Salomon Brothers

Raymond J. Iwanowski
(212) 783-6127
riwanowski@zip.sbi.com An Investor’s Guide to
Floating-Rate Notes:
Conventions, Mathematics
and Relative Valuation
TABLE OF C O N T E N T S PAGE

Introduction 1
Floating-Rate Note Pricing: Equations and Definitions 2
Price Sensitivities 7
Relationship Between Discount Margins on Floaters and Fixed Corporate Spreads 9
A Case Study: Eurodollar Perpetual Floaters 13
Comparing Floating Rate Notes of Different Indices: Basis Risk 17
Optionality 27
Conclusion 31

FIGURES

1. Prices Under Various Discount Margins and Assumed Index Levels 6

2. Corporate Floating-Rate Note Indication Levels, as of 10 Mar 95 10
3. Swap Spread Indication Levels — Fixed Rate versus Three-Month LIBOR as of 10 Mar 95 10
4. Net Cash Flow of Swapped Corporate Bond 10
5. Net Cash Flow of Buying a Floater and a Series of Eurodollar Futures Contracts 12
6. Prices on Eurodollar Futures contracts, as of 10 Mar 95 12
7. Treasury and Swap Rates, as of 7 Jul 95 15
8. Net Cash Flow of Swapped Perpetual Floater 15
9. Yield Spread of Swapped Perpetual over "Old" Thirty-year Treasury 16
10. Current Coupon Comparison of Corporate Floating-Rate Notes, as of 10 Mar 95 17
11. Net Cash Flows of LIB3 Floating-rate Note and a LIB3-Fed Funds Basis Swap 19
12. Comparison of Forward CMT2 to Forward LIB3, 10 Mar 95 20
13. Comparison of Average Spreads of Various Indices 22
14. Average Daily Spreads to Three-Month T-Bill for Various Time Periods 22
15. Correlations Between One Period Changes in Various Indices, Mar 82-Jan 95 23
16. Comparison of Prime Rate Changes to Fed Funds Changes in Different Interest Rate
Environments 25
17. Cash Flow of Portfolio Consisting of an Unfloored Floating-rate Note and a Long Floor 27

September 1995 Salomon Brothers

I N T R O D U C T I O N

Virtually every sector of the fixed-income market contains bonds with

floating-rate coupons. The popularity of these securities arises from the
reluctance of many investors to accept a long-maturity fixed bond,
particularly in periods when market participants fear rising interest rates.
Consider a hypothetical bond that is issued at a price of 100, accrues a
coupon of some easily observed index plus some prespecified spread
(which resets to the new index level whenever the index level changes
[instantaneous reset]), has no credit risk or embedded options, and the
timing of the payment of principal is certain. In this case, the intuition that
floating-rate notes have no price sensitivity and should always trade at or
near 100 is valid.
Of course, such bonds do not exist in today’s markets, and therefore, all
floating-rate securities (FRN) do have some degree of price sensitivity. In
reality, no floating-rate securities reset instantaneously, and there typically
is a lag before the index rate used in coupon accrual resets to the market
index rate. Corporate and emerging market floaters have exposure to
changes in credit spreads. Floating-rate mortgage-backed securities and
many products in the structured note market usually have embedded
options such as caps and floors. Mortgage-backed FRNs have the
additional feature of the prepayment option which affects the amortization
of principal. In addition to the usual effect that faster or slower
prepayments have on the performance of a mortgage bond, unexpected
prepayment speeds will affect the value of the embedded caps and floors.
Investors who purchase FRNs are usually aware of these issues. However,
it is not always easy to evaluate the implied risks and the appropriate
compensation for taking on these risks. In fact, given the nature of
floating-rate securities, it is not trivial to measure the compensation that
the investor is receiving for taking risks. A framework by which to
measure risk-return trade-offs is particularly important for investors that
have the flexibility to invest in FRNs across several different sectors.
The purpose of this research is twofold. First, we provide a reference that
describes in detail the various conventions and tools used in FRN markets.
Next, we discuss the relative valuation of different types of FRNs and
provide some frameworks by which to assess their risk/return trade-offs.
The text is structured as follows:
• In the first section, we explicitly define pricing relationships and
commonly used terms such as discount margin and reset margin. One
particularly useful formulation of the standard pricing equation allows us to
decompose any FRN into a par floater and a fixed annuity. We provide
examples that use the formulas and illustrate the sensitivities of these
measures to pricing assumptions (pages 2-6).
• We then describe various sensitivity measures such as effective duration,
partial duration and spread duration and compare these measures to those
of fixed-coupon bonds (pages 7-9).
• Next, we discuss the relationship between FRNs, fixed corporate bonds
and derivatives markets (pages 9-13).
• In the following section, we illustrate the various concepts discussed
thus far with an example using Eurodollar perpetual floating-rate notes
(pages 13-16).

Salomon Brothers September 1995 1

 • We then address the relationship between FRNs of different indices.
Topics discussed are: (1) the reasons for buying FRNs of a particular
index; (2) the use of derivatives markets to evaluate relative FRN values of
different indices; (3) some statistical methods to assess the propensity of
these indices to move together; and (4) an example that uses statistics to
evaluate "conventional wisdom" of the relationship between prime and
Treasury rates (pages 16-25).
• Finally, we discuss the valuation of embedded optionality in FRNs. We
address this topic with respect to two types of securities: floored-corporate
FRNs and floating-rate mortgage securities (pages 25-29).

FLOATING-RATE N O T E PRICING: EQUATIONS AND DEFINITIONS

In this section, we explicitly define the various pricing equations and

market conventions used in floating-rate security markets. This section is
designed to serve as a reference, although we will provide examples to
make the exposition more lucid. Readers who are already familiar with
these concepts should skip this section; however, we will refer to the
equations shown here in the subsequent text.
Consider the following bullet corporate floating-rate note (FRN) which
matures at the end of period T and pays a coupon which resets every
period to a particular rate, It, plus a specified spread called the reset
margin (RM). The note is noncallable and nonputable. The principal
amount is $100. For ease of explanation, we assume that the pricing date is
the reset date.
If I1,..., IT-1 were known at time 0, the pricing relationship of the floater
would be as follows:

P(0,T) =
 I0 + RM I + RM I + RM + 1 
100 * (1 + d(1) + S) + (1 +1d(2) + S)2 +...+ (1T−1+ d(T) + S)T ,

T I0 + RM 1 
= 100 * ∑(1 + d(i) + S) + (1 + d(T) + S) 
i=1
i T
(1)

where P(t,T) is the price of a floater with T periods until maturity observed
at period t,

RM = the reset margin. This margin is fixed at issuance,

d(t) = the discount rate of a risk-free $1 paid at time t, and
S = the discount spread of the corporate issuer.

Floating-rate notes typically are "set in advance" which means that the
index level that is applied to the coupon at time t is the observed index at
time t-1 (It−1).
Because I1,..., IT-1 are not observable at time 0, a projected coupon must be
substituted. We denote the series of projected coupons as X1,..., XT-1 (we
will discuss the determinants of the X’s later).

2 September 1995 Salomon Brothers

 The pricing equation is then modified to:

T X + RM 1 
∑(1 + d(i) + S)i + (1 + d(T) + S)T
i−1
P(0,T) = 100 *
 i=1  (2)

where X0 = I0 if pricing date is a reset date. For ease of explanation, we

make this assumption going forward.
The trading convention is to quote FRN prices in terms of a discount
margin (DM) which is analogous to the yield spreads on fixed corporate
bonds. The discount margin is defined as the spread over the appropriate
index that equates the price of the FRN to the present value of the
projected cash flows. In other words, the DM is defined as follows:

(1 + d(i) + S)i = (1 + I0 + DM)(1 + X1 + DM)...(1 + Xi−1+ DM),

for i = 1, ..., T. (3)

Therefore, Equation (2) can be rewritten in terms of the discount margin as

follows:

 I0 + RM X1 + RM
P(0,T) = 100 *  +
(1 + I0 + DM) (1 + I0 + DM)(1 + X1 + DM)

1 + XT−1 + RM 
+...+
(1 + I0 + DM)(1 + X1 + DM)...(1 + XT−1 + DM) 
(4)

Equation (4) is simply the standard pricing equation with the appropriate
discount rate for the issuer (d(i) + S) restated in terms of the projected
coupon indices (I0, X1,..., XT−1).
Example
Consider a new issue FRN that is indexed to three-month LIBOR (LIB3) +
20 basis points, has a maturity of two years, pays and resets quarterly, and
is priced at par at time 0 (P(0,T)=100). Suppose that on the pricing date,
LIB3=6.06% and for now, we make the assumption that LIB31= LIB32 =
... = LIB3T-1 = 6.06% and T=8 quarters.1 LIB3 FRNs typically pay on an
actual/360 basis and, for simplicity of this example, we assume that each
quarter has 91 days. Therefore, the quarterly coupon in our example is:

Coupont = 100 * (LIB3t-1+20 bp)*(91/360)

= 100 * (0.0606 + 0.0020)(91/360) = 1.58

1 We will discuss the use of alternative assumptions later in the report.

Salomon Brothers September 1995 3

 Note that the index and discount margin in the denominator are accrued on
the same basis as the coupon. We can rewrite Equation (4) as:

 (0.0606 + 0.0020) ∗ (91/360) +

(0.0606 + 0.0020) ∗ (91/360)
100 = 100 * (1 + ((0.0606 + DM) ∗ (91/360))) (1 + ((0.0606 + DM)(1 + 0.0606 + DM)(91/360)(91/360)))

1 + ((0.0606 + 0.0020) ∗ (91/360)) 

(1 + ((0.0606 + DM) ∗ (91/360)))(1 + ((0.0606 + DM) ∗ (91/360)))...(1 + ((0.0606 + DM) ∗ (91/360)))
+...+
(5)

The discount margin is the value of DM that solves Equation (5). In this
example, since the bond is priced at par, it is clear that the solution is
DM=0.0020=20 basis points. In fact, whenever a floater is priced at par,
RM=DM. However, if the price were to drop instantaneously to 99.75,
then substituting by 99.75 into the left hand side of Equation (5), we
obtain DM=33 basis points. The following rule applies to FRNs:
• If RM > DM, then Price > 100
• If RM < DM, then P < 100
• If RM = DM, then P = 100
The pricing formulas described above help to answer the following
questions:
• We assumed that future levels of LIB3 are equal to today’s level. This
is the convention in U.S. corporate, mortgage and asset-backed markets.
For a given DM, how does the calculated price change under different
assumed coupon levels? What is the appropriate assumption to use?
• What is the sensitivity of the FRN’s price to changes in the Treasury
yield curve (effective and partial durations)? What is the price sensitivity
of the FRN with respect to changes in DM (spread duration)?
• How are the reset margins of new issues or the discount margins of
secondary-market issues determined? How should the RMs and DMs differ
across indices (for example, prime versus LIB3) for the same issue?
In the following analysis, we provide some answers to these questions. In
evaluating these issues, it is useful to rewrite Equation (4) by adding
DM-DM=0 to each coupon:

I0 + RM + (DM − DM) X1 + RM + (DM − DM)

P(0,T) = 100 *  (1 + I0 + DM)
+
(1 + I0 + DM)(1 + X1 + DM)

1 + XT−1 + RM + (DM − DM) 

+...+
(1 + I0 + DM)(1 + X1 + DM)...(1 + XT−1 + DM) 
(6)

4 September 1995 Salomon Brothers

 Recombining terms in Equation (6) yields the following:

P(0,T) =

 I0 +DM X1 + DM 1 + XT−1 + DM 
100 * 
 (1+ I 0 + DM)
+
(1 + I 0 + DM)(1 + X 1 + DM)
+...+
(1 + I 0 + DM)(1 + X 1 + DM) ...(1 + X T−1 + DM)
+

 RM − DM RM − DM RM − DM 
(1 + I0 + DM) + (1 + I0 + DM)(1 + X1 + DM) +...+ (1 + I0 + DM)(1 + X1 + DM) ...(1 + XT−1 + DM)  (7)

Although at first glance, Equation (7) may appear to be an unnecessarily

long restatement of the pricing relationship, it actually provides some
valuable intuition in understanding FRNs. The first term in the brackets
represents a FRN whose reset margin is equal to its discount margin.
Therefore, the value of this quantity is par. The second term is the present
value of an annuity that pays the difference between the reset margin and
the discount margin. Therefore, Equation (7) simplifies into the following
expression:

P(0,T) = 100 + present value of an annuity that pays (RM − DM) (8)

The first payoff of viewing the pricing equation in this way is to confirm
the rules of premiums and discounts that were described above. If RM =
DM, the second term of Equation (8) is zero and the bond is priced at par.
If RM is greater than DM, the second term is positive and the price of the
floater is greater than par (premium). If RM is less than DM, the second
term is negative and the price of the floater is less than par (discount).
Sensitivity of Discount Margin Calculations to Assumptions
Equations (7) and (8) also provide a framework in which to evaluate the
questions posed earlier. Recall that the first question was: For a given DM,
how does the calculated price change under different assumed index
levels? We use an example to illustrate the answer to this question.
Example
Consider the following corporate FRN:
• Index — LIB3
• Reset Margin — 100 basis points
• Reset/Pay Frequency — Quarterly/Quarterly
• Maturity — 5 years
• For simplicity, assume that the settlement date is a coupon date.
Suppose we were to consider two sets of index projections:
(1) LIB30 = LIB31 = LIB3T-1 = 6.31%; and
(2) LIB3t = LIB3t-1 + 25bps.
(i.e., LIB30 = 6.31%, LIB31 = 6.56%,..., LIB3t-1 = 10.81%).
Case (2) was chosen arbitrarily to illustrate the differences across
assumptions, but these projections are similar to the forward LIB3 rates
implied by the swap curve in a sharply upward-sloping yield curve
environment.

Salomon Brothers September 1995 5

 We substitute the projections, parameters and various discount margins into
Equation (7) and solve for price. The results are shown in Figure 1.

Figure 1. Prices Under Various Discount Margins and Assumed Index Levels
Discount Assumed Index Projection
Margin (1) All LIB3 remain at current level (6.31%) (2) LIB3 increase 25bps each quarter
200 95.87 96.02
100 100.00 100.00
0 104.34 104.18

The calculated price is:

• the same under either assumption when the discount margin is equal to
the reset margin;
• greater under the rising index assumption when the discount margin
exceeds the reset margin (discount); and
• lower under the rising index assumption when the reset margin exceeds
the discount margin (premium).
The terms under the first set of brackets in Equation (7) will equal 100
regardless of the assumed index level. This is true because the discount
rates in the denominator of each term will exactly offset any increase or
decrease in assumed coupon. However, the values of the terms under the
second bracket will change under various assumptions of the index level.
Specifically, a higher assumed level of the index will increase the rate at
which the fixed annuity is discounted. We can use Equation (8) to define
the price premium as follows:

Premium = P(0,T) - 100 = Present Value of Annuity which pays (RM-DM)

 RM − DM RM − DM RM − DM 
= 100 * (1 + I0 + DM) + (1 + I0 + DM)(1 + X1 + DM) +...+ (1 + I0 + DM)(1 + X1 + DM)...(1 + Xt−1 + DM) (9)

As the projected index levels increase, the value of the premium decreases.
Because a discount can be viewed as a negative premium, increasing the
projected indices results in a lower discount and a higher price. This
example demonstrates that the price of the bond for a given DM is clearly
sensitive to the projected index level except for the special case in which
RM=DM. The longer the maturity of the bond and the greater the
difference between RM and DM, the more pronounced the differential
between coupon projection methods. This example illustrates that the
discount margin, although a convenient tool by which to quote prices on
FRNs, is a flawed measure of relative value.

6 September 1995 Salomon Brothers

PRICE SENSITIVITIES

Equations (7) and (8) also provide insight into our second question: What
is the sensitivity of the FRN’s price to changes in the Treasury yield
curve?
Effective Duration
Effective duration is commonly defined as the sensitivity of a bond’s price
to a parallel shift of the Treasury yield curve, keeping yield spreads
constant. If the spread between the index rate and Treasury rate is held
constant, this definition is equivalent to saying that effective duration
measures the sensitivity of the price of the FRN to changes in the level of
its index. Henceforth, we use these two definitions interchangeably.
Consider two cases:
• Case (1). Suppose P(0,T)=100. Then RM=DM and the annuity shown in
Equation (7) is equal to zero. Now, suppose I0 instantaneously increased
by a small amount to I0 + ∆I. After the next reset, coupons are increased
to reflect the higher level of I. Equation (7) will now be written as:

 I0 + DM T
 X1 + RM + ∆I + DM
New P = 100 *
(1 + I + ∆I + DM)
0
+∑

i=2 (1 + I
0 + ∆I + DM)...(1 + Xi−1 + DM)

1   1 + I0 + DM 
+...+
(1 + I0 + ∆I + DM)...(1 + XT + ∆I + DM) = 100 *
1 + I
0 + ∆I + DM .
(10)

The terms sum in this manner because the value at time 1 of all
subsequent projected cash flows is equal to par. Therefore, the effective
duration of a noncallable, nonputable bullet FRN is the modified duration
of a bond that matures at the next reset date.
Example
Recall the example from the previous section:
• Index — LIB3
• Reset Margin — 100 basis points
• Reset/Pay Frequency — Quarterly/Quarterly
• Maturity — 5 years
• Settlement date — Reset date.
Suppose that this FRN is priced at par.
If the index level increases by ten basis points instantaneously
(∆I = 0.0010), the new price is given by Equation (10) as follows:

1 + ((0.0631 + 0.01)(91/360))
New P = 100 *   = 99.9752.

1 + ((0.0631 + 0.001 + 0.01)(91/360)) (11)

Salomon Brothers September 1995 7

 Similarly, a ten-basis-point decrease in the index (∆I = 0.0010) gives us a
new price of 100.0243. Therefore, the average price change for a
ten-basis-point shift in the index level is equal to (100.0243 - 99.9752)/2 =
0.0248 = 0.0248% of purchase price. Since effective duration is defined to
be the price change for a 100-basis-point change in yields,
Effective Duration = 0.0248% x 10 = 0.248%.
Note that this is the modified duration of a zero coupon bond which
matures in 91 days and has a yield of 1.85% (0.0731)(91/360).
• Case (2). RM ≠ DM (the FRN is trading at a discount or a premium).
From Equation (7), we can view this bond as a "portfolio" of a long
position in a par floater and a long position in an annuity paying RM-DM
(or a short position in an annuity paying DM-RM if a discount). As shown
in Case (1), the long floater will have an effective duration equal to the
modified duration of a bond that matures at its next reset. The annuity
position will have a longer duration. In the case of a discount, the annuity
piece will reduce the effective duration of the "portfolio." If the annuity is
a significant enough portion of the "portfolio," the effective duration of the
discount floater actually may be negative. Unfortunately, in this case, the
method used to project the coupons will affect the effective duration
because the effective duration of the annuity piece will differ for different
index levels.
Example
Suppose the LIB3 floater described above had a dollar price of 85. Using
the portfolio approach, the floater can be decomposed into a long position
in a FRN priced at 100 and a short position in an annuity worth 15. The
present value of that annuity is given by the terms under the second set of
brackets in equation (7). We assume that all subsequent levels of LIB3 are
equal to the current level of 6.31%, and we can solve for the DM which
sets that annuity value equal to -15.2 In this case, the DM equals 492 basis
points. We have shown earlier that the effective duration of the par floater
piece is equal to 0.248 years. The effective duration of an annuity that
pays DM-RM (392 basis points) for five years is equal to 2.367 years.3
Using the portfolio approach and the fact that the effective duration of a
portfolio is equal to the market-value-weighted average of the components
of the portfolio, we can obtain the effective duration of the discount FRN
as follows:
Effective Duration = (100/85)*(0.25) - (15/85)*(2.37) = -0.12 years.
Partial Durations
A careful, quantitative analysis of partial durations requires detail that is
beyond the scope of this paper. However, several intuitive points can be
made from our simple pricing equations:
• If the coupon of the FRN is indexed to a short rate and the bond is
priced at par, then from Equation (10), the price will be sensitive only to
movements in the index, and other partial durations will be zero.
• If the bond is at a premium or a discount, we can take the "portfolio"
approach discussed above. Shifts of intermediate rates will affect the price
of the bond to the extent that the annuity in the portfolio is affected.

2 A short position in a portfolio is indicated by a negative present value.

3 Note that the duration of the annuity is dependent upon the discount rate and time to maturity but independent of the
size of the annuity payment.

8 September 1995 Salomon Brothers

 • Some FRNs are indexed to a longer maturity rate such as the two-year
constant maturity Treasury (CMT2). Even if they are priced at par, such
floaters are sensitive to shifts of various portions of the curve because their
prices are sensitive to shifts in the longer forward rates.
Spread Duration
We define spread duration as the price sensitivity of a FRN with respect
to changes in its DM. To derive some intuition of the spread duration, it is
easiest to consider equation (4). Recall that Equation (4) is given by the
following:

 I0 + RM X1 + RM
P (0,T) = 100 *  +
(1 + I0 + DM) (1 + I0 + DM)(1 + X1 + DM)
+...+

1 + XT−1 + RM 
(1 + I0 + DM)(1 + X1 + DM)...(1 + XT−1 + DM) 
(4)

It is clear from Equation (4) that any increase in DM affects the

denominator of Equation (4) but does not increase the numerator. In fact,
increasing the DM of a floater will have the same price effect as increasing
the yield on a fixed bond that pays the projected coupons out to the
maturity of the floater. Therefore, a longer-maturity FRN will have a
spread duration equal to the modified duration of a fixed bond paying the
projected coupons out to the maturity of the floater. The method used to
project the coupons will also affect this measure.
Example
The five-year LIB3 floater used in our previous examples has a spread
duration of 4.15 years when priced at par and the future levels of LIB3 are
assumed to be the current level (6.31%). This is the modified duration of a
bond that pays a quarterly coupon of 1.83% (7.31/4%) for five years.

RELATIONSHIP BETWEEN D I S C O U N T MARGINS ON FLOATERS AND

FIXED CORPORATE SPREADS

Figure 2 shows the discount margins of various types of corporate FRNs

observed on March 10, 1995. One immediate observation from Figure 2 is
that discount margins increase as the maturity of the FRN increases (term
structure). In the previous section, we showed that for a given reset
frequency, FRNs priced at par will have the same effective duration
regardless of their maturity. However, longer-maturity floaters have longer
spread durations. Therefore, at least a portion of the term structure of DMs
for a given issuer likely represents the compensation for taking on spread
duration risk. We now discuss how this term structure of DM is
determined by focusing on FRNs indexed to LIB3.

Salomon Brothers September 1995 9

 Figure 2. Corporate Floating-Rate Note Indication Levels, as of 10 Mar 95 (Quality: A2/A)
Discount Margin (bp)
Index
Maturity LIB3 TB3 Prime Fed Funds CMT2 CP1
One-Year +0 +23 -255 +25 +10 +5
Two-Year +10 +33 -250 +28 +18 +8
Five-Year +25 +55 -240 +30 +25 +15
Index Level
LIB3 6.31%
TB3 5.93
Prime 9.00
Fed Funds 5.93
CMT2 6.82
CP1 6.08
bp Basis points. LIB3 Three-month LIBOR. CMT2 Two-year Constant Maturity Treasury. CP1 One-month Commercial
Paper. TB3 Three-month Treasury Bill Yield.
Note: All rates quoted in CD equivalent except CMT2, which is quoted in bond equivalent.
Source: Salomon Brothers Inc.

Three-Month LIBOR (LIB3)-Indexed Floating-Rate Notes

Consider the following transaction:
• Purchase a newly issued fixed corporate bond with a maturity of five
years which is priced at 100 with a coupon of the five-year Treasury
(TSY50) plus spread (CS).
• Enter into a five-year interest rate swap in which the investor pays a
fixed-rate plus a swap spread (SS) and receives the three-month LIBOR
observed at reset date (LIB3t-1).4 Figure 3 shows the market levels of swap
spreads observed on March 10, 1995.5 Figure 4 shows the net cash flows
of this transaction.

Figure 3. Swap Spread Indication Levels — Fixed-rate versus Three-Month LIBOR,

as of 10 Mar 95
Spreads to Treasury (bps)
Maturity Receive Fixed Pay Fixed
One Year +30 +33
Two Year +26 +29
Three Year +26 +29
Four Year +26 +29
Five Year +25 +28
bp Basis points.
Source: Salomon Brothers Inc.

Figure 4. Net Cash Flow of Swapped Corporate Bond

Transaction Cash Flow at Quarter t=1 Cash Flow at Quarter t=2
Buy Fixed bond (TSY50+CS)/2
Interest Rate Swap
Receive Float +LIB30*Act/360 +LIB31*(Act/360)
Pay Fixed -(TSY50+SS)/2
Net Cash Flow +LIB30*Act/360 (LIB31*Act/360)+(CS-SS)/2
bp Basis points. LIB3 Three-month LIBOR. TSY5 Five-year Treasury yield.
Source: Salomon Brothers Inc.

4 Swaps are conventionally "set in advance," which means that the time t floating-rate payment is LIB3 observed at
time t-1.
5 Actual swap rates may deviate from the indication levels shown in Figure 3 because of a differential in credit risk
for the counterparties. See Pricing of Interest Rate Swap Default Risk, Eric H. Sorenson and Thierry F. Bollier,
Salomon Brothers Inc., October 1993 for a discussion on the determinants of this adjustment.

10 September 1995 Salomon Brothers

 The net cash flow approximates the cash flow on a LIBOR floater where
Reset margin (RM)=spread on fixed corporate bond-swap spread = CS-SS.
The swapped fixed bond differs from a FRN only by slight differences in
timing of cash flows because the synthetic will receive its "reset margin"
semiannually and the FRN will receive the reset margin quarterly.
Therefore, the fixed corporate bond and swap markets provide a good "rule
of thumb" as to where LIB3 floaters should trade. For corporate issuers
with similar credit quality to that implied by the swap market, this rule of
thumb should hold fairly closely. However, there likely will be some
deviation when the credit quality of the corporate issuer differs
substantially from the credit quality of the swap counterparty.6 Some
theoretical models of credit spreads provide analytic solutions for the price
of a FRN, thereby quantifying the difference between swapped fixed bonds
and corporate FRNs. However, a thorough discussion of these models is
beyond the scope of this analysis.7

Does this relationship On March 10, 1995, the following sets of prices were observed in the
hold in practice? market:

Yield Spread (bps)

(a) GMAC (Baa1/BBB+) 6.35% of 6/28/98= +68/Actual
Swap spreads: pay fixed-three year = +29
four year= +29
Corporate spread-swap spread= 68bps - 29bps = 39bps.
A GMAC FRN with a coupon of LIB3 plus 25 basis points and maturing
on 2/2/98 was trading at a discount margin of LIB3 plus 37.5 basis points.

Yield Spread (bps)

(b) Nations Bank (A2/A) 7.50% of 2/15/97= +40/Actual
Swap spreads: pay fixed two year= +29
Corporate spread - swap spread = 40bps - 29bps = 11 bps.
Nations Bank FRN with a coupon of LIB3 plus six basis points and
maturing in 11/96 was trading at a discount margin of 12 basis points.
In these two cases, the relationship between swapped fixed corporate bonds
and floaters held quite closely.
An Alternative Specification
Investors that are familiar with the swap market know that a close
relationship exists between the fixed swap rate and the yield on a strip of
Eurodollar futures contracts. In this analysis we will not provide a detailed
description of hedging swaps with Eurodollar futures contracts but will
briefly discuss the relationship between corporate floaters and Eurodollar
futures.8

6 For an extreme example, see Brady Bond Fixed-Floating Spreads — Forget History, Vincent J. Palermo, et al.,
Salomon Brothers Inc, December 15, 1993.
7 See "A Simple Approach to Valuing Risky Fixed and Floating-Rate Debt," Francis A. Longstaff and Eduardo S.
Schwartz, The Journal of Finance, July 1995, and "A Probabilistic Approach to the Valuation of General Floating-Rate
Notes with an Application to Interest Rate Swaps," Nicole El-Karoui and Helyette Geman, Advances in Futures and
Options Research, volume 7, JAI Press Inc., 1994.
8 For a detailed discussion of hedging swaps with Eurodollar futures, see Eurodollar Futures and Options, Galen
Burghardt, et al., Probus Publishing Company, 1991. For a discussion on the effect of convexity on the relative pricing
between swaps and Eurodollar futures, see "A Question of Bias," Galen Burghardt and Bill Hoskins, Risk, March
1995.

Salomon Brothers September 1995 11

 Suppose we entered into the following transactions:
• Purchase a newly issued floating-rate note with a maturity of five years
that is priced at 100 with a quarterly coupon of LIB3 plus a reset margin
(RM).
• Buy the series of Eurodollar futures contracts maturing each quarter
until the maturity of the FRN. The notional amount of the contracts equal
the par amount of the bond.
If a Eurodollar futures contract that matures at time t is held to maturity,
the payoff at time t is:
Ft,t − F0,t = 100 − LIB3t − [100 − FLIB30,t].
where Ft,T = the time t price of a Eurodollar futures contract that matures
at time T.
FLIB3t,T equals the yield on a T-period Eurodollar futures contract
observed at time t. This rate will be closely related to the forward LIB3t
but will differ slightly because of the "mark-to-market" feature of futures
contracts. Henceforth, we will use the terms "futures yield" and "forward
rate" interchangeably.
The net cash flows of these transactions can be approximated as shown in
Figure 5.

Figure 5. Net Cash Flow of Buying a Floater and a Series of Eurodollar Futures Contracts
Transaction Cash Flow at Any Quarter t
Buy Floater (LIB3t-1 + RM)*(Days/360)
Buy Futures Contract (Forward LIB3t-1 - LIB3t-1)/4
Net Cash flow Forward LIB3t-1 + RM + ε
ε The difference in interest as a result of day count conventions.

The description of the cash flows in Figure 5 is slightly imprecise because

of the following: (1) the maturity date on the contracts will not exactly
match the coupon dates of the bond; (2) the bond coupon is set in advance
to LIB3t-1 but the cash flow of the futures contract takes place at t-1; and
(3) the mark-to-market feature of futures contracts.
We essentially have created a synthetic fixed bond that pays the forward
rates plus a reset margin using prices that are observed at time 0.
Therefore, the synthetic should trade at a similar yield spread as actual
fixed bonds of the same issuer. Note that the partial durations are not
matched to those of a fixed-coupon-paying bond of the same maturity. If
the Eurodollar curve is upward sloping, the synthetic will have lower
coupons in the early quarters and higher coupons in later quarters.

Does this relationship

hold in practice? Figure 6. Eurodollar Futures Contract Prices, 10 Mar 95
Maturity Price Yield
6/19/95 $93.43 6.57%
9/18/95 93.20 6.80
12/18/95 92.98 7.02
3/18/96 92.91 7.09
6/17/96 92.84 7.16
9/16/96 92.80 7.20
12/16/96 92.72 7.28
3/17/97 92.76 7.24
6/16/97 92.72 7.28
9/15/97 92.70 7.30
12/15/97 92.63 7.37
3/16/98 92.64 7.36
Source: Salomon Brothers Inc.

12 September 1995 Salomon Brothers

 Recall from the earlier example that a secondary-market GMAC FRN with
approximately three years to maturity traded at a discount margin of 37.5
basis points. At the same time, three-year fixed GMAC bonds were trading
at a yield spread of 68 basis points.
Suppose GMAC issued a new floating-rate note with a dated date of
3/13/95 priced at 100 with a reset margin of 37.5 basis points. We test
how closely the theoretical relationship described above holds by
substituting the futures rate shown in Figure 6 for the appropriate coupon
each quarter and allowing the interest to accrue on an Actual/360 basis.
The calculated yield on this synthetic fixed bond was 7.57%, equal to the
three-year Treasury + 64 basis points.
Given the relationship between FRNs, swap markets and futures contracts,
one may argue that the appropriate method of projecting the coupons on
FRN is to the forward rates because an investor can effectively fix his
coupons to the forwards. Nonetheless, the convention in U.S. corporate,
mortgage and asset-backed markets has always been to use current LIBOR
levels as the projected coupon in calculating DM. It is likely that this
convention has persisted because of its simplicity and because of the
difficulty in practice of creating arbitrage opportunities from the
mispricings. If the market prices FRNs using a fixed index, then through
transactions like the previous two examples, investors occasionally can
synthetically create a "cheap" fixed corporate bond. However, it is
extremely unlikely that a fixed corporate bond of the same issuer, coupon
and maturity exists which an investor can costlessly short sell, and thus
collect arbitrage profits. Furthermore, as we saw in Figure 1, in most cases,
the differences in methodology will be small.

A CASE S T U D Y : EURODOLLAR PERPETUAL FLOATERS

One application that nicely demonstrates the points made in the previous
two sections is the case of perpetual FRNs, which are common in the
Eurobond market. A typical perpetual FRN will represent a promise to pay
the level of an index plus a prespecified spread (RM) forever. These
securities usually reset and pay frequently (for example, every three
months). In the mid-1980s, banks issued many of these secirities in the
Eurobond market at par. A 1984 article that discussed the popularity of
FRNs states, "Since an FRN coupon is reset to market levels every three or
six months, Eurodollar FRN investments are similar to rolling over funds
in the certificate of deposits market."9 A portfolio manager quoted in the
same article considered a floater portfolio as "sort of an insurance policy
against rising rates."10
One typical A2/A-rated perpetual issue has a coupon of six-month LIBOR
(LIB6) + 12.5 basis points, reset and paid quarterly. Most of the perpetual
issues are callable and some have floors, but for the purpose of this
example, we assume that these securities do not contain embedded options.
Like most bonds in this market, this issue currently trades at a substantial
discount. In this case, the bond has an offer price of 80.5.11 If floaters are
indeed an "insurance policy" or a similar investment to rolling CDs, what
caused such drastic price depreciation?

9 "The FRN Dilemma," Institutional Investor — International Edition, April 1984, pp. 215-218.
10 "The FRN Dilemma."
11 As of July 7, 1995.

Salomon Brothers September 1995 13

 For the purpose of this example, suppose that this security was issued at
par on August 1, 1985. LIB6 on this date was 8.25%. We can rewrite the
pricing Equation (4) for a perpetual as follows:

∞ Xi−1 + RM 
P = 100 * ∑(1 + I
 i=1 0 + DM)...(1 + Xt−1 + DM) (13)

Suppose we project the index to some constant level (Xi=I for all i). Then,
using the solution for an infinite geometric series, we obtain the following
analytic solution for Equation (13):

I + RM
Price of perpetual = 100 *
I + DM (14)

Equation (14) is consistent with our previous pricing expressions in the

sense that if RM=DM, then the price will equal 100 irrespective of the
projected index.
Despite the extremely long maturity (infinite), Equation (10) is still
appropriate to evaluate the effective duration of a perpetual FRN priced at
par. Therefore, the effective duration on August 1, 1985, was slightly less
than 0.25 years. In this sense, the security is a similar investment to rolling
certificates of deposit.
Yet, now consider the spread duration of these bonds. By differentiating
Equation (14) with respect to DM and dividing by the price, we obtain the
following expression:

1
Spread duration of perpetual FRN =
I + DM (15)

Using the August 1, 1985, LIB6 rate of 8.563% as the projected coupon,
Equation (15) becomes:

Spread duration of perpetual FRN =

1
= 11.51 years.
0.08563 + 0.00125 (16)

The long spread duration is the major risk of the perpetual FRN relative to
traditional short-term strategies, such as rolling CDs or buying short-dated
floaters. If LIB6 had increased by 100 basis points on August 1, 1985, the
perpetual would have experienced a price depreciation of only
approximately $0.25 since its effective duration is roughly 0.25 years.
However, an increase in the market discount margin of 100 basis points
because of either deteriorating credit or technical factors would have
resulted in a price depreciation of $11.51. In the case of perpetual floaters,
spreads widened and to the surprise of some investors, the price was quite
sensitive to it.

14 September 1995 Salomon Brothers

 The perpetual also provides a nice example of how an investor can use the
swap market in assessing relative value in the FRN market. Figure 7 shows
the Treasury and swap rates that were observed on the date that the
perpetual floater in our example was priced.

Figure 7. Treasury and Swap Rates, 7 Jul 95

Maturity Treasury Yield Swap Spreads Fixed Swap Rate
10 year 6.08% 42bp 6.50%
30 year 6.55 45 7.00
bp Basis points.
Source: Salomon Brothers Inc.

Suppose an investor purchased the perpetual floater at 80.50 on August 1,

1995, and entered into a thirty-year swap in which he pays LIB6 and
receives the fixed swap rate.12 The net cash flows for 30 years is given in
Figure 8.

Figure 8. Net Cash Flow of Swapped Perpetual Floater

Transaction Cash Flow at Time t
Buy Perpetual Floater +(LIB6t-1+0.125) * (Act/360)
Interest Rate Swap
Pay Float -LIB6t-1 * (Act/360)
Receive Fixed (TSY300+SS)/2
Net Cash Flow (TSY0+SS)/2+(0.125*(Act/360))
Act Actual number of days. LIB6 Six-month LIBOR. SS Swap spread. TSY30 30-year Treasury yield.

Therefore, the investor has locked in a fixed semiannual coupon of

approximately 7.00% + 0.125% = 7.125% for 30 years at a dollar price of
80.50. The synthetic differs from a fixed corporate bond in the sense that,
after 30 years, the bondholder does not receive principal but rather still
owns a perpetual stream of floating-rate cash flows. The following
methodology can be used to calculate the present value of this stream:
(1) Assume a LIB6 (LIB6') for the coupons from year 30 onward;
(2) Use the perpetual pricing formula in Equation (14) to obtain a value of
these cash flows at the end of year 30;
(3) Discount this value back to today.
This methodology allows us to obtain the yield on the synthetic by solving
the following equation:

Price =

60
(TSY300 + Swap Spread + RMperpetual FRN)/2
∑ (1 + yield/2)i
+ PV (perpetual which pays LIB6′ + RM)
i=1

where PV(.) = Present Value (17)

12 Assume July 7, 1995 Treasury and swap rates are prevalent on that date. The floating rate on plain-vanilla swaps is
usually LIB3. The difference on the fixed rate between a swap which receives either LIB3 and a swap which receives
LIB6 is typically not more than a few basis points. For simplicity, we assume that these rates are the same.

Salomon Brothers September 1995 15

 Of course, this calculation depends on our assumption of LIB6′, but the
first cash flow for which the assumption is important is 30 years hence.
Many investors are reluctant to enter into a 30-year swap because of credit
and liquidity issues. We can also use Equation (17) to calculate the yield
of a synthetic that is swapped for ten years by substituting a term of 20
semi-annual periods rather than 60. In this case, the yield on the synthetic
is more sensitive to the assumptions of LIB6′ because the assumed level
enters the pricing relationship only ten years hence.
Figure 9 shows the spread over the "old" 30-year Treasury of the perpetual
floater swapped for ten and thirty years under various assumptions for
LIB6 in the "tail."13

Figure 9. Yield Spread of Swapped Perpetual over 30-Year Treasury

Assumed LIB6 After Yield Spread of Perpetual Yield Spread of Perpetual
Maturity of Swap Swapped for Thirty Years Swapped for Ten Years
10.00% 252bp 330bp
7.00 226 191
6.50 221 164
5.75 214 121
3.00 184 -82
bp Basis points. LIB6 Six-month LIBOR.
Source: Salomon Brothers Inc.

The historical average over the past ten years of LIB6 was 6.55%.
An investor can use Figure 9 to compare the spreads of the swapped
perpetual to the spreads of long corporate fixed bonds. For a perpetual that
is swapped for 30 years, even after imposing extremely conservative
assumptions to the unswapped cash flows, the spread does not change
much. Of course, because of lesser liquidity, an investor should expect the
synthetic to trade at wider spreads than a fixed issue. Figure 9 enables the
investor to assess how much he is being compensated for accepting lower
liquidity.

13 Corporate bonds are conventionally not spread off the newest issued 30-year Treasury but the previous issue. In this
case, we use the yield of the 7.50% of 11/24.

16 September 1995 Salomon Brothers

COMPARING FLOATING RATE NOTES OF DIFFERENT INDICES: BASIS RISK

Recall Figure 2 which shows the indication discount margins of generic

A-rated FRNs across various indices as of March 10, 1995. For the
reader’s convenience, we reproduce this table below. These levels also
approximate where the reset margin on a generic new issue FRN of
A-rated credit quality would be set.

Figure 2. Corporate Floating-Rate Note Indication Levels, as of 10 Mar 95 (Quality: A2/A)

Discount Margin (bp)
Index
Maturity LIB3 TB3 Prime Fed Funds CMT2 CP1
One-Year +0 +23 -255 +25 +10 +5
Two-Year +10 +33 -250 +28 +18 +8
Five-Year +25 +55 -240 +30 +25 +15
Index Level
LIB3 6.31%
TB3 5.93
Prime 9.00
Fed Funds 5.93
CMT2 6.82
CP1 6.08
bp Basis points. CMT2 Two-year Constant Maturity Treasury. CP1 One-month Commercial Paper. LIB3 Three-month
LIBOR. TB3 Three-month Treasury Bill yield.
Note: All rates quoted in CD equivalent except CMT2, which is quoted in bond equivalent.
Source: Salomon Brothers Inc.

A quick addition of the various index levels to the appropriate new issue
reset margins show that, for the same credit quality, the current coupon on
FRNs across indices and maturities are not the same. Figure 10 shows a
comparison of FRNs of the various indices to a LIB3 floater. The previous
sections illustrate that the coupons typically increase across maturities to
compensate the investor for taking on additional spread duration risk. It is
important to understand the reasons that these FRNs may trade at different
current coupons for the same maturity FRN.

Figure 10. Current Coupon Comparison of Corporate Floating-Rate Notes, as of 10 Mar 95 (Quality:
A2/A)
Current Coupon versus LIB3 Floater (bp)
Index
Maturity TB3a Prime Fed Funds CMT2a CP1b
One-Year -23 +14 -13 +52 -15
Two-Year -24 +9 -20 +50 -23
Five-Year -17 +4 -33 +42 -34
a Adjusted to reflect the difference in day count convention. b Adjusted to reflect the difference in pay frequency.
CMT2 Two-year Constant Maturity Treasury Yield. CP1 One-month commercial paper. LIB3 Three-month LIBOR.
TB3 Three-month Treasury bill yield.
Source: Salomon Brothers Inc.

Four reasons that an investor may purchase a floater with a lower initial
coupon are:
(1) To avoid risk of deviating from benchmark. Many short-term
investors, such as money market funds and securities lending accounts, buy
short maturity floaters and hold them until maturity. As long as the bond
does not default, these funds earn a return of the index plus the reset
margin. The objective of the managers of such funds is to earn a modest
excess return over some benchmark (often some rate such as LIB3) while
maintaining a very low tolerance for substantive underperformance over

Salomon Brothers September 1995 17

 any period of time. Given their extreme risk aversion, the natural choice
for such fund managers is to purchase floaters which match their
benchmarks unless the current coupon advantage is sufficiently enticing.
(2) Institutional restrictions. Some investors are prohibited from buying
floaters of certain indices.
(3) More frequent reset. In the previous section, we showed that the
effective duration of a FRN without options priced at par is approximated
by the time to the next reset. Therefore, a short-term investor who has an
outlook of immediate Fed tightening may prefer a daily reset Fed funds
FRN or a weekly reset TB3 FRN rather than one indexed to LIB3 that
resets quarterly, even if the current coupon is lower.
(4) Speculation on relative movements between rates. Many investors
may accept a lower coupon on an FRN to benefit from an expected change
in the relationship between two rates (basis). For the various FRN shown
in Figures 9 and 10, this basis arises for a number of reasons:
• LIB3-TB3. This is the spread between two indices with the same
maturity, but LIB3 will always be the higher rate because it reflects the
rate on Eurodollar deposits that are subject to credit risk and have less
liquidity than Treasury bills. This spread, which is variable through time, is
often known as the "TED spread."14
• CMT2-LIB3, LIB3-Fed Funds. These spreads have two components:
(1) the difference between a longer-maturity rate and a shorter-maturity
rate; and (2) the difference between a credit-risky rate and a Treasury rate.
Thus, in addition, to the TED spread, the CMT2-LIB3 spread reflects the
term structure of Treasury yields from the two-year to three-month part of
the curve. We must emphasize that the decision between choosing a CMT2
quarterly reset floater over a LIB3 quarterly reset floater does not
constitute a decision to extend effective duration because the effective
duration is driven by the reset frequency. In fact, as we shall discuss in
more detail later, the decision about which of these indices to invest in
reflects a view on the slope of the Treasury curve rather than the level.
Similarly, the LIB3-Fed funds basis reflects the slope of the Treasury curve
on the very short end (overnight to three months) as well as the TED
spread.
• CP1-LIB3, Prime-LIB3. These bases are similar to CMT2-LIB3
spreads in that they comprise both the term structure of Treasury yields
and some credit spread (although not necessarily the same credit
component as that which drives the TED spread). These spreads differ
from the Treasury-LIBOR spreads discussed above because their statistical
properties differ and there is no futures market in which to hedge them.
Earlier, we showed that a strong relationship exists between the fixed
corporate bond, swap and FRN markets. There is also an active basis swap
market which links the LIBOR-indexed FRN market to FRNs of the same
issuer but indexed to alternative rates.
Consider the following transactions:
• Purchase a newly issued LIB3 floater with two years to maturity that is
priced at 100 and pays a quarterly coupon of LIB3 plus a reset margin
(RM).

14 The common usage of the term "TED spread" refers to the spread between prices on the Treasury bill futures
contract and the Eurodollar futures contract. This distinction is irrelevent for the purpose of this discussion, and
henceforth we will "misuse" the term as the difference between the spot rates.

18 September 1995 Salomon Brothers

 • Enter into a two-year basis swap in which the investor pays LIB3 and
receives the average daily Federal funds rate plus a basis swap spread
(BSS).

Figure 11. Net Cash Flows of a LIB3 FRN and a LIB3-Fed Funds Basis Swap
Transaction Cash Flow at Quarter t
Buy LIB3 Floater +(LIB3t-11+RMLIB3) * Act/360
Basis Swap
Receive Federal Funds+Basis Swap Spreads +(Avg. Daily Fed Funds+BSS) * Act/360
Pay LIB3 -LIB3t-11 * Act/360
Net Cash Flow +(Avg. Daily Fed Funds+RMLIB3+BSS) * Act/360
Source: Salomon Brothers Inc.

• This net cash flow approximates the cash flow on a Fed Funds floater
where
Reset margin (RMFF) = RMLIB3 + BSS. (18)
Understanding the Current Coupon Differential Between CMT2 and LIB3
Because an actively traded and liquid market exists for Eurodollar futures,
we can look to those markets to help determine the current coupon
differential between a FRN indexed to a longer Treasury rate and one
indexed to LIB3. Consider a FRN that promises to pay CMT2 plus a reset
margin (RMCMT2 ) for five years. In theory, the issuer could hedge the
exposure by selling the two-year Treasury rate forward each quarter over
the five-year life of the bond. However, since there is not an active market
in futures on the two-year Treasury yield, the hedge would involve a
complicated position of long and short Treasuries.
A more popular alternative is to hedge the exposure in the liquid
Eurodollar futures market. Although CMT2 is a two-year par rate and LIB3
is a three-month spot rate, the two-year forward rate can be expressed as a
function of a series of three-month forward rates. The number of each
contract needed to hedge this position can be determined by calculating the
partial duration of the CMT2 floater with respect to each of the
three-month forward rates. Although the determination of the number of
contracts needed to hedge the CMT2 exposure is beyond the scope of this
discussion, the following observations are important in understanding the
determinants of relative coupon differentials between CMT2 FRNs and
LIB3 FRNs.
• The typical hedge amounts for a five-year maturity CMT2 floater
consists of a short position in Eurodollar futures contract from quarter 0 to
quarter 7 and a long position in contracts from quarter 8 to quarter 27. The
long positions become much larger from quarter 20 to quarter 27.
Intuitively, the last promised cash flow is a two-year rate 4.75 years
forward. Therefore, this bond will be sensitive to forward rates all the way
out to the 6.75 year portion of the curve.
• Although the Eurodollar futures curve comprises the Treasury forward
rates and a forward TED spread, the issuer is exposed to changes in the
TED spread since the hedge is designed assuming constant spreads.
• Because the CMT2 floater can be hedged by futures contracts, the
current coupon differential between CMT2 and LIB3 floaters will reflect
differences in the forward CMT2-LIB3 spread. To be more specific,
ignoring the effect of convexity on the optimal hedge ratios and differential

Salomon Brothers September 1995 19

 option-adjusted spreads due to liquidity, the reset margin of a CMT2
floater is priced so that its present value equals that of a LIB3 floater of
the same maturity if the forward rates are realized.15
Figure 12 shows the spreads between forward CMT2 and forward LIB3
observed on March 10, 1995. This table is a good starting point for an
investor to assess the yield curve view that is embedded in the relative
coupons of these floaters. If the investor believes that future yield curves
will be steeper than those implied by the forwards shown in Figure 12,
then he should consider buying a CMT2 floater.16 Of course, other factors
such as liquidity also should enter into the decision.
Because the reset frequencies are equal and a parallel shift of the yield
curve will have approximately the same effect on all of the forwards, the
FRNs have equal sensitivity to small, parallel shifts of the yield curve in
either direction. However, the bonds have different sensitivities to
reshapings of the curve (partial durations). On March 10, 1995, substantial
flattening from the two-year to three-month part of the curve was priced
into the CMT2 floater, and this is why it receives a higher coupon at the
pricing date.

Figure 12. Comparison of Forward CMT2 to Forward LIB3, 10 Mar 95

Years Forward Forward CMT2 Forward LIB3 Difference
0.00 6.82% 6.31% 0.51%
0.25 6.95 6.48 0.47
0.50 7.06 6.77 0.29
0.75 7.13 6.99 0.14
1.00 7.17 7.08 0.09
1.25 7.23 7.15 0.08
1.50 7.24 7.19 0.05
1.75 7.26 7.27 -0.01
2.00 7.26 7.24 0.02
2.25 7.29 7.27 0.02
2.50 7.28 7.29 -0.01
2.75 7.29 7.35 -0.06
3.00 7.27 7.34 -0.07
3.25 7.28 7.37 -0.09
3.50 7.27 7.40 -0.13
3.75 7.28 7.43 -0.15
4.00 7.28 7.44 -0.16
4.25 7.31 7.45 -0.14
4.50 7.33 7.46 -0.13
4.75 7.38 7.48 -0.10
CMT2 Two-year Constant Maturity Treasury. LIB3 Three-month LIBOR.
Note: CMT2 forwards are calculated from the Salomon Brothers Treasury Model Curve. LIB3 forwards are calculated
from the swap curve.

The Use of Historical Data in Assessing Basis Risk

In comparing indices such as Fed funds, TB3 and Prime to LIB3, the term
structure of interest rates are not as important because the maturities
embedded in these rates are near the maturity of LIB3. Furthermore, for
Fed Funds and Prime, there is no liquid futures market with long maturity
contracts in which an issuer can hedge its financing rates by buying the
indices forward.17 Therefore, there are not "arbitrage" relationships to
dictate how these FRNs should trade relative to each other.

15 The pricing coupons of the CMT2 floater will differ from the forwards because of convexity. This error will
increase with the term of the index. We can account for the liquidity differentials by obtaining the present value after
discounting at the appropriate option-adjusted spread.
16 In assessing the relative coupons through time, one must account for the fact that LIB3 FRNs typically pay on an
actual/360 basis and the CMT2 pay on a 30/360 basis. On March 10, 1995, this differential was worth ten basis points
per annum.
17 Fed Funds futures contracts are traded on the Chicago Board of Trade but the open interest is small and the
contracts do not extend longer than one year in maturity.

20 September 1995 Salomon Brothers

 For example, consider an investor that is benchmarked to the Fed funds
rate and fears an imminent Fed tightening. He makes the decision to
purchase a two-year floater on March 10, 1995 and compares a daily reset
Fed funds floater to one of the same credit quality indexed to LIB3. We
define the relative coupon at any quarterly payment date t as:

Relative Coupont =
([LIB3t-1 + RMLIB3] − [Average Daily Fed Funds observed over period t + RMFF] * (days/360) =
([LIB3t-1 − Avg.Daily Fed Funds observed over period t] + [RMLIB3 − RMFF])* (days/360). (19)

If the Fed Funds rate does not change over the first quarter,

Relative Coupont = [(6.31% − 5.93%) +(0.10% − 0.28%)] * (92/360) ≈ 5 basis points. (20)

Should the investor accept the five-basis-point advantage and take on the
basis risk and the longer time to reset, given his views on Fed tightening?
If the Fed tightens, how long does it take for LIB3 to adjust?
Historical data can help the investor to assess such risk-reward trade-offs.
For example, he can compare current spreads between the indices to
historical average spreads to determine whether the basis is at historically
wide or tight levels. Furthermore, the data provide a measure of the
propensity of these rates to move together. Estimates of historical
volatilities and correlations between the indices allow the derivation of an
expected change in one index conditional on a specified change in the
other index.
A sufficient amount of data is available on these rates to provide good
estimates of the average levels of these spreads and their propensity to
move together. However, in practice, many statistical issues and problems
exist that must be addressed. They include:
• What frequency of data should be used to estimate the parameters?
• Over what time period should the estimate be calculated? In other
words, over what time period is the assumption of stationary parameters a
good one?
• Does the data warrant a more complicated time-varying parameter
model such as Generalized Autoregressive Conditional Heteroscedasticity
(GARCH)?18
• In comparing changes in rates, are only contemporaneous correlations
important or should we account for various lags?
Figure 13 shows a comparison of the average spreads between various
indices. These tables illustrate that the estimates can differ significantly
across measurement frequencies and across subperiods.

18 See How to Model Volatility: Grappling over GARCH, Joseph J. Mezrich, Robert F. Engle, Ashihan Salih, Salomon
Brothers Inc, September 1995.

Salomon Brothers September 1995 21

 Figure 13. Comparison of Average Spreads of Various Indices
Fed
Prime CP1 Funds LIB3 LIB1 TSY2 TB3
Average Rates Mar 82-Jan 95
Quarterly 9.35% 7.40% 7.59% 7.65% 7.52% 7.98% 6.80%
Monthly 9.29 7.16 7.34 7.54 7.42 7.91 6.74
Weekly 9.32 7.13 7.14 7.56 7.36 7.92 6.76
Daily 9.32 7.16 7.19 7.56 7.37 7.93 6.78
10 Mar 95 9.00 6.08 5.93 6.31 6.13 6.82 5.93
Average Spreads to 3-Month T-Bill
Quarterly 2.55% 0.60% 0.79% 0.85% 0.72% 1.18% x
Monthly 2.55 0.42 0.60 0.80 0.68 1.17 x
Weekly 2.56 0.37 0.38 0.80 0.60 1.16 x
Daily 2.54 0.38 0.41 0.78 0.59 1.15 x
10 Mar 95 3.07 0.15 0.00 0.38 0.20 0.89 x
CP1 One-month commercial paper. LIB1 One-month LIBOR. LIB3 Three-month LIBOR. TB3 Three-month Treasury bill
yield. TSY2 Two-year Treasury yield.
Source: Salomon Brothers Inc.

Figure 14. Average Daily Spreads to Three-Month T-Bill for Various Time Periods
Fed
Prime CP1 Funds LIB3 LIB1 TSY2
Full Sample 2.54% 0.38% 0.41% 0.78% 0.59% 1.15%
3/82-6/86 2.50 0.29 0.55 1.07 0.83 1.49
6/86-9/90 2.30 0.62 0.65 0.90 0.72 0.83
9/90-1/95 2.85 0.23 0.06 0.39 0.24 1.14
10 Mar 95 3.07 0.15 0.00 0.38 0.20 0.89
CP1 One-month commercial paper. LIB1 One-month LIBOR. LIB3 Three-month LIBOR. TSY2 Two-year Treasury yield.
Source: Salomon Brothers Inc.

Spreads of LIB1, LIB3 and CP1 over TB3 clearly have grown tighter
through time. Regardless of the subperiod or frequency of measurement,
one might have concluded on March 10, 1995, that prime spreads were
historically wide and TSY2 spreads were historically tight. Other spreads
are near their averages, at least over the previous five years.
If spreads between indices tend to be mean reverting, then one may
conclude that prime floaters are "rich" relative to LIB3 given that Figure
10 shows that the two securities were essentially receiving the same
coupon. As we discussed in detail earlier, the CMT2 floaters receive a high
current coupon because some flattening of the yield curve is priced into the
bond. However, historical spreads indicate that the CMT2-LIB3 spreads are
tighter than average historical levels, which may suggest that CMT2
floaters offer value.19
Using historical average spreads between indices does not address the issue
of how we incorporate the view of imminent Fed tightening into the
decision process. Correlations provide a measure of the propensity of two
random variables to move together. Figure 15 shows estimated standard
deviations and correlations between one period changes in various indices
for the full sample and the most recent one third of the sample. We will
show how these estimates can be used to incorporate an investor’s views
on Fed funds into the choice of FRNs.

19 Of course, even if an investor believes historical spreads are applicable today, he may prefer to hold prime floaters
because of the frequent reset or to not hold CMT2 floaters for reasons such as liquidity.

22 September 1995 Salomon Brothers

 Figure 15. Correlations Between One Period Changes in Various Indices
Fed
Mar 82-Jan 95 Prime CP1 Funds LIB3 LIB1 TSY2 TB3
Monthly Frequency
Prime 1.00
CP1 0.59 1.00
Fed Funds 0.34 0.50 1.00
LIB3 0.64 0.79 0.51 1.00
LIB1 0.60 0.76 0.54 0.93 1.00
TSY2 0.57 0.62 0.35 0.86 0.72 1.00
TB3 0.47 0.70 0.40 0.80 0.75 0.74 1.00
Standard Deviation 0.36 0.56 0.81 0.50 0.53 0.45 0.66
Weekly Frequency
Prime 1.00
CP1 0.10 1.00
Fed Funds 0.09 0.17 1.00
LIB3 0.27 0.27 0.33 1.00
LIB1 0.23 0.28 0.35 0.84 1.00
TSY2 0.12 0.19 0.26 0.66 0.52 1.00
TB3 0.09 0.19 0.32 0.61 0.49 0.71 1.00
Standard Deviation 0.16 0.70 0.39 0.23 0.24 0.18 0.22
Daily Frequency
Prime 1.00
CP1 0.06 1.00
Fed Funds 0.02 0.26 1.00
LIB3 0.04 0.52 0.16 1.00
LIB1 0.04 0.47 0.13 0.56 1.00
TSY2 0.03 0.18 0.07 0.27 0.20 1.00
TB3 0.04 0.19 0.08 0.24 0.17 0.60 1.00
Standard Deviation 0.07 0.09 0.30 0.11 0.12 0.08 0.10
Fed
Sep 90-Jan 95 Prime CP1 Funds LIB3 LIB1 TSY2 TB3
Monthly Frequency
Prime 1.00
CP1 0.55 1.00
Fed Funds 0.20 -0.19 1.00
LIB3 0.57 0.55 0.31 1.00
LIB1 0.49 0.39 0.33 0.88 1.00
TSY2 0.52 0.40 0.20 0.79 0.58 1.00
TB3 0.64 0.48 0.30 0.86 0.75 0.79 1.00%
Standard Deviation 0.28 0.32 0.69 0.31 0.42 0.32 0.25
Weekly Frequency
Prime 1.00
CP1 0.23 1.00
Fed Funds 0.15 0.23 1.00
LIB3 0.24 0.37 0.01 1.00
LIB1 0.12 0.40 0.04 0.74 1.00
TSY2 0.09 0.29 0.11 0.53 0.38 1.00
TB3 0.15 0.28 0.18 0.46 0.40 0.68 1.00%
Standard Deviation 0.14 0.19 0.34 0.13 0.18 0.13 0.08
Daily Frequency
Prime 1.00
CP1 0.14 1.00
Fed Funds -0.02 0.08 1.00
LIB3 0.08 0.37 0.00 1.00
LIB1 0.04 0.25 0.04 0.48 1.00
TSY2 0.02 0.19 0.06 0.21 0.12 1.00
TB3 0.07 0.21 0.10 0.21 0.11 0.59 1.00%
Standard Deviation 0.06 0.07 0.31 0.06 0.10 0.06 0.04
CP1 One-month commercial paper. LIB1 One-month LIBOR. LIB3 Three-month LIBOR. TB3 Three-month Treasury bill
yield. TSY2 Two-year Treasury yield.
Source: Salomon Brothers Inc.

Salomon Brothers September 1995 23

 We can make several observations from Figures 15 and 16:
• Across nearly all of the frequencies and subperiods, Fed funds is the
most volatile (highest standard deviation) index.
• In general, the correlations are smaller the more frequently the data are
sampled.
• Some of the estimated parameters change substantially over time. One
of the most difficult issues to resolve in the use of statistical measures to
assess financial risk is determining which set of data is appropriate to use
to estimate the parameters. On one hand, one would prefer the longest time
series of data possible in order to obtain more precise20 estimates. On the
other hand, if the true relationship between the variables has changed over
time, then some of the earlier observations will not be relevant in
estimating the new parameter.
Figures 13-15 show that the estimated distributions of LIB1 and LIB3 have
changed through time with the average spreads and volatilities decreasing
substantially since the late 1980s. Many market participants would argue
that a fundamental shift in the relationship between LIBOR rates and
Treasury rates has occurred. If this is true, then the estimated parameters
derived from the most recent subperiod would be the most applicable in
measuring the current basis risk.
Figure 15 also presents an example whereby it may be more appropriate to
use the full sample of data. For the last subperiod (September 1990 -
January 1995) and monthly frequency, the estimated correlation between
CP1 and Fed funds is -0.19. However, this estimate was driven by a large
negative correlation in late 1990 and 1991. This period had a large impact
on the subperiod estimates because of the small number of observations.
The estimated correlation between CP1 and Fed funds over the entire
period (March 1982 - January 1995) was 0.50. The 1991 period makes a
much smaller contribution to the estimate for the full sample. Many market
participants would argue that the full sample estimate appears more
reasonable in this case.
Prime-Treasury Spreads
One "conventional wisdom" is that banks raise their prime lending rate
quickly as Treasury rates increase but react much more slowly when
interest rates drop. This phenomenon would have positive implications for
prime floaters because the coupon would reset quickly to higher rates but
slowly to lower rates, thereby improving their performance relative to other
FRNs, keeping all else equal. Our simple statistics shown in Figure 15 do
not tell whether this conventional wisdom is substantiated because our
estimates represent an "average" of rising and falling interest rate
environments. Figure 16 gives average changes in rates for Prime and Fed
funds and correlations in different environments. The environments are
determined by whether the Fed funds rate rises (∆FF > 0) or falls
(∆FF <= 0) for the given period.

20 Precision refers to the degree of error in estimating the parameters.

24 September 1995 Salomon Brothers

 Figure 16. Comparison of Prime Rate Changes to Fed Funds Changes in Different Interest Rate
Environments
∆FF>0 ∆FF<=0
Correlation Average Average Correlation Average Average
(∆FF, ∆Prime) ∆Prime ∆FF (∆FF, ∆Prime) ∆Prime ∆FF
Mar 82-Jan 95
Monthly -0.06 0.05 0.48 0.30 -0.15 -1.10
Weekly -0.08 0.01 0.21 0.37 -0.03 -0.62
Daily -0.04 0.00 0.21 0.15 0.00 -0.23
Mar 82-Jun 86
Monthly -0.12 -0.01 0.66 0.39 -0.29 -0.92
Weekly -0.19 -0.03 0.34 0.13 -0.04 -0.36
Daily -0.01 0.00 0.25 0.05 -0.01 -0.20
Jun 86-Sep 90
Monthly 0.27 0.09 0.38 0.06 -0.04 -0.44
Weekly 0.33 0.03 0.13 0.09 -0.02 -0.12
Daily 0.02 0.00 0.16 0.01 0.00 -0.12
Sep 90-Jan 95
Monthly -0.22 0.06 0.40 0.22 -0.12 -0.46
Weekly 0.04 0.01 0.18 0.19 -0.02 -0.20
Daily -0.08 0.00 0.22 0.07 0.00 -0.15
∆FF One-period change in Fed funds rate. ∆Prime One-period change in prime rate.
Source: Salomon Brothers Inc.

The evidence in Figure 16 does not support the conventional wisdom. If

Prime rates moved with Treasury rates in rising rate environments but
remained "sticky" in falling rate environments, then we should see higher
correlations and substantially positive changes in the Prime rate in rising
rate environments. However, the data does not support the conjecture
except possibly in the June 1986 - September 1990 subperiod. In fact,
Figure 16 provides more support to the conjecture that the Prime rate falls
with decreases in Treasury rates but is "sticky" in rising rate environments.
Using the Statistics to Assess the Basis Risk
The average one-period change of each of these indices was close to zero
for all frequencies and periods. We can say that our unconditional
expectation of the change in any particular index over any frequency
equals zero. Given that the appropriate frequency and period has been
chosen, the estimated correlations allow an estimate of an expectation
conditional on a particular view. For example, if the investor believes that
the Fed will raise the Fed funds rate (FF) by 50 basis points within the
next week, what increase in LIB3 should he expect?
• The link between conditional expectations and correlations can be seen
in the standard regression equation:

y = a + βx. (21)

covariance (x,y)
β is defined as
std(x)2 (22)

covariance (x,y)
correlation (x,y) = ,
std (x) std (y)

where std (x) = standard deviation of x. (23)

Salomon Brothers September 1995 25

 Also, taking expectations of the standard regression equation shows that:

a = ȳ + βx̄,

where x̄ = the expected value of x and ȳ = the expected value of y. (24)

Substituting these equations into the standard regression equation yields the
following relationship:

correlation (x,y) ∗ std(y)

E(y conditional on x) = ȳ +  (x − x̄),
 std(x) 

where E(.) indicates expected value. (25)

We can substitute the estimated values from the table for weekly frequency
into Equation (25), to obtain the following relationship:

(0.328)(0.226)
E[change in LIB3] = 0 +  (change in FF − 0)
 0.390 

= 0.19 * change in FF. (26)

Suppose, as stated earlier, we want to investigate the case in which the

change in FF equals 50 basis points over the next week. Then, to get the
updated or "conditional" expectation, we substitute 0.50% for change in FF
in Equation (26) yielding the following result:
E[change LIB3 conditional on change in FF=0.5%]
= 0.19 * 0.50% = 0.095% = 9.5 basis points.
The statistic R2 provides a measure of the "goodness of fit" of the
regression model. For a univariate regression (one variable), R2 equals
correlation(x,y)2. Therefore, in our case, R2=(0.328)2=10.8%, implying that
the change in the Fed funds rate only "explains" 10.8% of the total
variation in LIB3.
In this example, we only considered contemporaneous correlations. To
better quantify basis risk, we also should consider lagged changes of
indices. A vector autoregression model21 estimates the effects of shocks to
a particular index and on its expected future levels and those of other
indices. An investor could use such a model to estimate relative cash flows
between two FRNs over time conditional on a particular change of one of
the indices.

21 For a technical definition and discussion of vector autoregression models, see Time Series Analysis, James D.
Hamilton, Princeton University Press, 1994.

26 September 1995 Salomon Brothers

OPTIONALITY

Thus far, we have only discussed FRNs that do not contain embedded
options. However, some corporate and asset-backed floaters are subject to
maximum or minimum coupons or have call provisions. Many structured
notes comprise fairly complicated option positions. Floating-rate mortgage
securities such as adjustable-rate mortgages (ARMs) or floating-rate CMOs
have maximum and minimum coupons and are subject to prepayment
options. In comparing these securities to noncall securities of the same
issuer or to each other, an investor must: (1) identify the option position
embedded in the bond; and (2) value the option position.
For some FRNs with embedded options, it is relatively straightforward to
decompose the security into a portfolio of a long position in a noncall FRN
and a position in some commonly traded option.
Example — Floored Corporate FRNs
Consider a FRN with the following characteristics:
• Rating — A3/A-,
• Maturity — 7/15/03,
• Coupon — LIB3+12.5 basis points,
• Minimum coupon — 4.35%,
• Reset/pay frequency — quarterly/quarterly.
This security can be decomposed into a portfolio of a long position in an
unfloored FRN with a coupon of LIB3 plus 12.5 basis points and a long
position in a LIBOR floor maturing on the same date with a strike equal to
4.225%. Figure 17 illustrates that the portfolio replicates the cash flows of
the floored floater.

Figure 17. Cash Flow of Portfolio Consisting of an Unfloored FRN and a Long Floor
Portfolio Cash Flow at Time t
Long Unfloored Floater LIB3t-1 + 0.125%
Long Floor With a 4.225% Strike Max[4.225% - LIB3t-1, 0]
Net Cash Flow Max[4.35%, LIB3t-1 + 0.125%]
FRN Floating-rate note. LIB3 Three-month LIBOR.
Source: Salomon Brothers Inc.

This approach is extremely useful in relative valuation. Suppose that, on

July 15, 1995 (exactly eight years from maturity), the market price of the
floored floater was 98. On this date, a fixed bond of the same issuer
maturing on February 1, 2003 was trading at a yield spread of 92 basis
points over the seven year Treasury. For simplicity, assume that this is also
the fair yield of a fixed bond which matures on July 15, 2003. On July 15,
1995, the fixed-rate on an eight-year swap was 6.45%. Therefore, the
floored floater could be swapped into a fixed coupon of 6.575% (6.45%
plus a 0.125% reset margin). Using a yield spread of 92 basis points, the
present value of the swapped floater (ignoring the value of the floor) is
97.85. Therefore, the implied price of the embedded floor is equal to 0.15
(98.00-97.85). Using an arbitrage-free term structure model, for an
eight-year floor with a strike equal to 4.225% and a dollar price of 0.15,
we obtain an implied volatility of 12.5%. Since the implied volatility of
offered floors of similar maturity and strikes on 7/15/95 was 23%, the
floored floater appears "cheap".

Salomon Brothers September 1995 27

 Several caveats apply to this type of analysis. One practical issue is that if
the floored FRN is less liquid than the portfolio of an unfloored floaters
and an over-the-counter (OTC) floor, then we should expect it to be
slightly cheaper to reflect higher transaction costs of unwinding the
position. Another issue is whether an investor can extract the "cheapness"
of the floored floater. Suppose an investor agreed with our analysis but
wants to purchase a FRN because he has a view that interest rates will rise
in the near term. He asks, "Even though the embedded floor appears cheap
at a premium of $0.15, why should I pay that premium when I have such
strong views that rates will rise?"
One answer that is often given is that the investor can extract the
cheapness today by writing an OTC 4.225% strike floor with the same
coupon dates and maturity as the FRN. Assuming the bid volatility of the
OTC floor was 21%, such a floor would provide the investor with an
up-front premium of $0.96. Therefore, he now created a synthetic
unfloored FRN at approximately 26/32 of a point "cheap" ($0.96-$0.15).
This is a completely valid argument if the investor’s intention is to keep
the long and short position until maturity and does not have to mark this
position to market. If the floors were to come into the money at any time
over the life of the bond, the difference between the floored floater coupon
and that of an unfloored floater would exactly offset any cash flow that the
investor would have to pay on the short floor.
However, suppose the investor had an investment horizon shorter than the
eight-year maturity of the FRN and either had to mark the combination to
market or trade out of the position at the short horizon. Suppose also that
the implied volatility on the OTC floors increased to 30%. There is no
mechanism to ensure that the embedded option in the FRN would increase
to reflect the higher market volatility. Therefore, the investor may have a
marked-to-market loss on the short floor position and no marked-to-market
gain on the FRN. Alternatively, suppose LIB3 decreased by 50 basis points
from its July 15 level of 5.81%. If the implied volatilities on the OTC
floors remained constant, then they would appreciate because the option
would be closer to the money. Once again, the FRNs may not necessarily
increase in price accordingly; they may just trade at even lower implied
volatilities.
These caveats do not invalidate the analysis as a measure of richness or
cheapness. Purchasing this FRN under the market conditions given in the
example still appears to be a cheap way of adding convexity because the
FRN should eventually experience price appreciation due to the changing
option value if rates decrease. This example simply illustrates that the
mispricing does not represent "arbitrage" opportunities.
Other types of FRNs contain much more complicated embedded option
positions. Many structured notes are bonds that pay a coupon that is some
function of LIB3. Although the component parts may not be as obvious as
in the previous example, these securities can often be decomposed into a
replicating portfolio in a similar manner.22 However, mortgage-backed
FRNs do not lend themselves to such a clean decomposition because of the
existence of the prepayment option.

22 For some examples, see Floating-Rate Securities: Current Markets and Risk/Return Trade-Offs in Rising Interest
Rate Environments, Raymond J. Iwanowski, Salomon Brothers Inc. April 6, 1994.

28 September 1995 Salomon Brothers

 Mortgage FRNs
A complete discussion of all of the relevant factors that influence the ARM
and floating-rate CMO markets would require much more detail and
analysis than we intend to devote in this piece. However, we will broadly
outline some of the main issues that an investor should consider in
comparing a mortgage-backed FRN to those of other asset classes,
particularly in the context of our discussion of optionality.
Like all mortgage-backed securities (MBS), floating-rate CMOs and ARMs
differ from corporate and government bonds in that the securities amortize
through time. Furthermore, although the securities are priced with respect
to some scheduled amortization, this schedule is based on projected
prepayments. Actual prepayments that deviate from scheduled prepayments
will cause the amortization schedule to change over time.
Actual prepayment speeds that are faster (slower) than pricing prepayment
speeds positively affect the total return performance of fixed
mortgage-backed securities when the bond is purchased at a discount
(premium) and negatively affect performance when the bond is purchased
at a premium (discount). This intuition also holds for floating-rate MBS.
However, suppose that the floating-rate mortgage is priced at or near 100.
Why should prepayment variability affect this bond? As long as spreads
remain the same, shouldn’t the investor be able to reinvest fast
prepayments back into another floater with the same spread to current
LIB3?
The main reason that prepayment variability affects the price of a
floating-rate MBS is the effect that prepayments have on the embedded
options. Specifically, both ARMs and floating-rate CMOs have maximum
and minimum coupons.23 ARMs also have periodic caps, which represent
the maximum amount the coupon could change at any particular reset.
Some investors attempt to "uncap" floating-rate CMOs by decomposing the
floating-rate MBS into an uncapped floater and a short position in LIBOR
caps with maturities equal to the average life of the CMO. A more
sophisticated approach to the same idea is to value a short position in an
amortizing cap where the amortization schedule is set to the amortization
schedule of the MBS at the pricing date. However, this "replicating
portfolio" idea misses the effect of prepayment variability. Although actual
prepayments are difficult to predict and the best forecasting models always
will have some error, it is quite predictable that, all other factors constant,
prepayments speed up when interest rates fall and slow when interest rates
rise. This phenomenon increases the value of the caps because the
"maturity" of the caps increases as they become nearer the money. This
effect decreases the value of the bond because the bond has an embedded
short position in the caps.
For example, suppose a floating-rate CMO with a reset margin of 40 basis
points and a 9.40% cap is priced on July 15, 1995 and had a
weighted-average life of five years. The market implied volatility for
five-year 9% OTC caps of 24.5% which results in a "value" for the cap,
ignoring extension risk, of $1.06.24 Now, suppose the LIBOR yield curve
experiences an instantaneous parallel increase in rates of 50 basis points.
Because the cap is 50 basis points closer to becoming "in-the-money," a

23 The floors are typically struck at very low levels, in some cases at 0%. For the ensuing discussion and without loss
of generality, we assume that the securities have caps but no floors.
24 Floating-rate CMOs typically are indexed to one-month LIBOR and pay monthly.

Salomon Brothers September 1995 29

 five-year cap would now be worth $1.45. However, suppose that the CMO
extends such that at the new yield curve levels, the market would be
pricing the CMO to a seven-year average life. Now, the price of a seven
year cap under the same volatility and the new yield curve is $2.87.25 This
example illustrates that "uncapping" the floating-rate MBS in this manner
overvalues the security because the method ignores the extension risk.
One method of comparing floating-rate MBSs to FRNs of other asset
classes is through the use of the option-adjusted spread (OAS)
methodology. Standard OAS methodology takes a term structure of interest
rates and a volatility assumption and values the security over a distribution
of possible interest rate paths. The parameters of the distribution are
calibrated such that the current term structure is correctly priced and
arbitrage opportunities are precluded. For mortgages, a prepayment function
is applied at each path, and therefore, the OAS calculation will incorporate
the effect of extension on the value of the caps. However, it must be
emphasized that the calculated OAS is a function of the assumed term
structure and prepayment models. Even the most accurate models will
capture the true distributions with some error. Therefore, a comparison of
the OAS of a complex optionable security such as a floating-rate MBS to a
FRN without options should incorporate some additional spread because of
"model risk."
Alternatively, the canonical decomposition methodology26 extends the
replicating portfolio idea to account for the negative convexity implied by
the extension risk of the caps. Specifically, this approach uses an
"arbitrage-free" term structure model to determine a portfolio of Treasury
bonds, caps and floors that replicate the cash flows of the floating-rate
MBS across all possible paths. The decomposition allows an investor to
evaluate the risks embedded in a complex security in the context of a
portfolio of very liquid instruments. In addition, this portfolio does not
require continuous rebalancing. Of course, like the OAS methodology, the
accuracy by which the cash flows of the portfolio replicate the cash flows
of the FRN along all paths is subject to the accuracy of the term structure
and prepayment model.
The mortgage example illustrates another important intuition in the
valuation of embedded options in FRNs. The spot LIB1 observed on July
15, 1995 was 5.88%. Given the 9.00% strike on the embedded caps, these
options are currently over 300 basis points "out of the money". An investor
that is familiar with option pricing theory may be surprised that options so
far "out of the money" would be valued so highly. In reality, caps
comprise a set of options on forward rates. Therefore, one should look at
the implied forward rates over the life of the cap to determine
"in-the-moneyness." In an upward sloping yield curve environment, the
options may be much nearer to the money than implied by the difference
between the strike and the spot LIB1.

25 This value is slightly overstated because longer caps typically get priced at lower volatility. At 22% volatility, the
value of the cap is $2.36.
26 See "Arbitrage-Free Bond Canonical Decomposition," Thomas S.Y. Ho, Global Advanced Technology and this
volume, April 1995.

30 September 1995 Salomon Brothers

CONCLUSION

Floating-rate securities have been, and will continue to be, popular in

practically all asset classes of fixed-income markets, particularly for
investors that are bearish on interest rates. Many of the features of these
markets make comparisons of the risks, and even the compensations for
taking risks, difficult to quantify. In this report, we present many of the
measures and conventions that are currently used in the various FRN
markets. We hope that these discussions will be useful as a reference that
makes these commonly used measures more concrete. We then describe in
the detail the risks that the various instruments are subject to and offer
frameworks in which an investor can assess these risks.

Salomon Brothers September 1995 31