# Calculating fra rates

Let's assume I constructed usd libor 3m curve setting 1m rate=3m rate (so the curve is flat between 1m-3m). Will 1x4 fra rates be good if calculated from such curve?

• Could you not just interpolate between spot 3m libor (aka 0mx3y) and a strip of Eurodollar futures (after adjusting for convexity, these would be Ym x 3m forwards)? These would provide the 3-month libor forward curve. – Helin Aug 5 '16 at 0:56

The 1x4 FRA rate is given by

$F(1,4) = \frac{12}{3} \left(\frac{(1+ 4/12 \times L(4))}{(1+ 1/12 \times L(1))}-1 \right)$

where $L(T)$ is the $T$-month Libor rate seen today.

Clearly $F(1,4)$ depends on the 1M and 4M LIBOR rates.

So if the market 1M rate $L(1)$ is below the market 3M rate $L(3)$ you will be understating the true FRA rate if you set $L(1)=L(3)$

• this is pre crisis approach. I was askign about post crisis (when you have multiple curves) – deceiver Aug 4 '16 at 20:20

The 1x4 FRA rate is where you can lock in 3 mo libor , 1 mo from now. To construct this rate , you must build a 3 mo libor curve. The first point in this curve is 0x3 libor , which is spot 3 mo libor. The next point on this curve is the next Eurodollar futures contract. These expire every month on the third Wednesday. Then you have to interpolate between the points you can observe. Today's value of 1 month libor is completely irrelevant to this calculation.

To value a 1x4 FRA you need the forward rate f(1,4) from a 3 Month yield curve. For that you need the discount factors at 1 month and 4 months. The 1 month discount factor can be extracted from the 1 month libor rate. But for the 4 month discount you can not use the 4 month libor. You can use swaps with 4 month left to maturity and payment frequency 3 month but these are not very liquid. Also often FRA quotes are used for these timebuckets below 1 year. of course you can also just interpolate between a 3 month zerorate from libor and a 1 year zerorate from swaps. That would be the most simple solution. Above 1 year swap rates are normally much more reliable than below.

• @dm63 provided the correct answer already. This is actually incorrect. You cannot calculate a 3m forward rate from a 1m LIBOR rate. Doing so would result in an inconsistency with the 1m3m basis. – Helin Aug 9 '16 at 21:08
• dm63 answer is a correct way of pricing a FRA. But the question was valuation of FRA with yieldcurves. That is something that people actually do, first building a 3 month libor yieldcurve, which means calculating discountfactors for all kind of maturities from all kind of instruments. For the 3 m curve use libor rates up to 3 month and then other instruments like futures, FRA or swap with pay frequency 3 m. Then calculate forwardrate f(1,4) from 1 and 4 month discount factors to price FRA. If the 4 m maturity of the curve was build from a future, you probably end up with dm63 method at the end – Ami44 Aug 9 '16 at 22:57
• The point is, you should build 3m libor curve, 1m libor curve, 6m libor curve, etc. as separate curves. The 1m libor rate has no place in building a 3m libor curve. FRA(1,4), being indexed against 3m libor, should be interpolated from the X-year forward 3-month LIBOR curve. The methodology you proposed is the pre-crisis approach. – Helin Aug 9 '16 at 23:02