# In Arrears Swap - what accrual period applies?

In Arrears Swap, the floating rate is reset and paid on the same date.

What accrual period is applied to compute the payment -

If the dates are t1, t2, t3 ...tn. (assume overlapping date schedules for reset, accrual start, accrual-end and payments)

Then, which accrual period applies to the floating rate set on t2

1. The trailing period, i.e. (t3-t2)*DCF or
2. The prior period accrual period, i.e. (t2-t1)*DCF

(The payment date for both 1 and 2 remains the same, i.e. t2).

Add-on question: pricing formulae for "In-Arrears Forward Rate Agreement" (IAFRA) -

(summation of IAFRA over all periods would give the "In-arrears Swap". I assume fixed coupon, K=0.)

1. Under 1 (i.e. natural accrual period is applied to the rate) -

$$IAFRA_1 = P(0,t2) \tau_{t2,t3} F(0,t2,t3) + P(0,t3) {\tau_{t2,t3}}^2 F(0,t2,t3)^2 \{\sigma(0,t2)^2 t2\}$$

where $$\tau_{t2,t3} = (t3-t2)*DCF$$

The above formula is from Brigo Mercurio's Book.

The first term is intuitive as it is simply the discounting of the estimated payoff (paid at t2).

The second is the convexity adjustment term (to correct the estimated payoff in first term, to the fair expectation of the payoff). Not fully intuitive, but the derivation steps prove it.

1. Under 2 (i.e. prior period accrual is applied to rate set at end of period) -

$$IAFRA_2 = \frac{t2-t1} {t3-t2} IAFRA_1$$

Is my formula, under 2, correct?

You can specify either. I have seen both, but most practitioner-ish references use 2: everything else remains the same (as a standard swap) but the rate index shifts one place to the right. So the accrual intervals and the intervals to which the rates naturally belong are disjoint. Easy to understand the logic when you recall that these products came about when people noticed that upward sloping yield curve implied higher rates in the future but the realised rates usually turn out to be different. So a fixed rate receiver gets better rates when they go against the expectation hypothesis. And it would be nice to have the accrual periods aligned to a standard swap conventions.

But as you said in one of the comments the accrual periods are deterministic so using 1 won’t cause too much trouble. Btw, which conventions would lead to a simpler convextiy formula?

Lastly, you said ‘the floating rate is reset and paid on the same date’. There would be a spot lag, usually 2 days, but there is no lag for GBP and some other commonwealth currencies.

• Using 1 would give a simpler formula. But more importantly from my perspective, it is far easier to understand the terms of trade, using 1. Under 1, the amounts paid are the same as a swap, only they get paid at start of the period (only a timing benefit, but no other difference). In this case, I can visually see that in arrears leg is more valuable than the swap flosting leg. I can beat the usual swap by re-investing for 1 more period. Under 2, it is messy to devise a strategy to beat usual swap. Ratio of year fracs needs to be applied, which makes notionals different in each period, etc. – bhutes May 2 '19 at 20:56
• Perfect! Good question btw – Magic is in the chain May 2 '19 at 21:04

It is 2 above. The logic is that the payment at the end of any given period is given by rate for that period * day count for that period (using the dates of that period). Changing the date of the rate set does not influence the accrual dates.

• Brigo and Metcurio, “Interest Rate Models - Theory and Practice” says it is 1. A rigorous mathematical derivation of pricing requires the rate to be paid for the same accrual period for which it is set. Albeit, both year fractions are known constants, so it would be simple to multilpy by the ratio (t3-t2) / (t2-t1) to get the desired accrual period used in pricing formula derivation. – bhutes May 2 '19 at 14:32
• Right but the question is not about how to calculate the rate. It is about how to compute the payment , given the rate. – dm63 May 2 '19 at 17:17
• Agree- you already answered the question. Supplementary question - can you refer to any standard text which gives me exact pricing formula, under 2? I am sort of hesitant applying my changes to Brigo’s formula (which assumes 1). I have no one to check my modified formula, even though the change looks quite simple. More so, as the payoff envisaged under 2, is less appealing / beautiful to me, I would like to see someone else handling the minor mess caused. – bhutes May 2 '19 at 21:07
• Re supplementary question, this would be of interest to many people so would be good if you can post a new question – Magic is in the chain May 2 '19 at 22:00