Question

A CEO is considering buying an insurance policy to cover possible losses incurred by marketing a new product. If the product is a complete failure, a loss of \(\$ 800,000\) would be incurred; if it is only moderately successful, a loss of \(\$ 250,000\) would be incurred. Insurance actuaries have determined that the probabilities that the product will be a failure or only moderately successful are .01 and \(.05,\) respectively. Assuming that the CEO is willing to ignore all other possible losses, what premium should the insurance company charge for a policy in order to break even?

Short answer

Answer: The insurance company should charge a premium of $20,500 for the policy in order to break even.

Step-by-step solution

Calculate the expected loss for each outcome

To find the expected loss for each outcome, we need to multiply the loss amount by the respective probability. Expected loss for failure = Loss amount for failure × Probability of failure Expected loss for moderate success = Loss amount for moderate success × Probability of moderate success

Calculate the total expected loss

Add the expected losses to find the total expected loss which the insurance company needs to cover. Total expected loss = Expected loss for failure + Expected loss for moderate success

Determine the break-even premium

The insurance company will need to charge a premium equal to the total expected loss to break even. The break-even premium is the total expected loss. Break-even premium = Total expected loss Now let's perform the calculations.

Calculate the expected loss for each outcome

Expected loss for failure = \(\$ 800,000 \times 0.01 = \$ 8,000\) Expected loss for moderate success = \(\$ 250,000 \times 0.05 = \$ 12,500\)

Calculate the total expected loss

Total expected loss = \(\$ 8,000 + \$ 12,500 = \$ 20,500\)

Determine the break-even premium

Break-even premium = Total expected loss = \(\$ 20,500\) The insurance company should charge a premium of \(\$ 20,500\) for the policy in order to break even.

Key concepts

Probability Theory

Probability theory is a branch of mathematics that deals with the likelihood of different outcomes. When discussing expected value in probability theory, it’s essentially about predicting the average result of an event if it were to be repeated many times. In our exercise, we have two possible outcomes: a complete failure or moderate success.
The probabilities given are 0.01 for failure and 0.05 for moderate success. These probabilities are used to weigh the possible financial loss associated with each outcome.
The expected value is calculated by multiplying each outcome's loss by its probability:

• Expected loss for failure: \(\\( 800,000 \times 0.01 = \\) 8,000\)
• Expected loss for moderate success: \(\\( 250,000 \times 0.05 = \\) 12,500\)

Adding these gives the total expected loss, which is a key step in predicting financial risk.Probability theory applies broadly beyond insurance, influencing fields like finance, science, and technology, where risk and uncertainty are common.

Insurance Mathematics

Insurance mathematics uses mathematical models to assess risk and determine premiums. When an insurance company calculates premiums, it considers the total expected losses and covers the risk by incorporating these calculations into the policy costs.
In this scenario, the expected loss is the sum of the expected costs from each scenario. For the policy to break even, the premium charged must equal this expected loss.
The steps shown in the exercise:

• Calculate loss for each outcome and its probability
• Add them up to find total expected loss
• Set premiums to cover these expected losses

Insurance mathematics ensures that the company neither loses money nor profits from uncertainty alone, maintaining fairness for both insurer and insured. Another factor often considered is the overhead cost, but in this simplified model, only direct calculation of potential claim payouts is considered to determine the break-even premium.

Risk Assessment

Risk assessment involves analyzing potential scenarios to understand possible financial risks and determine strategies to manage these risks. In our given exercise, risk assessment is crucial as it helps predict the potential losses of marketing a new product and setting a fair insurance premium.
Using probability and financial data, companies assess possible outcomes. Here, the potential losses are tied to the product's failure or moderate success rates and their consequences.
By evaluating:

• The probability of each outcome occurring
• The financial impact tied to these outcomes

Executives can decide whether the insurance policy’s premium is worth the potential risk if such scenarios unfold. Proper risk assessment not only advises the CEO on financial strategies but also guides the insurance company in pricing insurance products effectively. Mastery of risk assessment ensures sustainable business decision-making and financial planning.