Fitting Discount Curves with Flat Forwards (Chapter 21, Part 1)
Book: Fixed Income Securities: Tools for Today’s Markets | Author: Bruce Tuckman & Angel Serrat | ISBN: 978-0-470-89169-8
Previous: Mortgages, MBS, Prepayments, and OAS (Chapter 20) | Next: Piecewise Quadratics, Locality, and Curve Hedging (Chapter 21, Part 2)
Chapter 21 is the plumbing chapter. You have liquid benchmarks. You need to price a 13.4-year swap or a 7.3-year corporate bond with odd coupon dates. Nobody traded exactly those cash flow dates. You interpolate.
Three Goals for Any Curve
- Reprice your benchmark set exactly (deposits/FRAs, ED futures, swaps).
- Produce economically sane forwards (no wild sawtooth unless the market truly moved that way).
- Keep risk local: bumping the 5y swap should not rewrite 20y forwards through smoothing side effects.
Full term structure models (Part Three) can do this, but desks usually bootstrap a forward curve because it is fast, transparent, and good enough for fixed-cash-flow products.
Forward rates are the popular coordinate. They are traded and watched. They are non-overlapping segments you can bump independently. Spot and discount factors are derived quantities; small spot errors become large forward noise if you fit spots directly.
Bootstrapping in Plain Terms
Fit segment 1 to benchmark 1. Hold it. Fit segment 2 to benchmark 2. Repeat.
Chapters 1-2 did this with six-month forward steps off par swaps. Real markets have gaps: 10y and 12y swaps liquid, nothing at 11y. You extract the 2y forward starting at 10y from the 12y par rate given everything before 10y.
Granularity follows liquidity, not convenience.
Flat Forwards
Between knot dates t_{i-1} and t_i, assume constant instantaneous forward f_i. Then:
d(t_i) = d(t_{i-1}) x exp(-f_i x (t_i - t_{i-1}))
One new forward per new benchmark. Jumps at segment boundaries, but in practice the jumps are orderly, not circus-like.
Any forward over a span is a weighted average of flat forwards it crosses. Example: 4% forward from 10-12y, 4.25% from 12-15y implies the 3y forward from 11-14y is a blend, not a new free parameter.
Independence is the big win. Move the 5y segment, 20y forwards stay put until you choose to refit downstream benchmarks.
USD LIBOR Curve: May 28, 2010
Settlement Jun 2, 2010. Fit to first 10 ED futures, annual par swaps 3y-10y, then 12y, 15y, 20y, 25y, 30y, 40y.
Short-end flat forwards (continuously compounded) ramp from 0.59% (Jun-Sep 2010) to 2.76% (Dec 2012-Jun 2013). Each ED future pins one segment. ED deposit dates overlap, so segments cannot match deposit windows exactly. You split periods and take weighted-average forwards to match each future’s accrual window.
Pricing EDU0: one day at 0.5945%, 90 days at 0.8480%, blend to match market forward. Same logic chains through all ten contracts.
3y par swap at 1.6695%: project LIBOR on each accrual period from flat forwards, discount, sum fixed and floating legs. Both legs PV to 0.0490 per unit notional; with fictional principal, fixed side prices at par. Floating leg is not exactly par because accrual dates and LIBOR deposit dates misalign slightly (weekend/holiday shifts). Practitioners project forwards instead of assuming reset = par.
Continue: next segment Jun 3, 2013 to Jun 2, 2014 set to reprice 4y swap. Bootstrap until all benchmarks fit.
Figure 21.1 shows the resulting forward curve. Steps visible in the belly and long end. Short end segments are ~3 months, so jumps matter less there.
Segment Choice and Fed Meetings
Segments are chosen so each ED future solves one forward. One-month steps would be overkill and under-identified from ED alone.
Alternative: segment boundaries on central bank meeting dates. Forwards then read like policy expectations (noisy when risk premia live in the front end). Implementation gets messy when zero or two benchmarks sit between meetings.
Stub Rates
Common practice: use spot 3m LIBOR (or similar) for settlement to first ED period. Not detailed in the example but standard on every desk.
Two-Curve Note
Chapter focuses on one LIBOR discount curve. Chapter 17’s OIS discount + LIBOR forward setup uses the same bootstrap machinery twice. The technique transfers; the inputs change.
What Part 2 Adds
Flat forwards fit well and hedge locally. They are not pretty on a chart. Practitioners smooth them with piecewise quadratics that preserve benchmark pricing but link segments. That tradeoff between smoothness and locality is where curve construction gets opinionated.
Part 2 covers the quadratic smoother and a partial DV01 example that shows why your hedge ratios depend on how you built the curve.