Duration and Convexity: How Far a Bond Price Actually Moves

Modified duration is the percentage a bond price shifts for one percentage point of yield movement, and it draws a straight line against a relationship that curves. Small moves it handles. Large ones it misreads in a direction that can be predicted, always overstating the loss and understating the gain. Convexity puts a number on that curvature. Two bonds carrying identical modified durations still behave differently, and convexity is the whole of the difference.

What is being held still while the yield moves?

Every number below is the result of one experiment, and the experiment is worth naming before any of the numbers arrive. The schedule of payments is fixed. The convention used to discount them is fixed. Then the yield moves, nothing else moves at all, and what happened to the price is recorded. The experiment is the whole method, and each measure below is a different way of writing down the answer.

The same experiment runs in a kitchen without being called one. A sack of rice goes on the scale and the weight is read. One measure of rice comes out. The weight is read again. The difference is what one measure weighs, and it is a clean answer only because nobody added anything to the sack in between. Duration and convexity are one thing at a time measurements, and the moment a second thing moves alongside the yield, neither figure is the answer any more.

Two invented bonds carry everything that follows. The first is a ten year bullet bond: Rs 1,000.00/- of face, an 8.50 per cent annual coupon so Rs 85.00/- arrives each year, ten years to run, priced at par so its yield is also 8.50 per cent. The second is a zero coupon bondA bond with no coupons at all. One single amount arrives at maturity and nothing before it., introduced later for one reason that becomes obvious when it arrives. Neither carries a credit element of any kind, so every payment arrives in full and on the date it is due.

Now the compounding convention. Every price, yield and rate that follows is on annual compounding, one discounting period a year. The identical figures on a semi annual convention give different prices and different durations, so the convention is part of the answer rather than a footnote to it. Which convention actually attaches to a real instrument is decided by the Reserve Bank of India at rbi.org.in for government securities and by the Securities and Exchange Board of India (SEBI) at sebi.gov.in for corporate debt.

How is a waiting time built out of ten payments?

Duration Measures, and the one line that keeps them apart

There is more than one measure with the word duration in it, and the errors people make with them are not subtle. The two measures share a word and share almost nothing else, so the distinction is worth stating in full. A Macaulay durationThe weighted average waiting time of a bond's cash flows, where each weight is that payment's share of the price. Measured in years. is a weighted average waiting time and is measured in years; a modified durationThe percentage a price moves for each one percentage point of yield movement. It is a sensitivity, not a length of time. is a sensitivity, the percentage the price moves per one hundred basis pointsOne hundredth of one percentage point. Two hundred basis points is 2.00 percentage points. of yield movement, and is not a length of time at all. The measure intended belongs in the same sentence as the number, every single time.

Now build the first of them from the payments rather than accepting a formula. The bullet bond hands over ten amounts on ten dates. Discount each one at 8.50 per cent. Divide each of those present values by the total, and the share that results is the payment's weight. The ten weights add to one, being shares of a whole. Multiply each weight by the year the money arrives in. Add the ten products. The sum of those ten products is the MACAULAY duration.

Read the fourth column before the fifth. The nine coupons account for 52.01 per cent of the price between them, and the single Rs 1,085.00/- arriving in year ten accounts for 47.99 per cent on its own. A bond with ten years to run therefore has a waiting time of 7.1191 years rather than ten. Almost half the money is at the far end and slightly more than half is spread across everything before it, and the average of those two facts is where the answer lands.

The everyday version costs nothing to see. The average age of a queue is not the age of the last person standing in it. If nine people are near the front and one is at the back, the average sits well short of the back of the queue, and no amount of staring at the last person reveals where. A bond is that queue with money instead of people.

What does dividing by one plus the yield actually change?

Modified Duration, and why that division is there at all

A waiting time in years is a fact about a bond, and it is not yet an answer to the question anybody asked. The question is how far the price moves. Divide the MACAULAY duration by one plus the yield and the number stops describing time and starts describing a response. Seven point one one nine one divided by 1.085 is 6.5613, and that 1.085 is the entire distance between the two figures.

Now the consequence of confusing the two. A reader who picks up 7.1191 and uses it as the sensitivity is out by 8.5 per cent of their own answer, and out in the direction of exaggerating every move they estimate. On a 200 basis point calculation the error separates a claim of 14.238 per cent from a claim of 13.123 per cent, and the bond does neither. MACAULAY duration is measured in years. MODIFIED duration is measured in per cent per percentage point, and a MODIFIED duration is never written with the word years attached to it.

How wrong is the straight line, and which way is it wrong?

A move of 200 basis points is 2.00 percentage points. Apply the MODIFIED duration to it and the line offers 13.123 per cent of price movement either way. It has to. A straight line has one slope, and one slope cannot say two different things about two moves of the same size. Now reprice the bond properly at each new yield, discounting all ten payments again, and watch the estimate and the bond separate.

When the yield rises by 200 basis points the bullet bond loses 12.030 per cent, less than the line predicted. When the yield falls by 200 basis points it gains 14.378 per cent, more than the line predicted. Written in rupees on Rs 1,000.00/- of face: the line says Rs 868.7730/- at the higher yield and the bond is worth Rs 879.7045/-, an error of Rs 10.9315/-. The line says Rs 1,131.2270/- at the lower yield and the bond is worth Rs 1,143.7766/-, an error of Rs 12.5496/-. Both errors run the same way.

Take the rule out of that table and give it a line of its own. The MODIFIED duration line always overstates the loss and always understates the gain. It is not sometimes wrong and it is not randomly wrong. The line is wrong in a known direction on every reading ever taken, and a tool handed over without that direction has its failure mode filed off.

The reason is geometric and it is visible in the figure above. The real relationship between price and yield bends, and it bends so that the curve sits above the straight line on both sides of the point where they touch. Above the line on the right means the price after a rise in the yield is higher than the line claimed, so the loss is smaller. Above the line on the left means the price after a fall in the yield is higher than the line claimed, so the gain is bigger. One piece of geometry, two consequences, and they point in opposite directions only because the words loss and gain do.

What is the amount the line missed called?

Convexity, the measure of the bend itself

The gap has a name and the name is convexityA measure of how much the price to yield relationship curves. A second number that picks up what a straight line estimate leaves behind.. Convexity is computed from the same ten payments, with one change: instead of weighting each present value by the year t, weight it by t times t plus one, then divide by one plus the yield squared. Run that on the bullet bond and the answer is 58.4702. The figure is measured neither in years nor in per cent. Convexity is the coefficient on the squared term of an estimate, and its job is to bend the straight line the right way.

Adding convexity to the estimate is adding a second term. The first term is the straight line: the MODIFIED duration multiplied by the yield change. The second is half the convexity multiplied by the yield change squared, and because a squared number is positive whichever way the yield went, the second term pushes the estimate in the same direction both times. The second term is exactly the correction the arithmetic needed.

Put 200 basis points through it. The second term is half of 58.4702 multiplied by 0.02 squared. The result is 1.1694 per cent, and it is added on both sides. So the estimate moves from a symmetric 13.123 per cent to a loss of 11.9533 per cent on the rise and a gain of 14.2921 per cent on the fall, against actual moves of 12.0295 and 14.3777. Most of the gap has closed. About 0.076 of a percentage point survives on the rise and about 0.086 on the fall.

Now measure the error rather than only celebrating the improvement. The habit of measuring is what separates using a tool from believing one. The two term estimate is better than the one term estimate and it is still an estimate. The two term estimate is the first two pieces of an expansion that has more pieces in it, so a residue was always going to survive. Saying how much survives is more useful than presenting the second answer as the answer, and treating a two term estimate as exact repeats the original mistake one level further along.

Can two bonds share one modified duration and still behave differently?

Duration vs Convexity, set against each other on two bonds

Here is the pair the rest of this subject area keeps coming back to. Beside the ten year bullet bond, place a zero coupon bond whose maturity is set equal to the bullet bond's MACAULAY duration. A zero has only one payment, so its weighted average waiting time is simply the date that payment arrives on: its MACAULAY duration is its maturity. Set that maturity to 7.119062643353 years, price it at the same 8.50 per cent yield, and its MODIFIED duration comes out at 6.5613 as well. Not approximately. The second bond was built to produce that figure, so the match is exact by construction. Its price is Rs 559.47/- on Rs 1,000.00/- of face.

Two roundings sit behind that price, and a reader who checks the arithmetic deserves to know where a paisa went. The maturity above is the bullet bond's MACAULAY duration at full precision, 7.119062643353 years, and at that maturity the zero prices at Rs 559.4657/-, or the Rs 559.47/- used throughout. Recomputed on the rounded 7.1191 years, the price is Rs 559.4640/-, or Rs 559.46/-. The gap is about a sixth of a paise on Rs 1,000.00/- of face, and it moves the zero's convexity from 49.0986 to 49.0991. Both figures are defensible, and a reader who lands one paisa away can see exactly which rounding put them there.

A reader who has met only MODIFIED duration has been told, in effect, that these two bonds move alike. They do not. At a rise of 200 basis points in the yield the bullet bond loses 12.030 per cent and the zero coupon bond loses 12.193 per cent. At a fall of 200 basis points the bullet bond gains 14.378 per cent and the zero coupon bond gains 14.162 per cent. The bullet bond finishes ahead in both directions, by 0.164 percentage points on the rise and by 0.215 on the fall. Both bonds carry the same MODIFIED duration, so MODIFIED duration cannot see one basis point of the difference.

The difference is convexity, 58.4702 against 49.0986, a gap of 9.3716. And the reason is structural rather than accidental. A zero has all of its money on a single date, so there is no spread of payments for the bend to work on. Concentrating everything at one date is the least curved arrangement a set of payments can have, so for any given MODIFIED duration a zero carries the lowest convexity available. The bullet bond spreads ten payments across ten years around that same average, and the spread is what buys the bend.

One caveat belongs in the same breath as these numbers. Both bonds sit at the same 8.50 per cent yield above, and that is what makes the bullet bond's extra convexity look as though it costs nothing. Extra convexity is not free in a real market. Convexity is something people want, so it is paid for, and the more convex of two bonds would ordinarily be bought at a lower yield than the less convex one. No price is put on convexity at either yield above, so the arithmetic shows what these two bonds do and settles nothing about which one anybody should hold. State the arithmetic, name the caveat, stop.

Which other measures answer the same question about a different move?

Active Duration, and the Duration Gap that is the same subtraction

Everything so far has moved the yield on one bond. Changing what moves changes the measure, without changing the method at all. A holding of Rs 5,000 crore of fixed income carries a MODIFIED duration of 5.20 against a benchmark at 4.80. Subtracted, 0.40 is the active durationA holding's modified duration less its benchmark's. A difference between two sensitivities, and not a length of time.. The 0.40 is a difference between two sensitivities and inherits their nature, so it is not 0.40 of anything measured in years.

Put a rupee figure on it and the discipline this measure demands becomes obvious. A parallel rise of 100 basis points costs about 5.20 per cent of the holding, or Rs 260.00 crore. The benchmark part of that is 4.80 per cent of Rs 5,000 crore, or Rs 240.00 crore, and the remaining 0.40 per cent is Rs 20.00 crore. Add Rs 240.00 crore to Rs 20.00 crore and Rs 260.00 crore comes back. The two figures, Rs 260.00 crore and Rs 20.00 crore, differ by thirteen times, one of them is the whole position and the other is the decision, and naming which of the two is meant every single time is the entire discipline of this measure. Most of that exposure was never chosen; it arrived with the benchmark.

The word parallel is load bearing and is not decoration. Every figure in this block assumes the whole curve moves by the same amount at every point. A twist in the curve is a different move altogether, and none of these figures measures what a twist would cost this holding. Where a reader expects that number, the honest thing is to leave the cell empty and say why.

The duration gapThe difference between the modified duration of an institution's assets and the modified duration of its liabilities. is the identical subtraction pointed at a different pair. Instead of a holding and its benchmark, the pair is an institution's assets and its liabilities, and the MODIFIED duration of the second is subtracted from the MODIFIED duration of the first. The arithmetic does not change by a comma. The two sides change, and so does what the answer says: an active duration says how far a holding sits from what it is measured against, and a duration gap says how far one side of a balance sheet sits from the other.

Spread Duration, which holds the government rate still instead

One more variation, same method again. MODIFIED duration moves the whole yield. Spread durationThe price sensitivity to a move in the spread over the government rate, with the government rate itself held still. holds the government rate fixed and moves only the spread over it, then measures the price response to that alone. Spread duration is the same experiment with a different thing held still, and it is worked out under credit spreads. The three measures share one method: hold everything, move one thing, record the price.

How is one number split across the points of a curve?

Key-Rate Duration, one point on the curve at a time

MODIFIED duration carries an assumption that is easy to miss: that every point on the curve moves together. Real curves do not oblige. Key-rate durationThe price sensitivity to a move in one point on the curve, with every other point held still. drops that assumption and goes back to the method. Bump one point on the curve. Hold every other point exactly where it was. Reprice. Record the sensitivity that belongs to that point alone. Then do it again at the next point.

How to map Key-Rate Exposures across the ten dates

Run on the ten year bullet bond at each of the ten dates its payments arrive on, working from the flat 8.50 per cent curve used throughout, the method gives ten answers. The ten come out as 0.0722, 0.1331, 0.1840, 0.2261, 0.2605, 0.2881, 0.3098, 0.3263, 0.3383 and 4.4229. Two things about that list matter more than the list itself.

First, the ten add to 6.5613, exactly the bond's MODIFIED duration, so key-rate duration is not a rival measure but the same measure taken apart. Nothing was created by splitting it and nothing was lost. Second, 4.4229 of that 6.5613 sits at the ten year point on its own, or 67.4 per cent of the entire sensitivity at one single date. The ten bar map is what anybody means by mapping a key-rate exposure: one bar for each point on the curve, and the bars are nothing like equal.

The household version is close enough to be useful. A household with money going out on ten dates in the year is not equally exposed to each of them; if the annual insurance premium and the school fee both land in the same month, that month is where a shock actually reaches. The other nine months are real and they are small. A bond is that pattern with a curve instead of a calendar.

Which rate has actually been doing the discounting here?

Spot Rate, and the convenience behind a single yield

Every discounting above has used one single yield for all ten payments, and that single yield is a convenience rather than a description. A spot rateThe rate that discounts one single cash flow arriving at one stated future date. A full set of them across dates is what a curve is. is the rate that discounts a single payment arriving at one stated future date, and a full set of them across all the dates is what a curve actually is.

The ten payments of the bullet bond arrive on ten different dates, so strictly each one has its own SPOT rate, and the single yield to maturity is the one rate that happens to reproduce the same total. The key-rate map above had ten answers rather than one for exactly that reason: it went back to the dates. A forward rate is a completely different object that happens to sit at a similar number, so the word SPOT is load bearing and travels with every rate above. The two are separated properly under spot rates and forward rates.

What question are all of these measures built to serve?

Rate Thesis, and why naming one is not the same as holding one

A rate thesis is a position taken on where the general level of rates goes, carried through the MODIFIED duration a holder chooses rather than through which issuer they lend to. The definition ends there. A rate thesis and a credit view are two different questions asked about the same bond, they are answered with different measures, and the difference between them is covered separately.

Naming what a rate thesis is has nothing to do with holding one. A measure of sensitivity carries no view about where rates go, and neither of the two bonds was built to be held. Every figure is an arithmetic consequence of an invented schedule of payments, not an observation about any market anywhere.

Where does a rate move land on an income statement instead of a price?

Interest Coverage and Fixed-Charge Coverage, the borrower's side of the same move

Every measure above this point is about what happens to the price of a bond somebody holds. But a rate move has a second address, and it arrives there through the borrower. Rates rise, and debt that floats reprices while debt that matures has to be refinanced at whatever the new level is, so the interest bill goes with it. The same rise in rates that costs a holder price costs a borrower room in the income statement, and those are two arrivals of one event.

Interest coverageEarnings before interest and tax divided by the interest expense, both measured over the same accounting period. is earnings before interest and tax divided by the interest expense, both measured over the same accounting period, and it counts how many times over the interest bill is covered by what the business earned before paying it. Fixed-charge coverageThe same ratio with the other committed fixed payments added into the base, over the same period. widens the base of that same ratio to take in the other fixed payments the business has committed to, measured over that same period. A coverage figure quoted without its base and its period cannot be read at all, so both belong in the same sentence as the ratio every time.

And the honest limit. A coverage ratio needs an income statement, and neither bond has an issuer with one, so no coverage figure can be computed from these payments. The link is what matters: the measures above give what a rise in the yield does to a price, and these two give what the same rise does to the ability to pay the bill, and a reader who has only ever met the first half of that will keep being surprised by the second.

What does none of this establish?

None of these measures says where rates go. Every one of them is a response to a move, and there is nothing inside a response that settles whether the move happens. Silence about direction is a structural fact about what a sensitivity is.

None of them says whether an issuer pays. Every figure here was computed on bonds with no credit element at all, and adding one changes the question being asked rather than the arithmetic being done. None of them covers a curve that twists rather than shifts, with the single exception of the key-rate map, and that map was built on one bond rather than on the Rs 5,000 crore holding. And none of them prices convexity: the extra 9.3716 the bullet bond carries over the zero coupon bond is an arithmetic fact at one common yield, with no cost attached to it.