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Part 4: The demand model

Part 4: The demand model

Why you need one

The LR constraint and the volume constraint both depend on how many customers renew at the new rates. Without a demand model, you have to assume either:

  • Everyone renews regardless of price (unrealistic: volume does not change)
  • Lapse rates are fixed regardless of price (unrealistic: rates do not affect who stays)

Neither is right. A demand model tells the optimiser: if you raise this customer's premium by 5%, the probability they renew changes from, say, 68% to 65%. The optimiser accounts for this when computing expected LR and expected volume.

Without a demand model, the volume constraint is not meaningful (it is always satisfied unless you have a separate assumption about lapses), and the LR constraint is overoptimistic (it ignores the fact that rate increases cause lapses, which change the book composition).

The logistic demand model

The rate-optimiser library uses a logistic demand model. This is the workhorse specification for renewal probability in UK personal lines. The renewal probability for policy i at a price ratio p_i (new premium divided by market premium) is:

renewal_prob_i = sigmoid(intercept + price_coef * log(p_i) + tenure_coef * tenure_i)

where sigmoid(x) = 1 / (1 + exp(-x)) is the logistic function. The inputs are:

  • p_i = new_premium_i / market_premium_i: how expensive this policy is relative to what the customer could get elsewhere
  • log(p_i): the log of the price ratio. Using log makes the model multiplicative: a 10% increase from 100% to 110% of market has the same demand effect as a 10% increase from 90% to 99% of market
  • tenure_i: years the customer has been with the insurer. Longer-tenured customers are stickier — they are less price-sensitive

The key parameter is price_coef. It is negative (higher price, lower renewal probability) and is called the log-price semi-elasticity. A value of -2.0 means: a 1% increase in log price above market reduces the log-odds of renewal by 2 percentage points.

To understand what that means in practice: if a customer currently has a 60% renewal probability (logit of about 0.41), and we raise their price by 1% above market, the new logit is 0.41 + (-2.0 x 0.01) = 0.41 - 0.02 = 0.39, giving a new renewal probability of sigmoid(0.39) = 59.6%. That is a 0.4 percentage point reduction in renewal probability for a 1% price increase.

For UK motor, the relevant benchmarks from market research and published lapse analyses (e.g., Bain & Company UK motor loyalty studies 2018-2022) are:

  • PCW (price comparison website) channel: price semi-elasticity typically -1.5 to -3.0. PCW customers have already demonstrated they will shop around. They are the most price-sensitive segment.
  • Direct channel: -0.5 to -1.5. Direct customers have already chosen not to use a PCW. A modest rate increase is less likely to trigger a lapse.

These are starting points. You must calibrate the demand model against your own observed lapse data before using it in the optimiser.

What miscalibration looks like

If you use a PCW elasticity of -2.5 when your actual elasticity is -1.2, the optimiser will believe you have far less pricing power than you do. It will think that even a small rate increase causes a large volume loss, and it will constrain the rate action more than necessary. The frontier will show infeasibility at targets that are actually achievable.

If you use -0.8 when your actual elasticity is -2.0, the optimiser will overestimate pricing power. It will produce a frontier that claims 72% LR is achievable at 97% volume retention, but in practice the actual lapses will be much higher and the achieved LR will be worse than the model predicted.

The demand model must be calibrated before you run the optimiser. Exercise 1 includes a calibration check. In Part 11 we address the limitations of the logistic specification.