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FRM Exam Part I · Swaps

Swap Rates, LIBOR/OIS Spread and OIS Discounting

Updated 11 October 2026 · Fact-checked

A swap rate is the fixed rate that gives a new swap zero value. It usually sits above the Treasury yield of the same maturity. LIBOR has been replaced by near risk-free rates such as SOFR. Collateralized swaps are discounted at the OIS (overnight) curve. Zero rates come from bootstrapping swap rates.

Understand Swap Rates, LIBOR/OIS and Discounting

A swap rate is the fixed rate on a par swap, the rate at which the swap has zero value at the start. Think of it as a par yield for a bond whose floating leg is worth par on each reset date. That is why swap rates can be turned into discount factors and zero rates by bootstrapping: start with the shortest maturity and solve for one discount factor at a time.

A swap rate is not a Treasury yield. A swap spread is the swap rate minus the Treasury par yield of the same maturity. Historically the swap rate was higher because the floating leg was tied to bank borrowing rates, which carried bank credit and liquidity premiums. Treasuries also benefit from safe-asset demand and repo financing value. Swap spreads can be small, and at long maturities they can even be negative, so do not assume they are always positive.

For decades the floating leg was LIBOR, an unsecured interbank rate built from panel-bank submissions. The LIBOR-OIS spread is LIBOR minus the overnight indexed swap (OIS) rate of the same term. OIS is a fixed-for-compounded-overnight-rate swap, so it is close to risk-free. The spread was a gauge of bank credit and funding stress. It widened sharply in the 2007-2009 crisis. After the rate-rigging scandals and the thin underlying interbank market, regulators pushed markets to risk-free rates (RFRs): SOFR for USD, SONIA for GBP, €STR for EUR, SARON for CHF and TONA for JPY. USD LIBOR settings have mostly ceased publication.

SOFR is the Secured Overnight Financing Rate, based on overnight Treasury repo transactions. It is secured, so it has almost no bank credit premium. Swaps on RFRs pay the overnight rate compounded in arrears, so the floating payment is known only at the end of the period. Term versions such as Term SOFR exist but are derived from derivatives markets.

OIS discounting matters because most swaps are collateralized. Under a credit support annex (CSA), cash collateral earns roughly the overnight rate. The funding cost of a collateralized position is therefore the overnight rate, not LIBOR. So you discount collateralized cash flows at the OIS curve. In the older two-curve approach, forward rates were projected from the LIBOR curve and discounted at the OIS curve. In an RFR world, the same RFR curve can project and discount.

Key formulas to remember

Par swap rate (from discount factors)
s = (1 − d_n) ÷ Σ(τ_i × d_i)
d_i is the discount factor for payment date i, τ_i the accrual fraction. For annual payments τ = 1, so s = (1 − d_n) ÷ Σ d_i.
Bootstrapping the last discount factor (annual pay)
d_n = (1 − s_n × Σ_{i<n} d_i) ÷ (1 + s_n)
Use earlier discount factors already found. Start with d_1 = 1 ÷ (1 + s_1).
Zero rate from discount factor (annual compounding)
z_n = (1 ÷ d_n)^(1/n) − 1
With continuous compounding use z_n = −ln(d_n) ÷ n instead. Check which one the question wants.
Swap spread
Swap spread = swap rate − Treasury par yield (same maturity)
Quoted in basis points. 1 bp = 0.01%.
LIBOR-OIS spread
LIBOR-OIS spread = LIBOR (term) − OIS rate (same term)
A wider spread signals higher perceived bank credit and liquidity stress.
Compounded overnight rate over a period
Rate = [Π(1 + r_i × n_i ÷ 360) − 1] × 360 ÷ D
r_i is the overnight rate on day i, n_i the number of days it applies, D total days. 360 for USD SOFR (use 365 for SONIA).

How to solve Swap Rates, LIBOR/OIS and Discounting questions

Most questions on this topic ask you to compute a spread, find a discount factor or zero rate, or choose the right discount curve. Use this order.

  1. 1Identify what is asked: a spread, a discount factor, a zero rate, a PV, or a concept (why OIS, why SOFR).
  2. 2Note the compounding and payment frequency, the day count, and whether the swap rate is par and annual or semi-annual.
  3. 3For spreads, subtract like from like: same maturity, same term, same units. Convert to basis points at the end.
  4. 4For bootstrapping, start at the shortest maturity. Compute d_1, then use d_1 to get d_2, and so on, using the par condition.
  5. 5Convert discount factors to zero rates with the compounding stated, and check the result is sensible (close to the swap rate, same direction as the curve).
  6. 6For discounting questions, ask whether the trade is collateralized. If yes, discount at the OIS or RFR curve. If it is uncollateralized, funding cost matters and a spread to OIS applies.
  7. 7For concept questions, link the answer to credit risk, collateral and the secured nature of SOFR.
  8. 8Sanity check: discount factors fall as maturity rises, and a higher discount rate gives a lower PV.

Quickest way: Par condition shortcut

When to use it: Use when you are given par swap rates for consecutive annual maturities and need a discount factor or zero rate.

  1. Write d_1 = 1 ÷ (1 + s_1).
  2. Keep a running sum of discount factors found so far.
  3. Compute d_n = (1 − s_n × running sum) ÷ (1 + s_n).
  4. Take the zero rate with a calculator: z = (1 ÷ d)^(1/n) − 1, using the y^x key and 1/x.
  5. For concept MCQs, eliminate options that say LIBOR is risk-free or that OIS discounting is used because it is higher yielding.

Common mistakes in Swap Rates, LIBOR/OIS and Discounting

  • Treating LIBOR as a risk-free rate

    It was the standard benchmark, so it feels like the base rate.

    Fix: LIBOR was unsecured interbank borrowing and carried bank credit and liquidity premiums. OIS and SOFR are the near risk-free rates.

  • Saying SOFR is an unsecured rate or a term rate by nature

    Confusing SOFR with LIBOR, which had terms and was unsecured.

    Fix: SOFR is an overnight rate secured by Treasuries in repo. Term SOFR is a separate, forward-looking measure derived from derivatives.

  • Discounting collateralized swaps at LIBOR

    Old textbooks used one LIBOR curve for both projection and discounting.

    Fix: Collateral earns roughly the overnight rate, so discount at the OIS or RFR curve.

  • Using the swap rate as the zero rate

    Both are quoted as annual percentages, so they look interchangeable.

    Fix: The swap rate is a par rate. Bootstrap to get discount factors, then zero rates. The first-year zero rate equals the first-year par rate. For later maturities, zero rates are generally above par rates on an upward-sloping curve and below them on a downward-sloping curve.

  • Subtracting the wrong pair in a spread

    Mixing maturities or mixing LIBOR with Treasury yields.

    Fix: LIBOR-OIS uses the same term. Swap spread uses the same maturity. Compute the spread in the stated direction (swap rate minus Treasury yield, LIBOR minus OIS); the result can be negative.

  • Forgetting that a widening LIBOR-OIS spread means stress

    Memorizing the formula without its meaning.

    Fix: Wider spread means higher perceived bank credit risk or tighter funding liquidity.

Worked examples

Example 1

Annual-pay par swap rates are 3.00% for 1 year and 3.50% for 2 years. Find the 2-year discount factor and the 2-year zero rate (annual compounding).

Show the solution
  1. d_1 = 1 ÷ (1 + 0.03) = 0.970874.
  2. Par condition for the 2-year swap: 1 = 0.035 × (d_1 + d_2) + d_2.
  3. So d_2 = (1 − 0.035 × 0.970874) ÷ 1.035.
  4. 0.035 × 0.970874 = 0.033981, so the numerator is 1 − 0.033981 = 0.966019.
  5. d_2 = 0.966019 ÷ 1.035 = 0.933352.
  6. Zero rate: z_2 = (1 ÷ 0.933352)^(1/2) − 1 = (1.071406)^0.5 − 1 = 1.035087 − 1 = 0.035087.
  7. So the zero rate is 3.51%, slightly above the 3.50% swap rate.

Answer: d_2 ≈ 0.9334 and the 2-year zero rate ≈ 3.51% (annual compounding).

Example 2

A bank will receive USD 10,000,000 in one year on a fully collateralized swap. The 1-year OIS zero rate is 4.00% and the 1-year LIBOR-based rate is 4.50% (annual compounding). By how much is the PV higher when discounted at OIS rather than at LIBOR? Options: A) USD 23,004 B) USD 46,007 C) USD 50,000 D) USD 95,694

Show the solution
  1. Which curve is right? The trade is collateralized, so OIS is the proper discount curve.
  2. PV at OIS = 10,000,000 ÷ 1.04 = 9,615,385.
  3. PV at LIBOR = 10,000,000 ÷ 1.045 = 9,569,378.
  4. Difference = 9,615,385 − 9,569,378 = 46,007.
  5. Check with the exact identity: difference = 10,000,000 × (1.045 − 1.04) ÷ (1.04 × 1.045) = 50,000 ÷ 1.0868 = 46,007. Option C is just 0.5% of the notional with no discounting.

Answer: B) USD 46,007. Discounting at the lower OIS rate gives the higher PV.

Exam tips

  • Expect conceptual MCQs: why OIS is used, what the LIBOR-OIS spread signals, and what makes SOFR different from LIBOR. Learn one clean sentence for each.
  • For bootstrapping, set up the par condition first and calculate with the stored discount factors. Do not re-derive each time.
  • Read the compounding wording. Annual, semi-annual and continuous compounding give different zero rates from the same discount factor.
  • Remember direction: stress widens LIBOR-OIS, and a lower discount rate raises PV. Use these to eliminate wrong options quickly.
  • Do not assume the swap spread is always positive. If an option states that as a fact, treat it with suspicion.

Practice questions from Swaps

Swap Rates, LIBOR/OIS and Discounting: frequently asked questions

What is the LIBOR-OIS spread and why does it matter?

It is the term LIBOR minus the OIS rate of the same term. OIS is nearly risk-free, so the spread reflects bank credit and liquidity risk in unsecured interbank lending. It widens when funding markets are stressed.

Why are swaps discounted at OIS?

Most swaps are collateralized, and cash collateral earns roughly the overnight rate. The funding cost of the position is therefore the overnight rate, so the OIS curve is the appropriate discount curve. Using LIBOR would misstate PVs.

How does the transition from LIBOR to SOFR change swaps?

The floating leg now pays an overnight rate such as SOFR, compounded in arrears, instead of a term LIBOR fixing. SOFR is secured and nearly risk-free, so there is no bank credit premium in it. The same RFR curve can be used to project and to discount.

What is the difference between a swap rate and a Treasury yield?

A swap rate is the fixed rate that makes a swap worth zero at the start, and it historically reflected bank credit and liquidity premiums. A Treasury yield reflects government borrowing and safe-asset demand. The gap is the swap spread, and it can be positive or negative.

How are swap zero rates bootstrapped?

Start with the shortest swap, where the discount factor is 1 ÷ (1 + s). For each longer maturity, use the par condition and the earlier discount factors to solve for the new discount factor. Then convert each discount factor to a zero rate.