How Deductible Choice Affects Premium

Why Deductible Choice Changes Premium

A large deductible shifts part of each covered loss from the insurer to the insured. The expected insurer payment generally falls as the deductible rises, but the premium is not simply that expected payment: insurers also account for expenses and a risk margin. A sound comparison separates these components and applies the same pricing assumptions consistently across options.

The calculations below concern hypothetical policies. The expense allowances and risk-load methods are stated assumptions for each example; actual pricing methods and expense allocations depend on the insurer, product, and applicable requirements. The aim is to understand how each component affects the indicated premium, not to prescribe one universal pricing formula.

Definition: For a covered loss $X$ and a per-occurrence deductible $d$, the insurer's indemnity payment, assuming no policy limit, is $Y_d=(X-d)^+=\max(X-d,0)$. The expected insurer indemnity is $E[Y_d]$. A simplified indicated premium can be built from expected insurer payments, expenses, and a risk load. If fixed expense is $F$, the variable expense ratio is $e$ of premium, and the risk load is $R$, then $P=(E[L]+E[H]+F+R)/(1-e)$, where $L$ is aggregate insurer indemnity and $H$ is insurer-paid claim handling expense. This equation assumes the variable expense is a proportion of premium and that the listed components are not counted twice.

Separate Loss, Expense, and Risk Load

Expected insurer indemnity depends on the deductible and the covered-loss distribution. For a single occurrence it is $E[(X-d)^+]$. For $N$ occurrences, aggregate indemnity is $L=\sum_{j=1}^{N}(X_j-d)^+$; its expectation depends on both claim frequency and severity. A higher deductible typically reduces this expected amount, although the size of the reduction depends on how much probability lies above each deductible.

Expenses should be classified according to how they arise. Claim handling expense may be incurred on every reported claim, only on claims for which the insurer makes a payment, or according to another specified allocation. Fixed policy expenses do not necessarily change with the deductible. Acquisition expense or premium tax expressed as a percentage of premium is different again: its dollar amount changes when premium changes. State the basis before calculating.

A risk load compensates for uncertainty or other costs not represented by the expected loss alone. There is no single required risk-load formula for every large deductible program. A pricing exercise might prescribe a percentage of expected loss, a multiple of standard deviation, or another method. Use the stated method and apply it to the stated measure; do not silently substitute one method for another.

Deriving the Premium Equation

Suppose the insurer's expected claim cost is $E[L]+E[H]$, fixed expenses are $F$, and the risk load is $R$. If variable expenses equal a fraction $e$ of premium, the premium must fund all these items:

$$ P=E[L]+E[H]+F+R+eP $$

Subtract $eP$ from both sides and factor out $P$. This gives the indicated premium formula, provided $e<1$:

(1-e)P=E[L]+E[H]+F+R, \qquad P=\frac{E[L]+E[H]+F+R}{1-e}.

This derivation also shows why adding a percentage expense to the numerator as though it were a fixed dollar amount is incorrect. The expense is a share of the unknown premium, so it must be allowed for by dividing by $1-e$. If variable expense is instead defined as a percentage of loss cost, use that definition; the equation above would not apply unchanged.

Worked Example: Comparing Two Deductibles With Fixed and Variable Expenses

A hypothetical policy has exactly two claims in the year. Each claim independently has severity $X=\$20{,}000$ or $\$80{,}000$, with probability $0.5$ for each. There is no policy limit. Compare deductibles of $\$10{,}000$ and $\$30{,}000$. Insurer claim handling costs $\$1{,}000$ per reported claim, fixed policy expense is $\$3{,}000$, variable expense is $10\%$ of premium, and the risk load is $5\%$ of expected insurer indemnity. Find each indicated premium.

Method and assumptions. Calculate the insurer payment at each severity, average over the two equally likely outcomes, then multiply by the two certain claims. Both claims are reported, so expected handling expense is $\$2{,}000$ under either option. The risk load is the specified percentage of indemnity only.

At deductible $d=\$10{,}000$, payments per claim are $\max(20{,}000-10{,}000,0)=\$10{,}000$ and $\max(80{,}000-10{,}000,0)=\$70{,}000$. Thus expected payment per claim is $0.5(10{,}000)+0.5(70{,}000)=\$40{,}000$, and expected annual indemnity is $2(40{,}000)=\$80{,}000$. The risk load is $0.05(80{,}000)=\$4{,}000$.

The premium numerator is $\$80{,}000+\$2{,}000+\$3{,}000+\$4{,}000=\$89{,}000$. Therefore $P=89{,}000/(1-0.10)=\$98{,}888.89$, rounded to the nearest cent.

At deductible $d=\$30{,}000$, payments per claim are $\$0$ and $\$50{,}000$. Expected payment per claim is $0.5(0)+0.5(50{,}000)=\$25{,}000$, giving annual indemnity of $2(25{,}000)=\$50{,}000$. The risk load is $0.05(50{,}000)=\$2{,}500$. The numerator is $\$50{,}000+\$2{,}000+\$3{,}000+\$2{,}500=\$57{,}500$, so $P=57{,}500/0.90=\$63{,}888.89$.

Check. The premium difference is $\$98{,}888.89-\$63{,}888.89=\$35{,}000.00$. The numerator falls by $\$31{,}500$, and dividing by $0.90$ gives a premium reduction of $\$35{,}000$. The common mistake avoided is applying the variable expense ratio only to expected loss rather than solving for expense as a percentage of the full premium.

Risk Loads Reflect Variability as Well as Expected Loss

Two deductible options can differ in the variability of insurer payments as well as their means. For illustration, assume the annual claim count $N$ is Poisson with mean $\lambda$, and individual insurer payments $Y$ are independent and identically distributed, independent of $N$, with finite second moment. For aggregate indemnity $L=\sum_{j=1}^{N}Y_j$, the compound Poisson results are $E[L]=\lambda E[Y]$ and $\operatorname{Var}(L)=\lambda E[Y^2]$. These results rely on the stated frequency-severity assumptions; they need not hold for a fixed claim count or dependent claim payments.

The variance expression includes the variation in both claim count and payment size. In particular, it is not generally $\lambda\operatorname{Var}(Y)$. A risk load of $k\sqrt{\operatorname{Var}(L)}$, for a specified factor $k$, is one illustrative standard-deviation loading. It is a chosen pricing rule, not a universal requirement.

Justification: Conditional on $N=n$, the sum of $n$ independent payments has mean $nE[Y]$ and variance $n\operatorname{Var}(Y)$. Since a Poisson count has $E[N]=\operatorname{Var}(N)=\lambda$, the law of total expectation gives $E[L]=\lambda E[Y]$. The law of total variance gives $\operatorname{Var}(L)=E[N]\operatorname{Var}(Y)+\operatorname{Var}(N)(E[Y])^2=\lambda\{\operatorname{Var}(Y)+(E[Y])^2\}=\lambda E[Y^2]$.

Worked Example: Including a Standard-Deviation Risk Load

A hypothetical portfolio has Poisson claim frequency with mean $\lambda=4$ claims per year. Each ground-up claim is $\$10{,}000$, $\$20{,}000$, or $\$50{,}000$, with probabilities $0.5$, $0.3$, and $0.2$. Compare deductibles of $\$10{,}000$ and $\$30{,}000$. Claim handling costs $\$500$ per reported claim, fixed expense is $\$5{,}000$, variable expense is $8\%$ of premium, and risk load is $25\%$ of the standard deviation of aggregate indemnity. Assume every claim is reported, even if insurer indemnity is zero.

Method and assumptions. Work in thousands of dollars to calculate payment moments. Use the compound Poisson formulas above, then add expected handling expense, fixed expense, and the stated risk load before allowing for variable expense.

At $d=\$10{,}000$, the payment amounts are $0,10,40$ thousand dollars. The mean payment per claim is $0.5(0)+0.3(10)+0.2(40)=11$ thousand dollars. The second moment is $0.5(0^2)+0.3(10^2)+0.2(40^2)=30+320=350$ thousand-dollars squared. Therefore annual expected indemnity is $4(11)=44$ thousand dollars and annual variance is $4(350)=1{,}400$ thousand-dollars squared. Its standard deviation is $\sqrt{1{,}400}=37.4166$ thousand dollars, so the risk load is $0.25(37.4166)=9.3541$ thousand dollars.

Expected handling expense is $4(\$500)=\$2{,}000$. The premium numerator is $44+9.3541+2+5=60.3541$ thousand dollars. Thus $P=60.3541/0.92=65.6023$ thousand dollars, or $\$65{,}602.33$ to the nearest cent.

At $d=\$30{,}000$, payment amounts are $0,0,20$ thousand dollars. The mean payment per claim is $0.2(20)=4$ thousand dollars, and the second moment is $0.2(20^2)=80$ thousand-dollars squared. Annual expected indemnity is $4(4)=16$ thousand dollars; annual variance is $4(80)=320$ thousand-dollars squared. The standard deviation is $\sqrt{320}=17.8885$ thousand dollars, giving a risk load of $0.25(17.8885)=4.4721$ thousand dollars. Handling expense remains $\$2{,}000$ because all claims are reported. The premium is $(16+4.4721+2+5)/0.92=29.8610$ thousand dollars, or $\$29{,}861.02$.

Check. The higher deductible reduces expected indemnity by $\$28{,}000$ and the stated risk load by about $\$4{,}882.01$, while the other numerator components stay fixed. The resulting premium difference is $\$65{,}602.33-\$29{,}861.02=\$35{,}741.31$, with a one-cent possible display difference if rounded components rather than unrounded values are subtracted. The common mistake avoided is calculating the risk load from expected indemnity alone instead of from the specified aggregate standard deviation.

When Claim Expenses Also Depend on the Deductible

A deductible can affect claim expenses if the expense provision is tied to insurer-paid claims rather than all reported claims. For example, a stated model might charge the insurer a handling amount only when its indemnity is positive. That assumption changes expected expense with the deductible. It should not be generalized to situations where the insurer or its administrator handles claims below the deductible.

Worked Example: Handling Expense Triggered by an Insurer Payment

A hypothetical policy has exactly two independent claims. Each claim is $\$8{,}000$, $\$25{,}000$, or $\$60{,}000$, with probabilities $0.25$, $0.50$, and $0.25$. Compare deductibles of $\$10{,}000$ and $\$30{,}000$. The insurer incurs $\$900$ handling expense only when its indemnity is positive. Fixed expense is $\$4{,}000$, variable expense is $6\%$ of premium, and the risk load is $10\%$ of expected annual indemnity.

Method and assumptions. Compute indemnity and the probability of a positive payment per claim separately. Multiply each by two claims to obtain expected annual amounts. This treatment assumes handling expense is not incurred by the insurer on zero-payment claims.

For $d=\$10{,}000$, per-claim indemnities are $0,\$15{,}000,\$50{,}000$. Expected indemnity per claim is $0.25(0)+0.50(15{,}000)+0.25(50{,}000)=\$20{,}000$, so annual expected indemnity is $\$40{,}000$. A positive insurer payment occurs with probability $0.50+0.25=0.75$, so expected paid-claim count is $2(0.75)=1.5$. Expected handling expense is $1.5(\$900)=\$1{,}350$, and the risk load is $0.10(\$40{,}000)=\$4{,}000$. The premium is $(40{,}000+1{,}350+4{,}000+4{,}000)/0.94=\$52{,}500.00$.

For $d=\$30{,}000$, per-claim indemnities are $0,0,\$30{,}000$. Expected indemnity per claim is $0.25(30{,}000)=\$7{,}500$, so annual expected indemnity is $\$15{,}000$. A positive payment occurs with probability $0.25$; expected paid-claim count is $2(0.25)=0.5$, and expected handling expense is $0.5(\$900)=\$450$. The risk load is $0.10(\$15{,}000)=\$1{,}500$. The premium is $(15{,}000+450+1{,}500+4{,}000)/0.94=\$22{,}287.23$, rounded to the nearest cent.

Check. The premium difference is $\$52{,}500.00-\$22{,}287.23=\$30{,}212.77$. The higher deductible reduces the indemnity, risk load, and modeled handling expense, but not fixed expense. The common mistake avoided is charging handling expense on every reported claim despite the problem's explicit positive-payment condition.

Common Premium-Comparison Traps

Using ground-up losses as insurer losses. Apply the deductible to each loss before totaling insurer indemnity. The premium loss component is not the expected ground-up loss when the insured retains a deductible layer.
Subtracting the deductible from the mean loss. In general, $E[(X-d)^+]$ is not $\max(E[X]-d,0)$. The positive-part operation must be applied to each possible loss before taking its expectation.
Mixing expense bases. A fixed dollar expense, an expense per reported claim, and a percentage of premium are different items. Specify which claims trigger claim expense and whether a percentage is applied to premium or loss cost.
Changing the risk-load rule between options. A comparison is meaningful only if the same stated method is used consistently, unless the problem explicitly specifies different rules.
Calling the lower premium the lower total cost. A higher deductible generally transfers more loss to the insured. Premium comparison alone does not measure the insured's combined premium and retained-loss cost.

Key takeaway: For each deductible option, calculate expected insurer indemnity under that deductible, identify which expenses change and why, apply the stated risk-load rule, and solve for premium using the correct expense base. The resulting premium reflects the insurer's priced components; it does not by itself describe the insured's full cost of risk.

Check Your Understanding

For each hypothetical problem, follow the specified payment, expense, and risk-load assumptions.

A single claim is $\$5{,}000$, $\$20{,}000$, or $\$50{,}000$, with probabilities $0.2$, $0.5$, and $0.3$. With a $\$15{,}000$ per-occurrence deductible and no limit, find expected insurer indemnity. (Answer: $\$13{,}000$)
A policy has exactly three independent claims, each severity $\$10{,}000$ or $\$40{,}000$ with equal probability. There is a $\$20{,}000$ per-occurrence deductible and no limit. Find expected aggregate insurer indemnity. (Answer: $\$30{,}000$)
Expected annual indemnity is $\$32{,}000$, expected handling expense is $\$1{,}200$, fixed expense is $\$4{,}000$, and risk load is $\$3{,}000$. If variable expense is $10\%$ of premium, calculate indicated premium to the nearest cent. (Answer: $\$44{,}666.67$)
Annual claim count is Poisson with mean 2. Per-claim insurer payments are $0,\$10{,}000,\$30{,}000$, with probabilities $0.5,0.25,0.25$. Find the premium if handling expense is $\$400$ per reported claim, fixed expense is $\$2{,}000$, variable expense is $5\%$ of premium, and risk load is $20\%$ of aggregate indemnity standard deviation. Assume every claim is reported. Give the answer to the nearest cent. (Answer: $\$28{,}707.51$)
Option A has expected indemnity $\$50{,}000$, handling expense $\$500$, and risk load $\$5{,}000$. Option B has expected indemnity $\$32{,}000$, handling expense $\$200$, and risk load $\$3{,}000$. Both have fixed expense $\$3{,}000$ and variable expense $8\%$ of premium. Find Option A's premium minus Option B's premium, to the nearest cent. (Answer: $\$22{,}065.22$)

What You'll Learn