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FRM Exam Part I · Interest Rates

Types of Interest Rates and Compounding for FRM Part 1

Updated 11 October 2026 · Fact-checked

Interest rate types differ by credit risk and collateral: Treasury, repo and SOFR rates are near risk-free, while LIBOR was an unsecured bank borrowing rate. Compounding changes how a quoted rate is stated. Convert with Rc = m ln(1 + Rm/m) and Rm = m(e^(Rc/m) − 1), then compare rates on the same basis.

Understand Types of Interest Rates and Compounding

An interest rate is the price of borrowing money for a period. Two things change the rate you see: who is borrowing (credit risk) and whether the loan is backed by collateral. A rate that compensates for more risk is higher.

Treasury rates are the yields on debt issued by a government, such as US Treasury bills and bonds. They are treated as close to default-free in the home currency. They are often below other market rates, partly because of regulatory and tax treatment. Repo rates come from a repurchase agreement: one party sells a security and agrees to buy it back later at a higher price. The price difference is the interest. Because the security is collateral, the repo rate is very low risk and is usually close to the Treasury rate.

LIBOR was an average of the rates at which large banks said they could borrow unsecured. It carried bank credit risk. It has been discontinued, and SOFR (Secured Overnight Financing Rate) replaced it for US dollars. SOFR is based on overnight transactions secured by Treasuries, so it is nearly risk-free and has no bank credit premium. The fed funds rate is different: it is the unsecured overnight rate between US banks. Expect the exam to test what each rate measures and why LIBOR was higher than SOFR.

A quoted rate is incomplete without its compounding frequency. 6% per year compounded semiannually means 3% is earned every six months, so the money grows faster than 6% a year. You cannot compare two rates until they use the same frequency. The effective annual rate (EAR) is the usual common basis.

Continuous compounding is the limit as the frequency goes to infinity. Growth is then A × e^(R × n). Continuous rates are common in derivatives pricing because they make discounting and combining periods simple. Your job in most questions is to convert from one compounding basis to another, keeping the actual growth of money unchanged.

Key formulas to remember

Future value, m times per year
FV = A × (1 + Rm/m)^(m × n)
Rm is the annual quoted rate, m the compounding periods per year, n the years.
Future value, continuous
FV = A × e^(Rc × n)
Rc is the continuously compounded annual rate.
Effective annual rate
EAR = (1 + Rm/m)^m − 1
With continuous compounding, EAR = e^Rc − 1.
Periodic to continuous
Rc = m × ln(1 + Rm/m)
Use the natural log, not log base 10.
Continuous to periodic
Rm = m × (e^(Rc/m) − 1)
For annual compounding, set m = 1: R = e^Rc − 1.
Discount factor, continuous
PV = FV × e^(−Rc × n)
Reverse of continuous growth.

How to solve Types of Interest Rates and Compounding questions

Use this routine for any compounding or rate-type question.

  1. 1Identify what each rate is: Treasury, repo, SOFR, LIBOR or fed funds. Note whether it is secured or unsecured and its tenor.
  2. 2Write down the stated rate and its compounding frequency m. If none is given, check whether the question means annual.
  3. 3Decide the target basis: annual, semiannual, quarterly or continuous.
  4. 4Pick the formula. Periodic to continuous uses ln. Continuous to periodic uses e^x. Periodic to periodic goes through the EAR or through continuous.
  5. 5Plug in the rate as a decimal. Divide by m inside the bracket or the exponent where the formula says so.
  6. 6Compute with the calculator ln and e^x keys. Keep at least five decimals until the last step.
  7. 7Check the direction: for the same growth, the quoted rate falls as m rises: continuous < periodic with high m < periodic with low m < annual, and the annual rate equals the EAR.

Quickest way: Rank by EAR, or go through continuous

When to use it: When you must compare rates with different frequencies, or convert between two periodic bases, under time pressure.

  1. For comparing, compute (1 + R/m)^m − 1 for each option and choose by EAR. Skip any further conversion.
  2. For periodic to periodic, convert the first to continuous with Rc = m ln(1 + R/m), then convert to the new basis with m' (e^(Rc/m') − 1).
  3. Check direction: the quoted rate falls as m rises. Eliminate any option that goes the wrong way.
  4. For small rates, Rc is slightly below the annual rate. Use this to eliminate options before calculating.

Common mistakes in Types of Interest Rates and Compounding

  • Dividing a continuous rate by m before using it, as in e^(Rc/m × m n) with a periodic formula.

    Students blend the periodic and continuous formulas.

    Fix: Continuous growth is only e^(Rc × n). Dividing Rc by m appears only when converting to a periodic rate.

  • Treating a semiannual 6% as 6% effective per year.

    The quoted rate looks like the annual return.

    Fix: Compute EAR = (1 + 0.06/2)^2 − 1 = 6.09%. Always attach the frequency to the rate.

  • Using log base 10 instead of the natural log.

    The calculator has both LOG and LN keys.

    Fix: Use LN for Rc = m ln(1 + Rm/m).

  • Calling LIBOR or the fed funds rate risk-free, or calling SOFR unsecured.

    Students memorize that these are benchmark rates and stop there.

    Fix: LIBOR and fed funds are unsecured and carry bank credit risk. SOFR and repo rates are secured by Treasuries and are close to risk-free.

  • Treating the repo rate as the rate on a sale of the security.

    A repo is described as a sale followed by a repurchase.

    Fix: It is a collateralized loan. The repurchase price minus the sale price is the interest, so the repo rate is a borrowing rate for the cash provider.

  • Rounding early, for example ln(1.03) to 0.03.

    Students want clean numbers.

    Fix: Keep five decimals. Answer options can be only a few basis points apart.

Worked examples

Example 1

A bank quotes 6% per year compounded semiannually. Find the equivalent continuously compounded rate and the effective annual rate.

Show the solution
  1. Rm = 0.06 and m = 2, so the periodic rate is 0.06 ÷ 2 = 0.03.
  2. Rc = m × ln(1 + Rm/m) = 2 × ln(1.03).
  3. ln(1.03) = 0.029559, so Rc = 2 × 0.029559 = 0.059118, or 5.912%.
  4. EAR = (1.03)^2 − 1 = 1.0609 − 1 = 0.0609, or 6.09%.
  5. Check: e^0.059118 = 1.0609, which matches the EAR.

Answer: Rc ≈ 5.912% and EAR = 6.09%.

Example 2

A rate is 8% per year with continuous compounding. Find the equivalent annual rate and the equivalent rate compounded quarterly.

Show the solution
  1. Annual: R = e^0.08 − 1. e^0.08 = 1.083287, so R = 0.083287, or 8.329%.
  2. Quarterly: m = 4 and Rc/m = 0.08 ÷ 4 = 0.02.
  3. Rm = 4 × (e^0.02 − 1). e^0.02 = 1.020201, so e^0.02 − 1 = 0.020201.
  4. Rm = 4 × 0.020201 = 0.080804, or 8.080%.
  5. Check direction: continuous 8.000% < quarterly 8.080% < annual 8.329%, which is consistent with lower frequency needing a higher quote.

Answer: Annual equivalent ≈ 8.329%; quarterly compounded equivalent ≈ 8.080%.

Exam tips

  • Look at the compounding frequency before reading anything else. Many wrong options come from using the wrong m.
  • Know the secured versus unsecured split cold: Treasury, repo and SOFR are secured or government-backed; LIBOR and fed funds are unsecured.
  • Remember LIBOR has been discontinued and SOFR is the US dollar replacement. Questions on benchmark reform test why a nearly risk-free rate differs from a bank-credit rate.
  • Use ln and e^x keys. Practise until the conversion takes under a minute.
  • Sanity-check direction. For the same growth, the quoted rate falls as m rises: continuous < periodic with high m < periodic with low m < annual, and the annual rate equals the EAR. You can often eliminate options without computing.

Practice questions from Interest Rates

Types of Interest Rates and Compounding 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 Compounding: frequently asked questions

How do I convert continuous compounding to an annual rate?

Use R = e^Rc − 1. For Rc = 5%, e^0.05 = 1.051271, so the annual rate is 5.127%. For a periodic basis use Rm = m(e^(Rc/m) − 1).

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 implied in a repurchase agreement, a collateralized loan. Both are near risk-free, and the repo rate is typically close to the Treasury rate.

SOFR vs LIBOR: what do I need to know for FRM Part I?

LIBOR was an unsecured rate reflecting bank credit risk and has been discontinued. SOFR is based on overnight transactions secured by Treasuries, so it is nearly risk-free. SOFR replaced LIBOR for US dollar contracts.

What is the effective annual rate formula?

EAR = (1 + R/m)^m − 1, where R is the quoted annual rate and m the compounding periods per year. With continuous compounding, EAR = e^R − 1.