A prepayment ends two claims on the loan's coupon, the principal classes' and the IO strip's, and the base interest fraction splits the penalty between them.
In a conduit CMBS, a yield maintenance charge is split between the principal classes paid down by the prepayment and the interest-only classes using the base interest fraction: (class pass-through rate − discount rate) ÷ (mortgage rate − discount rate). On an illustrative $50.0M loan at 6.00 per cent prepaid against a 4.00 per cent Treasury yield, the $4.45M charge goes $445,182 to class A-SB, $2,314,948 to class A-2 and $1,691,692, 38 per cent, to the IO class. When Treasury yields rise, the IO class takes a larger share of a smaller charge.
Worked in full in CMBS and CRE CLOs by Julian R. Sterling, with every figure reproduced in a free workbook.See the book on Amazon →
Bond investors often assume that a prepayment penalty belongs to the bonds that get their money back. It does not, entirely. The loan's coupon funds two sets of claims, the coupon of the principal classes and the excess interest stripped into the X classes, and a prepayment ends both. The base interest fraction splits the compensation between them in proportion to what each loses. The formula below is the common form in conduit pooling and servicing agreements; each deal's definition, including whether the mortgage rate is gross or net of servicing, should be read for itself.
| Input | Value |
|---|---|
| Loan prepaid, interest-only | $50,000,000 |
| Mortgage rate | 6.00% |
| Remaining term to maturity | 5 years |
| Discount rate (Treasury yield used in the yield maintenance formula) | 4.00% |
| Principal distributed to class A-SB / pass-through rate | $10,000,000 / 5.00% |
| Principal distributed to class A-2 / pass-through rate | $40,000,000 / 5.30% |
Yield maintenance = balance × (mortgage rate − discount rate) × annuity factor at the discount rate.
$50,000,000 × 2.00% = $1,000,000 a year for 5 years, discounted at 4.00%: annuity factor 4.4518.
Charge = $1,000,000 × 4.4518 = $4,451,822, or 8.90 per cent of the loan. The full method, with monthly payments and the Treasury interpolation, is in how to calculate yield maintenance.
BIF = (pass-through rate − discount rate) ÷ (mortgage rate − discount rate), floored at 0 and capped at 1.
A-SB: (5.00% − 4.00%) ÷ (6.00% − 4.00%) = 1.00 ÷ 2.00 = 0.50.
A-2: (5.30% − 4.00%) ÷ 2.00 = 1.30 ÷ 2.00 = 0.65.
In Excel: =MAX(0,MIN(1,(PTR-Disc)/(MortRate-Disc))).
Each principal class receives: charge × (its principal this period ÷ total principal this period) × its BIF. The remainder goes to the IO classes.
| Class | Share of principal | Pro rata charge | BIF | Allocated |
|---|---|---|---|---|
| A-SB | 20% | $890,364 | 0.50 | $445,182 |
| A-2 | 80% | $3,561,458 | 0.65 | $2,314,948 |
| X-A (IO) | $1,691,692 | |||
| Total | 100% | $4,451,822 | $4,451,822 |
Class A-2 gets $40,000,000 back at par and loses a 5.30 per cent coupon it can only replace at 4.00 per cent: $520,000 a year. Over the 5 remaining years, discounted at 4.00 per cent, that is $2,314,948, exactly its allocation. Class A-SB loses $100,000 a year, worth $445,182, again exactly. The IO class loses the strip between the loan's 6.00 per cent and the classes' weighted 5.24 per cent, 0.76 per cent on $50,000,000, worth $1,691,692. The fraction works because both numerator and denominator are spreads over the same reinvestment rate.
The equality holds only when the bonds would otherwise have lived as long as the loan. A class that was due to be repaid sooner by other loans is over-compensated by the formula; one with a longer remaining life is under-compensated. That is the reason the fraction is a convention, not a valuation.
| Discount rate | Charge | BIF A-SB | BIF A-2 | A-SB | A-2 | IO | IO share |
|---|---|---|---|---|---|---|---|
| 3.00% | $6,869,561 | 0.67 | 0.77 | $915,941 | $4,213,331 | $1,740,289 | 25% |
| 4.00% | $4,451,822 | 0.50 | 0.65 | $445,182 | $2,314,948 | $1,691,692 | 38% |
| 5.00% | $2,164,738 | 0.00 | 0.30 | 0 | $519,537 | $1,645,201 | 76% |
| 5.50% | $1,067,571 | 0.00 | 0.00 | 0 | 0 | $1,067,571 | 100% |
| 6.00% | 0 | 0.00 | 0.00 | 0 | 0 | 0 | 0% |
The IO class's dollar share barely moves between 3.00 and 5.00 per cent, from $1,740,289 to $1,645,201, while the principal classes' share collapses. Once the Treasury yield passes a class's pass-through rate, that class can reinvest at a higher rate than its own coupon, loses nothing on the prepayment and gets nothing. That is why IO investors, who are hurt most by prepayments, care so much about the call protection in the loans: in a high-rate environment, they keep most of what yield maintenance pays.
The usual error is to allocate the whole charge to the classes receiving principal, pro rata. Class A-2 would then be credited with $3,561,458, overstating its prepayment income by $1,246,510, and the IO class with nothing, which makes a premium-priced X-A bond look far more exposed to voluntary prepayment than it is. The second error is to use the bond's own yield, or the deal's current spread, as the discount rate. The fraction uses the discount rate that set the charge, so both must come from the same Treasury yield. How the IO strip itself is built is worked in how the coupon on a CMBS X-A class is calculated.
Check before modelling: some deals allocate prepayment premiums that are a fixed percentage of the balance by the same fraction, others to the IO classes alone, and some use the net mortgage rate in the denominator. One line of the pooling and servicing agreement changes the split.
BIF = (pass-through rate − discount rate) ÷ (mortgage rate − discount rate). Here 0.50 and 0.65, sending 62 per cent of a $4,451,822 charge to the principal classes and 38 per cent to the IO. The cash flows of a whole conduit, class by class, are in the free deal model for this book.
It is the share of a prepayment premium a principal class receives: its pass-through rate less the discount rate, divided by the mortgage rate less the discount rate, between 0 and 1. A 5.30 per cent class on a 6.00 per cent loan prepaid at a 4.00 per cent Treasury yield has a fraction of 1.30 over 2.00, or 0.65.
Yes, usually the remainder after the principal classes take their base interest fractions. In the illustration the IO class receives $1,691,692 of a $4,451,822 charge, 38 per cent. Its share rises with Treasury yields: at a 5.50 per cent discount rate it receives the whole $1,067,571.
Because the fraction is floored at zero once the discount rate reaches the class's pass-through rate. A class paying 5.00 per cent that is repaid when Treasuries yield 5.00 per cent can reinvest at its own coupon and loses nothing, so the formula gives it nothing and the charge goes to the IO classes.
This article is one calculation from CMBS and CRE CLOs. 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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