Calculate simple whole life or annuity problems.

A whole life policy promises $1 at the moment (or year-end) of death, whenever that arrives. A whole life annuity pays $1 per period while the annuitant lives. Pricing either reduces to one move: take expectation of a discounted contingent cash flow.

Setup. Let $T_x$ be the future lifetime of a life age $x$ and $K_x=\lfloor T_x\rfloor$ the curtate (integer-years) lifetime. Survival is ${}_t{p}_x=P(T_x>t)$; deferred mortality is ${}_{k|}{q}_x={}_k{p}_x\cdot q_{x+k}$. Interest enters through $v=1/(1+i)$ discretely and $v^t=e^{-\delta t}$ continuously, where $\delta =\ln (1+i)$.

KEY: Every whole life EPV is the expectation of a single discounted indicator stream. Insurance discounts at the time of death. Annuities sum discount factors over survival times.

Whole life insurance, discrete. Pay $1 at the end of the year of death. The death year is $K_x+1$.

$$A_x=\sum _{k=0}^{\infty }{v}^{k+1}{}_{k|}{q}_x=\sum _{k=0}^{\infty }{v}^{k+1}{}_k{p}_x{q}_{x+k}$$

Whole life insurance, continuous. Pay $1 at the instant of death.

Common mistakes

• Forgetting to discount the death benefit one extra period in the discrete case. $A_x$ uses $v^{k+1}$, not $v^k$, because the benefit is paid at end of year of death. Dropping the extra $v$ overstates $A_x$ by a factor of $1+i$.
• Mixing $d$ and $\delta$. Discrete identity uses $d=i/(1+i)$. Continuous identity uses $\delta =\ln (1+i)$. Plugging $\delta$ into $A_x=1-d{\ddot{a}}_x$ breaks the relationship.
• Treating annuity-immediate as annuity-due. $a_x={\ddot{a}}_x-1$. Forgetting the subtraction overprices an immediate annuity by exactly $1.

Bottom line

• Whole life insurance EPV: $A_x=E[v^{K_x+1}]=\sum v^{k+1}{}_{k|}{q}_x$ (discrete) or ${\bar{A}}_x=E[v^{T_x}]=\int e^{-\delta t}{}_t{p}_x{\mu }_{x+t}dt$ (continuous).
• Whole life annuity-due EPV: ${\ddot{a}}_x={\sum }_{k=0}^{\infty }{v}^k{}_k{p}_x$, with the first payment at time 0.
• Fundamental identity: $A_x=1-d{\ddot{a}}_x$, with $d=i/(1+i)$; continuous version ${\bar{A}}_x=1-\delta {\bar{a}}_x$ with $\delta =\ln (1+i)$.
• Constant force $\mu$ plus force of interest $\delta$: ${\bar{A}}_x=\mu /(\mu +\delta )$ (mu on top), ${\bar{a}}_x=1/(\mu +\delta )$.

Exam shortcut

When the problem hands you both ${\ddot{a}}_x$ and an interest rate, compute $A_x=1-d{\ddot{a}}_x$ in one line rather than re-summing the insurance series. For any constant-force question, write ${\bar{A}}_x=\mu /(\mu +\delta )$ and ${\bar{a}}_x=1/(\mu +\delta )$ immediately; the benefit premium per unit is just $\mu$. When asked for variance of a level-benefit whole life, evaluate the same EPV at force $2\delta$, subtract the square of ${\bar{A}}_x$, and multiply by the face squared.

The full lesson (about 2,079 words, 14 min read) adds 2 worked examples, all 6 common mistakes, a self-check, free in the app.

Learning objectives

• A6

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