The prices come back
Run the recovered curve backwards and all ten original market prices reappear exactly. A curve that cannot reprice the bonds it was built from is not a curve.
VerificationFIXED INCOME / QUANTITATIVE FINANCE
Ten bonds, one curve.
The yield curve is the price of time — what it costs to borrow for one year, or five, or ten. It is never quoted directly; it has to be extracted from the prices of bonds trading in the market. This project extracts one, three independent ways, and checks the three against each other.
RATE / MATURITY
Ten quoted prices
one term structure
one term structure
1Y4Y7Y10Y
10Treasury bonds
1–10 years
1–10 years
3Bootstrap
methods
methods
0.0000Disagreement
between them
between them
5Equity tickers
in part two
in part two
The idea, in plain terms
No background needed. What follows is the whole problem in three short paragraphs.
Governments borrow money by selling bonds. A bond is a promise: pay a little each year, then repay the rest at the end. Thousands of them trade every day, and each has a price.
Hidden inside those prices is something nobody publishes directly — the interest rate for every future year. Working it out is called bootstrapping: use the one-year bond to pin down the one-year rate, then use that answer to unlock the two-year bond, and so on up the ladder. Each result pulls the next one up behind it.
This project does that with ten bonds, and recovers the full picture below.
The curve it found
Switch between the rates themselves and the discount factors they come from.
Two of these lines are what it costs to borrow up to a given year. The third — the forward rate — is the cost of borrowing during that single year alone. Hover any point to read the exact figures.
Rates in % by maturity
Borrowing is dearest around the two-year mark, then gets steadily cheaper — the market pricing in falling rates ahead.
HOW IT WORKS
One equation, three answers.
Line up all ten bonds and their payment schedules become a grid of numbers. Recovering the curve means solving that grid — and there is more than one honest way to do it.
THE PRICING IDENTITY
C · d = P
PAYMENTS × DISCOUNTS = PRICES
MatrixSolve the grid directly, in one step
SolverSearch for the values that fit best
IterativeStep up year by year, by hand
Three methods, one curve
Any single method can be quietly wrong and still look convincing. Three that share no machinery are unlikely to be wrong the same way — and these agree to four decimals.
ConsistencyCheaper money ahead
The forward curve falls from 5.21% to 2.08% — nearly three times the drop in the headline rates. It is the sharper signal of where borrowing costs are heading.
Interpretation- Python 3.9
- NumPy linear algebra
- SciPy optimisation
- numpy-financial IRR
- pandas panel data
- matplotlib + seaborn
- Jupyter
Read it in order
Theory first, results after — or jump straight to whichever you came for.
Five short chapters. The first three build the theory from scratch, assuming no background; the last two show what came out of it.
01
Discounting a bond
Discount factors, the cash-flow matrix, and why pricing ten bonds collapses into one matrix equation.
Read → 02Three ways to bootstrap
The linear solve, the global optimiser and the iterative recursion — how each works, and where they part company.
Read → 03Spot, yield and forward
Turning discount factors into the three quoted curves, and why the forward curve falls fastest.
Read → 04Results
The recovered curve, every number behind it, and an interactive view of both.
Read → 05Returns and correlation
Part two: daily prices, cumulative returns, correlation on returns, and a CSV round-trip.
Read → ↓The notebook
Every figure and table on this site comes from one Jupyter notebook. Download it and run the whole thing yourself.
Download ↓