Fixed income / MBS

MBS prepayment and WAL

A mortgage pool does not amortise on a schedule: borrowers refinance and move, and every one of those events returns principal early. The PSA model describes that with a ramp quoted as a multiple of a 6% terminal CPR. This runs the monthly cash flows, the weighted average life and the price — and shows why a premium pool is hurt by speed.

Terminal CPR6.00%Year-one CPR2.40%Price vs parPremium

Weighted average life

7.87 years

at 100 PSA, against 7.87 at 100

Price

100.000

per 100 of par at a 6.00% discount rate

Prepaid early

40.9%

$4,094,114 of the original balance

Total interest

$4,724,998

over the life of the pool

Annual cash flow

Interest, scheduled principal and prepayments, aggregated from the monthly schedule.

7.9 yr WAL

Annual cash flow
Year	Interest	Principal	Of which prepaid	Balance
1	$594,255	$254,327	$129,121	$9,745,673
2	$572,080	$505,030	$357,649	$9,240,643
3	$535,419	$709,694	$525,652	$8,530,949
4	$491,710	$732,353	$504,603	$7,798,596
5	$447,809	$730,872	$459,221	$7,067,724
6	$403,998	$729,394	$413,931	$6,338,330
7	$360,275	$727,918	$368,732	$5,610,412
8	$316,640	$726,446	$323,625	$4,883,966
9	$273,094	$724,976	$278,609	$4,158,990
10	$229,636	$723,510	$233,685	$3,435,481
11	$186,265	$722,046	$188,851	$2,713,435
12	$142,983	$720,585	$144,108	$1,992,849
13	$99,788	$719,128	$99,455	$1,273,721
14	$56,680	$717,673	$54,893	$556,048
15	$14,367	$556,048	$11,980	$0.00

The prepayment column is the whole story: it is principal that arrived early, and every dollar of it stops earning the coupon. A pool priced at a premium loses when prepayments speed up; a pool at a discount gains.

Balance decay

Faster than the amortisation schedule, because the prepayments sit on top of it.

The PSA ramp

CPR climbs from zero to 6% over thirty months and stays there; the multiple scales the whole ramp.

Canonical equations

The exact relationships implemented by this engine.

CPR(t) = 6% × min(t/30, 1) × speed · The 6% is the 100 PSA baseline.

1 − (1 − CPR)^(1/12) · Monthly survival equivalent.

SMM × (B − scheduled principal) · Applied after the scheduled payment.

Σ t·P_t / Σ P_t · In months, then divided by twelve.

Show model notation and computationOpen the symbolic form, the variable definitions, and the step-by-step derivation when you need the detail.

A clear path from inputs to results.

Step-by-step methodWorked exampleAssumptions explained

The formula

\mathrm{CPR}(t) = 6\%\times\min(t/30,1)\times\text{speed} \;;\; \mathrm{SMM} = 1-(1-\mathrm{CPR})^{1/12}

Reference: Public Securities Association prepayment model; the WAL and passthrough conventions follow Fabozzi, Fixed Income Analysis.

Worked example

$10m pool at 6% with 360 months remaining

Inputs

$10,000,000 original balance, 6% weighted-average coupon.
360 months remaining, 100 PSA.
6% discount rate.

Calculation path

1
The PSA ramp takes CPR from zero to 6% over thirty months and holds it there.
2
At 100 PSA the pool prepays 40.9% of its original balance early, $4.09 million.
3
That shortens the weighted average life to 7.87 years against a 30-year stated term.
4
The pool is fully retired in 178 months rather than 360.
5
At a discount rate equal to the coupon the price is exactly 100, because every prepayment returns par.
6
Total interest collected over the life of the pool is $4.72 million.

Result

WAL 7.87 years · price 100.000 · 40.9% prepaid early.

Interpreting the result

The invariance is the insight: at a discount rate equal to the coupon the pool prices at par whatever the prepayment speed, because a prepaid mortgage returns exactly par. The risk is in the spread — a pool bought at a premium loses when prepayments accelerate, and one bought at a discount gains. That is negative convexity.

Assumptions & limitations

The PSA ramp is applied from the first month of the projection, not from origination.

Prepayments are applied after the scheduled payment, on the surviving balance.

The coupon is the gross weighted-average coupon; servicing and guarantee fees are not stripped out.

No defaults, delinquencies or loss severities are modelled.

The discount rate is a single flat nominal rate, not a curve.

Frequently asked questions

What does 100 PSA mean?

It is the baseline: CPR rises linearly from zero at month one to 6% at month thirty, then stays at 6%. 150 PSA is one and a half times that curve at every point; 50 PSA is half.

What is negative convexity?

The price rises less than a comparable bond when rates fall, because falling rates cause refinancing and the pool is repaid at par just as it becomes valuable. The holder gives up upside without gaining equivalent protection on the downside.

What are the weaknesses of the PSA model?

It has no refinancing incentive — the same speed applies whether the coupon is 2% or 4% above the market — and no burnout, seasonality or borrower heterogeneity. Production models are functions of those variables, and PSA survives mainly as a quoting convention.

Why is the WAL so much shorter than the term?

Because prepayments return principal early. Even at 100 PSA, a 30-year pool has a weighted average life of under eight years — the stated maturity is almost never reached, and pricing off it would be badly wrong.

Why does the price not change with prepayment speed?

At a discount rate equal to the coupon it does not, because every cash flow including the prepayments is discounted at exactly the rate that makes par the fair value. Change the discount rate away from the coupon and the speed starts to matter.