Monotonicity And Table Consistency Checks

Why Check an ILF Table?

An increased limits factor (ILF) table translates a base-limit premium or expected loss to other policy limits. Before using a table, an actuary should check whether its values are consistent with the loss amounts they are intended to represent. A factor that is not normalized to the stated base limit, or that decreases at a higher limit, can signal a calculation, labeling, or data problem.

The checks in this tutorial assume that all entries refer to the same coverage, loss basis, and underlying claim severity distribution. In particular, the loss basis and any applicable coverage terms must not change from one row to another. These checks can reveal an inconsistency, but do not by themselves identify its cause.

Definition: Let $X$ be a nonnegative per-claim loss on a fixed loss basis, and let $L$ be a policy limit. The limited loss is $\min(X,L)$, and its expected value is $m(L)=E[\min(X,L)]$. For a base limit $b$ with $m(b)>0$, the increased limits factor at $L$ is $$ \operatorname{ILF}(L;b)=\frac{m(L)}{m(b)}. $$ Thus, the ILF at the base limit is one. The factor is meaningful only when the numerator and denominator use the same loss basis.

The central audit idea is to work with both the factors and the limited expected losses they imply. If the base expected limited loss is $B=m(b)$, a factor $r$ implies an expected limited loss of $rB$. Multiplying by the same positive $B$ preserves the ordering of factors, so the factor and loss checks can be compared directly.

Three Basic Consistency Checks

For a table of limits $L_0=b<L_1<\cdots<L_n$, first verify the base entry, then check the ordering of the factors, and finally translate the factors into implied losses. The first two checks are simple; the third makes the reasoning behind a suspected problem visible.

Check the base.
The factor at the stated base limit $b$ must be one, subject only to any explicitly stated rounding convention.

Check the direction.
At higher limits, the ILFs must be nondecreasing. A larger limit cannot reduce the claim payment on the same loss basis.

Recover the implied losses.
Multiply every factor by the same base expected limited loss $B$. Compute successive differences to see whether each additional layer has a nonnegative implied expected loss.

The base check catches misnormalization, a wrong base-limit label, or an accidental use of a different denominator. The monotonicity check catches a more direct contradiction: if the limit increases while the loss basis stays fixed, the capped payment for any individual claim stays the same or rises. The expected payment therefore cannot fall.

Theorem: For $X\ge 0$ and $L_2\ge L_1$, $\min(X,L_2)\ge \min(X,L_1)$ for every possible claim amount. Taking expectations gives $m(L_2)\ge m(L_1)$. If $m(b)>0$, division by $m(b)$ shows that $\operatorname{ILF}(L_2;b)\ge\operatorname{ILF}(L_1;b)$.

This is a pointwise argument: it does not require a particular severity distribution or an assumption that claims have a continuous distribution. For a claim of $80,000, for example, raising a per-claim limit from $50,000 to $100,000 raises the limited payment from $50,000 to $80,000. A claim below both limits is unchanged; no claim is paid less.

From Factors to Layer Losses

A useful second representation is the expected loss in the layer between two limits. For $L_2>L_1$, the additional limited loss is the difference between the amounts capped at those limits. Its expectation must be nonnegative.

m(L_2)-m(L_1) =E[\min(X,L_2)-\min(X,L_1)]\ge 0.

This identity follows by subtracting the two expected limited losses. Linearity of expectation makes the difference equal to the expected value of the claim-level difference. Because the higher cap is never below the lower cap, that claim-level difference is nonnegative for every $X$. The same check can be made directly from a table: if $B=m(b)$, the implied layer loss is $B(r_2-r_1)$, where $r_j$ is the ILF at $L_j$.

There is a stronger shape check when the table is intended to represent exact expected limited losses from a common severity distribution. The function $m(L)$ is concave in $L$: for each fixed claim amount $x$, $\min(x,L)$ is a concave function of $L$, and taking expectations preserves concavity. Consequently, as limits rise, the expected loss added per dollar of additional limit cannot increase. For equal-width limit steps, successive increases in $m(L)$ cannot get larger. This is a useful consistency check, but it is not the same as monotonicity: factors can increase and still imply a concavity violation.

In practice, published values may be rounded. A tiny apparent reversal or shape irregularity can result from rounding, especially if the underlying values are close. Check the table’s precision and stated conventions before treating a small discrepancy as decisive. Rounding does not explain a material inconsistency, and it does not change the underlying monotonicity requirement.

Worked Example: Detecting a Decrease in the Table

A hypothetical liability program has a base limit of $100,000. Its indicated table lists ILFs of 1.00 at $100,000, 1.24 at $250,000, 1.20 at $500,000, and 1.51 at $1,000,000. The expected limited loss at the base limit is $8,000. Check the table.

Step 1: Check the base entry. The factor at $100,000 is 1.00, so the table passes the normalization check.

Step 2: Check the ordering. The factor rises from 1.00 to 1.24, but then falls from 1.24 at $250,000 to 1.20 at $500,000. Since the limit rises, this decrease violates monotonicity.

Step 3: Calculate the implied limited expected losses. Multiply each factor by the common base value of $8,000:

\begin{aligned} m(100{,}000)&=1.00(8{,}000)=\$8{,}000,\\ m(250{,}000)&=1.24(8{,}000)=\$9{,}920,\\ m(500{,}000)&=1.20(8{,}000)=\$9{,}600,\\ m(1{,}000{,}000)&=1.51(8{,}000)=\$12{,}080. \end{aligned}

The implied increase from $250,000 to $500,000 is $\$9{,}600-\$9{,}920=-\$320$. A higher limit cannot produce a negative expected loss in the added layer. The table is therefore inconsistent as stated; the arithmetic alone does not establish which entry or label is wrong.

Method and mistake avoided. The method is to normalize-check, order-check, and then convert factors to dollar amounts. A common mistake is to notice that most entries rise and overlook the local decrease between the middle limits.

Worked Example: Checking Layer Increases

A hypothetical property portfolio uses a $100,000 base limit. The base expected limited loss is $6,000. The table gives factors 1.00, 1.30, 1.65, and 1.90 at limits of $100,000, $200,000, $500,000, and $1,000,000, respectively. Check monotonicity and compare the implied layer increases per $1,000 of added limit.

Step 1: Check the base and factor sequence. The base factor is 1.00. The factors are nondecreasing: $1.00\le1.30\le1.65\le1.90$.

Step 2: Convert the factors to implied expected limited losses.

\begin{aligned} m(100{,}000)&=1.00(6{,}000)=\$6{,}000,\\ m(200{,}000)&=1.30(6{,}000)=\$7{,}800,\\ m(500{,}000)&=1.65(6{,}000)=\$9{,}900,\\ m(1{,}000{,}000)&=1.90(6{,}000)=\$11{,}400. \end{aligned}

Step 3: Find each layer increase and its width. From $100,000 to $200,000, the increase is $\$7{,}800-\$6{,}000=\$1{,}800$ over 100 increments of $1,000, or $18 per $1,000 of limit. From $200,000 to $500,000, it is $\$9{,}900-\$7{,}800=\$2{,}100$ over 300 increments, or $7 per $1,000. From $500,000 to $1,000,000, it is $\$11{,}400-\$9{,}900=\$1{,}500$ over 500 increments, or $3 per $1,000.

All layer increases are positive, and the increase per $1,000 declines from $18 to $7 to $3. The table passes these monotonicity and concavity checks. These checks do not prove that the table is based on appropriate data or assumptions; they show that its values do not contradict these mathematical requirements.

Method and mistake avoided. Compare increases per unit of limit when limit intervals have different widths. Comparing the raw dollar increases alone would be misleading here: the $2,100 increase across $300,000 of limit is not a steeper increase than $1,800 across $100,000.

Worked Example: A Monotone Table That Fails the Shape Check

A hypothetical commercial coverage has a $100,000 base limit and base expected limited loss of $5,000. A table reports factors 1.00, 1.30, 1.48, and 2.00 at limits of $100,000, $200,000, $400,000, and $800,000. Determine whether monotonicity alone is enough to accept the values, and check the implied layer increases per $100,000 of added limit.

Step 1: Check normalization and monotonicity. The factor at the base is 1.00, and the sequence $1.00,1.30,1.48,2.00$ is increasing. The table passes both checks.

Step 2: Calculate implied limited losses.

\begin{aligned} m(100{,}000)&=1.00(5{,}000)=\$5{,}000,\\ m(200{,}000)&=1.30(5{,}000)=\$6{,}500,\\ m(400{,}000)&=1.48(5{,}000)=\$7{,}400,\\ m(800{,}000)&=2.00(5{,}000)=\$10{,}000. \end{aligned}

Step 3: Compare increases per $100,000 of limit. The increase from $100,000 to $200,000 is $1,500 over one $100,000 interval, or $1,500 per interval. From $200,000 to $400,000 it is $\$7{,}400-\$6{,}500=\$900$ over two intervals, or $450 per interval. From $400,000 to $800,000 it is $\$10{,}000-\$7{,}400=\$2{,}600$ over four intervals, or $650 per interval.

The per-interval increases are $1{,}500,\ 450,\ 650$ dollars. The last is greater than the preceding increase, contrary to concavity. The factors are monotone, but under the stated common-severity interpretation, the implied expected limited-loss curve fails the stronger shape check. This calls for investigation of the entries, their basis, or the assumptions behind them; it is not a reason to silently alter the table.

Method and mistake avoided. The method uses equal-width increments so that layer increases can be compared fairly. A common mistake is to accept any increasing ILF sequence without checking whether the implied limited losses have a compatible shape.

Interpreting a Failed Check

A failed mathematical check is a reason to investigate, not a diagnosis. Possible issues include an incorrect base limit, a transcription error, mismatched coverage or loss bases, or factors calculated using inconsistent denominators. Confirm that limits are ordered correctly and that every row refers to the same type of per-claim limit and coverage terms. Then recompute implied losses using one common base expected limited loss.

Do not confuse an ILF with a premium ratio that includes other changing components. The monotonicity result applies to expected limited losses under a fixed loss basis. A premium can include expenses, adjustments, or other features that do not follow the same ratio. Check what the table represents before applying a limited-loss consistency test to it.

When the entries are rounded, distinguish a small display-level irregularity from a material inconsistency. If the unrounded values are available, use them for the check. If not, report the observed issue and its size rather than claiming that rounding definitely explains it. In all cases, keep the base limit, factor, and implied dollar amount clearly identified.

Key takeaway: For a fixed loss basis and positive base expected limited loss, an ILF table must equal one at its base and must not decrease as limits rise. Converting factors to implied limited losses exposes the layer amounts; for exact limited expected losses from a common severity distribution, those amounts also have a concave pattern as limits increase.

Check Your Understanding

For each problem, assume a common per-claim loss basis across the stated limits. Use the stated base expected limited loss where needed.

A table has a $150,000 base limit and factors 1.00 at $150,000, 1.12 at $300,000, and 1.10 at $600,000. Does it pass the basic checks? (Answer: No. The base is normalized, but the factor falls from 1.12 to 1.10 as the limit rises.)
The base limit is $80,000, its expected limited loss is $4,000, and the factors at $80,000, $180,000, and $480,000 are 1.00, 1.25, and 1.55. What are the implied limited losses and layer increases? (Answer: Implied losses are $4,000, $5,000, and $6,200; the increases are $1,000 and $1,200.)
At limits of $100,000, $200,000, and $500,000, the factors are 1.00, 1.40, and 1.80. The base expected limited loss is $10,000. Calculate the increases per $100,000 of limit and state whether they are consistent with concavity. (Answer: Implied losses are $10,000, $14,000, and $18,000; increases are $4,000 over one interval and $4,000 over three intervals, or $1,333.33 per $100,000. The successive per-interval increase declines, so the values are consistent with concavity.)
A table has base expected limited loss $7,000 and factors 1.00, 1.20, and 1.50 at limits of $100,000, $300,000, and $500,000. Find the implied losses and increases per $100,000 interval. What does the shape check show? (Answer: Implied losses are $7,000, $8,400, and $10,500. Increases are $1,400 over two intervals, or $700 per interval, and $2,100 over two intervals, or $1,050 per interval. The later increase per interval is larger, so the values fail the concavity check.)
A table’s factors increase at every listed limit, but one factor is 1.03 at a stated base limit. What is the specific issue, and can monotonicity resolve it? (Answer: The table fails base normalization because the factor at the stated base should be 1.00. Monotonicity does not resolve the base-limit inconsistency.)

What You'll Learn