Finance semantics the language encodes

Several chapters have brushed against the same fact from different sides: the language's machinery encodes finance. The rate conversion in chapter 3, the due/mid placements in chapter 4, the TVM family in chapter 5 — each is a financial convention promoted into semantics. This chapter collects them and completes the set, because a modeler who knows which conventions the machinery applies can reconcile against any source; one who does not will spend afternoons chasing third-digit ghosts that were never errors at all.

Day counts: what a year is

"8% per annum" is incomplete until you say what a year is. The market's answers disagree on purpose: 30/360 (every month 30 days — bonds and mortgages, chosen so level periods accrue level interest), actual/360 (money markets — a real-day count over a short year, quietly yielding more than the quoted rate), actual/365 and actual/actual (real days over real years). Between two dates, these bases produce accruals that differ by real money at scale.

year_frac is where the model states its basis: the day-count-aware year fraction between two dates, the bridge from "a rate per annum" to "this period's accrual" on a daily-grain model. The discipline is the reconciliation rule from chapter 3 wearing its most common costume: a basis-point-sized disagreement with a source is a convention gap before it is an error. Find the source's basis, state the same one, and the gap closes exactly — or remains, and is now a finding.

Rate conversion: the compounding family

You have met this one twice — the discount conversion ((1+r)^(1/12) − 1, chapter 3) and the loan-document convention (r/12, chapter 5's exercise, worth −1,356 of NPV on a small loan). The general rule: annual rates convert to periodic rates geometrically when they compound, and by division when a document says so. Neither is "correct"; each is a claim, and the model's job is to state the one the deal makes.

The same geometry governs prepayment. A pool quoted at 8 CPR (8% of balance prepaying per year) does not lose 8%/12 monthly; the monthly rate is the one that compounds to 8% — 1 − (1 − 0.08)^(1/12), about 0.69% — and cpr_to_smm(0.08) states exactly that conversion by its market name (single monthly mortality). Wherever an annual rate meets a finer grid — discounting, prepayment, default rates, growth — the divide-versus-compound question is waiting, and the third digit of your reconciliation depends on the answer.

Placement as valuation: the annuity family, revisited

Chapter 4's due/mid/end trio is the machinery form of a valuation-course staple, restated here as semantics you can now defend in a committee. Payments in arrears (the default) are the ordinary annuity — debt service, most fees. Payments in advance (due) are the annuity due — rent, leases, insurance — worth more at the same total because every payment discounts one period less. Mid-period is the valuation convention for operating flows earned continuously, splitting the difference deliberately. When your NPV disagrees with a source by a factor suspiciously near one period's discounting, the placement convention is the first suspect — and the fix is one keyword, not a rebuild.

Declared schedules: depreciation as a table

Some finance is not a formula but a published table. US tax depreciation (MACRS) is the canonical case: the five-year schedule is 20%, 32%, 19.2%, 11.52%, 11.52%, 5.76% — six numbers from a revenue procedure, not derivable, only citable. macrs_rate(year, life) reads the table by its public name, so a tax shield stream says basis * macrs_rate(time.t / 12 + 1, 5)-shaped things and a reviewer checks it against the same table you read. The principle generalizes: when the source is a table, cite the table — a curve for a quoted deck (chapter 7), macrs_rate for the tax code — rather than fitting a formula that agrees this year and drifts the next.

Exit conventions: what a cap rate divides

A real-asset exit is one line — sale value = NOI / cap rate — hiding two conventions that move millions. Which NOI? Trailing (the year just ended) or forward (the year a buyer underwrites)? In a growing asset, forward NOI is higher, so the same cap rate on the two bases produces materially different proceeds — and offering documents choose deliberately. Which cap rate? The number is a market price, not a derived quantity, and its sensitivity is brutal at low rates: at a 5% cap, 25 basis points moves value by roughly five percent of the asset. The modeling consequences are direct: the NOI reference in an exit claim must name its window (the capstone's exit chapter does exactly this), and the cap rate belongs on the assumption page with a distribution around it (chapter 12) rather than buried as a constant — no single input in a real-estate model more deserves the Monte Carlo treatment.

Coverage: recompute, never average

The rule promised in chapter 16. A coverage ratio — DSCR, interest cover, any test of the form cash available over cash owed — is a ratio of sums, never a sum of ratios. A quarter's DSCR is that quarter's NOI over that quarter's debt service, and a year's DSCR is the year's NOI over the year's service — not the average of four quarterly ratios, which weights lean quarters wrongly and can show compliance across a quarter that breached. The averaged breach from chapter 16 is this rule violated by grain; it can equally be violated by arithmetic at the right grain. In model terms: compute coverage from rolled-up flows at the grain the covenant tests, and when a document tests quarterly, the model's grid must be at least quarterly (grain and semantics, meeting exactly where the chapter said they would).

The collected discipline

One habit unifies this chapter: every convention is a claim, so state it where a reviewer will look. The day-count basis in the expression that uses it; the rate conversion visible as arithmetic (0.072 / 12 says the document's convention out loud); the placement keyword on the schedule; the NOI window named; the cap rate on the assumption page. The machinery will apply whatever convention you state, exactly — which is precisely why the stating is your job, and why "the model is wrong" so often decodes to "two right models disagree about a convention neither wrote down." Yours writes it down.

Exercises

Then, on your own:

1. Compute by hand: at 8 CPR, what fraction of a pool survives 24 months? (Answer shape: (1−0.08)^2.) Confirm against your exercise model's factor field, and then explain to yourself why 1 − 0.08/12 × 24 disagrees — that explanation is this chapter in one breath.
2. Take chapter 12's Monte Carlo building and put the distribution where this chapter says it belongs: replace the exit-multiple LogNormal with a Normal on the cap rate (mean 5.5%, stdev 0.5%, clipped 4–7.5%) applied to a named NOI window. Compare the p05s. Cap-rate uncertainty is not symmetric in value space — you have just seen why.