# Accrual Swaps

PATRICK S. HAGAN BLOOMBERG LP 499 PARK AVENUE NEW YORK, NY 10022 PHAGAN1@BLOOMBERG.NET 212-893-4231 Abstract. Here we present the standard methodology for pricing accrual swaps, range notes, and callable accrual swaps and range notes. Key words. range notes, time swaps, accrual notes

1. Introduction. 1.1. Notation. In our notation today is always t = 0, and (1.1a) D(T ) = today’s discount factor for maturity T.

For any date t in the future, let Z(t; T ) be the value of $1 to be delivered at a later date T : (1.1b) Z(t; T ) = zero coupon bond, maturity T , as seen at t.

These discount factors and zero coupon bonds are the ones obtained from the currency’s swap curve. Clearly D(T ) = Z(0; T ). We use distinct notation for discount factors and zero coupon bonds to remind ourselves that discount factors D(T ) are not random; we can always obtain the current discount factors from the stripper. Zero coupon bonds Z(t; T ) are random, at least until time catches up to date t. Let (1.2a) (1.2b) These are deﬁned via (1.2c) D(T ) = e− T 0

f0 (T ) = today’s instantaneous forward rate for date T, f (t; T ) = instantaneous forward rate for date T , as seen at t.

f0 (T 0 )dT 0

,

Z(t; T ) = e−

T t

f (t,T 0 )dT 0

.

1.2. Accrual swaps (ﬁxed).

αj t0 t1 t2

…

tj-1

tj

…

tn-1

tn

period j

Coupon leg schedule Fixed coupon accrual swaps (aka time swaps) consist of a coupon leg swapped against a funding leg. Suppose that the agreed upon reference rate is, say, k month Libor. Let (1.3) t0 < t1 < t2 · · · < tn−1 < tn 1

Rfix

Rmin

Rmax

L(τ)

Fig. 1.1. Daily coupon rate

be the schedule of the coupon leg, and let the nominal ﬁxed rate be Rf ix . Also let L(τ st ) represent the k month Libor rate ﬁxed for the interval starting at τ st and ending at τ end (τ st ) = τ st + k months. Then the coupon paid for period j is (1.4a) where (1.4b) and (1.4c) θj = #days τ st in the interval with Rmin ≤ L(τ st ) ≤ Rmax . Mj αj = cvg(tj−1 , tj ) = day count fraction for tj−1 to tj , Cj = αj Rf ix θj paid at tj ,

Here Mj is the total number of days in interval j, and Rmin ≤ L(τ st ) ≤ Rmax is the agreed-upon accrual range. Said another way, each day τ st in the j th period contibutes the amount ½ αj Rf ix 1 if Rmin ≤ L(τ st ) ≤ Rmax (1.5) 0 otherwise Mj to the coupon paid on date tj . For a standard deal, the leg’s schedule is contructed like a standard swap schedule. The theoretical dates (aka nominal dates) are constructed monthly, quarterly, semi-annually, or annually (depending on the contract terms) backwards from the “theoretical end date.” Any odd coupon is a stub (short period) at the front, unless the contract explicitly states long ﬁrst, short last, or long last. The modiﬁed following business day convention is used to obtain the actual dates tj from the theoretical dates. The coverage (day count fraction) is adjusted, that is, the day count fraction for period j is calculated from the actual dates tj−1 and tj , not the theoretical dates. Also, L(τ st ) is the ﬁxing that pertains to periods starting on date τ st , regardless of whether τ st is a good business day or not. I.e., the rate L(τ st ) set for a Friday start also pertains for the following Saturday and Sunday. Like all ﬁxed legs, there are many variants of these coupon legs. The only variations that do not make sense for coupon legs are “set-in-arrears” and “compounded.” There are three variants that occur relatively frequently: Floating rate accrual swaps. Minimal coupon accrual swaps. Floating rate accrual swaps are like ordinary accrual swaps except that at the start of each period, a ﬂoating rate is set, and this rate plus a margin is 2

used in place of the ﬁxed rate Rf ix . Minimal coupon accrual swaps receive one rate each day Libor sets within the range and a second, usually lower rate, when Libor sets outside the range αj Mj ½ Rf ix Rf loor if Rmin ≤ L(τ st ) ≤ Rmax . otherwise

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