Understanding Two Different Constant Force of Mortality Models in Life Insurance

What Is the Constant Force of Mortality?

The constant force of mortality (CFM) is a mathematical assumption used in life insurance to model the probability of death at any age as a constant exponential rate. Under CFM, the survival function S(x) = e^{-\mu x}, where \mu is the force of mortality, stays unchanged over the interval considered. This simplification makes premium calculations tractable while still capturing the essential risk of mortality.

Why Use More Than One Constant Force Model?

Insurers often apply different CFM values for distinct age ranges or product types to reflect varying risk profiles. Using two separate constants—typically a lower \mu for younger ages and a higher \mu for older ages—balances accuracy with computational ease. The approach helps price term policies, whole‑life contracts, and annuities more realistically than a single universal constant.

Model 1: Single‑Segment Constant Force

In the single‑segment model, a single \mu applies to the entire lifespan considered. This is common in academic examples and simple term‑life pricing where the policy term is short.

Key characteristics

• Uniform death intensity across all ages.
• Easy closed‑form formulas for net premium, reserve, and surrender value.
• May under‑price policies that span wide age ranges.

Formula recap

Survival: S(x)=e^{-\mu x}Probability of death in (x, x+1]: q_x = 1 - e^{-\mu}.

Model 2: Dual‑Segment Constant Force

The dual‑segment model splits the age range into two intervals, each with its own constant force: \mu_1 for ages 0‑k and \mu_2 for ages k+1‑ω. The breakpoint k is chosen based on empirical mortality tables.

Advantages

• Better fits observed mortality curves, especially the steep increase after middle age.
• Retains analytical simplicity within each segment.
• Improves premium adequacy for long‑term contracts.

Mathematical expression

Survival function:S(x)=\begin{cases}e^{-\mu_1 x}, & 0\le x\le k \\ e^{-\mu_1 k}\,e^{-\mu_2 (x-k)}, & k< x\le \omega \end{cases}

Practical Impact on Premium Calculation

Premiums are derived from the expected present value (EPV) of future death benefits. Using the dual‑segment CFM, the EPV is the sum of two integrals, each with its own \mu. The result is a higher premium for policies covering ages beyond the breakpoint because \mu_2 > \mu_1.

Example comparison (illustrative)

Numbers are illustrative; actual rates depend on company experience and underwriting.

When to Choose Each Model

Single‑segment CFM works best for:

• Short‑term policies (e.g., 5‑year term).
• Preliminary pricing or educational purposes.
• Products where age‑specific risk variation is minimal.

Dual‑segment CFM is preferred for:

• Whole‑life or endowment policies covering a wide age span.
• Annuity pricing where longevity risk rises sharply after retirement age.
• Regulatory environments that require more granular mortality assumptions.

Limitations and Alternatives

Both constant‑force models are simplifications. Real mortality follows a Gompertz or Makeham curve, which captures the exponential increase in death rates with age more accurately. Advanced actuarial software now fits piecewise or parametric curves to large mortality datasets, reducing reliance on constant forces. Nevertheless, CFM remains a useful teaching tool and a baseline for quick calculations.

Key Takeaways

1. The constant force of mortality assumes a steady exponential death rate, simplifying actuarial formulas.2. A single constant (Model 1) is easy but may misprice long‑duration contracts.3. Splitting the force into two constants (Model 2) improves realism while preserving analytic tractability.4. Premiums, reserves, and policy values shift upward when the higher‑age force \mu_2 is larger.5. For precise pricing, insurers often move beyond constant forces to Gompertz, Makeham, or cohort‑based tables.