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An insurance premium is the amount charged by an insurer to provide coverage under an insurance policy. This is the cost you pay for being protected against potential risks or losses, such as damage in the event of an accident. Insurance companies calculate premiums based on several factors, including the level of risk they are taking on.
The goal of the insurer is to collect enough in premiums to cover anticipated payouts, administrative costs, and achieve a desired profit. For instance, if the expected payout (claims to be paid to policyholders) is estimated at $600, and the company wants a $100 profit, they will need to charge a total expected premium of $700.

The premium amount is heavily influenced by the probability distribution of loss events, meaning an insurer must carefully estimate the likelihood and extent of damages they will need to cover. This allows them to price their premiums competitively while covering expected costs.

A deductible policy involves a specified amount that the insured must pay out-of-pocket before insurance coverage takes effect. In simple terms, it is the portion of a claim that a policyholder must pay before the insurer pays the remaining expenses.

Deductible policies help reduce the cost of insurance by decreasing the likelihood and size of payouts. In the example, the insurance company sets a deductible of $500. This means for any claim that exceeds $500, the insured pays the initial $500, and the insurer provides coverage for the remaining amount.
Having a deductible:

• Encourages policyholders to prevent small claims or minor damages since they bear part of the cost.
• Promotes responsible behavior among insured individuals, potentially lowering risks.

For instance, in an accident causing $5,000 of damage, the payout from the insurer would be $5,000 - $500 = $4,500. This setup balances the insured's contribution and the insurer's risk.

Expected payout is a critical metric in insurance as it signifies the average amount an insurer anticipates paying out for claims on a policy. Rather than being a definite figure, it represents a weighted average calculated using the probabilities of different outcomes and the respective payouts.

We calculate the expected payout by summing the product of each payout's likelihood and its monetary value. For example, consider several possible damages: 0, 1000, 5000, and 10000. The expected payout (E[P]) can be calculated as follows: \[ E[P] = 0.8 \times 0 + 0.1 \times 500 + 0.08 \times 4500 + 0.02 \times 9500 = 600 \]Thus, the expected payout is $600 in this scenario. This figure helps insurance companies set premiums that secure enough revenue to cover potential claims.

Probability distribution is a statistical function that illustrates all the possible values and occurrences a random variable can take within a given range. In insurance, it helps in assessing the risk associated with different levels of potential losses.

By understanding the probability distribution of damage amounts, insurers can anticipate expected payouts accurately. This allows them to set only reasonable and profitable premium charges.
In the given scenario, damages and their probabilities are:

• 0 damage with a probability of 0.8
• 1000 damage with a probability of 0.1
• 5000 damage with a probability of 0.08
• 10000 damage with a probability of 0.02

The weighted nature of a probability distribution takes into account how likely each scenario is to occur. Understanding probability distribution is essential for designing financially sound insurance policies and is a core part of actuarial science.