{
  "deckName": "Exam FM — Annuities (Varying & Continuous)",
  "examCode": "Exam FM",
  "cards": [
    {
      "front": "What does the symbol $(Ia)_{\\overline{n|}}$ represent, and what is the size of each payment?",
      "back": "$(Ia)_{\\overline{n|}}$ 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 $0$.",
      "tag": "Increasing"
    },
    {
      "front": "State the closed-form present value of the increasing annuity-immediate $(Ia)_{\\overline{n|}}$.",
      "back": "$(Ia)_{\\overline{n|}}=\\dfrac{\\ddot{a}_{\\overline{n|}}-n\\,v^{n}}{i}$, where $\\ddot{a}_{\\overline{n|}}=\\dfrac{1-v^{n}}{d}$ is the level annuity-due.\nThink of it as a stack of $n$ level annuities deferred successively, summed via the annuity-due factor in the numerator.",
      "tag": "Increasing"
    },
    {
      "front": "Compute $(Ia)_{\\overline{10|}}$ at an annual effective rate $i=0.06$.",
      "back": "First $v=\\frac{1}{1.06}$, so $v^{10}\\approx 0.558395$ and $a_{\\overline{10|}}=\\frac{1-v^{10}}{0.06}\\approx 7.360087$.\nThe annuity-due is $\\ddot{a}_{\\overline{10|}}=a_{\\overline{10|}}(1.06)\\approx 7.801692$.\nThen $(Ia)_{\\overline{10|}}=\\dfrac{7.801692-10(0.558395)}{0.06}=\\dfrac{7.801692-5.583948}{0.06}\\approx \\mathbf{36.9624}$.",
      "tag": "Increasing"
    },
    {
      "front": "Verify the identity $(Ia)_{\\overline{n|}}+(Da)_{\\overline{n|}}=(n+1)\\,a_{\\overline{n|}}$ and explain why it holds.",
      "back": "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 $(n+1)\\,a_{\\overline{n|}}$.\nCheck at $i=0.06$, $n=10$: $36.9624+43.9985=80.9609$ and $(11)(7.360087)=80.9610$. ✓",
      "tag": "Increasing"
    },
    {
      "front": "An increasing annuity-immediate $(Ia)_{\\overline{10|}}$ is **deferred 5 years**. At $i=0.07$, find its present value today.",
      "back": "First value at time $5$ (one period before the first payment at time $6$): $(Ia)_{\\overline{10|}}$ at $7\\%\\approx 34.73913$.\nDiscount $5$ years: $\\times v^{5}=1.07^{-5}\\approx 0.712986$.\nPV $=34.73913(0.712986)\\approx \\mathbf{24.769}$.",
      "tag": "Increasing"
    },
    {
      "front": "Why does the increasing-annuity formula put $\\ddot a_{\\overline{n|}}$ (not $a_{\\overline{n|}}$) in the numerator of $(Ia)_{\\overline{n|}}$?",
      "back": "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.",
      "tag": "Increasing"
    },
    {
      "front": "What does $(Da)_{\\overline{n|}}$ represent, and what is its present-value formula?",
      "back": "$(Da)_{\\overline{n|}}$ is the **decreasing annuity-immediate** paying $n, n-1, \\ldots, 2, 1$ at the ends of periods $1$ through $n$.\nIts present value is $(Da)_{\\overline{n|}}=\\dfrac{n-a_{\\overline{n|}}}{i}$, where $a_{\\overline{n|}}=\\dfrac{1-v^{n}}{i}$.",
      "tag": "Decreasing"
    },
    {
      "front": "Compute $(Da)_{\\overline{10|}}$ at $i=0.06$.",
      "back": "With $a_{\\overline{10|}}\\approx 7.360087$ at $6\\%$:\n$(Da)_{\\overline{10|}}=\\dfrac{n-a_{\\overline{n|}}}{i}=\\dfrac{10-7.360087}{0.06}=\\dfrac{2.639913}{0.06}\\approx \\mathbf{43.9985}$.",
      "tag": "Decreasing"
    },
    {
      "front": "A decreasing annuity-immediate pays $n, n-1, \\ldots, 1$. At $i=0.10$, $n=20$, find $(Da)_{\\overline{20|}}$.",
      "back": "$(Da)_{\\overline{20|}}=\\dfrac{n-a_{\\overline{n|}}}{i}$. At $10\\%$, $a_{\\overline{20|}}=\\dfrac{1-1.10^{-20}}{0.10}\\approx 8.513564$.\n$(Da)_{\\overline{20|}}=\\dfrac{20-8.513564}{0.10}=\\dfrac{11.486436}{0.10}\\approx \\mathbf{114.86}$.",
      "tag": "Decreasing"
    },
    {
      "front": "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 $Y$.",
      "back": "$(Da)_{\\overline{20|}}=\\dfrac{20-a_{\\overline{20|}}}{0.08}$. At $8\\%$, $a_{\\overline{20|}}\\approx 9.818147$, so $(Da)_{\\overline{20|}}=\\dfrac{20-9.818147}{0.08}\\approx 127.2732$.\nLevel perpetuity PV $=\\dfrac{Y}{0.08}$. Setting equal: $Y=127.2732(0.08)\\approx \\mathbf{10.182}$.",
      "tag": "Decreasing"
    },
    {
      "front": "Define the **increasing perpetuity-immediate** $(Ia)_{\\overline{\\infty|}}$ and give its present value.",
      "back": "It pays $1, 2, 3, \\ldots$ forever at the ends of periods. Its present value is\n$(Ia)_{\\overline{\\infty|}}=\\dfrac{1}{i}+\\dfrac{1}{i^{2}}=\\dfrac{1}{i\\,d}$.\nThe $1/i$ is the level part; the $1/i^{2}$ captures the perpetual increase.",
      "tag": "Perpetuity"
    },
    {
      "front": "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.",
      "back": "This is $(Ia)_{\\overline{\\infty|}}=\\dfrac{1}{i}+\\dfrac{1}{i^{2}}=\\dfrac{1}{0.08}+\\dfrac{1}{0.08^{2}}=12.5+156.25=\\mathbf{168.75}$.\nEquivalently $\\dfrac{1}{i\\,d}$ with $d=\\frac{0.08}{1.08}\\approx0.074074$: $\\frac{1}{0.08\\times0.074074}\\approx168.75$. ✓",
      "tag": "Perpetuity"
    },
    {
      "front": "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).",
      "back": "$(I\\ddot{a})_{\\overline{\\infty|}}=\\dfrac{1}{d^{2}}=(Ia)_{\\overline{\\infty|}}\\,(1+i)$.\nAt $i=0.10$: $(Ia)_{\\overline{\\infty|}}=110$, so the due version is $110(1.10)=\\mathbf{121}=\\frac{1}{d^{2}}$ with $d=\\frac{0.10}{1.10}$. ✓",
      "tag": "Perpetuity"
    },
    {
      "front": "Why is there no finite present value for a strictly **decreasing perpetuity** $(Da)_{\\overline{\\infty|}}$?",
      "back": "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.",
      "tag": "Perpetuity"
    },
    {
      "front": "Derive the present value of the increasing perpetuity-immediate $(Ia)_{\\overline{\\infty|}}$ from the finite formula.",
      "back": "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 $n v^{n}\\to 0$.\nSo $(Ia)_{\\overline{\\infty|}}=\\dfrac{1/d}{i}=\\dfrac{1}{i\\,d}=\\dfrac{1}{i}+\\dfrac{1}{i^{2}}$ (using $\\frac{1}{d}=\\frac{1+i}{i}$).",
      "tag": "Perpetuity"
    },
    {
      "front": "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.",
      "back": "Split into a 9-year increasing piece plus a deferred level perpetuity of $100$.\nIncreasing $10,20,\\ldots,90$ over years 1–9: $10\\,(Ia)_{\\overline{9|}}$. At $5\\%$, $(Ia)_{\\overline{9|}}\\approx 33.23465$, giving $332.3465$.\nLevel $100$ from year 10 on: $\\dfrac{100}{0.05}\\,v^{9}=2000(1.05^{-9})=2000(0.644609)=1289.218$.\nTotal $\\approx 332.347+1289.218\\approx \\mathbf{1621.56}$.",
      "tag": "Perpetuity"
    },
    {
      "front": "Define the accumulated increasing annuity $(Is)_{\\overline{n|}}$ and relate it to $(Ia)_{\\overline{n|}}$.",
      "back": "$(Is)_{\\overline{n|}}$ is the accumulated value at time $n$ of payments $1,2,\\ldots,n$ made at the ends of periods.\n$(Is)_{\\overline{n|}}=(Ia)_{\\overline{n|}}\\,(1+i)^{n}=\\dfrac{\\ddot{s}_{\\overline{n|}}-n}{i}$, where $\\ddot{s}_{\\overline{n|}}=s_{\\overline{n|}}(1+i)$.",
      "tag": "Accumulated"
    },
    {
      "front": "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.",
      "back": "This is $(Is)_{\\overline{15|}}=(Ia)_{\\overline{15|}}\\,(1+i)^{15}$.\nAt $4\\%$: $\\ddot a_{\\overline{15|}}\\approx 11.563123$ and $v^{15}\\approx 0.555265$, so $(Ia)_{\\overline{15|}}=\\dfrac{11.563123-15(0.555265)}{0.04}=\\dfrac{3.234155}{0.04}\\approx 80.85389$.\nWith $(1.04)^{15}\\approx 1.800944$: $(Is)_{\\overline{15|}}=80.85389(1.800944)\\approx \\mathbf{145.61}$.",
      "tag": "Accumulated"
    },
    {
      "front": "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 $i=0.05$.",
      "back": "Decompose in PV: level $90$ plus increasing $10,20,\\ldots,150 = 10\\times(1,\\ldots,15)$, then accumulate by $(1.05)^{15}$.\nAt $5\\%$: $a_{\\overline{15|}}\\approx 10.379658$, $(Ia)_{\\overline{15|}}\\approx 73.66769$.\nPV $=90(10.379658)+10(73.66769)=934.169+736.677=1670.846$.\nAV $=1670.846(1.05)^{15}=1670.846(2.078928)\\approx \\mathbf{3473.57}$.",
      "tag": "Accumulated"
    },
    {
      "front": "A 10-year annuity-immediate pays $5,10,15,\\ldots,50$ (increasing by $5$). At $i=0.06$, find its present value.",
      "back": "Factor out the step: $5,10,\\ldots,50 = 5\\times(1,2,\\ldots,10)$, so PV $=5\\,(Ia)_{\\overline{10|}}$.\nAt $6\\%$, $(Ia)_{\\overline{10|}}\\approx 36.96241$, so PV $=5(36.96241)\\approx \\mathbf{184.81}$.",
      "tag": "Arithmetic"
    },
    {
      "front": "A 10-year annuity-immediate pays $100, 95, 90, \\ldots, 55$. At $i=0.06$, find its present value.",
      "back": "Decompose as a level $55$ plus a decreasing $5,10,\\ldots,50$: $100,95,\\ldots,55 = 55\\,a_{\\overline{10|}}+5\\,(Da)_{\\overline{10|}}$.\n$a_{\\overline{10|}}\\approx7.360087$, $(Da)_{\\overline{10|}}\\approx43.99855$.\nPV $=55(7.360087)+5(43.99855)=404.8048+219.9927\\approx \\mathbf{624.80}$.",
      "tag": "Arithmetic"
    },
    {
      "front": "An annuity-immediate pays $10, 11, 12, \\ldots$ for $10$ years. At $i=0.06$, find the present value.",
      "back": "Split payment $t$ as $9+t$: a level $9$ plus an increasing $1,2,\\ldots,10$.\nPV $=9\\,a_{\\overline{10|}}+(Ia)_{\\overline{10|}}$.\nAt $6\\%$: $9(7.360087)+36.96241=66.24078+36.96241\\approx \\mathbf{103.20}$.",
      "tag": "Arithmetic"
    },
    {
      "front": "General arithmetic annuity-immediate: first payment $P$, each subsequent payment increases by $Q$, for $n$ payments. State the present value.",
      "back": "$\\operatorname{PV}=P\\,a_{\\overline{n|}}+Q\\,\\dfrac{a_{\\overline{n|}}-n\\,v^{n}}{i}$.\nThe 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.)",
      "tag": "Arithmetic"
    },
    {
      "front": "An annuity-immediate has first payment $50$, then increases by $8$ per year for $12$ years. At $i=0.07$, find the present value.",
      "back": "Use $\\operatorname{PV}=P\\,a_{\\overline{n|}}+Q\\,\\dfrac{a_{\\overline{n|}}-n v^{n}}{i}$ with $P=50,Q=8,n=12$.\nAt $7\\%$: $a_{\\overline{12|}}\\approx 7.942686$, $v^{12}\\approx 0.444012$.\n$\\dfrac{7.942686-12(0.444012)}{0.07}=\\dfrac{7.942686-5.328142}{0.07}\\approx 37.35063$.\nPV $=50(7.942686)+8(37.35063)=397.1343+298.8050\\approx \\mathbf{695.94}$.",
      "tag": "Arithmetic"
    },
    {
      "front": "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).",
      "back": "Arithmetic with $P=500, Q=100, n=10$: $\\operatorname{PV}=P\\,a_{\\overline{10|}}+Q\\dfrac{a_{\\overline{10|}}-10 v^{10}}{i}$.\nAt $7\\%$: $a_{\\overline{10|}}\\approx 7.023582$, $v^{10}\\approx 0.508349$.\n$\\dfrac{7.023582-5.083493}{0.07}\\approx 27.71555$.\n$\\operatorname{PV}=500(7.023582)+100(27.71555)=3511.79+2771.56\\approx \\mathbf{6283.35}$.",
      "tag": "Arithmetic"
    },
    {
      "front": "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 $X$.",
      "back": "Set $(Ia)_{\\overline{15|}}=X\\,a_{\\overline{15|}}$ so $X=\\dfrac{(Ia)_{\\overline{15|}}}{a_{\\overline{15|}}}$.\nAt $6\\%$: $(Ia)_{\\overline{15|}}\\approx 67.26680$, $a_{\\overline{15|}}\\approx 9.712249$.\n$X=\\dfrac{67.26680}{9.712249}\\approx \\mathbf{6.926}$. (Sensible: below the arithmetic mean $8$, since big payments are back-loaded.)",
      "tag": "Arithmetic"
    },
    {
      "front": "What is a **geometric (compound-increasing) annuity**, and how is its present value computed efficiently?",
      "back": "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 $\\operatorname{PV}=\\dfrac{P}{i-g}\\left[1-\\left(\\dfrac{1+g}{1+i}\\right)^{n}\\right]$.",
      "tag": "Geometric"
    },
    {
      "front": "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.",
      "back": "Geometric with $P=1000,g=0.03,i=0.07,n=20$:\n$\\operatorname{PV}=\\dfrac{1000}{0.07-0.03}\\left[1-\\left(\\dfrac{1.03}{1.07}\\right)^{20}\\right]=25000\\left[1-(0.962617)^{20}\\right]$.\n$(0.962617)^{20}\\approx 0.466733$, so $\\operatorname{PV}=25000(0.533267)\\approx \\mathbf{13{,}331.66}$.",
      "tag": "Geometric"
    },
    {
      "front": "When does a **geometric perpetuity** converge, and what is its present value?",
      "back": "Payments $P, P(1+g), P(1+g)^{2}, \\ldots$ at the ends of periods converge only when $i>g$. Then\n$\\operatorname{PV}=\\dfrac{P}{i-g}$, where $P$ is the **first** payment (at time $1$).\nIf $i\\leq g$ the present value diverges. For a perpetuity-due, multiply by $(1+i)$.",
      "tag": "Geometric"
    },
    {
      "front": "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.",
      "back": "Geometric perpetuity: $\\operatorname{PV}=\\dfrac{P}{i-g}=\\dfrac{500}{0.06-0.02}=\\dfrac{500}{0.04}=\\mathbf{12{,}500}$.\n(Convergence holds since $i=0.06>g=0.02$.)",
      "tag": "Geometric"
    },
    {
      "front": "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.",
      "back": "When $i=g$ the formula $\\frac{P}{i-g}[\\cdots]$ is $\\frac{0}{0}$; instead each discounted payment is constant: $\\dfrac{(1+g)^{t-1}}{(1+i)^{t}}=\\dfrac{1}{1+i}$.\nSo PV $=n\\cdot\\dfrac{P}{1+i}=25\\cdot\\dfrac{1}{1.04}\\approx \\mathbf{24.0385}$.",
      "tag": "Geometric"
    },
    {
      "front": "A perpetuity-due pays $1$ now, then increases $3\\%$ per year forever. At $i=0.07$, find the present value.",
      "back": "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**:\nPV $=25(1.07)=\\mathbf{26.75}$.",
      "tag": "Geometric"
    },
    {
      "front": "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 $j$.",
      "back": "Equivalent rate: $1+j=\\dfrac{1+i}{1+g}=\\dfrac{1.10}{1.06}$, so $j\\approx 0.037736$.\n$\\operatorname{PV}=\\dfrac{P}{1+g}\\,a_{\\overline{12|}j}=\\dfrac{2000}{1.06}\\,a_{\\overline{12|}j}$.\n$a_{\\overline{12|}j}=\\dfrac{1-(1.037736)^{-12}}{0.037736}\\approx 9.50958$, so $\\operatorname{PV}\\approx 1886.792(9.50958)\\approx \\mathbf{17{,}942.61}$.",
      "tag": "Geometric"
    },
    {
      "front": "Define the **continuous annuity** $\\bar{a}_{\\overline{n|}}$ and give its present-value formula.",
      "back": "$\\bar{a}_{\\overline{n|}}$ is the present value of payments made continuously at a rate of $1$ per unit time over $n$ periods:\n$\\bar{a}_{\\overline{n|}}=\\displaystyle\\int_{0}^{n} v^{t}\\,dt=\\dfrac{1-v^{n}}{\\delta}$, where $\\delta=\\ln(1+i)$ is the force of interest.\nIt also equals $a_{\\overline{n|}}\\cdot\\dfrac{i}{\\delta}$.",
      "tag": "Continuous"
    },
    {
      "front": "At $i=0.05$, compute the continuous annuity $\\bar{a}_{\\overline{12|}}$.",
      "back": "$\\delta=\\ln(1.05)\\approx 0.048790$ and $v^{12}=1.05^{-12}\\approx 0.556837$.\n$\\bar{a}_{\\overline{12|}}=\\dfrac{1-v^{12}}{\\delta}=\\dfrac{0.443163}{0.048790}\\approx \\mathbf{9.0830}$.\nCheck: $a_{\\overline{12|}}\\approx 8.863252$ and $8.863252\\times\\frac{0.05}{0.048790}\\approx 9.0830$. ✓",
      "tag": "Continuous"
    },
    {
      "front": "A continuous annuity pays at rate $1$ for $n=15$ years under a **constant force of interest** $\\delta=0.06$. Find $\\bar{a}_{\\overline{15|}}$.",
      "back": "$\\bar{a}_{\\overline{15|}}=\\dfrac{1-e^{-\\delta n}}{\\delta}=\\dfrac{1-e^{-0.06(15)}}{0.06}=\\dfrac{1-e^{-0.9}}{0.06}$.\n$e^{-0.9}\\approx 0.406570$, so $\\bar{a}_{\\overline{15|}}=\\dfrac{0.593430}{0.06}\\approx \\mathbf{9.8905}$.",
      "tag": "Continuous"
    },
    {
      "front": "Relate the continuous annuity $\\bar{a}_{\\overline{n|}}$, the immediate $a_{\\overline{n|}}$, and the due $\\ddot{a}_{\\overline{n|}}$ at a common rate $i$.",
      "back": "All share the numerator $1-v^{n}$; only the denominator differs by timing:\n$a_{\\overline{n|}}=\\dfrac{1-v^{n}}{i}$ (end), $\\ddot{a}_{\\overline{n|}}=\\dfrac{1-v^{n}}{d}$ (start), $\\bar{a}_{\\overline{n|}}=\\dfrac{1-v^{n}}{\\delta}$ (continuous).\nSince $d<\\delta<i$, we get $\\ddot{a}_{\\overline{n|}}>\\bar{a}_{\\overline{n|}}>a_{\\overline{n|}}$ — continuous payment sits between immediate and due.",
      "tag": "Continuous"
    },
    {
      "front": "Define the **continuously increasing continuous annuity** $(\\bar{I}\\bar{a})_{\\overline{n|}}$ and give its present value.",
      "back": "Payment is made continuously at an **instantaneous rate equal to $t$** at time $t$ (the rate itself rises linearly). Its present value is\n$(\\bar{I}\\bar{a})_{\\overline{n|}}=\\displaystyle\\int_{0}^{n} t\\,v^{t}\\,dt=\\dfrac{\\bar{a}_{\\overline{n|}}-n\\,v^{n}}{\\delta}$.",
      "tag": "Continuous"
    },
    {
      "front": "At $i=0.05$, compute $(\\bar{I}\\bar{a})_{\\overline{12|}}$ (continuously increasing continuous annuity).",
      "back": "With $\\delta\\approx 0.048790$, $v^{12}\\approx 0.556837$, $\\bar{a}_{\\overline{12|}}\\approx 9.083031$:\n$(\\bar{I}\\bar{a})_{\\overline{12|}}=\\dfrac{\\bar{a}_{\\overline{12|}}-12\\,v^{12}}{\\delta}=\\dfrac{9.083031-6.682049}{0.048790}\\approx \\mathbf{49.2104}$.",
      "tag": "Continuous"
    },
    {
      "front": "What is the present value of a **continuously increasing continuous perpetuity** $(\\bar{I}\\bar{a})_{\\overline{\\infty|}}$, and evaluate it at $\\delta=0.04$.",
      "back": "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\n$(\\bar{I}\\bar{a})_{\\overline{\\infty|}}=\\dfrac{1}{\\delta^{2}}$.\nAt $\\delta=0.04$: $\\dfrac{1}{0.04^{2}}=\\dfrac{1}{0.0016}=\\mathbf{625}$.",
      "tag": "Continuous"
    }
  ]
}