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CAS Exam 5 · Expense Provisions in Rates · Tutorial 332 of 989

Integrated Multi-Class Expense Allocation Workshop

Learn to allocate portfolio overhead between classes, incorporate class-specific and premium-based expenses, and compare the resulting indicated rates.

Intermediate 12 min read

What You'll Learn

  • Separate directly assigned class expenses from shared fixed overhead
  • Allocate overhead using exposure counts or expected losses
  • Convert allocated overhead into a fixed expense per exposure
  • Calculate class indicated rates with class-specific variable expense ratios
  • Reconcile allocated expenses and compare the effects of alternative allocation bases

Why Multi-Class Expense Allocation Matters

A rate indication for several classes must account for expenses that are not all incurred in the same way. Some expenses vary with each class, while shared overhead must be assigned across the portfolio. The allocation changes the expense amount attributed to each class and can therefore change its indicated rate, even though the total overhead assigned to the portfolio stays the same.

This tutorial combines direct class expenses, shared fixed overhead, and expenses expressed as a percentage of premium. It compares two allocation bases: exposure count and expected losses. The calculations are prospective and assume that the selected expense amounts and ratios are appropriate for the period being rated. The framework is aligned with the CAS Exam 5 (Basic Ratemaking and Estimated Claim Liabilities) syllabus.

Definition: For class \(j\), let \(n_j\) be the number of exposures, \(L_j\) the expected loss cost per exposure, \(e_j\) the directly assigned fixed expense per exposure, and \(v_j\) the variable expense ratio applied to premium. Let \(H\) be shared fixed overhead for the portfolio. An allocation assigns a fraction \(w_j\) of \(H\) to class \(j\), where \(w_j \geq 0\) and \(\sum_j w_j=1\). The allocated overhead per exposure is \(h_j=Hw_j/n_j\).

Choose the Allocation Base Before Calculating Rates

An allocation base determines the weights \(w_j\). Under an exposure-count allocation, each class receives a share in proportion to its exposure count. Under an expected-loss allocation, each class receives a share in proportion to its total expected losses, \(n_jL_j\).

$$ \begin{aligned} w_j^{(\mathrm{exposure})} &= \frac{n_j}{\sum_k n_k}, & h_j^{(\mathrm{exposure})} &= \frac{H}{\sum_k n_k},\\ w_j^{(\mathrm{loss})} &= \frac{n_jL_j}{\sum_k n_kL_k}, & h_j^{(\mathrm{loss})} &= \frac{Hn_jL_j}{n_j\sum_k n_kL_k} =\frac{HL_j}{\sum_k n_kL_k}. \end{aligned} $$

With exposure allocation, all classes have the same allocated overhead per exposure. With expected-loss allocation, a class with higher expected loss cost per exposure receives more overhead per exposure. Neither basis is automatically correct in every situation: the selected basis should be consistent with the intended expense allocation approach and the available data.

Once overhead has been allocated, the indicated rate \(R_j\) is found by balancing expected losses and fixed expenses against the portion of premium remaining after variable expenses. The variable ratio \(v_j\) applies to that class’s indicated premium, so it must be grossed up rather than added as a fixed dollar amount.

$$ R_j=L_j+e_j+h_j+v_jR_j \qquad\Longrightarrow\qquad R_j=\frac{L_j+e_j+h_j}{1-v_j}, \quad 0\leq v_j<1. $$

The total expense provision per exposure at this rate is \(e_j+h_j+v_jR_j\). This includes the class’s direct fixed expense, its allocated overhead, and the variable expense calculated as a percentage of premium. It does not include a profit or contingency provision.

Assumptions and condition: This rate equation assumes that the loss cost and fixed expenses are stated per exposure, and that the variable expense ratio applies to the same premium base as the indicated rate. A finite positive rate requires \(v_j<1\). If expense percentages apply to different bases, such as different premium components, each component must be formulated using its own stated base before solving.

Worked Allocations and Rate Indications

Worked Example: Comparing Two Overhead Bases

A hypothetical portfolio has two classes. Class 1 has 1,000 exposures, a loss cost of $300, a direct fixed expense of $20 per exposure, and a variable expense ratio of 8%. Class 2 has 500 exposures, a loss cost of $600, a direct fixed expense of $35 per exposure, and a variable expense ratio of 12%. Allocate $180,000 of shared overhead first by exposure count and then by expected losses. Find each class’s indicated rate under both allocations.

Method. Calculate the class allocation weights, convert allocated overhead to dollars per exposure, and use the class-specific rate formula. This avoids treating shared overhead as if it were a direct expense with the same amount per exposure in every allocation scheme.

Step 1: Calculate expected losses. Class 1 has \(1{,}000 \times \$300=\$300{,}000\) of expected losses. Class 2 has \(500 \times \$600=\$300{,}000\). Total expected losses are $600,000; total exposures are 1,500.

Step 2: Allocate by exposure count. The exposure shares are \(1{,}000/1{,}500=2/3\) and \(500/1{,}500=1/3\). Thus Class 1 receives \(\$180{,}000(2/3)=\$120{,}000\), or $120 per exposure. Class 2 receives \(\$180{,}000(1/3)=\$60{,}000\), also $120 per exposure.

Step 3: Calculate exposure-allocation rates. Class 1 has a fixed cost of \(300+20+120=\$440\) per exposure. Its rate is \(440/(1-0.08)=440/0.92=\$478.26\). Class 2 has a fixed cost of \(600+35+120=\$755\) per exposure. Its rate is \(755/(1-0.12)=755/0.88=\$857.95\). Rates are rounded to cents only at the end.

Step 4: Allocate by expected losses. Since each class has $300,000 of expected losses, each receives half the overhead: $90,000. This is $90 per exposure for Class 1 and $180 per exposure for Class 2.

Step 5: Calculate expected-loss-allocation rates. Class 1’s fixed cost is \(300+20+90=\$410\) per exposure, so its rate is \(410/0.92=\$445.65\). Class 2’s fixed cost is \(600+35+180=\$815\) per exposure, so its rate is \(815/0.88=\$926.14\).

Check. Under either allocation, the sum of expected losses, direct fixed expenses, and overhead is \(\$600{,}000+\$37{,}500+\$180{,}000=\$817{,}500\). The premium remaining after variable expenses is also $817,500 in either case: for exposure allocation, \(1{,}000(440)+500(755)=\$817{,}500\); for expected-loss allocation, \(1{,}000(410)+500(815)=\$817{,}500\). The allocated overhead sums to $180,000 in both cases.

Worked Example: Calculating Each Class’s Expense Provision

A hypothetical insurer has three classes. Class A has 400 exposures, expected losses of $250 per exposure, direct fixed expenses of $18 per exposure, and a variable expense ratio of 10%. Class B has 200 exposures, expected losses of $500 per exposure, direct fixed expenses of $30 per exposure, and a variable expense ratio of 8%. Class C has 100 exposures, expected losses of $800 per exposure, direct fixed expenses of $50 per exposure, and a variable expense ratio of 5%. Allocate $140,000 of overhead by expected losses. Calculate the rate and total expense provision per exposure for each class.

Method. Use expected losses by class to determine the overhead shares, then calculate each indicated rate. The expense provision includes variable expenses only after the indicated rate is known, because those expenses are a percentage of premium.

Step 1: Find expected losses by class. Class A has \(400 \times \$250=\$100{,}000\), Class B has \(200 \times \$500=\$100{,}000\), and Class C has \(100 \times \$800=\$80{,}000\). The portfolio total is $280,000.

Step 2: Allocate overhead. Class A’s share is \(100{,}000/280{,}000=5/14\), so it receives \(\$140{,}000(5/14)=\$50{,}000\), or $125 per exposure. Class B has the same expected losses and also receives $50,000, or $250 per exposure. Class C’s share is \(80{,}000/280{,}000=2/7\), so it receives $40,000, or $400 per exposure. The allocations total $140,000.

Step 3: Calculate rates. For Class A, \(R_A=(250+18+125)/(1-0.10)=393/0.90=\$436.67\). For Class B, \(R_B=(500+30+250)/(1-0.08)=780/0.92=\$847.83\). For Class C, \(R_C=(800+50+400)/(1-0.05)=1{,}250/0.95=\$1{,}315.79\).

Step 4: Calculate expense provisions. Class A’s variable expense is \(0.10 \times 436.666\ldots=\$43.67\), so its total expense provision is \(18+125+43.666\ldots=\$186.67\) per exposure. Class B’s variable expense is \(0.08 \times 847.826\ldots=\$67.83\); its total provision is \(30+250+67.826\ldots=\$347.83\). Class C’s variable expense is \(0.05 \times 1{,}315.789\ldots=\$65.79\); its total provision is \(50+400+65.789\ldots=\$515.79\).

Check. For each class, the indicated rate equals its loss cost plus its expense provision: Class A, \(250+186.666\ldots=436.666\ldots\); Class B, \(500+347.826\ldots=847.826\ldots\); Class C, \(800+515.789\ldots=1{,}315.789\ldots\). The rate equation is satisfied before rounding.

Why the Gross-Up Formula Holds

The indicated rate must fund expected losses and fixed expenses, but the insurer does not retain the entire premium when some expenses are proportional to that premium. For one exposure in class \(j\), the premium is \(R_j\), and the variable expense is \(v_jR_j\). The amount left after variable expenses is therefore \(R_j-v_jR_j=(1-v_j)R_j\). To fund the loss cost and fixed expenses, that retained amount must equal \(L_j+e_j+h_j\).

$$ (1-v_j)R_j=L_j+e_j+h_j \quad\Longrightarrow\quad R_j=\frac{L_j+e_j+h_j}{1-v_j}. $$

This derivation also shows why variable expense ratios are not simply added to the loss cost as dollar amounts. The variable expense depends on the rate being calculated, so the equation must be solved for that rate.

Worked Example: How Allocation Changes Class Rates

A hypothetical two-class portfolio has 300 Class 1 exposures and 100 Class 2 exposures. Class 1 has a $200 loss cost, $15 of direct fixed expense per exposure, and a 5% variable expense ratio. Class 2 has an $800 loss cost, $40 of direct fixed expense per exposure, and a 10% variable expense ratio. Allocate $80,000 of shared overhead by exposure count and by expected losses. Compare the indicated rates.

Worked Example: How Allocation Changes Class Rates

Method. Compute each class’s overhead per exposure under both allocation bases, then gross up each class’s own fixed costs using its own variable expense ratio.

Step 1: Calculate the allocation weights. There are 400 exposures, so exposure shares are \(300/400=0.75\) and \(100/400=0.25\). Expected losses are \(300 \times \$200=\$60{,}000\) and \(100 \times \$800=\$80{,}000\), totaling $140,000. The expected-loss shares are \(60{,}000/140{,}000=3/7\) and \(80{,}000/140{,}000=4/7\).

Step 2: Allocate by exposure count. Class 1 receives \(0.75 \times \$80{,}000=\$60{,}000\), or $200 per exposure. Class 2 receives $20,000, also $200 per exposure. The rates are \(R_1=(200+15+200)/0.95=\$436.84\) and \(R_2=(800+40+200)/0.90=\$1{,}155.56\).

Step 3: Allocate by expected losses. Class 1 receives \((3/7)\times \$80{,}000=\$34{,}285.71\), or \(\$34{,}285.71/300=\$114.29\) per exposure. Class 2 receives \((4/7)\times \$80{,}000=\$45{,}714.29\), or \(\$45{,}714.29/100=\$457.14\) per exposure. The rates are \(R_1=(200+15+114.2857\ldots)/0.95=\$346.62\) and \(R_2=(800+40+457.1429\ldots)/0.90=\$1{,}441.27\).

Check and interpretation. Both schemes allocate exactly $80,000 of overhead. Under exposure allocation, the premium remaining after variable expenses is \(300(415)+100(1{,}040)=\$228{,}500\). Under expected-loss allocation, it is \(300(329.2857\ldots)+100(1{,}297.1429\ldots)=\$228{,}500\). Class 1’s rate falls by $90.23 under the expected-loss allocation, while Class 2’s rate rises by $285.71. The different movements reflect where the fixed overhead is assigned, not a change in total expected losses or fixed expenses.

Common mistake avoided: Do not apply one class’s variable expense ratio to another class. The denominator for Class 1 is \(1-0.05=0.95\); for Class 2 it is \(1-0.10=0.90\). Also, do not add the two class ratios and use 15% for both rates: those ratios apply to different class premiums, not to a single common premium base.

Common Allocation and Rate-Indication Traps

  • Allocating by the wrong total. Expected-loss weights use \(n_jL_j\), not \(L_j\) alone. The number of exposures matters because the allocation base is total expected loss by class.
  • Forgetting to convert dollars allocated into dollars per exposure. The rate equation uses per-exposure amounts. Divide a class’s allocated overhead by that class’s exposure count before adding it to the loss cost and direct expense.
  • Mixing up allocated overhead and total expenses. Shared overhead is one part of a class’s fixed expense provision. Direct expenses remain assigned to their class and should not be included in the shared overhead a second time.
  • Adding variable expense percentages as dollars. If variable expenses are \(v_j\) of the indicated premium, solve \(R_j=(L_j+e_j+h_j)/(1-v_j)\). For example, with a $440 fixed cost per exposure and an 8% variable expense ratio, \(440 \times 1.08=\$475.20\) is wrong. The correct rate is \(440/0.92=\$478.26\), since \(0.08 \times 478.26\) is approximately $38.26 and the remaining $440 funds fixed costs.
  • Assuming a different allocation changes portfolio fixed costs. A change in allocation changes the class distribution of overhead, not the total amount allocated. Class indicated rates may change even though the total underlying fixed expenses do not.
Key takeaway: First assign class-varying expenses directly and select a consistent base for shared overhead. Then express all fixed costs per exposure and solve each class rate using that class’s variable expense ratio. Confirm that allocated overhead reconciles to the portfolio total.

Check Your Understanding

For each problem, use the stated expense base and round dollar rates to cents only at the final step.

  1. A hypothetical portfolio has 120 exposures in Class A and 80 in Class B. It has $52,000 of shared overhead. How much overhead is assigned to each class under exposure-count allocation, and what is the overhead per exposure in each class? (Answer: Class A receives $31,200, or $260 per exposure; Class B receives $20,800, or $260 per exposure.)
  2. Class A has 100 exposures with a $300 loss cost; Class B has 200 exposures with a $450 loss cost. Allocate $60,000 of overhead by expected losses. How much goes to each class, and what is the amount per exposure? (Answer: Class A receives $15,000, or $150 per exposure; Class B receives $45,000, or $225 per exposure.)
  3. A class has a $420 loss cost, $24 of direct fixed expense per exposure, and $36 of overhead per exposure. Two variable expense components, 7% and 3%, both apply to the same premium base. Find the indicated rate. (Answer: The combined ratio is 10%; the rate is \((420+24+36)/0.90=\$533.33\).)
  4. A hypothetical portfolio has 100 Class 1 exposures with a $200 loss cost, $10 direct expense per exposure, and a 10% variable expense ratio. Class 2 has 200 exposures, a $500 loss cost, $20 direct expense per exposure, and a 5% variable expense ratio. Allocate $60,000 of overhead first by exposure count and then by expected losses. Find both class rates under both schemes. (Answer: Exposure allocation assigns $200 per exposure to each class, giving rates of $455.56 and $757.89. Expected-loss allocation assigns $100 per Class 1 exposure and $250 per Class 2 exposure, giving rates of $344.44 and $810.53.)

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