Question

You can insure a \(\$ 50,000\) diamond for its total value by paying a premium of \(D\) dollars. If the probability of loss in a given year is estimated to be .01 , what premium should the insurance company charge if it wants the expected gain to equal \$1000?

Short answer

Answer: The insurance company should charge a premium of approximately $1,515.15 to achieve an expected gain of $1,000 per year.

Step-by-step solution

Write down the given information

We are given the following information: - Value of the diamond: \$50,000 - Probability of loss in a given year: 0.01 - Desired expected gain for the insurance company: \$1,000

Develop the expected gain formula

We know the expected gain formula is as follows: Expected gain = (Probability of no loss)*(Premium) - (Probability of loss)*(Payout) Since we want the expected gain to be $1000, we can rewrite the formula as: 1000 = (Probability of no loss)*(Premium) - (Probability of loss)*(Payout)

Plug in the given probabilities

Since the probability of loss is 0.01, the probability of no loss would be (1 - Probability of loss) = 0.99. We also know that the payout would be the full value of the diamond, which is \$50,000, so we can plug these values into the formula: 1000 = (0.99)*(Premium) - (0.01)*(\$50,000)

Solve for the premium

We can now solve the equation for the premium (D): 1000 = (0.99)*D - 500 Adding 500 to both sides gives: 1500 = 0.99*D Dividing both sides by 0.99 gives: D ≈ \$1,515.15

Interpret the result

The insurance company should charge a premium of approximately \$1,515.15 for insuring the diamond to achieve an expected gain of \$1,000 per year.

Key concepts

Insurance Premium

An insurance premium is an amount of money you pay to an insurance company in exchange for coverage over certain risks. This payment typically occurs on a regular basis, like monthly or annually. In this exercise, the premium is the sum paid to protect a valuable item like a diamond against potential loss or damage.

For the insurer, setting the right premium is a balancing act. They must ensure it covers the risk of a payout while also providing them with a profit. Premiums are calculated based on information such as:

• The value of the item being insured
• The probability of a loss occurring
• The desired profit (or expected gain) for the insurer

In this scenario, the goal was to find the appropriate premium that results in an expected gain of $1,000 for the insurer.

Probability of Loss

The probability of loss refers to the likelihood that an insured event, such as the theft or destruction of the diamond in this exercise, will occur. It's a critical component in determining insurance premiums.

This probability is expressed as a number between 0 and 1, where 0 means there's no chance of loss and 1 indicates certainty of loss. In this example, a 0.01 probability of loss means there's a 1% chance that the diamond will be lost or damaged within a year. The insurance company uses this probability to calculate both risk and appropriateness of the premium.

Understanding probability helps insurers estimate their potential liability or payout, allowing them to adjust their premium rates effectively.

Expected Value Formula

The expected value formula in insurance deals with calculating the anticipated value of future gains or losses. It enables insurers to make informed pricing decisions for their policies. The formula in this exercise aims to establish what the insurance company stands to gain:

\[\text{Expected Gain} = (\text{Probability of No Loss} \times \text{Premium}) - (\text{Probability of Loss} \times \text{Payout})\]In the exercise, the expected gain is targeted to be \(1,000. The only unknown is the premium.

• Probability of no loss is calculated as 1 minus the probability of loss, giving us 0.99.
• Payout is the total value of the item, in this case, \)50,000.

This equation helps determine the premium by solving for it, ensuring the calculated amount provides the insurer with the desired gain while covering potential claims.