Recalculating Mods Under Alternative Split Points

Why Recalculate at Alternative Split Points?

Experience-rating plans can divide each claim into a primary component and an excess component at a specified per-claim split point. Changing that point changes how each claim is allocated, and it may also change the expected losses and component weights used in the modification. To compare two indicated modifications fairly, recalculate all of these quantities at each split point while holding the underlying experience and exposure basis consistent.

The purpose is not to change the total amount of the claims. A split point reallocates each claim between components; it does not, by itself, change the claim’s total loss. The modification can nevertheless change because the primary and excess components may receive different credibility weights.

Definition: For a ground-up claim loss $X$ and per-claim split point $c$, the primary loss is $P_c(X)=\min(X,c)$, and the excess loss is $Q_c(X)=\max(X-c,0)$. Thus $X=P_c(X)+Q_c(X)$. For a set of claims, actual primary and excess losses are the sums of these components claim by claim.

Here, “excess” means the part of an individual claim above the split point; it does not mean an excess-of-loss reinsurance recovery. The examples assume the claims are on the same loss basis as the plan, with no additional limits, exclusions, or adjustments. If a plan specifies a different basis, that basis must be applied consistently to both actual and expected losses.

The Modification Formula at Each Split Point

Use the plan’s component weights and ballast for the split point being evaluated. Let $A_P(c)$ and $A_X(c)$ be aggregate actual primary and excess losses, and let $E_P(c)$ and $E_X(c)$ be their corresponding expected losses. Let $W_P(c)$ and $W_X(c)$ be the primary and excess weights, and let $B(c)$ be the stabilizing ballast. Under the deviation form of the plan, the indicated modification is:

M(c)=1+\frac{W_P(c)\left[A_P(c)-E_P(c)\right]+W_X(c)\left[A_X(c)-E_X(c)\right]}{E_P(c)+E_X(c)+B(c)}.

The formula requires actual and expected losses to use the same exposure period, coverage basis, valuation basis, and split point. Weights and ballast must be those applicable to the calculation being made. A weight is dimensionless; each weighted deviation therefore remains in loss units, as does the denominator. If the plan supplies a different modification formula, its specified formula takes precedence.

Key takeaway: At every alternative split point, recalculate actual components, expected components, and any split-dependent weights or ballast. Do not carry a primary or excess total from one split point into a calculation for another.

Why the Allocation Changes but Total Loss Does Not

For any claim and any split point $c$, there are two cases. If $X\leq c$, then $P_c(X)=X$ and $Q_c(X)=0$. If $X>c$, then $P_c(X)=c$ and $Q_c(X)=X-c$. In either case, $P_c(X)+Q_c(X)=X$. Summing over claims gives the same identity for aggregate actual losses:

A_P(c)+A_X(c)=\sum_j X_j.

The same component identity applies to expected losses when they are split consistently. But preserving the sum does not preserve the weighted deviation in the numerator. If the weights differ, moving dollars from one component to the other changes the weighted result. This explains why the modification may rise, fall, or remain unchanged when the split point changes; there is no universal direction.

Set the split point.
Use the stated per-claim threshold, not a threshold applied once to aggregate losses.

Recalculate actual components.
For each claim, calculate the amount up to the split point and the amount above it; then sum each component.

Use matching expected components and plan values.
Obtain the expected primary and excess losses, weights, and ballast for that split point.

Calculate and compare.
Substitute into the modification formula, then compare the results using the same rounding convention.

Worked Comparisons

Worked Example: Moving a Split Point With Fixed Component Weights

A hypothetical account has three ground-up claims of $ \$12{,}000$, $ \$28{,}000$, and $ \$55{,}000$. Compare split points of $ \$25{,}000$ and $ \$40{,}000$. At either split point, the expected total is $ \$100{,}000$, the ballast is $ \$10{,}000$, and the weights are $W_P=0.40$ and $W_X=0.20$. All amounts below are in thousands of dollars.

Method and actual losses at $c=25$. Apply the split separately to each claim. The primary amounts are $12,25,25$, so $A_P(25)=12+25+25=62$. The excess amounts are $0,3,30$, so $A_X(25)=0+3+30=33$. The check is $62+33=95$, equal to the claim total $12+28+55=95$.

The expected components at this split are $E_P(25)=70$ and $E_X(25)=30$. Their deviations are $62-70=-8$ and $33-30=3$. The weighted deviation is $0.40(-8)+0.20(3)=-3.2+0.6=-2.6$. The denominator is $70+30+10=110$, giving $M(25)=1-2.6/110=0.9763636\ldots$.

Recalculate at $c=40$. The primary amounts are $12,28,40$, so $A_P(40)=80$. The excess amounts are $0,0,15$, so $A_X(40)=15$. Again, $80+15=95$, the same total actual loss. Expected primary and excess losses are $82$ and $18$, respectively. The deviations are $80-82=-2$ and $15-18=-3$, giving the weighted deviation $0.40(-2)+0.20(-3)=-0.8-0.6=-1.4$. The denominator remains $82+18+10=110$, so $M(40)=1-1.4/110=0.9872727\ldots$.

Comparison and check. To four decimal places, $M(25)=0.9764$ and $M(40)=0.9873$, an increase of $0.0109$. The higher split reallocates loss between components, but the aggregate actual loss remains $95$ thousand at both points. The mistake to avoid is treating the unchanged total as proof that the modification must be unchanged.

Worked Example: Recalculating When Component Weights Differ

A hypothetical portfolio has four claims of $ \$15{,}000$, $ \$25{,}000$, $ \$35{,}000$, and $ \$60{,}000$. Compare $c=20$ and $c=35$, with amounts in thousands of dollars. Expected total losses are $140$ at either split, and ballast is $20$. The plan specifies $W_P=0.60,\ W_X=0.25$ at $c=20$, and $W_P=0.60,\ W_X=0.25$ at $c=35$.

At $c=20$. The primary components are $15,20,20,20$, giving $A_P(20)=75$. The excess components are $0,5,15,40$, giving $A_X(20)=60$. Their total, $75+60=135$, matches the four claims’ total. Expected components are $E_P(20)=90$ and $E_X(20)=50$. Thus the deviations are $-15$ and $10$, and the weighted deviation is $0.60(-15)+0.25(10)=-9+2.5=-6.5$. The denominator is $90+50+20=160$; hence $M(20)=1-6.5/160=0.959375$.

At $c=35$. The primary components are $15,25,35,35$, giving $A_P(35)=110$. The excess components are $0,0,0,25$, giving $A_X(35)=25$; again, $110+25=135$. Expected components are $112$ and $28$, so the deviations are $-2$ and $-3$. The weighted deviation is $0.60(-2)+0.25(-3)=-1.2-0.75=-1.95$. The denominator is $112+28+20=160$, and $M(35)=1-1.95/160=0.9878125$.

Comparison and check. To four decimals, the modifications are $0.9594$ and $0.9878$, an increase of $0.0284$. The expected total and denominator are unchanged, but the component deviations differ after reallocation. The method avoids the mistake of calculating one aggregate actual-to-expected ratio and applying it to both components.

Worked Example: Allowing Weights to Change With the Split

A hypothetical account has claims of $ \$18{,}000$, $ \$32{,}000$, $ \$48{,}000$, and $ \$72{,}000$. Compare $c=30$ and $c=50$, with losses in thousands of dollars. At both points expected total losses are $160$, and ballast is $40$. At $c=30$, the weights are $W_P=0.50,\ W_X=0.40$; at $c=50$, they are $W_P=0.70,\ W_X=0.20$.

At $c=30$. The primary components are $18,30,30,30$, totaling $A_P(30)=108$. The excess components are $0,2,18,42$, totaling $A_X(30)=62$. Their sum is $170$, equal to the total of the claims. Expected components are $100$ and $60$, so deviations are $8$ and $2$. The weighted deviation is $0.50(8)+0.40(2)=4+0.8=4.8$. The denominator is $100+60+40=200$, and $M(30)=1+4.8/200=1.024$.

At $c=50$. The primary components are $18,32,48,50$, totaling $148$. The excess components are $0,0,0,22$, totaling $22$; $148+22=170$, as required. Expected components are $130$ and $30$, giving deviations $18$ and $-8$. The weighted deviation is $0.70(18)+0.20(-8)=12.6-1.6=11$. With denominator $130+30+40=200$, $M(50)=1+11/200=1.055$.

Comparison and check. To four decimals, $M(30)=1.0240$, $M(50)=1.0550$, and the increase is $0.0310$. Both the component allocation and the weights changed. Using the old weights at the new split would not implement the stated plan and would produce the wrong comparison.

Common Exam Trap: Applying the Split to the Aggregate

A per-claim split point is applied to each claim before summation. For example, with claims of $12,28,55$ and a split of $25$, the correct aggregate primary loss is $12+25+25=62$. A wrong calculation would first add the claims to get $95$, then set aggregate primary loss to $\min(95,25)=25$. That applies one split to the portfolio total, not one split to each claim. The corresponding “excess” would incorrectly be $70$ rather than the correct $33$.

Another trap is to update actual components but leave expected components at the original split. The numerator then compares quantities defined on different bases. Likewise, if the plan’s weights depend on the split point, reusing the old weights is not a valid recalculation. A careful solution labels every component by its split point and checks that actual primary plus actual excess equals total actual loss.

Exam-aligned takeaway: Comparing modifications under alternative split points is a full recalculation, not a simple adjustment to one component. Keep each claim’s total intact, match actual and expected components at each split, and use the plan’s corresponding weights and ballast. This treatment is aligned with the CAS Exam 8 (Advanced Ratemaking) syllabus.

Check Your Understanding

For each problem, use the deviation-form modification formula and retain full precision until the final answer. Amounts are in thousands of dollars unless stated otherwise.

Claims are $8,24,41$, with split point $20$. Expected primary and excess losses are $52$ and $21$; weights are $0.50$ and $0.30$, and ballast is $7$. Calculate the modification. (Answer: $M=0.9900$.)
For the same claims, use split point $30$, expected primary and excess losses $63$ and $10$, weights $0.40$ and $0.20$, and ballast $7$. Calculate the modification and its change from Problem 1. (Answer: $M=0.9975$, an increase of $0.0075$.)
Claims are $10,40,65$, and the split point is $25$. Expected primary and excess losses are $70$ and $35$; weights are $0.60$ and $0.25$, and ballast is $15$. Calculate aggregate actual components and the modification. (Answer: actual primary $60$, actual excess $55$, $M=0.9917$.)
Claims are $14,36,58$, and the split point is $30$. Expected primary and excess losses are $80$ and $30$; weights are $0.50$ and $0.20$, and ballast is $10$. Calculate the weighted deviation and modification. (Answer: weighted deviation $-2.2$; $M=0.9817$.)
Claims are $20,35,70$. At split $25$, expected components are $75,45$, weights are $0.40,0.30$; at split $45$, expected components are $95,25$, weights are $0.60,0.20$. Ballast is $20$ at both points. Compare the modifications. (Answer: $M(25)=1.0071$, $M(45)=1.0214$, an increase of $0.0143$.)

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