Actuarial Notes

Large Deductible Policy

Study guide: Exam 5

A Large Deductible Policy is a commercial policy — commonly workers compensation, also general and auto liability — whose per-occurrence deductible is high enough that the insured bears a significant share of its own losses (Werner's example is a \$1 million workers compensation deductible). It is a loss-sensitive arrangement: the insured's total cost depends on its own claims, and the premium covers only what the insurer still bears.

\text{Premium} = \frac{\text{Excess Losses} + \text{ALAE} + F + \text{CR} + \text{RM}}{1 - V - Q}

\text{Excess Losses} = E[X] \times \big(1 - \text{LER}(d)\big)

  • E[X] is expected ground-up loss, \text{LER}(d) the loss elimination ratio at deductible d (so 1 - \text{LER} is the excess ratio), F fixed expenses including the cost of processing deductibles, CR a credit risk provision, RM a risk margin, V the variable expense and Q the profit provision. This is Werner & Modlin's premium formula; otherwise the pricing follows ordinary deductible rating.
  • Werner's four additional considerations:
    • Claims handling — usually the insurer handles every claim, so the premium must cover handling cost even on claims that never pierce the deductible; if the insured handles them, price for the risk of leakage above the deductible.
    • Application of the deductible — to loss only, or to loss plus ALAE; the LERs must be computed on data consistent with the policy terms.
    • Deductible processing — where the insurer pays the whole claim and bills the insured back, the premium carries the cost of invoicing and monitoring and a provision for the insured's inability to pay; collateral rarely covers that credit risk fully.
    • Risk margin — losses above a large deductible are more uncertain than those below, so the profit margin may need to rise.
  • Thin data at high deductibles. An insured's own losses above a high d are too sparse to price from, so excess ratios come from size-of-loss distributions or fitted curves, and they are leveraged: with d fixed, the excess layer grows faster than ground-up severity.
  • Many programs also cap the insured's total reimbursements with an aggregate limit; the insurer's expected payments above it are an additional insurance charge, much like the charge for a retro maximum.
  • Contrast a self-insured retention, where the insured pays its own claims and the policy sits above them. The reserving counterpart — estimating and collecting what insureds owe back — is Deductible Recovery. Related mechanisms: Loss Sensitive Rating, Commercial Lines Rating.

Pricing a Workers Compensation Large Deductible Expected ground-up losses are \$2{,}400{,}000; \text{LER}(\$250{,}000) = 75\%. The insurer handles and pays every claim and bills back amounts under the deductible, which applies to loss only; ALAE is 9\% of ground-up losses and is not reduced by the deductible. Fixed expenses are \$60{,}000; deductible processing costs 3\% and credit risk 1.5\% of expected deductible reimbursements; the risk margin is 12\% of excess losses; V = 10\% and Q = 3\%.

Compute the premium and compare with a guaranteed-cost (no deductible) premium.

Answer

\begin{align*}
\text{Excess losses} &= \$2{,}400{,}000 \times 0.25 = \$600{,}000 \\
\text{Reimbursed} &= \$2{,}400{,}000 \times 0.75 = \$1{,}800{,}000 \\
\text{ALAE} &= 0.09 \times \$2{,}400{,}000 = \$216{,}000 \\
\text{Processing} &= 0.03 \times \$1{,}800{,}000 = \$54{,}000 \\
\text{Credit risk} &= 0.015 \times \$1{,}800{,}000 = \$27{,}000 \\
\text{Risk margin} &= 0.12 \times \$600{,}000 = \$72{,}000
\end{align*}
\begin{align*}
\text{Numerator} &= 600{,}000 + 216{,}000 + 60{,}000 \\
&\quad + 54{,}000 + 27{,}000 + 72{,}000 \\
&= \$1{,}029{,}000 \\
\text{Premium} &= \frac{\$1{,}029{,}000}{1 - 0.10 - 0.03} \\
&= \$1{,}182{,}759
\end{align*}

A guaranteed-cost policy would cost (\$2{,}400{,}000 + \$216{,}000 + \$60{,}000) / 0.87 = \$3{,}075{,}862. The insured's expected total outlay under the large deductible is \$1{,}182{,}759 + \$1{,}800{,}000 = \$2{,}982{,}759: in this model, variable expense and profit are no longer loaded on the retained layer, which outweighs the processing, credit and risk charges, and the insured keeps the cash for its retained losses until they are paid. In exchange it takes on the volatility of \$1.8 million of its own losses.

Leveraged Trend in the Excess Layer Three claims, ground-up: \$200{,}000, \$400{,}000 and \$900{,}000. The deductible is \$250{,}000 per occurrence. Severity inflation of 8\% applies to every claim. How do the insured's retained losses and the insurer's excess losses change?

Answer

\begin{align*}
\text{Excess before} &= 0 + 150{,}000 + 650{,}000 = \$800{,}000 \\
\text{Excess after} &= 0 + 182{,}000 + 722{,}000 = \$904{,}000 \\
\text{Change} &= 904 / 800 - 1 = +13.0\% \\
\text{Retained before} &= 200{,}000 + 250{,}000 + 250{,}000 = \$700{,}000 \\
\text{Retained after} &= 216{,}000 + 250{,}000 + 250{,}000 = \$716{,}000 \\
\text{Change} &= 716 / 700 - 1 = +2.3\%
\end{align*}

Ground-up losses grew 8\%; the insurer's layer grew 13\% and the insured's only 2.3\%, because every dollar of inflation on a claim already above d lands in the excess layer. A large deductible left unchanged for several years quietly shifts cost to the insurer, so both the excess losses and the deductible level need trending.