Relationship between Forward and Futures Prices

[Bruno]

Hi David,

Regarding the relationship between forward & futures prices (pg. 19) which is also addressed in Treasury bonds Futures & Eurodollar Futures (page 26), I would appreciate it if you could please clarify the following:

1) Both Eurdollar futures and T-bills are quoted in price (as opposed to FRA). As rates increase, prices fall. Therefore the long position on either a ED or T-bill future loses money as rates increase. Is this correct?

2) Eurodollar futures contracts are based on 90-day LIBOR, which is an "add-on" yield. What is an "add-on" yield? I assume that a 30-day Eurodollar futures contract is a 30-days futures contract on 90-day LIBOR. Is this correct?

3) How do we calculate the price of a ED futures contract (given the futures quoted price, Z)?

Contract price = $10,000*[100-(100-Z)*(0.25)].

I don't quite undertand this formula (where does it come from? notional is 1 million). How can I relate it to LIBOR? I mean the relationship between Z and LIBOR.

4) Fixed income values fall when interest rates rise, so rates and values are negatively correlated. Because of the mark-to-market feature of futures, when bond prices fall and funds are needed (margin call), borrowing costs are higher (interest rates). And when funds are generated (bond prices increase providing an excess margin), the reinvestment rate is lower. Therefore, since the correlation between bond prices and interest rates is negative, futures < forwards (pg. 19). I don't understand why the implied rate on the futures contract will be greater than the rate on the FRA. What am I missing?

Thanks

 

[Bruno]

David, I think that I finally understand the convexity bias (my previous question #4).

4) futures PRICES < forwards PRICES (pg. 19). So the implied RATE on the futures contract will be greater than the implied RATE on the FRA. Is this correct?

 

[David Harper CFA FRM]

Hi Bruno,

Great question. These are so relevant that i will, within the next 2-3 days, make sure to get an annotated EditGrid/XLS uploaded so these are concretely illustrated.

1) Both Eurdollar futures and T-bills are quoted in price (as opposed to FRA). As rates increase, prices fall. Therefore the long position on either a ED or T-bill future loses money as rates increase. Is this correct?

Correct the ED futures and Treasury futures quote in prices ($) whereas FRAs tend to be discussed in terms of rates (%). But this probably has more to do with fact that the former are exchange-traded and latter are OTC (i.e., as the economics are similar, we could price the FRA if we want to).

And, yes, also correct: "the long position on either a ED or T-bill future loses (gains) money as rates increase (fall)"

2) Eurodollar futures contracts are based on 90-day LIBOR, which is an "add-on" yield. What is an "add-on" yield? I assume that a 30-day Eurodollar futures contract is a 30-days futures contract on 90-day LIBOR. Is this correct?

See next, maybe easier with the example....

3) How do we calculate the price of a ED futures contract (given the futures quoted price, Z)?

Contract price = $10,000*[100-(100-Z)*(0.25)].

I don't quite undertand this formula (where does it come from? notional is 1 million). How can I relate it to LIBOR? I mean the relationship between Z and LIBOR.

If we take Hull's example 6.2, where the settlement price is 94.79, then:

10,000*[100-.25*(100-94.79)] = $986,975 is the contract price.

The .25 is 90/360 days. This effectively locks-in (100-94.79) 5.21% interest rate for the long position.

If the final price = 96, then
10,000*[100-.25*(100-96)] = $990,000; i.e., long gains +$3,025. This is the contract's design: the long should profit +$25 for each 1 basis point drop in rates.

In this case, the rate drops 121 basis points and the long gains: (5.21 to 4%) and 121*25 = $3,025.

"I assume that a 30-day Eurodollar futures contract is a 30-days futures contract on 90-day LIBOR"
Yes, that would be a contract for delivery in 30 days and always for a 90-day LIBOR rate.

Add-on yield refers to the fact that, the pricing of the contract (superficially) looks like a discounted (zero-type) instrument; i.e., 94.79 for $100 at delivery. But, if you take this example, the gain = 25%*(5.21% - 4%)*$1,000,000 [contract] = $3,025. So, it's not a discounted yield but rather an yield on the full $1MM contract.

4) futures PRICES < forwards PRICES (pg. 19). So the implied RATE on the futures contract will be greater than the implied RATE on the FRA. Is this correct?

Yes, this is CORRECT. Frankly, I have learned more about convexity bias thru the great folks at fixedincomerisk.com

They have a simpler way to make this point: Whereas Hull illustrates with daily settlement, as you have summarized above (nicely), says Sanjay et al "The difference arises because futures contracts are marked-to-market every day, which makes these contracts more volatile than forwards." I like that b/c it's easy to remember: mark-to-market implies higher volatility. Higher volatility, ceteris paribus, is greater risk so must be higher rate."

Hope that helps...I'll update with a note after i upload an illustrated example...

David

 

[Bruno]

David,

Thanks so much for your concise explanation. Your help is priceless! Bionic Turtle is a great buy.

For convenience to other users, I’ve summarized some important conclusions on this topic:

PRICING OF ED FUTURES CONTRACTS

I've realized that it is important to remember the price of a basis point because in the exam we can solve the exercise very fast as you have demonstrated above.

The price of 1bp is: $1,000,000*[0.0001*(90/360)]=$25

Anyway, we must know the contract pricing formula. $10,000 * [ 100- ((100-QFP)*(90/360)) ]


INTEREST RATE FUTURES vs FRAs

I've also noted that it is important to remember that if you want to lock-in an interest rate (hedge interest rate risk) you can either:

a) BUY (long) FRA
b) SELL (short) a Eurodeposit future or a T-bill future

and that:

Forward prices > Futures prices
Forward rates < Futures rates

Since the correlation between interes rates and bond prices is negative, the mark-to-market feature of futures penalizes futures in comparison to forwards. That is, a long position on a futures contract will receive a margin call when interest rates increase and the cash needed will be more expensive to get because of the higher interest rates. Also, when interest rates decrease and the long position receives excess margin, he/she can reinvest it at a lower interest rate.

It is easy to remember that the mark-to-market feature makes futures more volatile and consequently more risky (less price) than FRAs.


CONVEXITY BIAS

Remeber that in terms of RATES, Forward rates < Futures rates

The implied forward rates futures can be adjusted by the convexity bias:

Actual forward rate = Forward rate implied by futures - convexity bias

 

[nikogeorgiev]

I don't quite understand why the interest adjusted basis is a function of storage cost and convenience yield. Can anyone help me understand? Many thanks

PS. I didn't want to start a new thread

 

[Arka Bose]

this thread is nice, cleared a lot of concept. Thank you guys!