# Pricing in probabilities? watch

1. Hi there,

I read today that US interest rate futures moved to price in a 64% chance of a further half point cut by the Fed next month...

How can you look at a price and get a probability like that from it?

Cheers
2. (Original post by F_A_T_E)
Hi there,

I read today that US interest rate futures moved to price in a 64% chance of a further half point cut by the Fed next month...

How can you look at a price and get a probability like that from it?

Cheers
If 64/100 people think that there will be rate cut.

edit: I know it's a stupid answer, but that is all there is to it. Sentiment and speculation.
3. (Original post by uthinkilltellu)
If 64/100 people think that there will be rate cut.

edit: I know it's a stupid answer, but that is all there is to it. Sentiment and speculation.
that's not helpful - he asked how you could tell from the price, not y doing a survey.

OP, one way would to look at the difference between future and spot prices, and then look back at historical data. i'm sure it's possible to deduce a probability from theory too (i.e. not from history), but then you rely on 'assumptions'.
4. There are traded fed fund rate futures from which the rate can be calculated. Google "finding implied interest rates" or something, there was a WSJ article about it a while ago.
5. I'm just guessing but wouldn't it be along the lines of:

You assume the price of a future is the expected payout on the exercise date (i.e. weighted probabilities of event payouts). Then since we are dealing with futures we know exactly what the payout of different events are (eg. payout of fed increases by 1% is x).

So what we want are the probabilities. We chuck away the probabilities that the fed will change rates by 3/17% etc. So then we have a few likely changes that the Fed might make. So since we have lots of different people selling futures over-the-counter and on exchanges we can pick as many different futures prices as variables we have. So then we just essentially solve a system of n equations with n variables, where we have n 'non-neglible' events.

(This is a complete guess though and probably is flawed.)
6. (Original post by uthinkilltellu)
If 64/100 people think that there will be rate cut.

edit: I know it's a stupid answer, but that is all there is to it. Sentiment and speculation.
(Original post by Bathian)
I'm just guessing but wouldn't it be along the lines of:

You assume the price of a future is the expected payout on the exercise date (i.e. weighted probabilities of event payouts). Then since we are dealing with futures we know exactly what the payout of different events are (eg. payout of fed increases by 1% is x).

So what we want are the probabilities. We chuck away the probabilities that the fed will change rates by 3/17% etc. So then we have a few likely changes that the Fed might make. So since we have lots of different people selling futures over-the-counter and on exchanges we can pick as many different futures prices as variables we have. So then we just essentially solve a system of n equations with n variables, where we have n 'non-neglible' events.

(This is a complete guess though and probably is flawed.)
Don't know, don't guess.

Let's take a simplified example to make the explanation as clear as possible, and assume we are back in June 2007. The central banks set the base rate, say 4%. This means that banks lend to each other O/N cash at 4% + a basis. This was between 5-10bp for different markets back then, but pretty stable. After ECB hiked 25bp in June 2007, banks were lending to each other at 4.25%+6bp=4.31%.

There is a number of contracts that allow you to trade this rate, most notably O/N swaps. An O/N swap is indexed every day at the O/N cash rate, and is actually a more precise way of betting on the base rate compared to an interest rate future. Primarily this is because (as its OTC) you can trade the 5-Jul-07 to 2-Aug-07 swap, i.e. you have one meeting period in there, rather than 3-4 periods as is the case for futures.

Thefore, if it's 15th June 2007 today and the July-meeting O/N swap trades at 4.45%, backing out the implied probability is very easy. (Simplified version) Firstly, you strip away the 6bp basis, secondly you strip away the current rate to get 445-6-425=14bp. 14bp/25bp = 56%, which is the probability of a 25bp hike.

There's a few problems with this, e.g. whether this implies a 56% probability of a 25bp hike or a 28% of a 50bp hike, but generally it does the job. If you have these one-meeting-contracts trading for every meeting period, you can bootstrap the probability distribution fairly easily.

Two important points to take into accounts-
1) This doesn't work that well anymore. The basis, which used to vary pretty much by 1-2bp at most, except for quarter/fiscal year ends when it would vary by 4-16bp, has now dislocated and is hugely volatile. This explains why in Sep/Oct this year, Fed funds rate was set at 2% yet the actual cash traded between 0-7%. You can trade the basis in the market, and hence there's a forward curve for it, afaik, these days thoug, so you can strip it away - it's just a bit more complicated.

2) The entire exercise is more difficult for Fed (or any IR) futures, as these are typically for the IMM dates, e.g. something like 17-Sep to 17-Dec. One future with one price will therefore span 4 different meetings (Sep, Oct, Nov, Dec) and solving for the rates is a bit more complicated.

In general though, although people will talk about the "Fed futures implied rate", the rate is really take from the O/N swap market. The reason, I'm guessing, is that 'Fed futures' sounds less scary..
7. (Original post by Jan)
Thefore, if it's 15th June 2007 today and the July-meeting O/N swap trades at 4.45%, backing out the implied probability is very easy. (Simplified version) Firstly, you strip away the 6bp basis, secondly you strip away the current rate to get 445-6-425=14bp. 14bp/25bp = 56%, which is the probability of a 25bp hike.

There's a few problems with this, e.g. whether this implies a 56% probability of a 25bp hike or a 28% of a 50bp hike, but generally it does the job. If you have these one-meeting-contracts trading for every meeting period, you can bootstrap the probability distribution fairly easily.
I'm not really disputing this but it does seem rather messed up. Isn't this derivation of implied probability assuming either; the only other event under consideration is that the Fed doesn't change the rate; or, that the implied probabilities of all other events under consideration are zero? (or both I guess)

To clarify, surely in some sense we have;

14bp = (probability fed increases 25bp)*25 + (probability fed increases x(1)bp)*x(1) + (probability fed increases x(2)bp)*x(2) +...+ (probability fed increases x(n)bp)*x(n)

where x(i) are different rate changes that we consider. So for (probability fed increases 25bp) = 14/25 we have to have that all other terms are zero.

(Obviously present value it or whatever.)
8. (Original post by Bathian)
I'm not really disputing this but it does seem rather messed up. Isn't this derivation of implied probability assuming either; the only other event under consideration is that the Fed doesn't change the rate; or, that the implied probabilities of all other events under consideration are zero? (or both I guess)

To clarify, surely in some sense we have;

14bp = (probability fed increases 25bp)*25 + (probability fed increases x(1)bp)*x(1) + (probability fed increases x(2)bp)*x(2) +...+ (probability fed increases x(n)bp)*x(n)

where x(i) are different rate changes that we consider. So for (probability fed increases 25bp) = 14/25 we have to have that all other terms are zero.

(Obviously present value it or whatever.)
Hence the '(Simplified version)' qualification in my post..the point is that at any given instance, there are usually (or at least used to be) at most 2 alternatives to keeping rates on hold that had a significant probability of happening.

Even now, for most purposes, this approximation will be good enough. You can get a more precise view from the options market, depending on how liquid they are for a given currency.
9. all right.

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