Interest-rate curves and swaps

~10 min read

🎯 Interview frequent

Why it matters

After TVM, the yield curve is the single most important object in rates. A swap is the contract that defines a large part of that curve: exchanging fixed vs. floating on a notional that never itself exchanges. Interviewers want you to bootstrap discount factors from liquid instruments, explain par vs. forward vs. zero rates, and describe a vanilla IRS without waving your hands.

Core concepts

Several ‘rates’ that are not the same.
- Zero / spot rate \(z(T)\): the YTM of a zero maturing at \(T\). \(DF(T) = e^{-z(T)T}\) (continuous) or \(1/(1+z/m)^{mT}\).
- Par rate: the coupon that makes a vanilla bond (or the fixed leg of a swap) price at par.
- Simply compounded forward rate \(F(t; T, T+\delta)\): the rate, locked at \(t\), for a loan from \(T\) to \(T+\delta\). In a single-curve world, \(F = \frac{1}{\delta}\bigl(DF(T)/DF(T+\delta) - 1\bigr)\).
The discount curve is the primitive. Once you have \(DF(t)\) for all \(t\), every vanilla bond, FRA, and (single-curve) swap is a linear combination of DFs. YTMs and par rates are quotes; DFs are what you compute with.
Bootstrapping. Use the most liquid instruments covering successive maturities: deposits / OIS, FRAs or futures, then swaps. Each new instrument solves for the next unknown DF (or a short interpolating segment). Interpolation choice (log-linear on DF, monotonic cubic on forwards, etc.) is a modeling decision, not unique.
A vanilla interest-rate swap. Two legs on the same notional \(N\), no exchange of \(N\):
- Fixed: pay \(C \times \delta_i \times N\) on a schedule.
- Floating: pay \(L_{i-1} \times \delta_i \times N\) (LIBOR-legacy) or a compounded overnight (SOFR) over the period. At inception a par swap has \(C\) set so PV(fixed) = PV(float) \(\approx N(1 - DF(T))\) in the single-curve LIBOR-discounting world. After inception, a receiver swap gains when rates fall (like a long bond).
Why swaps exist. Transform floating-rate debt into synthetic fixed (or vice versa) without refinancing; hedge duration; take a view on the curve. Counterparties originally used comparative advantage in credit markets (Khan Academy’s swap videos); today they are a standard risk-transfer tool vs. a dealer/CCP.
Single curve vs. multi-curve (post-2008). You discount on the OIS / SOFR curve (the collateral / funding curve) and project floating legs on their own forward curves (3s, 6s, SOFR, etc.). The 3s-6s basis is a real instrument, not noise. For interviews: discount on OIS, project on the index the contract actually pays.
Duration of a swap. A par receiver ≈ long a fixed-rate bond, short a floating-rate bond (near par, low duration). So swap DV01 ≈ DV01 of the fixed leg. Curve shape still matters (see Bonds & duration/convexity).

Mental model

graph LR
    quotes["Market quotes: OIS, FRAs, swap par rates"] --> boot["Bootstrap DF(t)"]
    boot --> zeros["Zero curve z(T)"]
    boot --> fwds["Forward curve F(T,T+δ)"]
    boot --> pv["PV any vanilla: sum CF × DF"]

  Swap (payer of fixed):

  you -- C, C, C, ... C --> counterparty
  you <-- L1, L2, ... Ln -- counterparty
           (notional stays put)

  PV_payer = PV_float - PV_fixed
           = (high when forwards are high)

A yield curve is a snapshot of today’s discount factors by maturity, not a forecast of the path of the short rate (same warning as futures curves).

Interview questions

Given DFs \(DF(1y)=0.97\), \(DF(2y)=0.94\), what is the 1y1y simply compounded forward (annual)? Answer: \(F = DF(1)/DF(2) - 1 = 0.97/0.94 - 1 \approx 3.19\%\). Intuition: investing 2y vs. rolling 1y then 1y; the forward is the break-even second-year rate.
A 2y annual par swap rate is 4% (single-curve). What is the 2y par bond’s coupon, and why? Answer: 4%. In the single-curve world a par swap’s fixed leg is a par bond minus a floating bond worth par, so the par swap rate equals the par bond coupon. That’s why people say “the swap curve is the par curve.”
Why, after 2008, do we not discount a SOFR-collateralized swap on a LIBOR curve? Answer: Collateral earns (approximately) the overnight index. Discounting should use the curve consistent with that funding/collateral — OIS/SOFR — not the unsecured 3m LIBOR fixing the legacy floating leg projected. Mixing them double-counts credit/funding.
You receive fixed on a 10y par swap, then the whole curve rallies 10bp in parallel. What’s the P&L sign, and what’s a rough scale? Answer: Rates down → receiver wins (you locked a high fixed). Rough P&L \(\approx + DV01 \times 10\text{bp}\), and DV01 is about that of a 10y par bond on the notional (modified duration ~8–9 years times notional times 0.001). Shape risk remains if the move wasn’t parallel.
What’s the difference between a forward rate and the future short rate the market ‘expects’? Answer: The simply compounded forward is a no-arbitrage rate from DFs. Under risk-neutral (or T-forward) measures it is an expectation of a floating rate; under the real world it generally is not, because of risk premia and convexity (futures vs. forwards). Don’t say “the curve is the market’s forecast of Fed funds.”

Watch

Introduction to the yield curve — Khan Academy. What “the” Treasury yield curve actually plots.
Interest rate swap 1 — Khan Academy. Two counterparties exchanging fixed vs. floating on a notional.
Interest rate swap 2 — Khan Academy. After the swap, one party has synthetic fixed-rate debt, the other synthetic floating.