Table L

Table L is a table of insurance charges $\phi^*_D(r)$ and savings $\psi^*_D(r)$ that prices a per-occurrence limit $D$ and an aggregate limit together, as a single factor of expected unlimited loss. It is named for Skurnick's California Table L. Like Table M $_D$ it has three dimensions — expected size of the policy, entry ratio, and per-occurrence limit — so it is one table per limit.

$k = \frac{E - \mathrm{E}\{A_D\}}{E} = 1 - \frac{\mathrm{E}\{A_D\}}{E}$

$r = \frac{A_D}{E}$

$\phi^*_D(r) = \int_r^{\infty} (y - r)\, dF_D(y) + k$

$\psi^*_D(r) = \int_0^{r} (r - y)\, dF_D(y) = \phi^*_D(r) + r - 1$

$E$ is the expected unlimited aggregate loss, $A_D$ the actual aggregate loss with each occurrence limited to $D$ , and $F_D$ the distribution of $A_D/E$ . $k$ is the excess ratio for the per-occurrence limit — the fraction of expected loss above $D$ (called "ER" in the pre-2019 NCCI retro manual). Exam solutions often call $k$ the "LER": the share of loss the per-occurrence limit eliminates from the insured's losses. That is $1 - \text{LER}$ in the deductible sense of Loss Elimination Ratio, whose LER is the share below the deductible.
The entry ratio is limited loss over unlimited expected loss. The Table L charge is the average difference between a risk's unlimited and limited loss, plus its limited loss in excess of $rE$ , all as a ratio to expected unlimited loss. The $k$ term is the per-occurrence excess; the integral is the additional effect of the aggregate limit beyond the occurrence limit.
From a limited table. A charge $\phi_{\text{lim}}$ read from a Table M $_D$ , which is a ratio to expected limited loss, converts to a ratio to unlimited loss, for the same aggregate limit, by $\phi^* = \phi_{\text{lim}}(1 - k) + k$ . When the aggregate charge is already a ratio to expected unlimited loss, as the integral in Table L is, no $(1 - k)$ factor is applied: $\phi^* = (\text{that charge}) + k$ .
Shape. $\phi^*_D(0) = 1$ , because all of the expected unlimited loss is either above $D$ or above zero. $\phi^*_D(r) - k$ is a Table M charge of the limited distribution, so $\phi^*_D$ is decreasing and convex in $r$ . It never falls below $k$ : losses above the per-occurrence limit are always the insurer's, however high the aggregate. At and above the largest limited entry ratio the aggregate part is 0, and $\phi^*_D(r) = k$ .
With no loss limit, $k = 0$ , $F_D = F$ , and Table L reduces to Table M. Using the limited distribution $F_D$ in the integral is what keeps the per-occurrence and aggregate limits from double counting the losses they overlap on (see Lee Diagram); the alternative way of pricing a limited aggregate from an unlimited Table M is ICRLL.

The idea Table L asks one question — of every dollar of expected loss, how much falls to the insurer after both limits act? — and answers it per occurrence first. The fraction $k$ above the occurrence limit is gone before the aggregate is even counted, so the charge has a floor at $k$ and only the limited losses can pile up past the aggregate.

Building Table L from data

For each risk in the group, record both the unlimited and the limited aggregate loss.
Compute $k$ from the averages: $k = 1 - (\text{average limited})/(\text{average unlimited})$ .
Compute each risk's entry ratio as its limited aggregate loss over the group's expected unlimited loss. The average entry ratio is $1 - k$ .
Slice horizontally as for Table M: begin from the bottom row with 0, since no loss lies above the largest entry ratio, and work up, adding (difference in $r$ ) × (% of risks above $r$ ). This gives $\phi^*_D(r) - k$ ; add $k$ to every row.
Savings follow from $\psi^*_D(r) = \phi^*_D(r) + r - 1$ .

Claim-level unlimited data can be limited "as if" at any per-occurrence limit, so one body of data serves every table. Simulating a Table L needs claim counts and severities simulated separately, so the limit can be applied claim by claim.

Building Table L for a $\$50000$ Limit Ten risks have unlimited aggregate losses (in $000) of $20, 50, 60, 70, 80, 80, 90, 100, 150, 300$ and, with each occurrence limited to $\$50{,}000$ , limited aggregate losses of $20, 50, 60, 70, 80, 80, 90, 100, 120, 250$ . Find $k$ , $\phi^*_D(1.0)$ and $\phi^*_D$ at the top entry ratio.
Answer
 $\begin{align*} k &= 1 - \frac{92}{100} \\ &= 0.08 \end{align*}$ 
The entry ratios, limited loss over $\$100$ K, are $0.2, 0.5, 0.6, 0.7, 0.8, 0.8, 0.9, 1.0, 1.2, 2.5$ . Slicing from the top:
 $\begin{align*} \phi^*_D(2.5) - k &= 0 \\ \phi^*_D(1.2) - k &= 0 + 1.3 \times 10\% \\ &= 0.13 \\ \phi^*_D(1.0) - k &= 0.13 + 0.2 \times 20\% \\ &= 0.17 \\ \phi^*_D(1.0) &= 0.17 + 0.08 \\ &= 0.25 \end{align*}$ 
At the top entry ratio $\phi^*_D(2.5) = k = 0.08$ : the aggregate limit removes nothing more, but the per-occurrence limit still does. (Fisher et al., Exhibits 3.24–3.27.)

How it appears on the exam

Exam 8 gives unlimited and limited aggregate losses for a group of risks and asks for $k$ and a Table L charge or savings at an entry ratio.

It asks for the shape of $\phi^*_D(r)$ : its value at $r = 0$ , its limit as $r$ grows, and why it is convex.

It asks when the $(1 - k)$ adjustment applies, and how Table L compares with Table M and ICRLL.

Exam traps

Dividing by limited expected loss: the Table L entry ratio is limited loss over unlimited expected loss. Dividing by expected limited loss gives the Table M $_D$ entry ratio instead.

Forgetting to add $k$ : the horizontal slices give $\phi^*_D(r) - k$ ; without $k$ , the charge falls to 0 at high entry ratios.

Applying $(1 - k)$ twice: convert with $(1 - k)$ only a charge that is a ratio to expected limited loss.