Present Value Random Variables for Life Insurance, Endowment, and Annuity Payments | Free Exam FAM Lesson

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 $v = 1/ (1 + i) = e^{- δ}$ . For a life aged $x$ : $T_{x}$ = continuous future lifetime, $K_{x} = ⌊ T_{x} ⌋$ = 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 $K_{x} + 1$ .

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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 $(1 - v^{T_{x}}) / δ$ (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, $Z = v^{m i n (T_{x}, n)}$ .
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

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