David Bowman, Senior Associate Director
Board of Governors of the Federal Reserve
This information is provided for illustrative and educational purposes only. The views expressed in this presentation are solely those of the author and do not necessarily
represent those of the Federal Reserve, the Alternative Reference Rates Committee or its members or ex officio members.
Templates for Using SOFR
SOFR has a number of characteristics that LIBOR and
other similar rates like LIBOR that are based on
wholesale term unsecured funding markets do not:
• It is a rate produced by the Federal Reserve Bank
of New York (FRBNY) for the public good;
• It is derived from an active and well-defined
market with sufficient depth to make it
extraordinarily difficult to ever manipulate or
influence;
• It is produced in a transparent, direct manner and
is based on observable transactions, rather than
being dependent on estimates, like LIBOR, or
derived through models; and
• It is derived from a market that was able to
weather the global financial crisis and that the
ARRC credibly believes will remain active enough in
order that it can reliably be produced in a wide
range of market conditions.
However, SOFR is also new, and many are unfamiliar
with how to use it.
0
100
200
300
400
500
600
700
800
Secured
Overnight
Financing Rate
(SOFR)
Overnight
Bank Funding
Rate
Effective
Federal Funds
Rate
3-month T-bills 3-month GSIB
wholesale
funding
3-month AA
nonfinancial
CP
3-month A2/P2
nonfinancial
CP
Daily Volumes in U.S. Money Markets
Billions USD
$754 billion
$197 billion
$79 billion
Est. $13 billion
$1.1 billion
$343 million
$132 million
The Secured Overnight Financing Rate (SOFR)
Source: ARRC Second Report
2
4/16/2019
SOFR is published on every U.S.
business day at approximately
8:00am EST. Because the Fed has
the ability to correct and republish
this rate until 2:30pm New York
City Time each day, users may wish
to reference the rate after this
time (e.g. 3:00pm)
The SOFR rate published on any
day represents the rate on repo
transactions entered into on the
previous business day and the date
associated with each rate reflects
the date of the underlying
transactions rather than the date
of publication.
SOFR Publication
SOFR is published on the Federal Reserve Bank of New York’s
website (https://apps.newyorkfed.org/markets/autorates/sofr)
every U.S business day at approximately 8am EST. FRBNY’s
revision policies state that SOFR may be revised up to 2:30pm
EST.
SOFR is also available on Bloomberg and Reuters and can
additionally be accessed through an API offered by FRBNY
(https://www.newyorkfed.org/markets/effr-obfr)
The rate published each day represents the rates on overnight
repo transactions that were entered in to the previous business
day and that are to be repaid on the current business day. So, for
example, on April 16, the rate for transactions entered in to on
April 15 would be published.
This is similar to how the effective federal funds rate (EFFR) and
risk-free rates (RFRs) in other jurisdictions are published.
SOFR Published around 8am the next business day
SONIA Published at 9am the next business day
TONA Published at 10am the next business day
ESTER Will be published at 9am the next business day
SARON Published at 6pm the same business day
Table 3: The Publication Timing of the RFRs
3
0
1
2
3
4
5
6
7
1998 2000 2002 2004 2006 2008 2010 2012 2014 2016 2018
Three Month Compounded Effective Fed Funds Rate (EFFR) and
SOFR/Primary Dealer Survey Data
Quarterly Compound SOFR
Quarterly Compound EFFR
Percent
Source: FRBNY; staff calculations
SOFR Data
• FRBNY, in cooperation with the Office of Financial
Research, began publishing SOFR on April 3, 2018.
• Prior to the start of official publication, FRBNY
released data from August 2014 to March 2018
representing modeled, pre-production estimates
of SOFR that are based on the same basic
underlying transaction data and methodology that
now underlie the official publication.
(https://www.newyorkfed.org/newsevents/speeches/2017/fr
o171108)
• FRBNY has also separately released a much longer
historical data series based on primary dealers'
overnight Treasury repo borrowing
activity. (https://www.newyorkfed.org/markets/opolicy/op
erating_policy_180309)
• A forthcoming note I have written argues that the
historical survey data is an adequate proxy for
SOFR for risk modelling or other purposes
4
Three Key Basic Choices in Determining How to Use SOFR:
• Averaging: Compound or Simple
Compound averaging is used in OIS swaps and some futures. However, many loan and FRN systems currently use simple
averaging, largely because of historical precedent. There is some basis between the two types of averaging, although it is
generally small. Use of simple averaging may be an expedient to begin using SOFR, but most ARRC members tend to feel that
moving toward compounding over time is sensible since it better interest reflects the time value of money.
• Payment Notice: In Advance, In Arrears, or Hybrid
An in advance payment structure based on SOFR would reference an average of the overnight rates observed before the
current interest period began, while an in arrears structure would reference an average of the rates over current the interest
period and would only be fully known at the end of the interest period. An average overnight rate in arrears will reflect what
actually happens to interest rates over the period and will therefore fully hedge interest rate risk in a way that LIBOR or a
SOFR-based forward-looking term rate will not.
• Underlying Market: SOFR (U.S. Treasury Repo Market) or SOFR Derivatives (SOFR futures or OIS)
The U.S. Treasury Repo Market underlying SOFR is already deep and highly liquid. SOFR futures and OIS are growing but still at
early stages and are not yet deep or highly liquid enough to produce a robust, IOSCO-compliant rate). Many market
participants would prefer term rates based on derivatives, but at the same time, the ARRC and the FSB have warned that
people should not simply wait for term rates and that those who are able to move to SOFR should seek to do so if they can.
5
For derivatives, it is fairly clear that the market will be based on compound SOFR in arrears. For cash products, ARRC Working Groups
have so far gravitated toward four basic models of SOFR use:
• Published Simple or Compound Average of SOFR Set in Advance
Should require few or no changes to existing systems to use.
• Published Forward-Looking Term SOFR set in Advance
These rates may not come until 2021, but they should require few or no changes to existing systems to use.
• Simple Average of SOFR Set in Arrears
Would require few or manageable changes to existing systems to use
• Compound Average of SOFR Set in Arrears
Will require more changes to existing systems to use.
The Different Potential Versions of SOFR-Based Rates
6
Published SOFR Averages
1
1.5
2
2.5
3
3.5
Jan-18 Apr-18 Jul-18 Oct-18 Jan-19
Recent Movements in SOFR versus Averaged SOFR
SOFR
1-Month Average SOFR
3-Month Average SOFR
6-Month Average SOFR
Percent
Source: Federal Reserve Bank of New York; Federal Reserve Board staff calculations
• The Federal Reserve Bank of New York has
indicated that it plans to publish averages of
SOFR in the first half of next year.
• Averages of SOFR show very little or no impact
from these kinds of temporary, day-to-day
volatility that can be seen in overnight SOFR
around year/quarter-ends.
• A 3-month average of SOFR is less volatile than
3-month LIBOR, even over the last year end.
• FRBNY has not stated whether it will publish
compound or simple averages, but the
differences between the two choices would
typically be small.
• Further details on the average rates that FRBNY
would produce should follow, but from an
systems perspective, using these rates in
advance would be easily implemented.
0
2
4
6
8
10
12
2000 2002 2004 2006 2008 2010 2012 2014 2016 2018
Historical Basis Between Compound and Simple SOFR (bp)
Monthly Quarterly Semiannual
7
1.5
1.7
1.9
2.1
2.3
2.5
2.7
Jun-18 Jul-18 Aug-18 Sep-18 Oct-18 Nov-18 Dec-18 Jan-19 Feb-19 Mar-19 Apr-19 May-19
Comparing an Indicative SOFR Term Rate to EFFR OIS
3-Month Indicative SOFR Forward-Looking Term Rate
3-Month EFFR OIS
Source: Federal Reserve Bank of New York, CME, Bloomberg, and Federal Reserve staff calculations
Percent
• Federal Reserve staff members are
producing “indicative” forward-looking
term rates that are not meant to be used in
contracts and are not IOSCO compliant, but
may help provide a sense as to how the
term rates will behave (a link to this data,
which are periodically updated, is on the
ARRC’s website).
• The forward-looking term rates that the
ARRC envisions will effectively be segments
of the SOFR OIS curve, and as such should
behave much like EFFR OIS rates do today.
• The forward-looking term rates should also
be tightly linked to compound averages of
SOFR, just as EFFR OIS rates are tightly
linked to compound averages of EFFR.
Forward-Looking Term Rates
0
1
2
3
4
5
6
2001 2003 2005 2007 2009 2011 2013 2015 2017
Comparing EFFR OIS and Compounded Averages of EFFR
3-Month OIS
3-Month Compound Average
Percent
Source: Federal Reserve Bank of New York, Bloomberg; Federal Reserve Board staff calculations
8
Monthly SOFR Futures and a number of SOFR FRNs are based on simple averages of
SOFR in Arrears. FRN issuance systems had already been developed for the effective fed
funds rate based on simple interest, making it easier for initial SOFR FRNs to use simple
interest.
Loan systems that use overnight LIBOR, Prime, or the effective fed funds rate based on
simple interest are also already frequently in place if not all that regularly used.
Simple Averages of SOFR in Arrears
9
Three-month SOFR futures, SOFR OIS, and some recent SOFR FRNs are based on compounded averages of SOFR in
Arrears, but in general these systems are not yet in place for cash products
Where
d
b
= the number of business days in the interest period
d
c
= the number of calendar days in the interest period
r
i
= the interest rate applicable on business day i
n
i
= the number of calendar days for which rate r
i
applies (on most days, n
i
will be 1, but on a Friday it will generally be 3,
and it will also be larger than 1 on the business day before a holiday). This can also be stated as the number of
calendar days from and including business day i to but excluding the following business day.
N = the market convention for quoting the number of days in the year (in the United States, the convention is N = 360)
And i represents a series of ordinal numbers representing each business day in the period.
This term is to annualize
the compounded rate
Compound Averages of SOFR in Arrears
This term is to translate
the annualized overnight
rate Into an effective
daily/next business day
rate
10
Secured Overnight
Financing Rate
(Percent, Annualized)
Number of
Days Rate is
Applied
Effective Rate
(Not Annualized)
Principle
Principal +
Accumulated
Interest
Interest Charge for Next Business
Day
(Effective Rate*(Principal+Accumulated
Interest))
Monday, Jan 7, 2019 2.41 1 0.0241/360 = 0.006694% $1,000,000.00 $1,000,000.00 $66.94
Tuesday, Jan 8, 2019 2.42 1 0.0242/360 = 0.006722% $1,000,000.00 $1,000,066.94 $67.23
Wednesday, Jan 9, 2019 2.45 1 0.0245/360 = 0.006806% $1,000,000.00 $1,000,134.17 $68.06
Thursday, Jan 10, 2019 2.43 1 0.0243/360 = 0.006750% $1,000,000.00 $1,000,202.23 $67.51
Friday, Jan 11, 2019 2.41 3 3*0.0241/360 = 0.020083% $1,000,000.00 $1,000,269.74 $200.89
Monday, Jan 14, 2019 --- --- --- $1,000,000.00 $1,000,470.63
Payment Due
Monday, Jan 14, 2019
$1,000,470.63
Annualized Compound Rate of Interest:
= (360/7)*(.047064%) = 2.4204%
Compound Interest on a One-Week SOFR Loan of $1 Million Drawn on Jan 7, 2019
An Example of the ISDA Compound Average Formula
11
An Index would compound daily SOFR every day., similar to a price-level index It could serve as a trusted key allowing
desired.
Or, recursively
Taking the ratio of two Index values automatically calculates compounded interest over the period between the two dates
-1
Making Compound Calculations Easier – a SOFR Compound Index
12
Deciding Which Segments to Compound
One issue is whether to compound any margin or to only compound the rate and add margin separately
• Compound both rate and margin:
Pros: Economically pure – in theory, both rate and margin should compound
Cons: Harder to calculate, cannot rely on an Index to compound both rate and margin, will have some basis
relative to OIS
• Compound rate but not margin
Pros: Easy to calculate, can rely on an Index to compound the rate, will be fully hedged relative to OIS
Cons: Not economically pure
ARRC Working Groups have gravitated toward compounding the rate but not margin
13
Models for Using RFRs in Arrears
The FSB and National Working Groups are looking at several models for using overnight risk-free
rates in cash products. There are several different variants of both in Arrears and in Advance
conventions, as well as potential hybrid conventions that attempt to bridge the difference between
the two by allowing for advance notice while also allowing for complete or almost complete hedging
of contemporaneous rate movements.
• In Arrears
o Plain: Used averaged rate over current interest period, paid on last day of the period (day T)
o Payment Delay: Use averaged rate over current interest period, paid k days after day T (Note: ISDA’s
conventions for SOFR swaps use a 1-day payment delay)
o Lookback: Use averaged rate over current interest period lagged k days (a 3-5 day lookback has been used
in SONIA FRNs)
o Lockout: Use averaged rate over current period with last k rates set at the rate for day T-k (a 3-5 day
lockout has been used in most SOFR FRNs).
14
Day 1
(First Day of
Interest Period)
Day 2 … Day T-2 Day T-1
Day T
(Last Day of
Interest Period)
Day T+1
(First Day of
Next Period)
Day T+2
SOFR for
Day 1
Published
SOFR for
Day T-3
Published
SOFR for
Day T-2
Published
SOFR for
Day T-1
Published
SOFR for
Date T
Published
Plain Arrears
Use SOFR for
Day 1
Use SOFR for
Day 2
…
Use SOFR for
Day T-2
Use SOFR for
Day T-1
Use SOFR for
Day T
Payment Due
Use SOFR for
Day 1
Use SOFR for
Day 2
…
Use SOFR for
Day T-2
Use SOFR for
Day T-1
Use SOFR for
Day T
Payment Due
Use SOFR for
Day 1
Use SOFR for
Day 2
…
Use SOFR for
Day T-2
Use SOFR for
Day T-1
Use SOFR for
Day T-1
Payment Due
Use SOFR for
Day 0
Use SOFR for
Day 1
…
Use SOFR for
Day T-3
Use SOFR for
Day T-2
Use SOFR for
Day T-1
Payment Due
Models for Using SOFR in Arrears
Arrears with
Payment
Delay
Arrears with
1-Day
Lockout
Arrears with
1-Day
Lookback
OIS generally settle at
T+2
15
Convention #1: A lookback uses the rate from k days ago to calculate today’s interest owed. For example, in a 2-day lookback, if
today were Friday, one would use Wednesday’s rate in calculating today’s interest. A narrow definition might be taken to imply
that you should apply Friday’s weighting (n
i
= 3 since Friday covers three calendar days until payment is due) to Wednesday’s rate,
but a more sensible reading would be to apply Wednesday’s weighting to Wednesday’s rate, which has been called an
“observation shift”
Lookback (narrowly defined):
Lookback with observation period shift:
A lookback/observation shift will be fully hedged relative to OIS while a lookback with no observation shift will have some basis.
Convention #2: Is using the SOFR rate published today for the business day’s rate a lookback? The SOFR rate published today
represents that market rate for borrowing on the previous business day to be repaid today, and FRBNY post it as the rate for the
previous business day. OIS markets would not refer to this as a lookback, and some recent FRNs have taken the same convention,
but some of the early FRN issuances did call this a 1-day lookback. The payments and when they are to be made regardless, but
potential differences in what constitutes a lookback can cause confusion if not understood.
A Few Convention Issues for Lookback Structures
16
Pure Arrears:
Payment Delay:
Lookback (narrowly defined):
Lookback with observation period shift:
Payment Delay with interest period shift:
Lockout:
For Those Who Like Math
17
In Arrears: Lockout Versus Lookback
Payment Delays or Lookbacks with observation shift are consistent with ISDA compounding definitions and
more easily hedged and does not skip any interest days. A lockout does skip some days and has some basis to
the In Arrears model used in OIS swaps (below), On the other hand, for most of the interest period, the daily
interest rate will correspond to the most recent published value of the RFR, which may be important to certain
investors who do not have hedging needs.
-15
-10
-5
0
5
10
15
20
25
2000 2002 2004 2006 2008 2010 2012 2014 2016 2018
Basis between Quarterly Compounded 3-day Lockout vs Pure Arrears (bp)
18
Most cash product
issuances have used in
Arrears frameworks,
but there have been a
wide array of choices
between lookbacks,
payment delays, and
lockouts as well as
compounding versus
simple averaging.
Models of In Arrears (Continued)
SOFR FRNs SONIA FRNs
Swiss Working Group
FRN
Recommendations
OIS
In Arrears/In Advance
In Arrears In Arrears In Arrears In Arrears
Averaging
Generally simple
average, but several
recent issuances have
used compound
averages
Compound Average Compound Average Compound Average
Payment Delay
Generally none
(Payment due next
business day after
Accrual Period ends),
although one recent
issuance employed a
payment delay except
for the final payment
None (Payment due
next business day
after Accrual Period
ends)
None (Payment due
next business day
after Accrual Period
ends)
One business day
(Payment due two
business days after
accrual period ends)
Lookback
0-2 business days 5 business days 3-5 business days None
Lockout/Suspension Period
Generally 2 business
days
None
None
None
Comparing Typical Conventions for RFR-Based FRNs and OIS
19
In Arrears/In Advance (continued)
The amount of basis between In Advance and In Arrears depends on the frequency of interest periods. With a
one-month reset, the basis is comparable to the amount of basis between simple and compound averaging.
Even at 3- or 6-month resets the basis is limited and averages out to zero over longer periods of time.
-60
-50
-40
-30
-20
-10
0
10
20
30
40
1990 1992 1994 1996 1998 2000 2002 2004 2006 2008 2010 2012
Basis Spread between in Advance and In Arrears 5-Year Loan with Monthly
Payments (bp)
1MonthSpread 3MonthSpread 6MonthSpread
20
Hybrid Models
Hybrid Models mix an in Advance payment structure with in Arrears accrual of principal/interest owed:
Principal Accrual: Payments set In Advance, principal and interest accrue In Arrears
Interest Rollover: Payments set In Advance, any missed interest relative to In Arrears is rolled over
into the next payment period.
Either of the Hybrid Models can substantially further cut the basis relative to a pure In Arrears baseline, even for a
product with a less frequent reset such as 5/1 ARM, while still allowing borrowers to know their payments at the
start of the interest period
-10
-8
-6
-4
-2
0
2
4
6
8
10
1983 1986 1989 1992 1995 1998 2001 2004 2007 2010
Comparing Bases to In Arrears for the Hybrid 5/1 Mortgage Models
Principal Adjutments
Interest Rollover
Basis Points
Source: Federal Reserve Bank of New York, Haver; Federal Reserve Board staff calculations
21
Hybrid Models (cont’d)
These models don’t materially alter the cumulated payments that a borrower would make relative
to a basic Last Reset In Advance Product. They could be fairly easy to incorporate in to some
business loans, and from a systems perspective, all that would be needed is the ability for systems
to accrue interest and billing or principal accumulation accordingly.
40%
50%
60%
70%
80%
90%
100%
110%
1983 1986 1989 1992 1995 1998 2001 2004 2007 2010
Comparing Cumulated Payments due on Hybrid Models and an In Advance Mode in a 5/1 ARMl
Last Reset 6M Principal Adjustment Interest Rollover
Percent of Loan Amount
Source: Federal Reserve Bank of New York, Haver; Federal Reserve Board staff calculations
22
Questions?
23
