Inflation-Linked Bonds: Principal That Moves With Prices

An inflation-linked bond keeps its coupon rate frozen and lets the principal underneath that rate travel with a measured index of prices. Each coupon is therefore a fixed percentage of a base that has already moved, and what comes back at maturity is the moved principal rather than the face amount printed at issue. Only the base is indexed. The rate never is.

Which part of an inflation-linked bond is the part that moves?

Picture a shopkeeper who rents a stall from a landlord and the two of them agree that the rent will be one fiftieth of whatever the stall takes in a month. The share is written down once and nobody touches it again. In a slow month the landlord receives less; in a busy month, more. Nothing in the agreement changed between those two months. The takings the share was applied to changed, and the rupees followed.

The stall agreement is the whole instrument, transplanted into a bond. An inflation-linked bondA bond whose principal is moved with a measured index of prices while the coupon rate written on it stays fixed for the life of the bond. writes down a coupon rate at issue and never revisits it. The bond does revisit the principal that the rate is applied to, on a schedule set out in the terms. The principal is moved with a measured index of prices, and the result is called the indexed principalThe face amount after it has been moved with the index. The indexed principal is the base every coupon is worked out on, and it is also the amount handed back at maturity..

The rate does not change and the base does. The coupon amount therefore changes, and the coupon rate on the term sheet stays the figure it was on the day of issue. Because the words inflation-linked sit closer to the word coupon than to the word principal, almost every reader meeting this instrument for the first time gets that sentence backwards. The mistake is understandable. The term sheet carries one fixed percentage and one moving quantity, and the fixed percentage is the one with the word coupon beside it.

Set an ordinary government bond next to it and the contrast becomes a single row. An ordinary bond applies a fixed coupon rate to Rs 1,000.00/- on the first date, to Rs 1,000.00/- on the second and to Rs 1,000.00/- on the third, and hands back Rs 1,000.00/- at the end. Every one of those four amounts is knowable on the morning of issue. An inflation-linked bond applies its fixed coupon rate to a base that is Rs 1,000.00/- only at the start, and none of the later amounts can be written down until the index has done whatever it is going to do.

Where does the 5.00 per cent used here come from?

The 5.00 per cent is declared, and it comes from nowhere else. The answer is uncomfortable and it is the honest one, and it stands before any figure below leans on it. No measured rate of change in prices enters the arithmetic at any point.

How a measured rate of change in prices is compiled and how often it is released is set and published by the Reserve Bank of India, at rbi.org.in, with its data site at dbie.rbi.org.in as the route to any series. Which series the amount repaid on an inflation-linked government security is indexed to is likewise set by the Reserve Bank of India at rbi.org.in. A figure copied from either would be wrong the first morning the series moved and would carry no sign of it. Both authorities are named, and neither figure is written out.

In place of a measured figure stands an assumptionA figure declared as an input to arithmetic rather than taken from any measurement. Changing the figure changes the answers, and the simulation lower down allows exactly that. declared in the open: prices change by 5.00 per cent a year, every year, for three years. The 5.00 per cent is an input to arithmetic and nothing else, neither a forecast nor a measurement, and every number below that leans on it says so in the same sentence. Without its condition a figure is mostly the condition, so a figure carried away from here travels with the condition attached.

One more rule has to be fixed before any sum runs, and it belongs inside the arithmetic rather than in a note under 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.: one discounting period a year, so an amount due in three years at the three year SPOT rateThe rate for money placed today and returned at one stated future date. There is a different one for each date, and together they make a schedule. of 6.55 per cent a year is divided by 1.0655 three times over. The convention is not housekeeping. The identical six numbers on a semi-annual convention give different prices and a different set of FORWARD rates, and a reader who has not been told which convention is in use cannot reproduce a single sum here.

What does the principal actually do over the three years?

Start with Rs 1,000.00/- of face on a three year inflation-linked bond and let the declared assumption do its work. After one year the principal is Rs 1,000.00/- multiplied by 1.05, or Rs 1,050.00/-. After two years it has been multiplied by 1.05 again, giving Rs 1,102.50/-. After three years, a third multiplication takes it to Rs 1,157.625000/-. Nothing subtle is happening here, and that is the point: the whole of the indexation is one multiplication repeated once a year.

Now notice what that last figure is. The repayment is the first genuine departure from every bond priced so far. The amount repaid at maturity on an inflation-linked bond is not the face amount, and on the declared assumption of 5.00 per cent a year it is Rs 1,157.625000/- against a face of Rs 1,000.00/-. Every ordinary bond priced so far returned exactly what it borrowed. An indexed bond returns what it borrowed after the index has been applied to it.

The moved repayment creates an exposure worth naming plainly. The holder's final repayment has stopped being a fixed sum and has become an assumption-dependent one. Rs 1,157.625000/- written into a spreadsheet without the condition beside it is a number roughly one seventh assumption by construction, and nothing in the cell will ever say so. The habit that prevents it is small and it is not negotiable: the assumption travels in the same sentence as the figure, every single time.

How is a real rate built rather than quoted?

Every rate on the invented SPOT curve used throughout is a nominal rateA rate with the change in prices still inside it. A nominal rate reports how many more rupees arrive, not how much more those rupees will buy.. Its one year SPOT rate is 5.90 per cent a year, its two year SPOT rate 6.25 per cent and its three year SPOT rate 6.55 per cent, and each of the three says how many more rupees arrive, not how much more those rupees will buy. A real rateA rate with the change in prices divided out of it rather than subtracted off, so it describes purchasing power rather than rupee count. answers the second question, and getting from one to the other is a division rather than a subtraction.

The cleanest way to see it is to build the discount factors rather than the rates. A nominal discount factor is one divided by one plus the SPOT rate, raised to the number of years. On this schedule those come to 0.94428706 for one year, 0.88581315 for two and 0.82668420 for three. A real discount factorA nominal discount factor multiplied back up by the assumed change in prices over the same number of years, which takes the price change out of the discounting. is that same figure multiplied back up by the assumed change in prices over the same stretch of years, which cancels the part of the discounting that was only compensating for prices.

Run it three times and watch each one build. For year one, 0.94428706 multiplied by 1.05 gives 0.99150142. For year two, 0.88581315 multiplied by 1.1025 gives 0.97660900. For year three, 0.82668420 multiplied by 1.157625 gives 0.95699030. The three real factors, 0.99150142, 0.97660900 and 0.95699030, are the entire real schedule needed here, and every one of them holds only while the declared 5.00 per cent a year assumption holds. Their sum, taken at the eight places shown, is 2.92510072, and that sum is about to do a job.

The third year taken on its own shows the division even more starkly. Divide 1.0655 by 1.05, subtract one, and the answer is 1.476190 per cent a year. The three year REAL SPOT rate on this declared assumption is that figure, and not 6.55 less 5.00, which would be 1.55 per cent. The two answers stand 0.073810 percentage points apart, or 7.3810 basis pointsOne hundredth of a percentage point. So 0.073810 percentage points is 7.3810 basis points, and 1.9397 basis points is 0.019397 of a percentage point., and the gap is the cross term that the subtraction throws away.

Why does the real par coupon sit below the real SPOT rate?

Earlier working established how a par couponThe coupon that puts a bond's price exactly at its face amount on a given schedule of rates. A par coupon is solved out of the schedule rather than chosen. is derived. Nominal par pricing on this same schedule landed on 6.523518 per cent a year for three years. The move here is identical in shape and the only thing that has been swapped is the schedule of factors. Take one less the final real factor, divide by the sum of all three real factors, and the answer is the coupon rate that puts the bond at its face amount in real terms.

With the numbers in: one less 0.95699030 is 0.04300970. Divided by 2.92510072 that is 0.01470367, or 1.470367 per cent a year. Set against the three year REAL SPOT rate of 1.476190 per cent a year, the two sit 0.5824 basis points apart, with the coupon underneath. The two earlier coupons are discounted at the lower earlier rates and drag the average down, so a par coupon must sit below the SPOT rate of the same maturity on an upward sloping schedule, in real terms for exactly the reason it does in nominal terms.

The real par coupon and the real SPOT rate are the tightest pair on this schedule, and the temptation to round them into agreement is real. Resist it. The gap is a property of the two objects rather than an artefact of the arithmetic, and it is named here at full precision precisely so that a reader who reproduces the sum and gets 1.4704 does not think something has gone wrong. Rounding a pair like this into one number is how a genuine structural fact quietly becomes invisible.

Does the whole instrument still price at its face amount?

Now assemble the thing and find out. Apply the real coupon rate of 1.470367 per cent a year to the principal as it moves, year by year. In year one the principal is Rs 1,050.00/- and the coupon is Rs 15.438849/-. In year two the principal is Rs 1,102.50/- and the coupon is Rs 16.210791/-. In year three the coupon is Rs 17.021330/-, and it arrives alongside the moved principal of Rs 1,157.625000/-, so the last cash flow is Rs 1,174.646330/-.

Here is the moment that makes the rest of the working believable, and it repays a slow reading. The three amounts are nominal rupees arriving on three dates, and nominal rupees are discounted on the nominal schedule. Rs 15.438849/- divided by 1.0590 once, Rs 16.210791/- divided by 1.0625 twice, and Rs 1,174.646330/- divided by 1.0655 three times, added together, come to Rs 1,000.00000000/-.

On the declared assumption holding exactly, an inflation-linked bond and an ordinary bond issued at par on the same schedule are worth precisely the same amount today. The equality is worth sitting with, and it dissolves the most common hope people bring to this instrument. Indexation is not a route to more. Indexation is a rearrangement of which part of the payment is fixed and which part is allowed to travel, and the market prices that rearrangement at nothing on the day the assumption is taken as true.

What breaks when the assumption does not hold?

Moving the control puts the answer in view rather than in a sentence that has to be trusted. A higher assumed change in prices makes the moved principal larger on every date, every coupon amount grows with the principal it is applied to, and the three flows discounted on the unchanged nominal schedule no longer land on Rs 1,000.00/-. A lower one runs the same three things the other way. At 0.00 per cent a year the principal never moves at all and the price falls well below the face amount. A coupon of 1.470367 per cent a year is a thin payment to be discounting at nominal rates near 6.55 per cent.

The real coupon rate of 1.470367 per cent a year does not move at any position of that control. The rate was fixed at issue and it is applied to whatever the principal turns out to be. The coupon amounts travel and the coupon rate does not, the distinction drawn at the outset and the one most worth carrying away.

Now the honest limit on everything above, and it is not a footnote. No measured rate of change in prices, no series and no history appears here. The arithmetic can therefore show what a different input does, and it cannot settle which input is the right one. The choice of input belongs to the Reserve Bank of India at rbi.org.in and to the series it publishes, with its data site at dbie.rbi.org.in as the route to any level.

One writing rule holds the whole account together and it is worth stating before the last block. No rate goes up or down anywhere here. Up and down mean the price in one sentence and the yield in the next, and an account that lets the two slide together ends up saying the opposite of its own arithmetic without a single reader noticing. A movement is therefore written as a rise in the yield or a fall in the yield, every single time.

Why can a five year version not be built?

Because the schedule stops. The SPOT curve used throughout carries six points and nothing whatsoever between them: one year, two years, three years, five years, ten years and thirty years. There is no four year SPOT rate here, no nine year SPOT rate and no twenty nine year SPOT rate, and nothing at all shorter than one year. The missing rates are a fact about the record rather than a gap somebody forgot to fill.

So no line is drawn between two recorded points and no value is read off one. Interpolating would give a figure that depends on the method chosen. A straight-line reading and a curved reading would disagree, and two accounts working from the identical record would then print two different numbers for the same object. Where a reader expects a rate in between, the accurate statement is that the rate is not in this record. Drawing the gaps as gaps is more truthful than filling them, and it is the thing most pictures of a curve get wrong.

The one, two and three year SPOT rates are all recorded, so the three year inflation-linked bond above can be built. The four year SPOT rate the second coupon of a five year bond would need is simply not there, so a five year one cannot. Notice that this is not a limitation of the instrument. The record is what limits it, and the limit is stated in the open: what can be built is built, and what cannot is named.

One last labelling rule matters more here than anywhere else in the subject. 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. SPOT and FORWARD rates are different objects, and on this schedule they sit close enough to be merged by a careless reader. The one year one year FORWARD rate, worked out of the two SPOT rates behind it, is 1.0625 squared divided by 1.0590, less one, or 6.601157 per cent a year. The three year SPOT rate reads 6.55 per cent. The two stand 0.051157 percentage points apart, or 5.1157 basis points. Nothing has been moved to separate them, and nothing will be. The label does the work.