# Back of the Envelope

Observations on the Theory and Empirics of Mathematical Finance

## [PGP-I FM] Futures contracts

Forward Contracts

It turns out that all forward contracts we have discussed have some common features – in particular:

1. The farmer and the baker have to meet: Had the farmer (baker) not known the baker (farmer) s/he’ll still be faced with the risk of having to sell (buy) the wheat a lower (higher) price. So both counter-parties in the trade must find each other to even have a contract in the first place.
2. The contract is agreed upon before the delivery of the underlying asset (called the “deliverable instrument”): When the farmer and the baker meet, the farmer agrees to make the delivery of the wheat at some future date. It’s not an ‘on-the-spot’ transaction.
3. The time at which the delivery is to be made (called the “expiration date”) is agreed upon in advance: The farmer and the baker mutually agree upon the time (and often the location) at which the farmer will delver the wheat.
4. Forward contracts can be highly customized: The farmer and the baker may, for all they like, agree to the quantity of wheat till the last ‘gram’. They may settle that the baker will pick up the wheat from the farmer’s house at a particular time (say, before/after lunch) on the expiration date. Or the farmer may agree to deliver the wheat at a particular location at the baker’s house or warehouse. It’s a lot like you going to your tailor to get a custom suit stitched – right to the shape of buttons, it’s your choice.
5. The price at which the delivery is to be made (called the “delivery price”): Quite importantly, the farmer and the baker agree on the price at which the farmer will deliver the wheat to the baker on the expiration date (this is kinda really the whole point, isn’t it?).
6. No money changes hands at the time the contract is entered into.
7. Both parties cannot get out of the contract before the expiration date: By their very nature forward contracts are very illiquid.
8. At least legally, both the farmer and the baker are obligated to honour the contract: While this may not be an issue for a farmer and a baker in a village who know each other, this is of crucial importance in the real world.
9. In a primitive village economy it’s a farmer and baker coming together, but in the modern world, typically both the counter-parties are a financial institution, like a bank. (And they do so because most big banks deal with each other very frequently, and ‘know’ each other.)

Now this last couple of features above makes it little problematic for forward contracts. And why, pray this be so?

The problem is that if the expected price of wheat changes during the life of the contract (that is, before the delivery is made but after the contract is entered into), then both the farmer and the baker have an incentive to renege on their obligation. How?

Let’s say the farmer and the baker meet and agree on a price of 100, i.e. farmer agrees that she’ll sell the wheat to the baker at the end of, say, 1 month at 100 a bushel. And then, say, in a fortnight’s time both farmer and the baker find that there are lot of birthdays coming up in the village around the end of the month, and there will be a disproportionate demand for breads, i.e. the demand for wheat flour will be high.

Now, if our smart baker has already entered into the forward contract with the baker, he’ll be all smiles knowing that he has already locked-in the price at which he is going to procure the wheat. But the poor guy perhaps doesn’t know that our lady friend farmer has other plans. When the time comes, she may just, say, well too bad for you that we entered into a contract, but I am afraid I can’t sell you a bushel for just 100 , when the market price is 120. You might say that the farmer is spoiling her ‘trust quotient’, and she can be sure that she’ll have one less friend in the village, but, hey, it’s money we are talking about.

Now I am not saying she will do it, but there will be a mighty big temptation for her to do so. The fact that being a part of a community acts as natural enforcer doesn’t take away the problem. Again, consider the case, if for some reason at the end of a fortnight now both come to know that there’ll be lesser demand for wheat. Now it’s the baker who is faced with a dilemma.

So herein lies the problem. The fact that a forward contract requires both parties to honour their contract makes it difficult to operationalize it in the ‘real world’. While in a small village both parties may, and in all probability, will, know each other, in a place like Mumbai, even if you go and try hunting for a counter-party you may never find him/her in time because you’ll probably be stuck in a jam near Andheri (or these days even Bandra), or if you are going through an intermediary, your broker may just form a cartel with your counter-party and fleece you by offering a forward price that is clearly to your disadvantage (but which may still be the best you could do).

The problem, therefore, is not just about ‘trust’ or ‘honour’ . The fact that an unforeseen circumstance (say, a flood) can make one of the parties renege on their contract makes it practically a big problem. Not to mention the fact that even if the farmer wanted to get out of the contract, because say, she was robbed of all her wheat, and she needs the wheat for her consumption, she can’t do so even if it’s for a ‘right’ reason. The baker still would feel cheated as he is stranded if the farmer fails to deliver the wheat for whatever reason.

For people or entities who knew each other well they could still (and do) enter into forward contracts (for example, big banks who transact with each other on a daily basis they do have a fairly high turnover of outstanding forward contracts – but they ‘know’ each other and their businesses quite well), for a farmer and the baker (as for an exporter and an importer), there are these problems.

So, just like most good things in life, forward contracts also come with riders attached.

Let’s revisit our earlier example and make it concrete. Let’s say when the baker and the farmer meet after a fortnight, say, over a social occasion they agree that the fair expected price at the delivery time would be 120. That is, if they had settled on a forward contract on this day, both would agree that the fair price at the delivery time would be 120.

Our lady friend farmer, of course, isn’t too happy at the turn of the events and would prefer to sell the wheat in the market at the higher expected price in the market rather than deliver the wheat to the baker when the time comes at the agreed price of 100. She is definitely getting tempted to side with the devil on this one and deceive the baker.

But let’s say our lady friend is not bad at heart. She has good intentions, but is still concerned about her financial loss, and given a choice she would rather not lose baker’s friendship. If she expects the price of the wheat to still go higher than the today’s expected future price of 120, she may think, well alright the price in the future could still end up higher than 120, and maybe it’s a good idea to just gracefully ‘walk out of the relationship’ with the baker and just pay-up for his loss of 120 – 100 = 20.

It’s baker’s loss because it is the farmer who ‘wants out of the relationship’, and it is the baker who ends up ‘without a partner’ (ok, wheat) and so must be compensated. Or rather, more boringly, now the baker is stuck with the possibility of having to buy the wheat at the higher expected delivery price of 120 when he was promised wheat at 100.

So, if the farmer could pay up 20 to the baker, baker would say, ok, my friend I understand, but now that you’ve been honest and compensated me for the fact that I’ll have to enter into a new contract with another farmer, we are good. So long, and thanks for all the fish.

In fact, it’s also possible that the farmer and the baker may not have to ‘break-up’ at all. If the farmer is not sure that the wheat price is going to keep rising, and her best estimate is that it’ll end up near 120, then both could still ‘settle their disagreement’ if farmer pays up 20 to the baker. This is workable because if the expected price of the wheat, say, were to go down in a week’s time (but still before delivery), if the baker ‘honors the relationship’, and now pays to the farmer, it’ll be all good in the village. Everybody’s life can go on in the atmosphere of trust and good-will.

Imagine now that there are many such bakers and farmers in the village, and not enough social gatherings. This leaves us with a village with lot of worried bakers and farmers. And if they could not meet like our original couple, and if there were only a single rotten egg, it would lead to a vicious cycle of distrust and antipathy.

Human beings are good at resolving such situations by deciding on a neutral third party. In the village this would mean someone everybody knows and trusts, say someone from the gram panchayat. In that case, they may decide that they would monitor the expected future price every week after the forward contract is signed, and settle their profits and losses with the panchayat. So, if the price of wheat goes down, the bakers of the village would come and pay up their losses, and if the price of wheat goes up, the farmers would do the same. Existence of a neutral, trusted third-party again results in a nice (as against a lousy) equilibrium (I guess that is what, in some sense, a judicial system also ensures).

Of course, this arrangement in the village would work only if our panchayat member is a trusted fellow, and if for some reason either a baker or a farmer is unable to keep his/her end of the bargain, that is pay up the ‘margin’ of loss that is due, he should be able to compensate the other party still.

However, the first and an even bigger problem is that of existence. The question of a forward contract ‘coming into the world’ doesn’t even arise if the farmer (exporter) and the baker (importer) can’t meet. For them to even enter into a contract means they must find each other first. So there is a search (transaction) cost involved.

Fast forward this primitive village economy to the modern world, where as it is there is a deficit of trust for reasons of scarcity of time, resources, opportunities and what not, an entity like panchayat doesn’t suffice. But as it is often the case, left to their own devices, one can trust the financial market participants to come up with a workaround.

For a financial market participant the solution is very clear – create markets. Create a platform to bring the farmers and bakers of the world together. Having a common marketplace reduces transaction costs by bringing the participants together. And when finance people talk about markets, more often than not they have in mind a clearing exchange, and in the case of forward contracts they are called futures exchanges.

Futures Contracts

The way the exchange manages the ‘trust’ of the counter-parties is exactly the same way as the panchayat in the village. They settle the profits and losses ‘every so often’. But because in a city like Mumbai, one can’t afford to run around and find the person who refuses to settle his end of the bargain, and there are more crooks in Mumbai, the mechanism of “margining” in exchanges are run by strict rules.

But the fact that having a market, or an exchange, allows buyers and sellers to come together doesn’t solve the problem of default. If anything, perhaps, it exacerbates it. If even a few people started defaulting on the exchange, for lack of trust, the whole system will very quickly break down. So what is the solution?

At this point one of my colleagues draws a nice analogy of this problem with that of a pawnbroker’s in a big city. A pawnbroker is someone who gives loan to people who can’t get it otherwise, and his (a pawnbroker typically is a “he”) obvious worry is that his money won’t come back. And for a right reason – the kind of people who go to a pawnshop to borrow money are exactly the ones who have a bad (or no) credit history. And what is his solution? His solution is that he asks for a collateral of value much more than the value of the loan that he is giving out. (Contrast this with the problem of a moneylender’s in a village – would he ask for the same collateral as a pawnbroker in Mumbai?)

The solutions that markets came up for this problem is that they agreed to make the exchange as the counter-party for all forward contracts (either entered into by a farmer or a baker). And, like a pawnbroker, exchanges made a system of asking for collateral from the counter-parties in the forward contract, and this new kind of forward contract in which exchange was the intermediary counter-party for the farmer and the baker, they called the futures contracts (not “future”, but futures, with an “s”) – very similar to a forward, but different enough to warrant a new name.

The exchange-traded futures contracts offered almost everything what forward contracts did, but in a stroke of genius, exchanges came up with a way that prevented default exactly where it was most likely to occur – among traders who did not know each other. And what is truly a marvel of modern finance is that while banks have had a history of defaulting (and very much so in the recent past), very rarely have exchanges collapsed the way banks have.

As by now you should be able to guess, for futures contracts to be traded on an exchange they must be standardized. So, unlike forward contracts which could be customized right down to a T, futures contracts, by their nature of being traded on an exchange, must be standardized. In fact, it’s now a good time to list down the features of futures contracts:

1. Exchange Traded: Futures are exchange traded. By construction all futures contracts are traded on exchange, and as a consequence, just like stocks, everybody can see what is being traded at what price and so on. Unlike, forward contracts, then, they are not opaque and highly transparent. An implication is that futures contracts are highly liquid. At any point the buyers and sellers of the futures contract can buy from or sell to the exchange. This means that if a farmer ‘sold’ wheat on the futures exchange, if she now desires, she could unwind her contract by entering into an offsetting contract with the exchange at the going price. We’ll elaborate on this point later.
2. Standardization: Futures contracts are highly standardized. For a commodity/investment asset to be traded on the exchange, it has to standardized. So if wheat/rice is harvested in particular months, the delivery dates of the futures contracts will be closer to those dates – that is, for a specific asset futures contracts have a fixed/limited number of delivery dates. Not only delivery dates, often the location and the ‘grade’ of a good (different grades of wheat are grown and priced differently) is also pre-specified exactly by the exchange.
3. Anonymity: The farmer and the baker need never know about each other. For both the farmer and the baker, the counter-party is the exchange. That is, both the farmer and the baker could protect themselves against the risks they face by entering into forward contracts with the exchange that are mirror images of each other. So while exchange is the counter-party for both the farmer and the baker, on a net basis the exchange has exactly offset its exposure. That is, unlike in a forward contract the farmer and the baker are not dealing with each other at all. The consequence is that all the default risk is assumed by the exchange.

And how does an exchange protect itself from default by the farmer or the baker? This goes back to the ‘stroke of genius’ we referred to on part of the exchange. The key idea here is “margining”.

Futures Exchanges and “Margining“: The idea of Mark-to-Market

So the ‘stroke of genius’ by futures exchanges was to modify the system adopted by bakers and farmers in the village, and implement it in the context of the modern financial markets as this is what make futures contract really work in the market place. The key idea is Mark-to-Market.

1. Mark-to-Market: Exchanges made the system of settling profits and losses on part of the bakers and the farmers more systematic  ‘every so often’ and called it mark to market:

• Settlement every business day: First of all exchanges made ‘every so often’ specific and this is one of the crucial feature of futures contracts traded on the exchange. The profits and losses of the bakers and farmers are settled every business day. This makes sense, because it may be easy for a panchayat member to locate a particular baker or a farmer who fails to pay up at the end of a week or so, in modern cities this is not only prohibitive but also may not be possible.
• Anonymous trading: This above feature of marking to market, in fact, is what allows the exchange to become a counter-party to any entity. It could be anyone who needs the wheat, need not just be the baker. So, the counter-party may never know each other at all. And in today’s modern world, they could even be situated in different geographical regions. As long as they can transfer money to the exchange’s bank account when the margin payment is due that’s all what is needed. This is made possible because exchanges require settling the profits and losses in the contracts every day.
• Settlement price: Just like in the case of the baker and farmer who settled their profits and losses on the basis of the future expected price at the time of delivery (120), the futures contracts are also settled on the basis of the futures price prevailing on the settlement day. So once the futures contract has been entered into between the exchange and the baker, and the next day the futures price changes (because perhaps the spot price and the cost of carry have changed), the profits and losses are settled on the basis of the futures price prevailing on that day. Think of it this way. If the contract was signed yesterday, and futures price for the same delivery date today has gone up to 120, the farmer would have been better off had she got into agreement with the baker today rather than yesterday. So her loss based on changed futures price is 120 – 100 = 20. The amount/loss settled is referred to as the “margin money” or “variable margin” or “mark-to-market margin”.
• Settling profit and losses: Since futures exchange is the counter-party for both the baker and the farmer, when the price of the wheat goes up, the exchange gains vis-a-vis the farmer (collects 120 – 100 = 20 from the farmer), but loses viz-a-viz the baker (pays up  120 – 100 = 20 to the baker received from the farmer). But because both parties are obligated to settle their losses every day, exchange is in a position to transfer the proceeds it receives from the losing party.
• Obligation of the exchange: Profit and loss settlement by the exchange subsumes that the exchange is never going to default on its obligation. Even if the baker defaults, the exchange must still honor the farmer’s contract, and still ‘buy’ the wheat at the agreed futures price on the delivery date. (The fact that exchange never has to do so is another matter. See point 3 below.)

2. Initial Margin: Although daily mark to market helps reduce the default risk for the exchange, it does not completely eliminate it. So, just like a pawnbroker, exchanges also require both counter-parties to keep a collateral with the exchange. As you would expect, since the exposure of the exchange is only to the extent of the margin money every day, the collateral required by the exchange is just the maximum amount it expects the wheat price to fluctuate every day. This margin is called the initial margin, and both the baker (if the expected future price of wheat falls) and the farmer (if the expected future price of wheat rises) have to pay this to the exchange before they can enter into any futures contract with the exchange.

3. Early Close-out

• Default on the margin payment: If one of the counter-party fails to keep its side of the bargain, and does not pay up the margin money at the end of the business day, the exchange confiscates its initial margin, and closes out that side of the contract by entering into an offsetting contract with another counter-party. Since the futures are on standardized contracts and exchanges offer liquidity, it’s never a problem for the exchange to find another counter-party at the going futures price (remember that if there are not enough takers at a particular, the supply-demand dynamics ensures that the futures price would adjust). If the baker fails to pay up when the price of wheat has fallen down to say, 80, the exchange will first of all say good-bye to the baker’s initial margin and use some of that money to transfer the profit due to the farmer. Now because the exchange is stuck with the obligation of having to buy the wheat at 100, when the prevailing futures price is 80, the exchange will enter into a futures contract with another counter-party to sell the wheat at the prevailing futures price, i.e. it will find another baker. By doing so it has taken care of its obligation to buy the wheat from the farmer. From next business day on-wards, the original situation would return, as both the original farmer and the (new) baker would resume mark to market. Note that in the entire process, the farmer need never know about the default on the margin by the baker.
• Early exit: An advantage of this possibility of early closing out is that any counter-party can get out of the futures contract by paying up the mark to market margin. Let’s continue with the same example. So, if the next day after the contract has been entered into, the price of wheat has gone down to 80, the baker has reasons to worry that he may be in a soup if the price of wheat keeps falling (as he’ll have to pay up margin money every day). Having an exchange as the counter-party, and the fact that the futures contracts are liquid, gives the baker a choice to walk out of his contract without any impact to anyone as long as he keeps paying his margin money. So, in this case, he may just pay up his loss of 20 loss and enter into an offsetting contract with the exchange to sell wheat at 80 with the same time to delivery. So exchange, in its account books now, would just have an offsetting entry and baker can go live his life in peace without having to bother about the margin payments to the exchange. And if the price is going up, the farmer is free to do the same thing. If the price goes up to 120, she may just pay up her loss of 20 and walk out of the whole thing by just entering into an offsetting contract (to buy) with the exchange.
• Choice not to take delivery: Extend the idea of early exit to the delivery date, and its clear why both parties may choose not to take/give delivery of the wheat at all. On the day of the delivery, both parties may just choose to settle the last margin payment and at the same time enter into an offsetting futures contract with the exchange for delivery on the same date. So, ignoring time value of money, the net cash for each would be exactly the same as 100, and by doing so both parties have exactly hedged their exposure to the rise (for baker) or fall (for farmer) of wheat price

4. Hedging and Basis Risk: A natural question to ask at this point is why would all the bakers and farmers wish to come to exchange when they may need to hedge exposure to very specific kind of wheat when the exchange provides liquidity and contracts for only  very specific kind of wheat. The answer to this lies in correlation. If we are talking about the same commodity (wheat), it’s very likely that the price movements in different qualities and grades of wheat will be highly correlated. So even if prices of different grades of wheat may not move in perfect lock-step with each other, they would still be safely assumed to change very closely together.

So, in that case, say, if the price of the wheat traded on the exchange has gone down by $20\%$, it’s likely that the price of the wheat that the baker wants to hedge himself against would be trading say, at $23\%$ or $18\%$. In that case, if the baker has a futures contract with the exchange, while he is gaining as a counter-party with the exchange, in the markets he is losing. And wasn’t that his really worry in the first place. What if the price of wheat goes up in the future? By entering into the futures contract he is better of by $20$, but in the real market place he is worse off by either $23\%$ or $18\%$ depending on the ‘state of the world’. So on the ‘net basis’ he is either gaining by $20 - 18 = 2\%$ or losing by $23 - 20 = 3\%$.

If the two grades of wheat were moving in a perfect lock-step with each other his net gain/loss would have been zero. But because the futures exchange does not provide exactly the same kind of wheat that he wants to hedge, there is a residual exposure still left, but more-or-less he is still protected, because exposure in futures contract and the spot are highly negatively correlated, the diversification effect implies that the variance of his losses would be a lot lower compared to if he hadn’t hedged. The residual exposure because the asset to be hedged is not exactly the same as the instrument used for hedging is known as the basis risk.

We close this post with a simple example of marking to market.

Example

 Time t = 0 (F = 100) t = 1 (F = 80) t = 2 (F = 90) t = 3 (F = 110) t = 3 (Delivery at F = 110) Baker’s Inflow/Outflow – 80 – 100 = -20 90 – 80 = 10 110 – 90 = 20 -110 (Net = 0-20+10+20-110 = -100) Farmer’s Inflow/Outflow – 100 – 80 = 20 80 – 90 = -10 90 – 110 = -20 110 (Net=0+20-10-20+110 = 100) Exchange’s Net – -20 + 20 = 0 10 – 10 = 0 20 – 20 = 0 – 100 + 110 = 0

Needless to say, futures contract requires that the users have enough liquidity to meet daily margin payments. Liquidity constraints applicable to most retail investors means that it’s mostly companies or institutions that typically hedge using futures contracts.

Written by Vineet

October 25, 2016 at 12:39 am

Posted in Teaching: FM

## [PGP-I FM] Forward contracts

Forward on Stocks

Just like we could talk about interest rates and FX in the future, we are tempted to believe that perhaps we can do something of that sort with common stocks too. Well, indeed we can.

Locking-in a stock price in the future is not too different from locking-in the price of foreign exchange in the future. If anything, this is even simpler. By now, you should now how we would proceed in this case.

Since we want a stock in the future, what we would do is borrow money and buy the stock today. And if we don’t sell the stock we would have the stock for as long as we want. So by buying the stock today (say, at price $S_0$) and ‘carrying’ it forward, assuming no dividends, we have ensured that we’ll indeed have the stock at the end of the period as we wanted.

So what is our payable at the end of 1 period? Well, since we borrowed money to buy the stock, our payable is just the money that we owe to the bank at the end of the period. If the one-period rate is $r$, the our ‘forward price’ for holding the stock at the end of the period is just the amount that we return to the bank, that is:

$\boxed{F_S = \displaystyle S_0(1 + r)}$

And again, because people can do so, financial institutions offer forward contracts on stocks and indices that allows buyers/sellers to lock-in their buy/sell price in the future.

If the stock paid dividends too, then there is an extra income from the stock, and the forward price will have to be a little lesser than what it would have been otherwise, as ‘carrying’ a stock now gives dividends, and in that case we’ll have:

$F_S = \displaystyle (S_0 - D)(1 + r)$

where $D$ represents the present value of all dividends through the period.

Or if we use continuous compounding we would write it as:

$F_S = \displaystyle S_0e^{r - d}$

In general, for a contract expiring at time $t$, using continuous compounding, the forward price of stock will be:

$\boxed{F_S = \displaystyle S_0e^{(r - d)t}}$

(By the way, getting into a forward contract to buy a stock refers to ‘going long’, and getting into one to sell a stock refers to ‘going short’.)

Forward on Commodities

While financial institutions do provide a very useful service by allowing people to lock-in the prices of traded securities by offering forward contracts, as we have seen, assuming away transaction costs people could mighty well do that on their own. So, loosely speaking, at least for big players, a forward contract is redundant security as they can ‘replicate’ the payoff from a forward contract by buying/selling the stock today and borrowing/lending (just like we found the ‘fair’ price of FX and stock above).

The social utility of having forward contracts in the world really becomes apparent in the case of commodities.

Let’s take the example of a farmer and a baker in a small village. Our farmer sells wheat and the baker uses that wheat to bake and sell bread. Farmer worries that when the time comes to sell wheat the price would be low, and the baker worries the price will be high. If wheat were like a common stock, and if the baker wanted to buy wheat in the future, the baker would, just like in the example of forward contract on stock, buy stock today and ‘carry’ it forward and he would have locked-in his price.

But wheat is not a like a share, is it? First the wheat has to be available today for him to buy in the spot market and ‘carry’ it forward. If the farmer hasn’t harvested the wheat yet he can’t even do that. Even if the wheat has been harvested he may not have enough physical space or the wherewithal to store the wheat to ‘carry’ it forward.

Similarly, even if the farmer could ‘short’ the wheat today, if there is no wheat there is no way she can do that. The only way the baker and the farmer could achieve their objective of locking-in the price is by entering into a forward contract and agreeing on a price today.

And they can (and will) do so because when the baker and the farmer enter into the contract, their meeting together is a zero-sum game. Say, both agree that 100 is about the ‘fair’ price for wheat at the end of one period. Farmer worries the price of wheat at the end of 1 period is going to fall below 100, and the baker worries that at the end of the period the price is going to be more that 100.

While by entering into a forward contract they have reduce the uncertainty for them, if the actual price of wheat were to be more than 100 (say, 120), the farmer will probably curse herself for agreeing to sell the wheat at 100 when the actual price turned out to be 120. But the loss of farmer (by 20) is the gain of the baker. Baker will be thanking his stars that he entered into this contract, because now he is able to buy the wheat at 100 when the market price is 120. The farmer’s loss is exactly the baker’s gain in this case.

How do we find the forward price in this case – the ‘fair’ price they should agree upon when the contract is entered into?

It turns out we can still use the insights we developed when we were talking about common stock. While it maybe difficult for the baker to store the wheat at his place, he may yet hire the services of a professional warehouse and pay the storage cost (assuming the wheat is available in the market), and the baker will proceed exactly the way we did in the case of the common stock.

The baker would borrow money today, take the delivery of wheat today, store it in the professional warehouse and take it out at the end of the period at which time he’ll return the money to the bank. That is the baker will literally ‘carry forward’ the wheat. And the forward price for wheat would be just what we found in the case of stock, but now we’ll have an extra cost of physically ”carrying’ the wheat into the future, that is:

$F_C = \displaystyle S_0(1 + r) + \mbox{Storage Cost}$

But is it all? Can not baker use the wheat in the mean-time, if he runs out of his inventory’s ‘safety stock’ (the minimum amount of wheat he needs at all times)?

Yes, by holding wheat physically the baker is getting the ‘convenience’ of using the wheat if he needs to in emergency. Obviously, assuming no dividends, there is no direct utility to holding a share certificate. For one, we can’t ‘consume’ our share certificate the way we can consume wheat. So to that extent the ‘carrying’ cost of wheat is lower than it would have been otherwise. And we revise the forward price for wheat as:

$\boxed{F_C = \displaystyle S_0(1 + r) + \mbox{Storage Cost - Convenience Yield}}$

If we assume that the storage cost and convenience yield can be measured as a percentage of the amount/value of wheat to be stored and that loss of wheat due to any wastage is counted as part of the storage cost, using continuous compounding we could write this more succintly as:

$\boxed{F_C = \displaystyle S_0 e^{r + x - y}}$

where $x$ is the per-unit storage cost and $y$ is the per-unit convenience yield – all continuously compounded. As you may guess, typically it’s not at all obvious how to quantify convenience yield.

In general, for a contract expiring at time $t$, using continuous compounding, the forward price of a commodity will be:

$\boxed{F_C = \displaystyle S_0 e^{(r + x - y)t}}$

Time to GeneralizeThe Cost of Carry Model

If you notice carefully, what has been common to valuation of forward contracts across asset classes is that we are ‘replicating’ the payoff of a forward contract by ‘financially carrying’ the FX or the stock to the future time period, or in the case of wheat, physically carrying the wheat to the future time period.

A useful comparison is obtained by writing the forward price in each case by taking logs. Taking log of the three forward prices (and here continuous compounding is really a boon), we get:

\begin{aligned} ln(F_{FX}) &= ln(S_0) + r_{IN} - r_{f} \\ ln(F_{S}) &= ln(S_0) + r - d \\ ln(F_{C}) &= ln(S_0) + r + x - y \end{aligned}

That is, in each case the forward price is a sum of the spot price and carrying cost net of any yield.

We could interpret the borrowing cost as the cost of ‘financially carrying’ the FX or the stock forward. In case of wheat it’s the most obvious. We are literally borrowing money (financial ‘carry cost’) and physically carrying the wheat to the next time period. So in all cases, one could write the forward price as the sum of the spot price and a ‘Cost of Carry’, net of any income/yield that may be available while carrying the underlying asset.

In case of FX, that yield is the rate of return on USD when’carrying’ USD into the future. In the case of stock, it is the dividends, and in case of commodities it is an unobservable convenience yield.

To generalize, then, we could write as:

$\boxed{\mbox{Forward Price = Spot Price + Cost of Carry}}$

where the ‘Cost of Carry’ is to be interpreted as the net of any yield from carrying the underlying asset.

This way of finding the forward price is referred to as the ‘Cost of Carry’ model. As should be clear, it works best for financial/investment goods, and not as well for consumption goods. For consumption goods convenience yield is a fuzzy thing, highly time varying (you need oil in the winters more than in the summers) and hard to pin-down.

Written by Vineet

October 25, 2016 at 12:35 am

Posted in Teaching: FM

## [PGP-I FM] FX: Covered Interest Parity

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We’ve already been exposed to forward rates when we talked about the term structure of interest rates. Our work-horse model for introducing forward exchange rates isn’t too different from the toy model we used when introducing forward rates in the interest rate world.

Forward on Foreign Exchange: Covered Interest Rate Parity

Let’s say there is an exporter who has receivables, say 1 million USD at the end of 1 year. If she does not have natural offsetting payable around the same time, she is exposed to the exchange rate fluctuations. And unless her business is dealing with exchange rates, she is faced with a business headache. It turns out that there is a way that she can get rid of her headache and lock-in a rate today itself.

And the way to do so is not too unlike the way we went about introducing forward rates in the interest rate world. There we said that a borrower who needed money at the end of 1 year for one year could do so by borrowing for two years and lending for one year. Here we do something similar.

If she had a natural import expense, she would not be bothered about exchange rates. USD comes in from export and goes out to import. But she needn’t be too bothered. If she does not have a natural payable, it does not mean she cannot artificially create one. Good that it is not too difficult either.

One way to create such a payable is to, well, simply borrow the ‘right’ amount of USD today itself for 1 year. This ensures that she has a payable to the bank at the same time she gets money from the business. So assuming there are no capital controls, if she can borrow PV of 1 million USD from a US bank, at the end of 1 year the money from the business could go straight to the bank.

In the process she is left with PV of 1 million USD today, which can be immediately converted into INR. Since our focus is end of 1 year, we know how much this is worth then. This is exactly the the future value of money of the INR equivalent today.

So graphically:

(Click to zoom; Note the color coding. The green arrow represents the final receivable in INR after paying back the 1 million USD  due to the US bank from the business.)

That is, by borrowing USD and investing in INR, the exporter has locked-in the receivables in the domestic currency, and the effective ‘forward exchange rate’ for the exporter is:

$\boxed{F_{FX} = \displaystyle \frac{S_0(1 + r_{IN})}{1 + r_{f}}}$

or using continuous compounding as:

$F_{FX} = \displaystyle S_0e^{r_{IN} - r_f}$

where, $r_f$ denotes the foreign currency interest rate. If the financial institution offered a ‘forward price’ different from this price there’ll be possible arbitrage opportunity.

In general, for a contract expiring at time $t$, using continuous compounding, the forward price of FX will be:

$\boxed{F_{FX} = \displaystyle S_0e^{(r_{IN} - r_f)t}}$

For completeness sake, this is how an importer will lock in her forward price (as should be clear all signs/direction of arrows have reversed, and borrowings have become lending and vice-versa):

(Click to zoom; Note the color coding. The green arrow represents the final payable in INR after getting ‘back’ the 1 million USD  from the US bank due to the business.)

Written by Vineet

October 25, 2016 at 12:32 am

Posted in Teaching: FM

## [PGP-I FM] Term structure of interest rates

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In a previous post we introduced the notion of the term structure of interest rates. Here we try and understand the shape of the term structure.

There is considerable evidence that historically the shape of the term structure has exhibited the following patterns:

• More often than not the term structure has been upward sloping, i.e: $r_1 \le r_2 \le r_3 \le ... \le r_n$
• Movement in short-term and long-term rates are positively correlated (that is, they tend to move together)
• The term structure tends to be downward sloping in recessionary times (i.e. in those circumstances typically short-term rates are higher than the long-term rates)
• Corporate bond rates are higher than the sovereign rates – the difference between corporate bonds and the sovereign rates for a given maturity is called the spread
• Spread tends to widen during crises

We focus on the first three facts in this post. As always, we use a two-period set-up to try and understand the problem.

Let the one-period rate be $r_1$ and the two-period rate be $r_2$. From our discussion on forward rates we already know that embedded in these two rates is an implied one year forward rate $f_{12}$ from year one to year two such that:

$(1 + r_2)^2 = (1 + r_1)(1 + f_{12})$

Ignoring higher order terms, the above simplifies as:

\begin{aligned} 1 + 2r_2 + r^2_2 &= 1 + r_1 + f_{12} + r_1 f_{12} \\ 1 + 2r_2 &\approx 1 + r_1 + f_{12} \\ \mbox{ or } r_2 &\approx \frac{r_1 + f_{12}}{2} \end{aligned}

First thing to notice here is to see that the two-period rate $r_2$ is an average of the today’s one-period rate, $r_1$ and the one-period  ‘forward rate’, $f_{12}$. Secondly, if the term-structure has to be upward sloping, then in our two-period world the only way $r_2$ can be greater than $r_1$ is if $f_{12} > r_1$. Similarly, the term structure can be downward sloping iff $f_{12} < r_1$.

So if we can understand the relationship between $f_{12}$ and $r_1$, we would have explained the shape of the term structure.

Expectations Hypothesis

The expectations hypothesis of the term structure says that the term structure is upward or downward sloping only because people expect it to be so. That is, the reason the term structure is upward sloping, i.e. $f_{12} > r_1$ is because $f_{12}$ is just the expected value of the future one-year rate, $\mathbb{E}[r_{12}]$ and people expect that to be higher than today’s one-year rate, i.e. $f_{12} =\mathbb{E}[r_{12}] > r_1$. That is, the theory says that:

$(1 + r_2)^2 = (1 + r_1)(1 + \mathbb{E}[r_{12}])$

So one gets an upward sloping term structure if $\mathbb{E}[r_{12}] > r_1$ and vice-versa, i.e.

$\mathbb{E}[r_{12}] > r_1 \Rightarrow r_2 > r_1$

and

$\mathbb{E}[r_{12}] < r_1 \Rightarrow r_2 < r_1$

According to the expectations hypothesis, then, the forward rate $f_{12}$ captures the expected future rate $\mathbb{E}[r_{12}]$. The economic story, as we talked about in the class, is just that as people expect the rates to go up, they flock to borrow long-term right now itself and in the process demand for 2 years loans go up, causing the two year rate $r_2$ to go up.

Horizon Matters

Consider now an Investor A who wants to invest money for two years. She has two choices:

2. Buy a one-year bond and ‘roll-over’ at the end of first year till year two

If A selects option 1, then for every unit of money invested today, she will get $(1 + r_2)^2$ at the end of two years. On the other hand, if she selects option 2, then for every unit of money invested today, she can expect to get $(1 + r_1)(1 +\mathbb{E}[r_{12}])$ at the end of two years.

Now consider another Investor B who wants to invest money for one year. She also has two choices:

1. Buy a two-year bond and sell that bond at the end of one year

If B selects option 2, then for every unit of money invested today, she will get $(1 + r_1)$ at the end of one year. On the other hand, if she selects option 1, then for every unit of money invested today, at the end of year one she expects to get whatever the price of a two year bond would be at the end of year one. The expected price of a two year bond at the end of one year is just the PV of $(1 + r_2)^2$ discounted to end of year one at the expected one year rate prevailing at that time, i.e. she can expect to get $\displaystyle \frac{(1 + r_2)^2}{(1 +\mathbb{E}[r_{12}])}$ at the end of year one if she selects option 2.

If expectations hypothesis is true, that is $f_{12} = \mathbb{E}[r_{12}]$, then it essentially means that both the investors should be indifferent to the available options (why?). The upshot is that if expectations hypothesis is true, bonds of different maturities are perfect substitutes for each other – that is, time horizon of investment is irrelevant as far as investors are concerned.

(Another implication of $f_{12} =\mathbb{E}[r_{12}]$ is that if the forward rate is just the expected value of the future one-year rate, then on an average $f_{12}$ is as likely to be less than $r_1$ as it is likely to be more than $r_1$. So if it is only the expectations driving the shape of the term structure, then it is as likely to be upward sloping as it is to be downward sloping . Then, on an average over long periods of time we should expect to encounter an upward sloping term structure as often as a downward sloping term structure. But the evidence has been that the term structure has been upward sloping for better part of the last 100 years. Clearly, the expectations hypothesis cannot be the whole story.)

The question to ask, then, is are investors really indifferent to time horizons? Well, clearly not.

For investor A, option 1 is completely risk free, as the two-period rate today is known to be $r_2$. Option 2, however, for investor A comes with the risk of not knowing what the one-period rate would be at the end of one year, expectations notwithstanding. That is, there is no guarantee that the future realized $r_{12}$ will turn out to be equal to $\mathbb{E}[r_{12}]$. Similarly for investor B, option 2 seems to be less risky, as the one-period rate today is known to be $r_1$. Option 1, however, for investor B, comes with the risk of not knowing what the one-period rate would be at the end of one year.

I believe the point has been made. If the horizon matches the maturity of the investment, there is no risk in holding a bond, but if the horizon is different from the maturity of the investment, there is an interest rate risk even in holding a (default) risk-free bond. That is, a risk-free bond is free of risk only if held to maturity, otherwise if an investor plans to sell it off before maturity he/she runs the risk of taking a loss (if interest rates are high at the time of the sale).

But then, in aggregate, which is riskier – investing for one year or investing for two years?

Answering the question whether a one year bond is more risky than a two year bond is an empirical matter. Which kind of investors dominate the real world – the ones with an investment horizon of two years, or the ones with an investment horizon of one year? Like many other original insights on macroeconomics and finance, the answer to this question was also first given by John Maynard Keynes.

Liquidity Preference

Keynes argued that most people like to invest their money only for short horizons and are averse to locking-in their money for long horizons. That is, given a choice, most people prefer having a liquid cash position. And their demand for liquidity, as you will learn in your first year macro, comes from needing money for:

1. Day-to-day transactions (the transactions demand)
2. Emergency reasons (the precautionary demand)
3. Taking advantage of possible arbitrage opportunities in the future (the speculative demand)

That is, a preference for liquidity means that most people would rather lock-in their money for one year than for two years. Investing for two years means taking the risk of selling the investment off at a lower price if the money is required either for reasons of transactions or emergency or speculation. So, for long-horizon investments risk-aversion kicks in.

For risk-averse people, as we have already seen, utility of a sure payoff is more than the expected utility from the payoff, i.e. $U(\mathbb{E}[X]) > \mathbb{E}[U(X)].$ So risk-aversion implies that people would want a (liquidity) premium for taking the risk of having to sell off the two year investment at the end of one year at a loss, i.e. they would invest for two years only if there was an extra incentive for locking-in money for two years, i.e. they would want $r_2$ such that:

\begin{aligned} (1 + r_2)^2 &= (1 + r_1)(1 +\mathbb{E}[r_{12}]) + \mbox{ Liquidity Premium } (> 0) \\ \mbox{ i.e. } (1 + r_2)^2 &> (1 + r_1)(1 +\mathbb{E}[r_{12}]) \\ \Rightarrow f_{12} &>\mathbb{E}[r_{12}] \end{aligned}

That is, even if the expected one-year rate at the end of one year is the same as today’s rate, i.e. $\mathbb{E}[r_{12}] = r_1$, the sign of Liquidity Premium (> 0) implies that even then $f_{12} >\mathbb{E}[r_{12}] = r_1$, i.e. even if the rates are not expected to change at all the term-structure would be upward sloping.

In fact, even if the expected one-year rate at the end of one year is less than today’s rate, i.e. $\mathbb{E}[r_{12}] < r_1$, even then it is possible that the term structure would still be flat or even upward sloping depending on the extent of the Liquidity Premium.

A stronger consequence is that for a term structure to be downward sloping, the expected one-year rate at the end of one year has to be a lot less than today’s rate to counter the effect of Liquidity Premium. That is, the term structure being downward sloping is a very strong signal of $\mathbb{E}[r_{12}] < r_1$, i.e. recessionary expectations.

Why do we call $\mathbb{E}[r_{12}] < r_1$ recessionary expectations? Expected future interest rates lot lower than today’s imply that there will be a reduced demand for loanable funds in the future, which, in turn, points towards a slowing economy.

But is this all?

Can we say that the these two theories together explain the term structure completely? Well, of course not. But expectations and liquidity premium are two of the bigger forces in play driving the shape of the term structure.

For the most part, to the extent one can’t explain all the movements in the term structure to either expectations hypothesis or the liquidity premium, one can attribute the changes to the simple fact of demand and supply. Sometimes companies and institutions have very specific needs linked to short or long horizons, for either the reasons of working capital and/or investment requirements, and those concerns can sometimes dominate short-term movements in the term-structure.

Also, very often central banks deliberately play around with the amount of funds available in the money market to suit their policy needs, and thereby influencing the shape of the term structure. (Come to think of it, when a central bank makes a policy intervention, one of the things it is really doing is playing around with expectations.) So, yes, of course, it’s not all, but the liquidity preference theory does help put the data in perspective.

Written by Vineet

October 7, 2016 at 8:51 pm

## [PGP-I FM] Forward rates

Consider the following situation. A businesswoman wants to borrow money for 1 year, but she wants to start the loan at the end of 1 year, i.e. borrowing is planned in the future and not today. While she knows today’s term structure, so she is aware of today’s spots rates $r_1$, $r_2$ etc, she is exposed to the risk of not knowing what the rates would be a year from now.

Well, it turns out that using the bond/bank market she can still manage something which will lock-in her effective rate exactly, that is there would be no risk attached to it, and she can get on with her business. Let’s see how she can work this out.

If she can use the banks today, she will borrow money for 2 years and lend (deposit) all of that money to the bank for 1 year. And what has she achieved in the process? What she has got is that she has managed to get  cash inflow at the end of 1 year (money ‘coming back’ from one year lending) and cash outflow at the end of two years (original money to be returned to the bank at the end of two years). With the loan money coming from the bank and that money being put in the bank for shorter maturity means there is no money left over ‘today’ – which is exactly what she wanted.

So, how do the cash flows look like at the end of years 1 and 2. Given a million rupee loan, for example, her due to the bank in 2 years time is exactly the future value of that, i.e.

$\displaystyle FV_2 = (1 + r_2) ^ 2$

And inflow at the end of 1 year from the million rupee invested in the bank would be

$\displaystyle FV_1 = (1 + r_1)$

That is, she is in a way artificially created a one year loan for herself with the money $FV_1$ coming in at the end of the first year, and the amount $FV_2$ to be paid out at the end of two years – a one year loan starting at the end of one year. And what is the effective interest rate? Well, that is easily found as:

\begin{aligned} \mbox{Effective rate} &= \frac{FV_2}{FV_1} - 1 \\&= \frac{(1 + r_2)^2}{1 + r_1} - 1 \end{aligned}

It turns out that there is a specific name for this effective rate, and it is called the forward rate for the period 1 to 2, and we write it as $f_{12}$. So with this, we have the following result:

$\boxed{1 + f_{12} =\frac{(1 + r_2)^2}{1 + r_1}}$

or alternatively:

$(1 + r_2)^2 = (1 + r_1)(1 + f_{12})$

which can be pictorially represented is:

It turns out that forward rate as described above is only one example of a class of instruments called Forwards and Futures, and we’ll talk about such things at a fair bit of length separately. For now, we come back to the term structure discussion and use the idea of forward rates to understand the shape of the term structure.

Written by Vineet

October 7, 2016 at 8:46 pm

## [PGP-I FM] Duration

Consider the following bond pricing equation using yield to maturity:

$\displaystyle P = \frac{C}{1 + y} + \frac{C}{(1 + y)^2} + \frac{C}{(1 + y)^3} + \frac{C}{(1 + y)^4} + ... + \frac{C + 100}{(1 + y)^n}$

where $C$ is the annual coupon rate, $y$ is the yield to maturity, and $n$ is the maturity of the bond.

This formula follows straight from the discounted cash flow model we encountered earlier. The only difference is that instead of the dividends, we have known coupons from the Government of India. That is, we are in a world of (default) risk-free interest rates.

The above can be written in a short-hand as:

$\displaystyle P =\sum_{t = 1}^{n} \frac{C_t}{(1 + y)^t}$

where $C_t = C$ for $t = 1, 2, ..., n - 1$, and$C_t = C + 100$ for $t = n$.

I know the summation sign causes that queasy feeling in the stomach for at least a few of us, but, again, it’s a very useful short-hand. I can only say that more you use it, the better/more comfortable you’ll get at it. But, still, in deference to those of us who are not so comfortable at using the summation sign, let’s take the example of a simple two period bond.

Example: Consider two separate zero coupon bonds maturing at time $t = 1$ and $t = 2$, with cash flows $C_1$ and $C_2$ and prices $P_1$ and $P_2$ respectively. Assuming that the yield to maturity $y$ is the same for both, we may write :

\displaystyle \begin{aligned} P_1 &= \frac{C_1}{1 + y}, \hspace{1pc} P_2 = \frac{C_2}{(1 + y)^2} \end{aligned}

Let’s say our RBI governor Dr. Raghuram Rajan, raises the interest rates, as for example, a couple of days back by $25$ basis points, i.e. by $0.25\%$. The question we ask then is, by how much the PV of each cash flow changes. The natural way to address this question is to use the technology of simple calculus and we can write the change in $P_1$ w.r.t change in $y$ as:

\begin{aligned} \displaystyle \frac{dP_1}{dy} &= -\frac{C_1}{(1 + y)^2} = -P_1\frac{1}{(1 + y)} \\ \Rightarrow -\frac{dP_1/P_1}{dy} &= \frac{1}{1 + y} \end{aligned}

i.e. the percentage change in bond-price to small changes in $y$ of of a single cash-flow at time $1$ is $1/(1 + y)$. Since, this is also its time to maturity, the word duration is apt. That is,  the percentage change in a $1$ year zero coupon bond price to small changes in $y$ given by $1/ (1 + y) = \mbox{Duration} / (1 + y)$.

Similarly, one can write the change in the PV of the second zero coupon bond for small changes in $y$ as:

\begin{aligned} \displaystyle \frac{dP_2}{dy} &= -\frac{2C_2}{(1 + y)^3} = -P_2\frac{2}{(1 + y)} \\ \Rightarrow -\frac{dP_2/P_2}{dy} &= \frac{2}{1 + y} \end{aligned}

i.e. the percentage change in bond-price to small changes in $y$ of of a single cash-flow at time $2$ is $2/(1 + y)$. Again, we notice that it is related to its time to maturity, or as can now start calling it, duration. The duration of a single cash-flow at the end of time $2$ is $2$, and then the percentage change in the bond price to small changes in $r$ given by $2/ (1 + y) = \mbox{Duration} / (1 + y)$.

That is, irrespective of the cash-flow at time $1$ or time $2$, the percentage change in the bond price is given by $\mbox{Duration} / (1 + y)$. We can now generalize this result to say that the percentage change in the bond price to small changes in $y$ (compounding $k$ times a year) is given by:

$\displaystyle -\frac{dP/P}{dy} = \mbox{Modified Duration} =\displaystyle \frac{\mbox{Duration}}{(1 + y/k)}$

We define $\displaystyle \frac{\mbox{Duration}}{(1 + y/k)}$ as the Modified Duration. The modified duration represents the percentage change in the bond price to small changes in $y$.

Duration for a coupon-bearing bond can be thought of being as that average time all the money gets received, i.e. as the weighted average time to maturity as:

\begin{aligned} \displaystyle \mbox{Duration} &= \frac{1 \times PV_1 + 2\times PV_2 + 3 \times PV_3 + ... n \times PV_n}{PV_1 + PV_2 + PV_3 + ... + PV_n} \\&= \frac{1}{P} \Big(\frac{1 \times C_1}{1 + y} + \frac{2 \times C_2}{(1 + y)^2} + \frac{3 \times C_3}{(1 + y)^3} + ... \frac{n \times C_n}{(1 + y)^n} \Big) \\&= \frac{1}{P} \sum_{t = 1}^{n} \frac{t \times C_t}{(1 + y)^t} \end{aligned}

The formula for Modified Duration naturally extends for a set of cash-flows, and we can write the Modified Duration for a coupon bearing bond also as:

$\mbox{Modified Duration} =\displaystyle \frac{\mbox{Duration}}{1 +y/k}$

And now we can measure the percentage change in the bond price to small change in yields using:

\begin{aligned} \displaystyle -\frac{dP}{P} &= \mbox {Modified Duration } * dy \\&= \frac{\mbox{Duration}}{(1 + y/k)} * dy \end{aligned}

That is, if we know the Modified Duration/Duration of our bonds, to see the percentage change in the value of the bonds we only need multiply the Modified Duration of our bonds by the change in yields and we are done. No need to re-evaluate the price of all our bonds for new changed yields. And since yield changes by the central banks are typically of the order of 25 – 50 basis points only, using Modified Duration/Duration is not a bad first approximation (why do I say approximation?).

As should be clear, if yields are small, using either Modified Duration and Duration is ok – of course, using Modified Duration is more accurate.

Written by Vineet

October 7, 2016 at 8:40 pm

Posted in Teaching: FM

Tagged with , , , ,

## [PGP-I FM] Yield to maturity and bootstrapping

Till we began our discussion on bond markets, we had been working with the following pricing equation with a constant rate of interest, $r$:

$\displaystyle P = \frac{C}{1 + r} + \frac{C}{(1 + r)^2} + \frac{C}{(1 + r)^3} + \frac{C}{(1 + r)^4} + ... + \frac{C + 100}{(1 + r)^n}$

But clearly, in the real world we just don’t have one interest rate, but as a visit to any bank website will tell you, there is a separate rate for each maturity.

Consider a bond that matures at time $t = 2$, pays a cash flow $C_1$ at time $1$, and cash flow $C_2$ at time $2$. Then, if we knew the rate applicable for each maturity (called the spot rates), we could still price the bond easily by writing:

$\displaystyle P = \frac{C_1}{1 + r_1} + \frac{C_2}{(1 + r_2)^2}$

where $r_1$ is the one period annualized rate for $1$ year, and $r_2$ is the annualized rate for $2$ years. A constant rate that gives the same price as obtained using the correct spot rates is what the market participants call the Yield to Maturity or YTM. That is, finding YTM involves finding the price of the bond by using instead a constant rate , $y$, and this gives the following equation:

$\displaystyle P = \frac{C_1}{1 + y} + \frac{C_2}{(1 + y)^2}$

But which is the ‘right’ way? What comes first, the rate for each maturity $r_1$ and $r_2$ or the YTM $y$?

It is the spot rates which are the fundamental quantities, and not the YTM. In fact, just like IRR, YTM is nothing but a rate that makes the NPV from investment in a bond $0$. So, for our two-period bond example, $y$ is that rate which solves the following equation:

$\displaystyle -P + \frac{C_1}{1 + y} + \frac{C_2}{(1 + y)^2} = 0$

In fact, this is how YTM is defined.

Bootstrapping

In fact, even more fundamental than the rates $r_1$ and $r_2$ are the prices themselves. Because one buys and sells bonds, and not interest rates. This brings us to the idea of bootstrapping.

(In case you are curious, bootstrap refers to the strap/loop provided in the shoes/boots. It was designed to help people get in and out of the shoes easily. As this wiki entry will tell you, over time this came be known as the metaphor to ‘pull oneself up’ without outside effort – a kind of ‘self-sustaining force’.)

Say, we have zero-coupon bonds of maturity $1$ and $2$ available in the bond market, with prices $P_1 \mbox{ and } P_2$ respectively. Then, since the bonds are zero coupon bonds (i.e. there are no intervening coupons, and only a final cash flow, say $C$), we can write their prices are:

\begin{aligned} \displaystyle P_1 &= \frac{C}{1 + r_1} \\ P_2 &= \frac{C}{(1 + r_2)^2} \end{aligned}

If we observe $P_1 \mbox{ and }P_2$, then it’s clear that we can find the spot rates $r_1 \mbox{ and }r_2$ as:

\begin{aligned} \displaystyle r_1 &= \frac{C}{P_1} - 1 \\ r_2 &= \sqrt{\frac{C}{P_2}} - 1 \end{aligned}

So, if there are zero coupon bonds in the market, finding the spot rates $r_1 \mbox{ and } r_2$ is easy.

Unfortunately in most countries, including India, there does not exist an active market in zero coupon bonds across maturities. All long-term bonds issued by Government of India are coupon bearing bonds. In that case, as you would guess, extracting spot rates is not as easy.

Let’s again consider two bonds – but this time we consider coupon bearing bonds (a more realistic situation) rather than zero-coupon bonds.

Let’s call the price of coupon bond of maturity $1$ (with only a single cash flow $C_{11}$) as $P_1$, and the price of of coupon bond of maturity $2$ (with cash flow $C_{21}$ at time $1$ and cash flow $C_{22}$ at time $2$) as $P_2$. Then we have:

\begin{aligned} \displaystyle P_1 &= \frac{C_{11}}{1 + r_1} \\ P_2 &= \frac{C_{21}}{1 + r_1} + \frac{C_{22}}{(1 + r_2)^2} \end{aligned}

Since $P_1$ is traded, given its coupon $C_{11}$, we can still find $r_1$ as earlier as:

$\displaystyle r_1= \frac{C_{11}}{P_1} - 1$

What is not so obvious now is finding $r_2$, as instead of a single cash flow at time $2$, we have cash flows from a coupon bearing bond both at time $1 \mbox{ and } 2$. But since $P_2$ is known from the market, it turns out we can still find $r_2$. This is how:

Write

\begin{aligned} \displaystyle P_2 &= \frac{C_{21}}{1 + r_1} + \frac{C_{22}}{(1 + r_2)^2} \end{aligned}

Since $r_1$ is known from $P_1$, and $P_2$ is known from the market, given the cash flows $C_{21}$ and $C_{22}$ from the bond, the only unknown remaining is $r_2$, i.e. in:

$\displaystyle \underbrace{P_2}_{\mbox{known}} = \underbrace{\frac{C_{21}}{1 + r_1}}_{C_{21} \mbox{ known, } r_1 \mbox{ known from } P_1} + \underbrace{\frac{C_{22}}{(1 + r_2)^2}}_{C_{22} \mbox{ known, } \underline{r_2 \mbox{ unknown}}}$

the only variable unknown is $r_2$. Since $LHS = RHS$, we can now easily ‘bootstrap’ $r_2$ from the above equation.

After finding $r_2$, say, if we now had a third coupon bond with maturity $3$, we can find out $r_3$ similarly as:

$\displaystyle \underbrace{P_3}_{\mbox{known}} = \underbrace{\frac{C_{31}}{1 + r_1}}_{C_{31} \mbox{ known, } r_1 \mbox{ known from } P_1} + \underbrace{\frac{C_{32}}{(1 + r_2)^2}}_{C_{32} \mbox{ known, } r_2 \mbox{ known from } P_2} + \underbrace{\frac{C_{33}}{(1 + r_3)^3}}_{C_{33} \mbox{ known, } \underline{r_3 \mbox{ unknown}}}$

This process of successively backing out spot rates from bond prices like this is called bootstrapping.

That is, even if there aren’t any zero coupon bonds in the market, one can still extract the spot rates by the process of backing out, i.e. by bootstrapping, as above.

As should be clear, this process would only work if there are enough coupon bearing bonds across all maturities in the bond market. As even if one bond is missing “in between”, that would mean all other rates starting from that point would be indeterminate. So, for example, if there were no $2$ period bond, i.e. $P_2$ in the market, one couldn’t have extracted $r_2$. But not only that, in that case one couldn’t have extracted even $r_3$, as bootstrapping $r_3$ depends on knowing $r_2$.

The schedule of spot rates ($r_1 \mbox{ , } r_2 \mbox{ , } r_3 \mbox{ , ... } r_n$) for different maturities ($1 \mbox{ , } 2 \mbox{ , } 3 \mbox{ , ... } r_n$) is called as the Term Structure of Interest Rates.

Plotting $r_n$ for a given maturity $n$ gives us a curve which is known as the Zero Coupon Yield Curve (ZCYC) – or simply stated, just a yield curve (not to be confused with the plot of YTMs for different maturities).

So in practice when one doesn’t have ‘nice’ sequential set of coupon bonds, one is forced to use fit the term structure using curve fitting techniques.

Written by Vineet

October 7, 2016 at 8:28 pm

Posted in Teaching: FM