Mortgage-Backed Securities for Quants
A mortgage-backed security (MBS) passes through principal and interest from loans whose repayment timing is controlled partly by borrowers, making its cash flows an interest-rate-dependent option portfolio. Quantitative MBS work therefore combines loan amortization, behavioral prepayment modelling, pathwise interest-rate valuation, and liquidity-aware hedging.
From loan balance to pass-through cash flow
Each month, the pool receives scheduled interest, scheduled principal, and unscheduled prepayments. Conditional prepayment rate (CPR) is annualized; single monthly mortality (SMM) converts it into the fraction prepaid from the post-scheduled balance:
SMM_t = 1 - (1 - CPR_t)^(1/12)
prepay_t = SMM_t × (balance_t - scheduled_principal_t)
The pass-through investor receives net coupon and principal after guarantee and servicing fees. Real pools also have delinquency, buyout, and remittance conventions, so a generic cash-flow engine should label every assumption.
| Driver | Faster prepayment tendency | Portfolio consequence |
|---|---|---|
| refinance incentive | mortgage rate above current rate | premium amortizes sooner |
| turnover | home sales or relocation | rate-independent principal return |
| seasoning | mature loan age | ramp-up then stabilization |
| burnout | prior non-refinancers | lower future response |
| servicing/friction | high costs or poor access | slower response |
Negative convexity is the central exposure
When rates fall, borrowers refinance and return principal while reinvestment yields are low: duration contracts and price upside is capped. When rates rise, low-coupon loans remain: duration extends into a sell-off. Volatility raises the value of the borrower option and normally lowers MBS value.
def next_balance(balance, scheduled_principal, cpr):
smm = 1 - (1 - cpr) ** (1 / 12)
after_schedule = balance - scheduled_principal
prepaid = after_schedule * smm
return after_schedule - prepaid, scheduled_principal + prepaid
This code is deliberately mechanical. A usable CPR model estimates behavior by coupon, vintage, geography, borrower credit, loan size, and servicing channel. It must use the borrower’s refinance rate after costs, not merely a Treasury yield.
OAS and scenario valuation
Project cash flows across risk-neutral rate paths, forecast prepayment on each path, and solve the constant OAS that matches price:
P_market = E_Q[Σ CF_t(path) exp(-∫(r_path + OAS)dt)]
OAS is a residual after the model assigns value to rate and prepayment options. A wider OAS may be cheapness, but it can equally compensate for uncertain speeds, liquidity, settlement financing, or a misspecified volatility surface.
| Report | Question answered | Limitation |
|---|---|---|
| OAS | spread after modelled option cost | model dependent |
| effective duration | rate-shock sensitivity | changes by scenario |
| effective convexity | curvature of P&L | unstable near refinance thresholds |
| WAL | expected principal timing | not a risk measure |
Compare OAS only for securities valued under the same curve, volatility, prepayment specification, and settlement assumptions. Mortgage prepayment risk details validation by cohort and tail scenario.
Specified pools and basis risk
Agency TBA securities trade a generic delivery claim; specified pools add collateral characteristics and a pay-up. A pool with historically slower speeds can be valuable to an investor exposed to contraction, but its past behavior may not survive a new refinancing regime. Model pay-ups against a matched TBA hedge, include dollar-roll economics, and avoid treating reported speeds as independent observations when pools share servicer or geography.
Hedge duration with Treasuries, futures, or swaps, but the hedge ratio must change as MBS duration extends or contracts. Swaptions may offset convexity, while TBAs hedge mortgage basis more directly. In a broad sell-off, similar MBS holders can receive fixed or sell duration together, amplifying rates moves and degrading hedge liquidity.
Test hedges on portfolio cash-flow scenarios rather than current-coupon duration alone. This is most important after a large move, when TBA rolls, swap liquidity, and dealer balance-sheet capacity may deteriorate at the same time as the model requires a new hedge ratio.
Key takeaways
- MBS cash flows are modeled borrower behavior layered on contractual amortization.
- CPR must be segmented and validated across cohorts, rate regimes, and extension tails.
- Negative convexity causes contraction in rallies and extension in sell-offs.
- OAS comparisons require a common rate, volatility, prepayment, and liquidity framework.
- Hedge scenario duration and convexity, then stress market-feedback and basis risk.