DSCR is linear in one input and hyperbolic in the other three

Worked coverage arithmetic on a single fixed loan, with the implementation regression-tested against a published set of examples first. A permanent copy.

Debt service coverage ratio is the simplest underwriting number there is: net operating income divided by annual debt service. Because it looks like a division it gets treated like one, and deals get argued about as if moving any input a little moves the ratio a little. It does not. DSCR is exactly linear in NOI and it is nothing like linear in rate, term or principal, and one of those three has a ceiling that no amount of the input can push past. This note works all of that out on one concrete loan.

The implementation is tested before it is used

Everything below comes from the standard fixed-rate amortizing payment

pmt = P * r * (1+r)^n / ((1+r)^n - 1),   r = annual_rate / 12
annual_debt_service = 12 * pmt
DSCR = NOI / annual_debt_service

Before generating a single new figure, that implementation was run against a published set of 6 worked examples. Largest payment deviation across all 6 rows: 0.0035 USD. Largest DSCR deviation: 0.000451. Independently, each row’s recomputed ratio was passed through the published five-band verdict table and compared with the verdict on the row: 0 mismatches out of 6. That is the whole basis on which the rest of this page is allowed to make claims.

All six examples share one loan: 300,000 USD principal at 7.50% for 360 months. That gives a payment of 2,097.64 USD a month and an annual debt service of 25,171.72 USD. Hold that number; it is the only thing the rest of the arithmetic needs.

The verdict bands are NOI thresholds wearing a disguise

For a fixed loan, debt service is a constant, so DSCR is a straight line through the origin in NOI. Every threshold in a lender’s band table is therefore just a dollar figure of income, and it is more useful to know the dollar figure:

Because the relationship is linear, the steps are even: each 0.10 of DSCR is worth exactly 2,517.17 USD of NOI on this loan, at every point on the scale. That is the one place the intuition is right, and it is worth using. It also shows how narrow the interesting region is. The published example with the thinnest margin carries 25,200 USD of NOI against 25,171.72 USD of debt service — a cushion of 28.28 USD, or 0.11% of income. One vacancy, one insurance renewal, one roof, and the verdict changes. A ratio quoted to three decimals invites a confidence the underlying estimate does not support.

Extending the term has a hard ceiling

The standard move on a deal that does not cover is to stretch the amortization. It works, and it works less than people expect, because the payment does not go to zero as the term grows. Take the limit:

as n -> infinity,  (1+r)^n / ((1+r)^n - 1) -> 1
so                 pmt -> P * r
and    annual debt service -> 12 * P * r = P * annual_rate

An infinitely long amortization is an interest-only loan. On this loan that floor is 300,000 × 7.50% = 22,500.00 USD a year, and no term can go below it. Take the weakest of the published examples, 24,000 USD of NOI, currently a fail at DSCR 0.953:

So the entire range available to term extension on this deal, from a 180-month amortization all the way to never repaying principal at all, is DSCR 0.719 to 1.067. It crosses 1.00. It cannot reach 1.25, the “good” threshold, at any term whatsoever — that would need NOI of 28,125 USD even with no principal repayment at all. Whenever NOI is below target × P × annual_rate, term is not the lever, and knowing that before the term sheet saves a round trip.

The lever that is left is principal. Because debt service is proportional to principal at a fixed rate and term, the loan that gets 24,000 USD of NOI to DSCR 1.25 at 360 months is 228,828 USD — a paydown of 71,172 USD, or 23.7% of the original balance.

What 100 basis points actually costs

Rate enters through the payment formula, so DSCR falls as rate rises but not proportionally. Taking the example that currently underwrites as “good” at 33,500 USD of NOI:

From 7.50% to 8.50% the ratio falls 0.121, which is 9.1% of the ratio, and it moves the deal from “good” to “tight”. Note that the percentage cost of the next hundred basis points is not the same as the last, which is what “hyperbolic, not linear” means in practice: the payment is bounded below by the interest-only floor, so each additional increment of rate buys progressively less of the remaining distance to that floor while raising the floor itself.

Run it the other way and ask what rate this income can carry. Holding NOI at 33,500 USD, principal at 300,000 USD and the term at 360 months, DSCR 1.25 breaks at 8.153% and DSCR 1.00 breaks at 10.711%. The gap between merely covering debt service and clearing a common lender threshold is worth about 256 basis points on this deal. That is a more actionable number than the ratio itself, because rate is the input that is quoted to you and NOI is the one you argue about.

The part the arithmetic cannot help with

Everything above is exact given its inputs, and that is precisely the problem. The formula is definitional; the uncertainty is entirely in NOI. Whether a management fee, a reserve for capital expenditure, or an allowance for vacancy sits above or below the NOI line is a convention, and different lenders apply different ones to the same building. On this loan a 5% disagreement about NOI moves DSCR by 0.067 — comfortably more than the 0.10 that separates two of the verdict bands. Precision in the ratio is free; accuracy in the numerator is the entire job.

Two further caveats that are not arithmetic. Band tables vary by programme: the five-band table used here is one published set and a lender may draw the lines elsewhere, may size to a stressed rate rather than the note rate, or may qualify interest-only debt on an amortizing basis specifically to close the loophole in the term-extension section above. And the examples are illustrative scenarios, not quotes; nothing here is an offer of credit.