Underwriting is the discipline of deciding which insurance applications to accept and at what price, and actuarial risk scoring is the statistical engine behind that decision at scale. This simulation generates a stream of synthetic applicants, each described by real underwriting risk factors — age, prior claims history, insured asset value, and a regional risk tier — and scores every one with a logistic regression frequency model of the same mathematical form used in production actuarial pricing systems. The model's linear risk score is passed through a sigmoid function to produce a bounded claim probability, which is then combined with an asset-scaled severity estimate to produce an expected loss, and finally grossed up by a loading factor into a premium.
An adjustable risk threshold sets the underwriting cutoff: applicants whose modelled claim probability exceeds the threshold are declined, while everyone else is accepted at their computed premium. Because each accepted policy's realised outcome is then drawn from its true underlying probability rather than copied from the model's estimate, the simulation's running loss ratio and acceptance rate genuinely respond — with realistic statistical noise — to every change you make to the threshold or the loading factor, exactly as a real underwriting book would.
What is underwriting risk scoring, and why not just review every application by hand?
Underwriting is the process of deciding whether to accept an insurance application and at what price. A pure manual approach cannot scale, and human reviewers apply inconsistent standards across thousands of applications. Actuarial risk scoring instead fits a statistical model — historically a generalised linear model such as logistic regression — to estimate the probability and expected cost of a future claim from measurable risk factors (age, claims history, asset value, region). The score is consistent, auditable, and can be recalibrated as loss experience accumulates, which is why it underpins pricing and acceptance decisions at virtually every modern insurer.
How does the logistic risk model in this simulation actually compute a risk score?
Each synthetic applicant's raw factors (age, prior claims count, insured asset value, region risk tier) are standardised and combined into a linear score z = b0 + b1·age_z + b2·claims + b3·asset_z + b4·region, exactly as in a real generalised linear model. That z is passed through the logistic (sigmoid) function p = 1/(1+e^-z) to produce a probability of a claim in the coming policy period, bounded between 0 and 1. This is the same mathematical form used in real actuarial frequency models — it guarantees the output is a valid probability regardless of how extreme the input factors are, unlike a plain linear regression which could predict probabilities below 0% or above 100%.
How is the premium calculated from the risk score, and why does it depend on more than just claim probability?
Premium is built from expected loss, not probability alone. Expected loss = claim probability (frequency) × expected claim size (severity), and severity scales with the insured asset value — a claim on a larger policy costs more even at the same probability. The simulation then applies a loading factor on top of expected loss to cover underwriting expenses, capital costs, and profit margin, plus a small fixed expense charge per policy: premium = expected_loss × (1 + loading) + fixed_expense. This mirrors the real actuarial pricing identity used across property, motor and life insurance, where frequency and severity are modelled and priced separately before being combined.
The threshold sets the maximum modelled claim probability the underwriter will still accept. Raising the threshold accepts more applicants — acceptance rate goes up — but includes riskier policies whose realised claims push the loss ratio (claims paid divided by premium collected) higher. Lowering the threshold declines more marginal risk, shrinking the book of business (lower acceptance rate) but keeping the accepted population closer to the assumptions the pricing model was built on, which keeps the loss ratio nearer its target. There is no threshold that maximises both metrics at once — this simulation makes that tradeoff directly visible and adjustable in real time.
Loss ratio = total claims paid / total premium collected over a period. A loss ratio of 60% means 60 cents of every premium pound went to claims, leaving 40 cents for expenses and profit — this simulation's loading factor is calibrated so that a target loss ratio (typically 55–70%) leaves room for the built-in expense charge. A loss ratio far below target usually means the threshold is too strict and premiums are overpriced relative to risk (losing competitive business); a loss ratio well above target means claims are eating into the loading meant for expenses and profit, and the book is unprofitable at the current threshold and pricing.
Adverse selection is the tendency for the riskiest applicants — who know their own risk better than the insurer's model captures it — to be disproportionately represented among accepted policies when underwriting standards loosen. In this simulation, raising the threshold lets in applicants whose modelled probability is close to the cutoff; because the model score is an estimate, not a certainty, some of those marginal applicants have true risk noticeably higher than their score suggested. Watch the loss ratio: as the threshold rises, the accepted mix increasingly tilts toward the model's blind spots, and realised losses grow faster than the modelled expected loss would predict.
The risk score is a probability, not a forecast of any single outcome — an applicant scored at 12% claim probability will still generate a claim roughly one time in eight if you underwrite enough of them, purely from irreducible randomness. This simulation draws each accepted policy's actual outcome as a Bernoulli trial using its true underlying probability and a lognormal severity draw, so short-run realised loss ratio fluctuates around the model's expectation rather than matching it exactly tick by tick. This is exactly why real insurers underwrite large, diversified books and monitor loss ratio over rolling windows rather than judging the model by any single policy.
Because probability enters the expected-loss calculation multiplicatively with severity and then gets grossed up by a proportional loading factor, small increases in risk score near the high end translate into disproportionately larger premium increases than the same size increase near the low end — a compounding effect rather than a flat per-point surcharge. This convex relationship is deliberate: it reflects how insurers price tail risk more aggressively than routine risk, discouraging adverse selection by making the riskiest-but-still-acceptable applicants pay a premium that already reflects most of their added expected cost.