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FRN - Equal Coupon Rates & Equal Coupon Periods

When an FRN has equal coupon rates and equal coupon periods, the calculation of the dirty (or settlement) price of an FRN can be simplified by using an annuity formula to value almost all of the future coupon payments concurrently. The dirty price is given by:

Equation Template

P = dirty price (clean price plus accrued interest) of the FRN per 100 units face value.

v1 = the discount factor for the period between settlement and the payment date of the next nominal coupon.

c1 = the first/next coupon payment. For an odd first period, this amount may differ from the standard coupon payment. It may also equal zero if the bond is trading ex-coupon.

c2 = the coupon payment due on the next nominal payment date. For bonds with an odd first period, this amount may differ from the standard coupon payment.

c = the coupon payment per 100 units of face value, based on the constant coupon rate and the assumption that normal coupon periods have equal length.

cfn = the final coupon amount for a bond with an odd final period. This is zero for all other bonds.

n = number of full coupon periods remaining until redemption. If the bond has a long last coupon period, then n is determined as the number of full coupon periods between settlement and the penultimate coupon date.

v = the discount factor for a standard coupon period, again based on the assumption of coupon periods with equal length.

 

If it is assumed that the reference rate is constant for all future rate resets and that the length of all future normal coupon periods is equal, then we can further define the following for an FRN with no odd coupon periods:

Equation Template

 

d1 = the number of days between the settlement date and the first coupon payment date.

h = the number of coupon periods in a year.

r1 = the reference index rate observed at the beginning of the current coupon period.

rf = the forecast reference index rate for all future coupon periods.

ym = the required annual nominal yield to maturity, expressed as a decimal. With a given price for the FRN, this is the initial yield parameter that is solved for (see calculation of effective margin).

DIP = number of days in the period, based on the selected accrual method.

DIY = number of days in the year, based on the selected accrual method.

 

We also need to consider the input definitions when the FRN under consideration includes either an odd first coupon period or an odd last period. These are set out in the following tables. Note that the definitions for the odd first period are only required if the FRN is being valued during the first nominal coupon period. For valuation dates that do not fall within the odd first coupon period, these definitions are not required at all.

 

 

 

c1

 

c2

 

.

Short First Coupons

Equation Template

Equation Template

Long First Coupons

 

 

0

Equation Template

 

 

 

Coupon Payment Definitions for an FRN with an Odd Last Coupon Period (equal coupon payments):

 

 

cfn

 

 

.

 

Short Last Coupons

Equation Template

 

Long Last Coupons

Equation Template

 

 

 

 

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