Understanding Continuous Whole Life Insurance Through De Moivre's Theory

What Is Continuous Whole Life Insurance?

Continuous whole life insurance is a type of permanent coverage that provides a death benefit for the insured's entire lifetime, with premiums and benefits calculated on a continuous (often annual) basis rather than at discrete intervals. Unlike term policies, it never expires, and the cash value grows over time. The "continuous" aspect refers to the mathematical treatment of mortality and interest as smooth functions, which simplifies valuation and pricing.

De Moivre's Contribution to Actuarial Science

Abraham de Moivre (1667‑1754) was a French mathematician whose work laid the foundation for modern actuarial science. His most famous result, the "De Moivre law of mortality," assumes that a population's probability of dying is uniform over a fixed age interval. This linear mortality assumption enables closed‑form formulas for life‑contingent values, including continuous whole life insurance.

Key Assumptions in the De Moivre Model

The De Moivre model simplifies real‑world mortality with three core assumptions:

• All lives are uniformly distributed between ages x and ω (the limiting age, often set at 100).
• The force of mortality (μ) is constant: μ = 1/(ω‑x).
• Interest is compounded continuously at a constant rate δ.

Valuing a Continuous Whole Life Policy

Under the De Moivre framework, the present value (PV) of a continuous whole life insurance benefit of $1 payable at death is:

PV = \( \int_{0}^{\infty} e^{-\delta t} \; \mu \; e^{-\mu t} dt \)

Because μ and δ are constants, the integral simplifies to:

PV = \( \frac{\mu}{\mu + \delta} \)

This elegant result shows that the policy's value depends only on the ratio of the mortality rate to the combined mortality‑plus‑interest rate.

Example Calculation

Assume a 40‑year‑old with ω = 100, so μ = 1/(100‑40) = 0.0167. If the insurer uses a continuous interest rate δ = 0.03, the present value of a $100,000 death benefit is:

PV = 100,000 × (0.0167 / (0.0167 + 0.03)) ≈ $38,600.

Premium Determination

Premiums are set so that the expected present value of premiums equals the expected present value of benefits, plus expenses and profit loadings. For a level premium P paid continuously, the equation is:

P × \( \frac{1}{\delta} \) = PV + Loadings

Rearranging gives:

P = (PV + Loadings) × δ

Using the example above with no loadings, P ≈ $38,600 × 0.03 ≈ $1,158 per year, paid continuously (or equivalently as a series of very small, frequent payments).

Practical Advantages of the Continuous Approach

• Smoother Valuations: Continuous formulas avoid step‑wise jumps that occur with annual premiums.
• Analytical Simplicity: Closed‑form expressions reduce reliance on large mortality tables.
• Better Risk Management: Insurers can model cash‑flow timing more precisely.

Limitations of the De Moivre Model

While mathematically convenient, the uniform mortality assumption is unrealistic for most populations. Modern actuarial practice uses more granular life tables (e.g., the Society of Actuaries' 2020 CSO tables) and stochastic interest models. Nevertheless, De Moivre's framework remains a valuable teaching tool and a baseline for quick approximations.

Comparing Continuous Whole Life with Other Products

When to Consider a Continuous Whole Life Policy

Policyholders who value:

• Predictable, lifelong death protection.
• A cash‑value component that grows smoothly.
• Premiums that can be aligned with cash‑flow streams (e.g., payroll deductions).

are good candidates. Financial advisors often pair such policies with retirement income strategies because the cash value can be accessed via policy loans.

Key Takeaways

• De Moivre's uniform mortality law provides a simple, closed‑form valuation for continuous whole life insurance.• The present value of a $1 benefit equals μ / (μ + δ), linking mortality directly to interest rates.• Continuous premiums align cash‑flow timing but require careful expense loading.• Modern practice refines the model with detailed tables, yet the De Moivre approach remains an essential conceptual foundation.