{
  "deckName": "Exam FM — Common Traps",
  "examCode": "Exam FM",
  "cards": [
    {
      "front": "Valuing a $10$-payment annuity-**due** at $i=6\\%$ — the #1 error?",
      "back": "Using the annuity-immediate factor and forgetting the timing shift. Due payments come at the **start** of each period, one period earlier, so they're worth $(1+i)$ more: $\\ddot{a}_{\\overline{n|}}=(1+i)\\,a_{\\overline{n|}}$. Likewise $\\ddot{s}_{\\overline{n|}}=(1+i)\\,s_{\\overline{n|}}$. Trap: applying $a_{\\overline{10|}}$ when payments are at times $0,1,\\dots,9$.",
      "tag": "Timing"
    },
    {
      "front": "Quick check: how do the *subscripts* of immediate vs due annuities compare for the same $n$ payments?",
      "back": "Same number of payments, different valuation date. $a_{\\overline{n|}}$ values at one period before the **first** payment; $\\ddot{a}_{\\overline{n|}}$ values **at** the first payment. So $\\ddot{a}_{\\overline{n|}}=a_{\\overline{n|}}(1+i)=1+a_{\\overline{n-1|}}$ and $\\ddot{s}_{\\overline{n|}}=s_{\\overline{n+1|}}-1$. Memorize one identity each direction so you can convert under pressure.",
      "tag": "Timing"
    },
    {
      "front": "A loan is repaid with end-of-month deposits, but the problem says \"due.\" Default assumption?",
      "back": "Don't assume! If the cash flow timing isn't stated, **annuity-immediate** is the default on FM, but always re-read. The phrase \"payments at the beginning of each period,\" \"annuity-due,\" or \"first payment today\" signals due — multiply the immediate factor by $(1+i)$. Drawing a timeline of $t=0,1,2,\\dots$ before plugging in prevents the whole class of timing errors.",
      "tag": "Timing"
    },
    {
      "front": "Interest is \"$8\\%$ compounded quarterly.\" You discount with $v=(1.08)^{-1}$. What broke?",
      "back": "$8\\%$ is a **nominal** annual rate $i^{(4)}=0.08$, not effective. The per-quarter rate is $\\frac{i^{(m)}}{m}=\\frac{0.08}{4}=0.02$. Either work in quarters at $2\\%$, or convert to an annual effective rate $i=\\left(1+\\frac{i^{(m)}}{m}\\right)^{m}-1=1.02^{4}-1\\approx 8.24\\%$. Never discount an annual nominal rate as if it were effective.",
      "tag": "Rate conversion"
    },
    {
      "front": "Nominal **discount** $d^{(m)}$ vs nominal interest $i^{(m)}$ — the conversion slip?",
      "back": "They use *different* per-period operations. Effective: $\\left(1+\\frac{i^{(m)}}{m}\\right)^{m}=\\left(1-\\frac{d^{(p)}}{p}\\right)^{-p}=1+i=v^{-1}$. Trap: treating $d^{(m)}/m$ like an interest rate. The per-period **discount** factor is $\\left(1-\\frac{d^{(m)}}{m}\\right)$, applied as a *deduction in advance*, not $\\frac{1}{1+\\,\\cdot}$.",
      "tag": "Rate conversion"
    },
    {
      "front": "Payments are annual but interest converts **monthly** at $i^{(12)}=12\\%$. Which rate goes in $a_{\\overline{n|}}$?",
      "back": "Match the rate to the **payment** period. The annuity factor needs the *annual effective* rate over the payment interval: $i=\\left(1+\\frac{0.12}{12}\\right)^{12}-1\\approx 12.6825\\%$. Trap: plugging $1\\%$ (the monthly rate) or $12\\%$ (nominal) directly into a yearly-payment annuity. First convert the conversion-period rate to the payment-period effective rate.",
      "tag": "Mismatched periods"
    },
    {
      "front": "Monthly payments, but the yield is given as an **annual effective** $i=6\\%$. What's the fix?",
      "back": "Convert the annual effective rate **down** to the monthly effective rate before valuing: $j=(1+i)^{1/12}-1=1.06^{1/12}-1\\approx 0.4868\\%$. Then use $n=12\\times(\\text{years})$. Trap: dividing $6\\%$ by $12$ — that's only valid for a *nominal* rate, and an annual effective rate is not nominal.",
      "tag": "Mismatched periods"
    },
    {
      "front": "Perpetuity-immediate vs perpetuity-due present value — the constant people swap?",
      "back": "Immediate (first payment at $t=1$): PV $=\\dfrac{1}{i}$. Due (first payment at $t=0$): PV $=\\dfrac{1}{d}=\\dfrac{1+i}{i}=\\dfrac{1}{i}+1$. Trap: using $\\frac{1}{i}$ for a perpetuity starting **today**. Since $d=\\frac{i}{1+i}$, the due perpetuity is exactly $(1+i)$ times the immediate one.",
      "tag": "Perpetuities"
    },
    {
      "front": "Increasing perpetuity-immediate $1,2,3,\\dots$ — quoting $\\frac{1}{i^2}$. Right or wrong?",
      "back": "Wrong constant. The standard increasing perpetuity-immediate is $(Ia)_{\\overline{\\infty|}}=\\dfrac{1}{i}+\\dfrac{1}{i^{2}}=\\dfrac{1+i}{i^{2}}$, **not** $\\frac{1}{i^2}$ alone. The increasing perpetuity-**due** is $(I\\ddot a)_{\\overline{\\infty|}}=\\dfrac{1}{d^{2}}$. Memorize the due form as $\\frac{1}{d^2}$ and the immediate as $\\frac{1+i}{i^2}$ to avoid mixing them.",
      "tag": "Perpetuities"
    },
    {
      "front": "Is a bond at a premium or discount? You compare coupon rate to yield blindly. When does that fail?",
      "back": "It fails when redemption $C\\neq$ face $F$. The correct test is the sign of $Fr-Ci$ (coupon minus yield-on-redemption): **premium** if $Fr>Ci$, **discount** if $Fr<Ci$. Price form: $P=C+(Fr-Ci)\\,a_{\\overline{n|}}$. Comparing $r$ to $i$ directly only works when $C=F$.",
      "tag": "Bond premium/discount"
    },
    {
      "front": "Premium bond: which way does the book value move, and what's the amortized-coupon trap?",
      "back": "A **premium** ($P>C$, $Fr>Ci$) writes the book value **down** to $C$. Each coupon $Fr$ splits into interest $i\\cdot B_{t-1}$ plus a *write-down* of principal — the \"premium amortization,\" which is $(Fr-Ci)v^{n-t+1}$ and **grows** over time. Trap: treating the whole coupon as interest income, or moving book value the wrong direction (discount accretes **up**).",
      "tag": "Bond premium/discount"
    },
    {
      "front": "Amortized loan: the interest portion of payment $t$. The error that wrecks the split?",
      "back": "Interest in payment $t$ is $I_t=i\\cdot B_{t-1}$ — the rate times the **prior** (outstanding) balance, *not* the original loan. Principal repaid is $P_t=R-I_t$. For a level payment $R$, principal grows **geometrically**: $P_{t+1}=P_t(1+i)$, so $P_t=R\\,v^{n-t+1}$. Trap: using $i\\times$ original balance, or assuming a level principal split.",
      "tag": "Amortization"
    },
    {
      "front": "Outstanding balance after $k$ payments — prospective vs retrospective mix-up?",
      "back": "Both must give the same answer. **Prospective:** $B_k=R\\,a_{\\overline{n-k|}}$ (PV of remaining payments). **Retrospective:** $B_k=L(1+i)^{k}-R\\,s_{\\overline{k|}}$ (accumulated loan minus accumulated payments). Trap: using $a_{\\overline{n|}}$ or $s_{\\overline{n|}}$ instead of the *remaining* term $n-k$ / *elapsed* term $k$.",
      "tag": "Amortization"
    },
    {
      "front": "Sinking fund vs amortization: when is the **total cost** the same, and the common slip?",
      "back": "When the sinking-fund rate $j$ equals the loan rate $i$, total outlay equals the amortization payment exactly. The sinking-fund outlay is $L\\cdot i + \\dfrac{L}{s_{\\overline{n|}j}}$ (interest to lender plus deposit). Trap: forgetting the lender still gets $L\\cdot i$ **every** period — the deposit alone is not the total payment. If $j<i$, sinking fund costs **more**.",
      "tag": "Sinking fund"
    },
    {
      "front": "Sinking fund: the borrower's net balance over time — what surprises people?",
      "back": "Under the sinking fund method the **stated loan balance stays at $L$** the whole term (interest-only to the lender); only the *net* balance $L-\\text{(fund value)}$ declines. Contrast with amortization, where the outstanding balance itself drops each period. Trap: amortizing the loan when the problem specifies a sinking fund — the fund accumulates at rate $j$, separate from the loan at rate $i$.",
      "tag": "Sinking fund"
    },
    {
      "front": "Dollar-weighted vs time-weighted yield — which one is sensitive to cash-flow timing?",
      "back": "**Dollar-weighted** (money-weighted) is an IRR: it solves the equation of value, so large deposits before strong sub-periods inflate it — it *is* affected by timing and amount of external cash flows. **Time-weighted** chains sub-period returns $\\prod(1+r_k)-1$ and **removes** the effect of deposits/withdrawals, measuring the manager's performance. Trap: using the wrong one for the question asked.",
      "tag": "Weighted returns"
    },
    {
      "front": "Dollar-weighted return, simple-interest approximation — the formula people misremember.",
      "back": "$i\\approx\\dfrac{I}{A+\\sum_t C_t(1-t)}$, where $I=B-A-\\sum C_t$ is interest earned, $A$ is the start balance, and each contribution $C_t$ is weighted by the fraction of the year it's invested, $(1-t)$. Trap: weighting deposits by $t$ instead of $(1-t)$, or forgetting that a withdrawal is a **negative** $C_t$.",
      "tag": "Weighted returns"
    },
    {
      "front": "Time-weighted return with three sub-periods — the arithmetic trap.",
      "back": "**Multiply**, don't add: $1+i_{TW}=\\prod_{k}(1+r_k)$ where each $r_k=\\dfrac{\\text{value before next flow}}{\\text{value after previous flow}}-1$. Compute each sub-period return on the balance **just after** the cash flow that started it. Trap: averaging the sub-returns, or letting a deposit leak into the return numerator (it must reset the base, not count as gain).",
      "tag": "Weighted returns"
    },
    {
      "front": "Coupons are reinvested at a **different** rate $j\\neq i$. What yield are you actually solving for?",
      "back": "The realized (reinvestment) yield, not the bond's stated yield. Accumulate the coupon stream at the reinvestment rate: total at maturity $=Fr\\,s_{\\overline{n|}j}+C$, then solve $P(1+i')^{n}=Fr\\,s_{\\overline{n|}j}+C$ for $i'$. Trap: assuming coupons compound at the bond's yield $i$ — that's only true when $j=i$.",
      "tag": "Reinvestment"
    },
    {
      "front": "Annuity payments reinvested at rate $j$ while the annuity earns $i$ — the accumulated-value trap.",
      "back": "Two rates, two roles. Level payments of $1$ for $n$ years, each reinvested to time $n$ at rate $j$, accumulate to $s_{\\overline{n|}j}$ — this uses the **reinvestment** rate, not $i$. If only the interest is reinvested at $j$, the value is $n+i\\,(Is)_{\\overline{n-1|}j}$ style builds. Trap: blending the earning rate $i$ and reinvestment rate $j$ into a single $s_{\\overline{n|}}$.",
      "tag": "Reinvestment"
    },
    {
      "front": "A callable bond — pricing it to the **maturity** date. Why is that a trap for a premium bond?",
      "back": "For a **premium** bond (priced above redemption), the issuer calls **early** to your disadvantage, so you must price to the **earliest** call date — that gives the lowest price (yield-to-worst). For a **discount** bond, the worst case is the **latest** date (maturity). Rule: premium → assume earliest call; discount → assume latest. Price at the worst-case date and quote that price.",
      "tag": "Callable bonds"
    },
    {
      "front": "Yield-to-worst on a bond with several call dates — the shortcut that fails.",
      "back": "There's no formula shortcut: compute the price (or yield) at **every** possible redemption date and take the **minimum price** the investor would accept (equivalently the lowest yield). The premium/discount rule narrows it but you should still check each candidate when the call price varies by date. Trap: stopping at the first call date without confirming it's truly the worst.",
      "tag": "Callable bonds"
    },
    {
      "front": "Converting Macaulay duration to modified duration — the factor people drop.",
      "back": "Divide by $(1+i)$: $D_{mod}=\\dfrac{D_{mac}}{1+i}$. Macaulay is the PV-weighted average time $\\dfrac{\\sum t\\,v^{t}CF_t}{\\sum v^{t}CF_t}$; modified is $-\\frac{1}{P}\\frac{dP}{di}$. Trap: using $D_{mac}$ directly in the price-change estimate $\\Delta P\\approx -P\\,D_{mod}\\,\\Delta i$. Under a continuous force, $D_{mod}=D_{mac}$ — but only then.",
      "tag": "Duration"
    },
    {
      "front": "First-order price-change estimate — the sign and the duration type.",
      "back": "$\\Delta P\\approx -P\\cdot D_{mod}\\cdot\\Delta i$ — note the **minus** (price falls as yield rises) and that it's **modified** duration. With convexity: $\\Delta P\\approx -P\\,D_{mod}\\,\\Delta i+\\tfrac{1}{2}P\\,C\\,(\\Delta i)^2$. Trap: dropping the negative sign, or pairing the estimate with Macaulay duration instead of modified.",
      "tag": "Duration"
    },
    {
      "front": "Portfolio duration of two bonds — averaging the two durations. Correct?",
      "back": "Only by coincidence. Portfolio duration is the **present-value-weighted** average: $D_P=\\dfrac{\\sum_k P_k D_k}{\\sum_k P_k}$, weighting each bond's duration by its **price** (market value), not a simple mean. Trap: $(D_1+D_2)/2$ when the holdings have different values. Dollar durations *do* simply add: $\\sum_k P_k D_{mod,k}$.",
      "tag": "Duration"
    },
    {
      "front": "Redington immunization — the convexity condition people state backwards.",
      "back": "You need asset **convexity $\\geq$** liability convexity: $C_A\\geq C_L$ (assets more spread out around the duration point). The three conditions: (1) $PV_A=PV_L$, (2) $D_A=D_L$ (durations match), (3) $C_A\\geq C_L$. Trap: writing $C_A\\leq C_L$, or matching dollar duration but forgetting the present values must be equal first.",
      "tag": "Immunization"
    },
    {
      "front": "Redington vs **full** immunization — what's the scope difference?",
      "back": "**Redington** protects against *small* yield changes (local: matched PV and duration, asset convexity $\\geq$ liability). **Full immunization** protects against shifts of *any* size, but typically requires a single liability bracketed by asset cash flows on both sides (one before, one after the liability date), with matched PV and duration. Trap: claiming Redington covers large shifts — it only guarantees the local minimum.",
      "tag": "Immunization"
    },
    {
      "front": "Force of interest $\\delta$ vs effective rate $i$ — the conversion that trips people.",
      "back": "$\\delta=\\ln(1+i)$, equivalently $1+i=e^{\\delta}$ and $v=e^{-\\delta}$. Accumulation under constant force: $a(t)=e^{\\delta t}$. Trap: treating $\\delta$ as if it were $i$ (it's the *continuously-compounded* rate, always slightly **less** than $i$), or forgetting $\\delta=i^{(\\infty)}=\\lim_{m\\to\\infty}i^{(m)}$.",
      "tag": "Force of interest"
    },
    {
      "front": "Time-varying force of interest $\\delta_t$ — the integral people set up wrong.",
      "back": "Accumulation from time $a$ to $b$ is $\\exp\\!\\left(\\displaystyle\\int_{a}^{b}\\delta_t\\,dt\\right)$, and the discount factor is its reciprocal $\\exp\\!\\left(-\\int_{a}^{b}\\delta_t\\,dt\\right)$. Trap: plugging $\\delta_t$ in as $e^{-\\delta_t t}$ without integrating, or dropping/flipping the sign of the exponent. Always integrate the force over the interval first.",
      "tag": "Force of interest"
    },
    {
      "front": "Equation of value — the most common setup error before solving for $i$ or $n$.",
      "back": "Picking a **comparison date** but accumulating/discounting some terms to a *different* date. Every cash flow must be valued at the **same** point in time: $\\sum (\\text{inflows})\\,v^{t}=\\sum(\\text{outflows})\\,v^{t}$ at one chosen date. Trap: mixing dates, which silently changes the equation. The chosen date doesn't affect the answer — but it must be consistent across all terms.",
      "tag": "Equation of value"
    },
    {
      "front": "Equation of value — the **sign** convention that flips the answer.",
      "back": "Keep one consistent sign rule: inflows positive, outflows negative (or vice versa), then set the net PV to $0$: $\\sum_t C_t\\,v^{t}=0$. Trap: writing deposits and withdrawals with the same sign, or moving a term across the equals sign without negating it. For IRR, you're solving $NPV(i)=0$ with the signs intact.",
      "tag": "Equation of value"
    },
    {
      "front": "Solving for an unknown rate $i$ in an equation of value — what tool, and the verification trap?",
      "back": "Use the **BA II Plus cash-flow/TVM worksheet** or linear interpolation between two trial rates; there's rarely a closed form. Trap: accepting an interpolated $i$ without plugging it back to check $NPV\\approx 0$, and forgetting non-level cash flows can give **multiple** IRRs (sign changes in the flow stream). Verify the root and confirm the cash-flow signs change only as expected.",
      "tag": "Equation of value"
    },
    {
      "front": "Increasing vs level annuity — using $a_{\\overline{n|}}$ for $1,2,3,\\dots,n$. The fix?",
      "back": "Payments that **grow** need the increasing factor: $(Ia)_{\\overline{n|}}=\\dfrac{\\ddot{a}_{\\overline{n|}}-n\\,v^{n}}{i}$, not $a_{\\overline{n|}}$. For a level perpetuity it's $\\frac{1}{i}$; for the increasing perpetuity $\\frac{1+i}{i^2}$. Trap: treating an arithmetic-increasing stream as level, or confusing $(Ia)$ with $(I\\ddot a)$ (due version, divide by $d$).",
      "tag": "Varying annuities"
    },
    {
      "front": "Geometrically increasing annuity (payments grow at rate $g$) — the slick substitution people miss.",
      "back": "Payments $1,(1+g),(1+g)^2,\\dots$ valued at rate $i$ collapse to a level annuity at the **rate** $i^{*}=\\dfrac{1+i}{1+g}-1=\\dfrac{i-g}{1+g}$: PV $=\\dfrac{1}{i-g}\\left(1-\\left(\\frac{1+g}{1+i}\\right)^{n}\\right)=\\dfrac{a_{\\overline{n|}i^{*}}}{1+g}$. Trap: applying the arithmetic $(Ia)$ formula to a **geometric** grower, or fumbling the $i^{*}$ substitution.",
      "tag": "Varying annuities"
    },
    {
      "front": "Spot rates vs the yield to maturity — discounting a bond with one flat $i$. When is that wrong?",
      "back": "When you're given the **spot-rate curve** $s_1,s_2,\\dots$: each cash flow at time $t$ must be discounted at *its own* spot rate, $CF_t(1+s_t)^{-t}$, **not** a single YTM. The YTM is the one rate that reproduces that price. Trap: using a flat rate when a term structure is provided. Forward rates link spots via $(1+s_t)^{t}=(1+s_{t-1})^{t-1}(1+f_{t-1,t})$.",
      "tag": "Term structure"
    },
    {
      "front": "Method of equated time / replacing several payments with one — the rate-vs-time trap.",
      "back": "To replace payments $C_t$ at times $t$ with a single equal payment at time $\\bar t$, the **exact** date solves $\\sum C_t v^{t}=\\left(\\sum C_t\\right)v^{\\bar t}$. The *approximate* method of equated time uses the dollar-weighted average time $\\bar t=\\dfrac{\\sum t\\,C_t}{\\sum C_t}$ — which slightly **overstates** $\\bar t$ (true value is a bit earlier). Trap: quoting the approximation as exact.",
      "tag": "Equated time"
    }
  ]
}