then chooses ten prices to match it. The takings come first and the average is worked out of them afterwards. A bond yield is exactly that, with dates attached, and it loses exactly the same kind of detail: the average margin cannot say which item is carrying the shop, and a yield cannot say which date is carrying the price.

What does the coupon rate, also called the Nominal Yield, read?

The coupon rate is the rate the issuer fixed at issue and then stopped touching. Bond A's is 8.50 per cent a year, and its base is the face amount of Rs 1,000.00/-, never the price. Whatever anyone pays for Bond A tomorrow, the coupon is Rs 85/- a year. The 8.50 per cent is applied to Rs 1,000.00/- and to nothing else. Nominal yield is simply the older name for that same rate, and the two terms mean one thing.

The word nominal carries a second, completely different meaning in this subject, and one sentence must never be made to carry both. The second sense means not adjusted for the change in prices, and it is the sense used further down when a real yield is worked. So a nominal yield in the first sense is a coupon rate; a nominal rate in the second sense is any rate before the change in prices has been taken out of it. Which of the two is meant is stated every time here, and the same is worth insisting on from any other source.

A household example makes the split obvious. A fixed deposit paying 7.00 per cent a year is described by that 7.00 per cent whether it was opened with Rs 20,000/- or with Rs 2,00,000/-. The rate is a term of the deposit rather than a description of what the holder gets relative to what they paid. The coupon rate sense works exactly like that. Whether the deposit leaves the household better off once things in the shop have got dearer is the other sense entirely, and no amount of staring at the 7.00 per cent will answer it.

What does the Current Yield read, and what can it not see?

The current yield is the shortest measure here and the easiest to compute mentally. One year of coupon divided by the price. The division is the whole of it, and the base of the ratio is the price paid, not the face amount.

On Bond A the coupon is Rs 85/- a year, so at a price of Rs 1,000.00/- the current yield is Rs 85/- over Rs 1,000.00/-, or 8.5000 per cent a year of the price. Move the price and the ratio moves with it while the coupon sits still. At a price of Rs 1,079.4804/-, a premiumA price sitting above the face amount, so the buyer hands over more today than the issuer will hand back at the end. to the face amount, the current yield is 7.8742 per cent a year of the price. At a discount priceA price sitting below the face amount, so the buyer hands over less today than the issuer will hand back at the end. of Rs 931.2325/-, it is 9.1277 per cent a year of the price.

Time is what the current yield cannot see: there are no dates in it at all, so it does not know when the face amount comes back, or that it comes back at Rs 1,000.00/- whatever was paid. The blindness to time is the whole limitation, and it is a large one. A buyer who paid Rs 1,079.4804/- will be handed Rs 1,000.00/- at the end, so Rs 79.4804/- of what they paid never comes back as face amount, and the current yield has no way of telling them. A buyer who paid Rs 931.2325/- will be handed Rs 1,000.00/- at the end, so Rs 68.7675/- arrives on top of every coupon, and the current yield cannot see that either.

Consider a street vendor who buys a cart for Rs 40,000/- and takes Rs 900/- a day. Dividing Rs 900/- by Rs 40,000/- describes today honestly and says nothing about whether the cart is worth anything in three years. The current yield is that division. The current yield is a true statement about one year, and a silent one about every other year.

What does the Yield to Maturity read, and what does it quietly assume?

The yield to maturity is the measure most people mean when they simply say yield. The yield to maturity is the single rate that discounts every remaining payment, the coupons and the final face amount together, back to the price. On Bond A at Rs 1,000.00/- that rate is 8.50 per cent a year, and it is exactly the arithmetic of the very first table above, read backwards. There, 8.50 per cent was given and the price was the question. Here, the price is given and the rate is the question. Same eleven amounts, same convention, question turned around.

Because it uses every payment and every date, the yield to maturity is the only measure so far that can see the face amount coming back. Seeing the face amount is why a bond bought at a premium has a yield to maturity below its current yield, and a bond bought at a discount has one above it: the yield to maturity has priced in the pull towards Rs 1,000.00/- at the end, and the current yield has not.

Which assumption is hiding inside that arithmetic?

Turn the definition around one more time and it says something stronger than most readers expect. If the yield to maturity is the one rate that makes the discounted payments equal the price, then the arithmetic that produced it has treated every coupon as though it went straight back out at that same rate until the end. The only way a holder actually earns the quoted yield to maturity over the whole life is if every coupon received is placed back at that same rate until maturity, and nothing in this record supports any claim about what a coupon will really earn when it arrives.

So the assumption is named, and the cost of the assumption failing is worked further down. The rate a coupon actually earns after it arrives depends on whatever is available on the day it lands, and nobody knows that in advance.

Why do the coupon rate and the yield land on the same number at par?

Bond A's coupon rate is 8.50 per cent a year on Rs 1,000.00/- of face, and its yield to maturity is 8.50 per cent a year. The two figures are the same number, and it is very easy to read the match as a happy accident. It is not.

The price is Rs 1,000.00/-, exactly the face amount. A bond whose price equals its face amount returns exactly its coupon each year and hands back exactly what was paid at the end. So the one rate that discounts that stream back to that price has to be the coupon rate; there is nothing else for it to be. The equality is what the words at par mean, and printing 8.50 and 8.50 side by side without saying why they are the same number throws the point away.

The same arithmetic governs a household loan taken at 11.00 per cent where every instalment is pure interest and the principal is repaid in one lump at the end. The rate on the loan and the rate the household is actually paying are the same figure, and they are the same figure precisely because nothing was repaid early and nothing extra was handed over. Change either of those and the two numbers separate immediately.

What does the Yield to Call read, and when does it bind instead?

Some bonds give the issuer a right to repay before the final date. Where that right exists, the same arithmetic can be run to the earlier date and to whatever must be paid on it, instead of to maturity and the face amount, and the answer is called the yield to call.

Neither Bond A nor Bond B is callable. Bond A is a bullet that repays once at the end, and Bond B pays nothing until its final day. So the comparison below runs on a supposed call feature, named as a supposition every time it appears.

Suppose a bond carrying Bond A's exact terms had also given its issuer the right to repay at the end of year five at Rs 1,040.00/-, and suppose it were priced at Rs 1,079.4804/-. Run the arithmetic to maturity and the yield to maturity is 7.35 per cent a year: ten coupons of Rs 85/- and Rs 1,000.00/- at year ten, discounted back to that price. Run the same arithmetic to the supposed call instead, using five coupons of Rs 85/- and Rs 1,040.00/- at year five, and the yield to call is 7.241232 per cent a year.

Two yields, one price, one bond, and they differ by 0.108768 percentage points a year, or 10.8768 basis points. The reason it matters is not the size of the gap but who chooses: the holder gets whichever outcome the issuer picks, rather than the one that was quoted to them. An issuer holding a right to repay early will tend to use it when repaying early suits the issuer. Repaying early suits the issuer at exactly the moment it does not suit the holder. Everything in this section belongs to the supposition; Bond A itself carries no call and no early repayment. The full working of this comparison is covered separately.

What is a Real Yield, and why is subtracting the wrong arithmetic?

Every rate so far has been stated in rupees. A real rate asks a different question: after things in the shop have got dearer over the same stretch of time, what is left? A nominal rate counts rupees. A real rate counts what the rupees buy.

A measured rate of price change is published by an authority and moves month by month, and no level of it is quoted below. So the arithmetic below runs on an assumed 5.00 per cent a year over the same period as the yield, the word assumed is written beside it every time it is used, and changing that assumption changes the answer while changing nothing at all about the method. How a measured rate of price change is compiled and released is set by the Reserve Bank of India at rbi.org.in, with the data site at dbie.rbi.org.in as the route to any series.

How do Nominal Yield vs Real Yield differ on one set of figures?

Take the 8.50 per cent a year nominal yield and the assumed 5.00 per cent a year change in prices. The instinct is to subtract, giving 3.50 per cent. The correct arithmetic is a division: 1.085 divided by 1.05 is 1.0333333, so the real rate is 3.333333 per cent a year.

So the subtraction is out by 0.166667 percentage points, or about 17 basis points. The error is not random: it points the same way every time. Whenever the nominal rate is above the assumed change in prices, the subtraction overstates the real rate. The direction is a property of the algebra, not of these particular numbers.

Now change only the assumption and watch. A very common belief goes wrong at exactly this step. Hold the nominal at 8.50 per cent and raise the assumed change in prices to 6.00 per cent. Dividing gives 1.085 over 1.06, or 2.358491 per cent a year. Subtracting gives 2.50 per cent. The error is now 0.141509 points, smaller than before rather than larger.

The received wisdom says the shortcut gets worse as rates rise, and that is true only when both rates rise together. Here only one of them moved. Read the error expression again and the direction is obvious: raising the assumed change in prices while holding the nominal rate shrinks the gap between them faster than it grows the multiplier, so the error falls. Raise both, say to a 17.00 per cent nominal against an assumed 10.00 per cent, and the same expression gives 0.636364 points, far worse than either row above. The honest statement is that the shortcut is always too high, and how badly depends on both rates rather than on either one alone.

How do the Yield Measures compare when they are put on one instrument?

Six measures have now been defined one at a time. Putting them on one instrument at three different prices is where the differences stop being definitions and start being visible. So hold Bond A completely still and move only the price.

At a yield of 7.35 per cent a year the price is Rs 1,079.4804/-. At 8.50 per cent it is Rs 1,000.0000/-. At 9.60 per cent it is Rs 931.2325/-. The bond has not changed in any of the three columns: same ten dates, same Rs 85/- coupon, same Rs 1,000.00/- face amount, same annual compounding. Only what somebody paid has changed.

The ladder is the whole payoff of the comparison: above par the coupon rate is the highest of the three and the yield to maturity the lowest, below par that order reverses exactly, and at par all three read 8.5000. Nothing about the bond produced that pattern. The price produced it.

The reason the order flips is the one thing the current yield cannot see. Bought at a premium, Rs 79.4804/- of the amount paid will not come back as face amount, so the measure that counts every date has to read lower than the measure that counts one year of cash. Bought at a discount, Rs 68.7675/- arrives at the end on top of every coupon, so the measure that counts every date has to read higher. The coupon rate, meanwhile, ignores the price entirely and never moves at all.

One warning before that ladder becomes a rule of thumb. The ladder compares three measures on one bond. The ladder does not compare two bonds, and it says nothing whatever about which price is the better one to pay: the arithmetic is identical in all three columns and only the price differs. The detailed head to head comparisons, of the current yield against the yield to maturity and of the coupon against the yield, are covered separately.

Bond Price vs Yield: which way does it go, and in what shape?

Put the yield on one axis and the price on the other. The relationship slopes downwards, exactly as every reader expects. Dividing by a bigger number gives a smaller answer, and every term in the price sum is divided by one plus the yield. The part almost nobody predicts is that the line is bent rather than straight.

Work the two halves of one move away from par. Take Bond A's yield down 100 basis pointsHundredths of a percentage point. A hundred of them make one percentage point, which is why 220 of them and 2.20 percentage points are the same quantity written two ways. to 7.50 per cent a year and the price goes to Rs 1,068.6408/-, a gain of Rs 68.6408/-. Take it up 100 basis points to 9.50 per cent a year and the price goes to Rs 937.2120/-, a loss of Rs 62.7880/-.

The gain is Rs 5.8528/- larger than the loss on the same sized move in the opposite direction, and equal moves producing unequal results is only possible if the line is bent. On a straight line the two would match to the paisa. The two do not match, and the difference is not a rounding artefact: it is Rs 5.8528/- on a Rs 1,000.00/- instrument, worked from the two prices rather than assumed.

Only the shape is settled above, without a number on the steepness of that bend. The measures that quantify how far a price moves for a given move in yield are covered separately and come after this, for a reason worth stating: a reader who meets a sensitivity multiplier before they can compute the price itself has learned a multiplier without knowing what it multiplies. Direction and rough size are what these particular figures give; the measurement itself is covered separately.

What does a spread read, and what does a Spread Return isolate?

Everything so far has treated a yield as one thing. On a corporate bond it is not one thing. A corporate yield is a government rate for the same maturity plus an amount stacked on top of it, and the two parts move for different reasons.

Palash Cements Limited, an invented issuer and the only one named anywhere in this record, has a five year bond carrying 9.10 per cent a year, against a five year government SPOT rate of 6.90 per cent a year on the invented curve, and the same Rs 1,000.00/- of face used elsewhere here. Do the subtraction: 9.10 less 6.90 is 2.20 percentage points.

2.20 percentage points and 220 basis points are the same quantity written in two units, and neither is ever written as the other. A basis point is one hundredth of a percentage point, so a hundred of them make one point and 220 of them make 2.20 points. Writing 2.20 basis points, or 220 percentage points, is the error worth checking for hardest. The mistake is out by a factor of a hundred, and it reads perfectly fluently.

Separating the two limbs matters because a holder of Palash Cements paper is exposed to two quite different things at once. The government limb moves when the price of lending to anybody for five years moves. The 2.20 point limb moves when what a lender demands from this particular issuer moves. A price change driven by the first is a different event from a price change driven by the second, even when the two arrive on the same day and land in the same total. A spread return is the reading that isolates the second limb, and it is worked in full separately. The expected loss a spread implies is covered separately too.

How to calculate a Bond's Yield and Return

The procedure is the part worth coming back to, so it is set out as steps rather than prose. Nothing in it is new; it is everything above, in the order it would actually be done.

One. Write down every payment with the date it lands on. For Bond A that is Rs 85/- at the end of each of years one to nine, and Rs 1,085/- at the end of year ten. Eleven amounts, ten dates.

Two. State the compounding convention. Here it is annual: one discounting period a year. Write it beside the working, not underneath it. The same schedule on a different convention gives a different answer, and there is no way to tell from the numbers alone which one was used.

Three, to get a price from a yield. Each payment is discounted at that yield and the results are added. One pass of arithmetic, no searching, and the answer is exact. The first table above does exactly that.

Four, to get a yield from a price. There is no closed formula, so the answer is searched for. A rate is tried, everything is discounted at it, the total is compared to the price, and the rate moves the opposite way to the error. Suppose the price is Rs 931.2325/-. At 8.00 per cent the payments discount to Rs 1,033.550407/-, too high, so the rate rises. At 9.00 per cent the total is Rs 967.911711/-, still too high. At 10.00 per cent it is Rs 907.831493/-, now too low, so the answer sits between 9.00 and 10.00. Closing in, 9.60 per cent lands on Rs 931.232534/- exactly. A spreadsheet is doing precisely this when it looks instantaneous.

Five, to get a return. A yield is not enough, and this is where the procedure stops being mechanical. The assumption about the coupons after they arrive has to be stated, and so does the assumption about the price at the end. The two assumptions are the entire difference between a yield and a return, and step five is where a reader either states an assumption or hides one. The first four steps produce a number; the fifth decides whether the number means anything.