International Fixed Income
Topic IC: Fixed Income Basics Floaters
Introduction
• A floating rate note (FRN) is a bond with a coupon
that is adjusted periodically to a benchmark
interest rate, or indexed to this rate.
– Possible benchmark rates: US Treasury rates, LIBOR
(London Interbank Offering Rates), prime rate, ....
• Examples of floating-rate notes
– Corporate (especially financial institutions)
– Adjustable-rate mortgages (ARMs)
– Governments (inflation-indexed notes)
Floating Rate Jargon
• Other terms commonly used for floating-rate notes
are
–
–
–
–
FRNs
Floaters and Inverse Floaters
Variable-rate notes (VRNs)
Adjustable-rate notes
• FRN usually refers to an instrument whose coupon
is based on a short term rate (3-month T-bill, 6month LIBOR), while VRNs are based on longerterm rates (1-year T-bill, 2-year LIBOR)
Cash Flow Rule for Floater
• Consider a semi-annual coupon floating rate
note, with the coupon indexed to the 6month interest rate.
– Each coupon date, the coupon paid is equal to
the par value of the note times one-half the 6month rate quoted 6 months earlier, at the
beginning of the coupon period.
– The note pays par value at maturity.
• Time t coupon payment as percent of par:
ct t 0.5 rt / 2
Cash Flows
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Example
• What are the cash flows from $100 par of the
note?
– The first coupon on the bond is 100 x 0.0554/2=2.77.
– What about the later coupons? They will be set by the
future 6-month interest rates. For example, suppose the
future 6-month interest rates turn out as follows:
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Valuation
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PT 0.5
Time T cash flow

1+T 0.5 rT / 2
100(1 T 0.5 rT / 2)

1+T 0.5 rT / 2
 100
Valuation continued...
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PT 1
PT  0.5 + coupon

1+T 1 rT  0.5 / 2
100  100T 1 rT  0.5 / 2

1+T 1 rT  0.5 / 2
 100
Valuation continued...
• Working backwards to the present,
repeatedly using this valuation method,
proves that the price of a floater is always
equal to par on a coupon date. Why?
• The coupon (the numerator) and interest
rate (the denominator) move together over
time to make the price (the ratio) stay
constant.
Risk of a Floater
• A floater is always worth par on the next coupon
date with certainty…..So a semi-annually paying
floater is equivalent to a six-month par bond.
• The duration of the floater is therefore equal to the
duration of a six-month bond…..Their convexities
are the same, too. For example,
0. 5
duration  duration of 6 - month bond 
 0.4865
1  0.0554 / 2
0.52  0.5 / 2
convexity  convexity of 6 - month bond 
 0.4734
2
(1  0.0554 / 2)
Inverse Floating Rate Notes
• Unlike a floating rate note, an inverse floater is a
bond with a coupon that varies inversely with a
benchmark interest rate.
• Inverse floaters come about through the separation
of fixed-rate bonds into two classes:
– a floater, which moves directly with some interest rate
index, and
– an inverse floater, which represents the residual interest
of the fixed-rate bond, net of the floating-rate.
Cash Flows
Components of a 11% Fixed-Rate Coupon
Floater and Inverse Floater
12
10
8
Coupon
6
Inverse
Floater
4
2
0
2% 3% 4% 5% 6% 7% 8% 9%
Benchmark Interest rate
Fixed/Floater/Inverse Floaters
FIXED RATE BOND
Fltr.
Inv. Fltr
Inv.Fltr. in Even Split
• Suppose a fixed rate semi-annual coupon bond
with par value N is split evenly into a floater and
an inverse floater, each with par value N/2 and
maturity the same as the original bond.
• This will give the following cash flow rule for the
inverse floater:
– time t coupon, as a percent of par, is
ct 
( k t 0.5 rt )
2
Example: 2-yr Inverse Floater
• Consider $100 par of a note that pays 11% minus
the 6-month rate. (We can think of this as coming
from an even split of a 5.5% fixed rate bond into
floaters and inverse floaters.)
• The first coupon, at time 0.5, is equal to
– $100 x (0.11-0.0554)/2 = $2.73.
• The later coupons will be determined by the future
values of the 6-month rate.
Example Continued...
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Example Continued...
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Decomposition of Inverse Floater
• Clearly, the cash flows of an inverse floater
with coupon rate "k minus floating" are the
same as the cash flows of a portfolio
consisting of
– long twice the par value of a fixed note with
coupon k/2, and
– short the same par of a floating rate note.
• Inverse Floater(k) = 2 Fixed(k/2) - Floater
Valuation of Inverse Floater
• Price of Inverse Floater(k) = 2 x Price of Fixed(k/2) - 100
• For example, the 2-year inverse floater paying 11% minus
floating is worth
– 2 x price of 5.5% fixed rate bond minus 100.
• Using the zero rates from 11/14/95,
– r0.5=5.54%, r1=5.45%, r1.5=5.47%, r2=5.50%,
• To discount each of the cash flows of the 2-year 5.5%
fixed coupon bond, the bond is worth 100.0019.
• Therefore the inverse floater is worth
– 2 x 100.0019 - 100 = 100.0038
Interest Rate Risk
• Dollar Duration of an Inverse Floater(k) = 2
x Dollar Duration of Fixed(k/2) - Dollar
Duration of Floater
• Dollar Convexity of an Inverse Floater(k) =
2 x Dollar Convexity of Fixed(k/2) - Dollar
Convexity of Floater
Duration
• The dollar duration of $100 par of the 5.5%
fixed rate bond is 186.975, while its
duration is 1.87.
• This means the dollar duration of the
inverse floater paying 11%-floating is 2 x
186.975 - 48.65 = 325.30, which gives us a
duration of 325.30/100.0038 = 3.25!
– An inverse floater is roughly twice as
sensitive to interest rates as an ordinary
fixed rate bond.
Convexity
• The dollar convexity of $100 par of the
5.5% fixed rate bond is 448.76, while its
convexity is 4.49.
• This means the dollar convexity of the
inverse floater paying 11%-floating is 2 x
448.76 - 47.34 = 850.18, which gives us a
convexity of 850.18/100.0038 = 8.50.