Two Roads to a Rate: Pure Premium & Loss Ratio

The one question every rate must answer

In the previous guide you laid out the ratemaking goal and its raw materials: take a body of past experience, clean it up, and project it forward so that next year's price is *adequate* (it covers expected losses and costs), *not excessive*, and *fair* between policyholders. That last balance is exactly what rate adequacy and equity demand. This guide tackles the central calculation of the whole craft — the [[overall-rate-level-indication|overall rate level indication]]: by how much, up or down, does the *average* rate across the whole book need to move for next year?

Notice the word *indication*. An indication is the number the data *suggests*, the answer the past hands you when you ask it honestly. It is not yet the rate you will file or charge — management, regulators, competition, and the profit provision all have a say before the indication becomes the *selected* rate change. Keep that gap in mind: the methods below produce an indication, a recommendation grounded in evidence, not a verdict carved in stone.

Here is the pleasant surprise of this whole topic: there are exactly two standard ways to reach that indication, and they are *not* rivals. The pure premium method builds the right price from the ground up. The loss ratio method starts from the price you already charge and asks how far off it is. They are two roads to the same town — under matching assumptions they give the *same* answer — and a working actuary keeps both in the kit because each is the natural tool in a different situation.

Road one — pure premium: build the rate from scratch

The [[pure-premium|pure premium]] is the average loss per unit of exposure — losses (with loss adjustment expense) divided by earned exposure. It is also called the loss cost, and it is the bedrock of road one. The trick you already met in the loss-modelling rung is that the pure premium decomposes cleanly: pure premium = frequency × severity, where frequency is claims per exposure and severity is the average cost per claim. This frequency–severity split is what lets the pure premium method 'see inside' the loss and project its two halves separately — they move for very different reasons.

But the *raw* pure premium from last year's data is not yet usable, for two honest reasons. First, last year's losses are not finished: late-reported and still-open claims mean the figures must be developed to ultimate before you trust them — the loss-development triangles you met in reserving are exactly this step. Second, the world keeps moving: medical bills, car-repair costs, and jury awards drift year on year, so both losses *and* exposures must be trended forward to the future period the rate will actually cover. Skip developing losses and trend and your 'pure premium' describes a year that is already gone.

Once you have a *projected, ultimate* pure premium, the indicated rate is built by loading on the costs that pure premium leaves out. Fixed per-policy expenses are added on; the variable costs and the profit margin are made room for by dividing by the permissible loss ratio (the share of the rate you intend losses to consume — more on it shortly). The headline formula is short and worth memorising: indicated rate = (pure premium + fixed expense per exposure) ÷ permissible loss ratio. That single division turns the average loss into a full, gross price.

Road two — loss ratio: adjust the rate you already charge

Road two starts from the opposite end. Instead of building a price, the loss ratio method asks a simpler question: the rate I am *currently* charging — is it too high, too low, or about right? To answer it you compare the loss ratio you actually experienced against the loss ratio you *budgeted* to run at. That budgeted target is the [[permissible-loss-ratio|permissible loss ratio]]: the fraction of premium you deliberately set aside for losses and LAE after carving out room for expenses and profit. If your expenses and profit provision together claim 30 cents of every premium dollar, the permissible loss ratio is 70%.

The mechanism is a ratio of ratios. You take your *experience* loss ratio — projected losses divided by premium — and divide it by the permissible loss ratio. If experience came in at 77% but you only had room for 70%, the quotient is 77/70 = 1.10, an indicated rate change of +10%: losses are eating ten percent more of the premium than you planned, so the average rate must rise about ten percent to claw that room back. Come in at 63% against a 70% permissible, and 63/70 = 0.90 indicates a 10% *decrease*. The whole indication falls out of one honest comparison.

One subtlety makes or breaks the loss ratio method, and it is the most common mistake beginners make. The *premium* in that experience loss ratio must be restated as if every policy in the historical period had been sold at *today's* rates. Why? Because some of those policies were written under last year's rate level, before the increase you took six months ago. Comparing fresh losses to stale premium would overstate the loss ratio and demand a rate hike you have already taken. The fix is [[on-leveling-premium|on-leveling]] — putting all historical premium on the current rate level — and it is the indispensable partner of the loss ratio method.

Same town, two roads — and the permissible loss ratio that links them

The deep point — and the reassuring one — is that these are two faces of one calculation, not two theories that might disagree. Look again at the formulas. The pure premium method's divisor is the permissible loss ratio; the loss ratio method's *target* is the permissible loss ratio. The pure premium method works in *dollars per exposure* and produces an indicated *rate*; the loss ratio method works in *dimensionless ratios* and produces an indicated *change*. Feed them the same trended, developed, on-leveled data and the same permissible loss ratio, and they land on the identical answer. The choice between pure premium and loss ratio is a choice of *road*, not of destination.

Same book, same assumptions -> same indication.

  Permissible loss ratio (PLR) ......... 0.70

  ROAD 1 - Pure premium method
    projected pure premium (loss/exp) .. 210 per exposure
    fixed expense per exposure ......... 15
    indicated rate = (210 + 15) / 0.70 = 321.43 per exposure
    current avg rate ................... 292.21 per exposure
    indicated change = 321.43 / 292.21 - 1 = +10.0%

  ROAD 2 - Loss ratio method
    experience loss ratio (on-leveled) . 0.77
    indicated change = 0.77 / 0.70 - 1 = +10.0%

  Two roads, one +10% indication.

There is a close cousin worth a sentence. The [[expected-loss-ratio-method|expected loss ratio method]] runs the loss ratio idea in *reverse*: when a line is so new or so thin that its own experience is barely worth trusting, you take an assumed (expected) loss ratio from a benchmark or a related line and back into the rate from it. It is the loss ratio method wearing a different hat — useful precisely when the data you would normally lean on is not yet there.

Choosing the road — and where each one breaks

If they agree, why keep both? Because each needs different fuel, and one fuel is often missing. The pure premium method needs a well-defined exposure base — a clean count of car-years, payroll, or amounts of insurance — because it works in dollars *per exposure*. When exposures are crisply measured (auto, workers' compensation), it is the natural road, and it is the only road that can price a *brand-new* product that has no current rate to adjust at all. The loss ratio method needs no exposure base — only premium and losses — so it shines exactly where exposures are awkward or undefined.

1. Reach for the pure premium method when exposure is cleanly measured, when you are pricing a new product with no existing rate, or when you want to model frequency and severity separately to understand *why* the cost is moving.
2. Reach for the loss ratio method when exposure is hard to define (commercial property, where every building is unique), when you only have premium and loss totals, or when you simply want to fine-tune an existing rate rather than rebuild it.
3. Either way, do the unglamorous preparation first: develop losses to ultimate, trend losses and exposures forward, on-level the premium, and pin down the permissible loss ratio honestly. The method is the easy part; the preparation is where the truth lives.