Variance of an Insurance Present Value¶

Definition¶

For a discounted insurance benefit $Z$ , its variance is $\mathbb{E}[Z^2]-(\mathbb{E}[Z])^2$ $(E [Z])^{2}$ , provided the second moment is finite. For a unit benefit paid once on event $D$ at time $\tau$ , $Z=\mathrm{e}^{-\delta\tau}\indicator_D$ and $Z^2=\mathrm{e}^{-2\delta\tau}\indicator_D$ $e^{- 2 δ τ} 𝟙 1_{D}$ .

Interpretation¶

The second moment uses twice the force of interest because the discount factor is squared. Mortality and the covered event stay the same.

Decreasing curve λ/(λ+s) for λ = 0.02 marking 0.4 at force δ and 0.25 at force 2δ, with a dashed line at 0.4 squared = 0.16 and the gap of 0.09 bracketed as the variance.

For the Examples case (constant force of mortality $\lambda=0.02$ $0.02$ ), the actuarial present value (APV) of unit whole life insurance valued at force of interest $s$ is $\lambda/(\lambda+s)$ $s)$ , the blue curve. The first moment is its value at $\delta=0.03$ $0.03$ , namely $0.4$ ; the second moment is its value at $2\delta=0.06$ $0.06$ , namely $0.25$ , which lies above $(\bar A_x)^2=0.16$ $0.16$ . The gap $0.25-0.16=0.09$ is the variance.

Change $\lambda$ , $\delta$ or the contract: $\mathbb E[Z^2]$ is the same APV evaluated at force $2\delta$ , and the green gap between it and $(\mathbb E[Z])^2$ is $\operatorname{Var}(Z)$ .

Actuarial present value gives the first moment of the discounted benefit. Its variance also needs the APV of the squared payment, evaluated with the squared discount factor; the first moment alone does not determine risk.

Key Results¶

For a unit benefit, a second-moment symbol such as ${}^2\bar A_x$ means valuation at force $2\delta$ , not $(\bar A_x)^2$ .
If annual effective interest is $i$ , the double-force equivalent rate is $(1+i)^2-1$ $1$ , not $2i$ .
For benefit $B$ , the second moment is multiplied by $B^2$ .
If benefits vary with time, both the benefit function and the discount factor must be squared.

Examples¶

For unit whole life insurance with $\lambda=0.02$ and $\delta=0.03$ $0.03$ , the second moment is $0.02/(0.02+0.06)=0.25$ $0.25$ . Variance is $0.25-0.4^2=0.09$ $0.09$ .

Prerequisites¶

Sources¶

Life Insurance · Lecture 4

Aliases: second moment of insurance PV, double-force valuation, variance of present value, variance of a death benefit Keywords: 2A x minus A x squared, valuation at force 2 delta, (1+i)^2 - 1, Var(Z) = E[Z^2] - E[Z]^2, double force of interest Symbols: delta, lambda, tau

Last updated: 17 October 2026 21:53