Decomposing Portfolio And Segment Indications

Why Decompose a Portfolio Indication?

A portfolio-level indicated change summarizes the premium change needed across a book of business. Segment indications show how that need varies across rating groups. To make the two views consistent, each segment’s indication must be applied to its own current premium, then the indicated segment premiums must be added. A simple average of segment percentages generally does not perform this reconciliation.

This tutorial develops the exact arithmetic for that reconciliation. It is aligned with the CAS Exam 8 (Advanced Ratemaking) syllabus and focuses on translating between segment changes, current premium distribution, and a portfolio change.

Definition: For segment $s$, let $P_s$ be its current premium on the portfolio’s chosen basis, and let $g_s$ be its indicated percentage change, expressed as a decimal. Its indicated premium is $P_s(1+g_s)$. For a portfolio with total current premium $P=\sum_s P_s$, the portfolio indicated change is the percentage change from $P$ to the sum of the indicated segment premiums.

The premium basis must be consistent across segments: use the same period, exposure basis, and treatment of items such as premium adjustments. The formula describes the indicated change for the current portfolio distribution. It does not, by itself, predict what the distribution will be after customers respond to a rate change.

Premium-Weighted Reconciliation

Define the current premium weight of segment $s$ as $w_s=P_s/P$. These weights are nonnegative and sum to one when the segments partition the portfolio and all current premiums are included. The overall indicated change is the weighted average of segment changes, using these premium weights.

P_{\text{ind}}=\sum_s P_s(1+g_s)

g_{\text{portfolio}} =\frac{P_{\text{ind}}-P}{P} =\frac{\sum_s P_sg_s}{P} =\sum_s w_sg_s

The last equality follows because $\sum_s P_s=P$. Expanding the indicated premium gives $\sum_s P_s(1+g_s)=\sum_s P_s+\sum_s P_sg_s=P+\sum_s P_sg_s$. Subtracting $P$ leaves the dollar change $\sum_s P_sg_s$; dividing by $P$ gives the portfolio percentage change. This also explains why each segment’s contribution to the portfolio’s dollar change is $P_sg_s$.

Key result: A portfolio change is exactly the current-premium-weighted average of segment changes when each segment change is applied to its own current premium on a consistent basis. Segment indicated premiums must add to the portfolio indicated premium.

Exposure counts are not generally the right weights. If segment $s$ has $E_s$ exposures and average current premium $p_s$ per exposure, then $P_s=E_sp_s$. Therefore its portfolio premium weight is $E_sp_s/\sum_j E_jp_j$, not simply $E_s/\sum_j E_j$. Exposure-count weights coincide with premium weights only when the relevant average current premium is the same across segments (or when another special circumstance makes the weights equal).

Worked Reconciliation Examples

Worked Example: Combining a Positive and a Negative Segment Change

A hypothetical portfolio has $250,000 of current premium in Segment A and $750,000 in Segment B. The indicated changes are +4% for A and −2% for B. Calculate each segment’s indicated premium and the portfolio indicated change.

Method. Apply each percentage to that segment’s own current premium, then add the indicated premiums.

For A, the change is $0.04$, so the indicated premium is $$250{,}000(1+0.04)=$250{,}000(1.04)=$260{,}000$. The dollar increase is $10,000.

For B, the change is $-0.02$, so the indicated premium is $$750{,}000(1-0.02)=$750{,}000(0.98)=$735{,}000$. The dollar decrease is $15,000.

The current portfolio premium is $$250{,}000+$750{,}000=$1{,}000{,}000$. The indicated portfolio premium is $$260{,}000+$735{,}000=$995{,}000$. Thus the portfolio dollar change is $$995{,}000-$1{,}000{,}000=-$5{,}000$, and the percentage change is $-$5{,}000/$1{,}000{,}000=-0.005=-0.50\%$.

Check. The premium weights are 0.25 and 0.75. Their weighted change is $0.25(4\%)+0.75(-2\%)=1.00\%-1.50\%=-0.50\%$, matching the dollar calculation. The common mistake to avoid is reporting −1.00%: the segment dollar changes are +$10,000 and −$15,000, a net decrease of $5,000, not $10,000.

Worked Example: Why Exposure-Count Weights Can Mislead

A hypothetical book has 200 policies in Segment A, with average current premium of $500 per policy and an indicated change of +6%. Segment B has 100 policies, average current premium of $1,000, and an indicated change of −3%. Calculate the premium-weighted portfolio change and compare it with the exposure-count-weighted change.

Step 1: Calculate current segment premiums. Segment A has $200($500)=$100{,}000$ of current premium. Segment B has $100($1{,}000)=$100{,}000$. Total current premium is $$100{,}000+$100{,}000=$200{,}000$.

Step 2: Calculate indicated segment premiums. For A, $$100{,}000(1.06)=$106{,}000$. For B, $$100{,}000(0.97)=$97{,}000$. The indicated portfolio premium is $$106{,}000+$97{,}000=$203{,}000$.

The portfolio increase is $$203{,}000-$200{,}000=$3{,}000$. Dividing by current premium gives $$3{,}000/$200{,}000=0.015=1.50\%$.

Step 3: Compare with exposure-count weighting. The policy-count weights are $200/300=2/3$ and $100/300=1/3$. The count-weighted change is $(2/3)(6\%)+(1/3)(-3\%)=4\%-1\%=3.00\%$. This is not the portfolio premium change. Because the segments have different average premiums, equal segment premiums do not correspond to equal numbers of policies. Here, the premium weights are each $100{,}000/200{,}000=0.5$, giving $0.5(6\%)+0.5(-3\%)=1.50\%$.

Check. The dollar changes are +$6,000 and −$3,000, for a net increase of $3,000 on $200,000, or +1.50%. The common mistake is to use the 200-to-100 policy-count ratio as if it were the premium distribution.

Worked Example: Solving for a Segment Change That Meets a Portfolio Target

A hypothetical portfolio has $800,000 of current premium: $500,000 in Segment A and $300,000 in Segment B. The portfolio target is an indicated increase of 3.5%, and A’s segment indication is +5%. What indicated premium and percentage change must be assigned to B for exact reconciliation?

Step 1: Find the portfolio target premium. $$800{,}000(1.035)=$828{,}000$.

Step 2: Find A’s indicated premium. $$500{,}000(1.05)=$525{,}000$.

Step 3: Assign the residual indicated premium to B. B must contribute $$828{,}000-$525{,}000=$303{,}000$. Its current premium is $300,000, so its required dollar increase is $$303{,}000-$300{,}000=$3{,}000$. The required percentage change is $$3{,}000/$300{,}000=0.01=1.00\%$.

Check. The premium weights are $500{,}000/800{,}000=0.625$ for A and $300{,}000/800{,}000=0.375$ for B. Their weighted change is $0.625(5\%)+0.375(1\%)=3.125\%+0.375\%=3.50\%$. The segment indicated premiums also add to $$525{,}000+$303{,}000=$828{,}000$. A common trap is to calculate the target portfolio increase but fail to subtract A’s indicated premium before finding B’s residual.

From Exposure Data to a Reconciled Indication

When the data are provided as exposure counts and average current premiums, first convert them to current segment premiums. Only then should the percentage indications be applied. This prevents a change in one segment’s premium level from being confused with a change in its number of exposures.

Establish the common basis.
Confirm that current premiums and indicated changes refer to the same portfolio period and premium basis, and that the segments cover the intended portfolio.

Calculate current segment premiums.
If the data give exposure $E_s$ and average current premium $p_s$, calculate $P_s=E_sp_s$.

Apply each segment indication.
Calculate $P_s(1+g_s)$ for every segment, keeping decreases negative and increases positive in $g_s$.

Reconcile totals and percentage.
Add current premiums and indicated premiums separately. Divide the total dollar change by total current premium to obtain the portfolio change.

A useful arithmetic check is to calculate the result in two independent ways: add the segment indicated premiums, and compute $\sum_s w_sg_s$. If they disagree, check for a missing segment, an inconsistent denominator, a percentage entered as a whole number instead of a decimal, or a weight based on exposure count rather than premium.

Common Mistakes and Appropriate Interpretation

Averaging percentages without weights. The simple average gives each segment equal influence regardless of its current premium. It is appropriate only when the segment premiums are equal, or when the intended calculation explicitly assigns equal segment weights.
Using exposure shares automatically. Exposure shares are not premium shares if average premium differs by segment. Calculate $E_sp_s$ first when the information is given in exposures and average premiums.
Adding percentage changes. Segment percentages cannot be added to obtain a portfolio percentage. Add the segment dollar changes $P_sg_s$, then divide by total current premium.
Mixing denominators. An indication based on one premium level or exposure period cannot be reconciled directly with a portfolio total on another basis without first putting the quantities on a common basis.
Confusing a current-mix result with a future-mix result. The premium-weighted calculation holds the current segment distribution fixed. A later change in the portfolio’s segment mix can produce a different realized aggregate outcome.

Key takeaway: Reconcile indications in dollars first. Multiply each segment’s current premium by its indicated change, add the resulting segment premiums, and compare the total with current portfolio premium. The corresponding percentage calculation uses current premium weights—not policy counts unless policy counts imply those same weights.

Check Your Understanding

For each problem, use the stated current premium basis and distinguish premium weights from exposure-count weights where requested.

A hypothetical portfolio has $360,000 current premium in Segment A with an indication of +8%, and $540,000 in Segment B with an indication of −2%. Find the indicated premium for each segment and the portfolio change. (Answer: A indicated premium $388,800; B $529,200; portfolio indicated premium $918,000; change +2.00%.)
Segment A has 150 exposures at an average current premium of $800 and a +2% indication. Segment B has 250 exposures at an average current premium of $400 and a +5% indication. Find the portfolio change and the exposure-count-weighted change. (Answer: current premium $220,000; indicated premium $227,400; premium-weighted change +3.36%; exposure-count-weighted change +3.875%.)
A hypothetical portfolio has $1,200,000 current premium. Segment A accounts for $450,000 and has an indication of −1%; Segment B accounts for the remaining premium. If the portfolio target is +2%, what indicated premium and percentage change are required for B? (Answer: B current premium $750,000; B indicated premium $778,500; B change +3.80%.)
A book has 80 exposures at $1,200 average premium with a +3% indication, 120 exposures at $900 with a −1% indication, and 200 exposures at $600 with a +2% indication. Find the total indicated premium and portfolio change. (Answer: current premium $324,000; indicated premium $328,200; portfolio change +1.30%.)
Current premium shares are 20% for A, 50% for B, and 30% for C. Indications are +4% for A and −2% for B. What change must C have to produce a portfolio indication of +0.8%? (Answer: C indication +3.33%.)

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