Participation Rate: How Much of a Move Is Passed On

A participation rate measures how much of the reference asset's move reaches whoever holds the instrument. It is not a feature of the instrument at all. Take the outlay, subtract what the promise leg costs today, and divide the leftover by the price of one option leg. Move either of those two prices and the number moves with it.

Every figure below belongs to a packaged instrument, invented for teaching, written on an invented reference asset. Three inputs generate the whole of it. The reference asset has a spot price of Rs 2,000.00/-. Financing costs 6.50 per cent a year and the instrument runs for one year. One call contract carries a premium of Rs 180.00/-. The premium is given rather than worked out, since working it out would need a figure for how far the reference asset might move and the three inputs above do not supply one. The reference asset pays nothing at all while it is held. A payout during the year would change every amount below it.

A participation rate is computed out of two prices, and both of those prices belong to one particular day. A number quoted on the front of a document was struck on some particular day out of some particular pair of prices, and a reader arriving with that number in hand will find neither of them printed beside it.

What does a participation rate get applied to?

A participation rate is applied to the move in the reference asset, measured from the strikeThe level a move in the reference asset is measured from. Below it there is no move to measure and nothing to pass on.. The strike is the whole of the answer, and getting it wrong is the error that follows a reader for years.

The everyday version is closer than it looks. A cousin who runs a cloth shop offers a deal: whatever this festival season adds to his takings over last season, two thirds of that increase goes to the relative who helped him stock up. The two thirds is applied to the increase. The two thirds is not applied to the takings, not applied to the shop, and not applied to the amount handed over to help him stock up. Three quantities are in the room and the fraction touches exactly one of them. A reader who applies the two thirds to the wrong one produces a number that looks like money and means nothing.

A participation rateThe share of a move in the reference asset that a packaged instrument passes on to whoever holds it. The rate scales a move, never an amount anybody paid. is applied to the move in the reference asset measured from the strike, and to nothing else. The rate is not applied to the outlay of Rs 2,000.00/-. Nor is it applied to a profitWhat is left after everything paid has been counted and carried forward to the same date. A payoff becomes a profit only once that subtraction is done.. At the moment the rate is written down nothing has been earned by anybody, so it is not a share of anything anybody has already earned.

On these figures one distinction hides, so it has to be made in the same breath. The move being scaled is a move in the value of the reference asset, and that value is exposureThe value of the reference asset a contract is written on. Nobody has paid it and it does not change hands. Exposure is the quantity a move is measured against.: the size of the contract the option leg is written against. Nobody has paid that size and it never changes hands. The outlay of Rs 2,000.00/- is a different animal altogether. The outlay is an amount that actually moved, from the buyer to whoever sold the instrument. Both happen to read Rs 2,000.00/- in this example, and that agreement makes the distinction harder to see rather than easier.

Where does the number come from?

Two steps, and there is no third. Step one: the outlay less what the promise legA promise to pay a stated amount at a stated date. A promise leg does not move when the reference asset moves, and what it costs today is arithmetic on a rate. costs today leaves a budgetWhat is left of the outlay once the promise leg has been bought. The budget is the only part of the outlay that ever reaches the moving part of the instrument.. Step two: that budget divided by the price of one option legThe contract inside a packaged instrument whose value comes from the reference asset. Here it is a call struck at Rs 2,000.00/-. is the participation rate.

Run it. The promise leg here promises Rs 2,000.00/- at the end of one year, and what that costs today is arithmetic on a rate the record carries. Divide Rs 2,000.00/- by 1.065 and the cost today is Rs 1,877.9343/-. Subtract, and the outlay of Rs 2,000.00/- has Rs 122.0657/- left in it. The leftover is the budget. One whole option leg carries a premium of Rs 180.00/-, and that figure is given rather than computed. Divide: Rs 122.0657/- divided by Rs 180.00/- is 67.8143 per cent.

Which of the two prices came from where recurs below, so it is worth holding on to. The promise leg's cost is arithmetic on a rate and the option leg's premium is given rather than derived, and a participation rate is one of those divided by the other. Neither of them was priced from scratch here, and neither could be: pricing an option contract from scratch needs a figure for how far the reference asset might move, and no such figure is on hand.

Now the everyday version. The shape of this ratio is older than any instrument. A person has Rs 900/- left in pocket after the month's rent has gone out. One seat at the match costs Rs 1,200/-. How much of a seat does that buy? Three quarters of one, and that fraction is the whole of the story. The fraction is a ratio of two prices. Neither of the two numbers knows anything about the match, about who is playing, or about whether anybody scores. Both steps set one price against another. A participation rate is therefore a ratio of two prices, and it carries no information of its own about anything else at all.

A careful reader has already noticed one repeated figure, and a repeated figure that nobody explains gets read as a mistake. Three quantities in this guide read Rs 2,000.00/-, and they agree for three separate reasons. The spot price of the reference asset is Rs 2,000.00/- because that is what the reference asset costs. The strike is Rs 2,000.00/- because this option leg is struck at the money, and struck at the money is precisely what that means. And the outlay is Rs 2,000.00/- because this worked instance sizes everything against exactly one unit of the reference asset. Rs 2,000.00/- is therefore the scale the arithmetic runs against rather than a price anybody has asked for this instrument. Nothing was copied from one line into another.

Why is the number not a property of the instrument?

Because of the two steps just worked. The number is built out of two prices, so move either price and the number moves, without one word of the instrument's terms having changed.

Take them one at a time. A lower financing rate makes the promise leg cost more to buy today. A dearer promise leg leaves a smaller budget behind it, so the participation falls. An option leg that carries a bigger premium absorbs a bigger budget for the same share, so the participation falls again. Neither of those is a change to the instrument. The promise is still a promise of the same amount at the same date. The option leg is still a call struck at the same level. The price of one or both of them on the day somebody did the sum is what has changed.

A participation rate belongs to a date and two prices, not to a product. The consequence of that line is blunt: two instruments quoting different participation rates may be doing nothing more than quoting them on different days. Until both dates and both sets of prices are known, no comparison has started.

The everyday version again, and it is the same pocket and the same match. The Rs 900/- buys three quarters of a Rs 1,200/- seat today. Next week the seats are Rs 1,000/- and the same Rs 900/- buys nine tenths of one. Nothing about the buyer changed, nothing about the match changed, and the fraction moved by a sixth. Reporting the fraction without the date is reporting half a number.

What has to be given up for the number to rise?

There is one outlay and it has to cover both legs. The single outlay is the whole constraint, and everything below follows from it. The promise leg is the only other place the outlay is going, so the only way to enlarge the budget is to promise back less at the end.

Work it at both ends rather than describing it. At one end, the instrument promises the whole Rs 2,000.00/- back. The promise leg costs Rs 1,877.9343/-, the budget is Rs 122.0657/-, and the budget buys 67.8143 per cent of one option leg. At the other end, ask what the promise would have to be for the budget to buy one whole option leg. The budget would have to be Rs 180.00/-, so the promise leg would have to cost Rs 2,000.00/- less Rs 180.00/-, or Rs 1,820.00/-. Multiply Rs 1,820.00/- by 1.065 and the amount promised back would have to be Rs 1,938.30/-. The far end promises Rs 61.70/- less back at the end, and the participation is then 100.00 per cent of one option leg.

The two ends of that trade differ in exactly two figures, and one of them is bought with the other: Rs 61.70/- less promised back at the end is what buys the move from 67.8143 per cent to 100.00 per cent. The trade between the two ends is a budget constraintThe fact that one outlay has to cover both legs and cannot cover more, so anything added to one leg comes out of the other., the same shape as any other budget constraint, and it is a description of an arithmetic relationship rather than a menu of choices put in front of anybody.

Can the other end be checked a second way?

The other end can be checked a second way, and the second route is the reason the first can be trusted. Rs 61.70/- is the put premium of Rs 57.9343/- carried at 6.50 per cent for one year. The budget of Rs 122.0657/- is the same figure as the difference between the two premiums under the parity relationship, a relationship used here rather than rebuilt. The comparison is direct: Rs 180.00/- less Rs 57.93/- is Rs 122.07/-, and the unrounded parity difference is Rs 122.0657/-. The two routes differ by 0.43 of a paisa, and they differ for one reason: the put has been rounded to the paisaOne hundredth of a rupee. Rounding to the paisa is why two routes to the same figure can differ in the fourth decimal place. so that a person can use it. The relationship therefore holds to the paisa rather than exactly. A relationship claimed as exact, when the printed figures do not produce an exact equality, teaches a reader to stop checking.

There is a tidier way to hold the whole constraint in one line, and it falls out of the same arithmetic. The budget at any setting is the gap between Rs 2,130.00/- and the amount promised back, brought to today by dividing by 1.065. Test it. Promise Rs 2,000.00/- back and the gap is Rs 130.00/-. Divided by 1.065 that gap is Rs 122.0657/-. Promise Rs 1,938.30/- back and the gap is Rs 191.70/-. Divided by 1.065 that gap is Rs 180.00/-. The Rs 2,130.00/- in that sentence is the outlay of Rs 2,000.00/- carried at 6.50 per cent to the end of the year, and it is a cost of carry rather than anybody's opinion about where the reference asset is going.

What does the number not say about what a holder receives?

Most of what a reader wants to know is worth saying plainly rather than leaving to be discovered. A participation rate does not say how far the reference asset will move. The rate is a ratio of two prices struck today, and a distance is not one of the ingredients. The rate does not say how likely any move is either, because no probability and no distribution enters the division that produces it. And it does not say what a holder will receive.

The third of those is a confusion between two words rather than a missing figure. A participation rate scales a move into a contribution to the payoffWhat is owed at the end, before anything paid for it is counted. A payoff is gross of the price that was paid at the start., the amount owed at the end before anything paid for the instrument is counted. A holder actually cares about a profit. A profit counts the outlay that was paid at the start and carries it forward to the same date, so the two figures are compared at the same moment. A participation rate scales a move into a contribution to the payoff, and turning that payoff into a profit is a separate subtraction the rate knows nothing about.

Work it once so the shape is concrete, and read it as arithmetic about an obligation rather than as a statement that any level will be reached. Suppose the reference asset ends the year Rs 300.00/- above the strike. The payoff contribution is 67.8143 per cent of Rs 300.00/-, or Rs 203.4429/-. Add the Rs 2,000.00/- the promise leg pays and Rs 2,203.4429/- arrives at the end. Now the subtraction the rate knows nothing about: the outlay of Rs 2,000.00/- paid at the start, carried at 6.50 per cent to the same date, is Rs 2,130.00/-. The profit is Rs 73.4429/-. The participation rate touched exactly one of those four figures.

The control above carries one more thing worth sitting with, and it is the strongest argument in this guide against reading a participation rate as a verdict. At any setting of the control, the move above the strike at which the profit line crosses nil does not budge: it sits at Rs 191.70/- at every single setting. Rs 191.70/- is the premium of Rs 180.00/- carried at 6.50 per cent for the year, and the reason it never moves is the budget constraint itself. Whatever is given up at the end buys exactly its own worth of extra participation, so the two effects cancel at that one point. A number that changes from 67.8143 per cent to 100.00 per cent while a crossing point stays fixed is not describing how well anybody does. The number is describing a shape.

Is the share that is not passed on a charge somebody collects?

No, and this block exists because the opposite is the commonest thing written about this number anywhere.

Start with the arithmetic rather than the argument. The budget of Rs 122.0657/- bought 67.8143 per cent of one option leg at a premium of Rs 180.00/-. The rest of that leg, Rs 57.9343/- of premium and 32.1857 per cent of any move, was simply not bought. The rest was not deducted, not withheld, and not skimmed off something on its way to somebody. There was never enough in the budget to buy it in the first place.

Nobody collects the 32.1857 per cent that is not passed on. The 32.1857 per cent is not a fee, it is not revenue, and there is no party standing on the other side of it receiving anything. One question settles it in one line: who receives it? Name them. There is nobody to name. The quantity is a share of something that was never purchased, and a share of an unbought thing has no recipient by construction.

The everyday version makes it obvious. The pocket held Rs 900/- and the seat cost Rs 1,200/-. Nobody bought a quarter of a seat and paid somebody Rs 300/- for the privilege. The quarter of a seat left unbought is not a charge, it is a quarter of a seat sitting there unsold. Nobody holds that Rs 300/- because there was never Rs 300/- to give them.

Where is the packaging actually paid for?

A reader who has just been told that the unbought share is not a charge will immediately and correctly want to know where the charge is.

The shape of the answer is exact. An instrument of this kind is offered at a price. Its legs cost something to buy separately, at prices somebody else set, on the same date. The difference between those two figures is where the packaging is paid for. The difference is a subtraction rather than a number, and the subtraction holds whatever the two figures turn out to be.

Behind this worked instance there is no seller, no price asked and no fee, so the first term of that subtraction is simply absent and no amount stands in its place in either direction. Naming the hole is the honest move. Filling it would be an invention, and an invented charge on an invented instrument is exactly the sort of figure a reader might carry away and use on a real document.

Two further readings of the same figures go wrong in the same way. The Rs 2,000.00/- here is not a price anybody asked for the instrument: it is the scale this worked instance runs against, being exactly one unit of the reference asset, and it must not be treated as the first term of the packaging subtraction. And the cost of the parts worked out here is the cost at the prices in this record on one date, not a claim about what any leg costs anywhere. Two facts would fill the subtraction in, and both are obtainable from the person offering the instrument: the price actually asked, and what each leg costs on that same day.

How does somebody with a document in front of them use this number?

The arithmetic above is not an exercise. A person reading one of these documents does exactly this with a participation rate printed on the front of it, and the order matters because each step depends on the step before it.

1. Write down the quantity the rate is applied to, in the document's own words Not from memory, and not from any general account of these instruments. The sentence that says where the move is measured from, and over what period, is the one to copy, word for word. A rate applied to a move from a starting level is a different instrument to a rate applied to an average of levels, and the front of the document rarely distinguishes them.
2. Find the date the number was struck A participation rate is a ratio of two prices on one day. Without that day, two quoted numbers cannot be set beside each other at all, and a number quoted three weeks ago against prices that have since moved is a number about a day that has passed.
3. Ask what is promised back, and put the two figures side by side The participation rate and the amount promised back are one bar with a boundary in it, not two independent terms somebody chose. A higher rate beside a smaller promised amount is the constraint working exactly as it must, and neither figure means much read on its own.
4. Write who owes each amount beside it This is the column with no arithmetic in it and it is usually the one that decides the answer. A promise of a stated amount at a stated date is worth what the party making it is worth, and no amount of accurate discounting substitutes for that name. The words on the front of a document, whether they read principal protection or capital guarantee, are labels naming the shape of a promise. Neither of them names who owes it.
5. Write down what could not be filled in, as questions rather than as failures What the parts cost on the day, what the price asked is, what leaving before the end date would cost, and who stands behind the promise are all things somebody knows and can be asked. A list of four obtainable questions is a far more useful thing to leave a conversation with than a number a reader has talked themselves into trusting.

The sequence produces a description of what is owed and by whom, at a level of detail the front of the document does not carry. The description is not a verdict, and it was never going to be one. Three of the five steps end in a question addressed to somebody else.

Should two instruments be compared on this number?

No, and for a stated reason rather than out of caution. Two participation rates are two ratios struck on different days out of different prices, against promises of different amounts owed by different parties. Ranking them would need an outcome and a probability, and a ratio of two prices contains neither.

The second half of that answer matters more than the first, and it is the half a reader who has just learned how the parts add up will skip. A lower number is not evidence of anything being wrong. A higher number is not evidence of anything being right. Treating a low number as a case against an instrument is a judgement, and treating a high number as a case for one is the same judgement turned around. Neither follows from the arithmetic.

The honest version of the question a reader is really asking is answerable, but not by arithmetic. Four things would have to be known first. Who owes each amount, and what stands behind that promise. The cost of each set of parts on the same day, from the person offering it. The cost of leaving before the end date, under the terms actually written. And the circumstances of the person asking, invisible to any general treatment. Not one of those four can be read off a participation rate.

Who sets what a person must be told about how the amount they receive is worked out?

Everything above is arithmetic and description, and it holds wherever the arithmetic is carried out. The set of requirements attaching to offering one of these to a person in India does not travel, and each of those requirements is named below by its authority rather than by its current value.

Four of them arise here and not one is written out below. The first is what must be told to a person about how the amount they receive is worked out, and in which document. The second is what an instrument of this kind may be called, and what may be claimed in its name. The third is the levels at which contracts are made available, where a leg happens to be an exchange traded contract. And the fourth is how the amount owed at the end is arrived at, and by whom. Each of those is set by the Securities and Exchange Board of India, at sebi.gov.in, and where what the instrument references is a rate or a currency the equivalent arrangements sit with the Reserve Bank of India, at rbi.org.in.

Each of these changes. Each row below is therefore drawn empty, and a written-out value would be wrong rather than merely out of date on the day it changed. Route to the authority named inside the row and read the current version there.

The packaged instrument and the reference asset it is written on are invented.
Educational material. Not advice on any investment, tax, budget or market position.