{
  "deckName": "Exam FM — Interest Measurement & TVM",
  "examCode": "Exam FM",
  "cards": [
    {
      "front": "Define the **accumulation function** $a(t)$ and the **amount function** $A(t)$ for an investment.",
      "back": "$a(t)$ accumulates a single unit invested at time $0$, with $a(0)=1$. The amount function scales it by the initial deposit $k$: $A(t)=k\\cdot a(t)$, so $A(0)=k$.\nUnder compound interest $a(t)=(1+i)^{t}$; under simple interest $a(t)=1+it$.",
      "tag": "Accumulation & PV"
    },
    {
      "front": "Define the **effective rate of interest** $i_{n}$ over the $n$-th period in terms of the amount function $A(t)$.",
      "back": "$i_{n}=\\frac{A(n)-A(n-1)}{A(n-1)}$ — interest earned in the period divided by the balance at its **start**.\nFor compound interest $i_{n}=i$ is constant every period; for simple interest the effective rate declines each period because the denominator grows while the numerator stays fixed.",
      "tag": "Interest & discount rates"
    },
    {
      "front": "What is the **discount factor** $v$, and how does it relate to the effective rate $i$?",
      "back": "$v=\\frac{1}{1+i}=(1+i)^{-1}$. It is the present value of $1$ paid one period from now.\nDiscounting $n$ periods multiplies by $v^{n}=(1+i)^{-n}$. With $i=0.07$, $v=\\frac{1}{1.07}\\approx 0.93458$.",
      "tag": "Accumulation & PV"
    },
    {
      "front": "Give the four key relationships connecting the effective rate of discount $d$, the interest rate $i$, and $v$.",
      "back": "$d=\\frac{i}{1+i}=iv$, $d=1-v$, $i=\\frac{d}{1-d}$, and $1-d=v$.\nAlso the useful identity $i-d=id$ (interest and discount differ only by the product term).",
      "tag": "Interest & discount rates"
    },
    {
      "front": "What is the conceptual difference between the rate of **interest** $i$ and the rate of **discount** $d$?",
      "back": "Interest $i$ is paid at the **end** of the period on the amount at the start (interest in arrears). Discount $d$ is the same charge taken at the **start** of the period out of the amount due at the end (interest in advance).\nBecause $d$ is applied earlier on a smaller base, $d<i$ for any positive rate, with $d=\\frac{i}{1+i}$.",
      "tag": "Interest & discount rates"
    },
    {
      "front": "Accumulate $\\$5{,}000$ for $8$ years at an annual effective interest rate of $4.5\\%$.",
      "back": "Use $A(8)=5000\\,(1+i)^{8}$ with $i=0.045$.\n$1.045^{8}\\approx 1.422101$.\n$A(8)=5000\\times 1.422101\\approx \\$7{,}110.50$.",
      "tag": "Accumulation & PV"
    },
    {
      "front": "Find the present value of $\\$10{,}000$ payable in $6$ years at an annual effective rate of $7\\%$.",
      "back": "$PV=10000\\,v^{6}=\\frac{10000}{1.07^{6}}$.\n$1.07^{6}\\approx 1.500730$.\n$PV=\\frac{10000}{1.500730}\\approx \\$6{,}663.42$.",
      "tag": "Accumulation & PV"
    },
    {
      "front": "Given an effective annual discount rate $d=0.05$, find the equivalent effective annual interest rate $i$.",
      "back": "$i=\\frac{d}{1-d}=\\frac{0.05}{0.95}\\approx 0.052632$, i.e. about $5.2632\\%$.\nCheck: $d=iv=0.052632\\times\\frac{1}{1.052632}=0.05$. ✓",
      "tag": "Interest & discount rates"
    },
    {
      "front": "Given an effective annual interest rate $i=0.08$, find the equivalent effective annual discount rate $d$.",
      "back": "$d=\\frac{i}{1+i}=\\frac{0.08}{1.08}\\approx 0.074074$, about $7.4074\\%$.\nEquivalently $d=1-v=1-\\frac{1}{1.08}\\approx 0.074074$. Both routes agree.",
      "tag": "Interest & discount rates"
    },
    {
      "front": "Distinguish a **nominal** rate $i^{(m)}$ from an **effective** rate. What is the per-period rate when interest is convertible $m$-thly?",
      "back": "$i^{(m)}$ is an annual **nominal** rate compounded $m$ times per year; it is a label, never applied directly. The actual rate earned each subperiod is $\\frac{i^{(m)}}{m}$.\nThe annual effective rate follows from $1+i=\\left(1+\\frac{i^{(m)}}{m}\\right)^{m}$. The classic trap is using $i^{(m)}$ itself instead of $\\frac{i^{(m)}}{m}$.",
      "tag": "Nominal rates"
    },
    {
      "front": "A nominal annual interest rate of $6\\%$ is convertible monthly ($i^{(12)}=0.06$). Find the annual effective rate $i$.",
      "back": "Monthly rate $=\\frac{0.06}{12}=0.005$.\n$1+i=(1.005)^{12}\\approx 1.061678$.\n$i\\approx 0.061678$, i.e. about $6.1678\\%$ effective annually.",
      "tag": "Nominal rates"
    },
    {
      "front": "A nominal rate $i^{(4)}=0.08$ is convertible quarterly. Find the annual effective rate $i$.",
      "back": "Quarterly rate $=\\frac{0.08}{4}=0.02$.\n$1+i=(1.02)^{4}\\approx 1.082432$.\n$i\\approx 0.082432$, about $8.2432\\%$.",
      "tag": "Nominal rates"
    },
    {
      "front": "Write the conversion identity for a **nominal discount** rate $d^{(m)}$ convertible $m$-thly, and relate it to $i$ and $v$.",
      "back": "$\\left(1-\\frac{d^{(m)}}{m}\\right)^{m}=1-d=v=\\frac{1}{1+i}$.\nSo $1+i=\\left(1-\\frac{d^{(m)}}{m}\\right)^{-m}$. The per-period discount actually applied is $\\frac{d^{(m)}}{m}$, taken in advance each subperiod.",
      "tag": "Nominal rates"
    },
    {
      "front": "A nominal discount rate $d^{(2)}=0.06$ is convertible semiannually. Find the annual effective interest rate $i$.",
      "back": "Per-half-year discount $=\\frac{0.06}{2}=0.03$.\n$v=(1-0.03)^{2}=0.97^{2}=0.9409$.\n$1+i=\\frac{1}{0.9409}\\approx 1.062812$, so $i\\approx 6.2812\\%$.",
      "tag": "Nominal rates"
    },
    {
      "front": "State the **unifying chain** linking $i^{(m)}$, $i$, $d$, $d^{(m)}$, and the force of interest $\\delta$.",
      "back": "$\\left(1+\\frac{i^{(m)}}{m}\\right)^{m}=1+i=(1-d)^{-1}=\\left(1-\\frac{d^{(m)}}{m}\\right)^{-m}=e^{\\delta}$.\nEvery equivalent-rate question converts a given quantity to $(1+i)$ — the common hub — then back out to the target form.",
      "tag": "Equivalent rates"
    },
    {
      "front": "Define the **force of interest** $\\delta$ for a constant-rate investment and relate it to $i$ and $v$.",
      "back": "$\\delta$ is the instantaneous (continuously compounded) rate, with $\\delta=\\ln(1+i)$, equivalently $1+i=e^{\\delta}$ and $v=e^{-\\delta}$.\nAccumulating a unit for time $t$ gives $e^{\\delta t}$; discounting gives $e^{-\\delta t}$.",
      "tag": "Force of interest"
    },
    {
      "front": "An account earns an annual effective rate of $7\\%$. Find the equivalent **force of interest** $\\delta$.",
      "back": "$\\delta=\\ln(1+i)=\\ln(1.07)\\approx 0.067659$, i.e. about $6.7659\\%$.\nNote $\\delta<i$: continuous compounding needs a slightly lower instantaneous rate to reach the same annual growth.",
      "tag": "Force of interest"
    },
    {
      "front": "Invest $\\$3{,}000$ for $7$ years under a constant force of interest $\\delta=0.045$. Find the accumulated value.",
      "back": "Accumulation factor $=e^{\\delta t}=e^{0.045\\times 7}=e^{0.315}\\approx 1.370260$.\n$AV=3000\\times 1.370260\\approx \\$4{,}110.78$.",
      "tag": "Force of interest"
    },
    {
      "front": "Contrast **simple** and **compound** interest accumulation. When does simple interest accumulate to *more* than compound?",
      "back": "Simple: $a(t)=1+it$ (interest only on principal). Compound: $a(t)=(1+i)^{t}$ (interest on interest).\nFor $0<t<1$ simple interest gives the larger value; at $t=1$ they coincide; for $t>1$ compound dominates and the gap widens. Simple interest is normally used only for accumulation, not discounting, unless a problem states otherwise.",
      "tag": "Simple vs compound"
    },
    {
      "front": "Invest $\\$1{,}000$ for $3$ years. Compare the accumulated value under simple interest of $5\\%$ versus compound interest of $5\\%$.",
      "back": "Simple: $1000(1+0.05\\times 3)=1000\\times 1.15=\\$1{,}150.00$.\nCompound: $1000\\,(1.05)^{3}=1000\\times 1.157625=\\$1{,}157.63$.\nCompound earns $\\$7.63$ more — the interest-on-interest from the first two years.",
      "tag": "Simple vs compound"
    },
    {
      "front": "An investor lends $\\$800$ at a **simple** interest rate of $6\\%$ per year. What is the value after $5$ years, and what is the effective rate earned in year $5$?",
      "back": "Value $=800(1+0.06\\times 5)=800\\times 1.30=\\$1{,}040.00$.\nInterest each year is a flat $800\\times 0.06=\\$48$. The effective rate in year $5$ is $\\frac{48}{800(1+0.06\\times4)}=\\frac{48}{992}\\approx 0.04839$, i.e. $4.839\\%$ — lower than $6\\%$ because the base has grown.",
      "tag": "Simple vs compound"
    },
    {
      "front": "What does it mean for two interest/discount rates to be **equivalent**, and what is the general strategy to convert between any two?",
      "back": "Two rates are equivalent if they produce the **same accumulated value** over the same period. Strategy: convert the given rate to the annual effective $(1+i)$ hub, then convert $(1+i)$ to the target form using the unifying chain.\nNever add or average nominal rates — always work through accumulation factors.",
      "tag": "Equivalent rates"
    },
    {
      "front": "Find the nominal rate $i^{(2)}$ convertible semiannually that is equivalent to an annual effective rate of $10\\%$.",
      "back": "Per-half-year rate $=(1+i)^{1/2}-1=1.10^{0.5}-1\\approx 0.048809$.\n$i^{(2)}=2\\times 0.048809\\approx 0.097618$, i.e. about $9.7618\\%$ convertible semiannually.",
      "tag": "Equivalent rates"
    },
    {
      "front": "An annual effective rate is $6\\%$. Find the equivalent nominal discount rate $d^{(12)}$ convertible monthly.",
      "back": "$v=\\frac{1}{1.06}$, and $\\left(1-\\frac{d^{(12)}}{12}\\right)^{12}=v$.\nMonthly discount $=1-v^{1/12}=1-1.06^{-1/12}\\approx 1-0.995157=0.0048439$.\n$d^{(12)}=12\\times 0.0048439\\approx 0.058128$, about $5.8128\\%$.",
      "tag": "Equivalent rates"
    },
    {
      "front": "A nominal rate $i^{(12)}=0.09$ is convertible monthly. Find the equivalent nominal rate $i^{(4)}$ convertible quarterly.",
      "back": "Annual accumulation factor $=(1+\\frac{0.09}{12})^{12}=(1.0075)^{12}\\approx 1.093807$.\nQuarterly rate $=1.093807^{1/4}-1\\approx 0.022669$.\n$i^{(4)}=4\\times 0.022669\\approx 0.090677$, about $9.0677\\%$.",
      "tag": "Equivalent rates"
    },
    {
      "front": "What is **doubling time** under compound interest, and how does the **Rule of 72** approximate it?",
      "back": "Exact doubling time solves $(1+i)^{n}=2$, so $n=\\frac{\\ln 2}{\\ln(1+i)}$.\nThe Rule of 72 estimates it as $n\\approx \\frac{72}{100\\,i}=\\frac{72}{\\text{rate in percent}}$ — a quick mental approximation that is accurate for typical rates near $8\\%$.",
      "tag": "Doubling time"
    },
    {
      "front": "At an annual effective rate of $9\\%$, compare the exact doubling time with the Rule of 72 estimate.",
      "back": "Exact: $n=\\frac{\\ln 2}{\\ln 1.09}=\\frac{0.693147}{0.086178}\\approx 8.043$ years.\nRule of 72: $\\frac{72}{9}=8$ years. The approximation is excellent — within about half a month of the exact value.",
      "tag": "Doubling time"
    },
    {
      "front": "Express doubling time directly in terms of the **force of interest** $\\delta$. Apply it to an account with $i=5\\%$.",
      "back": "Continuous accumulation $e^{\\delta n}=2$ gives $n=\\frac{\\ln 2}{\\delta}$.\nWith $i=0.05$, $\\delta=\\ln 1.05\\approx 0.048790$, so $n=\\frac{0.693147}{0.048790}\\approx 14.207$ years.",
      "tag": "Doubling time"
    },
    {
      "front": "How long until $\\$1{,}000$ grows to $\\$1{,}500$ at an annual effective rate of $6\\%$?",
      "back": "Solve $1000(1.06)^{n}=1500\\Rightarrow 1.06^{n}=1.5$.\n$n=\\frac{\\ln 1.5}{\\ln 1.06}=\\frac{0.405465}{0.058269}\\approx 6.959$ years.",
      "tag": "Doubling time"
    },
    {
      "front": "State the relationship between the **real** rate $i_{\\text{real}}$, the **nominal** rate $i_{\\text{nom}}$, and the inflation rate $r$.",
      "back": "$1+i_{\\text{real}}=\\frac{1+i_{\\text{nom}}}{1+r}$, so $i_{\\text{real}}=\\frac{1+i_{\\text{nom}}}{1+r}-1=\\frac{i_{\\text{nom}}-r}{1+r}$.\nThe quick approximation $i_{\\text{real}}\\approx i_{\\text{nom}}-r$ understates the gap; use the exact ratio on the exam. (Here \"nominal\" means the actual money rate, not a compounding label.)",
      "tag": "Real vs nominal"
    },
    {
      "front": "An investment earns a nominal (money) return of $8\\%$ in a year with $3\\%$ inflation. Find the real rate of return.",
      "back": "$i_{\\text{real}}=\\frac{1.08}{1.03}-1\\approx 1.048544-1=0.048544$, i.e. about $4.85\\%$.\nThe rough subtraction $8\\%-3\\%=5\\%$ is close but overstates; the exact figure is $4.85\\%$.",
      "tag": "Real vs nominal"
    },
    {
      "front": "An investor requires a **real** return of $5\\%$ and expects inflation of $7\\%$. What nominal (money) rate of return must the investment earn?",
      "back": "$1+i_{\\text{nom}}=(1+i_{\\text{real}})(1+r)=1.05\\times 1.07=1.1235$.\nSo $i_{\\text{nom}}=0.1235$, i.e. $12.35\\%$ — more than the naive $5\\%+7\\%=12\\%$ because of the cross term.",
      "tag": "Real vs nominal"
    },
    {
      "front": "A fund grows by a force of interest $\\delta=0.06$ for the first $4$ years, then earns an annual effective rate of $5\\%$ for the next $3$ years. Accumulate $\\$1$ over the full $7$ years.",
      "back": "First segment: $e^{0.06\\times 4}=e^{0.24}\\approx 1.271249$.\nSecond segment: $(1.05)^{3}\\approx 1.157625$.\nTotal $=1.271249\\times 1.157625\\approx 1.47163$, so $\\$1$ grows to about $\\$1.4716$.",
      "tag": "Equivalent rates"
    },
    {
      "front": "A force of interest $\\delta=0.05$ applies. Find the equivalent annual effective interest rate $i$ and discount rate $d$.",
      "back": "$1+i=e^{\\delta}=e^{0.05}\\approx 1.051271$, so $i\\approx 5.1271\\%$.\n$d=1-v=1-e^{-\\delta}=1-e^{-0.05}=1-0.951229\\approx 0.048771$, i.e. about $4.8771\\%$.",
      "tag": "Equivalent rates"
    }
  ]
}