Question

Life insurance A life insurance company sells a term insurance policy to a$21$-year-old male that pays $100,000$if the insured dies within the next $5$years. The probability that a randomly chosen male will die each year can be found in mortality tables. The company collects a premium of $250$each year as payment for the insurance. The amount Y that the company earns on this policy is $250$per year, less the $100,000$that it must pay if the insured dies. Here is a partially completed table that shows information about risk of mortality and the values of $Y=$profit earned by the company:

(a) Copy the table onto your paper. Fill in the missing values of $Y$.

(b) Find the missing probability. Show your work.

(c) Calculate the mean $\mu Y$. Interpret this value in context.

Short answer

a). Missing value of $Y$:

b).

Step-by-step solution

Given Information (Part a)

Here is a partially completed table that shows information about the risk of mortality and the values of $Y=$ profit earned by the company

Explanation (Part a)

Profit $23$ years old

If the client dies at age $23$ , then the insurance company needs to pay out the face value of $\$100,000$.

However, since the client closed the life insurance when he was $21$, the client already made 3 payments of $\$250$each (one payment per year).

Thus the loss of the company is then the cost of $\$100,000$that the company needs to pay decreased by the income of the 3 payments.

$\$100,000-3\times \$250=\$99,250$

The remaining amount is $\$99,250$ and thus the company suffers a loss of $\$99,250$ on this client.

Explanation (Part a)

Profit $24$ years old

If the client dies at age $25$, then the insurance company needs to pay out the face value of $\$100,000$.

However, since the client closed the life insurance when he was $21$, the client already made $4$payments of $\$250$each (one payment per year).

Thus the loss of the company is then the cost of $\$100,000$that the company needs to pay decreased by the income of the 5 payments.

$\$100,000-4\times \$250=\$99,000$

The remaining amount is $\$99,000$ and thus the company suffers a loss of $\$99,000$ on this client.

Explanation (Part a)

Profit $25$years old

If the client dies at age $25$, then the insurance company needs to pay out the face value of $\$100,000$.

However, since the client closed the life insurance when he was $21$, the client already made $5$payments of $\$250$each (one payment per year).

Thus the loss of the company is then the cost of $\$100,000$ that the company needs to pay decreased by the income of the 5 payments.

$\$100,000-5\times \$250=\$98,750$

The remaining amount is $\$98,750$ and thus the company suffers a loss of $\$98,750$ on this client.

Explanation (Part a)

Profit $\$26+\$$years old

If the client dies at age $\$26+\$$, then the insurance company does not need to pay out the face value of $\$100,000$as the client died outside the range to which the contract applies.

However, since the client closed the life insurance when he was $21$, the client already made $5$payments of $\$250$each (one payment per year).

Thus the profit of the company is then the income of the $5$payments.

$5\times \$250=\$1,250$

Thus the company makes a profit of $\$1,250$ on this client.

Final Answer (Part a)

The missing values of $Y$:

Given Information (Part b)

Here is a partially completed table that shows information about the risk of mortality and the values of $Y=$ profit earned by the company

Explanation (Part b)

To find the probabilities of missing values:

$P(Y=-\backslash \$99,750)=0.00183$$P(Y=-\backslash \$99,500)=0.00186$$P(Y=-\$99,250)=0.00189$

$P(Y=-\$99,000)=0.00191$$P(Y=-\$98,750)=0.00193$

The sum of all probabilities in a probability distribution needs to be equal to l, thus the missing probability can then be determined by subtracting all known probabilities from $1 .$