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 2,841 words, 19 min read) adds 8 worked examples, all 6 common mistakes, a self-check, free in the app.

Learning objectives

• A6

Browse all free MAS-I lessons or jump into free MAS-I practice questions.