## Overview

In the June 2005 Newsletter issue, Callable Range Accrual Notes were featured as a Spotlight article. Drawing from this article, a range accrual note is a structured product where the coupon is linked to the performance of a reference index such as the six-month USD LIBOR. The coupon in each period is determined by the number of days the reference rate fixes within a pre-specified range. Since investors expose themselves to interest rate risk, the return from a range accrual note can be higher than fixed-rate deposits so long as the reference rate does not move outside the pre-specified ranges during the life of the note. A callable range accrual note is an extension of a range accrual note as it starts out as a range accrual note and after an initial lockout period, the note becomes callable on coupon payment dates.

This Spotlight article will focus on **pricing range accrual swaps** and **callable range accrual swaps**. A range accrual swap is a swap in which one leg pays an accrual coupon and the other leg is a standard floating leg. A callable range accrual swap is an accrual swap which gives the party paying the accrual coupon the right to cancel (or call back) the swap on any coupon date after the initial lock-out period. The accrual coupon can be either a fixed rate, in which case the swap is commonly referred to as a fixed rate accrual swap, or a floating rate, in which case the swap is commonly referred to as a floating rate accrual swap. In a fixed rate accrual swap, the fixed accrual coupon payoff is determined by the number of days the reference rate remains within a specified range. The accrual coupon rate is fixed and unless otherwise specified remains fixed for the life of the swap. Conversely, in a floating rate accrual swap, the floating rate is set at the start of each accrual period.

## FINCAD Analytics Suite Valuation

FINCAD Analytics Suite will be introducing new functions in its next major release that can be used to price range accrual swaps and callable range accrual swaps. Until the new functions are introduced, one can combine a number of functions to value accrual swaps.

As long as there is no spread added to the floating leg of the accrual swap, a swap can be accurately priced by combining two functions: one for the accrual leg and another for the floating leg. The callable range accrual note function can be used to price the accrual leg for a callable range accrual swap and a range accrual swap. The latter can be valued by setting the call and put price columns to zero in the callable range accrual note function. The floating leg of the swap can be valued by using one of the floating leg functions. If there is a spread, these functions can be used to approximate the value of the accrual swap.

You can access the functions by downloading the latest trial version. **To download a free trial of FINCAD Analytics Suite, contact a FINCAD Representative.**

Below is an example in which a callable range accrual swap is priced with a spread on the floating leg.

Independent of whether or not there is a spread on the floating leg or exchange of principal at maturity, a callable range accrual swap can be valued by replacing the call prices in the callable range accrual rate table for aaCallRangeAccNote_dgen_p with the floating leg values (stat 1 from aaFloatlg2_p) where the value date for each floating leg valuation call period is equal to the call date for that period. By replacing the call price with the floating leg value and comparing a range accrual leg to a floating leg on call dates, we are valuing the option to cancel a Range Accrual Swap, where the accrual leg payer has the right to cancel the swap. The "option price" output from aaCallRangeAccNote_dgen_p (stat 6) is then the value of the accrual leg payer's right to cancel the swap (as seen from the accrual leg receiver's position). (Note: if the accrual leg receiver has the right to cancel the swap, then we replace the put prices, not the call prices, with the floating leg values).

The clean price of a receiver's callable range accrual swap is the clean price of a (non-callable) receiver's Range Accrual note plus the price of payer's option to cancel the swap (as seen from the receiver's position) minus the clean price of the floating leg. The clean price of the range accrual leg including the option is given by stat 1 from aaCallRangeAccNote_dgen_p. The clean price of the floating leg is given by stat 1 from aaFloatlg2_p.

The clean price of the (non-callable) receiver's Range Accrual Swap is the clean price of the range accrual leg minus the clean price of floating leg. The clean price of the range accrual leg is given by stat 2 from aaCallRangeAccNote_dgen_p. The clean price of the floating leg is given by stat 1 from aaFloatlg2_p.

Note 1: To value the CRA Swap with or without a spread, run aaFloatlg2_p with mgn = spread or 0.

Note 2: To value the CRA Swap with exchange of principal at maturity, run aaCallRangeAccNote_dgen_p with "principal payment" = "notional principal amount" in the last row of the input rate table, and run aaFloatlg_p with redemp_val = 100.

Note 3: To value the CRA Swap without exchange of principal at maturity, run aaCallRangeAccNote_dgen_p with 0 in all rows of the "principal payment" column of the input rate table, and run aaFloatlg_p with redemp_val = 0.

Note 4: Be careful with date adjustments: the input rate table for aaCallRangeAccNote_dgen_p contains unadjusted dates. But aaFloatlg_p must be run using adjusted dates.

Note 5: Recall that the valuation of a CRA Swap using aaCallRangeAccNote_dgen_p is "approximate" if there is a spread on the floating leg, because aaFloatlg_p gives us the value of the floating leg (call price) on call/cancel dates based on today's yield curve instead of the yield curve at a call date node of the interest rate tree. If there is no spread, then the valuation is exact because accruing rates = discounting rates, which means the value of the floating leg is independent of the yield curve.

## Disclaimer

Your use of the information in this article is at your own risk. The information in this article is provided on an "as is" basis without any representation, obligation, or warranty from FINCAD of any kind, whether express or implied. We hope that such information will assist you, but it should not be used or relied upon as a substitute for your own independent research.

For more information or a customized demonstration of the software, contact a FINCAD Representative.