Net Premium and Gross Premium Policy Values

Problem Preview: $A_{40}=0.20$, $A_{50}=0.30$, $d=0.05$. The reserve at time 10 is $1-14.0/16.0=0.125$. For a $100,000 policy, the insurer holds $12,500 per survivor. Using $P_{50}$ instead of $P_{40}$ gives zero, the most common error.

HIGH-FREQUENCY: The prospective net premium reserve formula is the most frequently tested formula in all of Topic 10.

$${}_t{V}_x=A_{x+t}-P_x\cdot {\ddot{a}}_{x+t}$$

The premium $P_x$ is fixed at issue. At $t=0$: $A_x-P_x{\ddot{a}}_x=0$ under equivalence.

TRAP: The premium $P_x$ is locked at the issue age. Using the attained-age premium $P_{x+t}$ gives $A_{x+t}-P_{x+t}{\ddot{a}}_{x+t}=0$ at every duration, the most common reserve error.

$${}_t{V}_x=1-\frac{{\ddot{a}}_{x+t}}{{\ddot{a}}_x}$$

KEY: The annuity ratio ${}_t{V}_x=1-{\ddot{a}}_{x+t}/{\ddot{a}}_x$ is the fastest reserve shortcut on the exam, it requires no insurance values at all.

Common mistakes

• Using $P_{x+t}$ instead of $P_x$. The premium is fixed at issue. Using the attained-age premium gives $A_{x+t}-P_{x+t}{\ddot{a}}_{x+t}=0$. Trap: getting zero at every duration.
• Inverting the annuity ratio. Correct: $1-{\ddot{a}}_{x+t}/{\ddot{a}}_x$. Inverted: $1-{\ddot{a}}_x/{\ddot{a}}_{x+t}$ gives a negative reserve.
• Wrong age in the recursion. Stepping from ${}_{t}V$ to ${}_{t+1}V$ uses $q_{x+t}$, not $q_{x+t+1}$. The mortality applies to the year just starting.

Bottom line

• Prospective: ${}_{t}V=A_{x+t}-P_x\cdot {\ddot{a}}_{x+t}$ (future benefits minus future premiums at attained age, premium fixed at issue)
• Annuity ratio: ${}_{t}V=1-{\ddot{a}}_{x+t}/{\ddot{a}}_x$, fastest exam shortcut, no insurance values needed
• Recursion: $({}_{t}V+P)(1+i)=q_{x+t}+p_{x+t}\cdot {}_{t+1}V$ connects consecutive reserves
• Boundaries: ${}_{0}V=0$, ${}_n{V}^{\text{term}}=0$, ${}_n{V}^{\text{endowment}}=1$

Exam shortcut

Decide which formula to use immediately: prospective if you have $A$ and $\ddot{a}$ at both ages, annuity ratio if you have only annuities, recursion if you have consecutive reserves and mortality. Write ages vs. durations before computing, the most common error is mismatching $t$ and $x+t$. "Future minus future, both at the attained age, but the premium was locked at issue." "Hold, Collect, Earn, Pay out" for the recursion.

The full lesson (about 2,831 words, 19 min read) adds 6 worked examples, all 5 common mistakes, a self-check, free in the app.

Learning objectives

• 10c

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