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	In [[finance]], a '''range accrual''' is a type of derivative product very popular among structured-note investors. It is estimated that more than USD 160 billions of Range Accrual indexed on interest rates only have been sold to investors since 2004.<ref>[http://www.globalmtn-i.com Mtn-I Publication 2007]</ref> It is one of the most popular non-vanilla financial derivatives. In essence the investor in a range accrual is betting that interest rates will stay within a predefined range.

	~~== Payoff description ==~~

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	~~: <math>P \times \sum_{i=1}^{N} 1_{\text{index}(i) \in \text{Range}} \times \frac{1}{N}<~~/~~math>~~

	* index(''i'') is the value of the ~~index at the ''i''th observation date~~
	* ''N'' is the total number of observations within a period
	* ''P'' is the payout when the index is in the range

	~~If the observation frequency is daily~~, ~~the payoff could be more easily written as~~

	~~: <math>P \times \frac{n}{N}</math>~~

	~~where~~

	* ''n'' is the number of days a ~~specified index is within a given range~~
	* ''N'' is the total number of days of the ~~observation period~~
	* ''P'' is the payout for any ~~given day where the index is in the range~~

	~~The index could be an interest rate (e.g. USD 3 months Libor)~~, ~~or a FX rate (e~~.g. ~~EUR~~/~~USD) or~~ a ~~commodity (e~~.g. ~~oil price) or any other observable financial index~~.~~<br />~~
	~~The observation period can be different from daily (e~~.g. ~~weekly, monthly,etc.), though a daily observation is~~ the ~~most encountered~~.

	~~The receiver of the range accrual coupons~~ is ~~selling [[Digital option\|binary options]]. The value of these options~~ is ~~used to enhance the coupon paid~~.

	~~=== Example ===~~

	~~Let's take an example of~~ a ~~5 years range accrual note linked to USD 3 months Libor~~, with ~~range set as [1~~.~~00%; 6.00%] and~~ a ~~conditional coupon of 5.00%. Let~~'s ~~assume~~ the ~~note~~ to ~~start on January 1~~, ~~2009~~ and the ~~first coupon payment to happen on July 1, 2009~~.

	~~An investor who buys USD 100m~~ of ~~this note will have~~ the ~~following cash flows:~~

	* '''First coupon''' — Between January 1 and ~~July 1~~, ~~2009~~, ~~if USD 3m Libor fixes between~~

	1.~~00% and 6.00% for 130 days~~, ~~then~~ the ~~rate applied for the first semester~~ will be:

	:5.~~00% × 130~~/~~181 = 3~~.~~5912% ''(there are 181 days in total between January 1~~, ~~2009~~ and ~~July 1, 2009)''.~~
	:~~The coupon paid on July 1, 2009 would be: USD 100m × 3~~.~~5912% × 0~~.5 = ~~USD 1~~,~~795,600 ''(assuming 0.5 for the day-count fraction between January 1, 2009 and July 1, 2009)''~~

	* '''Second coupon''' - Between July 1, 2009 and January 1, 2010, if USD 3m Libor fixes between 1.00% and ~~6.00% for 155 days~~, ~~then~~ the ~~rate applied for the second semester will be~~:
	:5.~~00% × 155~~/~~184= 4~~.~~2120%.~~
	~~:The coupon paid on January 1~~, ~~2010 would~~ be~~: USD 100m × 4~~.~~2120% ×~~&~~nbsp;0.5 = USD 2~~,~~106,000 ''(assuming 0.5 for~~ the ~~day-count fraction between July 1, 2009~~ and ~~January 1, 2010)''~~.

	* For the ~~8 following coupons, the same methodology applies~~. ~~The highest rate investor will get is 5~~.~~00% and the lowest 0~~.~~00%.~~

	=~~== Different types~~ of ~~range accruals ===~~

	~~The payout (''P'' in our notation), for each day~~ the ~~index~~ is ~~in the range~~, ~~could~~ be ~~either~~ a ~~fix or variable rate.~~

	~~== Valuation and risks ==~~

	~~A range accrual can~~ be ~~seen as a strip of [[Digital option\|binary options]]~~, with a ~~decreasing lag between fixing date and payment date~~. ~~For this reason~~, it is ~~important~~ the ~~valuation model is well calibrated to~~ the ~~volatility term structure~~ of the ~~underlying~~, ~~at least at the strikes implied~~ by the ~~range~~.

	~~If furthermore~~ the ~~range accrual is [[Callable bond\|callable]]~~, ~~then~~ the ~~valuation model also needs~~ to take ~~into account the dynamic between the [[swaption]] and~~ the ~~underlying~~.

	~~Accrual swaps that monitor permanence of interest rates into~~ a ~~range~~ and ~~pay~~ a ~~related interest rate times~~ the ~~permanence factor also depend on correlation across different adjacent forward rates~~. ~~For the details see for example Brigo~~ and ~~Mercurio~~ (~~2001~~).

	~~== Market ==~~
	~~''(To be completed)''~~

	~~==References==~~
	~~{{reflist}}~~
	*{{cite book \| title = Interest Rate Models — Theory and Practice with Smile, Inflation and Credit\| author = [[Damiano Brigo]], [[Fabio Mercurio]] \| publisher = Springer Verlag \| year = 2001 \| edition = 2nd ed. ~~2006 \| isbn = 978-3-540-22149-4}}~~

	~~[[Category:Investment]]~~
	~~[[Category:Mathematical finance]]~~
	~~[[Category:Options (finance)]]~~

en>Shepelyansky at 13:19, 7 December 2013

2013-12-07T13:19:47Z

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