Portfolio Mix Changes and Class Rate Indications

Why Portfolio Mix Matters

A portfolio’s aggregate loss cost can change even when no class’s indicated loss cost changes. If exposure shifts toward classes with higher indicated costs, the aggregate rises because the portfolio contains a different mix of risks. Conversely, class indications can change while the exposure mix remains fixed. These are distinct sources of change, and a sound rate analysis separates them before interpreting the result.

This distinction matters when assessing a portfolio-level indication, comparing periods, or explaining why an aggregate rate level moved. The calculation is an aggregation of class results, not a substitute for the credibility methods used to produce those results. Here, each class indication is treated as an input; the focus is how changes in class weights and class costs combine at the portfolio level.

Definition: For class $j$, let $w_j$ be its share of the portfolio’s earned exposure, and let $C_j$ be its indicated loss cost per unit of that exposure. The weights satisfy $w_j \geq 0$ and $\sum_j w_j=1$. The aggregate portfolio loss cost is $P=\sum_j w_jC_j$. Use the same exposure basis and loss-cost definition for every class and comparison period.

For a comparison, use subscripts 0 and 1 for the earlier and later periods. Then $P_0=\sum_j w_{0j}C_{0j}$ and $P_1=\sum_j w_{1j}C_{1j}$. A change in $P$ can reflect a change in weights, class indications, or both. If the exposure shares are based on policy count in one period and earned car-years in another, the comparison is not on a consistent basis and the decomposition is not meaningful.

Separating the Two Effects

A sequential calculation can isolate the two sources of change. For example, first change the weights while holding class indications at their earlier values, then change the indications using the later weights. This gives an exact total change. However, if both weights and indications change, the attribution depends on which change is applied first: the effect of a class-cost change is weighted differently at the old and new mix.

A symmetric midpoint decomposition avoids privileging either sequence. It uses the average indication across periods to measure the mix effect and the average exposure share to measure the class-cost effect:

P_1-P_0 = \sum_j (w_{1j}-w_{0j})\frac{C_{0j}+C_{1j}}{2} + \sum_j (C_{1j}-C_{0j})\frac{w_{0j}+w_{1j}}{2}

Theorem: In the midpoint decomposition, the first sum is the mix effect and the second sum is the within-class indication effect. Together they equal the change in aggregate portfolio loss cost exactly. This convention divides the interaction between changing weights and changing costs equally between the two effects.

To see why the identity holds, consider one class and write $\Delta w_j=w_{1j}-w_{0j}$ and $\Delta C_j=C_{1j}-C_{0j}$. The class contribution to the aggregate change is $w_{1j}C_{1j}-w_{0j}C_{0j}$. The proposed two midpoint terms for that class sum to:

\Delta w_j\frac{C_{0j}+C_{1j}}{2} + \Delta C_j\frac{w_{0j}+w_{1j}}{2} = \frac{w_{1j}C_{1j}-w_{0j}C_{0j}+w_{1j}C_{1j}-w_{0j}C_{0j}}{2} = w_{1j}C_{1j}-w_{0j}C_{0j}

Summing this equality over all classes gives the stated decomposition. The weights in each period must sum to one, so $\sum_j(w_{1j}-w_{0j})=0$. As a result, the mix effect reflects a redistribution of exposure among classes rather than a change in the total exposure scale.

Worked Calculations

Worked Example: A Change in Mix With Unchanged Class Indications

A hypothetical insurer has two classes with indicated loss costs of $600 and $1,200 per earned exposure unit. In the earlier period, the classes have 700 and 300 earned exposure units; in the later period, they have 400 and 600. Class indications are unchanged. Calculate both portfolio loss costs and identify the source of the change.

Method. Convert the earned exposures to shares, then aggregate the class costs. Since costs are unchanged, the portfolio change is entirely a mix effect. This avoids interpreting a portfolio movement as a change in class pricing when the indications did not change.

Earlier total exposure is $700+300=1{,}000$. The earlier shares are $w_{01}=700/1{,}000=0.70$ and $w_{02}=300/1{,}000=0.30$. Thus:

P_0=(0.70)(600)+(0.30)(1{,}200)=420+360=780

Later total exposure is $400+600=1{,}000$. The later shares are $w_{11}=0.40$ and $w_{12}=0.60$, so:

P_1=(0.40)(600)+(0.60)(1{,}200)=240+720=960

The portfolio loss cost increases by $960-780=180$ per exposure unit. The mix effect confirms this result: class 1’s share changes by $-0.30$, while class 2’s share changes by $+0.30$. Therefore, the mix effect is $(-0.30)(600)+(0.30)(1{,}200)=-180+360=180$. The indication effect is zero because neither class cost changed.

Check. The class share changes sum to zero, and the entire $180 increase is the result of shifting exposure toward the class with the higher cost.

Worked Example: Decomposing Simultaneous Mix and Cost Changes

For two hypothetical classes, earlier shares are 0.60 and 0.40, and earlier indications are $500 and $1,000. Later shares are 0.50 and 0.50, and later indications are $560 and $910. Calculate the aggregate change and its midpoint mix and indication components.

Method. Calculate each aggregate using the weights and indications from that period, then apply the midpoint decomposition. This avoids assigning the entire change to whichever factor happens to be recalculated first.

The earlier portfolio loss cost is:

P_0=(0.60)(500)+(0.40)(1{,}000)=300+400=700

The later portfolio loss cost is:

$$ P_1=(0.50)(560)+(0.50)(910)=280+455=735 $$

The aggregate increase is $735-700=35$. For the mix effect, the average indications are $(500+560)/2=530$ for class 1 and $(1{,}000+910)/2=955$ for class 2. The share changes are $-0.10$ and $+0.10$. Thus:

\text{Mix effect}=(-0.10)(530)+(0.10)(955)=-53+95=42

For the within-class indication effect, the average shares are $(0.60+0.50)/2=0.55$ and $(0.40+0.50)/2=0.45$. The indication changes are $560-500=60$ and $910-1{,}000=-90$. Therefore:

\text{Indication effect}=(0.55)(60)+(0.45)(-90)=33-40.5=-7.5

Adding the two components gives $42+(-7.5)=34.5$, which does not equal the aggregate change of 35. That mismatch signals an arithmetic error: the class 2 mix contribution is $0.10(955)=95.5$, not 95. The corrected mix effect is $-53+95.5=42.5$, and $42.5-7.5=35$.

Check. The corrected decomposition reconciles exactly to the aggregate increase. The mix moved toward the more expensive class, while the indication changes, on balance, reduced the aggregate by $7.50 per exposure unit.

Worked Example: Aggregating Revised Credibility-Weighted Indications

A hypothetical portfolio has three classes. The table gives each class’s exposure share, a credibility factor $Z$, an observed class loss cost $L$, and a collective complement $M$, for two periods. For this calculation, each class indication is $C=ZL+(1-Z)M$. Calculate the class indications, portfolio costs, and midpoint decomposition.

Class	$Z$	$M$	$L_0$	$L_1$	$w_0$	$w_1$
A	0.50	$800	$1,000	$1,100	0.50	0.40
B	0.25	$1,000	$800	$900	0.30	0.35
C	0.40	$1,400	$1,500	$1,300	0.20	0.25

Method. First apply the stated credibility blend within each class; next aggregate using the period’s exposure shares; finally decompose the aggregate change. The common mistake avoided is applying portfolio weights to the observed costs before producing the class indications.

For class A, the earlier indication is $(0.50)(1{,}000)+(0.50)(800)=900$, and the later indication is $(0.50)(1{,}100)+(0.50)(800)=950$. For class B, the indications are $(0.25)(800)+(0.75)(1{,}000)=950$ and $(0.25)(900)+(0.75)(1{,}000)=975$. For class C, they are $(0.40)(1{,}500)+(0.60)(1{,}400)=1{,}440$ and $(0.40)(1{,}300)+(0.60)(1{,}400)=1{,}360$.

The earlier aggregate is $(0.50)(900)+(0.30)(950)+(0.20)(1{,}440)=450+285+288=1{,}023$. The later aggregate is $(0.40)(950)+(0.35)(975)+(0.25)(1{,}360)=380+341.25+340=1{,}061.25$. The aggregate increase is $38.25 per exposure unit.

Average indications for A, B, and C are $925, $962.50, and $1,400. The share changes are $-0.10$, $+0.05$, and $+0.05$, giving a mix effect of $(-0.10)(925)+(0.05)(962.50)+(0.05)(1{,}400)=-92.50+48.125+70=25.625$.

Average shares are 0.45, 0.325, and 0.225. The indication changes are $+50$, $+25$, and $-80$, giving an indication effect of $(0.45)(50)+(0.325)(25)+(0.225)(-80)=22.50+8.125-18=12.625$. The two effects sum to $25.625+12.625=38.25$, matching the aggregate change.

Check. Class C’s indicated cost declined, but its share also increased; the midpoint method accounts for both changes. All three exposure shares sum to one in each period.

Choosing and Interpreting the Decomposition

A sequential decomposition is also valid when its order is stated. In the second example, holding earlier indications fixed and changing the mix first gives a mix effect of $(-0.10)(500)+(0.10)(1{,}000)=50$. Repricing at the later mix then gives an indication effect of $(0.50)(60)+(0.50)(-90)=-15$. The sum is still 35. Reversing the order assigns the interaction differently. The midpoint decomposition averages those two orderings, so it is a useful neutral convention when an analysis needs a single split.

Neither attribution convention changes the actual aggregate loss costs. It changes only how the interaction is assigned between the two explanatory components. State the convention, preserve a consistent exposure basis, and verify that the components sum to $P_1-P_0$.

Common exam trap: Treating the entire change in aggregate loss cost as a change in class indications. If weights change, that conclusion can be wrong even when every class indication is unchanged. Calculate both period aggregates and explicitly separate the effects; do not infer the cause from the aggregate difference alone.

A useful diagnostic is to calculate the portfolio cost twice under controlled conditions: once using changed weights with indications held fixed, and once using changed indications with weights held fixed. This gives a clear sequential explanation, though the numerical attribution depends on the chosen order when both factors change. For a midpoint attribution, use the exact formula and check the reconciliation.

Check Your Understanding

For each hypothetical portfolio, calculate the requested aggregate change or decomposition. Exposure shares are on a consistent earned-exposure basis.

Two class indications remain $720 and $1,080. Their shares move from 0.65 and 0.35 to 0.50 and 0.50. What are the earlier and later portfolio costs, and what is the mix effect? (Answer: earlier $846, later $900; mix effect +$54 per exposure unit.)
Two classes have earlier indications $400 and $900 and later indications $460 and $840. Their shares move from 0.75 and 0.25 to 0.60 and 0.40. Find the aggregate change and midpoint effects. (Answer: aggregate change +$87; mix effect +$66; indication effect +$21 per exposure unit.)
Two class indications use $C=ZL+(1-Z)M$. For class A, $Z=0.30$, $M=\$700$, and observed costs are $900 then $1,000. For class B, $Z=0.50$, $M=\$1,100$, and observed costs are $900 then $800. Shares are 0.80 and 0.20, then 0.70 and 0.30. Calculate the class indications and aggregate change. (Answer: class indications A $760 then $790, B $1,000 then $950; aggregate $808 then $838, a $30 increase.)
Three class indications change from $500, $1,000, and $1,500 to $550, $900, and $1,600. Shares change from 0.50, 0.30, and 0.20 to 0.40, 0.40, and 0.20. Find the midpoint effects. (Answer: aggregate increases from $850 to $900; mix effect +$42.50 and indication effect +$7.50 per exposure unit.)
Two unchanged class indications are $600 and $1,200. Earlier earned exposures are 800 and 200; later earned exposures are 500 and 500. Calculate the portfolio costs and explain the change. (Answer: earlier $720, later $900; the $180 increase is entirely a mix effect.)

What You'll Learn