Yield Curve Bootstrap Forward Rate Calculator

Purpose and model boundary

This model bootstraps eight annual discount factors from a user-entered sequence of zero-rate and par-coupon instruments. It derives node zero rates, adjacent one-year forward rates, and an interpolated discount factor and zero rate at a selected maturity. It is a transparent single-curve educational model, not a dealer curve, executable market-data service, hedge-accounting valuation, collateral curve, or regulatory result.

Inputs and units

The fixed grid contains exactly eight rows. Each row has a text node label, quote type, maturity, and market quote. Quote type is Zero rate or Par coupon; maturity in row i must equal year i; market quotes are annual fractions from zero through one. Zero-rate convention is annual effective or continuous. The interpolation method is linear in zero rate or linear in log discount factor. Target maturity is measured in years and lies from 0.25 through 8.

All rows are user-entered synthetic curve observations. The workbook contains no government, swap, credit, or vendor curve.

Governing relationships

Let q_i be the quote at annual maturity t_i = i and let DF_i be the discount factor. For a zero-rate quote:

DF_i = (1 + q_i)^(-t_i) under annual-effective convention

DF_i = exp(-q_i × t_i) under continuous convention

For a unit-par annual coupon instrument, the recursive bootstrap is:

DF_i = [1 - q_i × sum(DF_j, j = 1...i-1)] / (1 + q_i)

The first par row therefore has an empty earlier-factor sum. A node zero rate recovered from a positive discount factor is:

z_i = DF_i^(-1/t_i) - 1 under annual-effective convention

z_i = -ln(DF_i) / t_i under continuous convention

The first one-year forward uses DF_0 = 1; subsequent forwards use adjacent nodes:

f_i = DF_(i-1) / DF_i - 1

For target maturity T*, let L = floor(T*), U = ceiling(T*), and w = (T* - L)/(U - L) when L != U. The workbook defines DF_0 = 1 and z_0 = 0 for a target below year one. Log-discount interpolation is:

DF(T*) = exp[ln(DF_L) + w × (ln(DF_U) - ln(DF_L))]

Linear-zero interpolation first calculates z(T*) = z_L + w × (z_U - z_L), then converts that zero rate to DF(T*) with the selected convention. The reported target zero rate is recovered from the interpolated discount factor.

Calculation sequence

1. Validate the two selectors, target maturity, and all eight complete ordered rows.
2. Process rows from year one through year eight. Convert a zero quote directly or solve a par quote using all previously bootstrapped factors.
3. Reject the curve if any discount factor is nonpositive or any required discount, zero, or forward value is nonnumeric.
4. Recover the eight node zero rates and the eight one-year forwards, including the spot-to-year-one forward.
5. Locate the target's surrounding nodes and interpolate by the selected method.
6. Report endpoint and target measures, minimum and maximum forward, average zero rate, and curve shape.
7. Draw the workbook-generated zero and forward rate series and apply the negative-forward status check.

Outputs and interpretation

Headline results show the final discount factor and zero rate, the year-seven-to-eight forward, and the target discount factor and zero rate. Supporting outputs include the first discount factor, minimum and maximum forward, arithmetic average node zero rate, and curve-shape label.

Curve shape compares adjacent zero rates with 1e-12 tolerance. A nondecreasing curve is Flat when the first-to-last difference is within 1e-10, otherwise Upward; an entirely nonincreasing curve is Downward; all other patterns are Mixed. The chart keeps zero and forward rates visually distinct while preserving workbook values.

Validation and status logic

The workbook evaluates status in this order:

Input and derived failures take precedence over the negative-forward check. Negative forwards are retained as valid analytical evidence; they are not overwritten or floored.

Assumptions and limitations

• Nodes are annual and ordered from year one through year eight.
• Par instruments pay annual coupons at the quoted par rate and redeem one unit at maturity.
• One curve supplies both discounting and forward inference.
• Interpolation is local. The workbook does not extrapolate beyond year eight.
• There is no calendar, day-count, settlement, stub, compounding-in-arrears, multi-curve, collateral, credit, liquidity, or instrument-convention adjustment.
• A curve built from synthetic entries is not suitable for trading or valuation without independent market-data and model review.

Restrictions and non-computing states

The request must retain the exact eight-row, four-column grid. Labels must be text, nonblank after cleaning, and at most 40 characters. Quote types must match the two declared options. Maturities must be the integers 1 through 8 in their matching row positions, and quotes must be from 0 through 1. Blank, partial, extra, or misordered rows are rejected. A par quote that implies a nonpositive final discount factor prevents a result.

Errors and warnings

This calculator refuses unknown options, malformed grid shape, blank cells, quoted numerics, or entries outside their allowed ranges before workbook calculation. Workbook NOT VALID separates row/intake failure from a nonpositive or nonfinite bootstrap. Workbook CHECK flags a computed negative forward without suppressing the curve. A connection or calculation-service failure is not evidence of a zero rate or a failed bootstrap.

Public par-yield maturity and presentation context is available from the U.S. Treasury Daily Treasury Par Yield Curve Rates. Discount-factor, zero-yield, and forward-rate curve context follows the Federal Reserve paper The U.S. Treasury Yield Curve: 1961 to the Present. No observation, table, chart, or proprietary curve is embedded.