The Discount Curve: Present Value Across Every Maturity

A discount curve is the collection of zero-coupon bond prices across every maturity: one number saying what one unit received at each future date is worth today. The curve is made of bond prices and of nothing else, and building one is nothing more than arranging those prices by maturity. Under the locked model the nine readings run from 0.974750 at six months down to 0.169551 at thirty years.

No new arithmetic is needed here. Every number that follows was already computed, one at a time, when the price of one unit arriving at a single future date was worked out from four chosen parameters. A starting rate of 5 per cent, a long-run level of 6 per cent, a speed of reversion of 0.5 a year and a rate volatility of 1 percentage point produce a price for any maturity that can be named, and nine of those maturities have already been named and priced.

Putting those nine numbers side by side, in order of maturity, produces the shape, and the shape is the only thing that was not visible before. The claim is worth sitting with. The word curve sounds like a new object and it is not one. Think of nine index cards, each carrying one number, worked out on nine separate afternoons and left in a drawer in no particular order. Lay them out on a table from soonest to latest and a line appears. No card changed. Nothing was added to the drawer. The arrangement did all the work, and the arrangement is free.

Nobody measured this curve. A curve is the object readers most readily assume somebody went out and observed, and every price on this one is instead a computed consequence of four chosen parameters. The four parameters and a calculator will reproduce all nine numbers to the last decimal place, and no measured curve has that property.

What is a discount curve?

A discount curveThe collection of zero-coupon bond prices across every maturity, read as one object rather than as a list. is a rule that answers one question for every date that can be named: what is one unit received on that date worth today? Name a maturityThe date the one unit arrives. On a curve it is the horizontal axis, and every point on the curve sits above one of them. and the rule hands back a single number between nought and one. Name a different maturity and it hands back a different one. The curve is that rule, considered as one object, rather than any single answer it gives.

Two things are true of it at once and both are ordinary. The curve is a function, in the plain sense that each maturity has exactly one price attached to it. And it is a collection, in the plain sense that the whole of it can be written down as a list of pairs when only nine maturities matter. Neither description is more correct than the other. Which one an analyst reaches for depends on whether the curve is to be drawn or consulted for a single value.

Most of that expression does not change as the maturity moves along the curve. Today's rate does not change: it is one number, 0.050000, and it is the same number at every maturity. The long-run level does not change. The speed does not change. The rate volatility does not change. Only the maturity moves. Nine points on this curve therefore cannot disagree with each other in the way nine separately quoted numbers could. They are nine readings of one machine at nine settings, not nine independent opinions.

Structural consistency is a stronger property than it looks. If somebody handed an analyst nine numbers and said they were prices at nine maturities, the analyst would have to check that they were consistent: that the ten year one was not somehow above the seven year one, that no pair implied something absurd about the stretch between them. Here no such check is needed. The consistency is structural. The four parameters brought it in, and no maturity chosen for evaluation can break it.

What is the curve made of, and what is it not made of?

The curve is made of zero-coupon bond prices and of nothing else. Every point on it is the price today of one unit at one future date, and the bond gives exactly that number. Building a curve is arranging those prices by maturity, and no second ingredient enters at any stage.

Now the harder half, the list of things the curve is not made of. The curve is not made of observations. Nobody went and looked at anything to produce these nine numbers, and no quotation from anywhere is involved. It is not made of a fit: nothing was adjusted to match anything, no error was minimised, and no data was consulted. And it is not made of guesses about the maturities in between. Evaluating the same expression at eleven and a half years returns an exact price for that maturity, so nobody has to draw a line between two neighbours and read off the middle.

The left half of the picture and the right half carry identical information. The picture is worth reading twice for that reason alone. Nine numbers in a column on the left. The same nine numbers, placed above their maturities, on the right. The count is the same on both sides. The shape on the right is not a tenth thing that appeared: it is what nine dots look like once they are no longer treated as a list. An arrangementA rearrangement of numbers already in hand. An arrangement can make a pattern visible, and it cannot add anything that was not in the numbers. can reveal, and it cannot add.

There is a household version of this. Somebody steps on a bathroom scale every Sunday for two months and writes each figure in a notebook, one per line. Eight numbers, no pattern, just a list. Plotted, a direction appears, or does not, and either way the scale never reported anything it had not been reporting eight weeks running. The plot did no weighing. The plot arranged what the weighing had already produced, and the only new thing in the room is the ability to see a slope.

What shape does the locked model give it?

The curve falls, and it never once turns back. From 0.974750 at six months it goes down through 0.949216 at one year, 0.898265 at two, 0.848492 at three, 0.754894 at five, 0.670468 at seven, 0.560610 at ten, 0.308325 at twenty and 0.169551 at thirty. The curve is monotoneMoving in one direction only. A monotone falling curve never rises anywhere along its length, not even slightly. falling across its whole length, which is the one shape property that can be stated without qualification.

The third column is only there to make the first two concrete. One unit is any unit at all, Rs 1/- for concreteness, and a hundred of them due in twenty years is worth Rs 30.8325/- today under this model. Nothing in that column is a new figure. The third column is the second column multiplied by a hundred, and the arrangement point holds one more time.

How the curve falls and that it falls are two different observations. Take the first of them now. Over the first six months the price gives up 0.025250. Over the six months after that it gives up 0.025534. Between twenty years and thirty, a stretch twenty times as long, it gives up only 0.138774, less than six times as much. In absolute terms the curve flattens out at the far endThe long maturities on a curve, where prices are small and each further year takes away less in absolute terms than an early year did. and there is simply not much left to give up.

In proportional terms it is a different story and a much steadier one. The ten year price multiplied by 0.549981 gives the twenty year price. The twenty year price multiplied by 0.549911 gives the thirty year price. The two factors agree to four decimal places, so the far end of this curve is settling into an almost constant proportional decay per decade even while the absolute steps are collapsing. Both facts are true of the same nine numbers, and a reader who has only looked at one of them has half the shape.

How does that shape compare with a flat five per cent?

Set the comparison up properly before drawing any conclusion from it. A flat curveA curve built from one constant rate applied to every maturity, so its shape carries no view about the future at all. is what comes of taking a single rate, applying it to every maturity, and letting nothing else in. A flat curve is the simplest possible curve, and it says the same thing about thirty years that it says about six months. The locked model starts from exactly the same 5 per cent, so a flat 5 per cent is the natural thing to hold it against.

At the same nine maturities the flat curve reads 0.975310, 0.951229, 0.904837, 0.860708, 0.778801, 0.704688, 0.606531, 0.367879 and 0.223130. Put the two sets side by side and the first fact is unanimous. The locked curve sits below the flat one at every single maturity, and there is no maturity anywhere along the axis where it does not.

Why below, everywhere? Because the locked model does not think the rate stays at 5 per cent. The rate is pulled toward 6 per cent, and the pull starts working immediately. Discounting a future unit through a stretch of time in which the rate is heading upward takes more away from it than discounting through a stretch in which the rate never moves. The whole of the gap's sign comes from that single disagreement about where the rate is going, and none of it comes from the rate being uncertain.

The last clause surprises people, so take the twenty year reading apart. Under the locked model the exponent that produces 0.308325 is made of three parts: today's rate contributes minus 0.099995, the pull toward the long-run level contributes minus 1.080005, and the uncertainty about the rate contributes plus 0.003400. The flat curve's exponent is simply minus 1.000000. Strip the uncertainty term out of the locked exponent and the price would be 0.307278 rather than 0.308325. The uncertainty is worth plus 0.001047 at twenty years, and it pushes the locked price up, toward the flat curve rather than away from it.

What does the comparison show that is not obvious?

Taken in order, the nine gaps do something to the mind that reads them. Minus 0.000560. Minus 0.002013. Minus 0.006573. Minus 0.012215. Minus 0.023906. Minus 0.034221. Minus 0.045920. Minus 0.059554. Eight readings and every one is wider than the one before it. By the eighth there is a trend, and a trend is a thing the mind finishes on its own before it has been given permission to.

There it is. The gap widens at eight consecutive readings and then, between twenty years and thirty, it turns and comes back to minus 0.053579. On the six-decimal readings the recovery is 0.005975, which is roughly a tenth of the widest gap given back over the last decade of the axis. Moved slowly through the twenties, the slider shows the bar in the readout column stop growing and start shrinking while the curve underneath it is still falling.

Two numbers are worth separating here. Among the nine readings the widest gap is at twenty years, at minus 0.059554. But the nine readings are just nine places somebody chose to look, and the true widest point of the gap sits a little past them, at 20.242391 years, where the gap reads minus 0.059560. The 20.242391 year figure is not one of the nine, and it comes from searching the gap function for its lowest point. The turning pointThe maturity at which the gap stops widening and starts narrowing. Past it, every further year of maturity closes the space between the two curves. is a property of the two curves, not of the maturities anybody happened to tabulate.

Why does the gap have to turn somewhere?

The turn is not an accident of these four numbers. The turn is forced, and the reason rules out ever extrapolating a gap between two falling curves. Both curves are heading toward nought, and two quantities that both approach nought must have a difference that also approaches nought. There is nowhere else for it to go. A gap that has been opening for twenty years is therefore living on borrowed time, whatever it looked like on the way out.

There is a version of this visible in a kitchen. Two buckets are filled and a hole is punched in each, a wider hole in one than in the other. At the start the water levels are level. The bucket with the wider hole falls faster and the difference between the two levels opens up, and a difference written down every minute would make a nice clean widening series. Then both buckets empty. The difference had been growing, and it ends at nothing. Two empty buckets cannot stand at different levels. The widening was real and the return to nought was never in doubt, and both statements were true at the same time from the first minute.

Push the maturity to a hundred years and the arithmetic says the same thing the buckets do. The locked model prices one unit at 0.002578 and the flat curve at 0.006738, so the gap has come back to minus 0.004160. Minus 0.004160 is 6.98 per cent of the widest reading, so more than nine tenths of the gap has already been given back. The gap is still negative, still shrinking, and heading to nought as slowly and as certainly as the two prices are.

One caution against the wrong lesson. The absolute gap turns, and the proportional comparison never does. The locked price divided by the flat price at each maturity gives 0.999426, 0.997883, 0.992736, 0.985808, 0.969304, 0.951439, 0.924290, 0.838114 and 0.759876, falling to 0.382663 at a hundred years. In proportional terms the locked model keeps pulling away from the flat curve at every maturity without exception, and it is only in rupees and paise per unit that the gap has a turning point. Both statements describe the same two curves. The two statements differ in what is being subtracted or divided, and a reader who has not said which one they mean will argue with somebody who means the other.

How does somebody working with a rate model actually read this shape?

Four habits, and each one follows directly from something above. The single most useful thing a curve does is show where a model's assumptions are doing the most work, and the comparison against a flat curve is how that place is found.

1. Compute the maturity required rather than reading it off a drawingAn eleven and a half year price is one evaluation of the same expression. Reading it off a plotted line between the ten year dot and the twenty year dot introduces an error that nothing in the model requires anybody to accept, and on this curve the ten to twenty stretch is where the shape bends most.
2. Hold the shape against a flat curve before believing anything about itThe nine locked prices on their own look unremarkable. Against a flat 5 per cent they say something specific: this model is asserting a rate that rises, and the assertion is worth minus 0.059554 per unit at twenty years. That is the size of the claim the four parameters are making.
3. Find where the disagreement is widest, because that is where the parameters matter mostAt six months the two curves differ by 0.000560 and any argument about the parameters is nearly irrelevant. At twenty years they differ by more than a hundred times as much. If somebody wants to challenge the long-run level or the speed, the twenty year end is where the challenge shows up and the six month end is where it hides.
4. Never continue the shape past the last maturity computedThis is the whole failure block in one line. The curve is cheap to evaluate at any maturity, so there is never a reason to guess at one, and the one place a guess feels safest, a long run of consistent movement, is exactly where this curve turns.

None of those four is a view about whether the model is right. A curve computed from four parameters reports what those four parameters imply and says nothing whatever about whether the parameters deserve to be believed. Reading a shape carefully and believing a shape are separate acts, and only the first is settled by the arithmetic above.

What does the curve not tell the reader?

The curve answers one question and answers it completely: what is one unit received at a stated date worth today, under this model. Anything that is not that question is not a curve question, however carefully anybody stares at the nine numbers.

The curve does not report what any instrument pays. A contract has terms, dates, conditions and a settlement rule, and no collection of present values contains any of that. The curve will happily give the worth of one unit at seven years, 0.670468, and it has nothing at all to say about what somebody agreed to hand over at seven years or under what circumstances.

The curve does not report that anything was observed. Every number here was computed, and a computed curve looks exactly like a measured one in print. The only defence against confusing the two is a label on the figure saying where its numbers came from.

The curve does not establish that the model is right. The shape is a faithful report of what four chosen parameters imply. All nine points come from the same four numbers, so a change in the long-run level moves the whole shape, and no mechanism lets the ten year price move while the six month price stays put. Building a curve this way is a strength and a constraint at once, and the curve itself is silent on whether the four numbers were well chosen.

The rate parameters, the curve, the comparison curve and every price shown here are invented.
Educational material. Not advice on any investment, tax, budget or market position.