Pricing Short-Term Insurance Contracts for Exam STAM
Explore the fundamentals of pricing short-term insurance, including frequency-severity models and premium components.
Frequency-Severity Framework
Short-term insurance pricing builds on separating claim frequency from claim severity. The pure premium equals expected frequency times expected severity: E[S] = E[N] times E[X]. Common frequency distributions include Poisson, negative binomial, and binomial. Severity distributions include lognormal, Pareto, gamma, and Weibull. This decomposition allows actuaries to model each component independently and apply trend factors, development factors, and credibility adjustments to each.
The power of this framework lies in its modularity. If claim frequency is increasing at 3% per year due to more drivers on the road, while severity is increasing at 5% per year due to rising medical costs, you can trend each component separately. This produces more accurate projections than trending aggregate losses as a single quantity.
Frequency Distributions in Detail
The Poisson distribution with parameter lambda is the starting point for modeling claim counts. It assumes E[N] = Var(N) = lambda, which implies that the variance-to-mean ratio equals 1. In practice, insurance data often shows overdispersion (variance exceeding the mean) because of parameter uncertainty or heterogeneity in the insured population. The negative binomial distribution handles overdispersion with E[N] = r*beta and Var(N) = r*beta*(1+beta), giving a variance-to-mean ratio of (1+beta) > 1.
The binomial distribution applies when there is a fixed maximum number of claims possible (for example, m policies each with claim probability q). It produces underdispersion with Var(N) = m*q*(1-q) < m*q = E[N]. On Exam STAM, the choice of frequency distribution is usually specified in the problem, but understanding when each is appropriate helps you verify that your setup makes sense.
Severity Distributions in Detail
Each severity distribution has characteristics suited to different types of losses. The exponential distribution has a constant hazard rate and a thin tail, making it suitable for small, routine claims. The Pareto distribution has a heavy tail and is commonly used for liability and large-loss lines. Its limited expected value E[X wedge d] = theta * [1 - (theta/(theta+d))^(alpha-1)] / (alpha-1) appears frequently on Exam STAM.
The lognormal distribution is versatile and commonly used when losses span several orders of magnitude. If Y = ln(X) follows a normal distribution with mean mu and variance sigma^2, then E[X] = exp(mu + sigma^2/2). The gamma distribution offers flexibility through its shape parameter alpha; with alpha = 1, it reduces to the exponential. Weibull distributions are useful when the hazard rate is increasing or decreasing over the loss range.
Premium Components
The gross premium includes the pure premium plus provisions for expenses, profit, and contingencies. Fixed and variable expenses are loaded differently. Fixed expenses (such as policy issuance costs) are expressed as a dollar amount per policy. Variable expenses (such as commissions and premium taxes) are expressed as a percentage of premium.
The expense-loaded premium can be derived using two equivalent methods. The pure premium method calculates: Premium = (Pure Premium + Fixed Expenses per Exposure) / (1 - V - Q), where V is the variable expense ratio and Q is the profit and contingency factor. The loss ratio method calculates: Premium = Expected Losses / Target Loss Ratio, where the Target Loss Ratio = 1 - V - Q - F/P (with F being fixed expenses). Both methods produce the same indicated premium when applied correctly.
Example Calculation
Suppose a book of auto insurance has expected claim frequency of 0.08 claims per car-year and expected claim severity of 4,500. The pure premium per car-year is 0.08 * 4,500 = 360. Fixed expenses are 50 per policy, the variable expense ratio is 25%, and the target profit margin is 5%. Using the pure premium method, the indicated premium is (360 + 50) / (1 - 0.25 - 0.05) = 410 / 0.70 = 585.71 per car-year.
To verify using the loss ratio method, the target loss ratio is 1 - 0.25 - 0.05 - 50/585.71 = 0.70 - 0.0854 = 0.6146. Then premium = 360 / 0.6146 = 585.71, confirming the result. In practice, slight rounding differences may occur, but the two methods are algebraically equivalent.
Policy Modifications
Deductibles, policy limits, and coinsurance modify the relationship between ground-up losses and insured losses. An ordinary deductible d means the insurer pays max(X - d, 0) per loss. This reduces both frequency (losses below d generate no payment) and severity (each payment is reduced by d). The expected cost per loss with a deductible is E[X] - E[X wedge d], where E[X wedge d] is the limited expected value at d.
A policy limit u caps each payment at u. With both a deductible d and a limit u, the insurer pays min(max(X - d, 0), u), and the expected insured cost per loss is E[X wedge (d+u)] - E[X wedge d]. Coinsurance at rate c means the insurer pays c times the amount otherwise payable. These modifications are tested extensively on Exam STAM, often requiring computation of limited expected values for specific parametric distributions.
Common Pitfalls
A frequent error is confusing "expected cost per loss" with "expected cost per payment." The expected cost per loss includes zeros for losses below the deductible, while the expected cost per payment conditions on a payment being made. The relationship is: E[cost per payment] = E[cost per loss] / P(X > d). Make sure you know which quantity the problem asks for.
Another pitfall is forgetting that trend factors compound. If frequency trend is 3% and severity trend is 5%, the aggregate trend is not 8% but rather (1.03)(1.05) - 1 = 8.15%. Over multiple years, the compounding effect becomes more significant.
Study Tips
Exam STAM tests your ability to combine frequency-severity models with expense loading, and to adjust premiums for policy modifications like deductibles and limits. Build a reference sheet of limited expected value formulas for each distribution family. Practice problems that layer multiple adjustments: start with ground-up losses, apply a deductible, add a limit, trend forward, load for expenses, and produce a final premium. Getting comfortable with this full pipeline is essential for exam success.