Study Willys Flashcards · spaced-repetition decks for actuarial exams willysflashcards.com →
Become an Actuary › Exams › Flashcards › Exam FM › Annuities: Varying & Continuous
Exam FM · ~10-12%

Exam FM: Annuities (Varying & Continuous) Flashcards

Non-level and continuous annuities for SOA Exam FM: arithmetic increasing and decreasing annuities $(Ia)$, $(Is)$ and $(Da)$, increasing and decreasing perpetuities, geometric (compound-growth) annuities valued by an adjusted rate, continuous annuities $\bar{a}_{\overline{n|}}$, continuously increasing annuities, and payments in arithmetic and geometric progression — with fully worked, recomputable calculations.

40 cards7 topicsFree · fact-checked · LaTeX math
Tap card or press Space to flip
Answer

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, 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 FM decks:

Annuities: Level Annuities (Level) Practice Annuities (Varying & Continuous) Practice Bonds Bonds Practice Cash-Flow Analysis

← All Exam FM decks

Browse all 40 cards as a list
  1. Increasing
    What does the symbol $(Ia)_{\overline{n|}}$ represent, and what is the size of each payment?
    $(Ia)_{\overline{n|}}$ is the present value, one period before the first payment, of an **increasing annuity-immediate** that pays $1, 2, 3, \ldots, n$ at the ends of periods $1$ through $n$. The payment in period $t$ equals $t$, and the whole stream is valued at time $0$.
  2. Increasing
    State the closed-form present value of the increasing annuity-immediate $(Ia)_{\overline{n|}}$.
    $(Ia)_{\overline{n|}}=\dfrac{\ddot{a}_{\overline{n|}}-n\,v^{n}}{i}$, where $\ddot{a}_{\overline{n|}}=\dfrac{1-v^{n}}{d}$ is the level annuity-due. Think of it as a stack of $n$ level annuities deferred successively, summed via the annuity-due factor in the numerator.
  3. Increasing
    Compute $(Ia)_{\overline{10|}}$ at an annual effective rate $i=0.06$.
    First $v=\frac{1}{1.06}$, so $v^{10}\approx 0.558395$ and $a_{\overline{10|}}=\frac{1-v^{10}}{0.06}\approx 7.360087$. The annuity-due is $\ddot{a}_{\overline{10|}}=a_{\overline{10|}}(1.06)\approx 7.801692$. Then $(Ia)_{\overline{10|}}=\dfrac{7.801692-10(0.558395)}{0.06}=\dfrac{7.801692-5.583948}{0.06}\approx \mathbf{36.9624}$.
  4. Increasing
    Verify the identity $(Ia)_{\overline{n|}}+(Da)_{\overline{n|}}=(n+1)\,a_{\overline{n|}}$ and explain why it holds.
    Adding the two streams payment-by-payment gives $t+(n+1-t)=n+1$ every period, i.e. a **level** annuity of $n+1$, whose PV is $(n+1)\,a_{\overline{n|}}$. Check at $i=0.06$, $n=10$: $36.9624+43.9985=80.9609$ and $(11)(7.360087)=80.9610$. ✓
  5. Increasing
    An increasing annuity-immediate $(Ia)_{\overline{10|}}$ is **deferred 5 years**. At $i=0.07$, find its present value today.
    First value at time $5$ (one period before the first payment at time $6$): $(Ia)_{\overline{10|}}$ at $7\%\approx 34.73913$. Discount $5$ years: $\times v^{5}=1.07^{-5}\approx 0.712986$. PV $=34.73913(0.712986)\approx \mathbf{24.769}$.
  6. Increasing
    Why does the increasing-annuity formula put $\ddot a_{\overline{n|}}$ (not $a_{\overline{n|}}$) in the numerator of $(Ia)_{\overline{n|}}$?
    Build $(Ia)_{\overline{n|}}$ as overlapping level annuities: a 1-unit annuity for $n$ years, plus another starting year 2, …, plus one starting year $n$. Summing their PVs telescopes to $\dfrac{\ddot a_{\overline{n|}}-n v^{n}}{i}$ — the **due** factor appears because each stacked annuity is valued from the beginning of its own period.
  7. Decreasing
    What does $(Da)_{\overline{n|}}$ represent, and what is its present-value formula?
    $(Da)_{\overline{n|}}$ is the **decreasing annuity-immediate** paying $n, n-1, \ldots, 2, 1$ at the ends of periods $1$ through $n$. Its present value is $(Da)_{\overline{n|}}=\dfrac{n-a_{\overline{n|}}}{i}$, where $a_{\overline{n|}}=\dfrac{1-v^{n}}{i}$.
  8. Decreasing
    Compute $(Da)_{\overline{10|}}$ at $i=0.06$.
    With $a_{\overline{10|}}\approx 7.360087$ at $6\%$: $(Da)_{\overline{10|}}=\dfrac{n-a_{\overline{n|}}}{i}=\dfrac{10-7.360087}{0.06}=\dfrac{2.639913}{0.06}\approx \mathbf{43.9985}$.
  9. Decreasing
    A decreasing annuity-immediate pays $n, n-1, \ldots, 1$. At $i=0.10$, $n=20$, find $(Da)_{\overline{20|}}$.
    $(Da)_{\overline{20|}}=\dfrac{n-a_{\overline{n|}}}{i}$. At $10\%$, $a_{\overline{20|}}=\dfrac{1-1.10^{-20}}{0.10}\approx 8.513564$. $(Da)_{\overline{20|}}=\dfrac{20-8.513564}{0.10}=\dfrac{11.486436}{0.10}\approx \mathbf{114.86}$.
  10. Decreasing
    A decreasing annuity-immediate pays $20,19,\ldots,1$ over 20 years; a level perpetuity-immediate pays $Y$ forever, both at $i=0.08$ with equal present value. Find $Y$.
    $(Da)_{\overline{20|}}=\dfrac{20-a_{\overline{20|}}}{0.08}$. At $8\%$, $a_{\overline{20|}}\approx 9.818147$, so $(Da)_{\overline{20|}}=\dfrac{20-9.818147}{0.08}\approx 127.2732$. Level perpetuity PV $=\dfrac{Y}{0.08}$. Setting equal: $Y=127.2732(0.08)\approx \mathbf{10.182}$.
  11. Perpetuity
    Define the **increasing perpetuity-immediate** $(Ia)_{\overline{\infty|}}$ and give its present value.
    It pays $1, 2, 3, \ldots$ forever at the ends of periods. Its present value is $(Ia)_{\overline{\infty|}}=\dfrac{1}{i}+\dfrac{1}{i^{2}}=\dfrac{1}{i\,d}$. The $1/i$ is the level part; the $1/i^{2}$ captures the perpetual increase.
  12. Perpetuity
    At $i=0.08$, find the present value of a perpetuity paying $1$ at the end of year $1$, $2$ at the end of year $2$, increasing by $1$ each year forever.
    This is $(Ia)_{\overline{\infty|}}=\dfrac{1}{i}+\dfrac{1}{i^{2}}=\dfrac{1}{0.08}+\dfrac{1}{0.08^{2}}=12.5+156.25=\mathbf{168.75}$. Equivalently $\dfrac{1}{i\,d}$ with $d=\frac{0.08}{1.08}\approx0.074074$: $\frac{1}{0.08\times0.074074}\approx168.75$. ✓
  13. Perpetuity
    Give the present value of an **increasing perpetuity-due** $(I\ddot{a})_{\overline{\infty|}}$ (payments $1,2,3,\ldots$ at the start of each period).
    $(I\ddot{a})_{\overline{\infty|}}=\dfrac{1}{d^{2}}=(Ia)_{\overline{\infty|}}\,(1+i)$. At $i=0.10$: $(Ia)_{\overline{\infty|}}=110$, so the due version is $110(1.10)=\mathbf{121}=\frac{1}{d^{2}}$ with $d=\frac{0.10}{1.10}$. ✓
  14. Perpetuity
    Why is there no finite present value for a strictly **decreasing perpetuity** $(Da)_{\overline{\infty|}}$?
    A perpetuity has infinitely many payments, but a decreasing arithmetic stream $n, n-1, \ldots$ would hit zero and turn negative after $n$ periods — it cannot continue forever as positive payments. So $(Da)_{\overline{\infty|}}$ is undefined; a 'decreasing perpetuity' must be reframed as level-minus-increasing over a finite horizon.
  15. Perpetuity
    Derive the present value of the increasing perpetuity-immediate $(Ia)_{\overline{\infty|}}$ from the finite formula.
    Start from $(Ia)_{\overline{n|}}=\dfrac{\ddot{a}_{\overline{n|}}-n v^{n}}{i}$. As $n\to\infty$, $\ddot{a}_{\overline{n|}}\to\dfrac{1}{d}$ and $n v^{n}\to 0$. So $(Ia)_{\overline{\infty|}}=\dfrac{1/d}{i}=\dfrac{1}{i\,d}=\dfrac{1}{i}+\dfrac{1}{i^{2}}$ (using $\frac{1}{d}=\frac{1+i}{i}$).
  16. Perpetuity
    A perpetuity-immediate pays $10$ in year $1$, $20$ in year $2$, increasing $10$ each year, capping at a level $100$ from year $10$ onward. At $i=0.05$, find the present value.
    Split into a 9-year increasing piece plus a deferred level perpetuity of $100$. Increasing $10,20,\ldots,90$ over years 1–9: $10\,(Ia)_{\overline{9|}}$. At $5\%$, $(Ia)_{\overline{9|}}\approx 33.23465$, giving $332.3465$. Level $100$ from year 10 on: $\dfrac{100}{0.05}\,v^{9}=2000(1.05^{-9})=2000(0.644609)=1289.218$. Total $\approx 332.347+1289.218\approx \mathbf{1621.56}$.
  17. Accumulated
    Define the accumulated increasing annuity $(Is)_{\overline{n|}}$ and relate it to $(Ia)_{\overline{n|}}$.
    $(Is)_{\overline{n|}}$ is the accumulated value at time $n$ of payments $1,2,\ldots,n$ made at the ends of periods. $(Is)_{\overline{n|}}=(Ia)_{\overline{n|}}\,(1+i)^{n}=\dfrac{\ddot{s}_{\overline{n|}}-n}{i}$, where $\ddot{s}_{\overline{n|}}=s_{\overline{n|}}(1+i)$.
  18. Accumulated
    Deposits of $1, 2, 3, \ldots, 15$ are made at the ends of years $1$ through $15$ into a fund earning $4\%$ effective. Find the balance just after the year-15 deposit.
    This is $(Is)_{\overline{15|}}=(Ia)_{\overline{15|}}\,(1+i)^{15}$. At $4\%$: $\ddot a_{\overline{15|}}\approx 11.563123$ and $v^{15}\approx 0.555265$, so $(Ia)_{\overline{15|}}=\dfrac{11.563123-15(0.555265)}{0.04}=\dfrac{3.234155}{0.04}\approx 80.85389$. With $(1.04)^{15}\approx 1.800944$: $(Is)_{\overline{15|}}=80.85389(1.800944)\approx \mathbf{145.61}$.
  19. Accumulated
    Payments of $100$ at the end of year 1 increase by $10$ each year for $15$ years. Find the accumulated value at year 15 at $i=0.05$.
    Decompose in PV: level $90$ plus increasing $10,20,\ldots,150 = 10\times(1,\ldots,15)$, then accumulate by $(1.05)^{15}$. At $5\%$: $a_{\overline{15|}}\approx 10.379658$, $(Ia)_{\overline{15|}}\approx 73.66769$. PV $=90(10.379658)+10(73.66769)=934.169+736.677=1670.846$. AV $=1670.846(1.05)^{15}=1670.846(2.078928)\approx \mathbf{3473.57}$.
  20. Arithmetic
    A 10-year annuity-immediate pays $5,10,15,\ldots,50$ (increasing by $5$). At $i=0.06$, find its present value.
    Factor out the step: $5,10,\ldots,50 = 5\times(1,2,\ldots,10)$, so PV $=5\,(Ia)_{\overline{10|}}$. At $6\%$, $(Ia)_{\overline{10|}}\approx 36.96241$, so PV $=5(36.96241)\approx \mathbf{184.81}$.
  21. Arithmetic
    A 10-year annuity-immediate pays $100, 95, 90, \ldots, 55$. At $i=0.06$, find its present value.
    Decompose as a level $55$ plus a decreasing $5,10,\ldots,50$: $100,95,\ldots,55 = 55\,a_{\overline{10|}}+5\,(Da)_{\overline{10|}}$. $a_{\overline{10|}}\approx7.360087$, $(Da)_{\overline{10|}}\approx43.99855$. PV $=55(7.360087)+5(43.99855)=404.8048+219.9927\approx \mathbf{624.80}$.
  22. Arithmetic
    An annuity-immediate pays $10, 11, 12, \ldots$ for $10$ years. At $i=0.06$, find the present value.
    Split payment $t$ as $9+t$: a level $9$ plus an increasing $1,2,\ldots,10$. PV $=9\,a_{\overline{10|}}+(Ia)_{\overline{10|}}$. At $6\%$: $9(7.360087)+36.96241=66.24078+36.96241\approx \mathbf{103.20}$.
  23. Arithmetic
    General arithmetic annuity-immediate: first payment $P$, each subsequent payment increases by $Q$, for $n$ payments. State the present value.
    $\operatorname{PV}=P\,a_{\overline{n|}}+Q\,\dfrac{a_{\overline{n|}}-n\,v^{n}}{i}$. The second factor $\dfrac{a_{\overline{n|}}-n\,v^{n}}{i}$ equals $(Ia)_{\overline{n|}}-a_{\overline{n|}}$, i.e. the PV of the pure $0,1,2,\ldots,(n-1)$ increase. (If $Q<0$ the stream decreases.)
  24. Arithmetic
    An annuity-immediate has first payment $50$, then increases by $8$ per year for $12$ years. At $i=0.07$, find the present value.
    Use $\operatorname{PV}=P\,a_{\overline{n|}}+Q\,\dfrac{a_{\overline{n|}}-n v^{n}}{i}$ with $P=50,Q=8,n=12$. At $7\%$: $a_{\overline{12|}}\approx 7.942686$, $v^{12}\approx 0.444012$. $\dfrac{7.942686-12(0.444012)}{0.07}=\dfrac{7.942686-5.328142}{0.07}\approx 37.35063$. PV $=50(7.942686)+8(37.35063)=397.1343+298.8050\approx \mathbf{695.94}$.
  25. Arithmetic
    A loan is repaid by an increasing annuity: payments $500, 600, 700, \ldots$ at year-ends for $10$ years. At $i=0.07$, find the loan amount (present value).
    Arithmetic with $P=500, Q=100, n=10$: $\operatorname{PV}=P\,a_{\overline{10|}}+Q\dfrac{a_{\overline{10|}}-10 v^{10}}{i}$. At $7\%$: $a_{\overline{10|}}\approx 7.023582$, $v^{10}\approx 0.508349$. $\dfrac{7.023582-5.083493}{0.07}\approx 27.71555$. $\operatorname{PV}=500(7.023582)+100(27.71555)=3511.79+2771.56\approx \mathbf{6283.35}$.
  26. Arithmetic
    Streams (A) increasing annuity-immediate $1,2,\ldots,15$ and (B) a level annuity-immediate of $X$ for 15 years have equal PV at $i=0.06$. Find $X$.
    Set $(Ia)_{\overline{15|}}=X\,a_{\overline{15|}}$ so $X=\dfrac{(Ia)_{\overline{15|}}}{a_{\overline{15|}}}$. At $6\%$: $(Ia)_{\overline{15|}}\approx 67.26680$, $a_{\overline{15|}}\approx 9.712249$. $X=\dfrac{67.26680}{9.712249}\approx \mathbf{6.926}$. (Sensible: below the arithmetic mean $8$, since big payments are back-loaded.)
  27. Geometric
    What is a **geometric (compound-increasing) annuity**, and how is its present value computed efficiently?
    Payments grow by a constant factor $(1+g)$ each period (e.g. salary or rent escalation). Rather than expand the series, discount at an **adjusted rate** $j$ where $1+j=\dfrac{1+i}{1+g}$, then $\operatorname{PV}=\dfrac{P}{1+g}\,a_{\overline{n|}j}$ (first payment $P$ at time $1$), or directly $\operatorname{PV}=\dfrac{P}{i-g}\left[1-\left(\dfrac{1+g}{1+i}\right)^{n}\right]$.
  28. Geometric
    An annuity-immediate pays $1000$ at the end of year $1$, growing $3\%$ per year, for $20$ payments. At $i=0.07$, find the present value.
    Geometric with $P=1000,g=0.03,i=0.07,n=20$: $\operatorname{PV}=\dfrac{1000}{0.07-0.03}\left[1-\left(\dfrac{1.03}{1.07}\right)^{20}\right]=25000\left[1-(0.962617)^{20}\right]$. $(0.962617)^{20}\approx 0.466733$, so $\operatorname{PV}=25000(0.533267)\approx \mathbf{13{,}331.66}$.
  29. Geometric
    When does a **geometric perpetuity** converge, and what is its present value?
    Payments $P, P(1+g), P(1+g)^{2}, \ldots$ at the ends of periods converge only when $i>g$. Then $\operatorname{PV}=\dfrac{P}{i-g}$, where $P$ is the **first** payment (at time $1$). If $i\leq g$ the present value diverges. For a perpetuity-due, multiply by $(1+i)$.
  30. Geometric
    A perpetuity-immediate pays $500$ at the end of year $1$ and grows $2\%$ per year forever. At $i=0.06$, find the present value.
    Geometric perpetuity: $\operatorname{PV}=\dfrac{P}{i-g}=\dfrac{500}{0.06-0.02}=\dfrac{500}{0.04}=\mathbf{12{,}500}$. (Convergence holds since $i=0.06>g=0.02$.)
  31. Geometric
    An annuity pays $1$ at the end of year $1$, growing $4\%$, for $25$ years, valued at exactly $i=0.04$ (so $i=g$). Find the present value.
    When $i=g$ the formula $\frac{P}{i-g}[\cdots]$ is $\frac{0}{0}$; instead each discounted payment is constant: $\dfrac{(1+g)^{t-1}}{(1+i)^{t}}=\dfrac{1}{1+i}$. So PV $=n\cdot\dfrac{P}{1+i}=25\cdot\dfrac{1}{1.04}\approx \mathbf{24.0385}$.
  32. Geometric
    A perpetuity-due pays $1$ now, then increases $3\%$ per year forever. At $i=0.07$, find the present value.
    Geometric perpetuity-due: $\dfrac{P}{i-g}=\dfrac{1}{0.07-0.03}=25$ is the PV one period before the first payment; multiply by $(1+i)$ because the first payment is **today**: PV $=25(1.07)=\mathbf{26.75}$.
  33. Geometric
    Use the adjusted-rate trick to value a 12-payment annuity-immediate of $2000$ growing $6\%$ per year at $i=0.10$, and state the equivalent rate $j$.
    Equivalent rate: $1+j=\dfrac{1+i}{1+g}=\dfrac{1.10}{1.06}$, so $j\approx 0.037736$. $\operatorname{PV}=\dfrac{P}{1+g}\,a_{\overline{12|}j}=\dfrac{2000}{1.06}\,a_{\overline{12|}j}$. $a_{\overline{12|}j}=\dfrac{1-(1.037736)^{-12}}{0.037736}\approx 9.50958$, so $\operatorname{PV}\approx 1886.792(9.50958)\approx \mathbf{17{,}942.61}$.
  34. Continuous
    Define the **continuous annuity** $\bar{a}_{\overline{n|}}$ and give its present-value formula.
    $\bar{a}_{\overline{n|}}$ is the present value of payments made continuously at a rate of $1$ per unit time over $n$ periods: $\bar{a}_{\overline{n|}}=\displaystyle\int_{0}^{n} v^{t}\,dt=\dfrac{1-v^{n}}{\delta}$, where $\delta=\ln(1+i)$ is the force of interest. It also equals $a_{\overline{n|}}\cdot\dfrac{i}{\delta}$.
  35. Continuous
    At $i=0.05$, compute the continuous annuity $\bar{a}_{\overline{12|}}$.
    $\delta=\ln(1.05)\approx 0.048790$ and $v^{12}=1.05^{-12}\approx 0.556837$. $\bar{a}_{\overline{12|}}=\dfrac{1-v^{12}}{\delta}=\dfrac{0.443163}{0.048790}\approx \mathbf{9.0830}$. Check: $a_{\overline{12|}}\approx 8.863252$ and $8.863252\times\frac{0.05}{0.048790}\approx 9.0830$. ✓
  36. Continuous
    A continuous annuity pays at rate $1$ for $n=15$ years under a **constant force of interest** $\delta=0.06$. Find $\bar{a}_{\overline{15|}}$.
    $\bar{a}_{\overline{15|}}=\dfrac{1-e^{-\delta n}}{\delta}=\dfrac{1-e^{-0.06(15)}}{0.06}=\dfrac{1-e^{-0.9}}{0.06}$. $e^{-0.9}\approx 0.406570$, so $\bar{a}_{\overline{15|}}=\dfrac{0.593430}{0.06}\approx \mathbf{9.8905}$.
  37. Continuous
    Relate the continuous annuity $\bar{a}_{\overline{n|}}$, the immediate $a_{\overline{n|}}$, and the due $\ddot{a}_{\overline{n|}}$ at a common rate $i$.
    All share the numerator $1-v^{n}$; only the denominator differs by timing: $a_{\overline{n|}}=\dfrac{1-v^{n}}{i}$ (end), $\ddot{a}_{\overline{n|}}=\dfrac{1-v^{n}}{d}$ (start), $\bar{a}_{\overline{n|}}=\dfrac{1-v^{n}}{\delta}$ (continuous). Since $d<\delta<i$, we get $\ddot{a}_{\overline{n|}}>\bar{a}_{\overline{n|}}>a_{\overline{n|}}$ — continuous payment sits between immediate and due.
  38. Continuous
    Define the **continuously increasing continuous annuity** $(\bar{I}\bar{a})_{\overline{n|}}$ and give its present value.
    Payment is made continuously at an **instantaneous rate equal to $t$** at time $t$ (the rate itself rises linearly). Its present value is $(\bar{I}\bar{a})_{\overline{n|}}=\displaystyle\int_{0}^{n} t\,v^{t}\,dt=\dfrac{\bar{a}_{\overline{n|}}-n\,v^{n}}{\delta}$.
  39. Continuous
    At $i=0.05$, compute $(\bar{I}\bar{a})_{\overline{12|}}$ (continuously increasing continuous annuity).
    With $\delta\approx 0.048790$, $v^{12}\approx 0.556837$, $\bar{a}_{\overline{12|}}\approx 9.083031$: $(\bar{I}\bar{a})_{\overline{12|}}=\dfrac{\bar{a}_{\overline{12|}}-12\,v^{12}}{\delta}=\dfrac{9.083031-6.682049}{0.048790}\approx \mathbf{49.2104}$.
  40. Continuous
    What is the present value of a **continuously increasing continuous perpetuity** $(\bar{I}\bar{a})_{\overline{\infty|}}$, and evaluate it at $\delta=0.04$.
    Letting $n\to\infty$ in $\dfrac{\bar{a}_{\overline{n|}}-n v^{n}}{\delta}$, both $\bar{a}_{\overline{n|}}\to\frac{1}{\delta}$ and $n v^{n}\to 0$, giving $(\bar{I}\bar{a})_{\overline{\infty|}}=\dfrac{1}{\delta^{2}}$. At $\delta=0.04$: $\dfrac{1}{0.04^{2}}=\dfrac{1}{0.0016}=\mathbf{625}$.