Why the Two Thresholds Matter
A deductible determines when an insurer begins paying; a payment limit determines when the insurer stops increasing its payment. Combining them creates three regions of ground-up loss: losses at or below the deductible produce no payment, larger losses produce a payment that increases with the loss, and sufficiently large losses produce the maximum payment. The key is to express both boundaries in terms of the original loss, not just the insurer’s payment.
In this tutorial, \(X\) denotes the ground-up loss before coverage modifications. The deductible \(d\) is the amount borne by the insured, and \(u\) is the maximum amount the insurer will pay per loss after applying the deductible. Assume \(X\geq 0\), \(d\geq 0\), and \(u>0\). These definitions distinguish a payment limit from a limit stated as the maximum ground-up loss eligible for coverage.
The Three Payment Regions
If \(X\leq d\), the loss does not exceed the deductible, so \((X-d)^+=0\) and the insurer pays zero. If \(d<X<d+u\), the loss exceeds the deductible but the amount above it is less than \(u\); the insurer pays \(X-d\). If \(X\geq d+u\), the amount above the deductible is at least \(u\), so the payment is capped at \(u\).
| Ground-up loss \(X\) | Insurer payment \(Y\) | Interpretation |
|---|---|---|
| \(0\leq X\leq d\) | 0 | Loss is absorbed by the deductible |
| \(d<X<d+u\) | \(X-d\) | Payment increases with the loss |
| \(X\geq d+u\) | \(u\) | Payment has reached its maximum |
The boundary values fit the same rule without a gap or jump. At \(X=d\), the payment is \(0\), which agrees with the first region. At \(X=d+u\), the amount above the deductible is exactly \(u\), so the payment is \(u\), which agrees with the capped region. When describing events, it is therefore convenient to use \(X\leq d\) for no payment, \(d<X<d+u\) for a strictly partial payment, and \(X\geq d+u\) for a maximum payment.
Why the Upper Threshold Is \(d+u\)
The payment limit applies to the amount the insurer pays, not to the ground-up loss itself. Before the cap takes effect, each additional dollar of ground-up loss above \(d\) adds one dollar to the insurer’s payment. The insurer reaches payment \(u\) only when the ground-up loss has first covered the deductible and then supplied a further \(u\) dollars.
This is also obtained directly from the capped part of the payment function. The payment is capped when the excess above the deductible is at least the maximum payment:
where the equivalence uses \(u>0\). Thus \(d+u\), rather than \(u\), is the ground-up loss at which the cap begins. This threshold is the central calculation in a combined deductible-and-limit contract.
Record the ground-up amount \(d\) that must be exceeded before the insurer pays.
If it is the maximum insurer payment, record that amount as \(u\).
The ground-up loss at which the maximum payment begins is \(d+u\).
Compare \(X\) with \(d\) and \(d+u\), then apply the corresponding payment rule.
Worked Applications
Worked Example: Applying a Deductible and a Maximum Payment
A hypothetical property policy has a deductible of \(d=\$600\) and pays at most \(u=\$2{,}400\) per covered loss after the deductible. For a particular claim, the ground-up loss is fixed at \(X=\$3{,}500\). Determine the payment and identify the applicable region. Treat the stated loss as a fixed amount (a degenerate loss distribution concentrated at \(\$3{,}500\)).
Method. First find the two ground-up thresholds. Payment begins above \(\$600\), and the upper threshold is $$ d+u=\$600+\$2{,}400=\$3{,}000. $$
The loss is \(\$3{,}500\), which satisfies \(X\geq \$3{,}000\). It is in the capped region. The payment calculation confirms this classification: $$ (X-d)^+=\max(\$3{,}500-\$600,0)=\$2{,}900, $$ $$ Y=\min(\$2{,}900,\$2{,}400)=\$2{,}400. $$
Check. The payment equals the policy maximum, and it does not exceed the ground-up loss. The common mistake to avoid is treating \(\$2{,}400\) as the ground-up threshold; it is the maximum payment, while the cap begins at \(\$3{,}000\).
Worked Example: Probabilities of the Three Regions Under a Uniform Model
Suppose a hypothetical ground-up loss has \(X\sim\operatorname{Uniform}(0,12{,}000)\), with all dollar amounts measured in dollars. A contract has deductible \(d=\$2{,}000\) and maximum payment \(u=\$3{,}000\). Find the ground-up thresholds and the probabilities of no payment, a strictly partial payment, and the maximum payment.
Method. The first threshold is \(d=\$2{,}000\). The second is $$ d+u=\$2{,}000+\$3{,}000=\$5{,}000. $$ Therefore the regions are \(X\leq \$2{,}000\), \(\$2{,}000<X<\$5{,}000\), and \(X\geq \$5{,}000\).
For a uniform distribution on \([0,12{,}000]\), the probability of an interval is its length divided by \(12{,}000\). Thus, $$ P(\text{no payment})=P(X\leq 2{,}000)=\frac{2{,}000}{12{,}000}=\frac16=0.1667. $$ The strictly partial-payment interval has length \(\$5{,}000-\$2{,}000=\$3{,}000\), so $$ P(\text{strictly partial payment}) =\frac{3{,}000}{12{,}000} =\frac14 =0.2500. $$ Finally, the maximum-payment region has length \(\$12{,}000-\$5{,}000=\$7{,}000\), giving $$ P(\text{maximum payment}) =\frac{7{,}000}{12{,}000} =\frac{7}{12} =0.5833. $$
Check. The probabilities sum to \(0.1667+0.2500+0.5833=1.0000\), allowing for rounding. Since the uniform model is continuous, including or excluding either single boundary point does not change these probabilities. The common mistake to avoid is using \(u=\$3{,}000\) as the upper ground-up threshold; the correct boundary is \(d+u=\$5{,}000\).
Worked Example: Threshold Probabilities Under an Exponential Model
Assume a hypothetical ground-up loss follows an exponential distribution with mean \(\$4{,}000\), so its rate parameter is \(\lambda=1/4{,}000\) per dollar and \(P(X>x)=e^{-x/4{,}000}\) for \(x\geq0\). A contract has \(d=\$1{,}000\) and \(u=\$2{,}500\). Find the thresholds and the probabilities of no payment, a strictly partial payment, and the maximum payment. Give probabilities to four decimal places.
Method. The payment begins above \(\$1{,}000\), and the upper ground-up threshold is $$ d+u=\$1{,}000+\$2{,}500=\$3{,}500. $$ Using the exponential survival probability, the probability of no payment is $$ P(X\leq 1{,}000)=1-e^{-1{,}000/4{,}000} =1-e^{-0.25} \approx 1-0.7788008 =0.2212. $$
The probability of a strictly partial payment is the probability that \(X\) is above \(\$1{,}000\) but below \(\$3{,}500\): $$ P(1{,}000<X<3{,}500) =P(X>1{,}000)-P(X\geq3{,}500). $$ The exponential distribution is continuous, so \(P(X\geq3{,}500)=P(X>3{,}500)\). Therefore, $$ P(1{,}000<X<3{,}500) =e^{-0.25}-e^{-3{,}500/4{,}000} \approx 0.7788008-0.4168620 =0.3619. $$ The probability of a maximum payment is $$ P(X\geq3{,}500)=e^{-0.875}\approx0.4169. $$
Check. The three probabilities add to \(0.2212+0.3619+0.4169=1.0000\). The common mistake to avoid is using the exponential survival probability at \(\$2{,}500\), the payment limit, to calculate the maximum-payment region. That region starts at the ground-up threshold \(\$3{,}500\).
Common Interpretation Traps
Contract wording must be read carefully because “limit” can describe different quantities. In the payment function used here, \(u\) is the maximum insurer payment after the deductible. A contract that instead states a maximum covered ground-up loss \(L\) has a different upper boundary: if payment is the loss above \(d\) only up to \(L\), the payment can be written \(\min((X-d)^+,(L-d)^+)\) when \(L\geq d\). In that wording, the insurer’s maximum payment is \(L-d\), not \(L\). Do not substitute a ground-up limit into the role of \(u\) without translating the contract first.
A second trap is to apply the deductible and the cap in the wrong order. Under the stated contract, first determine the amount above the deductible, \((X-d)^+\), and then cap that amount at \(u\). In general, \(\min(X,u)-d\) is not the same payment function: it can produce a negative value for small losses and does not represent an ordinary deductible followed by a maximum payment.
The threshold method is useful because it separates contract interpretation from the choice of a severity model. Once \(d\) and \(u\) are correctly identified, the payment regions are always determined by \(d\) and \(d+u\). A distribution for \(X\) is needed only if probabilities or other distributional quantities are requested. Correctly identifying the regions is aligned with the SOA Exam FAM (Fundamentals of Actuarial Mathematics) syllabus.
Check Your Understanding
For each problem, identify the relevant ground-up thresholds before determining the requested payment or probability.
- A hypothetical appliance policy has \(d=\$350\) and maximum payment \(u=\$1{,}150\). A fixed loss is \(\$1{,}400\). Identify its payment region and calculate the insurer payment. (Answer: \(d+u=\$1{,}500\); strictly partial-payment region; payment \(=\$1{,}050\).)
- A hypothetical contract has \(d=\$1{,}200\) and \(u=\$2{,}800\). For a fixed ground-up loss of \(\$4{,}000\), state the payment and explain its boundary classification. (Answer: \(d+u=\$4{,}000\); maximum-payment boundary; payment \(=\$2{,}800\).)
- Let \(X\sim\operatorname{Uniform}(0,10{,}000)\), with \(d=\$1{,}500\) and \(u=\$2{,}500\). Find the probability of a strictly partial payment. (Answer: \((\$4{,}000-\$1{,}500)/\$10{,}000=0.2500\).)
- Let \(X\) be exponential with mean \(\$2{,}000\), and let \(d=\$500\), \(u=\$1{,}500\). Find the probability of a maximum payment, to four decimal places. (Answer: \(P(X\geq\$2{,}000)=e^{-1}=0.3679\).)
- A contract says the insurer pays the loss above a \(\$700\) deductible, subject to a maximum insurer payment of \(\$2{,}300\). A fixed ground-up loss is \(\$3{,}100\). State the upper threshold and payment, and distinguish the payment limit from that threshold. (Answer: upper threshold \(=\$3{,}000\); payment \(=\$2{,}300\); the payment limit is \(\$2{,}300\), while the corresponding ground-up threshold is \(\$3{,}000\).)