The Zero-Coupon Bond: The Building Block of Rate Modelling

A zero-coupon bond is the present value today of one unit received at a stated future date. The bond is the primitive of rate modelling because a model produces that single number directly, where a rate is the same number restated in different units and a curve is a collection of those numbers across dates. Nothing else needs defining.

Everything built in rate modelling is built out of present values of one unit at a date. The rate is one of those present values rewritten in per cent per year. The curve is a set of them lined up by date. Neither is more fundamental than the thing it rearranges, and a model that produces the present values has already produced both. Everything downstream depends on getting that order the right way round.

Present value and the bond contract are both settled in a different subject area and arrive here already known. Narrowed down until only the mathematics is left, the object turns out to carry a great deal inside it before anybody restates it as anything else.

What is a zero-coupon bond, mathematically?

A zero-coupon bond is a number attached to a date, and that is the whole of it. Write down a future date, ask what one unit of money arriving on that date is worth today, and the answer to that question is the object. There is no issuer in the definition, no coupon schedule, no settlement convention, no credit standing and no market. Issuer, coupon, convention, credit and market all exist and all matter enormously, but every one of them is settled elsewhere. One number for one date is what crosses into the mathematics.

Two properties fix it completely. First, one unit arriving now is one unit, so on the date itself the value is one. Second, before that date the value is an average, taken under the pricing measure, of the accumulated discount along every path the rate could take between now and then. The zero-coupon bondThe present value today of one unit received at a stated future date. Nothing about any contract is part of the definition. is therefore not a forecast of one rate, it is an average of a discount factor over a whole distribution of rate paths.

There is a difference between a reading and the thing being read. A tape measure gives a length in centimetres. A different tape gives the same length in inches. The object has one size, and the two readings are two ways of saying it. In rate modelling the object is the present valueWhat a future amount is worth today. Here it is the whole of what a bond price is., and the rate is the reading in other units. The mathematics produces the object and leaves the choice of units to whoever wants to talk about it.

The averaging is the step most easily lost. The average of the discount over many rate paths is not the discount computed from the average rate path. The two are different numbers, and the gap between them is not an error. The gap is a real feature of the object, and further down it turns out to be one of the two factors the price is made of.

The maturityThe date the one unit arrives. Every bond price in this guide is labelled by its maturity and by nothing else. is the only label the object carries. In this subject area two zero-coupon bonds differ by their date and by nothing else. That is exactly why one model holding a handful of parameters can produce all of them at once.

Why is the bond the primitive and not the rate?

Because of what a model produces once the handle is turned. Four parameters go into a short-rate model, the average the definition above asks for is taken, and what comes out is a number between nought and one attached to a date. The number comes out directly. No conversion happened on the way, and nothing was chosen about how to express it.

Two things are usually done next. The logarithm of that number, divided by the elapsed time and with its sign changed, is a rate. The numbers lined up for many dates are a curve. Both operations take a present value in and give the same content back in a different arrangement. Neither adds information, and neither can be done first. A primitiveThe quantity a model produces directly, before anything is rearranged or renamed. Here it is the bond price. is whatever the model produces before anybody rearranges it, and here that is the present value.

The everyday version is a weighing scale. The scale reads a weight. The same weight can be expressed in kilograms or in pounds, and a whole column of weights can be written down for a whole shelf of objects. Nobody would say that pounds are more fundamental than kilograms, or that the column is more fundamental than any single reading in it. The object is the weight. Everything else is units and arrangement.

What does its price already contain?

More than most readers expect, and the surplus is the interesting part. Look again at the definition. The exponent holds an integral of the short rate over the whole stretch of time between now and the arrival date, and the expectation runs over every path that integral could take. So the price is not built from today's rate alone, and it is not built from any single guess about where the rate goes. The price is built from the entire distribution of the accumulated rate.

The distribution has a consequence with teeth. Because the discount function bends, the average of the discount over a spread of outcomes is not the discount of the average outcome. Spread the possible accumulated rates wider while keeping their average fixed and the average discount goes up, not sideways. Uncertainty about the rate raises the present value of one unit, even when it leaves the expected rate completely unchanged.

So a bond price already contains three things at once. The price contains where the rate sits today. The price contains where the rate is being pulled towards, and how fast. And it contains how uncertain the journey is. Three ingredients, one number, and the number gives the total without giving the split. Separating them takes the closed form the model provides.

How does a short-rate model produce that number?

By making the short rate a process with a pull, and then doing the integral the definition asks for. The mean reverting rate process is covered under mean reversion, where the speed of the pull is fixed at 0.5 a year, and it is carried into this guide unchanged. Every figure below is a computed consequence of four numbers a reader can check.

Here are the four. The short rateThe rate applying over the next instant. Everything in a short-rate model is built out of the path this quantity takes. stands at 5 per cent today. The rate is pulled towards a long-run level of 6 per cent. The speed of that pull is 0.5 a year. The rate carries a volatility of 1 percentage point a year in absolute terms. Those four numbers are the entire input list. There is no fifth number and there is no observation anywhere in it.

For a model of this shape the integral in the definition can be done in closed form, and the answer has a particular structure. The logarithm of the bond price turns out to be linear in today's rate. A model with that property is called affineA model whose log bond price is a straight line in today's short rate, which is what makes the price separate into two clean pieces., and the linearity is not decoration. The linearity is exactly what lets the price split into two factors that can be looked at separately.

Turn the handle at one year. The response term is 0.786939, so the factor that depends on today's rate is the exponential of minus 0.786939 times 0.05, which is 0.961417. The other factor is 0.987309. Multiply them and the present value of one unit at one year is 0.949216. Scaled up so the arithmetic is easy to feel, Rs 100/- arriving in one year is worth Rs 94.9216/- today under this invented model.

Compare that with the flat world. Hold the rate at 5 per cent forever with no pull and no uncertainty, and the same one year discount factorAnother name for the same number: what one unit at a future date is worth today. is 0.951229, or Rs 95.1229/- on the same scale. The two differ by 0.002013. The gap is small, it is entirely a consequence of the pull and the uncertainty, and it is the reason a rate needs a model of its own rather than a constant.

The whole set of maturities, from the same four numbers

Nothing in the arithmetic changes as the date moves out. The same four parameters give the present value at every maturity, and nine of them are set out below.

What are the two pieces of that price?

One factor answers to today's rate. The other does not answer to it at all. The shape shows why: a factor that carries today's rate in an exponent, multiplied by a factor that has no today's rate in it anywhere. Everything interesting about a bond price comes from those two factors behaving completely differently as the date moves out.

The response term has the surprise in it, so start there. The response term is a pure function of the maturity and the speed of the pull, and it has a closed form that fits on one line.

Now the other factor. Written out fully, the logarithm of the price separates into three named ingredients, and the second and third of them together are the factor that carries no dependence on today's rate.

At one year the three read minus 0.039347, minus 0.012784 and plus 0.000012. Added, they give minus 0.052119. The exponential of minus 0.052119 is 0.949216, the number in the table. At thirty years they read minus 0.100000, minus 1.680000 and plus 0.005400, summing to minus 1.774600 and giving 0.169551. The first ingredient has stopped moving between those two maturities while the second has grown by a factor of well over a hundred.

The contrast between those two ingredients is the clearest statement of what the pull does, and it is worth drawing. The rate-dependent factor is 0.961417 at one year, 0.905447 at ten years and 0.904837 at thirty. The rate-dependent factor has a floor, and the floor is the exponential of minus two times today's rate. The parameter-only factor is 0.987309, then 0.619153, then 0.187383, and it keeps falling.

Inside the parameter-only factor sits the ingredient readers least expect: the convexity termThe part of the price that comes from uncertainty about the rate rather than from its level. The convexity term depends on the parameters and the date alone.. The convexity term is the third ingredient in the formula above, it is positive at every maturity, and it does not care where today's rate sits. Set today's rate exactly on the long-run level of 6 per cent and the convexity term is entirely unchanged.

How big is it? At one year the convexity term lifts the price by 0.000012, a bit over a tenth of a basis point in rate units. By thirty years it lifts the price by 0.005400, worth 1.8 basis points on the same reading. And it too has a ceiling: it approaches exactly 2 basis points, being the squared rate volatility divided by twice the squared speed. Two different ingredients of the same price each run into their own ceiling, and both ceilings are made out of the speed of the pull.

Why does the response to today's rate stop growing?

Because the pull eats it. Today's rate matters to a bond only for as long as today's rate still has any bearing on where the rate actually is. Under a process with a pull, the influence of the starting level decays as time passes, and it decays at the speed of the pull. Added up across the whole life of the bond, that decaying influence is a decaying quantity, so the sum converges instead of growing.

The everyday version is a room with a heater on a thermostat. Ask how warm the room will be in ten minutes and today's temperature matters a great deal. Ask about tomorrow and it matters less. Ask about next month and the answer is whatever the thermostat is set to, and the current reading has stopped carrying any weight at all. The thermostat has taken over. A pull on a rate does exactly that to today's level, and the speed of the pull decides how quickly.

Look at what that cap does to the numbers. A rise of one percentage point in today's rate cuts the one year price by 0.783850 per cent. The same rise cuts the ten year price by 1.966923 per cent, and the thirty year price by 1.980132 per cent. The step from one year to ten is enormous. The step from ten to thirty is thirteen thousandths of a percentage point. The far end of this curve has stopped listening to today's rate.

The bottom bar is the part that catches people out twice. The proportional response keeps creeping up towards 2, but the money response is a proportion of a price that is itself shrinking, so it rises to a peak at about four and a half years and falls away after that. At thirty years the same one percentage point move takes 0.003357 off the price, less than half of what it takes off at four and a half. Two entirely correct statements about sensitivity point in opposite directions, and which one is right depends only on whether the question was asked in per cent or in money.

What can be built once the bond price is in hand?

Everything else in this reading order, and nothing that was not already in the numbers. The prices at many dates, stacked by date, are the curve. Any single price rewritten as a rate is a point on a rate curve. The ratio of two prices at two dates is the discount that applies between those two dates, and the rate version of that is a forward rate. Three constructions, one raw material.

Each of those constructions is covered separately. The direction of the arrow is what matters: the constructions all run outward from the bond price, and none of them runs back into it. The curve and the prices are the same numbers, so an analyst handed only the curve could recover every bond price on it. Handed only a description of the rate process, the analyst would still have to do the integral in the definition before anything could be quoted.

How does somebody reading a rate model actually use this?

By asking three questions of any number that arrives with the word rate attached to it, and none of the three needs a screen or a data feed. The first is whether the number at hand is a present value or a restatement of one. A household that is told the number is 5.211896 per cent and a household that is told the number is 0.949216 have been told exactly the same thing, but only the second has been shown the object. The percentage is friendlier and it hides the split.

The second question is which ingredient the number came from. A model output at one year of 0.949216 against a flat world figure of 0.951229 is a gap of 0.002013, and the whole gap is the pull and the uncertainty. Ask that question at thirty years and the answer is dramatically different: 0.169551 against a flat 0.223130, a gap of 0.053579, almost all of it from the pull towards 6 per cent rather than from anything happening today. The near end of a curve is mostly today's rate and the far end is mostly the parameters, and a single quoted number never says which.

The third question is the most useful when somebody is deciding what to do about a change in today's rate: over what stretch of maturities does the response actually still vary? On this invented curve the answer is roughly the first ten years. Beyond that the response is within seven thousandths of its ceiling and the maturities move as a block. Past a point there is no extra leverage on today's level to be had by going further out, and in money terms going further out gives back what it appeared to promise.

The everyday version is the thermostat again. How much opening a window right now changes the temperature depends heavily on whether the question is about the next few minutes or about next week. Past a certain horizon the thermostat setting, and not the window, decides the answer. Reading a curve without knowing where that horizon sits is reading it without knowing which questions it can still answer.