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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.

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  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 $\delta=0.04$: $\dfrac{1}{0.04^{2}}=\dfrac{1}{0.0016}=\mathbf{625}$.