Yield to Maturity and Yield to Call: Which One Binds

A yield to maturity is solved over every payment still owed, with the last of them landing on the maturity date. A yield to call runs that same arithmetic to an earlier date, and to whatever the issuer must hand over on it. One price gives two answers. Where the price paid sits above the amount repayable early, the second answer lands lower, and the lower one is what a holder should carry.

Everything below follows from one fact about who holds a decision. A right to repay early sits with the issuer and not with the holder, so the holder's arithmetic has to be run over somebody else's choice. Two final dates then become possible on a single bond, two yields drop out of a single price, and the holder receives whichever of the two the issuer would rather give. The rest is that addition, set out with every term showing, evaluated once, and then evaluated again with exactly one input moved.

What is a right to repay early, and whose right is it?

A right to repay early is a term written into a bond at issue which lets the issuer hand back the money before the final date, on a stated date and for a stated amount. Both the date and the amount are fixed when the bond is sold. Neither is negotiated afterwards, neither moves with anything, and neither belongs to the holder.

Everything that follows rests on the fact that the issuer holds this right and the holder does not. The holder cannot force an early repayment when it would suit the holder, and cannot refuse one when it would not. The holder is a passenger on a decision that somebody else makes, using the issuer's arithmetic rather than the holder's. In the language of contracts this is an optionA right that one side of a contract may use, with no duty at all to use it. The other side has to live with whichever way the choice goes. held by one side only, and the side holding it will exerciseTo actually use a right that a contract has granted, on the terms already written into it. it when using it helps them.

The same shape appears in an arrangement far outside bond markets. A shop takes a ten year lease on a corner unit, fits it out, builds a queue of regulars, and plans on ten years of that queue. Buried in the lease is a clause letting the landlord end the arrangement at the end of year five. The shop now has a ten year plan and a five year problem, and which of the two turns out to be real is not something the shop decides. Nothing about the shop changed when that clause was written. The clause changed who gets to say when the arrangement stops.

The household version runs the other way round. Setting it beside the lease shows what it feels like to be on the side that holds the right. When a household clears a home loan ahead of schedule, that is a prepaymentA borrower clearing a loan before its due date, the move a household makes when it settles a home loan early out of a bonus or a sale., and the household chooses it freely, usually because money came in or because a cheaper loan appeared. The lender simply receives the money back and has to find something else to do with it. A bond with a right of early repayment puts the household in the lender's chair. The borrower here is the issuer. The issuer decides, the holder receives, and the moment the issuer chooses is the moment that suits the issuer.

Two things about that clause are worth naming while the terms are still in plain words. The mirror right does exist as an idea, and it would be a put rightThe mirror image of the right worked through above: a right the lender holds to hand a security back early and be repaid. Neither constructed bond carries one. held by the holder rather than by the issuer, but nothing in this record carries one. And a bond may carry call protectionA stretch at the start of a bond's life during which an early repayment right cannot be used at all. Its length is written into the terms at issue., an opening stretch during which the right may not be used at all. Call protection changes which dates go into the arithmetic without changing the arithmetic itself.

An issuer's duty to disclose such a right, and the notice a repayment carries and to whom, are not arithmetic. The Securities and Exchange Board of India (SEBI) settles both at sebi.gov.in for corporate debt, and the Reserve Bank of India settles them at rbi.org.in where government securities are concerned. The table near the foot lists every item settled by a rule maker rather than by arithmetic.

Why is the bond worked below a supposition rather than a stated instrument?

Because there is not one. The fixed income teaching runs on two constructed instruments, and neither of them can be repaid early. Bond A runs ten years. Rs 1,000.00/- is what it owes at the end, and 8.50 per cent a year is the rate struck on that amount, paid once a year until then. Bond B is a zero coupon bond that hands over nothing at all until it matures. Making either of them repayable early would silently change every other treatment built on the pair. Nor does the credit teaching supply one: Palash Cements Limited, the only issuer named anywhere, borrows on a five year bond that also repays in a single amount at the end, and Palash Cements is given no early exit either.

So the arithmetic below runs on a supposition, and the supposition is stated in full before a single figure comes out of it. Suppose a bond written on Bond A's exact terms. Rs 1,000.00/- of face amount. Ten yearly dates. A rate of 8.50 per cent written into the contract. And suppose that same bond had handed its issuer one extra right: to take the money back once year five closed, for a stated amount. Bond A stays untouched, exactly the instrument every other treatment here is built on: one repayment, at the end, with no early exit available to anybody.

Naming an absence rather than quietly filling it is not a formality. A treatment that had simply made Bond A repayable early would have produced numbers that look identical to numbers derived honestly, and no reader could tell the two apart. The habit worth taking away is smaller than the rule: when a figure has no source, the source is stated in the same breath as the figure, and a reader can then decide how much weight to put on it.

What is the yield to maturity, and which date does it run to?

One rate, and only one, will carry every payment still owed back to the price being handed over, with the last of those payments landing on the maturity date. The rate that does this is the yield to maturity. A yield to maturity is not read off anything anywhere. Solving for it means holding the price fixed, holding each amount and the date it lands on fixed, and searching for the rate at which the divided down amounts total the price paid. Applied to a bond, it is the same object the valuation work calls an internal rate of returnThe one rate that pulls a run of dated amounts back to a single starting figure. Settled in the valuation work that sits before this., wearing a different name.

Two of its inputs come straight off the terms of the bond and are the only two the working below will move: the last date, and the amount handed over on that last date. Everything else stays where it is. Keep the last date and its amount in view. The second measure below is built by changing exactly those two and nothing else.

Take the price already established for Bond A, Rs 1,079.4804/-. The price is not asserted anywhere. Rs 1,079.4804/- is what Bond A's eleven amounts cost when each is discounted at 7.35 per cent a year, and 7.35 per cent is the level standing at the ten year node of the invented spot rate curve all the pricing here runs on. A spot rate is the rate at which one amount arriving on one stated future date is discounted today, quoted for that date and no other. Bond A owes Rs 1,000.00/- at the end of ten years, with 8.50 per cent a year struck on that amount along the way. The price divides cleanly into two pieces.

The two pieces close on the total exactly, with nothing left over, and the coupons carry 54.4210 per cent of that price against 45.5790 per cent carried by the face amount. Run the search back the other way and it returns what it started from: the rate that makes those eleven amounts discount to Rs 1,079.4804/- comes out at 7.3500 per cent a year. The return trip is the check worth doing every time. A price and a yield are the same statement written twice, so if the return trip does not land where it started, one of the two is wrong.

Why does every price here have to say how often the discounting clock ticks?

Because a rate on its own does not price anything. A rate prices something only once the frequency at which it is applied is known. Every sum below uses annual compounding. The clock then ticks once a year: an amount arriving in five years is divided by one plus the rate five times over, and never more often than that. Where a rate is stated and the ticking is not, a reader cannot reproduce the figure however carefully they add.

The size of the thing is easily shown. With Bond A's eleven amounts exactly as they are and the rate still 7.35 per cent a year, but the clock ticking twice a year instead of once, 3.675 per cent is applied every six months. A full year of discounting then costs 1.03675 multiplied by itself. The full year cost is 7.4851 per cent rather than 7.35, the same eleven amounts price at Rs 1,069.7141/-, and that price sits Rs 9.7663/- below the annual figure, out of nothing but the ticking. Nobody changed the bond, the dates, the coupon or the rate as written.

The same defect turns up on the zero coupon instrument as well. Bond B matures in 7.1191 years and pays once. The zero coupon bond prices at Rs 559.4640/- where its 8.50 per cent a year is applied annually, against Rs 552.8781/- where that identical rate is applied in half yearly steps. Two lines that look identical in print therefore set Rs 6.5859/- of difference against a Rs 1,000.00/- face. Subtracting the two rounded prices would give Rs 6.58/-, and that is not the gap. The honest figure comes from differencing the unrounded results before either is rounded, and it is Rs 6.5859/-.

What is the yield to call, and what exactly is being swapped?

The yield to call is not a new formula and it does not need learning as one. The yield to call is the sum set out just above, run with two substitutions. The discounting stops at the call date instead of at the maturity date, and the amount payable on an early repayment sits where the face amount used to sit. That is all.

Nothing else in the sum moves: the same coupons, the same price, the same once a year clock, the same search for a single rate. Holding everything else fixed settles what a yield to call is measuring, and that settles more than it looks. A yield to call is not a measure of a different bond. It is the same money, read to a different finishing line.

Why does one price on one bond give two different yields?

Now the supposition gets worked. Suppose, again, a bond on Bond A's terms whose issuer may close it out once year five ends, and suppose the amount payable on that closing out is Rs 1,000.00/-. Bond A has a single repayment at the end everywhere else and stays that way. Every figure below belongs to the supposition and to nothing else.

The holder hands over Rs 1,079.4804/- today. Run to year ten, with ten amounts of Rs 85/- and Rs 1,000.00/- of face at the finish, the search returns a single fitting rate of 7.3500 per cent a year. Run identically to the end of year five, with five amounts of Rs 85/- and Rs 1,000.00/- payable on the call date, it returns a single fitting rate of 6.5831 per cent a year. Nothing about the money handed over changed between the two searches. The finishing line moved, and the answer moved with it.

Put the two builds beside each other and the difference stops being mysterious. Ten years of discounting puts Rs 587.4641/- of the price into coupons and Rs 492.0163/- into the final repayment. Five years of discounting divides the same Rs 1,079.4804/- into Rs 352.4416/- of coupons and Rs 727.0388/- of ending amount. There is half as much coupon in the shorter row, so the coupon share falls from 54.4210 per cent of the price to 32.6492 per cent and the ending amount has to carry the rest.

The two are answers to different questions about the same money, so neither of them is wrong. One asks what rate the holder earns if the bond runs its stated course. The other asks what rate the holder earns if the issuer stops it at year five. Both are true statements about Rs 1,079.4804/-, and which of them describes the actual outcome is settled by somebody other than the holder.

Percentage points and basis points get muddled constantly, so the size of the difference is worth writing in both. Set 6.5831 against 7.3500 and the gap is 0.7669 percentage points. Cut a percentage point into a hundred pieces and one piece is a basis point, so the same gap is 76.69 basis points. The gap is one quantity written two ways, and a figure of 0.7669 basis points would shrink it a hundredfold.

Which of the two yields actually binds, and who decides that?

The part that matters now is not an arithmetic question at all. Both numbers are correctly computed. Only one of them describes what the holder will hold at the end. Which one that turns out to be is decided by the issuer, on the issuer's reasoning, in the issuer's interest.

An issuer repays early when repaying early is the cheaper way to carry the borrowing. Cheapness is the entire rule, and it has no sentiment in it. Where the rule lands the holder is worth stating plainly. The moment repaying early is cheap for the issuer is the same moment the bond is worth more to the holder than the amount the issuer is handing over. Cheap to repay and dear to hold are two descriptions of one situation. So the holder is quoted the larger of the two figures at the time of buying and handed the smaller one at the time it counts.

The asymmetry between the figure quoted and the figure delivered is the reason a convention exists for it. Where a bond can be repaid on more than one date, the yields to each of those dates are computed and the lowest of them is the one carried forward, and that lowest figure is what a careful list of holdings will show. The lowest of the yields is a convention rather than a discovery.

When an issuer would actually repay early is a separate question. The answer turns on where rates stand on the day the decision is taken, and every rate worked above is held still. An answer offered anyway would have to invent the evidence for its own claim. Inventing the evidence for a claim is exactly the failure a reader should be able to spot.

Why is a price above the amount repayable the case where this bites hardest?

Change one input and evaluate again. Keep the price at Rs 1,079.4804/-, keep the five coupons of Rs 85/-, keep the call date at the end of year five, and lift the amount payable on an early repayment from Rs 1,000.00/- to Rs 1,020.00/-. The yield to call moves from 6.5831 to 6.9144 per cent a year. The yield to call has climbed, and it still sits 0.4356 percentage points, or 43.56 basis points, under the 7.3500 per cent a year the ten year reading gives.

A checking reader gets misled at this step, so the size of that climb deserves one careful sentence. Subtracting the two printed figures gives 0.3313 percentage points. Differencing the two unrounded results gives 0.3312 percentage points, or 33.12 basis points, and that is the honest number. Both roundings happened to travel outwards, and the pair of them opened a gap of one hundredth of a point between the printed route and the true one. Where a difference is printed, it should be worked from the unrounded quantities and not from the figures already displayed.

The pattern under all of it is simple: while the price paid sits above the amount that would come back on an early repayment, the yield to call sits below the yield to maturity. The holder paid Rs 1,079.4804/- and Rs 1,000.00/- is what returns, so Rs 79.4804/- of what was handed over never comes back as principal and has to be made good out of coupons. Over ten years that is Rs 7.94804/- a year of ground to recover. Over five it is Rs 15.8961/- a year, out of coupons that are exactly the same size. Fewer years, same shortfall, so more of each coupon is spent making the shortfall good and less of it is left to count as a return.

Pushing the amount payable on a call upwards shows where it goes. At Rs 1,040.00/- the five year reading is 7.2412 per cent a year, still under. At Rs 1,046.7137/- the two readings meet exactly, both at 7.3500 per cent a year. Above that the five year reading is the higher of the two: at Rs 1,060.00/- it reads 7.5639 per cent a year, and at the Rs 1,079.4804/- price paid it reads 7.8742 per cent a year, with no principal lost at all because the amount paid is the amount that comes back.

The meeting point is forced arithmetic rather than a quirk, and it belongs to no particular bond. The call amount at which the two readings agree is exactly what the remaining payments are worth on the call date, discounted at the yield to maturity itself. Worked out: five further coupons of Rs 85/- discounted at 7.35 per cent a year come to Rs 345.2749/-, and Rs 1,000.00/- of face arriving five years after the call date comes to Rs 701.4388/-. The two together make Rs 1,046.7137/-, the amount the issuer would have to hand over at the end of year five to leave the holder neither better nor worse off than letting the bond run.

Read back into rupees, the meeting point makes the whole shape fall out of one comparison. Rs 1,046.7137/- is what the rest of the bond is worth on the call date. An issuer who has written Rs 1,000.00/- into the terms is buying back something worth Rs 1,046.7137/- for Rs 1,000.00/-, and the Rs 46.7137/- of difference is exactly what leaves the holder's hands when the right is used. With the written figure pushed up to Rs 1,079.4804/- the position turns over completely. The amount paid is the amount that returns, so no principal is lost at all, and the reading that comes back, 7.8742 per cent a year, is the plain cash measure of Rs 85/- against Rs 1,079.4804/- worked through separately.

The substitution is worked in words and figures, which is how it would have to be done against a real set of issue terms.

What happens when the price paid sits below the amount that would be repaid?

Everything so far has run on a price above the amount an early repayment would return. Turned round, the answer changes and the rule does not. Bond A has also been worked at Rs 931.2325/-, the cost of the same eleven amounts when each is divided down at 9.60 per cent a year. With the supposition exactly as it was, and Rs 1,000.00/- payable if the issuer closes the bond out once year five ends, both searches run again from that lower price.

The price was built at 9.60 per cent a year, so the ten year reading comes back at 9.6000 per cent by construction. The five year reading comes out at 10.3294 per cent a year. The order has turned over: the shorter reading is now the higher of the two, by 0.7294 percentage points, or 72.94 basis points. Nothing in the method changed. The price changed: Rs 931.2325/- was paid for something that would hand back Rs 1,000.00/- on a call, so an early repayment brings Rs 68.7675/- of gain forward by five years instead of pushing a Rs 79.4804/- shortfall into half the time.

The reversal is why the convention is worded as it is. The convention does not say the yield to call is the one to carry. Its wording is to compute the yield to every date the issuer can choose and carry the lowest of them, and on this second price the lowest is the ten year reading of 9.6000 per cent a year. A reader who memorised the first half of the rule and stopped would take the wrong figure here with complete confidence.

What does a right of early repayment actually take away from a holder?

Stated as arithmetic rather than as a warning, because as a warning it sounds like an opinion: what the right removes is years six to ten from the holder's own plan. Years six to ten are the whole of it. The coupons up to year five arrive on either branch, the money paid is the money paid, and the only thing in dispute is the back half of the arrangement.

The five years are not removed at random, and the pattern is the part worth carrying away: they go exactly when the side holding the right would rather have them back. The holder keeps the back half of the bond in the circumstances where the issuer is content to leave it there, and loses it in the circumstances where the issuer would rather not. Set beside the shop and its lease: the landlord ends the arrangement in the year the corner has become worth ending it in, and that is the same year the shop most wanted to stay.

Putting a rupee figure on that lopsidedness is a real question, and it is answered separately. A rupee figure needs a way of pricing a right whose value turns on how far rates might travel, and every rate worked above is held still. The direction is stated and the mechanism is drawn; the valuation belongs to the work that carries the machinery for it.

Who reaches for which of the two numbers, and what do they do with it?

A lender assessing a bond it may hold to the end reaches for both and files the lower one. Filing the lower one is not caution but bookkeeping: a plan built on the higher figure has assumed that somebody else will act against their own interest for five years, and no lending book is allowed to assume that. A rate without its date is half a fact, so the working note beside the figure names the date it was computed to.

An analyst filling a comparison sheet does the same thing one column further left. Every bond on the sheet that carries a right of early repayment gets a yield computed to each date the issuer may choose, and the lowest of those figures is the one that goes in the ranked column. A reader will ask what the bond pays if it runs its stated course, so the other figures are kept, beside the binding figure rather than in front of it. A term sheetThe short document setting out what a security promises: the dates, the amounts, and the rights each side holds. is where the dates and amounts behind that column come from, and a right of early repayment shows up there as two extra lines that a plain bond simply does not have: a date, and an amount.

A household meets the same question in a smaller shape and usually without the vocabulary. Two deposits are offered at what looks like the same rate, and one of them lets the institution close the arrangement early while the other does not. The two deposits are not the same offer, and the one that can be closed is worth less to the household by whatever the early exit takes away, even though the printed rate on the two is identical. The habit that transfers is to ask, of any promised rate, who is allowed to end this and on whose timetable.

An investor with a purpose for the money is the one for whom this matters most. A plan that needs a certain sum in year eight cannot be built on a bond somebody else may close in year five, whatever the sheet says the yield is. And the quoted priceThe figure a bond is shown at in a list. What that figure includes is a matter of convention rather than of arithmetic. that all of this arithmetic starts from is struck on conventions written by somebody other than the person doing the sums. One more reason, then, to know what a given figure was computed from.