the LIBOR Transition: Key Points and Implications

LIBOR Transition

Qinjun Su, Xiaocheng Liang, Zhengxi

Yan

Introduction of LIBOR

Transition

Background

●

IBORs were the most widely used benchmark rates for a wide range of

financial products

●

LIBOR Scandal in 2012

●

Recommendations to reform and use near risk-free rates (RFR) in 2014 by

the Financial Stability Board

Replacement Benchmarks

Replacement Benchmarks

Libor Cessation Timeline

Difference Between IBORs and RFRs

●

RFRs are based on short-term wholesale transactions for unsecured RFRs

and repo transactions for secured RFRs. Therefore, RFRs do not require a

credit spread

●

IBORs are forward-looking term rates while RFRs are backward-looking

overnight rates

LIBOR Fallback

Fallback Methodology Background

●

The methodology used in the calculation of the IBOR fallbacks was

determined through a series of market-wide consultations conducted by

ISDA.

●

Bloomberg constructed the rules that detail how to implement the

calculations by the methodology.

●

The selected methodology is fixed and cannot be altered without ISDA

conducting further consultations.

Two Parts of Fallback Rate

Adjusted Reference Rate:

Compounded in arrears over a period corresponding to the tenor to get the

adjusted reference rate.

Spread Adjustment:

Used to adjust for the different nature of rates.

They are all published for each weekday.

Fallback Rate

Adjusted Reference Rate

Calculation of Adjusted Reference Rate

Right Part of ARR

Standard compounding

Performs the compounding of the reference rate over an accrual period denoted

by the term AP.

Middle Part of ARR

Annualization Factor part

This is the number of times the accrual period goes into a year under count

convention of the reference rate.

Left Part of ARR

Adjusted Day count convention

Upper: IBOR day count

Lower: Reference Rate day count

To be specific

Spread Adjustment

Used to adjust for the different nature of rates.

It’s Median spread between IBOR and adjusted reference rate over a five-year

historical period.(before cessation)

Become fixed value when the IBOR cessation is triggered.

Median spread between the IBOR and ARR over 5-year period.(MP period)

When the cessation is triggered:

Spread Adjustment

Calculation of Spread Adjustment

Spread Adjustment

SOFR Market Convention

I

n

A

r

r

e

a

r

s

V

S

I

n

A

d

v

a

n

c

e

I

n

A

d

v

a

n

c

e

reference a value determined before the beginning

of the interest period.

I

n

A

r

r

e

a

r

s

reference a value determined at the end of the

interest period.

Basis created by different convention

Depend on:

●

Whether interest rates happen to be trending up or down over a given period?

The lender will face a comparable sort of basis relative to in arrears if rates

rise during the interest period.

●

How frequently payments are made?

Typically, less payment frequency leads to more basis.

I

n

A

r

r

e

a

r

s

Plain Arrears:

under a pure in arrears structure, the SOFR rate for each given day in the interest

period would be applied to calculate interest for that business day, and interest would be

paid on the first day of the next interest period

I

n

A

r

r

e

a

r

s

Payment Delay:

Interest is calculated in the same way as in a plain arrears framework, with the SOFR

rate for each given day in the interest period applied to calculate interest for that

business day, but interest is paid k days after the start of the next period.

I

n

A

r

r

e

a

r

s

Lockout or Suspension Period:

the SOFR rate applied for the last k days of the interest period is frozen at the rate

observed k days before the period ends.

I

n

A

r

r

e

a

r

s

Lookback:

For each day in the interest period, the SOFR rate from k business days earlier is

used to accrue interest.

I

n

A

r

r

e

a

r

s

Lookback without 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 interest date.

I

n

A

r

r

e

a

r

s

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.

I

n

A

d

v

a

n

c

e

Last Reset:

Use the averaged SOFR over the last interest reset

period as rate for current interest period.

(Similar to a lookback model and will more closely

match the structure of an OIS, though the payment

structure will be lagged.)

Last Recent:

Use the averaged SOFR from a shorter recent

period as rate for current interest period.

 (Likely to have less basis relative to the in

arrears average interest rate over the current

interest period. )

An in advance payment structure based on an overnight rate would reference an average of the

overnight rates observed before the current interest period began.

ARRC Conventions for Different Product

ISDA RFR Conventions

●

https://www.isda.org/a/bdigE/RFR-Conventions-and-IBOR-Fallbacks-

Product-Table-October-2021.pdf

Term SOFR

Introduction

CME Term SOFR:

●

A daily set of forward-looking interest rate estimates, calculated and

published for 1-month, 3-month, 6-month and 12-month tenors.

Publication:

●

Each day the New York Federal Reserve calculates and publishes SOFR.

●

Publication will occur at 5:00 am CT.

ARRC’s recommendation on the usage of Term SOFR

●

The ARRC continues to recommend overnight SOFR and SOFR averages for all

products, particularly in markets where we have seen that there can be

successful adoption of these rates such as floating rate notes, consumer

products including adjustable rate mortgages and student loans, and most

securitizations.

●

The ARRC also supports the use of the SOFR Term Rate in areas where use of

overnight and averages of SOFR has proven to be difficult, such as multi-lender

facilities, middle market loans, and trade finance loans.

●

The ARRC does not support the use of the SOFR Term Rate for the vast majority

of the derivatives markets, because these markets already reference SOFR

compounded in arrears and transitioning derivatives markets to the more robust

overnight risk-free rates (RFRs) is essential to ensure financial stability.

ARRC’s recommendation on the usage of Term SOFR

●

The ARRC  recommends that any use of SOFR Term Rate derivatives be

limited to 

end-user

 facing derivatives intended to hedge cash products that

reference the SOFR Term Rate.

The ARRC considers an end-user to be:

●

A direct party or guarantor, either a lender or borrower who have entered into

a SOFR Term Rate business loan

●

A dealer counterparty would not be considered an end-user under these

recommendations

ARRC’s recommendation on the usage of Term SOFR

●

The ARRC does not recommend the trading of SOFR Term Rate derivatives

in the interdealer market because such activity could undermine trading

activity in the underlying overnight SOFR derivatives that are needed to

construct the SOFR term rate itself and could, thereby, compromise the

robustness of the rate and its corresponding utility to market participants.

Input Data selection

Input Data:

●

One-month SOFR Futures (SR1): 13 (thirteen) consecutive months contracts

To calculate the final settlement of a one-month SOFR Future:

The simple arithmetic average of the daily SOFR rates of the calendar month is calculated.

●

Three-month SOFR Futures (SR3): 5 (five) consecutive months contracts

To calculate the final settlement of a three-month SOFR Future:

Simple interest is accrued to all non-Business Days based on the daily SOFR benchmark from the

preceding Business Day.

Compounded interest is accrued to all Bussiness Days based on the daily SOFR benchmark.

Input Data selection

Sampling Market Prices:

CME Term SOFR Reference Rates use executed transactions and executable bids and

offers in SOFR Futures, traded on the CME Designated Contract Market (DCM).

Selected Prices derived from:

●

Volume Weighted Average Prices (VWAP)

●

Midpoint of the bid/ask that are calculated based on a snapshot of executable

bid/ask prices at a random moment

Calculation Methodology

Assumption:

●

The overnight SOFR rates follow a piecewise constant step function and can

only jump up or down the day after FOMC Policy Rate announcement dates

and remains at those levels across all dates in between the FOMC Policy

Rate announcement dates.

●

CME One-month SOFR Futures7 (SR1) and CME Three-month SOFR

Futures8 (SR3) contracts provide estimates of values of overnight SOFR on

average over the specific contract reference periods; CME SOFR Futures do

not directly provide estimates of individual overnight SOFR rates.

Calculation Methodology

The overnight SOFR rate for date t can be computed as:

●

𝑴

𝒌

: the date of the k th FOMC policy rate announcement date that occurs on or after 𝒕

𝟎

●

𝜽

𝟎

: the initial overnight SOFR rate as of date 𝒕

0

●

𝜽

𝒌

: the jump size in overnight SOFR rate occurs on the day after the k-th FOMC policy rate

announcement date. A positive 𝜽

𝒌

 means the overnight SOFR rate jumps up after the k-th

FOMC policy rate announcement date 𝑴𝒌; a negative means the overnight SOFR rate jumps

down after the k-th FOMC policy rate announcement date 𝑴

𝒌

●

𝒇

𝒕;𝜽

: the overnight SOFR rate as of date 𝒕, where 𝚯 = (𝜽

𝟎

, … , 𝜽

𝒌

) and 𝑲 is the index of the last

relevant FOMC policy rate announcement date

●

𝟏{∙}: binary function returning 1 if the statement in the parenthesis is true and 0 otherwise

Calculation Methodology

Calculation Methodology

Optimization:

●

𝑃

m

1

 and 

𝑃

q

3

 : the observed blended prices of SR1 and SR3 contract with

reference month 𝑚 and reference quarter 𝑞, respectively

●

𝑃

m

1

 (Θ) and 

𝑃

q

3

 (Θ): the implied value of SR1 and SR3 contract with reference

month 𝑚 and reference quarter 𝑞, respectively

●

𝑤

m

1

 and 

𝑤

q

3

 : weighting parameters for pricing errors of SR1 and SR3 with

reference month 𝑚 and reference quarter 𝑞, respectively

●

𝜆: weighting parameter for penalty function.

Calculation Methodology

For SR1 contracts, whose reference month is not the current month (𝑚 > 0), the

implied value only depends on projected overnight SOFR rates:

●

T

𝑚

1

 : set of calendar days for the 𝑚-th month

●

N

𝑚

1

 : total number of calendar days in 𝑚-th month.

Calculation Methodology

For the SR1 contract, whose reference month is the current month (𝑚 = 0), the

implied value can be calculated using published SOFR fixings and projected

overnight SOFR rates:

●

𝑇

0

1+

 = { 𝑡 𝜖 𝑇

0

1

 | 𝑡  ≥ 𝑡

0

 }

●

𝑇

0

1-

 = { 𝑡 𝜖 𝑇

0

1

 | 𝑡  < 𝑡

0

 }

●

𝑟

𝑡

 : published SOFR fixing for date 𝑡

Calculation Methodology

For SR3 contracts, whose reference quarter is not the current quarter (𝑞 > 0), the

implied value only depends on projected overnight SOFR rates:

●

𝑇

𝑞

3

 : set of Business Days for the 𝑞 -th quarter;

●

𝑁

𝑞

3

 : total number of calendar days in 𝑞-th quarter

●

 𝑑

𝑡

 : the number of calendar days from date 𝑡  to its next Business Day

following the SIFMA US Holiday Schedule

Calculation Methodology

For the SR3 contract, whose reference quarter is the current quarter (𝑞 = 0), the

implied value can be calculated using published SOFR fixings and projected

overnight SOFR rates:

●

𝑇

q

3+

 = { 𝑡 𝜖 𝑇

q

3

 | 𝑡  ≥ 𝑡

0

 }

●

𝑇

q

3-

 = { 𝑡 𝜖 𝑇

q

3

 | 𝑡  < 𝑡

0

 }

●

𝑟

𝑡

 : published SOFR fixing for date 𝑡

Computing Term Rates

Term Rates are derived by compounding the overnight SOFR rates over one,

three, six and twelve months:

●

𝑇̃ (𝑇 ): the set of Business Days from the term start date to date 𝑇 days in the future. The

term rate will span the corresponding tenor (e.g., 1-month, 3-month, 6-month, 12- month

which is represented by 𝑇 days in the formula)

●

𝑡 : a Business Day in set 𝑇̃ (𝑇 )

●

𝑑

𝑡

 : the number of calendar days from date 𝑡 to its next Business Day following the SIFMA US

Holiday Schedule.

●

𝑓 (𝑡, Θ): the overnight SOFR rate as of date 𝑡

Term SOFR VS LIBOR

SOFR-based Curve

Construction

Abramov et al. (2020)

The Curve Building instruments

1.

SOFR spot rate

2.

1-month SOFR futures (SR1)

3.

3-month SOFR futures (SR3)

4.

SOFR swaps (SWP)

Data Input

We denote the front SR1 to be SR1-0, and following maturities to be SR1-1, SR1-2,

…

Same with SR3

Only SR1-1 and SR3-1 through SR3-6 will be used

SOFR swaps starting from 2 years through 40 years

Step 1 - spot SOFR rate

Record spot SOFR rate from FRB web page

https://www.newyorkfed.org/markets/reference-rates/sofr

Step 2 - SR1 futures

Step 3 - SR3 futures

Following prices can be calculated using,

Step 4 - SOFR Swaps

SOFR future curve

Extended Models

Xu (2022)

●

mean-reverting Gaussian model : 1-factor Hull-White Model

SOFR Short Rate Model

●

Under Ti- forward measure, x(t) follows SDE:

SOFR Swap and Swaption

●

Fix-floating swaps are priced the same as a vanilla LIBOR swap

●

Swaptions are similar to LIBOR except index and resetting

Lyashenko and Mercurio (2019)

●

Extend the classic interest rate modeling framework for forward risk-free

term rates

●

In particular, the extension of LIBOR Market Model (LMM) to the backward-

looking rates, the Forward Market Model (FMM)

●

https://www.risk.net/awards/7204391/quants-of-the-year-andrei-lyashenko-

and-fabio-mercurio

The Forward Rate Dynamics

g(t) is used to capture the behavior of volatility decay in the accrual period

The Generalized FMM

●

Previous equation defines the dynamics of each forward rate under the

corresponding Tj-forward measure

●

We then apply change of numeraire to find its drift under the following three

cases:

○

Risk-neutral measure Q

○

Classic spot-LIBOR measure Qd

○

Tk-forward measure

The Generalized FMM

The Generalized FMM

Differences between LMM and FMM

●

Model completeness

●

Better pricing of futures contracts

●

Easier extension to a cross-currency interest-rate model

●

More natural hybrid modeling

Piterbarg (2022)

●

CCPs discounting switch on swaptions

●

Impact of the benchmark reform on non-linear rates market

Discounting and Swaptions

●

After part of rates benchmark reform, CCPs have announced that their

collateral rates will switch from legacy overnight rates (EFFR) to replacement

rates.

●

CCPs have proposed a compensation mechanism to eliminate the potential

value transfer.

Compensation for Value Transfer

Denote the forward annuity and the swap rate under two discount curve:

Compensation for Value Transfer

The Value of swaption under LIBOR(f)/SOFR(s)

Compensation for Value Transfer

Simplify the the notions and set by:

Set the transfer value to 0:

Calculate the factor c: we should replace one FedFunds swaption with c SOFR

swaptions.

Caps

Background

Interest rate caps present another challenge for the benchmark reform, in this

case caused by term Libor rates being replaced by a daily compounded in- arrears

overnight rate.

Discussion is only applicable to USD and GBP

Caps

Value before reform

Replace the LIBOR Rate by compounded in-arrears daily rate with a spread:

r(s): overnight rate

Caps

Difference:

the rate is fixed at time T.

Whereas, in the latter it continues to stochastically evolve till T+tau

Caps

Consider one-period swaptions in the Libor World:

post-Libor World:

Caps

From A. Lyashenko and F. Mercurio (2019):

Then, we can derive the value of the caplet:

Disappearing Species?

●

Libor-In-Arrears

●

Range Accruals

Willems (2020)

●

To provide an extension of SABR model to model the dynamics for the

forward value of the compounded overnight rate over a certain accrual period

●

Before entering the period: standard SABR dynamics

●

After entering the period: the rate keeps evolving stochastically but its

volatility is gradually scaled down

Forward-looking and Backward-looking Caplet

Backward-looking SABR model

Forward-looking and Backward-looking Caplets

Backward-looking effective SABR parameters

Backward-looking effective SABR parameters

Reference

Alternative Reference Rate Committee (2018), Second report

https://www.newyorkfed.org/medialibrary/Microsites/arrc/files/2018/ARRC-Second-report

Alternative Reference Rate Committee (2016), Interim Report and Consultation

https://www.newyorkfed.org/medialibrary/microsites/arrc/files/2016/arrc-interim-report-and-

consultation.pdf?la=en

An Updated User’s Guide to SOFR 

https://www.newyorkfed.org/medialibrary/Microsites/arrc/files/2021/users-

guide-to-sofr2021-update.pdf

Abramov, Vilen and Zhou, Xianwen and Bian, Zhengye, SOFR Bootstrapping Modeling Methodologies and

Issues (w/Python and Excel Replicas of Bloomberg SOFR @ GitHub) (July 17, 2020). Available at SSRN:

https://ssrn.com/abstract=3654466 or http://dx.doi.org/10.2139/ssrn.3654466

Reference

Xu, Mingyang, SOFR Derivative Pricing Using a Short Rate Model (December 12, 2021).

Lyashenko, Andrei and Mercurio, Fabio, Looking Forward to Backward-Looking Rates: A Modeling

Framework for Term Rates Replacing LIBOR (February 6, 2019).

Lyashenko, Andrei and Mercurio, Fabio, Looking Forward to Backward-Looking Rates: Completing the

Generalized Forward Market Model (November 6, 2019).

Piterbarg, Vladimir, Interest Rates Benchmark Reform and Options Markets (February 14, 2020).

Willems, Sander, SABR Smiles for RFR Caplets (April 3, 2020).