Using A Given Demand Curve In Optimization

From a Demand Curve to Expected Policies

A supplied demand curve is useful only after its output has been interpreted correctly. If the curve gives the probability that an eligible customer accepts a quoted premium, it does not directly give the number of policies sold. To obtain expected policies, combine that probability with the number of eligible customers. For a renewal curve, first define the policies offered renewal; the expected number retained is calculated on that base.

This translation is a building block for pricing objectives: expected policy volume can be combined with quoted premiums, costs, or other specified terms. It does not, by itself, select an optimal price. Correctly identifying the curve’s output, its exposure base, and its price units is essential to using it properly. This skill is aligned with the CAS Exam 8 (Advanced Ratemaking) syllabus.

Definition: Let $q_j(p_j)$ be the supplied probability that eligible risk $j$ purchases or renews at quoted premium $p_j$. For $N$ eligible risks with the same purchase probability $q(p)$, the expected number of policies is $Nq(p)$. For risks with potentially different probabilities, the expected number is the sum of their individual probabilities.

The eligibility base matters. For new business, it might be the number of customers receiving a quote. For renewals, it is the number of current policies actually offered renewal. It is not automatically the total number of people in a market, all policies in force, or the number of quotes produced over a different time period. A quote-level acceptance probability should be multiplied by a count of eligible quotes, not by an unrelated exposure measure.

A probability curve returns a value between zero and one. Some demand curves instead return expected policy volume directly, perhaps as a function of price. In that case, do not multiply the curve output by a customer count again. Read the definition and units of the supplied function before calculating.

Why Multiplying by the Eligible Count Works

For each eligible risk $j$, define an indicator $I_j$ that equals 1 if the risk purchases and 0 otherwise. The indicator’s expected value is its purchase probability: $E[I_j]=q_j(p_j)$. The total number of policies is $N_{\text{sold}}=\sum_{j=1}^{N} I_j$. By linearity of expectation,

E[N_{\text{sold}}] =E\left[\sum_{j=1}^{N}I_j\right] =\sum_{j=1}^{N}E[I_j] =\sum_{j=1}^{N}q_j(p_j).

If every risk has the same probability $q(p)$, this reduces to $E[N_{\text{sold}}]=Nq(p)$. Independence is not required for this expected-count result; linearity of expectation holds even when purchase decisions are dependent. Independence or additional joint-distribution assumptions would be needed for some calculations of the variance or distribution of the total count, but not for its expectation.

An expected count need not be an integer. For example, an expectation of 72.4 policies does not mean that a fraction of a policy will be issued. It means that the average count over repeated comparable groups, under the stated model, is 72.4. The realized count is an integer and may differ from the expectation.

Worked Example: Aggregating Two Logistic Purchase Curves

A hypothetical insurer quotes a premium of $1,000 to two new-business segments. Segment A has 400 eligible quote recipients and supplied purchase probability $q_A(p)=1/[1+\exp(-(2-0.001p))]$. Segment B has 250 eligible quote recipients and supplied probability $q_B(p)=1/[1+\exp(-(1-0.002p))]$. Calculate expected policies by segment and in total. Use the unrounded probabilities for the counts.

Method. Evaluate each supplied probability at the quoted premium, then multiply by the corresponding eligible quote count. The premium is entered in dollars, as required by the curve coefficients.

For Segment A, the logistic score at $1,000 is $2-0.001(1000)=1$. Thus,

q_A(1000)=\frac{1}{1+\exp(-1)} =0.7310585786\ldots

The probability is 0.7311 to four decimal places. Using the unrounded value for the expected count,

E[N_A]=400\left(\frac{1}{1+\exp(-1)}\right) =292.4234314\ldots \approx 292.42.

For Segment B, the score is $1-0.002(1000)=-1$. Therefore,

q_B(1000)=\frac{1}{1+\exp(1)} =0.2689414214\ldots

This probability is 0.2689 to four decimal places. The expected count, again using the unrounded value, is

E[N_B]=250\left(\frac{1}{1+\exp(1)}\right) =67.23535535\ldots \approx 67.24.

Adding the unrounded expected counts gives $292.4234314\ldots+67.23535535\ldots=359.6587868\ldots$, or 359.66 expected policies to two decimal places. As a check, the total eligible quotes are $400+250=650$, and the aggregate expected count of about 359.66 is between zero and 650.

Common mistake avoided. Rounding the probabilities first gives $400(0.7311)=292.44$ and $250(0.2689)=67.225$, which rounds to 67.23 using conventional half-up rounding. Those are not the correctly rounded expected counts from the supplied curves. The correct counts use the unrounded probabilities: 292.42 and 67.24. Intermediate rounding can change the final answer.

Working with Different Curve Forms

A supplied purchase function need not be logistic. It may be linear over a specified price interval, tabulated by price band, or expressed in terms of a transformed price. The calculation is the same once the function’s output is confirmed to be a probability: evaluate at the stated quote and multiply by the eligible count. Do not extrapolate beyond a stated valid range unless the problem explicitly permits it. A linear approximation can produce values below zero or above one outside its intended range.

Price units require particular care. If a curve uses premium in hundreds of dollars, entering a premium in dollars without converting it changes the curve input. Similarly, distinguish annual premiums from monthly premiums. A price variable measured in annual dollars cannot be replaced by a monthly amount merely because both are monetary values.

Worked Example: A Linear Purchase Probability and Fixed-Price Revenue

A hypothetical insurer has 600 eligible prospects. Over the stated annual-premium range from $500 to $2,250, the supplied purchase probability is $q(p)=0.90-0.0004(p-500)$, where $p$ is the annual premium in dollars. At an annual quote of $1,250, calculate the expected policies and the corresponding expected premium volume at that fixed quote.

Step 1: Check the input and domain. The quote is $1,250 per year, so it is within the specified $500 to $2,250 range and is entered in dollars. The curve’s output at both ends of the range is between zero and one: $q(500)=0.90$ and $q(2250)=0.90-0.0004(1750)=0.20$.

Step 2: Evaluate the probability.

$$ q(1250)=0.90-0.0004(1250-500) =0.90-0.0004(750) =0.90-0.30 =0.60. $$

Step 3: Calculate expected policies.

E[N_{\text{sold}}]=600(0.60)=360.

Step 4: Use the expected count at the fixed quote. Expected premium volume, meaning the quoted premium multiplied by expected policies sold, is $1250(360)=\$450{,}000$. This is a value of the premium-volume term at $1,250, not a claim that $1,250 maximizes any objective. The check $0\leq360\leq600$ is satisfied.

Common mistake avoided. The 0.60 is a probability, not a percentage to be divided by 100 again. Also, the function specifies annual dollars; using a monthly premium of $104.17 as its input would not represent the stated $1,250 annual quote.

Retention Curves and Pricing Objectives

A retention curve is applied to the policies eligible to renew. If a portfolio has several segments with different renewal quotes or supplied retention probabilities, calculate each segment’s expected renewals separately and add them. The expected retention rate for the combined portfolio is expected renewals divided by policies offered renewal, provided the denominator is positive. This weighted result will generally not equal the simple average of segment retention probabilities.

Expected policy counts are inputs to objectives, not objectives in themselves. At a fixed quote, a simple premium-volume term for segment $s$ is $p_sE[N_s]$, or $p_sN_sq_s(p_s)$ when all $N_s$ risks in the segment share the same probability. An objective with policy-level costs might instead use a sum of expected revenues and costs, with each term matched to the risks that generate it. If purchasers differ in expected cost from the quoted population, the cost term must reflect the appropriate purchaser mix; multiplying by an unconditional portfolio average may then be inappropriate. The purpose here is to translate the supplied curve correctly, not to choose which objective to maximize.

Worked Example: Expected Renewals Across Two Segments

A hypothetical insurer offers renewal to 180 policies in Segment A and 120 in Segment B. At their respective renewal quotes, the supplied retention probabilities are 0.80 and 0.50. The annual premiums offered are $900 in A and $1,100 in B. Calculate expected renewals by segment, total expected renewals, the portfolio expected retention rate, and premium volume at these fixed quotes.

Step 1: Calculate expected renewals by segment. Segment A has $180(0.80)=144$ expected renewals. Segment B has $120(0.50)=60$ expected renewals.

Step 2: Aggregate and calculate the retention rate. Total expected renewals are $144+60=204$. The policies offered renewal total $180+120=300$, so the expected retention rate is $204/300=0.68$, or 68.0%. This is a weighted rate: the segment probabilities are weighted by the numbers offered renewal.

Step 3: Calculate premium volume using each segment’s own quote. Segment A contributes $144(\$900)=\$129{,}600$, and Segment B contributes $60(\$1{,}100)=\$66{,}000$. Total expected premium volume is $\$129{,}600+\$66{,}000=\$195{,}600$.

Check. Expected nonrenewals are $180-144=36$ in A and $120-60=60$ in B. The total is 96, and $204+96=300$, the full number offered renewal. The calculation avoids treating the unweighted average $(0.80+0.50)/2=0.65$ as the portfolio retention rate; that would be wrong because the segment sizes differ.

Checks Before Using the Output

Identify what the function returns.
Confirm whether it gives a probability, expected policy count, or another quantity. Multiply by an eligible count only when the function returns a per-risk probability.

Match the exposure base.
Use the number of prospects quoted for purchase probability or the number of policies offered renewal for retention probability. Keep the time period and unit of exposure consistent.

Evaluate the curve as specified.
Use the correct premium units and the supplied price. Check any stated range and confirm that a probability result is between zero and one.

Aggregate by segment and preserve precision.
Calculate each segment’s expected count, add the unrounded results, and round only for the final reported answer.

Place the result in the objective.
At a fixed quote, pair each expected count with the matching premium or other stated per-policy quantity. Do not mistake this evaluation for finding an optimizing price.

A frequent exam trap is to use the right probability with the wrong base. If a retention probability is 0.70 for policies offered renewal, multiplying it by all policies in force—including policies not offered renewal—overstates expected renewals. Another is to average segment probabilities without weighting by segment counts. A final trap is premature rounding: keep the curve output at full precision through the count and aggregate calculations, then round once at the end.

Key takeaway: For a supplied per-risk purchase or retention probability, expected policy volume is the sum of the relevant probabilities, or $Nq(p)$ when all $N$ eligible risks share one probability. Match the probability to its eligible base, respect the curve’s units and domain, and use the resulting expected count as an input to—not a substitute for—the specified pricing objective.

Check Your Understanding

For each problem, distinguish probability from count, use the stated eligible base, and keep sufficient precision until the final answer.

A hypothetical insurer quotes 320 prospects. A supplied curve gives purchase probability 0.625 at the quoted annual premium. What is the expected number of policies sold? (Answer: 200 expected policies)
For annual premiums from $400 to $2,000, a supplied curve is $q(p)=0.88-0.0003(p-400)$, with $p$ in dollars. There are 500 eligible prospects and the quote is $1,400. Find the purchase probability and expected policies. (Answer: $q=0.58$; 290 expected policies)
A logistic curve gives $q(p)=1/[1+\exp(-z)]$, and the supplied score at the quoted price is $z=\ln(4)$. If 100 customers are quoted, find the expected count. (Answer: 80 expected policies)
A renewal offer goes to 250 policies in Segment A and 150 in Segment B. Supplied retention probabilities are 0.42 and 0.73, respectively. Find total expected renewals and the portfolio expected retention rate. (Answer: 214.5 expected renewals; 53.625% expected retention)
A hypothetical insurer offers renewal to 200 policies in Segment A and 120 in Segment B. At fixed quotes, their supplied retention probabilities are $1/[1+\exp(-\ln(3))]$ and $1/[1+\exp(\ln(3))]$, respectively. The annual quotes are $800 and $1,200. Find expected renewals by segment and total premium volume at these quotes. (Answer: 150 in A and 30 in B, 180 total; $156,000 expected premium volume)

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