The Adaptive Decomposition Grammar

Most metric trees are drawn on a whiteboard. Someone writes the number the board cares about at the top, draws three or four boxes beneath it, and keeps going until the boxes start to look like things people do. Some of the lines are formulas. Some are hunches. Some split the number by region or customer, which shows where it sits and says nothing about why it moved. The bottom row is usually a set of nouns, such as "engagement", "quality" and "efficiency", that no system records and no one person can change. The drawing looks like analysis, but nothing can be computed from it, and when the top number moves, the tree cannot say which of its boxes moved it.

The Adaptive Decomposition Grammar treats a metric tree as something to be computed. It is one set of rules that breaks any top-level business metric, of any kind, in any industry, down to the measurements that teams act on day to day. Every link in the tree is one of three types: an exact formula that reconciles to the record, a causal claim with a sign, a lag and a grade of evidence, or an observable stand-in for something that cannot be measured directly. Every branch ends at one of five kinds of leaf. The kind of metric at the top chooses the shape of the tree, and the tree is stated for one perimeter, one period convention and one definition, fixed before any box is drawn.

The rules below are the top level of the grammar, with the reasoning behind each. The extension that turns a tree into a plan, with targets, initiatives and tracking, is not published here.

Five things fixed before any node is drawn

Two people who disagree about why a metric moved are often measuring different things under the same name. One counts revenue by invoice date and the other by delivery. One includes a subsidiary and the other does not. The grammar fixes five things first, and every node in the tree is measured inside them.

The entity

The perimeter the metric is measured over: a legal entity, a business unit, a site, a product line, with the consolidation rule where entities nest. Every node is measured over the same perimeter. A node measured over a narrower one, such as one site's costs under a company-wide margin, is a slice of the parent and cannot be one of its causes.

The period convention

A metric is a balance at an instant, a flow over a period, an average over a period, or a cohort matched to output that arrives later. Headcount at year end, hires during the year, average headcount in the denominator of an attrition rate, and marketing spend matched to customers acquired a quarter later are four different conventions. The root states its convention; every child that combines with it arithmetically inherits it or states its conversion; every causal child states how long the parent takes to respond.

The definition

Many metrics have several accepted definitions. Churn can count accounts or revenue; EBITDA can be adjusted or reported. Where the choice would not change the shape of the tree, the grammar takes the definition most common in that business model and discloses it. Where it would, the choice is made on stated grounds and recorded. Industry context, down to the industry code, chooses the formula variant and names the drivers in the industry's own terms.

The unit

Currency and scale, count, share, days or score. The unit at the top fixes the dimensional check for every arithmetic node beneath it, so a tree that adds days to dollars fails before anyone reads it.

The resolution

The organizational level at which leaves are owned. A general tree names functions and roles; a tree for one company binds each leaf to the person or team that holds the role. A tree drawn for a board uses the same rules and stops no earlier; the board's view is a collapse of the same tree.

The grammar

A metric $m$ is decomposed by one rule, applied to every node until every branch ends in a leaf. The rule has four cases, tried in this order:

1. Arithmetic, when an exact formula exists and passes the tests: $m=F(c_1,\dots ,c_n)$.
2. Causal, when no exact formula exists: $m\to \left\{\right(c_i,{\sigma }_i,{\lambda }_i,e_i\left)\right\}$, with at most five children.
3. Proxy, when $m$ is a construct that no system records: $m\approx \pi$.
4. Leaf, when $m$ is tactical, exogenous, state, parameter or reference: $m=\ell$.

The cases hold under the constraints that every arithmetic node reconciles in the base period, any residual is named and bounded, and each metric is expanded once and referenced after that. The effect of any leaf $\ell$ on the root $m_0$ then has a computed direction:

• $F$ is an exact formula: a sum, a difference, a product, a ratio or a roll-forward. Its children $c_1,\dots ,c_n$ combine to give the parent exactly.
• An arrow $\to$ is a causal link, used where no formula exists. Each causal child carries a sign ${\sigma }_i$, the direction of its effect with the other children held; a lag ${\lambda }_i$, the time from a move in the child to the parent's response; and an evidence grade $e_i$: measured from the entity's own record, estimated from a benchmark or study, or asserted from judgment. Causal children are ranked by effect size and controllability, at most five of them.
• $\approx$ ends a construct, such as onboarding friction or brand strength, at an observable proxy $\pi$.
• $\ell$ is a leaf, one of the five kinds described below.
• The sign of every edge on a path is +1 or −1. An added term, a factor or a numerator carries +1; a subtracted term or a divisor carries −1; a causal edge carries its stated sign. Multiplying along a path gives the direction in which the leaf moves the root.

The order of the cases

The cases are tried in order, and the order is the first rule of the grammar. A metric with an exact formula is never given a causal expansion, because a formula can be checked against the record and a causal claim can only be graded. A metric with no formula is never forced into one, because a formula that does not reconcile hides a definition error behind exact-looking arithmetic. A proxy appears only at the end of a branch, because a stand-in measurement explains nothing beneath it.

Shared levers and tensions

Once every edge has a sign, two things that are usually argued about in meetings become computations. A leaf that reaches the root by several paths with the same sign is a shared lever: moving it helps in more than one place. A leaf that reaches the root by paths with opposite signs is a tension: moving it helps in one place and hurts in another, and the tree shows both. The worked examples at the end compute one of each.

The kind of metric chooses the shape

The first thing the grammar asks about a metric is what kind of quantity it is. The kind decides the form of the first split, the way an industry decides which model sits inside a valuation. The tests run in order, and the first one that fires decides.

• Value. A price set by a market or a counterparty: the equity value of a business, a share price, a multiple. For the value of a business, the root form is the Adaptive Valuation Kernel, and the records that valuation rests on, the cash flows and the capital that produces them, continue as trees under this grammar. Any other price set outside the business is decomposed causally or ends as an exogenous leaf. It is never rebuilt from the business's own outputs.
• Stock. A balance at an instant, such as headcount, backlog, inventory, receivables or recurring revenue. The root form is the roll-forward: the opening balance, plus each inflow, minus each outflow. Headcount at year end is opening headcount plus hires minus departures.
• Duration. Time per unit, or an elapsed interval, such as days sales outstanding, cycle time or time to fill a role. The root form is the sum of the stage durations, with queues and waits counted as stages. Where a stock and a flow pass the split test below, Little's law, stock divided by flow, is used instead.
• Index. A score on a scale of its own, such as a net promoter score or an engagement index. The root form is the index's own published definition: net promoter score is the share of promoters minus the share of detractors.
• Rate. A share, yield, margin or conversion, such as on-time delivery, attrition, gross margin or equipment effectiveness. Dividing the numerator by the denominator usually restates the metric, so the root form is either the loss form, one minus the sum of the shares lost to each cause, or the factor form, a product of component rates.
• Intensity. An amount per unit of a base, such as cost per unit, revenue per employee or cost per ticket. The root form is the per-unit build-up: each component of the amount, expressed per the same unit. The definition of the unit is fixed in the frame.
• Flow. An amount accumulated over a period, such as revenue, cost, orders or hires. The root form is base times rate (revenue is units times average price), a build-up by mechanism or cost element, or a funnel, volume times stage conversion rates, where each stage has its own owner.

A metric defined against a plan, such as budget variance or quota attainment, is decomposed on its actual, and the plan figure is a leaf.

How the grammar thinks

Each rule below exists because a common way of drawing metric trees produces a predictable error, and the rule is what removes it.

Arithmetic is reconciled to the record

An exact link is a claim that the children produce the parent, and the claim can be checked. In the base period, the formula evaluated on the children's recorded values must reproduce the parent's recorded value, within the record's rounding. A formula that does not reconcile means a child was measured over a different perimeter or a different period, and the fix is to correct the definition. The grammar never closes the gap with an unnamed balancing figure.

Where completeness needs a residual, such as late shipments with no recorded cause, the residual is a child of its own, and its share of the parent is reported. A residual above a stated bound is a measurement gap, and the partition is not accepted as an explanation until the gap is closed.

A split must separate levers

A ratio can always be written as its numerator over its denominator, and a total as volume times average. Most of the time that restates the definition and adds nothing. Churn rate written as churned accounts divided by starting accounts has two children, and neither tells anyone what to do differently.

The grammar splits a metric only when the parts are moved by different levers or owned by different people. Customer acquisition cost divides into sales and marketing spend, which a budget owner controls, and new customers, which the funnel produces; the split passes. Where it fails, the metric is decomposed by its kind's loss form or causally instead.

Partitions run by mechanism; slices are views

A partition by cause, mechanism or process stage is a decomposition: late shipments by cause of lateness, scrap by defect type, cost by cost element. A partition by customer, region, product line or channel shows where a number sits and cannot say why it moved, so the grammar keeps those slices as views attached to a node and never makes them children. Where the mix of customers or products drives the number, mix enters the tree once, as one causal driver, and its arithmetic, the split of the change into a mix effect and a rate effect, lives in the view.

A cause partition assigns each unit to exactly one primary cause, with a residual for units that have none. The grammar also records who assigns the cause. Exit-interview reasons are given by the person leaving and tend to favour pay over management. Stop codes on a production line are entered by operators and tend to favour causes outside the operator's control. A cause partition that rests on self-reported causes says which way they are likely to lean, and names a way of capturing them with less bias.

A causal claim carries its evidence

Where no formula exists, the grammar draws a causal link and requires four things of it: its sign, its rank among the parent's causes, its lag, and the grade of the evidence behind it. A link measured on the entity's own record outranks one estimated from a benchmark, and both outrank one asserted from judgment. An asserted link is kept, because judgment is often all there is, but it is flagged as a hypothesis and named with the test that would grade it.

A parent takes at most five causal children, the strongest by effect size and controllability. One child is correct when one driver dominates, and the count is never padded to look thorough. A child observed less often than its parent, such as a quarterly survey under a monthly outcome, cannot explain the parent's month-to-month movements, and is kept only with a lag at least as long as its own cadence.

Every branch ends at one of five leaves

A tree is finished when every branch ends at one of five kinds of leaf.

• Tactical. A measurement read directly from a system of record, such as the CRM, the ledger, the ticketing system or a machine controller. One function or role moves it through its own activity, and it can be observed daily, weekly or monthly. Accountability is not the test. Churn is reported by customer success and produced by sales, product, support and billing together, so churn is an outcome, and its branch keeps going.
• Exogenous. A driver that matters and that no one inside the business can move, such as market growth, a benchmark interest rate or a commodity price. It is included only when it is among the strongest drivers.
• State. A quantity the period inherits: an opening balance, contracts up for renewal, deliveries already scheduled.
• Parameter. A number chosen inside the business that enters a formula as a standard, target, budget, allocation key or conversion factor.
• Reference. A metric already expanded elsewhere in the tree, pointed to by name, with no children of its own.

Each tactical leaf is tagged with its owner, its cadence and the system it is read from. Every leaf's value in the base period carries its source, so a tree can have a complete structure while its values are still being checked.

Parameters are leaves, so their gaming is visible

Planned production time, an ideal cycle time, a standard cost, a sales quota and an overhead allocation key are all numbers someone inside the business chooses. Each one enters a formula, and changing it changes the metric without changing anything that happened. A team that sets its own standard can meet it by loosening it.

The grammar makes every such number a leaf, tags it with the role that sets it, and states how that role could move the metric by moving the parameter, and who should set it instead. The check runs on the tree, where everyone can see it. The equipment-effectiveness example below puts a number on what it catches.

A driver appears once

A driver that matters in two places is expanded at its first occurrence and referenced at the second. It is never renamed slightly and drawn again under another parent, because two copies of one driver double its apparent weight and hide the fact that the two branches move together. A reference carries its own sign, which can differ from the first occurrence's, and that is how the tensions in a tree become visible. The rule is the grammar's version of a principle the Adaptive Valuation Kernel applies to risk: count each thing in one place.

A change in an allocation key is reclassification

An allocated amount, such as overhead absorbed into product cost or a shared service charged out to business units, is written as the pool times the allocation key. The pool has drivers of its own; the key is a parameter set by whoever owns the allocation method. When a unit's cost falls because the key changed and the pool did not, the cost moved to someone else. The grammar reports that movement as reclassification, on its own line, and never as improvement.

A tree must be able to explain a change

A tree that cannot attribute a movement in its root to its leaves is a picture. Given two periods, or a plan and an actual, the grammar divides the change in the root among the leaves. At a sum or difference the division is exact and additive. At a product or ratio it uses the logarithmic-mean index:

The contributions sum exactly to the change in the parent, with no interaction term left over, and the answer does not depend on the order in which the children are considered. Where a sequential order is used instead, the interaction is reported as its own line. At a causal node the change is divided through measured sensitivities, with the unexplained share reported; where no sensitivity has been measured, the node reports direction only.

A ratio is attributed as its numerator times the inverse of its denominator. A rate that improved because its base shrank is shown as exactly that.

A target is reversed into the movements it requires

Given a target for the root, the grammar solves each tactical leaf for the value that would reach the target with every other leaf held where it is, and places that value against the range the leaf has shown in the record. A target that needs one leaf to move further than it ever has, with nothing else changing, is infeasible on that leaf alone, and the tree shows that before anyone commits to it. The Adaptive Valuation Kernel reverses a price into the expectations it implies; this is the same check applied to an operating target.

Depth is capped, and every collapse is recorded

Most leaves land two to four levels below the root, and none deeper than five in any one tree. Where a branch would go deeper, the node at the cap becomes the root of a tree of its own, grafted back by name, or intermediate levels are collapsed and the collapse is recorded beneath the tree. Signs, shared levers and tensions are computed across the grafted whole, so a lever in one tree shows its tension with a leaf in another.

Templates keep every instance comparable

A general tree for a metric in a business model carries every driver a business of that kind might have. A tree for one company is filled from it. A branch that does not apply is set aside at zero and listed with the reason, and is never deleted. Every company's tree stays comparable with the general one and with every other company's.

A worked example: days sales outstanding

The figures are illustrative, chosen to show the arithmetic rather than taken from any company.

A distributor with annual revenue of $36.5 million, or $100,000 a day, has days sales outstanding (DSO) of 52. DSO is a duration, so the tree is a sum of the stages the money waits in: the best possible DSO (BPDSO) that the payment terms allow, and the average days delinquent (ADD) that customers take beyond them.

Best possible DSO nearly equals the weighted average payment terms $T$, but not exactly, because invoices go out after delivery and month ends fall unevenly. The grammar writes the near-identity exactly, as the terms plus a named adjustment $a$:

Days delinquent is partitioned by the cause of each late payment (disputes, missing documentation, the customer's own cash), with a residual for invoices whose cause was never recorded:

The residual is one day of fourteen, about 7 percent, small enough for the partition to count as an explanation. The branches then end at leaves with owners:

• Weighted average terms is a tactical leaf, owned by sales, set deal by deal and read from the contract records.
• The dispute days continue causally, through the invoice error rate, owned by billing, and the time to resolve a dispute, owned by billing and customer service together, with billing named as holding the main lever.
• The documentation days continue through the share of deliveries with proof of delivery captured at the door, owned by logistics.
• The customer-cash days are driven by the customers' own finances, an exogenous leaf, and by the credit limits the business grants, a parameter set by the credit function.

A year later DSO is 50. The tree is all sums, so the attribution is exact:

Billing's work on invoice errors took three days off, logistics took one, and sales gave two back by lengthening terms to win business. The headline improvement of two days, worth $200,000 of cash, combines a four-day improvement in collections with a two-day cost from sales. Payment terms lower cash collected and help win revenue, so in any tree that holds both, the terms leaf carries both signs.

A worked example: equipment effectiveness

A packaging line's overall equipment effectiveness (OEE) is a rate in factor form, the product of availability $A$, performance $P$ and quality $Q$:

Availability takes the loss form: one minus the share of planned production time lost to changeovers, breakdowns and material shortages, with a residual for stops nobody coded:

On 400 hours of planned production time, that is 340 hours of running.

What the parameter leaf catches

Planned production time is a parameter, set by production planning. Suppose planning reclassifies 20 hours of the month as planned maintenance. Planned time falls to 380 hours, the line still runs 340, and availability rises to 0.895. OEE rises from 0.727 to 0.765, almost four points, and the line has not made one more good unit. Because planned time is a leaf tagged with the role that sets it, the tree reports the change as a movement in a parameter, and records that the planning role can move the metric this way and that the standard should be set by someone the metric does not measure. The same check applies to the ideal cycle time behind performance, which industrial engineering sets, and which a loosened standard would also flatter.

Attributing a change across a product

Next month, with planned time unchanged, availability rises to 0.88 and quality to 0.97. OEE rises to 0.76824, a gain of 0.04149. How much of it is availability's?

The usual method credits the factors one at a time in some order, and the order changes the answer. Crediting availability first gives it 0.02565 and quality 0.01584. Crediting quality first gives quality 0.01530 and availability 0.02619. Both versions add up to the total, and they disagree about who earned what.

The logarithmic-mean index removes the order:

The two contributions sum to 0.04149, the whole change, with nothing left over, and the maintenance team and the quality team get the same answer whichever of them runs the numbers.

A worked example: a tension the tree computes

A distributor's gross profit is revenue minus cost of goods sold, with revenue as units times price and cost as units times unit cost. Price is the list price less the discount, and the depth of the discount, set deal by deal and read from the billing system, is a tactical leaf owned by sales. Discounting also wins volume, so the discount appears a second time, as a causal driver of units, where it is a reference to the first occurrence with a sign of its own.

There are three paths from the discount to gross profit:

• Through price: a deeper discount lowers price (−1), price raises revenue (+1), revenue raises gross profit (+1). The product is −1.
• Through units and revenue: a deeper discount raises units (+1, an estimated causal link), units raise revenue (+1), revenue raises gross profit (+1). The product is +1.
• Through units and cost: the same rise in units (+1) raises cost of goods sold (+1), which lowers gross profit (−1). The product is −1.

The discount reaches the root with both signs, so the grammar reports it as a tension, computed from the tree. Whether the deeper discount pays is then a number. At a list price of $100, a unit cost of $60 and a 10 percent discount, 1,000 units earn a gross profit of $30,000. At a 15 percent discount the margin per unit falls from $30 to $25, so volume has to reach 1,200 units, a 20 percent rise, just to stand still. Reversed this way, the proposal to deepen the discount becomes a claim that volume will rise by more than a fifth, and the evidence grade on the causal link says how well that claim is supported.

Four maps from four industries

The worked examples above are small enough to follow on the page. The four maps below apply the same rules to a software company's recurring revenue, a hospital's emergency department, an insurer's underwriting result and a hotel's room revenue. Each is a different kind of metric, so each starts from a different root form, and each is owned by a different set of functions. The figures are illustrative. The build that publishes this page checks every exact node against its children and stops if one does not reconcile.

On a wide screen, select a node to open or close it and to read its formula, its link to its parent, its owner and its computed effect on the root. Drag a node to move it, and use Rebalance to let the map settle again. On a phone, each map is an outline.

Recurring revenue at a software company

Recurring revenue is a stock, so the tree is a roll-forward. New revenue runs through the funnel, where each stage has its own owner. Product value realization cannot be read from any system, so it ends at a proxy, the share of licensed seats active each week. It appears three times: it drives expansion, it holds down contraction, and it holds down churn for product reasons. All three paths have the same sign, so the map marks it as a shared lever. The churn partition records who codes the reasons and which way they are likely to lean.

• Ending ARR$48.0M · Exact sum= Opening ARR + New ARR + Expansion ARR − Contraction ARR − Churned ARR

  • +Opening ARR$40.0M · State leaf
  • +New ARR$9.0M · Exact product= New logos × Average new-logo ARR

    • ×New logos150 · Exact product= Marketing-qualified leads × Lead-to-opportunity rate × Win rate

      • ×Marketing-qualified leads3,000 · Tactical leaf · Marketing
      • ×Lead-to-opportunity rate20% · Tactical leaf · Sales development
      • ×Win rate25% · Tactical leaf · Sales
    • ×Average new-logo ARR$0.06M · Causal

      • −Discount depth18% off list · Tactical leaf · Sales
      • +List price$75,000 a year · Parameter leaf · Pricing committee
  • +Expansion ARR$5.0M · Causal

    • +Product value realizationConstruct, ended at a proxy · Shared lever

      • ≈Weekly active share of licensed seats64% · Proxy · Product · Shared lever
    • +Customers' own headcount growth4% · Exogenous leaf · Shared lever
    • +Expansion opportunities worked120 · Tactical leaf · Account management
  • −Contraction ARR$1.5M · Causal

    • −Product value realization↺ Construct, ended at a proxy · Shared lever
    • −Customers' own headcount growth4% · ↺ Exogenous leaf · Shared lever
  • −Churned ARR$4.5M · Exact sum= Churn: product fit + Churn: price + Churn: customer acquired or closed + Churn: reason not recorded

    • +Churn: product fit$2.0M · Causal

      • −Product value realization↺ Construct, ended at a proxy · Shared lever
      • −Onboarding completed within 30 days71% · Tactical leaf · Customer success
    • +Churn: price$1.2M · Causal

      • +Renewal price uplift7% · Parameter leaf · Pricing committee
    • +Churn: customer acquired or closed$0.9M · Exogenous leaf
    • +Churn: reason not recorded$0.4M · Residual

Length of stay in an emergency department

Length of stay is a duration, so the tree is a sum of the stages a patient waits through. The frame uses the mean rather than the median, because stage means add up to the total and medians do not. Time from decision to departure is split by the shares of patients admitted and discharged, which keeps the sum exact, and the admission rate then reaches the root by two paths of opposite sign. For admitted patients the longest wait is boarding, and most of its causes sit outside the emergency department: discharge timing on the inpatient units, bed turnaround and the inpatient census.

• ED length of stay280 min · Exact sum= Arrival to provider + Provider to disposition decision + Decision to departure

  • +Arrival to provider30 min · Causal

    • −Provider hours scheduled at peak36 hours a day · Tactical leaf · ED medical director
    • +Arrivals per hour at peak9 · Exogenous leaf
    • −Low-acuity patients sent to fast track55% · Tactical leaf · ED nursing
  • +Provider to disposition decision150 min · Causal

    • +Lab turnaround55 min · Tactical leaf · Laboratory
    • +Imaging turnaround70 min · Tactical leaf · Radiology
    • +Specialist consult response65 min · Tactical leaf · Medical staff office
  • +Decision to departure100 min · Exact sum= Admitted patients' share + Discharged patients' share

    • +Admitted patients' share79 min · Exact product= Admission rate × Boarding time, admitted patients

      • ×Admission rate25% · Exogenous leaf · Opposing paths
      • ×Boarding time, admitted patients316 min · Causal

        • −Inpatient discharges before noon22% · Tactical leaf · Hospital medicine
        • +Inpatient occupancy at midnight93% · State leaf
        • +Bed turnaround after discharge95 min · Tactical leaf · Environmental services
    • +Discharged patients' share21 min · Exact product= Discharge rate × Discharge processing time

      • ×Discharge rate75% · Exact sum= 1 − Admission rate

        • −Admission rate25% · ↺ Exogenous leaf · Opposing paths
      • ×Discharge processing time28 min · Tactical leaf · ED nursing

The combined ratio of an insurer

The combined ratio is a rate, built from three cost shares of earned premium. Each share is written in factor form, so frequency, severity and premium per policy appear as separate levers with separate owners. Premium per policy is the divisor in all three, so the discounts and credits applied at quote reach the combined ratio three times, each time raising it. The days taken to close a claim raise both the cost of the claim and the cost of handling it, and the map marks both as shared levers.

• Combined ratio97% · Exact sum= Loss ratio + Adjustment expense ratio + Underwriting expense ratio

  • +Loss ratio60% · Exact product= Claim frequency × Average claim severity ÷ Earned premium per policy

    • ×Claim frequency0.05 per policy-year · Causal · Shared lever

      • +Weather and catastrophe eventsExogenous leaf · Shared lever
      • −Risk selectionConstruct, ended at a proxy · Shared lever

        • ≈New policies bound within guidelines88% · Proxy · Underwriting · Shared lever
    • ×Average claim severity$12,000 · Causal

      • +Days to close a claim38 days · Tactical leaf · Claims · Shared lever
      • −First contact within 24 hours81% · Tactical leaf · Claims
      • +Repair and medical cost inflation6% · Exogenous leaf
    • ÷Earned premium per policy$1,000 · Causal · Shared lever

      • +Filed rate level+4% · Parameter leaf · Actuarial pricing · Shared lever
      • −Discounts and credits applied14% of premium · Tactical leaf · Underwriting · Shared lever
  • +Adjustment expense ratio9% · Exact product= Claim frequency × Adjustment expense per claim ÷ Earned premium per policy

    • ×Claim frequency0.05 per policy-year · ↺ Causal · Shared lever
    • ×Adjustment expense per claim$1,800 · Causal

      • +Days to close a claim38 days · ↺ Tactical leaf · Claims · Shared lever
      • +Claims sent to independent adjusters30% · Tactical leaf · Claims
    • ÷Earned premium per policy$1,000 · ↺ Causal · Shared lever
  • +Underwriting expense ratio28% · Exact sum= Commission ratio + Other underwriting expense ratio

    • +Commission ratio15% · Parameter leaf · Distribution
    • +Other underwriting expense ratio13% · Exact product= Other underwriting expense per policy ÷ Earned premium per policy

      • ×Other underwriting expense per policy$130 · Causal

        • −Policies per underwriter2,400 · Tactical leaf · Underwriting operations
        • −Quotes issued with no manual touch46% · Tactical leaf · Product and technology
      • ÷Earned premium per policy$1,000 · ↺ Causal · Shared lever

Revenue per available room at a hotel

Revenue per available room is an intensity: rooms revenue per room-night available. It splits through rooms sold into occupancy and average daily rate, and the split passes because one responds to volume and the other to price. The hotel's rate position against its competitors appears under both, lowering occupancy and raising the average rate, so the map marks it as a tension. Group room-nights already contracted for the month are a state leaf, because the month inherits them.

• RevPAR$120 · Exact product= Occupancy × Average daily rate

  • ×Occupancy75% · Causal

    • −Rate position against competitorsindex 1.06 · Tactical leaf · Revenue management · Tension
    • +Group room-nights on the books2,100 · State leaf
    • +City-wide demandExogenous leaf · Shared lever
    • +Guest satisfactionConstruct, ended at a proxy

      • ≈Average online review score4.3 of 5 · Proxy · Rooms division
  • ×Average daily rate$160 · Causal

    • +Rate position against competitorsindex 1.06 · ↺ Tactical leaf · Revenue management · Tension
    • +Transient share of rooms sold68% · Tactical leaf · Revenue management
    • +City-wide demand↺ Exogenous leaf · Shared lever

What the grammar does not claim

It does not find causes in data. A causal link in the tree is a claim with a grade of evidence, and the grammar keeps the claim, the grade and the test that would improve it in plain view. A tree full of asserted links is a well-organized set of hypotheses, and it says so.

It does not choose the metric. Which number belongs at the top is a decision for the people running the business. Once that decision is made, the grammar breaks it down by the same rules wherever it sits.

It does not replace the people who own the leaves. A tree can name the function that moves a measurement and the system that records it. Whether that function has the time, the skill and the authority to move it is a question for management, which the tree makes easier to ask.

Its parts are not new. The DuPont decomposition of return on equity, the factor form of equipment effectiveness, Little's law, the roll-forward, variance analysis by mix and rate, and the logarithmic-mean index from the study of energy use are all established, each with a literature behind it. The grammar holds them in one set of rules, with one order of precedence, so that each is used where it applies and nowhere else, and so that any tree built with it can be computed, reconciled to the record and checked by someone who did not draw it.