Interest Rate Curves

The day basis (day-count convention) for the money market of a currency can differ from the FX day basis used when computing forward points for that currency. These must not be conflated.

Given a discount factor \(DF\) over \(D\) calendar days, the discrete (money market) rate is:

\[r = \left[ \left( \frac{1}{DF} \right)^{365/D} - 1 \right] \times 100\]

where 365 is replaced by the appropriate day basis for the currency.

A continuous rate \(r_c\) converts to a discrete rate as:

\[r_d = \left( e^{r_c/100} - 1 \right) \times 100\]

Internal implementations prefer continuous rates because they are additive in log space; market quotations are always discrete.

A spot interest rate for maturity \(M\) is an interest rate payable on a spot loan of maturity \(M\) that accumulates interest to maturity. Discount factors therefore reflect discounting to the spot date, not to today — a convention that must be tracked when computing present values.

Because different instruments embed different coupon structures, bootstrapped curves are normalised to zero coupon (ZC) curves — continuous rates where every tenor carries the same economic meaning (yield of a zero-coupon bond). This allows discount and projection curves built from different instruments to be compared and combined consistently.

A deposit is a single-period instrument: cash is placed (lent) at the near leg and returned with interest at the far leg. Deposits are the primary instrument for the very short end (O/N through to 6M), quoting rates directly as LIBOR or equivalent fixings. The instrument defines:

• Near leg: cash outflow on start date
• Far leg: return of principal plus interest on maturity date
• Rate: fixed on trade date; day basis is currency-specific

An FRA is like a deposit that starts in the future. It locks in a fixed rate for a specified future period against the floating fixing at settlement. FRAs have zero value at initiation (fixed rate equals expected floating rate). They are used to bootstrap the curve in the 3M--12M region where deposits become illiquid.

An IRS exchanges two streams of interest payments — typically fixed vs. floating — on a notional principal over multiple periods. The fixed rate (the swap rate) is set at inception to make the trade NPV-zero:

\[PV_{\text{fixed}} = \sum_{i=1}^{n} N \cdot r_{\text{fixed}} \cdot \Delta t_i \cdot DF(t_i)\] \[PV_{\text{float}} = \sum_{j=1}^{m} N \cdot r_{\text{fwd},j} \cdot \Delta t_j \cdot DF(t_j)\]

IRS are the primary pillar instrument for tenors from 1Y to 30Y. Each currency has a standard IRS description specifying payment frequency and day count (e.g. GBP: 6M period, ACT/365).

Key IRS types:

• Fixed/Floating, same currency: the vanilla swap; floating leg typically LIBOR or SOFR
• Fixed/Floating, cross-currency: fixed rate on one currency notional vs. floating on another; introduces FX risk
• Floating/Floating, same currency (Basis Swap): two different floating indices (e.g. 3M LIBOR vs 6M LIBOR); used to capture tenor basis
• Floating/Floating, cross-currency: two floating indices in different currencies; carries both IR and FX risk

Exchange-traded quarterly contracts (Eurodollar, Short Sterling) priced off 3M rates at IMM dates. Used to fill the 6M--2Y gap where liquid deposit and swap quotes are sparse. The bootstrapper takes the nearest IMM quarterly futures and discounts back to derive the curve.

An OIS exchanges a fixed rate for a compounded overnight rate (Fed Funds, EONIA, SONIA, SOFR, etc.) over a given period. Post-2008, OIS curves became the standard collateralised discount curve. The OIS rate represents near-risk-free funding; the spread over OIS is the basis adjustment for uncollateralised exposure (relevant for FVA and MVA calculations).

A discount curve tells us the time value of money; a projection curve tells us what future floating fixings are expected to be. A system maintains multiple projection curves per currency — one for each floating tenor (3M LIBOR, 6M LIBOR, etc.) — because basis spreads mean 3M and 6M projections diverge. All projected cashflows are discounted using the funding (discount) curve.

Local methods use only the immediately bracketing pillars; changing one pillar does not affect distant tenors.

• Log-Linear (Linear in log-DF): Discount factor changes by a constant ratio every day within an interval. Daily forward rates are constant within each interval. Simple and stable; the industry default for the short end.
• Zero-Linear (Linear in zero rate): Linear in \(-\ln(DF)/t\). Daily forward rates change at a constant rate within each interval.

Both perform well in low-curvature regimes.

Non-local methods take into account the full shape of the curve, producing smoother results but introducing the risk that changing one pillar affects distant tenors.

• Cubic Spline: Piecewise cubic polynomial fitted globally. Produces smooth forward curves but can generate oscillations and negative forward rates in extreme cases.
• Monotone Spline: A shape-preserving variant of the cubic spline that prevents oscillations while maintaining smoothness.

A single curve may use different interpolation methods at different tenor ranges (e.g. log-linear at the short end, cubic spline beyond 2Y). A curve split tenor (typically 5Y) separates the short-end and long-end spline segments, preventing changes at one end from propagating to the other.

It is critical that all systems consuming the same curve agree on the interpolation method; different methods will produce different forward rates at off-pillar dates.