Present Value Random Variables for Life Insurance, Endowment, and Annuity Payments

Problem Preview: An insurer issues a whole life policy to a 40-year-old. If she dies at 41, $Z = v^1 = 0.952$. If at 95, $Z = v^{55} = 0.067$. The present value of the benefit is not a number, it is a random variable. Every pricing formula starts here.

HIGH-FREQUENCY: Every PV random variable on FAM is a function of either $T_x$ (continuous) or $K_x$ (annual). Knowing which drives which contract type is the first step.

Fix discount factor $v = 1/(1+i) = e^{-\delta}$. For a life aged $x$: $T_x$ = continuous future lifetime, $K_x = \lfloor T_x \rfloor$ = curtate future lifetime.

KEY: $T_x$ drives continuous contracts. $K_x$ drives annual contracts. Mismatching the lifetime variable to the contract type is the most common source of errors.

Continuous: $Z = v^{T_x}$. Annual: $Z = v^{K_x+1}$. The "+1" appears because the benefit is paid at the end of the year of death, time $K_x+1$.

Common mistakes

• Using $K_x$ when the problem calls for $T_x$. Benefit at moment of death = $T_x$. Benefit at end of year = $K_x$. Trap: a value close to but not equal to the correct answer.
• Forgetting the "+1" in annual insurance. The PV is $v^{K_x+1}$, not $v^{K_x}$. Omitting the +1 inflates the answer by $1+i$.
• Confusing annuity-due and annuity-immediate. Due uses $d$ in denominator, makes $K_x+1$ payments. Immediate uses $i$, makes $K_x$ payments. Swapping $d$ and $i$ changes the answer by $1+i$.

Bottom line

• Insurance PV: $Z = v^{T_x}$ (continuous) or $v^{K_x+1}$ (annual). The "+1" matters: benefit is paid at end of year of death.
• Annuity PV: $Y = (1 - Z)/d$ (due) or $(1 - v^{T_x})/\delta$ (continuous). Annuity is a linear function of insurance.
• Term insurance: $Z = 0$ if insured survives. Pure endowment: $Z = 0$ if insured dies. Endowment: always pays, $Z = v^{\min(T_x, n)}$.
• Endowment = term + pure endowment; pure endowment alone carries no death benefit.

Exam shortcut

When you see a contract description, identify three things: (1) benefit at moment of death, end of year, or $m$-thly? This sets $T_x$, $K_x$, or $K_x^{(m)}$. (2) Term limit or whole life? (3) Insurance or annuity? Once classified, the PV random variable writes itself. "TKM: T = continuous, K = annual, M = m-thly." "Endowment = Term + Pure." "Z to Y: divide out the d."

The full lesson (about 2,275 words, 15 min read) adds 4 worked examples, all 5 common mistakes, a self-check, free in the app.

Learning objectives

• 9a

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