How to Analyse a Bond Ladder That Is Already Built

Reading a ladder prepared by somebody else is seven steps in a fixed order. The analyst checks that the dates are spread. The analyst counts where the money sits and names the base counted on. The length is reduced to one cost weighted figure. The dates become waits. The arrival schedule is written. What the holding cannot settle is sorted. Then it is described, and there it stops.

Somebody slides a statement across the table. Six holdings, six dates, one borrower, and a question: what is this holding? Designing and pricing a ladder from scratch is a different job, covered separately. The rungs are fixed, the prices were paid, and every figure below was settled before the reader arrived. The entire task is to read what is already there without adding a single assumption to it, and the seven steps below exist so that two people reading the same statement write down the same things.

Reading without adding an assumption is stricter than it sounds. A reader who has just met a schedule of rates will want to know what the holding would be worth if those rates moved, and will reach for an answer. A reader who sees a gap between two dates will want to fill it. A reader who sees a thirty year date at the bottom of the list will describe the holding as long. All three instincts produce sentences that sound like analysis and are not, and the procedure below is arranged so that each of them meets a step that says no.

Every rate used in this guide comes from an invented SPOT curve carried across this sequence, with six recorded horizons and nothing between them. Every price in this guide was struck on annual compoundingOne discounting period a year. An amount due in three years is divided by one plus the yearly rate, three times over, rather than by a half-yearly rate six times.. The same six rates on a half-yearly convention give a different price for every single rung, so the convention is stated inside the arithmetic rather than in a note beneath it.

What are the seven steps for reading a ladder prepared by somebody else?

Seven, run in order, and no step may start before the one above it has produced its answer. Step one is a check rather than a measurement. Unusually, nothing is measured until the object has qualified. Steps two through five each produce one figure or one list. Step six sorts. Step seven writes.

1. Ask whether it is a ladder at allThe dates are written out in order and looked at before anything else on the statement is read. Two conditions: the holdings fall due at several different dates rather than at one, and those dates are spread rather than clustered.
   Checking: are there several distinct dates, and are they spread across years rather than months?
2. Find where the money sits, and name the baseDivide each rung's price by the total price to get its share. Write the base into the same sentence as the share, every time.
   Checking: does every share in the record say whether it was struck on cost or on face?
3. Answer how long the ladder is, in one numberWeight each horizon by what that rung cost and add. Set the result beside the date of the last rung. Most readers give that date instead.
   Checking: is the horizon weighted by price rather than by face or by nothing at all?
4. Read the waits rather than the rungsTurn six dates into the stretches between them, starting from the day the money left. A list of dates and a list of waits describe the same holding and read nothing alike.
   Checking: do the waits add back to the date of the last rung?
5. Write the schedule of what comes back and whenAmount by amount, date by date, with the total set against what was paid. Do not divide the two.
   Checking: does every line carry a date, and does the total match the face across all rungs?
6. Sort the readings into those the holding supports and those it does notTwo lists. Anything in the second list gets a row with the reason for the emptiness written inside it.
   Checking: is every unsupported row carrying a reason rather than a blank or a plausible figure?
7. Report it as a description, and stopSay what is there. Do not say whether it is well built, whether a rung is misplaced, or whether anything should change.
   Checking: could any sentence in the report be read as advice about what to hold?

Notice what the right hand side of that drawing does not contain. The drawing carries no row for the value of the holding after a rate move, no row for the earnings on money that has come back, and no row for whether the shape is a sensible one. Those are the three things a reader most wants and the three things a ladder statement cannot settle on its own. They are not omitted by accident; step six collects them and hands them back with a reason attached.

What is actually in the holding, rung by rung?

Six rungsOne holding inside a ladder, identified by the date it falls due. Six rungs means six separate dated claims held at once., each promising Rs 1,000.00/- of face, one at each of the six recorded horizons of the invented SPOT curve this sequence works from. The rung that falls due in one year cost Rs 944.287063/- at the one year SPOT rate of 5.90 per cent a year. The two year rung cost Rs 885.813149/- at the two year SPOT rate of 6.25 per cent. The three year rung cost Rs 826.684201/- at the three year SPOT rate of 6.55 per cent. The five year rung cost Rs 716.327252/- at the five year SPOT rate of 6.90 per cent. The ten year rung cost Rs 492.016324/- at the ten year SPOT rate of 7.35 per cent. And the rung that falls due in thirty years cost Rs 111.078974/- at the thirty year SPOT rate of 7.60 per cent.

Every one of those six rates carries the word SPOT, and that is not decoration. A SPOT rate is the rate for money placed today and returned at one stated future date. A rung is exactly that arrangement. A FORWARD rate is a different object. The invented curve carried here places one FORWARD rate close enough to one SPOT rate that a reader meeting them unlabelled will merge the two, so the reporting step comes back to the difference.

The rounding residue, named where a reader will meet it

Add the six printed prices in the column above and the result is Rs 3,976.206963/-, one micro-rupee below the Rs 3,976.206964/- printed on the total line. Nothing is wrong. Each of the six prices was rounded to six decimals for printing, six roundings each shed a fraction, and the total was rounded once from the unrounded sum of Rs 3,976.206963953952/-. The record's total is the correct one, and the column's is the one that can be reproduced with a calculator from the printed figures. Every column here is printed to six decimals, so every column carries a residue in its last place, and each residue is named where it appears rather than quietly tidied away.

The reason for saying so is that a reader who adds the column, lands one digit away and cannot see why will conclude the mistake was theirs. That is the worst thing a teaching text can do, and it is entirely avoidable by saying which rounding produced which figure. A column that adds exactly comes from rounding the six prices to four decimals instead: Rs 944.2871/-, Rs 885.8131/-, Rs 826.6842/-, Rs 716.3273/-, Rs 492.0163/- and Rs 111.0790/- add to Rs 3,976.2070/-, and the unrounded total gives the same figure at four decimals. The check closes at four decimals and carries a residue at six. Both statements are true, and a report should say which precision it is working at.

Where does the money sit, and on which base is it counted?

Step two. And it opens with a question that is not rhetorical: a share of what? A share of costOne rung's price divided by the total price of the ladder, as against one rung's face divided by the total face. The two produce different numbers for the same rung. and a share of face are both correct, both easy to compute, and they answer entirely different questions about the same holding.

On face, this ladder is perfectly even: every rung promises Rs 1,000.00/- out of Rs 6,000.00/-, so each is one sixth, 16.6667 per cent, by construction. On cost, nothing is even. The rung that falls due in one year took 23.7484 per cent of the money and the rung that falls due in thirty years took 2.7936 per cent, and the three nearest rungs took 66.82 per cent between them.

The shares of cost as printed add to 99.9999 per cent rather than to 100.0000. The rounding residue named earlier is wearing different clothes: six figures rounded to four decimals shed a fraction each. Round the same six shares to two decimals instead, giving 23.75, 22.28, 20.79, 18.02, 12.37 and 2.79, and they add to exactly 100.00. Every one of the six shares is a slice of the same denominator, so the underlying six add to one exactly. A share column that fails to add to its whole at some sensible precision is the first sign the base moved partway down the column.

Think of a household saving through six recurring deposits that all mature for the same amount but were started at six different times. Counted by what each will hand back, the six are identical. Counted by how much of this month's salary each one is currently absorbing, they are nothing like identical, and the second count is the one that shows where the household's money actually is right now. A ladder read by face and a ladder read by cost are the same two counts, on the same holding, in the same week.

How long is this ladder, in a single number?

Step three, and it is the step a reader is most often asked to produce out loud. Somebody wants one figure. Give them the cost weighted average horizonEach rung's horizon multiplied by that rung's share of cost, added across the ladder. It reports the average date of the money rather than the average of the dates.: weight each horizon by what that rung cost, add, and this ladder comes to 4.283010 years.

Now set that beside the two figures a reader reaches for instead. The longest rungThe last date on a ladder. It reports where the final rupee of face is, which is a much smaller question than where the money is. falls due in thirty years. The plain average of the six dates is 8.5 years, and weighting by face gives the same figure. One plus two plus three plus five plus ten plus thirty is fifty one, and fifty one over six is 8.5. Three numbers, three different questions, one holding. Only the cost weighted figure answers where the money is. In that figure alone, a rupee that was never spent cannot vote.

The last column repays slow reading, and it holds the surprise of the whole exercise. The rung that falls due in ten years contributes 1.237401 years to the answer, more than any other single rung. The rung that falls due in thirty years sits three times further out and had so little money put into it that it contributes only 0.838077 years. A horizon three times as distant, carrying two thirds as much weight in the answer. The printed column adds to 4.283009 against the unrounded 4.283010. The six decimal residue is at work again, and it is not a discrepancy.

What do the waits between the rungs tell a holder?

Step four, and it is the step that changes the description of this ladder more than any other. A list of six dates reads as a tidy structure. Turn the same six dates into the stretches between them and the tidiness disappears.

Count from the day the money left. The first waitThe stretch between two consecutive arrivals, counted from the day the ladder was paid for. It is what a holder actually lives through, as against the dates, which are what the statement lists. runs one year, from the payment to the first rung. Then one year to the second rung, one year to the third, two years to the fifth, five years to the tenth, and twenty years to the last. Six waits: one, one, one, two, five and twenty years.

Six dates leave five gaps between them, so a reader who counts only the gaps finds five waits rather than six. The gap count leaves out the stretch a holder feels first, the one between handing the money over and getting any of it back. Five gaps add to twenty nine years rather than thirty, so the closing check fails. When the waits start at the day of payment, the column adds to the date of the last rung, and the analyst knows from that total that nothing was dropped.

Why the waits are uneven, and why no rung may be added inside one

The waits on this ladder are uneven because the horizons the invented SPOT curve records are uneven. Six nodesA horizon at which a rate is actually recorded. A rung can be priced at a node and nowhere else, because a rate is needed and only the nodes carry one. are carried: one, two, three, five, ten 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.

So no line is drawn between the recorded points to read a value off it, and that refusal has a reason beyond tidiness. Interpolating produces a figure that depends on the method chosen: a straight line between two points and a curve fitted through several give different answers at the same horizon, so two readers working from the identical six rates would print two different prices for the identical rung and both would be right by their own method. Where a reader expects a rate between two recorded horizons, the rate is simply not in the record. Drawing the gaps as gaps is more honest than filling them.

What comes back, on what dates, and is the gap a return?

Step five writes the schedule of returnsThe dated amounts a holding pays back, written date by date rather than summarised. Each line carries an amount and the date it arrives.. On this ladder it is the simplest table in this guide: Rs 1,000.00/- at the end of year one, Rs 1,000.00/- at the end of year two, Rs 1,000.00/- at the end of year three, Rs 1,000.00/- at the end of year five, Rs 1,000.00/- at the end of year ten and Rs 1,000.00/- at the end of year thirty. Six equal amounts landing on six unequal dates.

Six thousand rupees of face arrives in total, against Rs 3,976.206964/- handed over at the start, and the difference is Rs 2,023.793036/-. Somebody will want to divide that difference by the cost and call the result the ladder's return. Do not. A return belongs to a stated period, and this difference spans thirty years of periods at once, so dividing it by the cost produces a number attached to no stretch of time at all.

One reading the schedule does support, and it is worth having: the running total of what has arrived first passes what was paid at the fifth year rung. By the end of year three the ladder has returned Rs 3,000.00/-, below the Rs 3,976.206964/- paid. By the end of year five it has returned Rs 4,000.00/-, above that cost by Rs 23.793036/-. The crossing is arithmetic on the amounts and the dates, nothing more. Money that arrived in year one and money that arrived in year five are not the same thing, so the comparison claims nothing about whether the holder is ahead and does not pretend the two arrivals are alike.

Which readings does this ladder refuse to support?

Step six sorts. Everything a reader might want to say about this holding goes into one of two lists, and the sort is mechanical rather than a matter of judgement: can the figure be computed from the dates, the prices and the recorded rates alone, or does it need something that is not there?

Supported, and each computed above: the six dates, what each rung cost, the share of cost with its base named, the cost weighted average horizon of 4.283010 years, the six waits, and the schedule of what arrives when. Not supported, and each written as an empty cellA row in a report that carries the reason it could not be filled, written where the figure would have gone. It is output rather than unfinished work. with its reason inside: the value of the ladder after a rise in the yield or a fall in the yield, the earnings on each rung's money once it has come back and been put to work again, and the cost of a rung at any horizon the record does not carry.

Look hard at the second empty row. People fill it without noticing. Rs 1,000.00/- arrives at the end of year one and something happens to it next. Whatever that something is, it earns at a rate ruling on a date a year from now, and no such rate exists in this record. The record carries only the rate for money placed today and returned in a year, a SPOT rate for an entirely different transaction. Substituting one for the other is not an approximation; it is answering a question that was not asked.

One writing rule runs underneath all of this. The word up is never attached to a rate. A movement is written as a rise in the yield or a fall in the yield, every time. In one sentence up would mean the price and in the next it would mean the yield, and a text that mixes the two ends up asserting the opposite of its own arithmetic without anybody catching it.

How is a ladder reported without smuggling a recommendation into it?

Step seven writes the thing somebody else will read. A ladder report is a description and it ends where description ends. It states what the dates are. It states where the money sits and on which base the count was made. It gives the cost weighted average horizon. It gives the waits. It gives the schedule. It names the readings that are not available and why.

Four sentences must not appear, and each of them fails for the same reason: it is a judgement about what the holder wants or about where rates are going, and neither of those is in a ladder or in this record. That the ladder is well spread. That the last rung is too long. That the near end is too heavy. That anything about it should be changed. Every one of those sounds like the conclusion a careful reader would draw, and the resemblance is exactly what makes them hard to keep out.

The labelling rule the report cannot be written without

Every rate that appears anywhere in the report 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. Two different objects, and on the invented curve carried here they sit close enough together to be merged by a reader who meets them unlabelled.

Now look at how close that lands to something else on the same curve. The one year FORWARD rate for the year beginning a year from today works out at 6.601157 per cent a year. The three year SPOT rate reads 6.55 per cent a year. The two sit 0.051157 percentage points apart, or 5.1157 basis points, and they are completely different objects describing completely different transactions. Forwards land near spots on any smooth schedule of rates, and pulling them apart would make the record a fiction; the label does the separating instead.

Who reads a ladder this way, and at what moment?

Three people, at three different moments, and the procedure earns its keep differently for each.

A treasury officer at an institution reads a ladder the week a holding is inherited, either because a colleague left or because two books were merged. The dates arrive on a statement and the prices arrive from somewhere else, exactly the circumstance the failure block describes. The reason step two comes before step three is this reader: the shares of cost have to be struck before the horizon can be, and a horizon computed from the dates alone is the wrong number produced confidently.

An analyst writing a note about somebody else's holding reads it because the note will outlive the conversation. An analyst writing for others gets the most out of step six. The note travels and the caveats do not travel with it unless they are rows. Six months later a colleague picks up the note, sees a filled row for value after a rate move, and has no way of knowing the figure was supplied rather than found. An empty row with a reason inside survives that journey; a plausible figure does not.

A household reads it at the kitchen table, and the everyday shape is the one worth holding onto. A couple saving for a wedding four years out has money in six places that mature on six dates. Somebody asks how long their savings are tied up. The last deposit might be a long-dated certificate holding a small amount, so its date is not the honest answer. It is the date the bulk of the money comes free, and the only way to get at it is to weight each date by how much money is actually parked there. That is step three, run with a calculator at a kitchen table, and it is the same arithmetic to the last decimal.

All three readers share a moment where a number is about to be said out loud and there is no time to derive it. A fixed order earns its keep there, because it produces the same answer whoever runs it and whenever they run it. The procedure is not clever. It is repeatable, and on a holding somebody else built, repeatable is worth more than clever.

The whole read, in one artefact

Here is what step seven produces for this ladder, written out. Nothing in it is new; every row was computed above, and the rows that could not be filled carry their reasons rather than a space.