A default overlay replaces the promised payments with expected ones: survival-weighted payments plus a recovery from the house two years after each default.
To price default risk into a performing note, replace the promised payments with expected ones: weight each payment by the probability the borrower is still paying, add the expected recovery from the house after each default, net of foreclosure costs and delay, and discount the result at your required yield. On a fictional $100,000, 8 per cent note priced at $83,567 for a 10 per cent yield, a 3 per cent annual default rate takes $1,158 off the price at 71.4 per cent LTV and $4,589 at 105.3 per cent. The equity, not the default rate, decides the discount.
Worked in full in The Mortgage Note Investor by Julian R. Sterling, with every figure reproduced in a free workbook.See the book on Amazon →
The usual way to price a performing note discounts every scheduled payment at the target yield, as if the borrower will pay to the last month. That is the price of a promise. A note buyer is buying a promise from someone who may stop keeping it, secured by a house that may or may not cover the debt. A default overlay puts both into the cash flows, so the yield you solve for is the yield you expect to earn, not the one you earn only if nothing goes wrong.
| Input | Value |
|---|---|
| Unpaid balance / note rate | 100,000 / 8.00% |
| Remaining term, months | 240 |
| Payment / net of 30 servicing | 836.44 / 806.44 |
| Property value / LTV | 140,000 / 71.4% |
| Annual default rate on surviving loans | 3% |
| Months from default to cash from the house | 24 |
| As-is haircut / selling costs | 10% / 7% |
| Foreclosure, holding and repair costs | 15,000 |
| Required yield | 10% |
Prepayment is left out so that the default effect is seen alone; it is worked in how an early payoff changes a note's yield.
Step 1: the clean price. Discount the 240 net payments of 806.44 at 10 per cent: 83,567, or 83.57 per cent of UPB.
Step 2: turn the annual default rate into a monthly hazard. 1 − (1 − 3%)1/12 = 0.254 per cent of surviving loans default each month. Survival to maturity is 54.4 per cent, so 45.6 per cent of loans like this one default at some point over twenty years.
Step 3: the recovery on a default in month m. The debt has grown by 24 months of interest by the time the house is sold; the lender recovers the lower of that debt and the net house value, less costs, 24 months later.
Net house = 140,000 × (1 − 10%) × (1 − 7%) = 117,180
Recoverym = min( balancem × (1 + 8%/12 × 24), 117,180 ) − 15,000, received at month m + 24
Default in month 12: debt 113,760 on a balance of 98,069, recovery 98,760, which is 100.7% of the balance
Step 4: the expected cash flow and its price.
E[CFt] = St × 806.44 + Σ (defaults in month t − 24) × Recovery
Price = NPV10%(E[CF]) = 82,409, or 82.41% of UPB, 1,158 below the clean price
Paid the clean 83,567, the expected yield is 9.77 per cent: a drag of 23 basis points
Expected collections fall from 193,546 to 173,910, yet the price moves only 1,158. On a note bought at 83.57 per cent and secured by a house that nets more than the debt, most defaults return about the full balance, the accrued interest roughly paying the foreclosure costs. What they cost is two years of waiting.
| Annual default rate | Ever defaults | Price at 10% | % of UPB | Yield at the clean price |
|---|---|---|---|---|
| 0% | 0.0% | 83,567 | 83.57% | 10.00% |
| 1% | 18.2% | 83,107 | 83.11% | 9.92% |
| 3% | 45.6% | 82,409 | 82.41% | 9.77% |
| 5% | 64.2% | 81,935 | 81.93% | 9.66% |
| 10% | 87.8% | 81,344 | 81.34% | 9.47% |
| House value | LTV | Net house | Recovery on a month-12 default, % of balance | Price at 10% | Discount for default | Yield at the clean price |
|---|---|---|---|---|---|---|
| 140,000 | 71.4% | 117,180 | 100.7% | 82,409 | 1,158 | 9.77% |
| 125,000 | 80.0% | 104,625 | 91.4% | 81,889 | 1,678 | 9.67% |
| 110,000 | 90.9% | 92,070 | 78.6% | 80,615 | 2,952 | 9.43% |
| 95,000 | 105.3% | 79,515 | 65.8% | 78,978 | 4,589 | 9.11% |
Read the two tables together. Raising the default rate from 3 to 10 per cent at 71.4 per cent LTV takes 1,065 off the price; a lower house value that moves LTV from 71.4 to 90.9 per cent at a constant 3 per cent takes 1,794. Combine the two bad cases, a 10 per cent default rate on the 95,000 house, and the justified price is 71,650 while the clean price earns only 7.12 per cent.
The common mistake is to price default from the borrower alone: a payment history, a credit score, a gut feeling, turned into a flat haircut on the price. The overlay shows why that misfires. The borrower sets how often defaults happen; the house sets what each one costs. A flat 5 per cent off the price is 4,178 on this note, more than three times the protection a 3 per cent default rate justifies at 71.4 per cent LTV, and still short of the 4,589 it justifies at 105.3 per cent. Get a value on the house before arguing about the borrower.
The second mistake is to forget the lag. The same recovery received in month 12 rather than month 24 after default is worth materially more at 10 per cent, which is why the overlay belongs in the same month-by-month model as the price, not in a separate expected-loss line. The base price itself is built in how to price a performing note for a target yield.
The Mortgage Note Investor carries a default overlay and a protection test on its performing note; the workbook is among the free companion files for the book.
It is a second layer on the pricing model that replaces promised payments with expected ones. Each month a share of surviving borrowers stops paying; their loans produce a recovery from the house, net of foreclosure costs, after a lag. On the fictional note a 3 per cent annual default rate, a monthly hazard of 0.254 per cent, means 45.6 per cent of loans default at some point over 240 months.
Because on a note bought at a discount, a default with full recovery mostly costs time, not principal. At 71.4 per cent LTV a borrower defaulting in month 12 leaves a recovery of 98,760 after costs, 100.7 per cent of the balance, and a 3 per cent default rate costs only 23 basis points of yield. At 105.3 per cent LTV the recovery is 65.8 per cent and the yield falls to 9.11 per cent.
Compute it rather than apply a flat haircut. On the fictional note at a 10 per cent target, a 3 per cent annual default rate justifies 1,158 off at 71.4 per cent LTV, 1,678 at 80.0 per cent, 2,952 at 90.9 per cent and 4,589 at 105.3 per cent. A flat 5 per cent haircut would overpay protection on the first and underpay it on the last.
This article is one calculation from The Mortgage Note Investor. The book takes the same case from first principles to the decision, chapter by chapter, and every figure it prints is a live formula in the free companion workbooks.
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