2. Overview of Rates

from fixedincome2025 import table

Overview

• We have only seen US Treasury yield. There are many more different rates in the market

• Assignment: Please read through Sections 4.1 - 4.3 of Options, Futures, and Other Derivatives 11th Edition by John Hull, Sankarshan Basu.

Fed Funds Rates

• At end of date each day, US banks are required by the regulator to maintain certain cash level (the reserve) depending on the business

• A bank can have more/less than they are required

• This leads to an interbank overnight lending market and the interest rate, determined by supply and demand, is the federal funds rate

• Central bank is in the market and can control the target rate by buying US treasuries from participating banks

• Fed funds rates are unsecured

SONIA and ESTER

• Other countries have similar systems and unsecured overnight rates

• In UK, it’s the sterling overnight index average (SONIA)

• In eurozone, it’s the euro short-term rate (ESTER)

Repo Rates

• A repurchase agreement (repo) is a contract for bank A to sell a security to bank B and promise to buy it back next day at a slightly higher price

• It is a secured lending transaction

  • slightly lower than fed funds rates

• There are overnight repos (the most common) and longer “term repos”

• The underlying securities can be

  • U.S. Treasury securities (most frequently used) due to their safety and market liquidity

  • Corporate bonds

  • Municipal bonds

  • Mortgage-backed securities (MBS), often used in term repo

SOFR (Daily)

• The secured overnight financing rate

• Volume-weighted median average of the overnight repo rates in the US

The Risk-Free Rate

• In financial modeling, when a risk-free rate is needed, SOFR is usually used

• Why not US treasury?

• Traders think US Treasury yield is too low to be used as the risk-free rate as

  • Treasuries must be purchased by financial instritutions to fulfill regulatory requirements, which increases the demand of the Treasuries, driving the yield down

  • The amount of capital a bank is required to hold to support an investment in the Treasuries is much smaller than other low-risk investments

  • Tax advantages: Coupons from the Treasuries are exempt from state and local income taxes

Referencing Rates (Index Rates)

• Many interest rate products need an index rate

• 5/1 ARM mortgage:

  • Rates are fixed to be \(5.37\%\) for 5 years and floating after (meaning mortgate rates are different every month after 5 years)

  • The floating rate is SOFR plus a fixed margin

  • If your margin is \(2.5\%\) and the SOFR then is \(3.75\%\), then you’ll be charged \(3.75\% + 2.5\% = 6.25\%\)

• The SOFR in this story is the index rate

LIBOR

• Retired since 2021 but still an important concept in rates modeling

• A daily survey published at 11:00 am LON time

• The participating banks answer this question: What’s the rate you want to charge today on a 3M lending?

• The median average of the answer is today’s 3M LIBOR rate

• There were other tenors, like 1M LIBOR, 6M LIBOR rates surveyed and published daily

• Unsecured

• No lending actually happened

LIBOR Scandal

• The LIBOR scandal is a series of fraudulent practices related to LIBOR that came in light in 2012

• Manipulation

  • Some banks engaged in fraudulent and collusive practices in submitting LIBOR rates, intentionally raising or lowering their bids to influence borrowing costs and profit

• Impact

  • LIBOR was widely used as a reference rate for many financial products, including mortgages and derivatives. Bank manipulation of LIBOR has a direct impact on consumers and financial markets worldwide

  • Statistics show that between 2000 and 2009, LIBOR rates rose on the first day of each month, increasing the burden on homeowners’ monthly payments

• Subsequent Developments

  • In June 2012, Barclays was fined \(\$453\) million, and UBS fined \(\$1.5\) billion, for manipulating LIBOR

  • The UK authorities’ investigation lasted 7 years and concluded in Oct 2019

  • The scandal paved the way for the LIBOR cessation

SOFR (Compounded Daily Rate; Term Rate)

• If you keep borrowing at the overnight SOFR, and roll it to 3M, what’s interest rate charge?

• It’s approximately

  \[\frac{1}{\tau}\left(\prod_j (1+\Delta t_j r_j) - 1\right),\]

• \(r_j\): overnight SOFR

• \(\Delta t_j\): one day’s year fraction, for example \(1/360\) for week days and \(3/360\) for a regular weekend

• \(\tau\): year fraction of the term, for example \(0.25\) for 3M

• SOFR term rate is currently the most common and important index rate by far

• Unlike LIBOR, SOFR is backward looking, which makes rates models different now

SOFR

In the compounding formula

\[\frac{1}{\tau}\left(\prod_j (1+\Delta t_j r_j) - 1\right),\]

The term

\[\prod_j (1+\Delta t_j r_j)\]

is a more general version of

\[\left(1+ \frac{r}{365}\right)^{365},\]

which is close to \(e^r\), a quantity slightly larger than one, as it’s the terminal account value after rolling daily CD 365 times in a year. See the discussion on simple and continuous compounding in the previous set. The \(-1\) is to compute the rate and \(1/\tau\) is to annualize it.

CME Term SOFR

• Term SOFR is a forward looking index rate different from SOFR

• Sometimes client wants a forward looking rate so that they can prepare funds ahead of time

• Quote implied from the SOFR futures, published daily by CME (more on SOFR futures later, which is also a CME product)

• When market anticipate a rate cut, longer terms can have lower rates, like the data below

table('term_sofr')

	Term SOFR 1M	Term SOFR 3M	Term SOFR 6M	Term SOFR 12M
09-17-2025	4.13359 %	4.02304 %	3.85385 %	3.59516 %
09-16-2025	4.13584 %	4.02570 %	3.85447 %	3.60384 %
09-15-2025	4.14288 %	4.02330 %	3.84833 %	3.60702 %

• Data from here

CMT and CMS

• A CMT (Constant Maturity Treasury) rate is for example the 10Y US Treasury yield (zero rate) observed daily

• Notation is \(R(t, t+\text{10Y})\)

• Can also be 1Y, 2Y, 30Y, etc.

• CMS is Constant Maturity Swap, say 10Y swap rate observed daily

• More on interest rate swaps and swap rates later