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FRM Exam Part I · Properties of Interest Rates

Treasury Rates, Repo Rates, SOFR and the Risk-Free Rate

Updated 11 October 2026 · Fact-checked

Reference rates are benchmark interest rates used to price loans and derivatives. The main ones are Treasury rates, overnight repo rates and SOFR, which replaced LIBOR for USD contracts. The risk-free rate in derivatives pricing is the rate on a riskless investment, now usually an overnight rate such as SOFR or OIS.

Understand Types of Interest Rates and Reference Rates

An interest rate is the price of borrowing money for a period. Different rates exist because borrowers differ in credit risk, loan term and whether collateral backs the loan. A reference rate (benchmark) is a standard rate that contracts point to, for example 'SOFR plus 1.5%'.

Treasury rates are what a government pays to borrow in its own currency. Because governments are seen as very unlikely to default on such debt, Treasury yields have been used as a proxy for risk-free rates. Dealers, however, find them a poor pure risk-free proxy. Financial institutions hold Treasuries to meet regulatory liquidity requirements, and in the US Treasuries are exempt from state tax. Both effects raise demand, which pushes prices up and yields down. So Treasury rates generally sit somewhat below other near-risk-free rates.

Repo rates come from a repurchase agreement. One party sells a security today and agrees to buy it back later at a slightly higher price. The price difference is the interest. Since the security is collateral, the loan is almost risk-free. An overnight repo rate is the rate for a one-day repo. Repo rates are close to risk-free. They are typically close to Treasury bill rates and can be above or below them depending on market conditions.

LIBOR was an average of rates at which large banks said they could borrow unsecured from each other. It contained bank credit risk, and it was based on estimates rather than actual trades. After manipulation scandals and thin interbank trading, regulators moved markets away from it. LIBOR was replaced by overnight risk-free rates (RFRs). For USD, this is SOFR (Secured Overnight Financing Rate), based on actual overnight repo transactions secured by Treasuries. Other currencies have equivalents, such as SONIA (GBP) and €STR (EUR).

In derivatives pricing, the risk-free rate is used to discount cash flows and to set forward prices. Before the crisis, LIBOR and swap rates were widely used. Now the market discounts with overnight rates, and OIS (overnight indexed swap) rates are the standard. An OIS swaps a fixed rate for the compounded overnight rate over the period. The key exam point is that SOFR is secured, nearly risk-free and overnight, while LIBOR was unsecured, included bank credit risk and had set term tenors.

Key formulas to remember

Repo interest (simple, Actual/360)
Repurchase price = Sale price × (1 + r × days ÷ 360)
USD repo is quoted on an Actual/360 basis. Interest = Sale price × r × days ÷ 360.
Compounded overnight rate (daily compounding in arrears)
Compounded rate = [Π (1 + rᵢ × dᵢ ÷ 360) − 1] × 360 ÷ D
rᵢ is the overnight rate for day i, dᵢ the number of calendar days it applies, D the total days in the period. This is how SOFR-based loans and OIS accrue.
Spread over the risk-free rate
Rate on a risky instrument = Risk-free rate + Spread
The spread compensates for credit, liquidity and term risk.
Discounting with the risk-free rate (continuous)
PV = FV × e^(−r × T)
r is the continuously compounded risk-free rate and T is time in years.

How to solve Types of Interest Rates and Reference Rates questions

Questions on this topic are mostly conceptual, with some short calculations. Use the same routine each time.

  1. 1Identify what is being asked: a definition, a comparison of rates, or a calculation.
  2. 2Classify each rate named: secured or unsecured, overnight or term, government or bank based.
  3. 3Link credit risk to the rate: unsecured bank rates such as LIBOR include bank credit risk; repo and SOFR are collateralised and nearly risk-free.
  4. 4For LIBOR transition questions, recall the reason (manipulation, thin trading) and the replacement (overnight RFRs such as SOFR).
  5. 5For calculations, write the day count first (Actual/360 for USD money-market rates) and the days in the period.
  6. 6Compute with the formula, keeping the rate as a decimal.
  7. 7Check that the result is sensible: overnight repo interest on a large amount for one day should be small.

Quickest way: Classify, then compare spreads

When to use it: Use it for multiple-choice questions that ask which rate is higher, lower or closest to risk-free.

  1. Ask: is the rate secured or unsecured? Secured rates are lower.
  2. Ask: does it include bank credit risk? If yes, it is higher than the matching OIS or SOFR rate.
  3. Treasury rates are generally somewhat below other near-risk-free rates because of demand and tax effects, but do not assume a strict ordering against repo, OIS or SOFR.
  4. Eliminate options that call LIBOR secured, or SOFR unsecured or forward-looking by nature.
  5. For repo interest, multiply amount × rate × days ÷ 360.

Common mistakes in Types of Interest Rates and Reference Rates

  • Saying LIBOR was based on actual transactions.

    Students assume benchmarks always come from trades.

    Fix: LIBOR relied on bank submissions of estimated borrowing costs. SOFR is based on actual repo transactions.

  • Treating SOFR as unsecured.

    Because it replaced LIBOR, it is assumed to be a bank borrowing rate.

    Fix: SOFR is secured by Treasury collateral, so it has almost no bank credit risk.

  • Assuming Treasury rates are exactly the risk-free rate.

    Textbooks often use Treasuries as the example of a risk-free asset.

    Fix: Treasury rates are a proxy. Regulation, demand and tax make them lower than other near-risk-free rates, which is why OIS is preferred for derivatives discounting.

  • Using a 365-day basis for USD repo interest.

    Students default to the usual year length.

    Fix: USD money-market and SOFR conventions use Actual/360 unless stated otherwise.

  • Thinking a repo is a sale with no obligation to return.

    The word 'sell' misleads.

    Fix: A repo is a collateralised loan: the seller must repurchase at a higher price, and the difference is interest.

Worked examples

Example 1

A dealer sells Treasury securities for $50,000,000 in a 3-day repo at an overnight-style repo rate of 4.80% per year (Actual/360). What is the repurchase price?

Show the solution
  1. Interest = 50,000,000 × 0.048 × 3 ÷ 360.
  2. 50,000,000 × 0.048 = 2,400,000.
  3. 2,400,000 × 3 = 7,200,000.
  4. 7,200,000 ÷ 360 = 20,000.
  5. Repurchase price = 50,000,000 + 20,000 = 50,020,000.

Answer: $50,020,000

Example 2

Overnight rates for two consecutive days are 5.00% and 4.80%, each applying for 1 day on an Actual/360 basis. What is the annualised compounded rate over the two days?

Show the solution
  1. Day 1 growth factor = 1 + 0.05 × 1 ÷ 360 = 1.00013889.
  2. Day 2 growth factor = 1 + 0.048 × 1 ÷ 360 = 1.00013333.
  3. Product ≈ 1.00013889 × 1.00013333 ≈ 1.00027224.
  4. Compounded return over the period ≈ 0.00027224.
  5. Annualise: 0.00027224 × 360 ÷ 2 = 0.049003, or about 4.90%.

Answer: About 4.90% per year

Exam tips

  • Know the three contrasts: secured versus unsecured, actual trades versus submissions, overnight versus term.
  • Expect a conceptual question on why LIBOR was discontinued and what replaced it. Name SOFR for USD.
  • Remember the risk-free rate used for discounting derivatives has moved from LIBOR and swap rates to OIS and overnight rates.
  • For rate comparisons, the firm point is that unsecured bank rates such as LIBOR are higher than secured rates such as repo, SOFR and OIS. Treasury rates are generally somewhat below other near-risk-free rates, but do not assume a strict order among them.

Practice questions from Properties of Interest Rates

Types of Interest Rates and Reference Rates in other exams

The same ground in other exams, if you are preparing for more than one or want another angle on it.

Types of Interest Rates and Reference Rates: frequently asked questions

What is the difference between a Treasury rate and a repo rate?

A Treasury rate is the yield on government debt. A repo rate is the interest on a short-term loan secured by collateral, often Treasuries. Both are low risk, and repo rates are typically close to Treasury bill rates, sometimes above and sometimes below.

What is the risk-free rate in derivatives pricing?

It is the rate earned on a riskless investment, used to discount cash flows and to price forwards and options. In current practice it is usually an overnight rate such as SOFR or the OIS rate rather than LIBOR.

Why was LIBOR replaced by SOFR?

LIBOR rested on bank estimates, not actual trades, and was manipulated. Interbank unsecured lending also became thin. SOFR is built from real overnight repo transactions, so it is harder to manipulate.

Is SOFR forward-looking?

The basic SOFR is an overnight rate observed daily. Loans and derivatives usually compound it over the period in arrears. Term versions based on futures also exist.