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Assuming that the age of 40 does not matter and the key is to determine the reserve needed when the insured reaches age 50, the problem involves calculating the actuarial present value (APV) of a combination of insurance contracts. The contracts include a whole life insurance with a benefit of $1000 and a 10-year term insurance with the same benefit. The total APV of this combined contract is given as the sum of their individual APVs, which amounts to 531.14. This calculation considers the timing of premiums and benefits, adjusting for the fact that after 10 years, the insured only needs coverage for an additional 10 years.

At time t=10, the insured pays only for the remaining period, and the APV of the premium annuity starting at that point is calculated based on the discounted value of future premiums. The calculation shows that for P=66, the APV for the premium is approximately 435.182 after necessary deductions. Such calculations require understanding the actuarial present values, discount factors, and how insurance contracts are modeled in discrete time settings. The formulas for the APV incorporate standard actuarial functions such as annuity-due values and life contingencies, reflecting the probability-weighted expected future payments.

For the specific case of a fully discrete whole life insurance issued at age 65, the retrospective formula at the end of ten years mirrors the original formulation because the same value 10V applies when analyzing the contract at that point. Using the prospective formula for the standard insurance, which involves actuarial functions and the current reserves, we find that the actuarial value at this point is approximately 0.183, consistent with the calculation for age 35. This demonstrates how actuarial formulas can be applied consistently across different ages, provided the actuarial assumptions remain constant.

The calculation of the net premium reserve at the end of 20 years involves determining the surrender value (SV) of the insurance policy. Using the provided premium P=3, the reserve is computed as SV= P* (some factor derived from the actuarial present value formulas). The resulting reserve value of approximately 3.889 reflects the accumulated benefits and premiums paid over time, adjusted for mortality and other contingencies. This example underscores the importance of precise actuarial modeling in ensuring sufficient reserve levels for insurance companies to meet future obligations.

Paper For Above instruction

The calculation of insurance reserves involves careful application of actuarial principles, notably the concept of actuarial present value (APV). When considering the insurance policy that matures or changes at a certain age—such as age 50 in this case—the primary goal is to determine the reserve necessary at that age to meet future obligations. The problem involves a combination of two types of insurance: a whole life policy and a 10-year term policy, both with benefits of $1000. The aggregate APV at the relevant age can be derived by summing the APVs of these individual contracts, factoring in the timing of benefits and premiums.

The calculation shows that the total APV of the combined insurance at age 50 is approximately 531.14. This involves discounting future benefits and premiums using actuarial interest rates and mortality assumptions. One essential aspect is the treatment of premiums paid over different periods. For example, after 10 years, the remaining premium payments are discounted back to the valuation date, leading to the computation of the APV of the premium annuity beginning at that time. This calculation is crucial for determining the reserve, as it accounts for the future cash flows the insurer must be prepared to meet.

Furthermore, the analysis of a standard fully discrete whole life insurance issued at age 65 illustrates the application of the prospective formula to compute reserves at different ages. Since the same 10-year value applies in both the original and retrospective contexts, the formula simplifies to calculating the value based on standard actuarial functions. The resulting value, approximately 0.183, reflects the current actuarial reserve necessary at age 35 when considering the policy issued at age 65 and evaluated ten years later. This demonstrates the consistency and transferability of actuarial formulas across different time frames and ages.

In addition, the calculation of the net premium reserve at the end of 20 years emphasizes the role of surrender values and accumulated premiums. Using the given premium P=3, the reserve, which is effectively the surrender value (SV), is computed as a product of the premium and an actuarial factor, resulting in approximately 3.889. This calculation underscores how reserves evolve over time as premiums are paid, and benefits accrue, reflecting the typical actuarial approach to ensuring sufficient reserves for future policyholders.

Overall, the analysis demonstrates the critical importance of actuarial science in insurance mathematics, involving the application of discount functions, mortality assumptions, and the structure of insurance contracts. Actuaries must carefully evaluate future liabilities and premiums to maintain financial stability and meet policyholder obligations. The calculations also highlight the interconnectedness of different types of insurance policies and the importance of consistent valuation methods across different ages and policy terms.

References

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