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 gives a 50% chance of profit on any single policy, the insurer relies on the law of large numbers.

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 $-P_x/d$.

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

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

If $K_x<n$: $L_0=v^{K_x+1}-P\cdot {\ddot{a}}_{\overline{K_x+1|}}$. If $K_x\ge n$: $L_0=-P\cdot {\ddot{a}}_{\overline{n|}}$ (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 2,349 words, 16 min read) adds 5 worked examples, all 5 common mistakes, a self-check, free in the app.

Learning objectives

• 10a

Browse all free Exam FAM lessons or jump into free Exam FAM practice questions.