Expenses

Study guides: Exam FAM, Exam ALTAM, Exam 8, Exam CP-321, Exam GI-101

Expenses are an insurer's costs of providing insurance other than the loss payments themselves. They come in two kinds: loss adjustment expense (LAE), the cost of settling claims, and underwriting expense, the cost of acquiring and servicing policies. In individual risk rating what matters is which expenses move with the insured's own losses. Only those are recalculated when a policy is retrospectively rated after it expires.

\text{Expenses} = \text{LAE} + \text{Underwriting Expenses}

R = (\text{BP} + \text{LCF} \times L) \times \text{TM}

• Components. LAE splits into ALAE (defence and experts, charged to a specific claim) and ULAE (the overhead of the claims department). Underwriting expenses are commissions and brokerage, other acquisition, general expenses, and premium taxes, licences and fees. Together they drive the expense ratio.
• Fixed or variable. In a rate, variable expenses rise with premium and fixed expenses are a dollar amount per exposure or per policy (Expense Provisions). Much of the cost of servicing an account doesn't grow with its size, so the expense ratio falls as premium grows. Large risks get premium discounts for that reason.
• Where expenses sit in a retro plan (Exam 8). R is the retrospective premium, L the insured's actual limited losses, \text{LCF} the loss conversion factor and \text{TM} the tax multiplier.
  • The loss conversion factor carries the LAE, which tracks losses, so it responds to actual experience.
  • The tax multiplier carries premium taxes and assessments, which are levied on whatever premium is finally charged.
  • The basic premium \text{BP} carries the remaining expenses (acquisition and general) plus the net insurance charge for the plan's maximum and minimum. It does not respond to losses.
• The basic premium formula. \text{BP} = e - (\text{LCF} - 1)\,E + \text{LCF} \times I. Here e is the expense provision excluding taxes, E is expected losses and I is the net insurance charge in loss dollars. Subtracting (\text{LCF}-1)E removes the LAE that the conversion factor will collect, so it isn't charged twice.

Which Expenses Respond to a Good Year A retro plan has standard premium \$500{,}000, expected losses E = \$325{,}000 and an expense provision excluding taxes (LAE included) of e = \$110{,}000. The terms are \text{LCF} = 1.12, net insurance charge I = \$30{,}000 and \text{TM} = 1.03. Actual limited losses come in at \$250{,}000, inside the plan's limits.

Find the basic premium and the retrospective premium, and say which expense provisions responded.

Answer

\begin{align*}
\text{BP} &= 110{,}000 - 0.12 \times 325{,}000 + 1.12 \times 30{,}000 \\
&= 110{,}000 - 39{,}000 + 33{,}600 \\
&= 104{,}600
\end{align*}

\begin{align*}
R &= (104{,}600 + 1.12 \times 250{,}000) \times 1.03 \\
&= 384{,}600 \times 1.03 \\
&= 396{,}138
\end{align*}

The LAE collected is 0.12 \times 250{,}000 = \$30{,}000, against \$39{,}000 expected, so it responded to the good year. Taxes of 0.03 \times 384{,}600 = \$11{,}538 responded to the lower premium. The \$104{,}600 basic premium was fixed at inception.

Check: on average the plan's limits remove I of losses, so E[R] = (104{,}600 + 1.12 \times 295{,}000) \times 1.03 = \$448{,}050. That equals (e + E) \times \text{TM}, so the plan is balanced.

Expense Ratio by Policy Size An insurer's costs are \$400 per policy plus 15\% of premium. Compare the expense ratio for a \$5{,}000 policy and a \$500{,}000 account.

Answer

\begin{align*}
\text{Small} &= \frac{400}{5{,}000} + 0.15 \\
&= 23.0\% \\[4pt]
\text{Large} &= \frac{400}{500{,}000} + 0.15 \\
&= 15.08\%
\end{align*}

A flat 23\% load would overcharge the large account by 0.0792 \times 500{,}000 = \$39{,}600. Large-risk rating plans therefore grade the expense provision by size.