Increased Limits

Study guides: Exam ASTAM, Exam 5, Exam 8, Exam GI-301

Increased Limits Factors (ILFs) are the multiplicative factors that convert a basic-limits loss cost into the loss cost for a higher policy limit, reflecting the additional expected loss in the layer above the basic limit.

$\text{ILF}(L) = \frac{E[X \wedge L]}{E[X \wedge B]}$

$\text{Layer } (M, L] \text{ cost} = \text{Basic LC} \times \left[\text{ILF}(L) - \text{ILF}(M)\right]$

$B$ is the basic limit (ISO uses $\$100{,}000$ for general liability and commercial auto liability), so $\text{ILF}(B) = 1.000$ by construction and $\text{ILF}(L) > 1$ for $L > B$ . $E[X \wedge L]$ is the limited expected value, the expected loss capped at $L$ .
Werner's indicated ILF is the ratio of losses and LAE at the two limits, assuming all underwriting expenses are variable and the variable expense and profit provisions do not vary by limit. Actuaries may vary the profit provision by limit for the volatility of high limits — a departure Werner notes from that assumption, not part of his formula. Palmer's fuller formulation puts it inside the factor: $\text{ILF}(L) = \dfrac{\text{LAS}(L) + \text{ALAE}(L) + \text{ULAE}(L) + \text{RL}(L)}{\text{LAS}(B) + \text{ALAE}(B) + \text{ULAE}(B) + \text{RL}(B)}$ , where the risk load covers process and parameter risk, both of which vary by limit.
ILFs are estimated from a size-of-loss distribution — empirically, building the limited average severity layer by layer so that losses censored at policy limits introduce no bias, or from a fitted curve (lognormal, Pareto, truncated Pareto); ISO's mixed exponential method and GLMs are also used. Data at high limits is sparse and volatile, which is why ILFs are calculated from a broader body of experience (class group, state group or countrywide) and applied to class-, state- and territory-specific basic limit loss costs.
Consistency. Since $\text{LAS}(k) = \int_0^k S(x)\,dx$ (Palmer), each additional dollar of limit at $x$ adds $S(x)$ — the probability a loss exceeds $x$ — which cannot rise as $x$ rises. So the marginal indemnity cost per dollar of additional limit must not increase as the limit rises; an indemnity ILF table whose marginal cost increases is not consistent with any size-of-loss distribution.
ILFs are leveraged by inflation. As losses grow, more of them pierce the basic limit and the excess layers grow faster than the ground-up trend — so ILF tables must be reviewed, not left in place, through inflationary periods.
The assumptions behind Werner's ILF matter: all underwriting expenses are variable and the variable expense and profit provisions do not vary by limit; frequency and severity are independent; and frequency is the same whatever limit is chosen. The last fails for some lines — personal auto insureds who choose very high limits tend to have lower accident frequencies — which is one reason a GLM, which does not assume equal frequency, can give different (even counter-intuitive) factors.

Pricing a Higher Limit The basic limit is $\$100{,}000$ , with $\text{ILF}(100\text{K}) = 1.00$ , $\text{ILF}(300\text{K}) = 1.25$ , $\text{ILF}(500\text{K}) = 1.35$ , $\text{ILF}(1\text{M}) = 1.45$ . The basic-limits loss cost is $\$800$ .

Price the $\$500$ K policy, and the $\$200$ K excess of $\$300$ K layer.
Answer $\text{Loss cost at } \$500\text{K} = \$800 \times 1.35 = \$1{,}080$

The layer between $\$300$ K and $\$500$ K:
 $\begin{align*} \text{Layer cost} &= \$800 \times (1.35 - 1.25) \\ &= \$80 \end{align*}$ 
Note how little the top of the tower costs: the first $\$100$ K costs $\$800$ , the next $\$200$ K costs $\$200$ , and the $\$200$ K above that costs $\$80$ . Each successive layer is cheaper per dollar of limit — which is exactly the consistency property.

Note also that Werner's ILF applies to the whole rate only under its all-expenses-variable assumption. Fixed expenses do not rise with the limit, so where they are loaded into the rate the final premium rises by less than the ILF.

Testing an ILF Table for Consistency A filing proposes:

Limit ILF

$\$100$ K $1.00$

$\$250$ K $1.30$

$\$500$ K $1.45$

$\$1$ M $1.75$

Is the table consistent?
Answer Marginal cost per $\$1{,}000$ of additional limit:
 $\begin{align*} \$100\text{K} \to \$250\text{K}: \; \frac{0.30}{150} &= 0.00200 \\[6pt] \$250\text{K} \to \$500\text{K}: \; \frac{0.15}{250} &= 0.00060 \\[6pt] \$500\text{K} \to \$1\text{M}: \; \frac{0.30}{500} &= 0.00060 \end{align*}$ 
The marginal cost falls from $0.00200$ to $0.00060$ , then stays flat. Flat is acceptable only in a degenerate case: it implies the expected loss in the $\$500$ K– $\$1$ M layer is the same per dollar as in the $\$250$ K– $\$500$ K layer, so $S(x)$ could not fall anywhere between $\$250$ K and $\$1$ M — no loss could end in that range.

If these are indemnity-only factors, either the $\$1$ M factor is too high or the $\$500$ K factor is too low. If they include a risk load, which Palmer describes as increasing with the policy limit, the load can legitimately lift the higher factors — so the test should be run on the indemnity component before the load is added.