# Fed Fund Swap Nuances

A few people have recently asked me about the details of USD Fed Fund Swaps. I collected some of my answers together in this post for convenience.

### Fed Fund Swap and OIS Swap differences

Fed Fund Swaps and OIS swap are easily confused at first glance, both are swaps involving the Federal Funds rate, both have slightly complicated coupons, and both are used to construct the USD OIS curve. However the main differences are;

Characteristic Fed Fund Swap OIS Swap
Structured Coupon Weighted Average of Fed Fund Rate Compounded Fed Fund Rate
Swap Structure Basis Swap (3M Fed Fund v 3M LIBOR) Fixed Float Swap (Zero Coupon or Annual Fixed Coupon)
Maturity Medium/Long Dated Short Dated
ISDA Floating Rate Option Code USD-Federal Funds-H.15 USD-Federal Funds-H.15-OIS-COMPOUND

### Weighted Average Rate

The structured coupon rate is the weighted average fed fund rate over the calculation period, where the rate is fixed on each business day in the calculation period. The weight of a particular fixing is simply the accrual period length of the fed fund rate, so that in general a fed fund rate on a Friday would have a heavier weight (3 days of accrual), than the rate on any other day (1 day of accrual). One exception to this natural weighting occurs on the last fixing of the calculation period, typically it is slightly heavier because of the Reset Cut-Off.

### Reset Cut-Off

The term ‘Reset Cut-Off’ denotes the date of the last fixing before the payment date. Often it is defined relative to the end of the calculation period, for example, ‘2 New York business days’. It is easy to understand why it is needed by considering an example. Suppose the calculation period end date and payment date is Friday 8-Feb-2014, one might naturally consider including the Fed Fund overnight rate from Thursday 7-Feb-2014 to Friday 8-Feb-2014, however the Fed Fund rate is not published until the morning of Friday 8-Feb-2014, in which case, there may not be enough time to determine the average rate for the period and confirm the cashflow amount. For this practical reason, one would typically not include the rate from 7th-8th, and simply apply an extra day’s weight to the rate of 6th-7th, yielding a reset cutoff of 2 business days.

### Pricing

Let $$R_i$$ denote the Fed Fund rate between $$t_i$$ and $$t_{i+1}$$ where $$i=0,1,2,…,n$$. Suppose the payment date is $$T>t_n$$.

Concentrating on one specific fixing one can easily see a problem, the Fed Fund rate start and end dates $$t_i$$ and $$t_{i+1}$$ are quite different to the payment date, $$T$$, of the swap coupon, in which case, a convexity correction similar to that found in in-arrears swaps is induced. More details are found in the recent article Valuation of Arithmetic Average of Fed Funds Rates and Construction of the US Dollar Swap Yield Curve, Katsumi Kevin Takada, September 30, 2011 However, it is common to ignore this correction especially when the calculation period is small, and simply determine the forward for each individual Fed Fund rate and discount from the payment date.

To speed up valuations, one can use the following approximation,

$$\sum w_i R_i \approx \int r(t) dt = \ln(D_0) – \ln(D_n)$$

where, $$D_i$$ denotes the discount factor at time $$t_i$$. Notice this approximation not only avoids computing the daily forward rates, one doesn’t even need to generate all of the fixing dates, the first and last fixing is enough.