Maximizing Whole Life Insurance: Finding The Elusive Q X

Whole life insurance is a type of permanent life insurance that remains in effect for the entirety of the insured's life, provided that premium payments are maintained. In the context of a fully discrete whole life insurance policy, the calculation of q x, which represents the probability of death within a given time frame, is essential for determining the associated financial risks and premiums. This value is influenced by various factors, including age, health, and risk factors, and plays a crucial role in the overall evaluation of life insurance policies.

Calculating the standard deviation of S

To calculate the standard deviation of S, we first need to understand what standard deviation is and what data we are working with. Standard deviation is a statistical measure of diversity or variability in a data set. It is a way to quantify how spread out a set of data is. A low standard deviation indicates that the data points are generally close to the mean or average value, while a high standard deviation indicates greater variability in the data points or higher dispersion from the mean.

In the context of the given problem, we are dealing with a fully discrete whole life insurance policy with a value of 1000 issued to (x). We are given the following information:

• 2 Ax = 0.08
• Ax = 0.2
• The annual premium is determined using the equivalence principle.
• S is the sum of the loss-at-issue random variables for 100 such independent policies.

To calculate the standard deviation of S, we need to follow these steps:

• Calculate the mean (average) of the 100 independent policies. This is done by adding up the values of all the policies and dividing by 100.
• For each policy, calculate the deviation from the mean by subtracting the mean from the value of that policy.
• Square the deviations from the mean for each policy.
• Calculate the variance by taking the sum of the squared deviations and dividing it by the number of data points (in this case, 100).
• Take the square root of the variance to obtain the standard deviation of S.

By following these steps and substituting the given values, we can calculate the standard deviation of S for this particular discrete whole life insurance scenario.

It is important to note that standard deviation provides valuable information about the variability and dispersion of data in a given set. In the context of insurance and risk assessment, standard deviation can help quantify the level of uncertainty and risk associated with a particular scenario or investment.

Calculating the value of L

To calculate the value of L, the loss random variable at the time of issue, we need to use the provided information:

• Death benefits are payable at the moment of death.
• Annual premiums of 380 are payable annually for the first 5 years.
• I = 0.04
• Death occurs on March 31, 2011.

Firstly, we need to calculate the total amount of premiums paid. Given that the annual premium is 380 and there are 5 payments, we can multiply these values to find that the total premiums paid are 1900.

Next, we need to determine the time that has passed since the policy was taken out. The policy was a 10-year term insurance policy taken out on January 1, 2003. Given that death occurred on March 31, 2011, we can calculate that approximately 8.25 years have passed.

Now, we can use the formula for the present value of an ordinary annuity to find the value of L. The formula is:

> PV = [[1 - (1 + i)^-n] / i] x P

Where PV is the present value, i is the interest rate, n is the number of periods, and P is the payment amount.

Plugging in the values, we get:

> L = [[1 - (1 + 0.04)^-8.25] / 0.04] x 380

Calculating this expression will give us the value of L, the loss random variable at the time of issue.

Performing these calculations will allow us to determine the value of L and complete the calculation as specified in the homework assignment.

Calculating the benefit reserve

To calculate the benefit reserve for a life insurance policy, there are a few key components to consider. Firstly, we need to understand the concept of life insurance and how it provides risk mitigation in the event of the policyholder's death. The insurance company (insurer) will provide financial compensation to the insured (policyholder) or their beneficiaries in the event of their death. This is known as the death benefit.

Now, let's delve into the calculation of the benefit reserve, which represents the amount of money the insurance company sets aside to fulfil its obligations to the policyholder. The benefit reserve is calculated using net and cost premiums. The insured pays a series of benefit premiums to the insurer, which is equivalent to the sum that will be paid out upon the insured's death or survival to the maturity date. This creates a liability for one party and an asset for the other, and this balancing item is the benefit reserve.

For a fully discrete whole life insurance policy, the benefit reserve can be calculated using the following formula:

> Benefit reserve = Annual premium * (Number of years / Number of payment periods per year)

For example, if the annual premium for a policy is $1,000, and the policy has monthly payment periods, the benefit reserve for one year would be:

> Benefit reserve = $1,000 * (1 year / 12 months) = $83.33

This calculation assumes that the insurance company receives the premium at the beginning of the year and pays out the benefit at the end of the year.

In another example, let's consider a two-year term insurance policy with a face value of $400. Given the following information:

• I = 0.1 (interest rate)
• Qx = 0.17 (probability of death in the first year)
• Qx+1 = 0.25 (probability of death in the second year)

To calculate the benefit reserve for a life who has survived to the end of the first policy year, we can use the following formula:

> Benefit reserve = Face value * (1 - qx)

Plugging in the values, we get:

> Benefit reserve = $400 * (1 - 0.17) = $332

So, the benefit reserve for this policy, after the insured has survived the first year, is $332.

These calculations are essential for insurance companies to ensure they have sufficient reserves to meet their future obligations to policyholders, especially in the event of unexpected claims or higher-than-expected mortality rates.

Determining the annual premium

The annual premium for a whole life insurance policy can be determined using the equivalence principle. This principle allows for the calculation of the standard deviation of the sum of loss-at-issue random variables for a set number of independent policies. In the given example, we are calculating the annual premium for 100 such policies, each with a value of 1000.

The first step is to identify the given values. We are given that 2 Ax = 0.08, Ax = 0.2, and we want to determine the annual premium, which we can denote as P.

Next, we need to understand the concept of the equivalence principle. This principle states that the expected value of the sum of loss-at-issue random variables is equal to the sum of the expected values of each random variable. In mathematical terms, this can be represented as:

> E [S] = E [X1] + E [X2] + ... + E [Xn]

Where:

• E [S] is the expected value of the sum of loss-at-issue random variables (S)
• E [X1], E [X2], ..., E [Xn] are the expected values of each individual random variable (X1, X2, ..., Xn)

In our case, we are dealing with 100 independent policies, so n = 100.

Now, we can apply the given values to the formula. The expected value of each random variable (E [Xi]) is equal to the probability of that value occurring multiplied by the value itself. So, for our given values:

> E [X1] = 2 Ax * 0.08 = 0.16

> E [X2] = Ax * 0.2 = 0.2

Plugging these values into the formula, we get:

> E [S] = 0.16 + 0.2 + ... + 100*E [X100]

Since we are calculating the standard deviation, we need to find the variance first. The variance of the sum of independent random variables is equal to the sum of their variances. So, we need to calculate the variance of each random variable (Xi) and then sum them up.

The variance of a random variable (Xi) is calculated as the expected value of the square of the deviation of that random variable from its expected value. Mathematically, it can be represented as:

> Var(Xi) = E [(Xi - E [Xi])^2]

Applying this formula to our given values:

> Var(X1) = E [(X1 - E [X1])^2] = E [(0.08 - 0.16)^2] = 0.0032

> Var(X2) = E [(X2 - E [X2])^2] = E [(0.2 - 0.2)^2] = 0

Calculating the variance for each random variable and summing them up, we can find the variance of S.

Finally, to determine the annual premium, we can use the standard deviation formula, which is the square root of the variance. Taking the square root of the variance of S will give us the standard deviation, which represents the annual premium for the whole life insurance policy.

In this case, the annual premium (P) would be the square root of the sum of the variances of each random variable (Xi), which can be denoted as Var(Xi). So, the formula for the annual premium would be:

> P = √(Var(X1) + Var(X2) + ... + Var(X100))

By following these steps and calculating the variances and sums accordingly, we can determine the annual premium for the given whole life insurance policy using the equivalence principle.

Understanding the equivalence principle

The equivalence principle is a key concept in insurance and finance, particularly when it comes to calculating premiums and benefits for life insurance policies. In the context of your question, the equivalence principle is applied to determine the annual premium for a fully discrete whole life insurance policy.

This principle establishes a relationship between the expected present value of future benefits and the expected present value of future premiums. In simple terms, it ensures that the value of future benefits matches the value of future premiums, creating a balanced insurance contract.

In the given scenario, you are provided with a fully discrete whole life insurance policy with a face value of $1000 issued to an individual. To calculate the annual premium using the equivalence principle, you need to consider the expected present value of future benefits and future premiums.

The expected present value of future benefits takes into account the probability of the insured individual's death in each year of the policy and the corresponding death benefit payable. On the other hand, the expected present value of future premiums considers the probability of the insured individual surviving each year and the annual premium amount.

By applying the equivalence principle, you can set the annual premium such that the expected present value of future benefits equals the expected present value of future premiums. This ensures that the insurance company collects sufficient premiums to cover the potential payout in the event of the insured's death.

In summary, the equivalence principle is a fundamental concept in insurance that ensures a fair and balanced contract between the insurer and the insured. By applying this principle, insurance companies can determine appropriate premium rates that align with the expected benefits over the life of the policy. This calculation takes into account mortality rates, investment income, and other factors to ensure the solvency and sustainability of the insurance provider.

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