How to Analyse a Bond's Price Sensitivity, Step by Step

Eight steps, in this order. Write down the schedule and the compounding convention. Discount every dated amount and turn each present value into a weight. Add the weighted years for the MACAULAY duration. Divide by one plus the yield for the MODIFIED duration. Compute convexity off the same present values. Estimate the price change, reprice in full, then record the error with its sign and list what stayed unmeasured.

Almost every figure in a sensitivity calculation is squeezed out of one table, and almost every wrong figure in one arrived through that table rather than through the arithmetic performed on it. So the order that matters here is not the order of the formulas. It is the order that puts the inputs on paper first and then hands each computed figure a test of its own. A bad number then announces itself at the step that made it, instead of four steps downstream. That is the whole design. Why a MACAULAY duration weights the way it does, and what convexity is measuring, are settled under those subjects themselves and are used here rather than rebuilt. A sensitivity run has three parts: a running order, a recording format at each stage, and a specific test each stage can and cannot pass.

What gets written down before any measure is computed?

Step one produces no measure at all, and people skip it for exactly that reason. The analyst writes down the workingThe intermediate lines somebody wrote on the way to an answer, as opposed to the answer sitting on its own at the bottom. before there is any working to do: every dated amount the bond promises, the number of years until each one arrives, the yield as a rate for one year, and the compounding convention that yield is quoted on.

Think of a household writing out a loan repayment schedule on the back of an envelope before deciding whether to prepay. Nobody argues about the arithmetic. The argument is about whether the twelfth instalment was Rs 14,000/- or Rs 14,400/-, and about whether the bank quoted the rate monthly or yearly. The loan letter settles the argument, and nothing else can. Once the envelope is written, the letter goes back in the drawer and every later number inherits whatever the envelope says.

The recording format is a table with one row per dated amount, sitting under a header line that carries the yield, the compounding convention, and whether the price is struck on a coupon date. The convention belongs on the header rather than in somebody's head because the identical amounts on a different convention give a different price, a different MACAULAY duration and a different convexity, and a run that never wrote the convention down cannot be reproduced by anybody, including the person who did it.

Here is step one on the ten year bullet bond that carries the rest of this guide. Ten dated amounts of Rs 85.00/- each, with Rs 1,000.00/- added onto the tenth. A yield of 8.50 per cent for one year. The clock here ticks once a year. The price is struck on a coupon date, so no accrued interest sits inside a single figure below.

The header line is not decoration but the only place on the whole run where a figure can be compared against something outside the run. Everything after the header line compares the run against itself.

How does the discounting and weighting work, and what does the weight column actually test?

Step two adds three columns to the table already written down. Discount every dated amount at the yield. Add the ten present values together and call the total the price. A weight is then one of those present values set against the total of all of them. The division turns a rupee figure into a share of the price. The last column multiplies each share by the year it belongs to.

The whole run of step two on the bullet bond is printed below rather than described, so that a reader can add it up by hand.

Adding the third column by hand lands on Rs 1,000.0000/- with nothing left over. A total that clean does not happen at every yield and cannot be assumed at any other one. The bond is at par, so the price it prints equals the face amount behind it.

Now the test, and it is worth slowing down here because most people believe it does something it cannot do. The weight column foots to one whatever bond it belongs to. A share of a total, added to every other share of that same total, comes back to the whole of it. So a wrong amount, a wrong date or a wrong yield shifts the parts and the total in step with each other, and the column still foots to one. The test does not see input errors. The one thing it sees is a divisor that did not come out of the column above it.

A divisor from somewhere else turns up more often than might be expected. Somebody prices the schedule, then divides by a price carried in from a quote screen, a previous run, or the face amount because the bond happened to be at par last week. Put a column that foots to Rs 1,000.0000/- over a carried-in Rs 995.72/- and the weights add to 1.0043. Nothing about the schedule is wrong. The divisor simply came from somewhere else, and the run has just said so for the cost of one addition.

There is a second thing the column can report, and this run shows it. Rounded to six places the ten weights add to 1.000001, not to 1.000000. Nine of the ten roundings went the same way. The weighted year column has the same shape: it foots to 7.119062 as printed against 7.119063 carried unrounded. Neither residualWhatever is left over once every accounted-for part has been taken away. is an error and neither gets forced away. The parts are printed as they round, the total the test is actually run on is printed alongside them, and each is labelled for what it is. A reader who adds the printed column by hand and lands one millionth away deserves to know that the difference was already understood.

How do the weighted years add into a MACAULAY duration?

Step three is one addition. Sum the weight times year column, and what falls out is the MACAULAY duration. On the bullet bond that column runs 0.078341, 0.144407, 0.199642, 0.245335, 0.282644, 0.312602, 0.336131, 0.354055, 0.367108 and 4.798797, and the total is 7.119062 as printed, 7.1191 years once rounded.

The recording format is the figure, the word MACAULAY sitting beside it, and the unit, and the unit is years. Not a percentage, not a multiple. Years. Write it any other way and the next person to pick the number up has to guess, and the guess they make is the expensive one.

The test at this step is that the answer lands strictly inside the dates on the schedule, and on any bond paying something before maturity it lands strictly below the maturity. A MACAULAY duration equal to the maturity says the bond pays nothing at all until the end. A MACAULAY duration above the maturity is not a surprising finding but an arithmetic impossibility. An average of ten dates cannot sit past the latest of them. Here 7.1191 lies between one year and ten, and it lies below ten because nine payments arrive before the tenth date, so the test passes.

The reach of that test is worth being clear about. The range test catches a date that is not on the schedule at all: a year typed as 100, a decimal point that went missing, a maturity carried over from a different instrument. The same test does not catch an amount sitting on the wrong row. Put the Rs 1,000.00/- repayment on year eight rather than year ten and the MACAULAY duration reads 6.3628 years. A reading of 6.3628 sits comfortably inside one to ten, and the test says nothing whatever.

How does the division for the MODIFIED duration work, and why record both?

Step four is one division, and the divisor is a single period's yield with one added to it. Put 7.1191 over 1.085 on the bullet bond and out comes 6.5613.

The recording format is the figure, the word MODIFIED beside it, and the unit. The unit says how many per cent the price shifts for each percentage point the yield shifts, and it carries no measure of time whatever. The two figures look alike. The pair reads 7.1191 and 6.5613, the two came out of the same table one line apart, and one of them is measured in years. The other is not. A pair that close is the single most reliable place in this whole run for a number to be picked up and used as though it were the other one.

The test here is the neatest in the run. The two figures check each other. Multiply the MODIFIED figure back by one plus the yield and the MACAULAY figure has to come back: 6.5613 times 1.085 returns 7.1191. Report a MACAULAY duration of 7.1191 years alongside a MODIFIED duration of 6.9000 and the pair fails on sight. Multiplying 6.9000 by 1.085 gives 7.4865 and not 7.1191.

Unlike step two, this test does reach outside the run. The divisor, one plus the yield for one period, is a separate input from the discounting, so a figure from a different compounding convention can walk in here and be caught. Somebody discounting once a year and then dividing by one plus half the yield produces exactly that mismatch, and the pair of durations refuses it.

How is convexity computed, and when is it worth the arithmetic?

Step five reuses the present values already sitting in the table. Weight each one by its year multiplied by that year plus one, add the lot, then set the total over the price and over the square of one plus the yield. On the bullet bond that gives 58.4702.

The recording format is the figure and the schedule it was computed on. Convexity belongs to a particular set of dated amounts and travels badly without them. Two things test the figure. Convexity has to come out positive on any instrument whose schedule is fixed. And it has to be larger for a schedule whose amounts are spread across many dates than for one whose whole weight sits on a single date.

The second test can be shown rather than asserted. A second bond on this run was built for exactly that comparison. The zero coupon bond pays nothing until it matures, and its maturity was fixed at the point where the bullet's MACAULAY duration lands. Its MODIFIED duration is therefore 6.5613 as well, identical by construction rather than by luck. Same MODIFIED duration, and convexity of 49.0986 against the bullet's 58.4702. Every rupee of the zero sits on one date, and the bullet's is spread across ten. The two readings stand 9.3716 apart on figures that report the same sensitivity.

Here is where the caveat has to go, in the same breath as the numbers rather than parked underneath them. Both bonds here are priced off the same 8.50 per cent. Nobody charged anything for the difference in curvature. Out in a market somebody would. A schedule that bends the holder's way is a thing buyers compete for, and competition of that kind turns up as a lower yield rather than as a line on an invoice. So the arithmetic above says what the two schedules do, and it does not say what either one costs. The measurement ends exactly where the price of it would begin, and pricing that difference is a separate subject.

One paise, explained in advance

The zero coupon bond stops paying at the moment the bullet's MACAULAY duration is used up. Read off the table above it is 7.1191 years; carried further, the division gives 7.11906264 years. Discount Rs 1,000.00/- back over the first of those and the result is Rs 559.4640/-. Over the second, Rs 559.4657/-. The zero's price above is Rs 559.47/-, so the longer figure was the one used. A reader working only from the printed table lands on Rs 559.46/- and has made no mistake at all. The same choice shifts the zero's convexity from 49.0986 to 49.0991. Both readings are set down here so nobody has to work out which one is wrong.

When the second term earns its arithmetic

Convexity costs a column of multiplications and a division, and there is no rule that it must always be paid. The size of the move being estimated is what decides, and this run can put a number on that rather than leaving it to taste. Taking the straight line on its own shows how far it misses.

At a move of one basis point the straight line lands Rs 0.000292/- away from the true repriced figure on Rs 1,000.00/- of face. A miss that size is under three hundredths of a paise. At five basis points it is Rs 0.0073/- away, still under a paise. At 200 basis points it is Rs 10.9315/- away. Forty times the move produced roughly 1,498.2 times the miss, so the miss grows far faster than the thing being measured, and that is the whole reason a threshold exists at all. Somebody sizing a one basis point move can skip the second term and lose nothing anybody could act on. Somebody sizing 200 basis points cannot.

How do the measures turn into an estimated price change?

Step six is where the table stops and the estimate begins. Set the MODIFIED duration against the move written as percentage points, and that product is the first reading. Then halve the convexity, apply it to the square of the same move written as a decimal, and add what comes out to the first reading to get the second.

Take a yield that rises by 200 basis points, the same move written another way as 2.00 percentage points. The slopeHow steeply one quantity changes against another right at the point being measured, before any bending is allowed for. reading is 6.5613 times 2.00, or 13.1227 per cent of the Rs 1,000.00/- the bond starts at, so Rs 868.7730/-. The curvature reading adds half of 58.4702 times 0.02 squared, or 1.1694 percentage points. The fall comes to 11.9533 per cent and the price to Rs 880.4671/-.

One thing about that first figure is worth flagging. A reader checking it will land a paise away and wonder where the slip lies. The 13.1227 per cent came from the MODIFIED duration carried unrounded at 6.56134806, not from the 6.5613 printed a step earlier. Feeding the printed figure in instead gives 13.1226 per cent and a price of Rs 868.7740/-, one paise adrift. A displayed figure is a display and not an input, and on this run the difference surfaces in the very first rupee amount the estimating produces.

The recording format is both readings, the move written in basis points and in percentage points, the base the percentage is taken on, and the rupee figures beside the percentages. The base matters because a percentage with no base attached is not a quantity anybody can act on, and here the base is the Rs 1,000.00/- price the run started from.

Step six also carries a labelling duty and no test. Both outputs are estimates. A straight line laid against a bending relationship always sits on one side of it, so the first reading is known in advance to overstate a loss and understate a gain. The second is closer and is still not the answer. The curvature term is one further term out of a longer run of them rather than the run itself, an expansionA way of writing a bending relationship as a run of terms, the first doing most of the work and each one after it doing less. cut short. Neither figure gets to be called the price change.

How does the full repricing work, and why do it every single time?

Step seven throws the estimates away and starts again. Take the schedule from step one, put all ten dated amounts through the discounting again at the new yield, add what comes back, and record the new price alongside the actual percentage change against the original base.

At 10.50 per cent for one year the bullet bond prices at Rs 879.7045/-, a fall of 12.0295 per cent of the Rs 1,000.00/- it started at. The same figure is quoted as 12.030 per cent to three places.

Step seven is a separate calculation and not a check on step six, and it gets performed every time rather than only when an estimate looks surprising. The distinction sounds pedantic and is not. A repricing performed only when something looks odd is a repricing whose absence says nothing. A wrong estimate that looks reasonable never triggers it. A repricing performed always yields an error figure on every single run, and it is that unbroken series of error figures, not any one of them, that shows whether the estimating is any good.

The test at this step is a boundary case. Put a move of nothing through the repricing and the starting price has to come back out. If it does not, something in the repricing differs from the pricing, and it is better to find that with a move of nothing than with a move of 200 basis points.

How is the error recorded, and what does its sign show?

Step eight writes down the difference between each estimate and the repricing, in percentage points and in rupees, and it writes down which side of the repricing the estimate fell on. On the rise, the first reading was Rs 10.9315/- away and the second Rs 0.7625/- away in the other direction. The curvature term closed about 93.02 per cent of the gap and left the rest.

The sign of that first error is the cheapest input test in the entire run. On a fixed schedule the sign is not free to be anything. A straight line laid against a relationship that bends this way always sits below the truth on a rise in the yield, which makes the first reading pessimistic, and always sits below it on a fall too, which makes the first reading low. Any other sign is not a discovery about the bond but a wrong figure in the run, pointing to where to look.

The sign test is genuine and it reaches outside the run. The sign depends on the direction of the move and on which base the percentage was taken against, and both are separate inputs. A sign slip on the move, or a percentage taken on the new price rather than the old one, shows up here and nowhere earlier.

What does the whole run look like on a fall in the yield?

The procedure does not change direction, so running it the other way is the shortest possible demonstration that the order is doing the work rather than the arithmetic. Every input above stays where it is. Only the move flips.

The first row read across shows the thing the whole sequence exists to install. The straight line predicts 13.1227 per cent in both directions, perfectly symmetrical. A straight line has no way of being anything else. The repricings are 12.0295 per cent and 14.3777 per cent, quoted to three places as 12.030 and 14.378. The bond loses less than the line says and gains more than it says, every time, and the second reading recovers about 93.02 per cent of the first gap and about 93.18 per cent of the second.

Which mistakes does this run catch, and which does it never see?

Every test above is a test of an identity. The weights add to one because of how a weight is defined. The two durations differ by exactly one plus the yield because of how the second is built from the first. Convexity is positive because every term inside it is a positive amount multiplied by a positive year. None of these is a fact about the bond, and that is precisely why they are useful: an identity fails only when something has entered the run from outside it.

So the procedure does something narrower than it first looks: step one is the only step that can catch a wrong input, and every step after it catches a figure that came from outside the run. Those are different things. A wrong input flows consistently through everything downstream, satisfies every identity on the way, and comes out the other end wearing a full set of passed tests.

Take the concrete case. Put the Rs 1,000.00/- repayment on year eight instead of year ten and see what happens. The weights still add to one. Weights always do. The MACAULAY duration reads 6.3628 years, inside one to ten, so the range test is silent. The MODIFIED duration still comes back to the MACAULAY figure when multiplied by 1.085. The division was performed on whatever figure it was handed. Convexity is still positive. The sign of the final error is still right. Six tests, six passes, and the wrong bond has been measured beautifully.

Who actually runs this, and what do they do with the output?

A lender funding ten year assets out of deposits that reprice inside a year runs this on the asset side to find out how much the value of what it holds moves for a given move in rates. The output is not a view about rates. The output is a size. A treasury committee that knows the size can decide whether to do something about it; one that does not know the size is deciding in the dark and calling it judgement.

An analyst writing up a bond runs it because the two readings tell different stories about the same instrument, and quoting only the first one is how a reader ends up expecting a symmetric move that the instrument is not going to deliver. The audit trailThe written record of where each figure came from, kept so that a stranger can follow it without having to ask the author. matters more here than the headline: a note that shows the ten dated amounts and the two column totals can be argued with, and a note that shows 6.5613 with nothing behind it can only be believed or ignored.

A household comparing two savings products that both lock money away is running a cruder version of the same procedure without the vocabulary. How much am I promised, on which dates, and what happens to what this is worth if rates move while my money is stuck. Step one is the whole of what they need and it is the step they most often skip. Reading the schedule off the document is boring and nobody feels clever doing it.

All three have the same thing in common: the useful output of the run is the error figure at step eight, not the estimate at step six. The estimate reports what the tool said. The error shows how much that tool can be trusted the next time no repricing is available. Somebody who has run this fifty times knows the size of move at which their straight line stops being good enough, and that knowledge is the actual product.

What has the run not measured?

Step eight closes with a list, and on any honest run the list is longer than the findings. The list is not modesty. The list is the difference between a figure that can be used and a figure that will be misused. A sensitivity number quoted without its list travels as though it covered everything the list names.

Read those four rows against what the run did produce and the proportion is uncomfortable. Discomfort is the point. The run measured how far one invented price moves when one yield applied to every date changes by a stated amount, and it measured how badly two estimating tools miss while doing it. The measurement is a real output, and also a narrow one, and its narrowness is only visible if somebody writes the list down.

There is a judgement of materialityWhether a difference is large enough that somebody would actually do something differently because of it. hiding in each of those rows, and the run cannot make it on anybody's behalf. Whether a curve moving unevenly matters depends on what else is being held against this. Whether the dated amounts arrive depends on who promised them. The run's job is to name the row, not to fill it.

A written run puts every intermediate figure in view at once, in the order the steps produce them. A reader who doubts the last line can then walk back up to the line that caused it, which a panel recomputing on a drag never allows.

The ten year bullet bond and the zero coupon bond are invented.
Educational material. Not advice on any investment, tax, budget or market position.