Three views of one object — From Zero to the Curve

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Worked examples

Read the claim, then run it and check the machine agrees. One at a time — nothing here is taken on trust.

Example 1 of 3 Round-trip the pillars

example.py

import math
import QuantLib as ql

today = ql.Date(9, 7, 2026)
dc = ql.Actual365Fixed()
pillars = [(1, 0.048), (2, 0.045), (5, 0.042)]
dates = [today] + [today + ql.Period(y, ql.Years) for y, _ in pillars]
dfs = [1.0] + [math.exp(-z * dc.yearFraction(today, d)) for (_, z), d in zip(pillars, dates[1:])]
curve = ql.DiscountCurve(dates, dfs, dc)
curve.enableExtrapolation()

for (y, z), d in zip(pillars, dates[1:]):
    t = dc.yearFraction(today, d)
    print(f"{y}y node: t = {t:.6f}   zero = {-math.log(curve.discount(t)) / t:.6%}   (quoted {z:.1%})")

z(t) = −ln df(t)/t recovers each quoted pillar to machine precision — the df↔rate conversion from Module 0, now applied to a whole curve. Note the node times: 1.000000, 2.002740, 5.002740. The 2028 leap day is in there, and pretending t = 2 exactly would already cost you in the sixth decimal.

Example 2 of 3 The forward hiding between the pillars

example.py

import math
import QuantLib as ql

today = ql.Date(9, 7, 2026)
dc = ql.Actual365Fixed()
dates = [today] + [today + ql.Period(y, ql.Years) for y in (1, 2, 5)]
dfs = [1.0] + [math.exp(-z * dc.yearFraction(today, d)) for z, d in zip((0.048, 0.045, 0.042), dates[1:])]
curve = ql.DiscountCurve(dates, dfs, dc)

t1, t2 = 1.0, dc.yearFraction(today, dates[2])
f12 = (math.log(curve.discount(t1)) - math.log(curve.discount(t2))) / (t2 - t1)
print(f"1y->2y forward = {f12:.4%}   (1y zero: 4.8000%)")
growth_direct = 1.0 / curve.discount(t2)
growth_rolled = (1.0 / curve.discount(t1)) * math.exp(f12 * (t2 - t1))
print(f"grow to t2 directly: {growth_direct:.10f}")
print(f"grow to t1, reinvest at the forward: {growth_rolled:.10f}")

The curve quietly quotes 4.2008% for money placed from year one to year two — well below the 4.8% you get for year one itself. The two growth routes match to every printed digit: that equality is the no-arbitrage derivation, verified numerically. An inverted zero curve is exactly this: forwards below spot rates, the market paying you less and less for each additional year of commitment.

Example 3 of 3 Your formulas vs. the library's

example.py

import math
import QuantLib as ql

today = ql.Date(9, 7, 2026)
dc = ql.Actual365Fixed()
dates = [today] + [today + ql.Period(y, ql.Years) for y in (1, 2, 5)]
dfs = [1.0] + [math.exp(-z * dc.yearFraction(today, d)) for z, d in zip((0.048, 0.045, 0.042), dates[1:])]
curve = ql.DiscountCurve(dates, dfs, dc)

mine_z = -math.log(curve.discount(2.0)) / 2.0
ql_z = curve.zeroRate(2.0, ql.Continuous).rate()
mine_f = (math.log(curve.discount(1.0)) - math.log(curve.discount(2.0))) / 1.0
ql_f = curve.forwardRate(1.0, 2.0, ql.Continuous).rate()
print(f"zero(2.0):    mine {mine_z:.12f}   QuantLib {ql_z:.12f}")
print(f"fwd(1,2):     mine {mine_f:.12f}   QuantLib {ql_f:.12f}")

curve.zeroRate(t, ql.Continuous) and curve.forwardRate(t1, t2, ql.Continuous) are the library's converters — and they agree with your two one-liners to ~1e-15. There is nothing inside those methods but the log formulas you now own. (The ql.Continuous flag matters: ask for annual compounding and you'll get the same price in a different dialect.)

Check the concept

One question at a time. Unsure? Revisit the lecture, then answer.

Question 1 of 4

Zero rate vs forward rate, in one sentence:

the zero is the average growth from today; the forward is the marginal growth over a future slice the zero is for bonds, the forward is for swaps the forward is the market's forecast of where the zero will be they are equal whenever the curve is arbitrage-free

The challenge

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