The Expense Approval Rate and the Queue Behind It

Why this matters

An expense approval rate looks like a statement about your people's spending, and it is mostly a statement about how fast somebody works a queue. A shop that reads it the obvious way calls a meeting about what the crew is buying, when the actual finding is that the person who reviews expenses was on holiday for a week. The two problems have different owners, different fixes and different costs, and the headline figure cannot tell them apart, because its denominator counts an undecided expense and a rejected one identically.

Three states in the denominator, two of which are decisions

The rate is approved expenses over all expenses dated in the window. An expense in that window is in one of three states:

• Approved. In the numerator and the denominator.
• Rejected. In the denominator only.
• Pending, awaiting review. In the denominator only, exactly like a rejection.

That third state is the whole article. A pending expense is not a judgement about anything - nobody has looked at it - and it pushes the figure down by the same arithmetic that a rejection does. The general failure here, that a rate counting outcomes in the same window as the opportunities understates itself and understates worse on short windows, is not re-derived in this card. See related: Estimate Conversion Rate and the Cohort Problem.

What is specific to this figure is that the fix is not a longer window. It is a different denominator, because unlike a sale that may simply never happen, every pending expense is going to be decided. The outcome is not uncertain, it is merely late, and that means you can split the metric cleanly rather than waiting the problem out.

The identity that explains every reading

Write the states as A approved, R rejected, P pending.

headline rate   = A / (A + R + P)
policy rate     = A / (A + R)
decided share   = (A + R) / (A + R + P)

headline rate   = policy rate  x  decided share

That product is the sentence to keep. The headline is two unrelated things multiplied together: how often you say yes, and how much of the window anybody has looked at. One is set by your spending rules. The other is set by your review cadence and by which day you happen to read the report. Nothing in the figure separates them, so a reader who does not do the division is reading a spending number and an admin number fused into one.

Note what the identity does not say. The gap in percentage points between the headline and the policy rate is not the pending share; they are different quantities and they will not match. In the case below the gap is 15.3 points and the pending share is 16.7 percent of the window's expenses. Do the multiplication instead of subtracting and hoping.

One shop, one day, three windows

A shop's expenses arrive at a steady rate of about 4 a day. Review happens weekly, every Friday, and the reviewer clears everything submitted up to that Friday. The owner reads the figure on the last day of a 30-day month, which falls 5 days after the last review.

Expenses dated in the month: 120. The most recent 5 days, about 20 of them, have not been reviewed yet. Of the 100 that have been decided, 92 were approved and 8 rejected.

• Headline rate: 92 of 120, 76.7 percent.
• Policy rate: 92 of 100 decided, 92.0 percent.
• Decided share: 100 of 120, 83.3 percent.
• Check the identity: 0.920 times 100 over 120 is 0.767.

The owner reading 76.7 percent concludes that nearly a quarter of what the crew spends is being refused, and goes looking for a spending problem. There is no spending problem. Nine times in ten that a claim is decided, it is approved, and the 20 undecided items are 0 to 5 days old, which for a weekly cycle is a healthy queue, not a backlog.

Now hold the shop, the policy and the cadence completely still and change only the window read on that same day, assuming the 92.0 percent policy rate holds across the whole period:

Same shop, same day, same behaviour, three answers spanning 14.0 points. The pending tail is a fixed 20 items because it is set by the review cadence and the arrival rate, not by the window, so it is a large share of a short window and a rounding error in a long one. Notice also that the longest window still reads 1.3 points under the 92.0 percent policy rate. The headline approaches the policy rate as the window lengthens and never reaches it while any queue exists at all.

The same month, read a fortnight later

Leave the window alone and move the reading date instead. Two weeks on, the 20 stragglers have been decided: 19 approved, 1 rejected.

The month now holds 111 approved and 9 rejected, nothing pending. Headline rate 111 of 120, 92.5 percent. The same month that read 76.7 percent now reads 92.5 percent, a swing of 15.8 points produced by the passage of time and nothing else.

One detail in there is worth more than the swing. The late cohort approved at 19 of 20, 95.0 percent, against the early cohort's 92 of 100, 92.0 percent. The queue is not a random sample of the month: the claims that sit longest are often the ones missing a receipt or a site, which get chased and then approved. So the policy rate itself drifted, from 92.0 percent to 92.5 percent across the same month, purely because the population it was computed over finished arriving.

The reading rule that follows: read this figure only on a window whose end sits at least two review cycles in the past. On a weekly cycle that is 14 days. The month read at its own last day breaks that rule, which is why it returned 76.7 percent; read 14 days later it returns 92.5 percent, and that is the number that means something. If you need a figure for the month that just ended, you are asking for it too early, and the honest answer is the policy rate plus the queue's size and age stated beside it.

The two numbers to keep instead

Split the metric and both halves become actionable, with a clear owner each.

The policy rate, approved over decided. This is the real question about spending and about whether your limits are set where people actually spend. There is no defensible cross-trade benchmark for it and anybody quoting one is guessing, because it is a function of your own limits, your receipt rules and whether techs can buy at a counter at all. Benchmark it against your own trailing six months and investigate a move of more than a few points, or any two consecutive moves in the same direction however small, since a slow drift never trips a single-period threshold.

The queue, as a count and an age. Pending items as a share of the window, and the median age of an item at the moment it is decided. With a weekly cycle and steady arrivals, an item waits between 1 and 7 days depending on when it landed, so a median age at decision under 7 days is the target and nothing should sit past 14 days. A median above the cycle length is the tell that cycles are being skipped rather than that volume rose. That 14-day stop is the same two-cycle clock the attachment card sets for tying an expense to a job, deliberately, because both acts get harder at the same rate once nobody remembers the purchase.

Age is the half that carries a real cost, and it is not an accounting cost. A pending expense is usually a technician's own money, advanced to the shop and waiting. Three weeks of that and the next counter purchase gets made on a personal card with no intention of claiming it, or does not get made at all and the job waits for a part. The approval rate never sees either outcome.

When a high rate is the bad news

This figure is one of the few you can improve by doing less work. Approve everything on sight and the policy rate reads 100 percent forever.

So a rate pinned at or very near 100 percent, with zero rejections across several review cycles, is not a well-behaved crew. It is a review that has stopped being a control, and the thing it was controlling is now uncontrolled. The point of a review is not to catch thieves, which is rare; it is to catch the charge on the wrong job, the duplicate submission, the personal item entered in good faith and the purchase that should have gone on a supplier account at a better rate. Those exist in every shop's expense flow at some small rate, and a review that never finds one is not finding them.

Treat a policy rate above about 99 percent, sustained across three or more review cycles with no rejections at all, the same way you would treat a sharp fall: as a signal to sample. Pull ten decided items at random and re-review them properly. If all ten stand up, the crew is genuinely disciplined and you can say so with evidence. If two do not, you have learned that the figure has been reporting on the reviewer.

Reading a move, and what separates the causes

When the headline moves, four things can have caused it, and each leaves a different fingerprint across the split figures.

Work the table from the right-hand column. The queue age and the rejection count are cheap to pull and between them they separate all four cases, which the headline rate on its own cannot do at all. The first row is an admin problem and it belongs to whoever owns the review. The second is a policy question and it belongs to whoever set the limit. The third is a mix question and usually needs nothing done. The fourth is the only one that looks like good news on the report.

References

• Trade-standard practice: a fixed weekly review cycle, with the queue's size and median age reported beside the rate rather than folded into it
• See related: The Close Rate Improved Because They Stopped Quoting, which owns the general rule that a ratio cannot be read without the counts underneath it
• See related: Estimate Conversion Rate and the Cohort Problem, for why a rate whose outcomes land after its window understates itself
• See related: Expense per Job Only Sees Expenses Somebody Attached to a Job, for what the approved subset then feeds
• See related: Expense Category Concentration and the Other Bucket, for what the approved spend looks like once it is grouped