The formula, the Excel one-liner, the two discounting errors and how the premium moves with rates and remaining term.
Yield maintenance is the present value, discounted at the treasury yield, of the monthly difference between the loan's note rate and that treasury yield over the remaining term, subject to a minimum. On an illustrative $40.0M interest-only loan at 6.25 per cent with six years left and a 4.00 per cent treasury, the premium is $4,793,808, 11.98 per cent of the balance. The two common shortcuts get it wrong in opposite directions: $5,400,000 if the gap is not discounted, $4,493,398 if it is discounted at the note rate.
Worked in full in How to Read a Real Estate Loan Agreement by Julian R. Sterling, with every figure reproduced in a free workbook.See the book on Amazon →
A borrower wants to sell a building and repay its fixed-rate mortgage early. The loan agreement allows prepayment on payment of a yield maintenance premium, defined in the usual way: the present value of the lost interest differential, discounted at the yield on the treasury security whose maturity matches the remaining term. The borrower needs the payoff figure before agreeing a sale price. All figures are illustrative, and the treasury yield is an assumption, not a market quote.
| Input | Value |
|---|---|
| Outstanding balance, interest-only | $40,000,000 |
| Note rate | 6.25% |
| Treasury yield matched to remaining term | 4.00% |
| Months to maturity | 72 |
| Minimum premium | 1% of balance |
The lost differential. The lender loses the gap between what the loan pays and what it can earn by reinvesting in treasuries: 6.25 less 4.00, or 225 basis points. On $40.0M that is $900,000 a year, $75,000 a month.
Discount it. The stream runs for 72 months and is discounted at the treasury yield, divided by 12 for monthly payments. The annuity factor is 63.92.
YM = Balance × (note − treasury) / 12 × [1 − (1 + treasury/12)−n] / (treasury/12)
= 75,000 × 63.92 = 4,793,808
Excel: =MAX(PV(4%/12, 72, -40000000*(6.25%-4%)/12), 1%*40000000)
Apply the floor. The premium is the greater of the calculation and 1 per cent of the balance, $400,000. Here the calculation wins comfortably. The payoff is the balance plus the premium, $44,793,808, plus accrued interest to the prepayment date.
For an interest-only loan, $44,793,808 is also the present value at 4.00 per cent of every remaining payment, coupons and balloon. Yield maintenance at a flat treasury yield leaves the lender exactly as well off as if it had kept the loan and funded it with treasuries, which is why it converges with defeasance.
| Method | Premium | % of balance |
|---|---|---|
| Differential × years, undiscounted | 5,400,000 | 13.50% |
| Discounted at the treasury yield | 4,793,808 | 11.98% |
| Discounted at treasury + 50 bp | 4,724,698 | 11.81% |
| Discounted at the note rate | 4,493,398 | 11.23% |
The undiscounted figure overstates the premium by $606,192, 13 per cent. Discounting at the note rate understates it by $300,410: it uses the rate the lender is losing, not the one it can reinvest at. Some agreements do discount at treasury plus a spread, which is borrower-friendly, and the definition clause is the only place to find out which applies.
The premium is driven by two things the borrower does not control on the day it wants to sell: the treasury yield and the time left.
| Treasury | 2 years | 4 years | 6 years | 8 years |
|---|---|---|---|---|
| 3.00% | 6.30% | 12.24% | 17.83% | 23.09% |
| 4.00% | 4.32% | 8.30% | 11.98% | 15.38% |
| 5.00% | 2.37% | 4.52% | 6.47% | 8.23% |
| 6.00% | 1.00% | 1.00% | 1.26% | 1.59% |
| 6.50% | 1.00% | 1.00% | 1.00% | 1.00% |
With six years left, each 100 basis point fall in the treasury from 4.00 per cent adds $2,336,352 to the payoff. That is the shape a borrower should keep in mind: yield maintenance is most expensive precisely when rates have fallen, which is when refinancing looks most attractive. Once the treasury reaches the note rate, only the floor is left, and the loan is effectively prepayable at a 1 per cent fee.
The most common mistake is not arithmetic but reading. Borrowers model the formula and miss the definitions that move it: the treasury is the one "most nearly equal" to the remaining term or to the average life, interpolated or not; the yield may be converted from semi-annual to monthly; the discount rate may be treasury flat or treasury plus a spread; the premium may run to maturity or to an open period a few months earlier; and the floor may be 1 per cent or more. On this loan the spread alone is worth $69,110, and running to an open prepayment date three months before maturity would cut the term and the premium with it.
The second mistake is checking the payoff only when the sale is agreed. A premium of $4.79M on a $40.0M loan is a price input, and it belongs in the hold-or-sell analysis from the start, alongside the refinancing test in what NOI a loan needs to refinance at maturity.
The book reads prepayment and the extension as clauses with a price, and the free workbook for this case includes a one-page loan model and a working document on prepayment and the extension. The covenant side of the same agreement is in why prepaying to cure a covenant costs eleven times a deposit.
Premium = balance × (note rate minus treasury yield) / 12 × the annuity factor for the months remaining at the treasury yield divided by 12, with a floor, often 1 per cent of the balance. On $40.0M at 6.25 per cent against a 4.00 per cent treasury with 72 months left, the factor is 63.92 and the premium is $4,793,808.
Because the premium compensates the lender for reinvesting at the treasury yield instead of the note rate. The smaller the gap, the smaller the loss. With six years left, the premium on the example loan is 11.98 per cent at a 4.00 per cent treasury, 6.47 per cent at 5.00, and only the 1 per cent floor once the treasury reaches the 6.25 per cent note rate.
No, but they converge. Defeasance buys a portfolio of government securities that replicates the remaining payments; yield maintenance pays the lender the present value of the lost spread in cash. For an interest-only loan discounted at a flat treasury yield the cost is the same: here the remaining payments are worth $44,793,808 at 4.00 per cent, $4,793,808 above the balance.
This article is one calculation from How to Read a Real Estate Loan Agreement. 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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