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## Friday, December 13, 2013

### FRA/ED-Futures - discount to fwd settlement date

--Example (from Jeff's lecture notes)--
Assume on 12 Nov you buy (borrow) a 3x9 FRA struck at 5.5% (paying 5.5%) on 1M notional. On 12 Feb, 6M Libor turns out to be 5.74% , compensation due to you =

\$1M x (0.0574-0.055) * 180/360 / (1 + 0.0574*180/360) = \$1166.52
--------Notation -------
Libor fixing date = 12 Feb

"accrual end date" (my terminology) = 12 Aug.
settlement could be either before or (occasionally) after the 6M loan tenor. This example uses (more common) fwd settlement.
disc factor from 12 Aug to 12 Feb = 1/ (1 + 0.0574 * 180/360)
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Note the "interest due date" is always end of the 6M accrual period. Since we choose fwd settlement, we discount that cashflow to the fixing date.

annualized interest Rate difference = 5.74 %- 5.5%
pro-rated  interest Rate difference = (0.0574-0.055) * 180/360
difference in interest amount (before discounting) = \$1M x (0.0574-0.055) * 180/360. This would be the actual settlement amount if we were to settle after the 6M loan period. Since we choose fwd settlement ...

discounting it from 12 Aug to 12 Feb = \$1166.52
------------
Now we come to the differences between FRA and ED Futures.
1) a simple difference is the accrual basis. ED futures always assumes 90/360 exactly. FRA is act/360.
2) Another simple difference is, ED Futures always uses 3M libor, so our example must be set on Mars where ED futures are 6M-Libor-based.

3) The bigger difference is the discounting to fwd settlement date or fixing date.
- EDF gets away without the PV discounting. It takes Libor rate as upfront interest rate like in Islamic banking. Since Libor turns out to be 5.74% but you "bought" at 5.5%, the difference in interest amount is, under EDF, due immediately, without discounting to present value.
- the payout, or price, is linear with the Libor rate L.
- this is essentially due to daily mark-to-market margin calculation
* FRA takes Libor rate as a traditional loan rate, where interest is due at end of loan period.
** under late settlement, the amount is settled AFTER the 6M, on the proper "interest due date". (Linear with L)
** under fwd settlement, the amount is settled BEFORE the 6M, but PV-discounted. This leads to a non-linear relationship with libor rate and convexity adjustment.