Interest-Rate Risk and Reinvestment Risk: Opposite Signs

Two risks sit on the same bond, and they are hurt by opposite things. Interest-rate risk is what a rise in the yield takes off the price of whatever is still held. Reinvestment risk is what a fall in the yield takes off the cash that has already arrived and has to be put back to work. Held for long enough, one very nearly pays for the other.

A bondholder is not holding one thing. The bondholder holds a stream of amounts that keep turning up on dated occasions and have to be put somewhere the moment they do, and separately a claim on whatever has not turned up yet, whose worth on any given day is whatever the yield of that day makes it. One rate move strikes both. The move strikes them in opposite directions. Which of the two dominates the outcome is settled by nothing about the bond at all: it is settled by how long the holder stays.

What is a bondholder actually exposed to, before either risk gets a name?

The two exposures are easier to see than they are to define, so the mechanics come first and the vocabulary afterwards. Take the ten year bullet bond used throughout: an invented instrument, Rs~1,000.00/- of face amount, an annual coupon of 8.50 per cent, ten annual payment dates, priced at par so its yield is also 8.50 per cent, on annual compounding throughout. The instrument has no issuer.

The next input is a date on which the money is needed: the horizon. Everything that happens between today and that date falls into exactly two buckets, and the split runs through the middle of the bond's own schedule of payments.

Whatever arrives before the horizon has to be put back to work at whatever rate the market offers on the day it arrives, and whatever has not arrived by the horizon has to be sold at whatever price the yield of that day produces. Carrying forward and selling are two different exposures running on two different clocks. The first one is counting forward from each payment date to the horizon. The second one is counting backward from each unpaid amount to the horizon. Most short treatments of rate sensitivity describe only the second. A reader who has met duration and nothing else therefore believes a rise in the yield is simply bad news.

Something close to this happens in an ordinary shop with a long lease. The shopkeeper worries about what the lease would fetch if it had to be handed over tomorrow, and separately about what to do with each month's takings as they come in. The lease and the takings are two worries with two different answers, and they respond to opposite conditions: a quiet high street depresses the value of the lease and does nothing whatever to the takings already banked. The bond version is tidier only because both sides are the same rate.

What is interest-rate risk, for a reader who has never had it defined?

Interest-rate risk is the risk that a change in yields changes the price of a bond. The definition stops there. Everything else follows from the arithmetic of discounting, covered separately. A price is the present value of a set of dated amounts. A higher rate applied to those amounts makes every one of them worth less today, so the price falls. A lower rate makes every one of them worth more, so the price rises.

Put a number on it. Move the yield on the ten year bullet bond by 200 basis points in one instant, from 8.50 per cent to 10.50 per cent, and reprice the ten amounts in full on annual compounding. The price falls from Rs~1,000.00/- to Rs~879.7045/-, a fall of 12.030 per cent. Move it the other way, to 6.50 per cent, and the price rises to Rs~1,143.7766/-, a gain of 14.378 per cent. Both figures were worked from the schedule rather than from a sensitivity, and a full repricing does not answer equally in the two directions.

Interest-rate risk bites at exactly one moment: when the holder has to realiseTo turn a holding into actual money by selling it, rather than waiting for it to pay out on its own schedule. the bond before its remaining amounts have arrived. The amounts still due are paid out at their stated size whatever the yield is doing, so a holder who never sells never meets this risk in the form of a loss. The measure that sizes the exposure is the MODIFIED duration, covered separately. For this bond it is 6.5613, and it reads as a price change of about 6.5613 per cent for each 100 basis points of yield.

And what is reinvestment risk, built from nothing?

Reinvestment risk is the risk that the rate available for putting arriving cash back to work is not the rate the holder was counting on. Reinvestment risk has nothing to do with the price of anything. The exposure is entirely about what happens to money after it has been received.

The ten year bullet bond pays Rs~85.00/- on each of the first nine anniversaries and Rs~1,085.00/- on the tenth. The Rs~85.00/- arriving on the first anniversary does not sit still. The coupon is put somewhere, and where it is put earns a rate, and that rate is whatever is on offer in year one rather than the 8.50 per cent that was on offer on the day the bond was bought. Then whatever it grows into earns again in year two, and so on to the horizon. A fall in the yield does no damage at all on the day it happens and then quietly takes something off every single one of those growth steps for the rest of the holding period.

The size of it depends on how much of the final outcome is made of interest on interest rather than of the coupons themselves. Over a short hold, almost none of it is: three coupons banked and barely grown contribute very little. Over a long hold it becomes the larger part of the answer. Hold this bond for its full ten years and the unchanged outcome is Rs~2,260.9834/-, of which Rs~1,000.00/- is the face amount returned and Rs~765.00/- is the nine coupons before the last one plus the last one's own coupon. The remaining Rs~495.9834/- is nothing but coupons on coupons, and every rupee of it is exposed to the reinvestment rate rather than to any price.

The everyday version is a household that has just finished paying off a loan and now has a fixed amount free every month. Nobody worries about the price of anything. The worry is that the place they were going to put that monthly amount pays less this year than it did last year, and that the shortfall repeats every month for years. Reinvestment risk with the labels taken off looks like that, and note which way it points: the reinvestment exposure is hurt by a fall in rates while the price exposure is hurt by a rise. The opposition of signs is the entire reason the two can be set against each other.

What happens to the same bond at two different holding periods?

A reader told that two exposures offset will not believe it from the sentence. The two columns have to be in front of them. So take the ten year bullet bond, buy it at par, let the yield move once and immediately and then stay exactly where it landed, and hold. Each coupon is put back to work at the new yield from the moment it arrives. Whatever is left of the bond on the horizon date is sold at the new yield. The comparison is against the unchanged outcome for that same horizon: Rs~1,000.00/- grown at 8.50 per cent a year for that many years, on annual compounding.

Hold for six years. The unchanged outcome is Rs~1,631.4675/-. After a 200 basis point rise the holder ends with Rs~1,601.4394/-, and that is Rs~30.0281/- short. After a 200 basis point fall the holder ends with Rs~1,668.9328/-, and that is Rs~37.4653/- ahead. So at six years the price exposure is running the show, and a rise is bad news exactly as a reader who has only met duration would expect.

Now hold for eight years and change nothing else. The unchanged outcome is Rs~1,920.6043/-. After the same 200 basis point rise the holder ends with Rs~1,955.3975/-, and that is Rs~34.7932/- ahead. After the same 200 basis point fall the holder ends with Rs~1,892.9453/-, and that is Rs~27.6590/- short. Nothing about the bond changed between those two rows, and nothing about the rate move changed either: only the holding period moved, and the sign of the exposure reversed.

Where do the two effects trade places, and what is that number?

If a rise hurts at six years and helps at eight, then somewhere between the two there is a horizon at which it does neither. The balancing horizon is not a guess and it is not a convention. The horizon falls out of the bond's own schedule. For the ten year bullet bond it is 7.1191 years, the bond's own MACAULAY duration.

A MODIFIED duration and a MACAULAY duration are two different objects, and a bare duration written anywhere near a number hides which of the two is meant. Here is what that distinction is for. A MODIFIED duration is a sensitivity, a price change per 100 basis points, and it is not a length of anything; a MACAULAY duration is a weighted averageAn average in which some items count for more than others. Each item is multiplied by its own weight, the products are added, and the total is divided by the sum of the weights. waiting time measured in years, and the crossing is what that length is a length of.

Which makes the naming rule practical rather than pedantic. The two figures for this bond are 6.5613 and 7.1191. The two figures describe the same instrument, they are one short division apart, and only one of them can be compared with a date on a calendar. A holding period set against the wrong one compares a price sensitivity with a number of years. The category error produces a plausible looking answer and survives on it.

A small precision point sits underneath this. The horizon that produces the balance is the MACAULAY duration carried at full length, 7.119062643353 years, and 7.1191 is that number rounded to four places for display. The horizon arithmetic worked at exactly 7.1191 puts the unchanged outcome at Rs~1,787.4251/- rather than Rs~1,787.4196/-. The horizon arithmetic runs on the unrounded length, and the rounded version sits alongside wherever the difference would otherwise leave a reader unable to reproduce a figure from the ones beside it.

What does the whole horizon table look like when every row is worked?

Here is the arrangement in full, on five horizons, worked by rolling the ten year bullet bond's own ten dated amounts forward rather than by quoting any rule. The setup is stated once and applies to every row. Buy at par, Rs~1,000.00/- of face amount, an 8.50 per cent annual coupon, ten years, annual compounding. Let the yield move once and immediately by 200 basis points and then leave it alone. Put each coupon back to work at the new yield from the moment it arrives. Sell whatever is left of the bond at the new yield on the horizon date. Compare against the unchanged outcome for that same horizon: the base named in every row of the table below.

Read the table twice, once across and once down. Across, each row says what the identical rate move did at that one horizon. Down, the two right-hand columns tell the story: below the MACAULAY duration the price exposure dominates and a rise in the yield hurts, above it the reinvestment exposure dominates and a rise in the yield helps, and at the MACAULAY duration the two are so nearly equal that both directions finish with a small surplus.

The seven year row is worth pausing on. A 200 basis point rise is a large move, and it leaves the holder Rs~0.5517/- short of the unchanged outcome on a base of Rs~1,770.1422/-. The shortfall is three hundredths of one per cent. The bond fell 12.030 per cent in price on the day and seven years later almost none of that is visible in the answer.

Two more figures belong with the table and both were solved from the same arithmetic rather than read off it. The line for a rise crosses the unchanged line at a horizon of 7.0171 years and the line for a fall crosses it at 7.2203 years. Anywhere between those two the holder is no worse off whichever way the yield went, and 7.1191 sits inside the band. Subtracting the two printed endpoints gives 0.2032 years; the unrounded crossings are 0.203263 apart, which is 0.2033. Both roundings are honest and they disagree in the fourth place, so the band width is quoted here as 0.2033 from the unrounded solve and the arithmetic on the printed figures is named rather than hidden.

Why is the balance nearly and not exactly?

The row in the middle of the table repays a second look, because most short treatments round it away and it is the most interesting line here. At a horizon of 7.1191 years a 200 basis point rise leaves the holder Rs~3.3330/- ahead of the unchanged outcome, and a 200 basis point fall leaves them Rs~3.3709/- ahead. Both figures are positive, so the balance point is not a tie at all: it is a small surplus in both directions.

The size comes before the meaning. The base is the unchanged outcome at that horizon, Rs~1,787.4196/-, and Rs~3.33/- on that base is 0.19 per cent. Nineteen hundredths of a per cent is not a large number. The sign is what matters. A genuine offset would leave the holder short in one direction and ahead in the other, and this one does neither.

The surplus has a name. The surplus is convexity, measured at 58.4702 for this bond and covered separately, where convexity explains why a full repricing beats the straight line estimate in both directions. The same curvature that put the bond ahead of the straight line on both sides of a rate move is what puts the holder ahead of the unchanged outcome on both sides of the balance point; it is one property arriving in a different costume. Rounding the surplus to zero would hide the fact that convexity and this surplus are one property met twice.

The caveat that travels with this pair of bonds belongs right here, in the same breath as the advantage. Both bonds sit at the same 8.50 per cent yield, and that equality is what makes the ten year bullet bond's extra convexity look as though it costs nothing. Convexity does not come free in a real market. Curvature is something buyers want, so it is charged for, and a more convex bond would ordinarily be bought at a lower yield than a less convex one carrying the same MODIFIED duration. The charge a market puts on convexity is a separate subject. The arithmetic reports what the two schedules do; which one anyone should hold is a separate question.

What does the zero coupon bond do differently?

The cleanest case on this subject is the other half of the pair. The zero coupon bond is an invented instrument bought at the same 8.50 per cent yield on annual compounding, with its maturity set equal to the ten year bullet bond's MACAULAY duration. Its own MACAULAY duration is therefore its maturity, its MODIFIED duration is 6.5613 and identical to the bullet's by construction, and its convexity is 49.0986.

Set its horizon at the date it matures and both exposures vanish. The zero pays nothing before that date, so there is no arriving cash and no reinvestment exposure at all. The bond matures on that date, so nothing is left to sell and there is no price exposure either. The holder receives Rs~1,000.00/- whatever the yield did in between. The offset here is exact rather than near, and the reason it is exact is that there was nothing to offset in the first place.

The exactness has a price. The zero achieves it by carrying the lowest convexity available to any bond of that MACAULAY duration, and that is precisely what cost it against the bullet in the full repricings: on a 200 basis point rise the bullet falls 12.030 per cent while the zero falls 12.193 per cent, a difference of 0.164 percentage points, and on a 200 basis point fall the bullet gains 14.378 per cent against the zero's 14.162 per cent, a difference of 0.215 points. The exact offset and the missing curvature are the same fact seen from two sides. Neither instrument is recommended over the other.

Who actually uses this, and what do they write down?

The people who use this arithmetic are the ones with a date they cannot move. A household saving for a wedding seven years out is the cleanest example. The money has to be there in seven years, not in five and not in nine, and the two things that could go wrong pull in opposite directions: rates rise and the holding they would have sold is worth less, or rates fall and every amount they receive between now and the wedding earns less than they had penciled in. Setting the holding's MACAULAY duration against the seven years is the arithmetic that makes those two worries offset instead of stacking.

The institutional version is the same shape with more zeroes and a longer list of dates. A desk with a known payout schedule works out the MACAULAY duration of what it holds and the MACAULAY duration of what it owes, and watches the difference rather than either figure on its own. The desk's written record is short: the horizon each pot of money is for, the MACAULAY duration of the holdings against it, the gap between the two in years, and the date the check was last run. The date of the last check matters most and goes missing most often, for reasons the failure block below sets out.

Notice what is not on that list. There is no view about where rates are going. The whole point of setting a horizon against a MACAULAY duration is that it does not require one, and a desk that has an opinion about rates would express it somewhere else entirely. Setting a horizon against a MACAULAY duration is a technique for people who have decided not to have a view. Holding no view is exactly why the technique survives being wrong about the direction.

What does this arrangement not protect against?

The list of things it does not do is longer than the thing it does, and it is worth reading slowly because every item is a way for a satisfied holder to be surprised.

The arrangement assumes one immediate move of the same size at every point on the curve, and then nothing further. No path of rates through time stands behind it, and no scenario in which one part of the curve moves differently from another, so no claim is made here about repeated moves, gradual moves or a curve that changes shape.

The arrangement assumes each coupon really is put back to work, at the new yield, promptly. A coupon that sits idle for four months has not earned what the arithmetic credited it with. The arrangement assumes the holder's horizon is fixed and known. A fixed horizon is a strong assumption about a person rather than a property of a bond, and horizons move for reasons that have nothing to do with rates. The arrangement assumes every dated amount arrives. Whether a dated amount arrives is a credit question, covered separately, and absent from both bonds above.

And it is a property of one schedule at one instant. The MACAULAY duration drifts as time passes and drifts again as the yield moves. Work it on the same bond a year later, with nine payment dates left and the yield back at 8.50 per cent, and it is 6.6392 years rather than 6.1191, so a year of calendar has consumed only about half a year of waiting time. Work it at a yield of 10.50 per cent with all ten dates still to come and it is 6.9139 years; at 6.50 per cent it is 7.3202. The figure moves and nobody sends a notice, so a horizon set against a MACAULAY duration today is not set against it tomorrow.