Payment Functions for Layered Coverage

Introduction

A layer of coverage insures only a specified band of a ground-up loss. The insurer pays nothing until the loss passes the layer’s attachment point; after that, its payment rises with the loss until the layer is exhausted. A single function captures all three regions: no payment, a payment that increases dollar for dollar, and a capped payment.

The key is to distinguish the layer’s upper endpoint from its maximum payment. If a layer attaches at $a$ and exhausts at $b$, its maximum payment is $b-a$, not $b$. This distinction is central when interpreting contract wording such as “$L$ excess of $a$.”

Definition: Let $X$ be a nonnegative ground-up loss. A layer attaches at $a$ and exhausts at $b$, where $0 \leq a \leq b$. Its width is $b-a$, and the insurer’s payment for the layer is $$ Y=\min\bigl((X-a)^+,\,b-a\bigr), $$ where $z^+=\max(z,0)$. The attachment point is the loss amount below which the layer pays zero. The exhaustion point is the loss amount at which the layer has reached its maximum payment, $b-a$.

Reading the Payment Function

The positive-part term $(X-a)^+$ measures how much of the loss lies above the attachment. The minimum caps that amount at the layer width. Equivalently, the payment can be read region by region:

Y= \begin{cases} 0, & X\leq a,\\ X-a, & a<X<b,\\ b-a, & X\geq b. \end{cases}

At exactly $X=a$, the payment is zero. At exactly $X=b$, it is $b-a$, the full width. Thus the formula treats the boundary values consistently: the payment begins increasing immediately above attachment and reaches its cap at exhaustion.

In contract shorthand, “$L$ excess of $a$” means that the layer attaches at $a$ and has width $L$. Its exhaustion point is therefore $a+L$. In the notation above, $b=a+L$. Conversely, if a contract specifies the attachment $a$ and exhaustion $b$, the layer width is $b-a$. The point $b$ is measured on the ground-up loss scale; it is not itself the insurer’s payment cap.

Why the Formula Works

The formula follows directly from the three possible positions of $X$ relative to the layer. Before attachment, there is no loss in the layer. Within the layer, the payment is the portion of the loss between $a$ and $X$. Beyond exhaustion, the layer cannot pay more than its full width.

Below or at attachment.
If $X\leq a$, then $X-a\leq0$, so $(X-a)^+=0$. The minimum of zero and the nonnegative layer width is zero.

Between attachment and exhaustion.
If $a<X<b$, then $(X-a)^+=X-a$. Also $X-a<b-a$, so the cap does not bind and $Y=X-a$.

At or above exhaustion.
If $X\geq b$, then $X-a\geq b-a$. The minimum is therefore $b-a$, the layer’s full width.

These cases establish the piecewise form and explain why the capped positive-part expression is useful: one formula handles losses below, inside, and above the layer without requiring a separate contract rule for each region.

Worked Examples

Worked Example: Evaluating a Layer Across Several Loss Amounts

A hypothetical insurer covers a layer attaching at $$18,000$ and exhausting at $$63,000$. For illustration, the ground-up loss $X$, in thousands of dollars, has a discrete severity model with support and probabilities $12$ (0.10), $18$ (0.20), $37$ (0.30), $63$ (0.25), and $94$ (0.15). The probabilities sum to 1. There is no discounting or mortality adjustment. Find the layer payment at each possible loss amount.

Method. Use $a=18$ and $b=63$, in thousands. The layer width is $$ b-a=63-18=45. $$ Thus $Y=\min((X-18)^+,45)$. The severity probabilities are not needed to evaluate the payment function at each support point.

At $X=12$, the loss is below attachment: $$ Y=\min((12-18)^+,45)=\min(0,45)=0. $$ At $X=18$, the loss is exactly at attachment: $$ Y=\min((18-18)^+,45)=0. $$ At $X=37$, the loss is within the layer: $$ Y=\min(37-18,45)=\min(19,45)=19. $$ At $X=63$, the layer is exactly exhausted: $$ Y=\min(63-18,45)=\min(45,45)=45. $$ At $X=94$, the loss is above exhaustion: $$ Y=\min(94-18,45)=\min(76,45)=45. $$

The payments, in thousands of dollars, are $0,0,19,45,45$, respectively. Check: no payment exceeds the layer width of 45, and losses at or above 63 produce the full 45. The common mistake avoided here is treating the attachment point itself as a loss that generates a positive payment.

Worked Example: Translating an Excess Limit into an Exhaustion Point

A hypothetical commercial policy provides $$30,000$ of coverage excess of a $$25,000$ attachment. A covered ground-up loss is $$46,800$. Suppose $X$ follows a Uniform distribution on $[0,100{,}000]$ dollars; the specified loss is a realization under that severity model. There is no discounting or mortality adjustment. Find the layer payment.

Method. “$$30,000$ excess of $$25,000$” gives width $L=30{,}000$ and attachment $a=25{,}000$. Therefore the exhaustion point is $$ b=a+L=25{,}000+30{,}000=55{,}000. $$ The payment function is $Y=\min((X-25{,}000)^+,30{,}000)$.

Substitute $X=46{,}800$: $$ (X-a)^+=(46{,}800-25{,}000)^+=21{,}800. $$ Since $21{,}800<30{,}000$, the cap does not bind, and $$ Y=\min(21{,}800,30{,}000)=\$21{,}800. $$

Check: the loss lies between the $$25,000$ attachment and the $$55,000$ exhaustion point, so the payment equals the amount of loss in that interval, $$46,800-\$25,000=\$21,800$. No probability calculation is required to evaluate a payment for a specified realization. The common mistake avoided is adding the limit to the attachment incorrectly or using the attachment as the layer’s width.

Worked Example: Allocating a Loss Across Adjacent Layers

A hypothetical risk program has three adjacent layers: $$15,000$ excess of $$25,000$, $$35,000$ excess of $$40,000$, and $$50,000$ excess of $$75,000$. A ground-up loss is $$137,000$. For context, let $X$, in thousands of dollars, have a discrete severity model with support $10,37,62,91,137$ and probabilities $0.10,0.20,0.25,0.25,0.20$, respectively. There is no discounting or mortality adjustment. Find each layer’s payment for the specified loss and the amount outside the layers.

Method. Convert each excess limit into an exhaustion point. The first layer attaches at 25 and has width 15, so it exhausts at $25+15=40$. The second attaches at 40, has width 35, and exhausts at $40+35=75$. The third attaches at 75, has width 50, and exhausts at $75+50=125$. All amounts in the calculations below are in thousands of dollars.

For $X=137$, the first layer pays $$ Y_1=\min((137-25)^+,15)=\min(112,15)=15. $$ The second layer pays $$ Y_2=\min((137-40)^+,35)=\min(97,35)=35. $$ The third layer pays $$ Y_3=\min((137-75)^+,50)=\min(62,50)=50. $$ The total paid by the three layers is $15+35+50=100$, or $$100,000$.

The amount below the first attachment is $$25,000$, and the amount above the third exhaustion point is $$ 137-125=12, $$ or $$12,000$. Check: the ground-up loss reconciles as $25+15+35+50+12=137$, and the layers together pay $$100,000$. The common mistake avoided is applying the third layer to the whole loss rather than only to the loss above its own attachment.

Interpreting Layer Boundaries and Contract Terms

The same function can be written using width $L$ instead of exhaustion point $b$. If the layer is $L$ excess of $a$, then $b=a+L$, and $$ Y=\min\bigl((X-a)^+,L\bigr). $$ This is not a different payment rule; it is the same rule with the width stated directly. It is often convenient when the contract gives the attachment and the layer limit rather than both endpoints.

When interpreting a layer, identify three quantities before substituting a loss: the attachment $a$, the exhaustion point $b$, and the width $b-a$. The attachment and exhaustion are expressed in ground-up loss dollars. The width is the greatest amount the insurer can pay under that layer. For a loss between the endpoints, the insurer pays only the part above attachment; for a loss at or beyond the upper endpoint, the insurer pays the full width.

For adjacent layers, each payment function uses its own attachment and width. A loss must be evaluated against each layer separately. A layer above another does not begin paying merely because the lower layer has paid something; the loss must pass that upper layer’s attachment point. Likewise, a loss beyond the last layer’s exhaustion can leave an amount outside all the stated layers.

Common Mistake: Using the Exhaustion Point as the Cap

Suppose a layer attaches at $$25,000$ and exhausts at $$55,000$. Its width is $$30,000$. For a $$70,000$ loss, the correct payment is $$ Y=\min((70{,}000-25{,}000)^+,55{,}000-25{,}000) =\min(45{,}000,30{,}000) =\$30{,}000. $$

A tempting but incorrect calculation is $\min(70{,}000-25{,}000,55{,}000)=\$45,000$. That calculation uses the exhaustion point $55{,}000$ as though it were the insurer’s maximum payment. The corrected cap is $b-a=55{,}000-25{,}000=30{,}000$. A quick check is that payment from a single layer can never exceed its stated width.

Key takeaway: A layer attaching at $a$ and exhausting at $b$ pays $\min((X-a)^+,b-a)$. Read $a$ and $b$ as points on the ground-up loss scale, and $b-a$ as the maximum payment.

Check Your Understanding

For each problem, use the layer’s own attachment and width. Amounts are in dollars unless stated otherwise.

A layer attaches at $$9,000$ and exhausts at $$24,000$. Find its payment for a $$31,000$ loss. (Answer: $$15,000$)
A layer is $$60,000$ excess of $$40,000$. Find its payments for losses of $$25,000$, $$75,000$, and $$130,000$, respectively. (Answer: $$0$, $$35,000$, and $$60,000$)
A layer attaches at $$17,000$ and exhausts at $$42,000$. A covered loss is $$37,500$. Find the payment and state whether the cap binds. (Answer: $$20,500$; the cap does not bind)
A hypothetical treaty covers $$24,000$ excess of $$16,000$. For a $$51,000$ loss, identify the exhaustion point and payment. (Answer: exhaustion at $$40,000$; payment $$24,000$)
A program has three adjacent layers: $$10,000$ excess of $$20,000$, $$25,000$ excess of $$30,000$, and $$35,000$ excess of $$55,000$. For a $$96,000$ loss, find each layer’s payment and the amount below the first attachment or above the last exhaustion. (Answer: payments $$10,000$, $$25,000$, and $$35,000$; $$20,000$ below the first attachment and $$6,000$ above the last exhaustion)

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