Instrument repricing
P = sum_t CF_t DF(0,t)Each new pillar is solved so the curve exactly reprices the selected market instrument under its conventions.
Foundations
Yield Curve Bootstrapping
Use a sequence of traded rates to solve for discount factors and build a zero-coupon curve.
Mental model
Bootstrapping extracts discount factors from traded instruments one maturity at a time.
Bootstrapping starts from the shortest-maturity instruments and works forward in time.
At each maturity it uses known discount factors from earlier pillars to solve for the next discount factor.
Explore the mechanism
How can each new market instrument reveal the next discount factor without changing earlier solved pillars?
This graphic shows three quoted par yields and the bootstrapped zero-coupon curve. Bootstrapping solves for discount factors sequentially so that all bond prices are exactly matched.
D(1Y): 0.9804 · D(2Y): 0.9423 · D(3Y): 0.8876
Numbers
1Y: par ≈ 2.00% · zero r₁ ≈ 2.00% · D(1Y) ≈ 0.9804
2Y: par ≈ 3.00% · zero r₂ ≈ 3.02% · D(2Y) ≈ 0.9423
3Y: par ≈ 4.00% · zero r₃ ≈ 4.05% · D(3Y) ≈ 0.8876
Note: if par inputs are extreme, the bootstrapping equations can become ill-posed (red labels near 2Y/3Y).
Interpretation
The grey dashed line is the par curve you observe in the market. Bootstrapping uses the 1Y instrument to pin down D(1Y), then uses that discount factor to solve the 2Y bond equation for D(2Y), and so on.
The blue line is the zero curve implied by these discount factors. In an upward-sloping environment, zero rates tend to sit below par rates at the same maturity, because coupons are partly paid earlier at lower discount rates. The key mental model: you are turning a small set of traded yields into a full set of discount factors that pricing models can use consistently.
UNDERSTAND
What to notice
The result is a set of zero-coupon rates and discount factors consistent with observed par yields.
The quantitative structure
Notation
P- observed instrument price
CF_t- cash flow at time t
DF(0,t)- discount factor for time t
A small example
Solving a second discount factor
A two-year annual coupon instrument costs 100 and pays 5 after one year and 105 after two years. The one-year discount factor is 0.97.
- Present value the first coupon: 5 * 0.97 = 4.85.
- The remaining present value is 100 - 4.85 = 95.15.
- Solve DF(0,2) = 95.15 / 105 = 0.9062.
ResultThe implied two-year discount factor is approximately 0.9062.
Where it appears
- constructing discount and projection curves
- pricing fixed-income instruments
- reconciling market quotes with zero rates
Where the intuition breaks
What this view leaves out
- A bootstrapped curve depends on instrument selection, conventions, interpolation, and quote quality; exact fit at pillars does not guarantee sensible behavior between them.
Common mistakes
“Bootstrapping estimates the statistically most likely yield curve.”
Not quite.It solves a curve that reprices selected market instruments under stated conventions; it is not a statistical forecast.
Where this concept sits
Follow the strongest prerequisites, applications, extensions, and contrasts.
Connections for Yield Curve Bootstrapping
Yield Curve BootstrappingYOU ARE HERE
Built onCurve Conventions and Day-CountBootstrapped curves depend on the market's day-count, compounding and calendar conventions.
SupportsDiscounting and Present ValueA bootstrapped discount curve supplies the discount factors used to turn future cash flows into present value.
Used byBlack-76 Pricing for Forwards and FuturesBlack-76 requires discount factors consistent with the relevant maturity.
Used byXVA (CVA/FVA) MechanicsXVA calculations require term structures for discounting, funding and often credit-related quantities.
SupportsDuration and Convexity (Intuition)Curve construction provides the rate term structure against which bond sensitivities are measured.
Built on · Curve Conventions and Day-Count
Bootstrapped curves depend on the market's day-count, compounding and calendar conventions.
Explore concept →