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 insure the $50,000 diamond and expect a gain of $1,000 per year.

Step-by-step solution

Understand the expected value formula

The expected value (EV) formula can be used to calculate the expected gain or loss from a given situation. In this case, we will use it to find the premium (D) that the insurance company should charge. The formula for expected value is given by: EV = Σ [P(x) * x] where P(x) is the probability of each outcome and x is the monetary value associated with each outcome.

Set up the expected value equation for the given insurance scenario

In this problem, we have two possible outcomes: the diamond is either lost with a probability of 0.01 or not lost with a probability of 0.99. If the diamond is lost, the insurance company pays out $50,000 and if not, they receive the premium D. So the expected value equation becomes: EV = P(Loss) * -($50,000) + P(No Loss) * D

Plug in the given probability and expected gain

We know that the probability of loss is 0.01 and we want the expected gain for the insurance company to be $1,000. So, we can plug these values into the expected value equation: \(1,000 = (0.01) * -(\)50,000) + (0.99) * D

Solve for the premium, D

Now we just need to solve the equation for D: \(1,000 = -(\)500) + (0.99) * D $1,500 = 0.99D D = \(\frac{1,500}{0.99} = \$1,515.15\)

Interpret the result

The insurance company should charge a premium of approximately \(1,515.15 to insure the \)50,000 diamond and expect a gain of $1,000 per year.

Key concepts

Expected Value

The concept of expected value (EV) is central to understanding insurance premium calculation and many other financial decisions. In its essence, the expected value is a measure of the center, or average, of a probability distribution and represents what one can expect to happen on average over a large number of trials.

Let's simplify this through an example related to our insurance problem. An insurance company assesses the risk of a diamond being lost and uses this risk to calculate a fair premium. In our exercise, the expected value is the insurance company's expected gain from insuring the diamond. The formula for calculating EV incorporates all possible outcomes, their monetary values, and their probabilities. Formally it is expressed as:

EV = \(\sum [P(x) \times x]\)

where \(P(x)\) is the probability of each outcome and \(x\) is the value associated with that outcome. In terms of an insurance premium, the EV helps insurers set a price that balances their expected payouts against their income from premiums plus their desired profit margin.

Probability

Probability is a way of quantifying the likelihood of an event occurring. It’s a number between 0 and 1, where 0 means the event is impossible and 1 means the event is certain. In insurance, understanding probability is fundamental to estimating how often an event, like the theft or loss of a diamond, might happen.

Mathematically, probability is calculated by dividing the number of ways an event can occur by the total number of possible outcomes. For our diamond insurance problem, there are two outcomes: the loss of the diamond (which has a small probability, 0.01) and the diamond remaining safe (with a high probability, 0.99).

The insurance company uses this probability to gauge risk and set premiums accordingly. Essentially, the lower the probability of an adverse event happening, the lower the premium can be. In contrast, higher risk leads to higher premiums to cover the potential payouts. The company aims to ensure that the premium they set will cover the expected cost of claims over time, considering the probability of each outcome.

Statistical Analysis

Statistical analysis in the context of insurance involves using a wide array of mathematical techniques to analyze past data and make predictions about future events. This analysis is crucial for determining risk and setting premiums that are fair and profitable for the insurance company.

Insurers gather large amounts of data on claims, losses, and policyholder behavior. They then use statistical models to assess the frequency and severity of claims associated with different risks. For example, in our diamond insurance problem, the company might study a large number of cases involving diamond losses to validate the 0.01 probability of loss.

Moreover, statistical analysis can reveal trends and patterns that might not be immediately apparent. It allows insurers to create more accurate risk profiles and segment their customer bases into different risk categories. Predictive models based on statistical methods can help insurers adjust their policies in anticipation of changing risk landscapes, ensuring they remain competitive and financially stable over time.