Insured Loss An insurance policy is written to cover a loss,...

Insured Loss An insurance policy is written to cover a loss, $X$ where $X$ has a uniform distribution on $[0,1000] .$ At what level must a deductible be set in order for the expected payment to be 25$\%$ of what it would be with no deductible? Choose one of the following. (Hint: Use a variable, such as $D,$ for the deductible. The payment is 0 if the loss is less than $D,$ and the loss minus $D$ if the loss is greater than $D .$ Source: Society of Actuaries. $$ \begin{array}{llll}{\text { a. } 250} & {\text { b. } 375} & {\text { c. } 500} & {\text { d. } 625} & {\text { e. } 750}\end{array} $$

Insured Loss An insurance policy is written to cover a loss, $X$ where $X$ has a uniform distribution on $[0,1000] .$ At what level must a deductible be set in order for the expected payment to be 25$\%$ of what it would be with no deductible? Choose one of the following. (Hint: Use a variable, such as $D,$ for the deductible. The payment is 0 if the loss is less than $D,$ and the loss minus $D$ if the loss is greater than $D .$ Source: Society of Actuaries.
$$
\begin{array}{llll}{\text { a. } 250} & {\text { b. } 375} & {\text { c. } 500} & {\text { d. } 625} & {\text { e. } 750}\end{array}
$$

Key Concepts

Loss Payment Function with Deductible

The loss payment function with a deductible modifies the claim payment such that losses below the deductible amount result in no payment, while losses above the deductible result in a payment equal to the loss minus the deductible. Understanding this function is essential for calculating how the deductible impacts the insurer’s expected payout, and ultimately, determining the deductible level required to meet a specified reduction in expected payment.

Uniform Distribution

The uniform distribution is a probability distribution where all outcomes within a given interval are equally likely. In this context, the loss is uniformly distributed over the interval [0,1000], meaning that any loss value within that interval has the same probability of occurring. This concept is fundamental in understanding the probability model used for analyzing the insurance loss scenario.

Deductible

A deductible is the amount subtracted from the loss before an insurance payment is made. It represents the portion of a loss that the policyholder is responsible for paying out-of-pocket. In the problem context, the deductible reduces the insurer's expected payout by only allowing payments for losses above a certain threshold.

Expected Value Calculation

Expected value calculation is a key statistical concept used to determine the average outcome of a random variable. When analyzing insurance payments with a deductible, the expected value of the payment is computed by integrating the amount paid over the probability distribution of losses. This calculation helps in setting parameters, such as deductibles, to achieve target outcomes like a specific reduction in expected payment.

Transcript

00:01 Okay, so the first step for this question is let's figure out what the expected loss is, because that's what the expected payout of this insurance policy would be if there was no deductible.

00:12 So since our variable is uniform from zero to a thousand, we know the expectation is just the middle, which would be 500, and 25 % of that would be 125.

00:22 So we want to find the deductible such that the expected payout is $125.

00:32 Okay.

00:33 So how i'm going to say we do this is we think about what where, what the payout with deductible looks like.

00:51 So if x is less than the deductible d, the payout is zero.

01:01 Else if x is between d and 1 ,000, which is our upper limit, the payout is whatever the loss.

01:10 Minus the deductible.

01:12 So if we think about expectation, just as the weighted average, we're adding up each payout times the probability of that loss occurring.

01:42 So for this problem, and as we do this on a continuous variable, this becomes, you know, an integral, right, calculus stuff.

01:50 So what we want to do this is, and as we do this on a continuous variable, this becomes, you know, an integral, right? calculus stuff.

01:52 So what we want to do this.

01:52 And as we want to do this is, to do is we want to integrate over the whole interval that is important to us, which since the payout is zero, we don't care about that part.

01:59 So we only care about d up to a thousand.

02:03 And we want to take the payout, which we just said is x minus the deductible times the probability of that loss occurring.

02:11 Well, since the loss is uniform 0 to 1 ,000, we know the density function for the loss is just 1 over 1 ,000.

02:24 So if we go ahead and do this integration, we can move the 1 over 1 ,000 out front.

02:30 We'll get x squared over 2 minus dx, and we want this whole thing to equal 125, remember...

Please give Ace some feedback