14 — Curve Construction and Forward Rates¶

Free learning pack¶

1. MIT OCW Portfolio Theory I–III
   https://ocw.mit.edu/courses/15-401-finance-theory-i-fall-2008/pages/video-lectures-and-slides/portfolio-theory/
2. resources/fixed_income.md
3. reference/fixed_income/curve_construction.md, forward_rates.md

Do not search for more material until these are insufficient.

PREDICT¶

A 1-year par yield is 4.80% and a 2-year par yield is 4.50% (lower — an inverted curve). Is the bootstrapped 2-year zero rate higher or lower than the 2-year par yield of 4.50%? Why might they differ at all for a bond that pays a coupon before maturity?

Formula (bootstrapping)¶

For consecutive annual-pay par bonds, solve shortest-to-longest maturity:

100 = sum_{k=1}^{n-1} coupon / (1+z_k)^k + (coupon+100) / (1+z_n)^n

where z_1 .. z_{n-1} are already known from earlier steps and you solve for z_n.

In [ ]:

import numpy as np
import pandas as pd

curve = pd.read_csv("../../data/mock_curve.csv")
par_yields = curve.loc[curve.tenor_years <= 5, "treasury_par_yield"].to_numpy()
print(par_yields)  # 1Y..5Y par yields

# MANUAL FIRST:
# bootstrap the zero rate for each maturity in order (1Y, 2Y, 3Y, 4Y, 5Y).
# For each maturity n, discount the earlier coupons using the zero rates
# you already solved, then solve for z_n from the remaining price.
zero_rates = None
print(zero_rates)

import numpy as np import pandas as pd curve = pd.read_csv("../../data/mock_curve.csv") par_yields = curve.loc[curve.tenor_years <= 5, "treasury_par_yield"].to_numpy() print(par_yields) # 1Y..5Y par yields # MANUAL FIRST: # bootstrap the zero rate for each maturity in order (1Y, 2Y, 3Y, 4Y, 5Y). # For each maturity n, discount the earlier coupons using the zero rates # you already solved, then solve for z_n from the remaining price. zero_rates = None print(zero_rates)

PREDICT (forward rate)¶

The curve above is inverted at the short end. Do you expect the 1y-forward starting in 1 year (the "1y1y" forward) to be above or below today's 1-year zero rate?

Formula (forward rate)¶

f = ((1+z2)^t2 / (1+z1)^t1)^(1/(t2-t1)) - 1

In [ ]:

# Using the zero_rates you just bootstrapped (z_1 at t=1, z_2 at t=2):

# MANUAL FIRST:
# compute the 1y1y forward rate (the rate for year 2, given what you
# already know about year 1).
forward_1y1y = None
print(forward_1y1y)

# Using the zero_rates you just bootstrapped (z_1 at t=1, z_2 at t=2): # MANUAL FIRST: # compute the 1y1y forward rate (the rate for year 2, given what you # already know about year 1). forward_1y1y = None print(forward_1y1y)

Experiment¶

Re-run bootstrapping with a steeper input curve (e.g. par yields rising 0.30% per year instead of falling). Compare how the zero curve and the 1y1y forward change shape.

Reference¶

reference/fixed_income/curve_construction.md
reference/fixed_income/forward_rates.md

Promote¶

Use src/pm/fixed_income/curve.py (bootstrap_zero_rates, interpolate_zero_rate, forward_rate) only after your own implementation.

Test¶

pytest tests/test_fixed_income.py

ORAL CHECK¶

Explain to a PM why forward rates are a break-even, not a forecast — and what it means for a trade if realized rates come in exactly at the forwards.

Try /tutor fixed-income curve construction for an adaptive walkthrough.