Bond Ladder: Six Rungs of One Borrower, Priced Out

A bond ladder is a holding of one borrower spread over several maturity dates instead of one, so money returns at intervals rather than in a single lump. A promise gets cheaper the further out the date it falls due. Give every rung the same Rs 1,000.00/- of face at six recorded horizons and the rungs cost wildly different sums, Rs 3,976.206964/- between them.

Nobody builds a ladder by choosing prices; everybody builds it by choosing face amounts, and every misdescription of a ladder afterwards comes from carrying on describing it in the units it was chosen in rather than the units it was paid in. A face amount is a decision. A price is a consequence. The two are only the same number at a horizon where the rate is nil, and no rate on the schedule used here is nil.

The instrument below takes the two figures a builder actually has, a face amount at each rung and a rate at each rung's own horizon, and prices the ladder rung by rung. The instrument opens on the six rungs worked through below, and it can be driven into the misdescription named at the foot of it.

Price a ladder, rung by rung

What makes a set of holdings a ladder rather than a collection?

Two conditions, and there is no third. The holdings mature at several different dates rather than at one, and those dates are spread across a range rather than bunched together. The two conditions are the entire definition of a bond ladderHoldings of one borrower falling due at several spread out dates rather than at a single date., and everything else people attach to the word is commentary on it rather than part of it.

The definition carries no rule about how much sits at each date, no claim about what the arrangement achieves, and no view about which dates are worth choosing. A ladder is a description of a shape, not an argument for one.

The shape is older than any bond market. A household holds six insurance covers and renews one every year, in a different month, rather than letting all six fall due in the same week. Nothing about the household changes across those six renewals; what changes is how long it has been since each cover was last looked at, so a decision comes round regularly instead of arriving all at once. Substituting a repayment for a renewal gives a ladder.

The finance version rests on one observation that is easy to skate past. The schedule is not a list of six separate rates. The schedule records one borrower at six different horizons, and the six rates are what that single borrower is charged for money returned at six different dates. Nothing about the borrower changes between the one year point and the thirty year point; what changes is how long the lender waits. A ladder built on one borrower is therefore a clean object to price: everything that varies across the six rungsOne holding in a ladder, identified by the date on which it falls due. is time.

Where does each rung's price come from, and what does it cost?

Input one is the face amountWhat a rung promises to repay on its due date, which is the number a builder usually picks first. of the rung, the number the builder chooses. A ladder is built that way in practice: somebody decides how much should come back on each date, and here that decision is Rs 1,000.00/- at every rung. The face amount is read off the terms of the holding, never off a market, and it is never solved for.

Input two is the SPOT rateThe rate for money placed today and returned at one stated future date, with nothing paid in between. at the rung's horizon, read off a schedule rather than chosen. The schedule used throughout is an invented SPOT curve carrying six points and nothing else: 5.90 per cent at one year, 6.25 at two, 6.55 at three, 6.90 at five, 7.35 at ten and 7.60 at thirty. The curve is not any market's schedule and it is not a forecast. How a benchmark government curve is built and published is settled by the Clearing Corporation of India Limited at ccilindia.com.

The third thing is not an input but a convention. Every rate and every price here is struck on annual compoundingOne discounting period a year, so an amount due in three years is divided by one plus the rate three times.: one discounting period a year, so an amount falling due in three years at the three year SPOT rate of 6.55 per cent is divided by 1.0655 three times over. The convention is not housekeeping. The same six rates read on a half yearly convention give six different prices, and a reader who is not told which convention is in force cannot reproduce a single sum here. Which convention a published yield is stated on is set by the Reserve Bank of India at rbi.org.in.

Run it six times. At the one year SPOT rate of 5.90 per cent the rung costs Rs 944.287063/-. At the two year SPOT rate of 6.25 per cent, Rs 885.813149/-. At the three year SPOT rate of 6.55 per cent, Rs 826.684201/-. At the five year SPOT rate of 6.90 per cent, Rs 716.327252/-. At the ten year SPOT rate of 7.35 per cent, Rs 492.016324/-. At the thirty year SPOT rate of 7.60 per cent, Rs 111.078974/-. Six identical promises, six prices that share almost nothing.

Now add them, in order. The running total reads Rs 944.287063/-, then Rs 1,830.100212/-, then Rs 2,656.784413/-, then Rs 3,373.111665/-, then Rs 3,865.127990/-, and the ladder closes at Rs 3,976.206964/- against Rs 6,000.00/- of face, a difference of Rs 2,023.793036/-. Watch where it gets to and where it stops: after three rungs it is already two thirds of the way, and the last three rungs, reaching from year five out to year thirty, add barely a third between them.

Why does equal face at every rung buy such unequal amounts of money?

Because the builder chose the promise and the schedule chose the price. Every rung here promises exactly Rs 1,000.00/-, so any description that counts face amounts reports a perfectly even arrangement: six rungs, one sixth apiece, evenly spread from one year out to thirty. Count what was paid instead and the evenness disappears completely.

Run the division six times and the share of costOne rung's price over the total price of the ladder, as against one rung's face over the total face. comes out at 23.7484 per cent for the one year rung, 22.2778 per cent for the two year rung, 20.7908 per cent for the three year rung, 18.0153 per cent for the five year rung, 12.3740 per cent for the ten year rung and 2.7936 per cent for the thirty year rung. The six shares add to the whole of the Rs 3,976.206964/- paid. Adding up is not by itself a check: shares built on face would read one sixth six times over and would also add to the whole. A column adding up proves nothing until the base it was struck on is named.

Here is the arithmetic underneath that spread. A promise thirty years out is divided by 1.0760 thirty times over and a promise one year out is divided by 1.0590 once, so the thirty year rung buys exactly the same promise as the one year rung and takes under an eighth of the money to do it. Set against its own face instead, the thirty year rung at Rs 111.078974/- costs a shade under a ninth of the Rs 1,000.00/- it will repay. Two comparisons, two answers, both worth having. A share figure that does not name its base is unreadable.

Where does the money in this ladder actually sit?

One number settles it. Weight each horizon by what the rung at that horizon cost, add the six products, and divide by what the whole ladder cost.

The six products are Rs 944.287063/-, Rs 1,771.626298/-, Rs 2,480.052603/-, Rs 3,581.636261/-, Rs 4,920.163244/- and Rs 3,332.369223/-, each one a horizon in years multiplied by a cost in rupees. Added as printed they come to Rs 17,030.134692/- of rupee years, a millionth of a rupee below the unrounded sum because each product is rounded before it is printed. Divide by the Rs 3,976.206964/- the ladder cost and the cost weighted average horizonEach horizon weighted by what that rung cost rather than by what it promises to repay. comes to 4.283010 years.

The picture suggests something else entirely. Six rungs, the furthest thirty years out, and the money averages a bit over four years of waiting. The average lands there because the three nearest rungs carry 66.82 per cent of the cost between them, so a ladder built on equal face amounts is dominated by its short end no matter how far its long end reaches. The thirty year rung brings the largest horizon to the numerator and one of the smallest costs, and the two do not cancel: 2.7936 per cent of the money cannot drag an average far, however distant its date.

The cost weighted average horizon is the figure to reach for when somebody asks how long a ladder is, and it is almost never the figure they are given. The longest rung is the one that gets remembered and repeated. A thirty year rung is memorable. Rs 111.078974/- is not.

Which dates does this ladder not have, and can they be filled in?

A ladder returns money on the dates its rungs fall due and on no other date. The ladder priced above pays at one, two, three, five, ten and thirty years: no year four, no year seven, no year twenty, and nothing at all inside the first twelve months. The honest reading is that a ladder does not smooth money out across time; it concentrates money on the dates it was built for, and the gapsA stretch between two rungs where the ladder returns nothing and the schedule records no rate. are as much a part of the structure as the rungs.

Why these dates and not tidier ones? Because of an absence in the recorded schedule. The invented SPOT curve carries six points and nothing between them: no four year rate, no nine year rate, no twenty nine year rate, and nothing shorter than a year. A rung with no rate has no price, so a rung can only sit where a rate sits.

The tempting move is to draw a line between the recorded points and read a value off it wherever one is wanted. The move is not made here, and the reason is not fussiness: a straight line reading and a curved reading disagree, so two desks working from one schedule would print two different prices for one object. Drawing the gaps as gaps is more honest than filling them, and it is the thing most pictures of a curve quietly get wrong.

A second consequence follows. The rungs here are unevenly spaced because the recorded horizons are: one year, another, another, then a two year stretch, then a five year stretch, then twenty years of nothing. Almost every drawing of a ladder shows evenly spaced rungs instead, and why these cannot be evened out is taken up further down.

What comes back, and when, and is the difference a return?

The schedule of arrivals is short enough to say in one breath. Rs 1,000.00/- at the end of year one, another at the end of year two, another at the end of year three, then a two year wait for the fourth, a five year wait for the fifth and twenty years for the sixth. Six arrivals, Rs 6,000.00/- in all, against Rs 3,976.206964/- paid on day one, a difference of Rs 2,023.793036/-.

The difference is the interest earned on six separate claims over six different lengths of time, and it is not a return. A return needs a period, and this figure spans thirty of them. The arithmetic is easy: Rs 2,023.793036/- divided by Rs 3,976.206964/- gives 50.8976 per cent, a number that looks like an answer and belongs to no stretch of time at all. Over what? The one year rung finished its work in twelve months while the thirty year rung had not started paying, so dividing one side by the other divides a thirty year quantity by a same day quantity.

The same shape turns up in a household. A cousin lends Rs 5,000.00/- to a neighbour who repays over one year and Rs 5,000.00/- to a nephew who repays over eight, then adds the two profits and divides by Rs 10,000.00/-. The two arrangements never ran alongside each other, so nobody would call that an annual return. Six rungs stacked across thirty years are the same objection multiplied.

The difference decomposes into six honest per rung amounts, each attached to its own stretch of time: Rs 55.712937/- over one year, Rs 114.186851/- over two, Rs 173.315799/- over three, Rs 283.672748/- over five, Rs 507.983676/- over ten and Rs 888.921026/- over thirty. The six amounts add to Rs 2,023.793036/-, and that is the check that closes. Each one has a period attached, so each one can be turned back into the rate it came from. The total has no period, so the total cannot.

How is the ladder read one rung at a time?

The control below draws two readings of whichever rung is selected: what it is by face, which never changes, and what it is by cost, which changes at every position. Moved from one end to the other, the top bar refuses to budge while the bottom bar collapses. The contradiction, held on one screen, is the whole of this guide.

Every figure the instrument opens on also stands as plain text here, so a reader who never touches it has the whole worked case: the one year rung is 16.6667 per cent of the ladder by face, cost Rs 944.287063/- at the one year SPOT rate of 5.90 per cent, and carries 23.7484 per cent of the Rs 3,976.206964/- paid, against a cost weighted average horizon of 4.283010 years. Push the control to the far end and the face bar has not moved a pixel while the cost bar has fallen to 2.7936 per cent, at a price of Rs 111.078974/-.

What can a single schedule of rates never say about a ladder?

Almost everything written anywhere about ladders concerns what they do when rates move. Answering that on one recorded schedule would mean inventing the second schedule the question asks about.

The schedule carries one set of six SPOT rates and no second set. Pricing this ladder after a rise in the yield or a fall in the yield across the schedule would take a second set that was never recorded. The amount earned when a rung falls due and the money is placed again needs the rates ruling on the dates the rungs fall due, and the schedule records none of them. The instrument above refuses in those words rather than estimating. The cell for what a ladder does when rates move is drawn below and left empty, with the reason written inside it.

One writing rule holds the whole thing together. No rate in this guide ever goes up or down; a move is written as a rise in the yield or a fall in the yield, every time. Up and down mean the price in one sentence and the yield in the next, and an account that lets the two senses mix ends up asserting the opposite of its own arithmetic with nobody noticing. Whether a ladder is better or worse than a single holding turns on that same missing scenario, so the comparison cannot be made from one schedule either.

Why are the rungs unevenly spaced, and could a tidier ladder be priced?

Here is the one sense in which this ladder is complete. Every rung sits at a horizon the schedule carries, so every price here can be rebuilt from six SPOT rates and one face amount. The completeness is stronger than it sounds. The four year and six year SPOT rates are not on the schedule, so a tidier ladder with rungs at one, two, three, four, five and six years would look better on a slide and could not be priced here at all. The neater picture would have been the less honest one.

Which brings in the labelling rule this guide cannot be written without. Every rate carries the word SPOT or the word FORWARD. A SPOT rate is for money placed today and returned at one stated future date; a FORWARD rate is for money placed at one future date and returned at a later one. Different objects entirely, and on this schedule they sit close enough together to be merged by a reader who meets them unlabelled. A quoted FORWARD rate looks like somebody's opinion about the future and a derived one is visibly arithmetic. How close the two sit shows in deriving the one year one year FORWARD rate from the two SPOT rates that already contain it.

The equality holds because both sides describe the same two years on the same recorded schedule with the same annual compounding, so writing it assumes nothing about the future. Rearranging for the unknown is ordinary algebra.

Put the two numbers in. One plus 0.0625 multiplied by itself gives 1.12890625. Divide by 1.0590 for 1.06601157. Take one away and the one year one year FORWARD rate is 6.601157 per cent a year. Set it beside the three year SPOT rate of 6.55 per cent. The FORWARD rate sits 0.051157 percentage points above the SPOT rate, or 5.1157 basis points, and the two are completely different objects. On any smooth schedule FORWARD rates land near SPOT rates, so the label does the work of separating them.

Note what the FORWARD rate does not do here: it does not price a rung. Every rung is priced off a SPOT rate at a horizon the schedule records, and the FORWARD rate appears only to show that the two objects are told apart by their labels rather than by their sizes.

Who actually runs this arithmetic, and what do they do with it?

Four people, and each one wants a different column out of the same table.

A household treasurer wants the schedule of arrivals and nothing else. School fees fall due in three years and a roof needs replacing in five, so the question is which dates money lands on and the answer is the face column: Rs 1,000.00/- at year one, another at year two, another at year three, then five, then ten, then thirty. Naming the base here means saying that face is the right base for that question.

Whoever has to say how much money is committed and for how long wants the cost column and the cost weighted average horizon. The cost reading reverses the picture: 66.82 per cent of the money paid sits in the first three rungs and the arrangement averages 4.283010 years rather than anything close to thirty, so a description leading with the thirty year rung is describing 2.7936 per cent of the money.

A lender looking at the arrangement as security wants the same cost column for a different reason. The worth of a holding today is a different question from what it repays later, and the two differ by Rs 2,023.793036/- here. Whoever reports the arrangement onward wants both columns side by side with the base named on each. The failure described above is a base that went unnamed between one desk and the next.

A price and a schedule of arrivals do not say what a ladder achieves. The arithmetic above describes what this ladder costs, what it promises, when it pays, and how the money inside it is distributed. Every claim about what a ladder protects against would need a scenario in which the schedule of rates moves, and no such scenario is recorded.

The schedule of SPOT rates used throughout and Palash Cements Limited are invented.
Educational material. Not advice on any investment, tax, budget or market position.