Probit Models for Binary Demand

Why Use a Probit Demand Model?

For many insurance pricing decisions, demand is recorded as a binary outcome: a customer either purchases a quoted policy or does not, or renews a policy or does not. A probit model represents the probability of that outcome as a smooth function of price and other observed characteristics. Its link function is the cumulative distribution function of the standard normal distribution.

The model is useful when the objective is to estimate a purchase probability at a particular price, or to quantify how that probability changes as price changes. A coefficient is not itself a change in probability: the probability effect depends on the customer’s fitted index. This tutorial is aligned with the CAS Exam 8 (Advanced Ratemaking) syllabus.

Definition: Let $Y_i=1$ if customer $i$ purchases or renews, and $Y_i=0$ otherwise. A probit demand model specifies $$ P(Y_i=1\mid \mathbf{x}_i)=\Phi(\eta_i), \qquad \eta_i=\mathbf{x}_i^\mathsf{T}\boldsymbol{\beta}, $$ where $\mathbf{x}_i$ is the vector of observed predictors, $\boldsymbol{\beta}$ is the coefficient vector, and $\Phi$ is the standard normal cumulative distribution function. Its density is $\phi(z)=\frac{1}{\sqrt{2\pi}}\exp(-z^2/2)$.

Model Specification and Fitting

For example, if $P_i$ is a quoted premium in dollars and $R_i$ indicates whether a customer is a renewal, one model is

P(Y_i=1\mid P_i,R_i)=\Phi(\beta_0+\beta_P P_i+\beta_R R_i).

A negative $\beta_P$ means that increasing price lowers the fitted index, and therefore lowers purchase probability, when the other predictors are fixed. The size of the probability change is not equal to $\beta_P$; it depends on the index through $\phi$.

The standard fitting method is maximum likelihood. Assume the outcomes are conditionally independent Bernoulli observations, given the predictors, and that each conditional purchase probability follows the specified probit model. For fitted probabilities $q_i=\Phi(\mathbf{x}_i^\mathsf{T}\boldsymbol{\beta})$, the likelihood is

L(\boldsymbol{\beta})=\prod_{i=1}^{n}q_i^{Y_i}(1-q_i)^{1-Y_i}.

The maximum-likelihood estimate chooses coefficients that maximize this likelihood, or equivalently its logarithm. In practice, numerical optimization is used. A fitted coefficient table is not enough on its own to interpret demand: one must also specify the customer profile and price at which a probability or marginal effect is being calculated.

The probit model can also be motivated with a latent index. Suppose an unobserved purchase propensity is $Y_i^*=\eta_i+\varepsilon_i$, where $\varepsilon_i$ has a standard normal distribution, and the customer purchases when $Y_i^*>0$. Then

P(Y_i=1\mid \mathbf{x}_i) =P(\varepsilon_i>-\eta_i) =1-\Phi(-\eta_i) =\Phi(\eta_i).

The last equality follows from the symmetry of the standard normal distribution. The variance of the latent error is fixed at one to identify the scale of the index; multiplying every index coefficient and the error scale by the same positive constant would leave the purchase probabilities unchanged.

Worked Example: Fit a Probit Model to Two Price Groups

In a hypothetical quote study, 72 of 100 customers offered a monthly premium of $80 purchase, while 50 of 100 offered $100 purchase. Fit a probit model using an indicator $H$ that equals 1 for the $100 offer and 0 for the $80 offer. State what the fitted coefficient does, and identify a limitation of this design.

Method. Use a saturated two-group probit model, $\operatorname{P}(Y=1\mid H)=\Phi(\alpha+\gamma H)$. There are two parameters for two observed group probabilities, so maximum likelihood fits each group’s observed purchase proportion.

At $H=0$, the observed proportion is $72/100=0.72$. Therefore $\widehat{\alpha}=\Phi^{-1}(0.72)\approx 0.5828$. At $H=1$, the observed proportion is $50/100=0.50$, so the fitted index is $\widehat{\alpha}+\widehat{\gamma}=\Phi^{-1}(0.50)=0$. Thus

\widehat{\gamma}=0-\widehat{\alpha}\approx -0.5828.

The fitted probabilities check against the data: $\Phi(0.5828)\approx0.72$ for the $80 group, and $\Phi(0.5828-0.5828)=\Phi(0)=0.50$ for the $100 group. The negative indicator coefficient represents a lower fitted index for the high-price group. It is not a per-dollar coefficient: $H$ records a change between two offer groups, not a continuous price change. Consequently, this model does not supply a derivative with respect to each dollar of premium. The common mistake avoided here is dividing $-0.5828$ by the $20 price difference and treating the result as a general continuous price slope; two price points do not establish that linear index relationship across other prices.

Marginal Effects and Price Elasticity

For a continuous price predictor $P$, write the index as $\eta=\beta_0+\beta_P P+\mathbf{z}^\mathsf{T}\boldsymbol{\beta}_z$, holding the other predictors $\mathbf{z}$ fixed. Differentiating the fitted probability gives the price marginal effect:

\frac{\partial q}{\partial P} =\frac{\partial \Phi(\eta)}{\partial \eta}\frac{\partial \eta}{\partial P} =\phi(\eta)\beta_P, \qquad q=\Phi(\eta).

This follows from the chain rule: the derivative of $\Phi$ is its density $\phi$, and the derivative of the linear index with respect to price is $\beta_P$. Since $\phi(\eta)>0$ for every finite $\eta$, the marginal effect has the same sign as $\beta_P$. Its magnitude varies with the index, even when the price coefficient is constant.

The point price elasticity of purchase probability is the proportional change in probability divided by the proportional change in price. For a positive price and probability, it is

\varepsilon_P =\frac{\partial q}{\partial P}\frac{P}{q} =\beta_P P\frac{\phi(\eta)}{\Phi(\eta)}.

A negative elasticity means a local price increase is associated with a lower purchase probability. This is a point elasticity: it describes the local derivative at the specified price and profile. For a finite price change, calculate and compare the two probabilities directly rather than treating the derivative as an exact change.

Worked Example: Calculate a Probit Probability and Price Elasticity

Suppose maximum likelihood estimation on hypothetical monthly quote data gives the probit model $q=\Phi(1.20-0.012P)$, where $P$ is the monthly premium in dollars. For a new customer quoted $P=\$80$, calculate the purchase probability, the price marginal effect, and the point elasticity. Then compare them with a logistic model evaluated at the same index.

Probit index and probability. The fitted index is $\eta=1.20-0.012(80)=1.20-0.96=0.24$. Standard normal values give $\Phi(0.24)\approx0.5948$ and $\phi(0.24)\approx0.3871$. Thus the fitted purchase probability is approximately $0.5948$, or 59.48%.

Marginal effect. The price coefficient is $\beta_P=-0.012$ per dollar. Therefore

\frac{\partial q}{\partial P} =(-0.012)(0.3871)\approx-0.004645.

At this profile, a small $1 premium increase corresponds locally to a decrease of about 0.004645 in purchase probability, or 0.4645 percentage points. This is a local approximation, not an exact finite-change prediction.

Elasticity. Using the fitted probability and price,

\varepsilon_P \approx\frac{80}{0.5948}(-0.004645) \approx-0.6249.

The local interpretation is that a 1% increase in price corresponds to approximately a 0.6249% decrease in purchase probability at this point.

Logistic comparison. For an illustrative logistic calculation at the same index, the probability is $q_L=1/(1+\exp(-0.24))\approx0.5597$. The logistic marginal effect is $\beta_Pq_L(1-q_L)$, so it is approximately $(-0.012)(0.5597)(0.4403)=-0.002957$. Its point elasticity is approximately $(80/0.5597)(-0.002957)=-0.4226$. These results differ because the two links assign different probabilities and slopes to the same numerical index. This is a comparison at a common index, not a claim that coefficients from separately fitted models are directly interchangeable. The common mistake avoided is using $\phi(\eta)$ as the probability: the probability is $\Phi(\eta)$, while the density $\phi(\eta)$ is used in the derivative.

Effects Depend on the Customer Profile

Because the probit marginal effect contains $\phi(\eta)$, it varies across customers with different fitted indices. A reported effect should therefore state whether it is evaluated at a particular profile, at average covariates, or averaged over the observed profiles. These quantities need not be the same: in general, the average of $\phi(\eta_i)$ differs from $\phi$ evaluated at an average index.

Worked Example: Compare an Average Marginal Effect With an Effect at Average Covariates

Consider the hypothetical fitted model $\eta=1.00-0.010P+0.50R$, where $P$ is the monthly premium in dollars and $R=1$ denotes a renewal customer. Calculate the average price marginal effect for two profiles, both quoted at $100, and compare it with the effect evaluated at the average value of $R$.

Profile-specific effects. At $P=100,R=0$, the index is $1.00-1.00+0=0$. Hence $\phi(0)=0.3989$ and the marginal effect is $(-0.010)(0.3989)=-0.003989$ per dollar. At $P=100,R=1$, the index is $0.50$. Since $\phi(0.50)\approx0.3521$, the marginal effect is $(-0.010)(0.3521)=-0.003521$ per dollar.

Average marginal effect. Give each profile equal weight. The average is

\frac{-0.003989+(-0.003521)}{2} =-0.003755 \text{ per dollar}.

Effect at average covariates. The average renewal indicator is $(0+1)/2=0.5$. Substituting $P=100,R=0.5$ gives the average index $\eta=0.25$. Since $\phi(0.25)\approx0.3867$, the effect at average covariates is $(-0.010)(0.3867)=-0.003867$ per dollar. It differs from the average marginal effect of $-0.003755$. The distinction matters when summarizing a heterogeneous set of fitted responses; the two calculations answer different questions.

The values check the expected direction: both effects are negative because the price coefficient is negative. The common mistake avoided is assuming that the effect at the average profile must equal the average of profile-specific effects.

Practical Checks and Common Traps

A careful probit calculation keeps three quantities distinct: the index $\eta$, the probability $\Phi(\eta)$, and the density $\phi(\eta)$. The index is an unbounded linear predictor; the probability lies between zero and one; and the density converts a change in index into a local change in probability.

Do not read a probit coefficient as a probability change. Multiply the relevant coefficient by the density at the profile’s index to obtain a continuous-predictor marginal effect.
Use the correct price units. If the model uses price in hundreds of dollars, its price coefficient is per hundred dollars. Convert the derivative before describing a one-dollar change or forming an elasticity.
Do not confuse a marginal effect with a finite change. For a proposed change from $P_0$ to $P_1$, compute $\Phi(\eta(P_1))-\Phi(\eta(P_0))$. A derivative is a local rate, not an exact difference over a large interval.
Do not compare coefficients without considering the link and scale. Probit and logistic models use different link functions. Similar-looking coefficients do not imply identical probabilities or marginal effects.
Check the sign and units. With price measured in dollars, $\partial q/\partial P$ is a probability change per dollar; elasticity is dimensionless. A negative price coefficient should produce a nonpositive marginal effect.

Key takeaway: A probit model maps a linear index to purchase probability with $\Phi$. Its price marginal effect is $\beta_P\phi(\eta)$, and its point elasticity is $\beta_P P\phi(\eta)/\Phi(\eta)$. Both depend on the customer profile and the price units used in the fitted model.

Check Your Understanding

For each problem, treat the stated fitted model as given and show the intermediate steps in your own calculation.

A fitted model is $q=\Phi(0.30-0.015P)$, with $P$ in dollars. At $P=20$, calculate the index, purchase probability, and price marginal effect. (Answer: $\eta=0$, $q=0.5000$, and $\partial q/\partial P\approx-0.005984$ per dollar.)
A fitted probit model has index $\eta=1.10-0.010P$. At $P=\$60$, calculate the probability and point price elasticity using $\Phi(0.5)\approx0.6915$ and $\phi(0.5)\approx0.3521$. (Answer: $q\approx0.6915$, $\partial q/\partial P\approx-0.003521$, and $\varepsilon_P\approx-0.3055$.)
A model has $\eta=-0.40+0.20R-0.008P$, where $R=1$ for a renewal and $P$ is in dollars. For a renewal quoted at $P=\$75$, calculate the index, probability, marginal effect, and elasticity using $\Phi(-0.8)\approx0.2119$ and $\phi(-0.8)\approx0.2897$. (Answer: $\eta=-0.8$, $q\approx0.2119$, $\partial q/\partial P\approx-0.002318$, and $\varepsilon_P\approx-0.8203$.)
At $P=\$50$, compare the probability, marginal effect, and elasticity from a probit model with index $0.40$ and price coefficient $-0.020$ with those from a logistic model evaluated at the same index. Use $\Phi(0.40)\approx0.6554$ and $\phi(0.40)\approx0.3683$. (Answer: probit $q\approx0.6554$, marginal effect $\approx-0.007366$, elasticity $\approx-0.5618$; logistic $q\approx0.5987$, marginal effect $\approx-0.004805$, elasticity $\approx-0.4013$.)
A fitted model has price coefficient $-0.010$. Two equally weighted profiles have indices $-1$ and $1$ at their quoted prices. Calculate their average marginal effect using $\phi(-1)=\phi(1)\approx0.2420$, then compare it with the effect at the average index, zero. (Answer: average marginal effect $\approx-0.002420$ per dollar; effect at the average index $\approx-0.003989$ per dollar.)

What You'll Learn