23 — Effective Duration and MBS Negative Convexity¶

Free learning pack¶

1. resources/fixed_income.md
2. reference/fixed_income/effective_duration.md, mbs_convexity.md

Do not search for more material until these are insufficient.

PREDICT (effective duration as a technique)¶

For a plain vanilla bond whose cash flows don't change with yield, should bump-and-reprice "effective duration" give roughly the same answer as the closed-form modified duration? Why is effective duration still useful if so?

Formula¶

EffectiveDuration = (Price_down - Price_up) / (2 * Price_base * bump)

In [ ]:

from pm.fixed_income.bond import bond_price
from pm.fixed_income.duration import modified_duration

ytm, face, coupon_rate, years, frequency = 0.05, 100.0, 0.05, 5.0, 2
bump = 0.0001

# MANUAL FIRST:
# price the bond at ytm, ytm+bump, and ytm-bump using bond_price, then
# compute effective duration from those three prices. Compare it to
# modified_duration(ytm, face, coupon_rate, years, frequency) - they
# should nearly match.
effective_dur = None
print(effective_dur, modified_duration(ytm, face, coupon_rate, years, frequency))

from pm.fixed_income.bond import bond_price from pm.fixed_income.duration import modified_duration ytm, face, coupon_rate, years, frequency = 0.05, 100.0, 0.05, 5.0, 2 bump = 0.0001 # MANUAL FIRST: # price the bond at ytm, ytm+bump, and ytm-bump using bond_price, then # compute effective duration from those three prices. Compare it to # modified_duration(ytm, face, coupon_rate, years, frequency) - they # should nearly match. effective_dur = None print(effective_dur, modified_duration(ytm, face, coupon_rate, years, frequency))

PREDICT (extension/contraction)¶

A pool's WAC is 6%. If market rates fall to 4% (strong refinancing incentive) versus rise to 8% (none), which scenario gives the shorter weighted average life?

Formula¶

CPR = refinancing_incentive_cpr(wac, market_rate)

SMM = single_monthly_mortality(CPR)

Apply SMM to the scheduled amortization, then compute WAL on the resulting total-principal cash flows.

In [ ]:

import numpy as np
from pm.fixed_income.mbs import mortgage_amortization_schedule

wac, balance, months = 0.06, 1_000_000.0, 360
beginning, scheduled, interest, _ = mortgage_amortization_schedule(balance, wac, months)
times = np.arange(1, months + 1) / 12

# MANUAL FIRST:
# for market_rate in [0.04, 0.08]: compute cpr, smm, apply_prepayment to
# get total_principal, then weighted_average_life(times, total_principal).
# Confirm WAL is shorter at 4% than at 8%.
wal_at_4pct = None
wal_at_8pct = None
print(wal_at_4pct, wal_at_8pct)

import numpy as np from pm.fixed_income.mbs import mortgage_amortization_schedule wac, balance, months = 0.06, 1_000_000.0, 360 beginning, scheduled, interest, _ = mortgage_amortization_schedule(balance, wac, months) times = np.arange(1, months + 1) / 12 # MANUAL FIRST: # for market_rate in [0.04, 0.08]: compute cpr, smm, apply_prepayment to # get total_principal, then weighted_average_life(times, total_principal). # Confirm WAL is shorter at 4% than at 8%. wal_at_4pct = None wal_at_8pct = None print(wal_at_4pct, wal_at_8pct)

What this does not show¶

This confirms cash-flow timing shifts (extension/contraction) but does not demonstrate MBS negative convexity in price terms — naively discounting these cash flows at a flat market rate doesn't correctly price the prepayment option (it misses that principal returns at par, capping upside vs. an option-free bond). That needs OAS — see reference/fixed_income/oas.md and reference/fixed_income/mbs_convexity.md for why it isn't built here. Read both before the oral check.

Not implemented here: non-agency/CMBS¶

Tranche waterfalls need machinery beyond this repo's scope. Read reference/fixed_income/non_agency_overview.md for what they are and why.

Experiment¶

Recompute WAL at 5% and 5.9% (both still below the 6% WAC, so both should show some incentive). Is the WAL response smooth or does it have a kink?

Reference¶

reference/fixed_income/effective_duration.md
reference/fixed_income/mbs_convexity.md

Promote¶

Use src/pm/fixed_income/mbs.py (effective_duration, refinancing_incentive_cpr, single_monthly_mortality, weighted_average_life) only after your own implementation.

Test¶

pytest tests/test_mbs.py

ORAL CHECK¶

Explain to a PM the difference between "this pool's average life shortens when rates fall" (which we just showed) and "this pool's price upside is capped when rates fall" (which needs OAS) — why doesn't proving the first one prove the second?

Try /tutor fixed-income mbs convexity for an adaptive walkthrough.