Willys Flashcards Download
Become an Actuary › Exams › Flashcards › Exam FAM › Premiums & Reserves Practice
Exam FAM · practice

Exam FAM — Premiums & Reserves Practice Flashcards

Thirty exam-realistic multiple-choice problems on SOA Exam FAM premiums and reserves — net level premiums by the equivalence principle (term, endowment, limited-pay, fully continuous, and semicontinuous), the loss-at-issue random variable and its variance, prospective and retrospective net premium reserves including the $1-\ddot a_{x+t}/\ddot a_x$ shortcut, the year-by-year reserve recursion with its savings/cost-of-insurance split, and expense-loaded gross premiums — each built from explicit $A$, $\ddot a$, $q_x$, and $i$ inputs with a fully worked solution.

8 free sample30 total · in appFree · fact-checked · LaTeX math
Tap card or press Space to flip
Answer

Unlock the full set

You're studying a free 8-problem sample. All 30 Premiums & Reserves practice problems — plus every other Exam FAM subject and spaced-repetition scheduling — are built into the Willys AI Flashcards & Quizzes app. 14-day free trial, then $14.99.

Get the app — free 14-day trial

Every deck is built into the Willys app

All of these decks — including the full practice problem banks — come built into Willys AI Flashcards & Quizzes for iPhone & iPad (Mac version coming soon), with FSRS + SM-2 spaced repetition, streaks, and exam-date cram mode. 14-day free trial, then $14.99. To load a deck in the app: Settings → Library → Browse, then pick your exam and deck.

Download on the App Store

More Exam FAM decks:

Aggregate Loss Models Aggregate Loss Models Practice Coverage Modifications Coverage Modifications Practice Credibility Credibility Practice

← All Exam FAM decks

Browse all 30 problems as a list
  1. Equivalence principle
    A $3$-year term insurance of $\$100{,}000$ on $(x)$ pays the death benefit at the end of the year of death and is funded by a level net annual premium under the equivalence principle. Mortality is $q_x=0.03$, $q_{x+1}=0.04$, $q_{x+2}=0.05$, and $i=0.06$. Calculate the net annual premium. (A) $\$3{,}715$ (B) $\$4{,}204$ (C) $\$4{,}333$ (D) $\$5{,}845$ (E) $\$11{,}268$
    **Answer: (A).** With $v=\dfrac{1}{1.06}$, survival is ${}_1p_x=0.97$ and ${}_2p_x=0.97(0.96)=0.9312$. APV of benefits $=100{,}000\big[v(0.03)+v^2(0.97)(0.04)+v^3(0.9312)(0.05)\big]$ $\approx 100{,}000[0.028302+0.034532+0.039093]=10{,}192.6$. APV of premiums (annuity-due over $3$ years) $=1+v(0.97)+v^2(0.9312)\approx 2.743859$. $P=\dfrac{10{,}192.6}{2.743859}\approx \$3{,}715$. Dividing the undiscounted benefit APV (summing $100{,}000\,q$ over survivors without time value, $\$11{,}536$) by the annuity gives $\$4{,}204$ (B); using an annuity-immediate for premiums (dropping the time-$0$ payment) inflates $P$ toward $\$5{,}845$ (D).
  2. Equivalence principle
    A $2$-year endowment insurance of $\$1{,}000$ on $(x)$ pays $\$1{,}000$ at the end of the year of death or $\$1{,}000$ on survival to time $2$, funded by a level net annual premium. Mortality is $q_x=0.05$, $q_{x+1}=0.06$, and $i=0.06$. Calculate the net annual premium. (A) $\$45.27$ (B) $\$453.32$ (C) $\$470.76$ (D) $\$498.84$ (E) $\$892.67$
    **Answer: (C).** $v=\dfrac{1}{1.06}$ and ${}_1p_x=0.95$. APV death benefit $=1000\big[v(0.05)+v^2(0.95)(0.06)\big]\approx 1000[0.047170+0.050730]=97.900$. APV pure endowment $=1000\,v^2\,(0.95)(0.94)\approx 794.767$. Total APV benefits $=97.900+794.767=892.667$. APV premiums $=1+v(0.95)\approx 1.896226$. $P=\dfrac{892.667}{1.896226}\approx \$470.76$. Dividing the full benefit APV $\$892.67$ (E) by the wrong annuity, or omitting the pure-endowment survival benefit entirely (treating it as a $2$-year term, giving $\approx \$51.6$ per $\$1{,}000$ which scales toward the small distractors), are the common slips.
  3. Equivalence principle
    A $2$-year term insurance of $\$5{,}000$ on $(x)$ pays at the end of the year of death and is funded by a level net annual premium. Mortality is $q_x=0.10$, $q_{x+1}=0.15$, and $i=0.04$. Calculate the net annual premium. (A) $\$480.77$ (B) $\$592.29$ (C) $\$615.98$ (D) $\$625.00$ (E) $\$673.08$
    **Answer: (B).** $v=\dfrac{1}{1.04}$ and ${}_1p_x=0.90$. APV benefits $=5000\big[v(0.10)+v^2(0.90)(0.15)\big]=5000[0.096154+0.124815]\approx 1104.84$. APV premiums $=1+v(0.90)\approx 1.865385$. $P=\dfrac{1104.84}{1.865385}\approx \$592.29$. Using an annuity-immediate for premiums (omitting the time-$0$ premium) understates the denominator and pushes $P$ up toward $\$673$ (E); not discounting the benefits gives a larger numerator and $\$625$ (D).
  4. Equivalence principle
    A $3$-year endowment insurance of $\$1{,}000$ on $(x)$ pays $\$1{,}000$ at the end of the year of death, or $\$1{,}000$ on survival to time $3$. Mortality is $q_x=0.02$, $q_{x+1}=0.03$, $q_{x+2}=0.04$, and $i=0.05$. Calculate the level net annual premium. (A) $\$289.55$ (B) $\$300.10$ (C) $\$310.09$ (D) $\$322.46$ (E) $\$866.88$
    **Answer: (C).** $v=\dfrac{1}{1.05}$. Survival: ${}_1p_x=0.98$, ${}_2p_x=0.98(0.97)=0.9506$, ${}_3p_x=0.9506(0.96)=0.912576$. APV death $=1000\big[v(0.02)+v^2(0.98)(0.03)+v^3(0.9506)(0.04)\big]\approx 1000[0.019048+0.026667+0.032849]=78.564$. APV pure endowment $=1000\,v^3\,(0.912576)\approx 788.314$. Total APV benefits $\approx 866.878$. APV premiums $=1+v(0.98)+v^2(0.9506)\approx 2.795692$. $P=\dfrac{866.878}{2.795692}\approx \$310.09$. Misapplying ${}_3p_x$ (using $0.94$ instead of compounding all three survival factors) in the endowment term gives the nearby distractors.
  5. Net premiums
    A fully discrete whole life insurance of $\$250{,}000$ on $(x)$ has $A_x=0.19780$ and $\ddot a_x=16.8000$. Calculate the net annual premium. (A) $\$2{,}943$ (B) $\$3{,}324$ (C) $\$14{,}881$ (D) $\$33{,}220$ (E) $\$49{,}450$
    **Answer: (A).** By the equivalence principle $P_x=\dfrac{A_x}{\ddot a_x}=\dfrac{0.19780}{16.8000}\approx 0.0117738$ per $\$1$. For a $\$250{,}000$ benefit: $250{,}000\times 0.0117738\approx \$2{,}943$. Inverting the ratio ($\ddot a_x/A_x$ per dollar) is nonsensical here; the tempting wrong move is $A_x\times\text{face}=0.19780(250{,}000)=\$49{,}450$ (E), which is the APV of the benefit, not the annual premium.
  6. Net premiums
    A fully discrete $20$-year endowment insurance of $\$50{,}000$ on $(x)$ has $A_{x:\overline{20|}}=0.42400$ and $\ddot a_{x:\overline{20|}}=10.5933$. Calculate the net annual premium. (A) $\$1{,}060$ (B) $\$1{,}780$ (C) $\$2{,}001$ (D) $\$2{,}500$ (E) $\$21{,}200$
    **Answer: (C).** For an endowment, premiums are paid over the $20$-year temporary annuity-due, so $P_{x:\overline{20|}}=\dfrac{A_{x:\overline{20|}}}{\ddot a_{x:\overline{20|}}}=\dfrac{0.42400}{10.5933}\approx 0.0400253$ per $\$1$. For $\$50{,}000$: $50{,}000\times 0.0400253\approx \$2{,}001$. Using a whole-life-style lifetime annuity in the denominator, or reporting $A_{x:\overline{20|}}\times\text{face}=\$21{,}200$ (E, the benefit APV), are the usual errors.
  7. Net premiums
    A $\$100{,}000$ whole life insurance on $(x)$ is paid up after $15$ years (a $15$-pay whole life). The net basis gives $A_x=0.30$ and $\ddot a_{x:\overline{15|}}=9.2$. Calculate the net annual premium. (A) $\$2{,}000$ (B) $\$3{,}261$ (C) $\$4{,}500$ (D) $\$6{,}522$ (E) $\$30{,}000$
    **Answer: (B).** The benefit is whole life (numerator $A_x$), but premiums are paid only for $15$ years, so the denominator is the $15$-year temporary annuity-due: ${}_{15}P_x=\dfrac{A_x}{\ddot a_{x:\overline{15|}}}=\dfrac{0.30}{9.2}\approx 0.0326087$ per $\$1$. For $\$100{,}000$: $100{,}000\times 0.0326087\approx \$3{,}261$ per year for $15$ years. Because $\ddot a_{x:\overline{15|}}<\ddot a_x$, this exceeds the lifetime-pay premium. Using a full whole-life $\ddot a_x$ (much larger than $9.2$) would understate the premium below the keyed value.
  8. Net premiums
    A fully continuous whole life insurance of $\$1{,}000$ on $(x)$ has $\bar A_x=0.32$ and force of interest $\delta=0.06$. Calculate the continuous net premium rate $\bar P(\bar A_x)$ per year. (A) $\$19.20$ (B) $\$28.24$ (C) $\$33.33$ (D) $\$47.06$ (E) $\$53.33$
    **Answer: (B).** First $\bar a_x=\dfrac{1-\bar A_x}{\delta}=\dfrac{1-0.32}{0.06}=\dfrac{0.68}{0.06}\approx 11.3333$. $\bar P(\bar A_x)=\dfrac{\bar A_x}{\bar a_x}=\dfrac{0.32}{11.3333}\approx 0.0282353$ per $\$1$. For $\$1{,}000$: about $\$28.24$ per year, paid continuously. Check via $\dfrac{\delta\bar A_x}{1-\bar A_x}=\dfrac{0.06(0.32)}{0.68}\approx 0.0282353$. Computing $\delta\bar A_x$ alone gives $\$19.20$ (A); using $\dfrac{\bar A_x}{1-\bar A_x}\cdot\delta\cdot(\ldots)$ slips produce the larger distractors.