Indicated Rates And Optimized Customer Prices

Two Different Pricing Questions

A cost-based indication and an optimized customer price answer different questions. An indication asks what premium is needed to cover expected costs and a specified provision for profit or risk. An optimized price asks what price best serves a stated business objective when customers may respond differently to different prices. The indicated amount is therefore a useful benchmark, but it is not automatically the price that maximizes an objective involving demand.

This distinction is aligned with the CAS Exam 8 (Advanced Ratemaking) syllabus. The key is to identify the unit of exposure, the basis of each cost, the target provision, and the demand response before comparing prices.

Definition: An actuarial indicated rate is a cost-based premium rate derived from expected losses and other covered costs, together with specified expense and profit or risk provisions. An optimized customer price is a price selected to maximize or otherwise improve a stated objective after accounting for the supplied relationship between price and customer response. The two amounts can coincide, but there is no general reason they must.

Calculating an Indicated Premium

An indication needs a consistent basis. For example, if expected losses and fixed expenses are measured per issued policy, the indicated premium is also calculated per issued policy. If a target profit provision and variable expenses are percentages of premium, both are applied to the indicated premium rather than added as fixed dollar costs.

Suppose $L$ is expected loss and loss-adjustment expense per policy, $F$ is fixed expense per policy, $v$ is the variable expense ratio, and $t$ is the target provision as a percentage of premium. Under the assumption that all remaining premium after these components is available for $L+F$, the indication $P_{\mathrm{ind}}$ satisfies:

P_{\mathrm{ind}} = L+F+vP_{\mathrm{ind}}+tP_{\mathrm{ind}}

Rearranging gives the indicated premium formula, provided $1-v-t>0$:

P_{\mathrm{ind}} = \frac{L+F}{1-v-t}

This formulation assumes that the listed costs and provisions are complete for the purpose of the indication and that the variable expense and target provision are each proportional to premium. If a provision is a fixed dollar amount, it belongs in the dollar-cost portion of the equation instead. If the denominator is zero or negative, this formula does not yield a finite positive indication for positive costs; the assumed expense and target percentages leave no positive share of premium to fund those costs.

Why the denominator appears: The premium must fund $L+F$, but the variable expense and target provision consume fractions $v$ and $t$ of that same premium. The amount left to fund $L+F$ is $P_{\mathrm{ind}}-vP_{\mathrm{ind}}-tP_{\mathrm{ind}}=(1-v-t)P_{\mathrm{ind}}$. Equating this remainder to $L+F$ and dividing by $1-v-t$ gives the formula.

Worked Example: Calculating an Indicated Premium

A hypothetical insurer estimates expected loss and loss-adjustment expense of $480 per issued policy. It also expects a catastrophe risk provision of $20 and fixed policy expenses of $40, both per issued policy. Variable expenses are 10% of premium, and the target profit provision is 5% of premium. Calculate the indicated premium per policy.

Method. Use the cost-based indication. Every dollar amount in the numerator is per issued policy, and the two percentage provisions are applied to premium.

Step 1: Add the per-policy dollar costs and provisions. The total is $480+20+40=\$540$ per issued policy.

Step 2: Identify the premium share available for those costs. The variable expense ratio plus the target provision is $0.10+0.05=0.15$. Thus $1-0.15=0.85$ of premium remains.

Step 3: Calculate the indication. $P_{\mathrm{ind}}=540/0.85=\$635.2941\ldots$, or $635.29 per policy, rounded to the nearest cent.

Check. At $635.29, variable expenses are approximately $63.53 and the target provision is approximately $31.76. The remaining premium is approximately $635.29-63.53-31.76=\$540.00$, matching the per-policy costs after rounding.

Common mistake avoided. Do not add 10% and 5% as dollar amounts or divide by $1+0.10+0.05$. Both provisions are percentages of premium that reduce the share available to pay the dollar costs.

How Demand Changes the Price Decision

An indicated rate does not, by itself, describe whether a customer will accept a quote. Let $q(p)$ denote the probability that an eligible quote results in an issued policy when the offered price is $p$. If expected losses and fixed expenses are stated per issued policy, they are incurred only when a policy is issued. Expected contribution per eligible quote is then:

q(p)\bigl[p-L-F-vp\bigr]

This expression is appropriate when the objective is expected contribution and $L$, $F$, and the variable expense rate $v$ all use the stated per-issued-policy basis. A different objective can produce a different selected price. For example, optimizing expected premium revenue does not account for costs in the same way as optimizing expected contribution. No unique “optimized price” exists until the objective and the relevant demand response are specified.

The formula also makes the exposure basis important. If a cost is instead an expected amount per eligible quote, do not multiply that cost by the probability of issuance: it is already on the quote basis. Convert all components to a common basis before combining them.

Worked Example: Comparing an Indication With Candidate Prices

A hypothetical insurer has expected loss and loss-adjustment expense of $500 per issued policy and fixed expenses of $30 per issued policy. Variable expenses are 8% of premium, and the target profit provision is 6% of premium. The insurer has 1,000 eligible quotes and receives the following supplied purchase probabilities: 0.55 at $600, 0.42 at $660, and 0.25 at $720. Use expected contribution as the stated objective. Find the indicated premium and identify the best of these candidate prices.

Method. First calculate the cost-based indication. Then calculate expected contribution per eligible quote at each candidate price and multiply by 1,000 quotes.

Step 1: Calculate the indication. Per-issued-policy costs are $500+30=\$530$. The premium share remaining after the variable expense and target provision is $1-0.08-0.06=0.86$. Therefore, $P_{\mathrm{ind}}=530/0.86=\$616.2791\ldots$, or $616.28 per policy, rounded to the nearest cent.

Step 2: Calculate contribution conditional on issuance. At price $p$, the contribution per issued policy before any target provision is $p-500-30-0.08p=0.92p-530$.

At $600, this is $0.92(600)-530=552-530=\$22$. At $660, it is $0.92(660)-530=607.20-530=\$77.20$. At $720, it is $0.92(720)-530=662.40-530=\$132.40$.

Step 3: Apply the purchase probabilities. At $600, expected contribution per eligible quote is $0.55(22)=\$12.10$, or $1{,}000(12.10)=\$12{,}100$ across the quotes. At $660, it is $0.42(77.20)=\$32.424$ per quote, or $\$32{,}424$ across the quotes. At $720, it is $0.25(132.40)=\$33.10$ per quote, or $\$33{,}100$ across the quotes.

Conclusion and check. Among the three candidates, $720 gives the greatest expected contribution: $33,100. It is above the $616.28 indication, but that does not make the two calculations inconsistent. The indication covers expected costs and the stated target provision on a per-policy basis; the candidate price comparison additionally reflects the supplied purchase probabilities and the stated objective.

Common mistake avoided. Do not compare only the contribution per issued policy. The $720 price has the largest such contribution here, but its purchase probability is lower. The objective is expected contribution across eligible quotes, so price and purchase probability must both enter the calculation.

Keep Quote Costs and Policy Costs Consistent

When a problem gives demand per eligible quote but costs per issued policy, the conversion probability connects the two bases. For a quote at price $p$, expected issued-policy costs are incurred with probability $q(p)$. For example, if losses and fixed expenses are per issued policy, expected contribution per quote is $q(p)$ times the conditional contribution per issued policy.

If expected loss is instead given per eligible quote, it is already unconditional with respect to issuance. In that case it should be subtracted once per quote, not multiplied by $q(p)$. Fixed expenses might still be per issued policy, while variable expenses are usually proportional to premium on an issued policy. Write each component with its basis before doing the arithmetic.

Worked Example: Evaluating a Supplied Customer Price

For a hypothetical product, expected loss and loss-adjustment expense are $420 per issued policy, fixed expenses are $25 per issued policy, variable expenses are 7% of premium, and the target provision is 5% of premium. There are 800 eligible quotes. A demand-response analysis supplies a customer price of $540 and estimates a 0.36 probability of issuance at that price. Calculate the indicated premium and the expected contribution at the supplied price. Determine whether the expected contribution exceeds the target provision on the expected premium volume.

Method. Calculate the indication from the per-policy cost basis. For the supplied price, calculate expected issued policies, then calculate contribution and the target amount on the expected premium volume.

Step 1: Calculate the indicated premium. Per-policy dollar costs are $420+25=\$445$. The share of premium remaining is $1-0.07-0.05=0.88$. Thus $P_{\mathrm{ind}}=445/0.88=\$505.6818\ldots$, or $505.68, rounded to the nearest cent.

Step 2: Calculate expected issued policies. The expected number is $800(0.36)=288$ policies. Expected premium is $288(540)=\$155{,}520$.

Step 3: Calculate contribution. Variable expense per issued policy at $540 is $0.07(540)=\$37.80$. Contribution per issued policy is $540-420-25-37.80=\$57.20$. Expected total contribution is $288(57.20)=\$16{,}473.60$.

Step 4: Compare with the target amount. The target provision is $0.05(155{,}520)=\$7{,}776$. Expected contribution exceeds that amount by $16{,}473.60-7{,}776=\$8{,}697.60$.

Check. Expected losses are $288(420)=\$120{,}960$, fixed expenses are $288(25)=\$7{,}200$, and variable expenses are $0.07(155{,}520)=\$10{,}886.40$. Subtracting these amounts from expected premium gives $155{,}520-120{,}960-7{,}200-10{,}886.40=\$16{,}473.60$, as calculated.

Common mistake avoided. The 800 quotes are not 800 issued policies. Apply the 0.36 issuance probability to find expected policy counts and to weight per-policy costs; do not charge every quote the full per-policy loss and fixed expense.

Interpretation and Common Traps

The indication and an optimized price should be compared only after their assumptions and bases are clear. A difference between them may reflect customer response, a different objective, a different cost basis, or a constraint on the available prices. It does not by itself establish that either calculation is wrong. Conversely, calling a price “optimized” does not show that it meets a target provision; the expected result should be checked against the indication or target when that comparison matters.

Mixing units. A loss cost per eligible quote cannot be added directly to fixed expense per issued policy without conversion or an explicitly quote-based calculation.
Applying conversion probabilities to the wrong amounts. Apply the probability to costs incurred only when a policy is issued. Do not apply it again to an amount already stated per eligible quote.
Treating the indication as an optimization result. The indication follows from costs and provisions. A price selected using demand also depends on the specified objective and customer response.
Assuming there is one universally optimal price. Different objectives can select different prices even with the same demand response. State the objective before interpreting a selected price.

In practice, indications are cost benchmarks rather than automatic customer-level quotes. A demand-informed price is meaningful only in relation to the response estimates and objective used to select it. Both should be evaluated with consistent exposure units, explicit assumptions, and an appropriate check of expected costs and provisions.

Check Your Understanding

For each problem, keep the stated quote or policy basis explicit. Round dollar indications to the nearest cent.

A hypothetical book has expected loss and loss-adjustment expense of $360 per issued policy and fixed expenses of $24 per issued policy. Variable expenses are 9% of premium and the target provision is 6% of premium. What is the indicated premium? (Answer: $451.76 per policy)
A product has expected costs of $700 per issued policy and fixed expenses of $50 per issued policy. Variable expenses are 10% of premium and the target provision is 5% of premium. Calculate its indicated premium. (Answer: $882.35 per policy)
For 500 eligible quotes, purchase probabilities are 0.60 at a price of $500 and 0.40 at a price of $600. Expected loss is $400 per issued policy, fixed expense is $20 per issued policy, and variable expense is 10% of premium. Which candidate gives greater expected contribution, and what is that contribution across the quotes? (Answer: $600; $24,000 expected contribution)
A hypothetical insurer receives 2,000 eligible quotes. At a supplied customer price of $350, the issuance probability is 0.30. Expected loss is $250 per issued policy, fixed expense is $15 per issued policy, and variable expenses are 8% of premium. What is expected contribution per eligible quote and across all quotes? (Answer: $17.10 per quote; $34,200 total)
Expected loss is $12 per eligible quote, already including the effect of whether a policy issues. There are 1,000 quotes; the issuance probability at $300 is 0.40. Fixed expenses are $20 per issued policy, and variable expenses are 10% of premium. What is expected contribution per eligible quote? (Answer: $88 per quote)

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