Future Loss Random Variables for Life Insurance and Annuities

Problem Preview: Given $A_{45} = 0.25$, ${}^{2}A_{45} = 0.09$, $d = 0.05$. The variance of the loss is $(0.09 - 0.0625)/(0.75)^2 = 0.04889$. For a $100,000 policy, the standard deviation is $22,110. The equivalence principle sets expected loss to zero. A single policy can have an asymmetric loss distribution, so its probability of profit must be calculated from the lifetime model.

HIGH-FREQUENCY: The loss at issue $L_0$ and its expression as a function of future lifetime appear on nearly every FAM sitting.

$$L_0 = \text{PV(benefits)} - \text{PV(premiums)}$$

KEY: $L_0$ is linear in $v^{K_x+1}$, which enables closed-form variance. The slope is $1 + P_x/d$ and the intercept is $-P_x/d$.

$$L_0 = v^{K_x+1} - P_x \cdot \ddot{a}_{\overline{K_x+1\vert}} = v^{K_x+1}\left(1 + \frac{P_x}{d}\right) - \frac{P_x}{d}$$

$$L_0 = v^{T_x}\left(1 + \frac{\bar{P}}{\delta}\right) - \frac{\bar{P}}{\delta}$$

If $K_x < n$: $L_0 = v^{K_x+1} - P \cdot \ddot{a}_{\overline{K_x+1\vert}}$. If $K_x \geq n$: $L_0 = -P \cdot \ddot{a}_{\overline{n\vert}}$ (insurer keeps all premiums, pays no benefit).

Common mistakes

• Confusing $K_x$ and $K_x+1$. The benefit PV is $v^{K_x+1}$, not $v^{K_x}$. Using $K_x$ overstates the PV by $1+i$.
• Using $a$ instead of $\ddot{a}$ for premiums. Premiums in fully discrete policies are paid at the beginning of each year, annuity-due.
• Forgetting the piecewise structure of term insurance. Term loss has two cases: death within the term, and survival. Omitting the survival case (where $L_0 < 0$) gives wrong $E[L_0]$.

Bottom line

• $L_0 = \text{PV(benefits)} - \text{PV(premiums)}$; positive means the insurer loses money.
• $L_0$ is linear in $v^{K_x+1}$ (discrete) or $v^{T_x}$ (continuous), with slope $1+P/d$, enabling closed-form variance.
• Variance shortcut: $\text{Var}(L_0) = ({}^{2}A_x - A_x^2)/(1-A_x)^2$ under the equivalence principle.
• ${}^{2}A_x$ uses interest rate $(1+i)^2-1$, not $2i$.

Exam shortcut

When you see "loss at issue," identify fully continuous ($T_x$, $\bar{a}$, $\delta$) or fully discrete ($K_x$, $\ddot{a}$, $d$) first. Write the general form $L_0 = Z - P \cdot Y$, then substitute. For variance, go directly to $({}^{2}A - A^2)/(1-A)^2$ if the equivalence principle applies. "BMP: Benefits Minus Premiums." "Two A minus A squared, over one minus A squared", say it until it sticks.

The full lesson (about 2,390 words, 16 min read) adds 5 worked examples, all 5 common mistakes, a self-check, free in the app.

Learning objectives

• 10a

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