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Appendix 1. Simple versus Compound Interest
The ARRC conventions recognize that either simple or compound interest can be charged when using SOFR in arrears. As discussed in the User’s Guide to SOFR, although compound interest will more accurately reflect the time value of money and will match the payment structure in derivatives and debt market, simple interest is in some ways operationally easier to implement, because daily interest accruals only depend on the principal outstanding at the time of accrual, while daily accruals under compound interest will additionally depend on the amount of unpaid interest (or, as discussed in Appendix 3, the cumulative compound rate of interest rates from the start of an interest period)
The ARRC expects that market participants will choose between simple or compounded interest, depending on the circumstances of each loan; however, many members of the ARRC’s business loans working group expressed a preference for simple interest in arrears over compound interest in arrears for syndicated U.S. dollar business loans. Those who held this preference noted that the basis between simple and compound interest has historically been very small, and even in higher interest rate periods was a few basis points, as shown in the figure below:^1
As shown in the next figure, compared to the basis between 1-month and 3-month LIBOR, which is relevant for loans that allow the borrower to move between different LIBOR tenors, the basis between simple and compound interest is essentially de minimis.
(^1) As noted in the User’s Guide to SOFR, the difference between compound and simple interest depends on the level of interest rates, because compounding interest charged on unpaid accrued interest will be smaller when interest rates are low, and it will depend on the length of the interest reset, because compound interest increases with the length of the interest period.
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1998 2001 2004 2007 2010 2013 2016 2019
Basis Between Compound and Simple SOFR
Monthly Compound - Simple Basis Quarterly Compound - Simple Basis
In addition to recognizing that the basis between simple and compound interest is fairly small, many working group members also recognized that loan and loan-trading systems are already able to handle simple in arrears loans, and believed that while vendors will offer systems to calculate compound interest (and some have already), that the need to update operational systems to allow compounding would take up time and resources that could be devoted to instead transitioning away from LIBOR more quickly.
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1998 2001 2004 2007 2010 2013 2016 2019
Compound/Simple Bases vs LIBOR Basis
Monthly Compound - Simple Basis Quarterly Compound - Simple Basis Basis Between 3-Month and 1-Month LIBOR
Basis Points
early prepayment is possible, as it is in many loans, and would be difficult to integrate with trading amongst lenders.
For these reasons, the ARRC has preferred a lookback structure, and we focus on those conventions below. To explain what a lookback is, we first describe the payment structure of a loan without a lookback, and then describe the different versions of lookback structure that the ARRC considered.
In the terminology of these conventions, the interest date is the date that interest is applied for, while the observation date is the date that the SOFR rate is pulled from. Under compound interest, the daily SOFR rate is compounded across business days and the given SOFR rate applied over the number of calendar days until the next business day.^3 The distinction between business days and calendar days isn’t as important with simple interest, but under compound interest it is a more important operational consideration.
Without a lookback, the interest date is equal to the observation date and interest is charged based on the SOFR rate that the Federal Reserve Bank of New York publishes for that business day. In the example below, interest on July 2 would be based on the SOFR rate for July 2 and would be applied for 1 day until Wednesday, July 3; interest on July 3 would be based on the SOFR rate for July 3 and because since July 4 is a holiday, Wednesday’s rate would be applied for 2 business days until Friday, July 5.
(^3) SOFR is published on government securities trading days, as established by the Securities Industry and Financial Markets Association (SIFMA).
Mon, Jun 24, 2019 2.39 1 Tue, Jun 25, 2019 2.41 1 Wed, Jun 26, 2019 2.43 1 Thu, Jun 27, 2019 2.42 1 Fri, Jun 28, 2019 2.5 3 Mon, Jul 1, 2019 2.42 1 Tue, Jul 2, 2019 2.51 1 Wed, Jul 3, 2019 2.56 2 Fri, Jul 5, 2019 2.59 3 Mon, Jul 8, 2019 2.48 1 Tue, Jul 9, 2019 2.45 1
FRBNY SOFR DATA
DATE
RATE (PERCENT)
Calendar Days Until Next Business Day
No Lookback: The date that the SOFR rate is pulled from (the observation date) is the same date that interest is applied (the interest date) and applies until the next business day following the interest date.
Example: The rate for July 2 is applied on July 2 for one day, while the rate on July 3 is applied on July 3 for two days.
Because the SOFR rate for any given day is published on the following business day (the day that payment would be due on an overnight repo transaction or would be due on an overnight loan), without a lookback or some other convention to give more time for payment, the borrower would have at most a few hours to make final payment. We look at each of the different potential lookback structures in turn, but some readers may wish to only focus on the recommended convention of a lookback without observation shift.
Lookback without observation shift
A lookback gives counterparties more notice by applying the SOFR rate from some fixed number of business days prior to the given interest date. If the lookback is for k days, then the observation date is k business days prior to the interest date. In a lookback without an observation shift, all other elements of the calculation are kept the same and the reference to a previous SOFR rate is the only change made.
Continuing the example, using a 5-day lookback without observation shift in calculating interest for Tuesday, July 2, the SOFR rate for June 25 (5 business days prior to July 2) would be applied for 1 business day until Wednesday July 3, while in calculating interest for Wednesday, July 3, the SOFR rate for June 26 (5 business days prior to July 3) would be applied for 2 business days until Friday, July 5.
Mon, Jun 24, 2019 2.39 1 Tue, Jun 25, 2019 2.41 1 Wed, Jun 26, 2019 2.43 1 Thu, Jun 27, 2019 2.42 1 Fri, Jun 28, 2019 2.5 3 Mon, Jul 1, 2019 2.42 1 Tue, Jul 2, 2019 2.51 1 Wed, Jul 3, 2019 2.56 2 Fri, Jul 5, 2019 2.59 3 Mon, Jul 8, 2019 2.48 1 Tue, Jul 9, 2019 2.45 1
DATE
RATE (PERCENT)
Calendar Days Until Next Business Day
FRBNY SOFR DATA
Lookback: The date that the SOFR rate is pulled from (the observation date) is k business days before the date that interest is applied (the interest date) and applies until the next business day following the interest date.
Example of a 5-business day lookback: The rate for June 25 is applied on July 2 for one day, while the rate on June 26 is applied on July 3 for two days.
Lookback with observation shift
A lookback with observation shift also applies the SOFR rate from some fixed number of business days prior to the given interest date, but in contrast to a lookback without a shift, it applies that rate for the number of calendar days until next business date following the observation date.
Continuing the example, using a 5-day lookback without observation shift in calculating interest for Tuesday, July 2, the SOFR rate for June 25 (5 business days prior to July 2) would be applied for 1 business day until Wednesday July 3, while in calculating interest for Wednesday, July 3, the SOFR rate for June 26 (5 business days prior to July 3) would be applied for 1 business day.
A lookback with observation shift is one of the conventions that has been recommended by the ARRC for floating rate notes (FRNs).^4 However, syndicated loans have several complicating features that FRNs do not – principal can typically be repaid at any time, and syndicated loans are frequently traded between lenders and they do not trade clean.
The fact that principal may be repaid or that a lender may trade out of a loan before the end of an interest period makes implementing an observation shift more difficult in the loan market. For instance, in the example above, on July 3 interest is only charged for one day even though it would be two days until interest was paid. A lender who bought in to the loan on July 3 and sold out on July 5 may consider
(^4) This convention is described under Two-Day Backward Shifted Observation Period and No Lockouts in the ARRC’s SOFR Floating Rate Notes Conventions Matrix. See https://www.newyorkfed.org/medialibrary/Microsites/arrc/files/2019/ARRC_SOFR_FRN_Conventions_Matrix.pdf.
Mon, Jun 24, 2019 2.39 1 Tue, Jun 25, 2019 2.41 1 Wed, Jun 26, 2019 2.43 1 Thu, Jun 27, 2019 2.42 1 Fri, Jun 28, 2019 2.5 3 Mon, Jul 1, 2019 2.42 1 Tue, Jul 2, 2019 2.51 1 Wed, Jul 3, 2019 2.56 2 Fri, Jul 5, 2019 2.59 3 Mon, Jul 8, 2019 2.48 1 Tue, Jul 9, 2019 2.45 1
FRBNY SOFR DATA
DATE
RATE (PERCENT)
Calendar Days Until Next Business Day Lookback^ with observation shift:^ The date that the SOFR rate is pulled from (the observation date) is k business days before the date that interest is applied (the interest date) and is applied for the number of calendar days until the next business day following the observation date.
Example of a 5-business day lookback with observation shift: The rate for June 25 is applied on July 2 for one day, and the rate on June 26 is applied on July 3 for one day.
that they have been less than fully compensated given that they have provided some amount of principal for two days but only receive interest for one day. Or consider what was meant to be a monthly loan that began on July 8 but was repaid the next day. Under a 5-business day lookback with observation shift, the borrower would be charged for three day’s interest based on the SOFR rate for Friday June 28, even though they had only borrowed money for one day and should therefore only be charged for one day’s interest.
Without trading or without early repayment, these discrepancies would average out and would be inconsequential. Because principal is constant in FRNs (and because they trade clean, meaning that the purchaser receives the full coupon), an observation shift is more easily implementable. With trading and the possibility of early repayment, these kinds of discrepancies may be more problematic, and the ARRC Business Loans Working Group members felt that a lookback with observation shift would not be the most appropriate convention for the syndicated loan market.^5
Other Potential Lookback Conventions
The Business Loans Working Group did discuss several variants of the observation shift that could avoid some of these problems, although they ultimately did not recommend them for syndicated loans: an interest-period weighted shift, a simple-imputed shift, and a compound-imputed shift. We briefly outline each:
Interest-Period Weighted Shift
As discussed above, the effective SOFR rate is used to calculate daily accruals. Without a lookback, the effective rate is
𝑖𝑖 (^) 𝑡𝑡 =
𝑟𝑟 (^) 𝑡𝑡 × 𝑛𝑛𝑡𝑡 𝑁𝑁
and with a k -day lookback but no observation shift, the effective rate is
𝑟𝑟 (^) 𝑡𝑡−𝑘𝑘 × 𝑛𝑛𝑡𝑡 𝑁𝑁
As discussed further in Appendix 3, with no lookback or a lookback without observation shift, the unannualized cumulative compound rate of interest is
𝑈𝑈𝑈𝑈𝑈𝑈 (^) 𝑡𝑡 = ��(1 + 𝑖𝑖 (^) 𝑡𝑡)
𝑡𝑡
𝑏𝑏=
− 1 �
(^5) An analogy would be the difference between renting an apartment and staying at a hotel. Under a rental agreement, rent is the same each month even though some months have 28 days and others have 31 days, but the differences average out and people feel free to ignore them. In contrast, someone staying at a hotel is much more likely to take offense if they are charged for 3 days but only stayed 1 day or if they are charged a weekend rate when they stayed on a weekday.
For FRNs, these choices matter less, because principal is constant and so any differences between whether, for example, one interest period has 89 days and another has 91, will tend to average out quickly. In a loan that can be repaid on held for only a short period of time, the calculations may not average out, although a borrower (or lenders) may not place much importance on hedging a loan that they could quickly repay or sell.
Another consideration that led the ARRC not to recommend an interest-period weighted shift is that it can be difficult to implement a daily floor under this convention. Daily accruals may in some circumstances be negative even if SOFR rates are positive or floored. The spreadsheet ARRC BWLG Examples - Other Lookback Options.xlsx demonstrates how to calculate an interest-period weighted shift, and also provides an example of a negative daily accrual under this convention.
Simple-Imputed Shift
The problems with using an observation shift in the syndicated loan market arise when the number of calendar days between two observation dates are different than the number of calendar days between the corresponding interest dates. With a 5-day lookback, this would only occur around holidays. One way around this problem is to impute (or “fill-in”) rates for those holiday dates. The most straightforward way to do this is to apply the rate observed for the date immediately preceding the holiday.
The spreadsheet ARRC BWLG Examples - Other Lookback Options.xlsx also demonstrates how to calculate an a simple imputed shift. In the screen shot of the spreadsheet shown above, this convention would require calculating interest for July 4 (the calculation itself could take place on July 5, but interest would be compounded separately for July 3 and July 4) using a 5-day lookback to June 27, and in calculating interest for July 11, it would impute a rate for July 4 by using the July 3 rate. With these two rates filled in, the number of days in the observation period for a 5-day lookback would equal the number of days in the interest period.
Interest Date (t)
Observation Date (t-5)
Relevant SOFR Print (rt-5)
days
rate applie s (n (^) t-5)
SOFR Effective Rate
SOFR Cumulative Compounde d Effective Rate Mon, July 1, 2019 Mon, June 24, 2019 2.39% 1 0.00664% 0.00664% Tue, July 2, 2019 Tue, June 25, 2019 2.41% 1 0.00669% 0.01333% Wed, July 3, 2019 Wed, June 26, 2019 2.43% 1 0.00675% 0.02008% Thu, July 4, 2019 Thu, June 27, 2019 2.42% 1 0.00672% 0.02681% Fri, July 5, 2019 Fri, June 28, 2019 2.50% 3 0.02083% 0.04765% Mon, July 8, 2019 Mon, July 1, 2019 2.42% 1 0.00672% 0.05437% Tue, July 9, 2019 Tue, July 2, 2019 2.51% 1 0.00697% 0.06135% Wed, July 10, 2019 Wed, July 3, 2019 2.56% 1 0.00711% 0.06846% Thu, July 11, 2019 Wed, July 3, 2019 2.56% 1 0.00711% 0.07558% Fri, July 12, 2019 Fri, July 5, 2019 2.59% 3 0.02158% 0.09718%
Shift with Simple Imputation (holidays imputed)
While this convention does have somewhat less basis relative to a standard SOFR OIS swap than a lookback without observation shift, the differences are very slight – typically less than a basis point. At the same time, implementing this would require nontrivial changes to vendor and lender systems, and the modest improvement in basis did not seem sufficient to warrant such changes.
Compound-Imputed Calendar Shift
This would be essentially the same as method a simple-imputed shift, but rather than taking the last day’s rate (which is akin to a simple interest concept) this convention would impute an implied daily compound rate based on the rate from the previous business day. To do this, if the rate on the previous business day before a holiday was r and there were n calendar days until the next business day, then the imputed daily compounded rate would be
𝑟𝑟̅ = (1 + 𝑛𝑛 × 𝑟𝑟)
1 𝑛𝑛 − 1
This convention has slightly less basis than a lookback without observation shift relative to a standard OIS swap, but as with the simple-imputed shift, the reduction in basis is slight and adopting the convention would require nontrivial changes to vendor and lender systems. Additionally, BWLG members believed that it would be difficult to explain how the imputed rate had been calculated.
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Basis Relative to ISDA Compound SOFR
No Observation Shift Simple Imputed Shift
Basis Points
General Case: Compound Balance
Whereas under simple interest daily accrued interest depends only on the outstanding principal for that day:
𝐴𝐴 (^) 𝑡𝑡+1 = 𝐴𝐴 (^) 𝑡𝑡′^ + 𝑖𝑖 (^) 𝑡𝑡 ∙ 𝑃𝑃𝑡𝑡
with compound interest, daily interest accrual is charged both on outstanding principal and on accumulated unpaid interest:
(2) (^) 𝐴𝐴 (^) 𝑡𝑡+1 = 𝐴𝐴 (^) 𝑡𝑡′^ + 𝑖𝑖 (^) 𝑡𝑡 ∙ [𝑃𝑃𝑡𝑡 + 𝐴𝐴 (^) 𝑡𝑡′^ ]
This formula is the basic definition of compound interest – interest is charged both on outstanding principal and accumulated unpaid accrued interest. Within the ARRC Business Loans Working Group, this equation (2) has been termed the “Compound Balance” approach to calculating compound interest. The Compound Balance approach can be applied regardless of whether principal changes or whether some portion of interest is repaid during an interest period.
The daily interest accrual under the Compound Balance approach is simply calculated by applying the appropriate day’s SOFR rate to outstanding principal and accrued unpaid interest:
𝐴𝐴 𝑡𝑡+1 − 𝐴𝐴 𝑡𝑡′^ = 𝑖𝑖 𝑡𝑡 ∙ [𝑃𝑃𝑡𝑡 + 𝐴𝐴 𝑡𝑡′^ ]
The ARRC has published a spreadsheet ARRC BWLG Compounding Methods Examples.xlsx containing examples of the different methods of calculating compound interest. A screenshot of the worksheet with an example of compound balance is shown below. Implementation requires keeping track of accumulated interest as shown in the screenshot:
A F G H I J K (^1) p
3 Effective Rate
Accumulated Unpaid Interest Before Paydown (A (^) t )
Accumulated Unpaid Interest After Paydown (Aʹt )
Daily Base Interest Accrual 4
Interest Date (t)
Principal (Pt) 12 July 9, 2019 $100,000,000.00 0.00681% $56,400.74 $56,400.74 $6,892. 13 July 10, 2019 $100,000,000.00 0.00683% $63,210.14 $63,210.14 $6,809. 14 July 11, 2019 $100,000,000.00 0.00669% $70,047.79 $70,047.79 $6,837. 15 July 12, 2019 $100,000,000.00 0.01967% $76,746.92 $76,746.92 $6,699. 16 July 15, 2019 $90,000,000.00 ($9,642.87) 0.00683% $96,428.68 $86,785.81 $19,681. 17 July 16, 2019 $90,000,000.00 0.00686% $92,941.74 $92,941.74 $6,155. 18 July 17, 2019 $90,000,000.00 0.00686% $99,123.12 $99,123.12 $6,181.
SOFR Rate and Principal/Paydown Information Compound Interest Accrual Calculations
Interest Paydown (PDt)
Daily Accrual Cell K18 = H17(F17+J17)*
Accumulated Unpaid Interest Before Paydown Cell I18 = J17+ H17(F17+J17)*
Accumulated Unpaid Interest After Paydown Cell J18 = I18 + G
Special Case: Compound Rate
While the Compound Balance approach can be applied generally, as discussed further in Box A.1, the “Compound Rate” approach should only be employed under specific conditions:
a) Principal remains constant within an interest period, or b) If some portion of principal is repaid, then a corresponding proportion or accrued interest is repaid at the same time.
Under the specific conditions, the general formula can be simplified to the (non-annualized) version of ISDA’s formula for Compound SOFR
(3) (^) 𝐴𝐴 (^) 𝑡𝑡+1 = UCR𝑡𝑡 ∙ 𝑃𝑃𝑡𝑡
Where the term UCR𝑡𝑡 = [∏^ 𝑡𝑡𝑏𝑏=1( 1 + 𝑖𝑖𝑏𝑏 )− 1 ]^ is called the Unannualized Cumulative Compound Rate.
Daily accrual can be calculated directly using this equation and equation for 𝐴𝐴 (^) 𝑡𝑡′
𝐴𝐴 (^) 𝑡𝑡+1 − 𝐴𝐴 (^) 𝑡𝑡′^ = A𝑡𝑡+1 − A𝑡𝑡 − 𝑃𝑃𝑃𝑃𝑡𝑡
but market participants have tended to prefer a variant of this calculation, the “Noncumulative Compound Rate” approach, which recognizes that the required relationship between that amount of interest paid down and any reduction in principal implies that this calculation for daily accrued interest can be simplified to:
𝐴𝐴 (^) 𝑡𝑡+1 − 𝐴𝐴 (^) 𝑡𝑡′^ = (𝑈𝑈𝑈𝑈𝑈𝑈 (^) 𝑡𝑡 − 𝑈𝑈𝑈𝑈𝑈𝑈 (^) 𝑡𝑡−1)^ ∙ 𝑃𝑃𝑡𝑡
A F G H I J 1
3 Effective Rate
Unannualized Cumulative Compound Rate (UCRt )
Noncumulative Compound Rate Daily Base Interest Accrual
4
Interest Date (t)
Principal (Pt ) 12 July 9, 2019 $100,000,000.00 0.00681% 0.06321% $6,892. 13 July 10, 2019 $100,000,000.00 0.00683% 0.07005% $6,809. 14 July 11, 2019 $100,000,000.00 0.00669% 0.07675% $6,837. 15 July 12, 2019 $100,000,000.00 0.01967% 0.09643% $6,699. 16 July 15, 2019 $90,000,000.00 ($9,642.87) 0.00683% 0.10327% $19,681. 17 July 16, 2019 $90,000,000.00 0.00686% 0.11014% $6,155. 18 July 17, 2019 90,000,000.00 0.00686% 0.11701% 6181.
SOFR Rate and Principal/Paydown Information Compound Interest Accrual Calculations
Interest Paydown (PD (^) t )
Unannualized Cumulative Compound Rate Cell I17 = (1+H17)(1+I16) - 1*
Daily Accrual Cell J18 = (I17 – I16) F*
The first formula is ISDA’s definition for Compound SOFR, and the second is a similar formula based on Box A1. Conditions Where the Compound Rate Approach Correctly Accrues Interest
The ISDA Compound SOFR formula was created for a standard OIS swap, which have constant principal over an interest period., but the formula can be applied to a loan or other instrument for which principal is constant within an interest period.
The equation isn’t too difficult to derive when principal is constant. First note that in the absence of any interest paydown, 𝐴𝐴 (^) 𝑡𝑡′^ = 𝐴𝐴 (^) 𝑡𝑡 and that if principal is constant (𝑃𝑃𝑡𝑡 +1 = 𝑃𝑃𝑡𝑡 = 𝑃𝑃) then the compound balance equation (1) can be written as: (𝐴𝐴 (^) 𝑡𝑡+1 + 𝑃𝑃) = (1 + 𝑖𝑖 (^) 𝑡𝑡) (^) ∙ (𝐴𝐴 (^) 𝑡𝑡 + 𝑃𝑃) Using this, and the fact that at the start of an interest period there is no unpaid accrued interest (𝐴𝐴 1 = 0), one can recursively solve for accrued interest: (𝐴𝐴 2 + 𝑃𝑃) = (1 + 𝑖𝑖 1 )𝑃𝑃 (𝐴𝐴 3 + 𝑃𝑃) = [(1 + 𝑖𝑖 2 )(1 + 𝑖𝑖 1 )]𝑃𝑃 (𝐴𝐴 4 + 𝑃𝑃) = [(1 + 𝑖𝑖 3 )(1 + 𝑖𝑖 2 )(1+ 𝑖𝑖 1 )]𝑃𝑃 ⁞ (𝐴𝐴 (^) 𝑡𝑡+1 + 𝑃𝑃) = �� (1 + 𝑖𝑖 (^) 𝑏𝑏)
𝑡𝑡 𝑏𝑏=
�𝑃𝑃
Which gives equation (3), 𝐴𝐴 (^) 𝑡𝑡+1 = [∏^ 𝑡𝑡𝑏𝑏=1(1 + 𝑖𝑖 (^) 𝑏𝑏)− 1]𝑃𝑃
If the borrower repays some portion of principal within an interest period, then the compound rate methods will not generally calculate accrued interest correctly. However, there is a special case, when a proportionate share of accrued interest is repaid along with principal, in which the compound rate approach will still calculate the correct amount of accrued interest. To see that, assume that there has been no principal paydown or interest repayment up to time t , so that the compound rate formula holds up to that time. If the share of principal paid down is α, then: 𝑃𝑃𝑡𝑡 = (1 − 𝛼𝛼)^ ∙ 𝑃𝑃𝑡𝑡 − and f or the compound rate equations to correctly accrue interest, it would require that the same share of accrued interest was also paid back at the same time so that: 𝐴𝐴 (^) 𝑡𝑡′^ = (1 − 𝛼𝛼) ∙ 𝐴𝐴 (^) 𝑡𝑡 With this proportionate interest paydown we have: 𝐴𝐴 (^) 𝑡𝑡′^ = (1 − 𝛼𝛼) ∙ ��^ (1 + 𝑖𝑖 (^) 𝑏𝑏)
𝑡𝑡− 𝑏𝑏=
− 1 � 𝑃𝑃𝑡𝑡 −
= �� (1 + 𝑖𝑖 (^) 𝑏𝑏)
𝑡𝑡− 𝑏𝑏=
− 1 � 𝑃𝑃𝑡𝑡
Using this with the compound balance equation 𝐴𝐴 (^) 𝑡𝑡+1 = (1 + 𝑖𝑖 (^) 𝑡𝑡)^ ∙ (𝐴𝐴 (^) 𝑡𝑡′^ + 𝑃𝑃𝑡𝑡 )^ − 𝑃𝑃𝑡𝑡 gives
𝐴𝐴 (^) 𝑡𝑡+1 = (1 + 𝑖𝑖 (^) 𝑡𝑡)^ ∙ ��^ (1 + 𝑖𝑖 (^) 𝑏𝑏)
𝑡𝑡− 𝑏𝑏=
� 𝑃𝑃𝑡𝑡 − 𝑃𝑃𝑡𝑡
= ��^ (1 + 𝑖𝑖 (^) 𝑏𝑏)
𝑡𝑡 𝑏𝑏=
− 1 � 𝑃𝑃𝑡𝑡
Appendix 4. Floors
Although interest rate floors in derivatives and FRNs would generally be implemented as a floor on the compound average rate determined at the end of the period, the ARRC is recommending a daily floor for SOFR in arrears syndicated loans. Because these loans can be repaid early and because trading in syndicated loans is not clean, without a daily floor it is possible that the amount a borrower would pay in the instance of an early payment or the amount a lender would earn if they did not hold the loan for the entire interest period would be less than the intended floor.^9
While this convention is different from some other products, it is straightforward to implement in a new SOFR in arrears loan. With either simple or compound interest in arrears, the daily SOFR rate would be floored at the desired rate level and then interest (either simple or compound) would be accrued on the floored rate while any accompanying margin would then be added as a simple interest accrual.
For legacy LIBOR loans falling back to SOFR, because LIBOR will convert to a spread-adjusted SOFR rate that takes in to account the historic differences between LIBOR and SOFR, the floor should be adjusted. If there is a floor for LIBOR, then the ARRC recommends that comparable floor for daily SOFR should be
(1) SOFR Floor = Legacy LIBOR Floor - ARRC Spread Adjustment
Under simple interest, it is straightforward to see that this will ensure that sum of the floored Daily Simple SOFR rate and the ARRC’s recommended spread adjustment will not fall below the original LIBOR floor, and it is equivalent to flooring the spread-adjusted Daily SOFR rate at the original LIBOR floor.
Although it is more involved to prove, this will also ensure that the combined annualized daily rate of accrual on Daily Compounded SOFR and the ARRC’s recommended spread adjustment does not fall below the legacy LIBOR floor. Per the recommended treatment of the spread adjustment in a LIBOR loan that transitioned to Daily Compounded SOFR, the legacy loan falling back would compound the daily SOFR rate but would apply the spread adjustment as simple interest.^10 In order to implement this, the Daily Compounded SOFR rate would first be floored according to equation (1) and then compound interest would be calculated using this floored Daily Compounded SOFR rate, while the spread adjustment and any margin accompanying the loan would accrue as simple interest.
The ARRC has published an accompanying spreadsheet ARRC BWLG Daily Floor Examples.xlsx that demonstrates how to calculate compound accrued interest on a legacy loan under these proposed conventions. The spreadsheet provides an example of a zero LIBOR floor and uses ESTR data rather
(^9) For a SOFR loan in advance, either based on the SOFR averages published by the Federal Reserve Bank of New York or a potential forward-looking term SOFR rate, the floor would not need to be daily because the same in advance rate would be used throughout the interest period. (^10) The ARRC’s recommended spread adjustment to compound SOFR in arrears for business loans will be based on the five- year historical median of the difference between a given tenor of LIBOR and the corresponding compound average of SOFR. As such, the spread adjustment should not itself be compounded, because it already reflects the difference between LIBOR and a compound average of SOFR. Including the spread adjustment as part of the compound interest calculations of the loan would in essence compound that amount twice.
