Types of Interest Rates

Open a trading floor's rates screen and you will see not one interest rate but dozens: a Treasury curve, a SOFR curve, a repo curve, a prime rate, a fed funds rate, a swap curve — each ticking independently. Every model you have used so far — cost-of-carry forward pricing among them — has quietly assumed a single number, "the" risk-free rate r. In practice a desk has to choose which of these screens' numbers plays that role, for which purpose, and the honest answer is: it depends who is asking, and it has changed within living memory. This module builds the machinery — the rate curve, not a single rate — that the rest of fixed-income and derivatives pricing runs on. This first lesson sorts out what each quoted rate actually measures, and fixes the compounding convention we will use throughout: continuous compounding.

Four rates, four different things

Treasury rates. The yield on debt issued by a national government (in the US, Treasury bills, notes and bonds). Because the government can, in the last resort, tax or print its own currency to repay, Treasuries are treated as having essentially zero default risk — the traditional textbook "risk-free rate." But Treasuries are not the funding rate a bank actually faces: specific issues can trade "special" in the repo market (everyone wants that exact bond as collateral, pushing its yield down below where pure credit risk would put it), so Treasury yields are a slightly idiosyncratic benchmark, not a clean funding cost.

Repo rates. A repurchase agreement (repo) is a sale of a security today with a promise to buy it back tomorrow at a slightly higher price — economically, a short-term loan collateralized by the security (almost always a Treasury). Because the lender holds collateral, repo rates sit very close to risk-free, and the overnight Treasury-repo market is enormous — hundreds of billions of dollars change hands in it every single day.

LIBOR (historical). The London Interbank Offered Rate was, for decades, the default benchmark for floating-rate loans and derivatives: a panel of large banks was asked each morning what rate they believed they could borrow unsecured from another bank, and the answers were averaged. It was unsecured (no collateral, so it embedded some credit risk) and — critically, as the vignette below explains — it was a survey, not a record of actual trades.

SOFR. The Secured Overnight Financing Rate is LIBOR's designated replacement for US dollar markets (the last LIBOR panels stopped publishing in mid-2023). SOFR is built directly from the overnight Treasury-repo transactions above — a transaction-based, collateralized rate published daily by the Federal Reserve Bank of New York. It is nearly risk-free like a Treasury rate, but — being overnight only — has no built-in term structure of its own; a term SOFR rate (3-month, 6-month, …) has to be built up separately, largely from SOFR futures prices, a construction we return to at the end of this module.

Rate	Secured?	How it is set	Typical role
Treasury	N/A (sovereign issuer)	Auctioned, then traded	"Risk-free" benchmark curve
Repo	Yes — Treasury collateral	Actual overnight transactions	Short-term secured funding
LIBOR (retired)	No	Panel survey of banks	Legacy loan / derivatives benchmark
SOFR	Yes — Treasury repo	Actual transaction volumes	Current USD derivatives / loan benchmark

Notice the pattern: the market has moved, over time, toward rates that are secured (so credit risk is nearly zero) and transaction-based (so the number reflects what actually happened, not what someone guesses might happen). That shift is really the whole story of the LIBOR-to-SOFR transition.

Because it was a promise, not a fact. Each submitting bank was asked a hypothetical: "at what rate could you borrow from another bank right now?" On days when the actual interbank market was thin (as it increasingly was after 2008 — banks stopped lending much to each other unsecured), that question had no real answer, and submitters had to estimate.

An estimate that determines trillions of dollars of loans and derivatives is a standing temptation. In 2012, investigators found that traders at several major banks had been nudging their firm's LIBOR submissions up or down for years to profit on their own derivatives positions — the LIBOR-rigging scandal, which led to billions of dollars in fines and several criminal convictions. Regulators drew the obvious conclusion: a benchmark this important should be built from something that actually happened. SOFR — an average rate over real, observable overnight repo transactions — was the answer, and by 2023 the transition away from LIBOR was complete for essentially every major currency and tenor.

The convention we'll use everywhere: continuous compounding

Quoted market rates come in different compounding frequencies — a bond might quote a semiannual yield, a bank deposit a monthly one. Comparing them directly is like comparing prices in different currencies. Rather than track every convention, this course fixes one and converts everything into it: continuous compounding, where an amount grows smoothly at every instant rather than in discrete jumps.

A(T) = A(0)\,e^{RT}

compared with compounding m times a year at quoted rate R_m,

A(T) = A(0)\left(1 + \frac{R_m}{m}\right)^{mT}.

Continuous compounding is the limit m \to \infty of the discrete formula, and it is exactly the convention the cost-of-carry argument already used for forward prices — this whole module simply extends that same e^{rT} machine to a rate that varies with maturity. Setting the two growth factors equal and solving gives the conversion in each direction:

R_c = m\,\ln\!\left(1 + \frac{R_m}{m}\right), \qquad R_m = m\left(e^{R_c/m} - 1\right).

Worked example. A bond desk quotes a yield the way US Treasury notes are conventionally quoted: semiannual compounding (m=2), at R_2 = 5.00\%. Convert to the continuous rate this course will actually compute with:

R_c = 2\ln\!\left(1 + \frac{0.05}{2}\right) = 2\ln(1.025) = 2 \times 0.024693 \approx 0.04939.

So a semiannual 5.00% is a continuous 4.94% — continuous compounding always reports the smaller number for the same growth, because it is compounding more often. Every zero rate, forward rate and yield in the rest of this module is quoted this way unless stated otherwise.

Two traps worth naming early, because both resurface for the rest of the module.

Zero default risk is not zero risk. A US Treasury bond has (essentially) no risk that the issuer fails to pay — but its price still moves whenever interest rates move, sometimes by a lot for a long-maturity bond. That price sensitivity is interest-rate risk, and it is very real for a "risk-free" bond; measuring it precisely is exactly what duration and convexity are for, later in this module.
Mixing compounding conventions. Plugging a semiannual bond-equivalent yield straight into a formula built for e^{RT} — without converting first — is one of the most common numerical slips in the subject, and it is silent: the formula still returns a number, just the wrong one. Convert every quoted rate to continuous compounding before it goes into any formula in this course.