Liability Insurance

Study guide: Exam DISC-RM

Liability insurance pays, on the insured's behalf, the sums the insured becomes legally obligated to pay as damages to a third party — for bodily injury or property damage caused by a covered activity, such as owning or using an auto — up to the policy limit, and provides the insured's legal defence. It is third-party cover: the claimant is not the policyholder, and the amount is set by tort law and the courts rather than by an inspection of the insured's own loss.

$\text{Indemnity} = \min(D, L)$

$\text{Insurer cost} = \min(D, L) + \text{Defence costs}$

$D$ is the damages for which the insured is legally liable and $L$ the limit of insurance. Under the business auto liability coverage, defence and other supplementary payments are paid in addition to the limit, and the duty to defend ends once the limit is used up in judgments or settlements; some professional liability forms put defence inside the limit instead.
Limits. A combined single limit applies one $L$ per accident to all bodily injury and property damage together; split limits (per person, per accident, property damage) apply separate caps. Higher limits are priced with increased limits factors.
Liability can attach without fault of the insured's own, through vicarious liability for an employee driver, and several defendants may share it (Joint and Several Liability). The insured is typically "anyone using a covered auto" with permission, not only the named insured.
Liability is the classic long-tail line: injuries are litigated and settled over years, so IBNR and development dominate its reserves — unlike first-party physical damage (Short Tail Insurance).

Combined Single Versus Split Limits A delivery van hits a car. Three occupants are injured, with damages of $\$450{,}000$ , $\$380{,}000$ and $\$300{,}000$ , and the other driver's $\$120{,}000$ car is destroyed. Defence costs are $\$90{,}000$ . How much does the insurer pay in damages, and how much falls on the insured, if the policy has (a) a $\$1{,}000{,}000$ combined single limit, (b) split limits of $\$250{,}000$ / $\$500{,}000$ / $\$100{,}000$ ?
Answer Total damages are $\$1{,}250{,}000$ .

(a) One limit for the whole accident:
 $\begin{align*} \text{Insurer pays} &= \min(1{,}250{,}000,\ 1{,}000{,}000) \\ &= \$1{,}000{,}000 \\ \text{Insured retains} &= \$250{,}000 \end{align*}$ 
(b) Each injured person is capped at $\$250{,}000$ , the accident at $\$500{,}000$ , property damage at $\$100{,}000$ :
 $\begin{align*} \text{BI} &= \min(250{,}000 \times 3,\ 500{,}000) \\ &= \$500{,}000 \\ \text{PD} &= \min(120{,}000,\ 100{,}000) \\ &= \$100{,}000 \\ \text{Insured retains} &= 1{,}250{,}000 - 600{,}000 \\ &= \$650{,}000 \end{align*}$ 
In both cases the $\$90{,}000$ of defence is paid on top of the damages. The combined single limit leaves the insured with $\$400{,}000$ less exposure for the same accident.

What a Higher Limit Costs Liability claims per vehicle are Poisson with mean $0.05$ , and claim size is Pareto with $\alpha = 2$ , $\theta = 200{,}000$ , so $E[X \wedge u] = 200{,}000\left[1 - \frac{200{,}000}{200{,}000 + u}\right]$ . Find the expected loss cost per vehicle at a $\$100{,}000$ basic limit and at $\$1{,}000{,}000$ , and the increased limits factor.
Answer
 $\begin{align*} E[X \wedge 100{,}000] &= 200{,}000\left(1 - \tfrac{2}{3}\right) \\ &= 66{,}667 \\ E[X \wedge 1{,}000{,}000] &= 200{,}000\left(1 - \tfrac{1}{6}\right) \\ &= 166{,}667 \\ \text{ILF} &= \frac{166{,}667}{66{,}667} \\ &= 2.50 \\[4pt] \text{Loss cost at } \$100\text{K} &= 0.05 \times 66{,}667 \\ &= \$3{,}333 \\ \text{Loss cost at } \$1\text{M} &= 0.05 \times 166{,}667 \\ &= \$8{,}333 \end{align*}$ 
Ten times the limit costs two and a half times the loss cost: the heavy tail puts most of the expected loss above the basic limit.

How it appears on the exam

DISC-RM (Business Auto Underwriting Considerations – Liability Coverage): a written or scenario question on what an underwriter weighs for the liability exposure of a commercial fleet — factors such as vehicle type and weight, how and how far the vehicles are driven, the industry, driver selection and records, and vehicle maintenance — and why each moves expected liability losses.

Expect a described account and a question on which feature most raises the liability hazard, or which underwriting action (driver standards, a higher retention, a limit change) addresses it.

Pairs with the business auto form's insuring agreement: who is an insured, what a covered auto is, and that defence is in addition to the limit.

Exam traps

Treating defence as eroding the limit. Under the business auto liability coverage it is paid on top; an answer that subtracts defence from the $\$1{,}000{,}000$ has applied a professional-liability-style form.

Applying split limits per claimant only. The per-accident cap binds after the per-person caps: $3 \times 250{,}000 = 750{,}000$ is cut to $\$500{,}000$ .

First-party versus third-party. Damage to the insured's own vehicle is physical damage coverage, not liability; adding it to the liability claim is a classification error.