Comparing Premium and Retained Loss Cost

Why Compare Premium and Retained Loss?

A lower premium does not necessarily mean a lower cost for an insured. A higher deductible can reduce the premium while increasing the amount the insured pays directly toward claims. To compare the alternatives on the same basis, combine the annual premium with the expected annual loss retained by the insured. This is a cost comparison, not a prediction of the exact amount the insured will pay in a particular year.

The calculation below concerns a per-claim deductible: for each covered claim, the insured pays the loss up to the deductible and the insurer pays any covered amount above it. The premium is treated as a fixed annual amount for each deductible option. The method is aligned with the CAS Exam 8 (Advanced Ratemaking) syllabus.

Definition: For a nonnegative covered claim loss $X$ and a per-claim deductible $d$, the insured's retained payment for that claim is $R_d(X)=\min(X,d)$. If the annual premium at deductible $d$ is $P_d$, and there are $N$ claims with losses $X_1,\ldots,X_N$, annual insured cost is $C_d=P_d+\sum_{j=1}^{N}R_d(X_j)$. Its expected value is the annual premium plus expected annual retained loss.

This definition assumes that each claim is subject to the stated deductible and that there is no aggregate deductible, reimbursement cap, or other contract term changing the insured's payment. If the contract includes such a term, calculate the retained payment under the actual contract first, then add its expected annual value to the premium. The premium and retained loss must also be measured for the same coverage period and loss basis.

Expected Annual Retained Loss

Suppose the annual claim count $N$ has expected value $\lambda$, and individual claim severities are identically distributed as $X$, independent of the claim count. Then the expected annual retained loss at deductible $d$ is $\lambda E[R_d(X)]$. Independence between claim severities is a conventional assumption for the examples; for this expectation formula, the key conditions are that the expected count is finite and the severity distribution is the same for each claim and independent of the count.

E[C_d]=P_d+\lambda E[\min(X,d)]

The formula does not require $N$ to be Poisson. It requires only the expected claim count and the stated relationship between claim count and severity. A Poisson model is one common way to specify frequency, but only $E[N]=\lambda$ enters this expected-cost calculation.

Why the Frequency-Severity Formula Holds

Condition on the number of claims. Given $N=n$, the retained payments are $n$ terms, each with expected value $E[\min(X,d)]$. Therefore, the conditional expected retained total is $nE[\min(X,d)]$. Taking expectation over the claim count gives the expected annual retained loss.

\begin{aligned} E\left[\sum_{j=1}^{N}R_d(X_j)\right] &=E\left[E\left[\sum_{j=1}^{N}R_d(X_j)\mid N\right]\right]\\ &=E\left[N E[R_d(X)]\right]\\ &=E[N]E[R_d(X)]\\ &=\lambda E[\min(X,d)]. \end{aligned}

For nonnegative losses, a useful way to find the expected retained payment is to integrate the survival probability over the deductible layer. Since $\min(X,d)$ cannot exceed $d$, its expectation is the area under the probability that the retained payment exceeds each amount from zero to $d$.

E[\min(X,d)]=\int_0^d P(X>x)\,dx=\int_0^d S_X(x)\,dx

For a discrete severity distribution, it is often simpler to evaluate $\min(X,d)$ at each possible loss and take the probability-weighted average. The integral formula is especially convenient for continuous distributions. In either case, compare the resulting expected retained loss with the premium; do not compare premiums alone.

Worked Comparisons

Worked Example: Two Deductibles With Discrete Claim Sizes

A hypothetical policy has $N\sim\operatorname{Poisson}(6)$ claims per year. Independent claim severity is $1,000 with probability 0.30, $3,000 with probability 0.40, and $8,000 with probability 0.30. The annual premium is $24,000 with a $2,000 per-claim deductible and $18,000 with a $5,000 per-claim deductible. Compare expected annual insured costs.

Method. For each option, calculate the probability-weighted retained payment per claim, multiply it by the expected annual claim count of 6, and add that option's premium.

At the $2,000 deductible, retained payments for severities $1,000, $3,000, and $8,000 are respectively $1,000, $2,000, and $2,000. Thus:

E[R_{2000}(X)]=0.30(1{,}000)+0.40(2{,}000)+0.30(2{,}000) =300+800+600=\$1{,}700.

Expected annual retained loss is $6(\$1,700)=\$10,200$. The total expected annual cost is $\$24,000+\$10,200=\$34,200$.

At the $5,000 deductible, the retained payments are $1,000, $3,000, and $5,000. Therefore:

E[R_{5000}(X)]=0.30(1{,}000)+0.40(3{,}000)+0.30(5{,}000) =300+1{,}200+1{,}500=\$3{,}000.

Expected annual retained loss is $6(\$3,000)=\$18,000$. Total expected annual cost is $\$18,000+\$18,000=\$36,000$. The $2,000 deductible has the lower expected cost by $\$36,000-\$34,200=\$1,800$. The premium falls by $6,000 at the higher deductible, but expected retained loss rises by $7,800, more than offsetting that premium reduction. The calculation is checked by comparing the two complete totals: $34,200 and $36,000.

Common mistake avoided: choosing the $5,000 deductible because its premium is lower. The premium saving alone does not account for the retained claim payments.

Worked Example: A Continuous Severity Distribution

A hypothetical insured expects 4.5 claims per year. Each covered claim severity is uniformly distributed from $0 to $10,000, independently of claim count. The annual premium is $31,000 with a $4,000 deductible and $25,000 with a $7,000 deductible. Compare expected annual insured costs.

Method. For $X\sim\operatorname{Uniform}(0,10{,}000)$, the survival probability is $P(X>x)=(10{,}000-x)/10{,}000$ for $0\le x\le10{,}000$. Integrate this probability from zero to the deductible.

For the $4,000 deductible:

\begin{aligned} E[\min(X,4{,}000)] &=\int_0^{4{,}000}\frac{10{,}000-x}{10{,}000}\,dx\\ &=\frac{1}{10{,}000}\left[10{,}000x-\frac{x^2}{2}\right]_0^{4{,}000}\\ &=\frac{40{,}000{,}000-8{,}000{,}000}{10{,}000} =\$3{,}200. \end{aligned}

Expected annual retained loss is $4.5(\$3,200)=\$14,400$, so total expected annual cost is $\$31,000+\$14,400=\$45,400$.

For the $7,000 deductible:

\begin{aligned} E[\min(X,7{,}000)] &=\int_0^{7{,}000}\frac{10{,}000-x}{10{,}000}\,dx\\ &=\frac{70{,}000{,}000-24{,}500{,}000}{10{,}000} =\$4{,}550. \end{aligned}

Expected annual retained loss is $4.5(\$4,550)=\$20,475$, and total expected annual cost is $\$25,000+\$20,475=\$45,475$. The $4,000 deductible has the lower expected cost by $75.00. The premiums differ by $6,000, while expected retained losses differ by $\$20,475-\$14,400=\$6,075$; their net difference is $75. The result is checked by the totals, $45,400 and $45,475.

Common mistake avoided: multiplying the deductible by the expected claim count as if every claim reached the deductible. The uniform severity model includes claims below each deductible, so the expected retained amount must account for the severity distribution.

Worked Example: A Larger Premium Reduction Can Still Lose

A hypothetical account expects 8 claims per year. Independent severity is $2,000 with probability 0.25, $5,000 with probability 0.50, and $12,000 with probability 0.25. The premium is $42,000 at a $3,000 deductible and $29,000 at an $8,000 deductible. Which choice has lower expected annual cost?

Method. Calculate retained severity at each possible claim amount, then apply the expected claim count and add the corresponding premium.

At $3,000, the retained amounts are $2,000, $3,000, and $3,000. The expected retained amount per claim is:

0.25(2{,}000)+0.50(3{,}000)+0.25(3{,}000) =500+1{,}500+750=\$2{,}750.

Expected annual retained loss is $8(\$2,750)=\$22,000$. Total expected annual cost is $\$42,000+\$22,000=\$64,000$.

At $8,000, the retained amounts are $2,000, $5,000, and $8,000. The expected retained amount per claim is:

0.25(2{,}000)+0.50(5{,}000)+0.25(8{,}000) =500+2{,}500+2{,}000=\$5{,}000.

Expected annual retained loss is $8(\$5,000)=\$40,000$. Total expected annual cost is $\$29,000+\$40,000=\$69,000$. The $3,000 deductible is lower by $\$69,000-\$64,000=\$5,000$, despite its higher premium. The arithmetic checks because the higher deductible saves $13,000 in premium but adds $18,000 in expected annual retained loss.

Common mistake avoided: treating a $13,000 premium reduction as a $13,000 reduction in total cost. The insured's expected retained payments must be included for both options.

Interpreting the Comparison

For two deductible options, the decision rule can be written directly in terms of their differences. Let $d_H$ be the higher deductible and $d_L$ the lower deductible, with premiums $P_H$ and $P_L$. The higher deductible has lower expected cost exactly when its premium saving is greater than the increase in expected annual retained loss.

P_L-P_H > \lambda\left(E[\min(X,d_H)]-E[\min(X,d_L)]\right)

If the two sides are equal, the options have equal expected costs under the assumptions used. If the premium saving is smaller, the higher-deductible option has the greater expected cost. This comparison is specific to expected cost: it does not establish which option is preferable for an insured who also considers variability, cash-flow capacity, or other contract terms.

Common Calculation Traps

A particularly important distinction is between gross claim severity and the amount retained by the insured. Under a per-claim deductible, the insured does not pay every dollar of every claim; the retained payment is capped at the deductible. For example, using the first worked example's $2,000 option, the expected gross severity per claim is:

0.30(1{,}000)+0.40(3{,}000)+0.30(8{,}000) =300+1{,}200+2{,}400=\$3{,}900.

A wrong calculation would add $6(\$3,900)=\$23,400$ of gross expected annual claims to the $24,000 premium, producing $47,400. That is not the insured's cost under this deductible: the insured retains only $\$1,700$ in expectation per claim, not the full $\$3,900$. The correct calculation is $\$24,000+6(\$1,700)=\$34,200$.

Apply the deductible claim by claim. For each possible loss, use $\min(X,d)$, not the full loss and not the deductible automatically.
Keep annual units consistent. Multiply expected retained payment per claim by the expected number of claims per year, then add an annual premium.
Compare total costs, not just premiums. A premium ranking can differ from an expected-cost ranking.
Use the actual contract payment rule. Aggregate deductibles, caps, or other provisions can change the retained amount from the simple per-claim formula.

Key takeaway: For a fixed annual premium and an uncapped per-claim deductible, expected annual insured cost is $P_d+\lambda E[\min(X,d)]$. Calculate this amount separately for each option, and select the lower expected-cost alternative only when expected cost is the comparison criterion.

Check Your Understanding

For each problem, calculate the expected annual retained loss before adding the premium. Amounts are hypothetical; report dollar answers to the cent.

A policy has expected annual claim count 3. Claim severity is $1,000 or $5,000, each with probability 0.50. At a $2,000 deductible, the annual premium is $14,000. Find expected annual insured cost. (Answer: $18,500.00)
A hypothetical insured expects 5 claims per year, with severity uniformly distributed from $0 to $6,000. Its annual premium is $16,000 at a $3,000 deductible. Find expected annual insured cost. (Answer: $27,250.00)
A policy has expected annual claim count 7. Severity is $2,000 with probability 0.20, $4,000 with probability 0.50, and $9,000 with probability 0.30. At a $4,000 deductible, the annual premium is $26,000. Find expected annual insured cost. (Answer: $51,200.00)
A hypothetical account expects 4 claims per year. Severity is $1,000 with probability 0.25, $4,000 with probability 0.50, and $10,000 with probability 0.25. The annual premium is $20,000 at a $2,000 deductible and $13,000 at a $6,000 deductible. Which option has lower expected annual cost, and by how much? (Answer: The $2,000 deductible is lower by $1,000.00; expected costs are $27,000.00 and $28,000.00, respectively.)
A hypothetical insured expects 6 claims per year. Severity is $2,000 or $8,000, each with probability 0.50. Compare a $2,000 deductible with a $6,000 deductible. By how much must the higher-deductible option's annual premium be lower for the two options to have equal expected annual costs? (Answer: $12,000.00.)

What You'll Learn