A Comprehensive Guide to Creating New Insurance Products for Startup Founders Introduction

Example for illustrative purposes

Scenario

• Warranty Coverage: A car manufacturer offers a warranty on its vehicles, covering engine repairs due to mechanical failures within the first 3 years or 36,000 miles, whichever comes first.
• Historical Data: The car manufacturer has records of engine repair claims made by customers over the past several years.
• Goal: Calculate the probability that a customer will make a claim for engine repairs during the warranty period.

Data Variables:

• Total Number of Vehicles Sold: Let's assume the car manufacturer has sold 100,000 vehicles in the last year.
• Number of Engine Repair Claims: From historical data, they find that 2,000 engine repair claims were made under warranty during the first 3 years or 36,000 miles.

Probability Calculation

The probability of a customer making an engine repair claim during the warranty period can be calculated as follows:
Probability of Claim=Number of Claims/Total Number of Vehicles Sold

In this case:
Probability of Claim=2,000/100,000=0.02

So, the probability of a customer making an engine repair claim on their car warranty is 0.02, or 2%. This means that, based on historical data and the current vehicle sales rate, there is a 2% chance that a customer will make an engine repair claim during the warranty period.

Data Variables

• Total Number of Engine Repair Claims: From historical data, the car manufacturer found that there were 2,000 engine repair claims made under warranty during the first 3 years or 36,000 miles.
• Total Cost of Engine Repairs: They also have data on the total cost of these repairs. Let's say the total cost was $2,000,000.

Severity Calculation: The severity of loss, often denoted as S, can be calculated as follows:
S=Total Cost of Claims/Number of Claims.

In our example:
S=$2,000,000/2,000=$1,000

So, the severity of loss for engine repair claims under this car warranty product is $1,000 on average. This means that, based on historical data, the average cost of an engine repair claim made by a customer during the warranty period is $1,000.

Data Variables

• Total Number of Engine Repair Claims: From historical data, the car manufacturer found that there were 2,000 engine repair claims made under warranty during the first 3 years or 36,000 miles.
• Total Cost of Engine Repairs: They also have data on the total cost of these repairs, which was $2,000,000.
• Policy Limit: The car manufacturer's warranty policy specifies a policy limit for engine repairs, which is $5,000 per claim.
• Indemnity Function Calculation: The indemnity function, often denoted as k, represents the amount the insurer will reimburse the policyholder in case a specific type of incident (engine repair claim) occurs. In this case, it's incident-specific since we're focusing on engine repair claims.

To calculate the indemnity function, we need to consider the policy limit. If the cost of a repair exceeds the policy limit, the insurer only pays up to the policy limit, and the policyholder is responsible for the remainder. If the cost is below the policy limit, the insurer pays the full cost.

Here's how we calculate the indemnity function for this car warranty product:

For each engine repair claim (let's call it claim k), the indemnity function I_k(X) is defined as:
Ik(X) =
    if Xk ≤ $5,000 then X = $5,000
    else if Xk > $5,000 then X = $5,000

In this formula:

• \(X_k\) represents the cost of the \(k\)-th engine repair claim.
• If \(X_k\) (the cost of the claim) is less than or equal to $5,000 (the policy limit), the indemnity is $5,000 because that's the maximum amount covered.
• If \(X_k\) is greater than $5,000, the indemnity is equal to the actual cost of the repair (\$X_k).

This indemnity function ensures that the policyholder is reimbursed up to the policy limit in case of engine repair claims, while any costs exceeding that limit are not covered by the warranty. It's important to note that such functions can vary widely based on the terms and conditions of the insurance policy or warranty.

Data Variables

• Total Number of Engine Repair Claims: From historical data, the car manufacturer found that there were 2,000 engine repair claims made under warranty during the first 3 years or 36,000 miles.
• Total Cost of Engine Repairs: They also have data on the total cost of these repairs, which was $2,000,000.
• Policy Limit: The car manufacturer's warranty policy specifies a policy limit for engine repairs, which is $5,000 per claim.

Retained Loss Calculation: The retained loss is essentially the portion of the repair cost that the policyholder is responsible for. It's calculated as the difference between the total cost of engine repairs and the total payout by the insurance company.

Here's how you calculate the retained loss:

Calculate the Total Payout by the Insurance Company

• In our example, the insurance company pays $5,000 for each engine repair claim that falls within the policy limit.
• So, the total payout by the insurance company for all 2,000 claims is:
  2,000 claims×$5,000=$10,000,000

Calculate the Total Cost of Engine Repairs

We already know from our data that the total cost of engine repairs is $2,000,000.

Calculate the Retained Loss

The retained loss is the difference between the total cost of engine repairs and the total payout by the insurance company:
Retained Loss=Total Cost of Engine Repairs−Total Payout by the Insurance Company
Retained Loss=$2,000,000−$10,000,000=−$8,000,000
(Note: The negative sign indicates that the insurance company paid out more than the total cost of repairs. This can happen if many claims were below the policy limit, and the insurance company paid the full policy limit for each.)

In this case, the retained loss is -$8,000,000, which means that the insurance company paid out $8,000,000 more than the total cost of engine repairs. This could be due to a combination of factors, including claims that were well below the policy limit and claims that reached the policy limit.

Data Variables

• Total Number of Engine Repair Claims: 2,000 claims.
• Total Cost of Engine Repairs: $2,000,000.
• Policy Limit: $5,000 per claim.

Assumptions

• Let's assume that the insurance company wants to achieve a target loss ratio of 80%. This means that they want premiums to cover 80% of expected losses, with the remaining 20% accounting for administrative costs and profit.
• The loss ratio is calculated as:
  Loss Ratio=Total Payout by the Insurance Company/Premiums Collected.
• Loss ratio of 80% implies that the insurance company intends to pay out 80% of collected premiums as claims.

Premium Calculation

We'll start by calculating the expected losses and then determine the premiums needed to achieve the target loss ratio.

Calculate Expected Losses

• The expected loss for each claim can be calculated as the average cost of engine repairs:
  $2,000,000/2,000claims=$1,000.
• So, the expected total losses for 2,000 claims are:
  2,000claims×$1,000=$2,000,000.

Calculate Premiums Needed for an 80% Loss Ratio

• We know that the loss ratio is the ratio of total losses to premiums collected. In this case, we want the loss ratio to be 80%, so:
  Loss Ratio=0.80.
• We can rearrange the formula to calculate premiums:
  Premiums Collected=Total Losses/Loss Ratio.
• Substituting in the values:
  Premiums Collected=$2,000,000/0.80=$2,500,000.

So, the insurance company needs to collect $2,500,000 in premiums to achieve an 80% loss ratio. This amount should be sufficient to cover the expected losses of $2,000,000 and leave a margin for administrative costs and profit.

To justify the expected loss ratio of 80%, the insurance company will need to set the premium for each policy accordingly. The premium per policy would depend on factors like the car's make and model, driving history, and other risk factors. The total premiums collected from all policyholders should add up to $2,500,000, ensuring that expected losses are covered while achieving the desired loss ratio.

Data Variables

• Total Number of Engine Repair Claims:
  2,000 claims.
• Total Cost of Engine Repairs:
  $2,000,000.
• Calculation of Incident-Specific Loss Severity:
  We'll calculate the loss severity for an individual engine repair claim.
• Calculate the Average Loss Severity per Claim:
  The average loss severity per claim is calculated by dividing the total cost of engine repairs by the total number of claims.
  Average Loss Severity = Total Cost of Engine Repairs / Total Number of Claims
Average Loss Severity = $2,000,000 / 2,000 claims
Average Loss Severity = $1,000 per claim

This means that, on average, for each engine repair claim made under the warranty, the insurance company pays out $1,000. This is the incident-specific loss severity for engine repair claims in this example.

It's important to note that this is an average, and actual claim amounts may vary from claim to claim. Insurance companies use such averages to help set premiums and determine the financial reserves needed to cover potential claims.

Examples:

• Kolmogorov-Smirnov Test:: this test checks if a dataset follows a specific distribution (e.g., normal) or not. Actuaries use it to assess if the loss data conforms to a known distribution.
• Chi-Square Test: Used to compare observed and expected frequencies within contingency tables. Actuaries might use it for testing relationships between variables)

Examples

• Log-Normal Distribution Model: Frequently used for modeling loss severities as it accommodates positively skewed data, which is common in insurance claims.
• Pareto Distribution Model: Sometimes employed to model extreme loss events or high-value claims.
• Generalized Linear Models (GLMs): These models are versatile and can be adapted to different loss severity distributions, depending on the type of data).

After selecting a distribution model (e.g., log-normal), actuaries estimate the parameters of that distribution (e.g., mean and standard deviation) to best fit the data. For example, they may fit a log-normal distribution to actual loss data.

• Akaike Information Criterion (AIC): Actuaries use AIC to compare different models' goodness of fit. Lower AIC values indicate better-fitting models.
• Bayesian Information Criterion (BIC): Similar to AIC, BIC is used for model comparison. It also penalizes complex models.
• Out-of-Sample Validation: Actuaries often divide their dataset into training and validation sets. They build models on the training set and validate them on the unseen validation set to assess predictive accuracy.

• Analysis of Variance (ANOVA): Actuaries might use ANOVA to assess whether adding more variables to a model improves its explanatory power. For example, it can test if adding policyholder age as a variable significantly enhances the model's ability to explain variance in losses.
• Chi-Square Goodness-of-Fit Test: Used to assess if observed data fits a hypothesized distribution. For instance, actuaries might check if actual claim counts match the expected counts under a certain distribution.
• Residual Analysis: Actuaries often examine the residuals (the differences between actual and predicted values) to identify patterns or trends that the model may have missed