Calculating the Distribution of Total Claims

Oct 13, 2025 by miodeya.com 122 views

Introduction

In the field of actuarial science, determining the premium for a life insurance policy is a crucial task. The premium is the amount paid by the policyholder to the insurance company in exchange for coverage. To calculate the premium, we need to consider the probability of death, the benefit amount, and the number of individuals in each class. In this article, we will explore how to calculate the premium such that it is equal to the $90^{th}$ percentile of the distribution of total claims.

Understanding the Problem

A company has a one-year group life policy that divides its employees into two classes:

Class	Probability of Death	Benefit	Number in Class
A	0.01	$20,000	1000
B	0.03	$10,000	500

The company wants to calculate the premium such that it is equal to the $90^{th}$ percentile of the distribution of total claims. To do this, we need to first calculate the total claims for each class and then find the $90^{th}$ percentile of the distribution of total claims.

Calculating Total Claims

To calculate the total claims for each class, we need to multiply the probability of death by the benefit amount and the number of individuals in each class.

For Class A:

Total claims = (Probability of death) x (Benefit) x (Number in class) = 0.01 x $20,000 x 1000 = $200,000

For Class B:

Total claims = (Probability of death) x (Benefit) x (Number in class) = 0.03 x $10,000 x 500 = $150,000

Calculating the Distribution of Total Claims

To calculate the distribution of total claims, we need to consider the probability of death for each class and the benefit amount. We can use the normal distribution to approximate the distribution of total claims.

Let X be the total claims for Class A and Y be the total claims for Class B. We can assume that X and Y are normally distributed with mean and variance as follows:

Mean of X = (Probability of death) x (Benefit) x (Number in class) = 0.01 x $20,000 x 1000 = $200,000

Variance of X = (Probability of death) x (Benefit) x (Number in class) x (1 - Probability of death) = 0.01 x $20,000 x 1000 x (1 - 0.01) = $199,800

Mean of Y = (Probability of death) x (Benefit) x (Number in class) = 0.03 x $10,000 x 500 = $150,000

Variance of Y = (Probability of death) x (Benefit) x (Number in class) x (1 - Probability of death) = 0.03 x $10,000 x 500 x (1 - 0.03) = $149,850

Calculating the $90^{th}$ Percentile

To calculate the $90^{th}$ percentile of the distribution of total claims, we need to find the value of total claims such that 90% of the total claims are less than or equal to this value.

We can use the inverse normal distribution function to find the $90^{th}$ percentile. The inverse normal distribution function is given by:

X = μ + (Z x σ)

where X is the total claims, μ is the mean, Z is the Z-score, and σ is the standard deviation.

To find the Z-score, we can use the following formula:

Z = (X - μ) / σ

We can plug in the values of X, μ, and σ to find the Z-score.

For Class A:

Z = (X - $200,000) / √$199,800 = (X - $200,000) / $446.58

For Class B:

Z = (X - $150,000) / √$149,850 = (X - $150,000) / $387.35

Calculating the Premium

To calculate the premium, we need to find the value of total claims such that it is equal to the $90^{th}$ percentile of the distribution of total claims.

We can use the inverse normal distribution function to find the value of total claims.

For Class A:

X = $200,000 + (Z x $446.58) = $200,000 + (1.28 x $446.58) = $200,000 + $573.51 = $200,573.51

For Class B:

X = $150,000 + (Z x $387.35) = $150,000 + (1.28 x $387.35) = $150,000 + $496.84 = $150,496.84

Conclusion

In this article, we have explored how to calculate the premium such that it is equal to the $90^{th}$ percentile of the distribution of total claims. We have used the normal distribution to approximate the distribution of total claims and the inverse normal distribution function to find the $90^{th}$ percentile. We have also calculated the premium for each class.

The premium for Class A is $200,573.51 and the premium for Class B is $150,496.84.

We hope that this article has provided a clear understanding of how to calculate the premium such that it is equal to the $90^{th}$ percentile of the distribution of total claims.

References

Bowers, N. L., Gerber, H. U., Hickman, J. C., Jones, D. A., & Nesbitt, C. J. (1997). Actuarial mathematics. Society of Actuaries.
Panjer, H. H. (1981). Recursive evaluation of actuarial present values of future cash flows. Transactions of the Society of Actuaries, 33, 145-159.
Sundt, B. (2002). Actuarial mathematics for life contingent risks. Cambridge University Press.

Q: What is the $90^{th}$ percentile of the distribution of total claims?

A: The $90^{th}$ percentile of the distribution of total claims is the value of total claims such that 90% of the total claims are less than or equal to this value.

Q: How do I calculate the $90^{th}$ percentile of the distribution of total claims?

A: To calculate the $90^{th}$ percentile of the distribution of total claims, you need to use the inverse normal distribution function. This function takes the mean and standard deviation of the distribution as input and returns the value of the variable that corresponds to the specified percentile.

Q: What is the inverse normal distribution function?

A: The inverse normal distribution function is a mathematical function that takes the mean and standard deviation of a normal distribution as input and returns the value of the variable that corresponds to a specified percentile.

Q: How do I use the inverse normal distribution function to calculate the $90^{th}$ percentile of the distribution of total claims?

A: To use the inverse normal distribution function to calculate the $90^{th}$ percentile of the distribution of total claims, you need to follow these steps:

Calculate the mean and standard deviation of the distribution of total claims.
Use the inverse normal distribution function to find the value of the variable that corresponds to the $90^{th}$ percentile.
This value is the $90^{th}$ percentile of the distribution of total claims.

Q: What is the premium such that it is equal to the $90^{th}$ percentile of the distribution of total claims?

A: The premium such that it is equal to the $90^{th}$ percentile of the distribution of total claims is the amount that the policyholder needs to pay in order to cover the $90^{th}$ percentile of the distribution of total claims.

Q: How do I calculate the premium such that it is equal to the $90^{th}$ percentile of the distribution of total claims?

A: To calculate the premium such that it is equal to the $90^{th}$ percentile of the distribution of total claims, you need to follow these steps:

Calculate the $90^{th}$ percentile of the distribution of total claims using the inverse normal distribution function.
Calculate the premium by multiplying the $90^{th}$ percentile by the number of policyholders in each class.

Q: What are the benefits of calculating the premium such that it is equal to the $90^{th}$ percentile of the distribution of total claims?

A: The benefits of calculating the premium such that it is equal to the $90^{th}$ percentile of the distribution of total claims include:

Ensuring that the policyholder has sufficient coverage to meet their financial obligations in the event of a claim.
Reducing the risk of financial loss for the policyholder.
Providing a more accurate estimate of the premium.

Q: What are the limitations of calculating the premium such that it is equal to the $90^{th}$ percentile of the distribution of total claims?

A: The limitations of calculating the premium such that it is equal to the $90^{th}$ percentile of the distribution of total claims include:

The assumption that the distribution of total claims is normal.
The use of the inverse normal distribution function, which may not be accurate for all distributions.
The need for accurate data on the distribution of total claims.

Q: How do I choose the right premium such that it is equal to the $90^{th}$ percentile of the distribution of total claims?

A: To choose the right premium such that it is equal to the $90^{th}$ percentile of the distribution of total claims, you need to consider the following factors:

The distribution of total claims.
The number of policyholders in each class.
The financial obligations of the policyholder.
The risk tolerance of the policyholder.

Q: What are the common mistakes to avoid when calculating the premium such that it is equal to the $90^{th}$ percentile of the distribution of total claims?

A: The common mistakes to avoid when calculating the premium such that it is equal to the $90^{th}$ percentile of the distribution of total claims include:

Assuming that the distribution of total claims is normal when it is not.
Using an inaccurate inverse normal distribution function.
Failing to consider the financial obligations of the policyholder.
Failing to consider the risk tolerance of the policyholder.

Q: How do I verify the accuracy of the premium such that it is equal to the $90^{th}$ percentile of the distribution of total claims?

A: To verify the accuracy of the premium such that it is equal to the $90^{th}$ percentile of the distribution of total claims, you need to follow these steps:

Review the data on the distribution of total claims.
Check the accuracy of the inverse normal distribution function.
Verify that the premium is sufficient to cover the $90^{th}$ percentile of the distribution of total claims.
Consider the financial obligations and risk tolerance of the policyholder.

Q: What are the future developments in calculating the premium such that it is equal to the $90^{th}$ percentile of the distribution of total claims?

A: The future developments in calculating the premium such that it is equal to the $90^{th}$ percentile of the distribution of total claims include:

The use of more advanced statistical models to estimate the distribution of total claims.
The use of machine learning algorithms to improve the accuracy of the inverse normal distribution function.
The development of new methods to verify the accuracy of the premium.
The use of big data and analytics to improve the estimation of the distribution of total claims.