Quick answer: mortgage prepayment models estimate the probability that borrowers repay principal earlier than the contractual schedule. The core mathematics can be expressed with hazard rates and survival curves: at each period, the model estimates the chance of prepayment among loans that have survived without prepaying up to that point. Refinancing incentives, loan age, borrower characteristics, housing turnover, transaction costs and changing credit conditions all affect that hazard. Because prepayment changes the timing of mortgage cash flows when interest rates move, it also creates the negative-convexity behaviour associated with many mortgage-backed securities.

A mortgage has a schedule, but the borrower owns an option to change the schedule.

Why this is a mathematics problem

Mortgage prepayment combines conditional probability, survival analysis, competing risks, nonlinear interest-rate sensitivity and cash-flow simulation. It is a useful example of why a contractual formula can be correct and still fail to describe realised cash flows.

The scheduled payment equation may be perfectly correct, as explored in How Banks Calculate Loan Repayments: Amortisation, Compound Interest and Recurrence Relations. Prepayment adds a behavioural branch: some loans disappear from the schedule early.

1. Start with survival

Let h_t be the conditional probability that a loan prepays in period t, given that it has not already prepaid before t. This is a discrete-time hazard rate.

If prepayment is the only exit being modelled, the probability of surviving without prepayment through k periods is:

S(k) = ∏_t=1^k(1 − h_t).

If the monthly prepayment hazard were a constant 1%, survival after 12 months would be approximately 0.99¹² ≈ 88.6%. The corresponding probability of having prepaid by then would be about 11.4%.

2. CPR and SMM: two industry ways to express prepayment speed

Mortgage analytics often express prepayment using Single Monthly Mortality (SMM) or Conditional Prepayment Rate (CPR). A simplified conversion is:

CPR = 1 − (1 − SMM)¹², and SMM = 1 − (1 − CPR)^1/12.

If SMM is 1% per month, the annualised CPR is about 11.36%, not 12%, because the surviving balance shrinks through the year. This is another recurrence-style lesson: percentages applied sequentially do not usually add linearly.

3. Refinancing incentive is powerful, but not sufficient

A borrower with a 6% mortgage may have a strong economic incentive to refinance if comparable new mortgages fall to 4%. A common model therefore includes a refinancing-incentive variable related to the difference between the borrower’s existing mortgage rate and the current market rate.

But a positive rate spread does not guarantee prepayment. Refinancing has transaction costs. The borrower may have moved into a weaker credit position. The property may have insufficient equity. Documentation, servicing frictions, loan size and expected time in the home can all change the economics.

Federal Reserve research has repeatedly used refinancing incentives as an important predictor while also showing that realised prepayment can be constrained by borrower and market conditions. See The Federal Reserve’s Portfolio and its Effects on Mortgage Markets.

4. Competing risks: prepayment is not the only exit

A mortgage can terminate through voluntary payoff or refinancing, but it can also terminate through default and liquidation. In a competing-risk framework, the total exit hazard can be represented as the combination of multiple cause-specific hazards.

Federal Reserve work on home-equity borrowers uses a discrete-time hazard structure in which the total hazard is the sum of default and payoff hazards, with survival equal to the product of one minus the total hazard through time. See End of the Line: Behavior of HELOC Borrowers Facing Payment Changes.

This matters because a model that treats every early termination as “prepayment” can confuse improving credit behaviour with distress-related exit.

5. Seasoning: new loans do not behave like seasoned loans

Immediately after origination, many borrowers are unlikely to refinance because closing costs have just been paid and the loan was recently priced. As months pass, prepayment propensity may rise. This age effect is often called seasoning.

A model may represent seasoning with a ramp, spline, age buckets or a nonlinear function. The correct shape is empirical rather than universal. Different origination vintages and products can season differently.

6. Burnout: the pool remembers who already left

Suppose mortgage rates fall sharply and many borrowers with the strongest ability and incentive to refinance leave the pool. The loans remaining afterward are not a random sample of the original pool. They may contain borrowers who are less responsive, face higher transaction costs or cannot refinance easily.

This selection effect is commonly called burnout. If the same refinancing incentive appears again later, the surviving pool may prepay more slowly than it did the first time. A model based only on current rate spread can therefore overpredict prepayment because it ignores the path the pool has already travelled.

7. A simplified prepayment hazard model

A teaching model might use a logistic form:

h_t = 1 / (1 + e^−z_t)

where z_t could combine refinancing incentive, loan age, current loan-to-value ratio, borrower credit score, loan balance, seasonality, geography and burnout measures.

The logistic function is useful because it converts any real-valued score into a probability between 0 and 1. But the form does not make the inputs correct. The model still needs calibration, validation and challenge.

8. Turning hazard into cash flow

Start with scheduled amortisation. Calculate contractual interest and principal for each future month.
Estimate the prepayment hazard. Use current borrower, loan and market variables.
Apply the hazard to the surviving balance. This produces expected unscheduled principal.
Reduce the future pool. Prepaid loans no longer generate later interest or scheduled principal.
Repeat month by month. The remaining balance becomes the next period’s state.
Discount the resulting cash flows. Use the relevant valuation curve and spread assumptions.
Shock rates and re-run. Lower rates can increase refinancing; higher rates can slow it and extend duration.
Compare scenario values. The cash-flow path itself changes with the interest-rate path.

9. Why mortgages create negative convexity

For an ordinary option-free bond, falling yields usually increase price and rising yields decrease price, with positive convexity providing favourable curvature. Mortgage cash flows can behave differently because the borrower holds a prepayment option.

When rates fall, borrowers refinance more, returning principal earlier just when investors would prefer to keep the higher-coupon asset. The security’s duration shortens and price appreciation is capped. When rates rise, refinancing slows, principal remains outstanding longer and duration can extend. This asymmetry produces the negative-convexity behaviour associated with many mortgage-backed securities.

The Federal Reserve’s Trading and Capital-Markets Activities Manual describes this call-like prepayment option and its effect on uncertain maturity and duration. See Residential Mortgage-Backed Securities, section 4110.1.

10. Interest-rate paths matter, not just today’s rate

Two scenarios can end at the same mortgage rate and still produce different surviving pools. If rates first fall sharply, many refinance, and then rise again, the pool remaining at the final rate is different from a pool that reached the same final rate without the earlier refinancing wave.

This path dependence is one reason option-adjusted spread models often simulate many interest-rate paths and recompute prepayments on each path. The valuation problem is not merely “discount fixed cash flows at stochastic rates.” The cash flows themselves are stochastic and rate-dependent.

11. Non-economic prepayment still matters

Borrowers prepay for reasons unrelated to refinancing profits. Homes are sold. Families relocate. Life events occur. The Federal Reserve manual summarises several idiosyncratic causes as the “five Ds”: death, divorce, destruction, default and departure. This is a useful reminder that an elegant interest-rate model cannot explain every termination.

12. Failure modes

Refinancing-only thinking. The model ignores housing turnover and life events.
Burnout blindness. Surviving borrowers are treated as if they were still the original population.
Credit-constraint blindness. A borrower has economic incentive but cannot qualify for a new loan.
Regime drift. A model calibrated before a major policy or underwriting change is reused unchanged.
Rate-path blindness. Only the current spread is used even though the pool’s prior rate history changed its composition.
Product mixing. Fixed-rate, adjustable-rate, different geographies or servicing channels are pooled without evidence.
Competing-risk confusion. Defaults are misclassified as voluntary prepayment.
Static duration. Interest-rate risk is measured as if the mortgage cash-flow schedule never changes.

13. Diagnostics and falsifiers

Does realised CPR rise monotonically with refinancing incentive?
Do seasoned loans behave differently from new loans?
Does the same rate incentive produce slower prepayment after a previous refinancing wave?
Are low-equity or lower-credit borrowers less responsive to incentives?
Do model residuals cluster by geography, servicer or origination vintage?
Does the model separate voluntary payoff from default?
Do simulated durations shorten when rates fall and lengthen when rates rise?
Does a small change in the prepayment function produce a large valuation change?

Suppose someone claims, “Prepayment is determined by the spread between the mortgage coupon and the current market rate.” A falsifier would be two pools with similar coupons and current rates but persistently different prepayment speeds because one pool is burned out, credit constrained or geographically different. The rate spread matters, but it is not the whole state.

14. Verification and update triggers

compare predicted and realised SMM/CPR by incentive bucket;
backtest across origination vintages and rate cycles;
separate refinancing, turnover and default where data allow;
stress transaction costs and credit-availability assumptions;
re-estimate seasoning and burnout after major refinancing waves;
validate cash-flow projections against actual principal returns;
re-run valuation when mortgage rates, home prices or underwriting regimes change materially;
inspect whether model errors systematically favour one valuation conclusion.

Connections across the finance-and-banking algorithms lane

Loan amortisation and recurrence relations — the contractual schedule prepayment modifies.
Yield-curve algorithms — the discount and forward-rate structure used in mortgage valuation.
Duration, convexity and scenario shocks — the wider interest-rate-risk framework into which mortgage optionality feeds.
Credit-scoring algorithms — a related use of logistic probability models, but for a different target event.

Research anchors

The deeper lesson

Mortgage prepayment shows what happens when human choice sits inside a financial contract. The scheduled cash-flow machine is deterministic until the borrower exercises an option. Survival analysis turns that behaviour into conditional probabilities. Scenario modelling feeds those probabilities back into valuation. The strongest model is therefore not the one that predicts every borrower; it is the one that states its behavioural assumptions clearly, measures when they fail, and updates when the population changes.

Educational note: This article explains mortgage mathematics and public fixed-income concepts. It is not mortgage advice, investment advice, trading advice or a recommendation regarding any financial product.

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How Banks Model Mortgage Prepayment: Hazard Rates, Survival Curves, Refinancing Incentives and Negative Convexity