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Home / Financial Tools / Actuarial Life Contingencies & Reserve Calculation Engine

Graduate-Level Modeling · Actuarial Science & Life Contingencies

Actuarial Life Contingencies & Reserve Calculation Engine

A life insurance premium isn't looked up in a table. It's solved from a survival curve, an interest rate, and one equation that has to balance exactly, and this tool shows you every step.

How To Use This Model

Reading This Tool

Set a Gompertz-Makeham mortality law, an interest rate, and a policy design, and this tool derives the actuarial present values, net level premium, and full reserve schedule from first principles.

Every number here is generated from the closed-form survival function, not a lookup table, so you can see exactly how a change in mortality assumptions or interest rate flows all the way through to the premium a policyholder pays and the reserve an insurer has to hold.

Mortality Law (Gompertz-Makeham)

Policy Design

The maximum attained age for all calculations is capped at 130, where survival probability is effectively zero under any realistic Gompertz-Makeham parameter set.

Mortality Profile At Issue Age

Annual Mortality Rate qx

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Curtate Life Expectancy

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Probability Of Surviving 10 Years

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Probability Of Surviving 30 Years

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Survival Probability From Issue Age Forward

Annual Mortality Rate By Attained Age

Premium & Reserve At Current Duration

Whole Life

APV Of Benefits (Per $1 Face)

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Annuity-Due Factor

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Net Level Annual Premium

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Reserve At Duration t

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Reserve Build-Up Over The Policy Term

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Net Level Premium By Policy Type (Same Age, Mortality & Interest)

Side-By-Side Comparison

Why The Reserve Has To Start At Exactly Zero

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What The Makeham Constant Is Actually Capturing

The Gompertz term alone would imply near-zero mortality risk at very young ages, which isn't realistic, accidents, violence and other age-independent causes of death set a floor under mortality at every age. The Makeham constant adds that floor, which is why even a healthy young adult has a small but non-trivial annual mortality rate baked into these calculations.

Why Endowment Insurance Costs So Much More Than Term

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Why The Reserve Trajectory Differs So Much By Policy Type

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Actuarial Science & Life Contingencies

The Core Formulas

Force Of Mortality: μ(x) = A + B·Cx (Gompertz-Makeham)
Survival: tpx = exp[−At − BCx(Ct−1)/ln C]
Annuity-Due: äx = Σk=0n vk·kpx
Whole Life Insurance: Ax = Σk=0n−1 vk+1·kpx·qx+k, identity: Ax = 1 − d·äx
Net Level Premium: P = Ax / äx, Reserve: tVx = Ax+t − P·äx+t

When To Actually Use This Model

  • Teaching life contingencies, the mathematical foundation of actuarial science, in an actuarial mathematics or insurance course.
  • Building intuition for how mortality assumptions and interest rates jointly determine a life insurance premium and its reserve.
  • Cross-checking a simplified reserve calculation against a full actuarial valuation system before relying on either.

Key Assumptions & Limitations

  • Real insurers use standardized mortality tables (e.g., CPM2014 in Canada) built from actual population experience, not a smooth parametric law, Gompertz-Makeham is a teaching and approximation tool, not the industry's production mortality basis.
  • This tool ignores expenses, lapses, and selection effects (mortality improvement right after underwriting), all of which materially affect real premium calculations.
  • Reserves shown are net level premium reserves, real statutory and GAAP reserves apply additional margins and expense loadings beyond the pure equivalence-principle calculation shown here.

Foundational References

Bowers, N. L., Gerber, H. U., Hickman, J. C., Jones, D. A., & Nesbitt, C. J. Actuarial Mathematics. Society of Actuaries.

Dickson, D. C. M., Hardy, M. R., & Waters, H. R. Actuarial Mathematics for Life Contingent Risks. Cambridge University Press.

Need the pension funding side of actuarial work too?

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