{
  "deckName": "Exam FM — Bonds",
  "examCode": "Exam FM",
  "cards": [
    {
      "front": "Define the standard bond notation: $F$, $r$, $Fr$, $C$, $i$, $n$, and $g$.",
      "back": "$F$ = **face (par) value**, the amount on which coupons are computed.\n$r$ = **coupon rate per period**; the periodic coupon is $Fr$.\n$C$ = **redemption value** paid at maturity (often $C=F$, but not always).\n$i$ = **yield rate per period** used to discount.\n$n$ = **number of coupons** remaining.\n$g$ = **modified coupon rate** $=\\frac{Fr}{C}$, the coupon as a fraction of the redemption value.",
      "tag": "Notation & basics"
    },
    {
      "front": "Distinguish **par value**, **face value**, and **redemption value** on a bond.",
      "back": "**Face value** = **par value** $F$ — the printed amount that coupons are calculated on ($Fr$).\n**Redemption value** $C$ is what the issuer actually pays at maturity. Usually $C=F$ (redeemable at par), but a bond can redeem at a premium ($C>F$) or discount ($C<F$).\nCoupons always use $F$; the maturity payment and the premium/discount test use $C$.",
      "tag": "Notation & basics"
    },
    {
      "front": "State the **basic price formula** for a bond priced at yield rate $i$ per period.",
      "back": "$P = Fr\\,a_{\\overline{n|}} + C v^{n}$, where $a_{\\overline{n|}}=\\frac{1-v^{n}}{i}$ is the annuity-immediate factor and $v=(1+i)^{-1}$.\nThe price is the present value of the coupon stream $Fr$ plus the present value of the redemption $C$.",
      "tag": "Pricing formulas"
    },
    {
      "front": "Price a $\\$1{,}000$ par bond with annual coupons at $8\\%$, redeemable at par in $10$ years, at an annual yield of $6\\%$.",
      "back": "Here $Fr=80$, $C=1000$, $i=0.06$, $n=10$.\n$a_{\\overline{10|}}=\\frac{1-1.06^{-10}}{0.06}\\approx 7.36009$ and $1.06^{-10}\\approx 0.558395$.\n$P = 80(7.36009) + 1000(0.558395) \\approx 588.807 + 558.395 = \\$1{,}147.20$.\nSince $P>C$ the bond sells at a **premium** (coupon rate $>$ yield).",
      "tag": "Pricing formulas"
    },
    {
      "front": "State the **premium/discount form** of the bond price and what it reveals.",
      "back": "$P = C + (Fr - Ci)\\,a_{\\overline{n|}}$.\nThe sign of $Fr - Ci$ decides the case: if $Fr>Ci$ the bond sells at a **premium** ($P>C$); if $Fr<Ci$ it sells at a **discount** ($P<C$); if $Fr=Ci$ it sells at $C$ (par-equivalent). The quantity $Fr-Ci$ is the level per-coupon premium/discount being amortized.",
      "tag": "Premium & discount"
    },
    {
      "front": "When does a bond sell at a **premium** versus a **discount**? Give the rate comparison.",
      "back": "Compare the coupon to the yield (relative to the redemption value):\n**Premium** when $Fr>Ci$, i.e. the coupon rate exceeds the yield. Then $P>C$ and the book value **amortizes down** to $C$.\n**Discount** when $Fr<Ci$, i.e. the coupon rate is below the yield. Then $P<C$ and the book value **accretes up** to $C$.\nWhen $C=F$ this is simply coupon rate vs yield rate.",
      "tag": "Premium & discount"
    },
    {
      "front": "Define **premium** and **discount** as dollar amounts in terms of $P$ and $C$.",
      "back": "**Premium** $= P - C$ (used when $P>C$): the excess paid above redemption, recovered by writing book value down.\n**Discount** $= C - P$ (used when $P<C$): the shortfall below redemption, recovered by writing book value up.\nEither way, over the bond's life the total write-up/write-down equals $|P-C|$, bringing book value to exactly $C$ at maturity.",
      "tag": "Premium & discount"
    },
    {
      "front": "Price a $\\$1{,}000$ par bond, $5\\%$ annual coupons, redeemable at par in $8$ years, at a yield of $7\\%$. Premium or discount?",
      "back": "$Fr=50$, $C=1000$, $i=0.07$, $n=8$.\n$a_{\\overline{8|}}=\\frac{1-1.07^{-8}}{0.07}\\approx 5.971299$ and $1.07^{-8}\\approx 0.582009$.\n$P = 50(5.971299) + 1000(0.582009)\\approx 298.565 + 582.009 = \\$880.57$.\nSince $Fr=50 < Ci=70$, it is a **discount** bond; discount $=1000-880.57=\\$119.43$.",
      "tag": "Pricing formulas"
    },
    {
      "front": "A bond has semiannual coupons at an annual rate of $7\\%$, par $\\$1{,}000$, $10$ years to maturity, yield $8\\%$ convertible semiannually. Find the price.",
      "back": "Convert everything to the **semiannual period**: coupon $=\\frac{0.07}{2}\\cdot 1000 = 35$, yield $i=\\frac{0.08}{2}=0.04$, and $n=10\\times 2 = 20$ periods.\n$a_{\\overline{20|}}=\\frac{1-1.04^{-20}}{0.04}\\approx 13.590326$ and $1.04^{-20}\\approx 0.456387$.\n$P = 35(13.590326)+1000(0.456387)\\approx 475.661 + 456.387 = \\$932.05$.\nIt is a **discount** bond since the $3.5\\%$ periodic coupon is below the $4\\%$ periodic yield.",
      "tag": "Pricing formulas"
    },
    {
      "front": "Why must you always work in the **per-period** rate when pricing a bond with semiannual coupons?",
      "back": "The price formula discounts each coupon one *period* at a time, so $i$, $r$, and $n$ must all be on the same coupon period. With semiannual coupons you **halve** the annual coupon and the annual yield and **double** the number of years to get $n$.\nExam yields are usually quoted as a nominal annual rate convertible semiannually, so the periodic yield is that nominal rate $\\div 2$ — not an annual effective rate.",
      "tag": "Pricing formulas"
    },
    {
      "front": "State **Makeham's formula** for the price of a bond and define each symbol.",
      "back": "$P = K + \\frac{g}{i}\\,(C - K)$, where $K = C v^{n}$ is the present value of the redemption payment and $g=\\frac{Fr}{C}$ is the modified coupon rate.\nThe term $\\frac{g}{i}(C-K)$ is the present value of the coupon stream. Makeham is handy when $K$ is known or when many bonds share the same $g$ and $i$.",
      "tag": "Pricing formulas"
    },
    {
      "front": "Use **Makeham's formula** to price a $\\$1{,}000$ par bond, $7\\%$ annual coupons, redeemable at par in $15$ years, yield $5\\%$.",
      "back": "$C=1000$, $g=\\frac{70}{1000}=0.07$, $i=0.05$, $n=15$.\n$K = 1000(1.05)^{-15} = 1000(0.481017)\\approx 481.017$.\n$P = K + \\frac{g}{i}(C-K) = 481.017 + \\frac{0.07}{0.05}(1000-481.017)$\n$= 481.017 + 1.4(518.983)\\approx 481.017 + 726.576 = \\$1{,}207.59$.",
      "tag": "Pricing formulas"
    },
    {
      "front": "A $\\$1{,}000$ par bond pays $6\\%$ annual coupons and is redeemable for $\\$1{,}050$ in $12$ years. At a $6\\%$ yield, find the price.",
      "back": "Coupons use the face: $Fr=60$. The redemption uses $C=1050$.\n$a_{\\overline{12|}}=\\frac{1-1.06^{-12}}{0.06}\\approx 8.383844$ and $1.06^{-12}\\approx 0.496969$.\n$P = 60(8.383844) + 1050(0.496969)\\approx 503.031 + 521.817 = \\$1{,}024.85$.\nNote the bond is above par even though coupon rate $=$ yield, because $C>F$.",
      "tag": "Pricing formulas"
    },
    {
      "front": "When $C\\neq F$, which value drives the **premium/discount test** — the coupon rate or $Ci$?",
      "back": "Use $Ci$, not the coupon rate on face. The bond is at a premium when $Fr>Ci$ and a discount when $Fr<Ci$.\nExample: $F=1000$, coupon $\\$60$ ($Fr=60$), $C=900$, $i=6\\%$. Then $Ci=900(0.06)=54$. Since $Fr=60>54$, the bond is at a **premium relative to $C$** even though the $6\\%$ coupon equals the $6\\%$ yield on face.",
      "tag": "Premium & discount"
    },
    {
      "front": "Define **book value** $B_{t}$ of a bond and the prospective formula for it.",
      "back": "The book value $B_{t}$ is the bond's value at yield $i$ just **after** the $t$-th coupon — the price of the remaining cash flows.\nProspectively $B_{t}=Fr\\,a_{\\overline{n-t|}} + C v^{n-t}$, valuing the $n-t$ remaining coupons plus redemption at the original yield. At $t=0$ this is the purchase price $P$; at $t=n$ it equals $C$.",
      "tag": "Book value & amortization"
    },
    {
      "front": "State the **book value recursion** and the split of a coupon into interest earned and principal adjustment.",
      "back": "Recursion: $B_{t}=B_{t-1}(1+i) - Fr$.\nInterest earned in the period $= i\\,B_{t-1}$.\nPrincipal adjustment $= Fr - i\\,B_{t-1}$. For a **premium** bond this is positive and is the **write-down** (book value falls); for a **discount** bond $Fr<iB_{t-1}$ so the adjustment is negative — a **write-up** (book value rises).",
      "tag": "Book value & amortization"
    },
    {
      "front": "Give the formula for the **amortization of premium** in the $t$-th coupon (the write-down).",
      "back": "Write-down in coupon $t$ $= (Fr - Ci)\\,v^{\\,n-t+1}$.\nIt is positive for a premium bond ($Fr>Ci$) and grows geometrically by $(1+i)$ each period as maturity nears. The sum of all write-downs over the life equals the original premium $P-C$, bringing $B_{n}$ to exactly $C$.",
      "tag": "Book value & amortization"
    },
    {
      "front": "For a $\\$1{,}000$ par bond, $9\\%$ annual coupons, $10$ years, yield $7\\%$ (price $\\$1{,}140.47$), find the interest earned and the write-down in the **first** coupon.",
      "back": "$B_{0}=1140.47$, $i=0.07$, $Fr=90$.\nInterest earned $= iB_{0}=0.07(1140.47)\\approx \\$79.83$.\nWrite-down (principal adjustment) $= Fr - iB_{0}=90 - 79.83 = \\$10.17$.\nCheck via the formula $(Fr-Ci)v^{\\,n}=(90-70)(1.07)^{-10}=20(0.508349)\\approx \\$10.17$. New book value $B_{1}=1140.47-10.17=\\$1{,}130.30$.",
      "tag": "Book value & amortization"
    },
    {
      "front": "For the same $9\\%/7\\%/10$-year premium bond, find the write-down in the **third** coupon directly.",
      "back": "Use $(Fr-Ci)\\,v^{\\,n-t+1}$ with $Fr-Ci=90-70=20$, $i=0.07$, $n=10$, $t=3$, so the exponent is $10-3+1=8$.\n$1.07^{-8}\\approx 0.582009$.\nWrite-down $=20(0.582009)\\approx \\$11.64$.\nIt exceeds the first coupon's $\\$10.17$ write-down, confirming write-downs grow by $(1+i)$ each period.",
      "tag": "Book value & amortization"
    },
    {
      "front": "Give the formula for the **accumulation of discount** in the $t$-th coupon (the write-up).",
      "back": "Write-up in coupon $t$ $= (Ci - Fr)\\,v^{\\,n-t+1}$.\nIt is positive for a discount bond ($Ci>Fr$) and, like the premium case, increases geometrically by $(1+i)$ each period. The cumulative write-up over the life equals the original discount $C-P$, lifting the book value to $C$ at maturity.",
      "tag": "Book value & amortization"
    },
    {
      "front": "A $\\$1{,}000$ par bond, $4\\%$ annual coupons, $8$ years, yield $6\\%$ (price $\\$875.80$). Find the interest earned and the discount accumulated in the first coupon.",
      "back": "$B_{0}=875.80$, $i=0.06$, $Fr=40$.\nInterest earned $= iB_{0}=0.06(875.80)\\approx \\$52.55$.\nThe coupon paid is only $40$, so the shortfall is added to book value: write-up $=52.55-40=\\$12.55$.\nCheck: $(Ci-Fr)v^{\\,n}=(60-40)(1.06)^{-8}=20(0.627412)\\approx \\$12.55$. New book value $B_{1}=875.80+12.55=\\$888.35$.",
      "tag": "Book value & amortization"
    },
    {
      "front": "How do you find the **book value after the 5th coupon** of a $9\\%/7\\%/10$-year par bond without building the whole table?",
      "back": "Use the prospective formula on the remaining $n-t=5$ coupons: $B_{5}=Fr\\,a_{\\overline{5|}} + C v^{5}$ at $i=7\\%$.\n$a_{\\overline{5|}}=\\frac{1-1.07^{-5}}{0.07}\\approx 4.100197$, $1.07^{-5}\\approx 0.712986$.\n$B_{5}=90(4.100197)+1000(0.712986)\\approx 369.018 + 712.986 = \\$1{,}082.00$.\nBeing a premium bond, $B_{5}$ sits between the price $1140.47$ and the redemption $1000$.",
      "tag": "Book value & amortization"
    },
    {
      "front": "Define **yield to maturity (YTM)** for a bond.",
      "back": "The YTM is the single per-period interest rate $i$ that makes the present value of all the bond's cash flows equal to its current price — i.e. the internal rate of return solving $P = Fr\\,a_{\\overline{n|}} + C v^{n}$.\nIt assumes the bond is held to maturity and that all coupons are reinvested at that same rate $i$.",
      "tag": "Yield to maturity"
    },
    {
      "front": "A zero-coupon bond redeeming for $\\$1{,}000$ in $10$ years is bought for $\\$600$. Find its annual effective yield.",
      "back": "With no coupons, $600 = 1000(1+i)^{-10}$, so $(1+i)^{10}=\\frac{1000}{600}=1.\\overline{6}$.\n$1+i = 1.6667^{1/10}$. Since $\\ln 1.6667 \\approx 0.510826$, divide by $10$ to get $0.051083$, and $e^{0.051083}\\approx 1.05241$.\n$i \\approx 5.241\\%$ annual effective.",
      "tag": "Yield to maturity"
    },
    {
      "front": "How do you solve for the YTM of a coupon bond on a BA II Plus, and why is iteration needed?",
      "back": "The price equation $P=Fr\\,a_{\\overline{n|}}+Cv^{n}$ cannot be solved for $i$ in closed form because $i$ appears inside both the annuity factor and the power. Use the calculator's **TVM keys** (enter $N$, $PV=-P$, $PMT=Fr$, $FV=C$, then compute $I/Y$) or the **bond/IRR worksheet**, which iterate numerically.\nRemember to interpret $I/Y$ as the per-coupon-period rate when coupons are not annual.",
      "tag": "Yield to maturity"
    },
    {
      "front": "A $\\$1{,}000$ par bond redeemable at par in $10$ years sells for $\\$1{,}050$ at a $6\\%$ yield. Find the annual coupon rate.",
      "back": "Solve $1050 = Fr\\,a_{\\overline{10|}} + 1000\\,v^{10}$ at $i=6\\%$.\n$a_{\\overline{10|}}\\approx 7.360087$, $1.06^{-10}\\approx 0.558395$.\n$Fr = \\frac{1050 - 1000(0.558395)}{7.360087} = \\frac{1050 - 558.395}{7.360087}=\\frac{491.605}{7.360087}\\approx 66.79$.\nCoupon rate $=\\frac{66.79}{1000}\\approx 6.68\\%$. (Coupon $>$ yield, consistent with the premium price.)",
      "tag": "Yield to maturity"
    },
    {
      "front": "A $\\$1{,}000$ par bond pays $6\\%$ annual coupons, matures in $10$ years, and sells for $\\$1{,}000$ at a $6.5\\%$ yield. Find the redemption value $C$.",
      "back": "Solve $1000 = 60\\,a_{\\overline{10|}} + C v^{10}$ at $i=6.5\\%$.\n$a_{\\overline{10|}}=\\frac{1-1.065^{-10}}{0.065}\\approx 7.188830$, $1.065^{-10}\\approx 0.532726$.\n$C = \\frac{1000 - 60(7.188830)}{0.532726}=\\frac{1000 - 431.330}{0.532726}=\\frac{568.670}{0.532726}\\approx \\$1{,}067.47$.",
      "tag": "Yield to maturity"
    },
    {
      "front": "Distinguish a bond's **coupon rate**, its **yield rate**, and its **current yield**.",
      "back": "**Coupon rate** $r$ is fixed by the contract and sets the cash coupon $Fr$.\n**Yield rate** $i$ (YTM) is the market discount rate that equates price to discounted cash flows; it moves with price.\n**Current yield** $=\\frac{Fr}{P}$ (annual coupon over price) — a quick income ratio that ignores the pull-to-redemption and is *not* the YTM (it lies between the coupon rate and the YTM for a discount/premium bond).",
      "tag": "Yield to maturity"
    },
    {
      "front": "What is a **callable bond**, and what is the **price-to-worst** principle for guaranteeing a yield?",
      "back": "A callable bond lets the **issuer** redeem early on one of several call dates. To guarantee at least a target yield $i$ regardless of when the issuer calls, the investor prices the bond at every candidate redemption date and pays the **minimum** of those prices — the \"price to worst.\" Paying that lowest price ensures the realized yield is at least $i$ for any call outcome.",
      "tag": "Callable bonds"
    },
    {
      "front": "For a callable bond selling at a **premium**, which call date should you use to price to the worst case, and why?",
      "back": "Use the **earliest** possible call date.\nAt a premium the investor is paying above redemption and recovering it through coupons; an **early** call cuts off coupons and gives the least time to amortize the premium, producing the **lowest** price (worst yield). Pricing to the earliest call guarantees the target yield even if the issuer calls right away.",
      "tag": "Callable bonds"
    },
    {
      "front": "For a callable bond selling at a **discount**, which call date is the worst case for the investor?",
      "back": "Use the **latest** possible date (usually final maturity).\nAt a discount the investor benefits from the pull-up to redemption, so **delaying** redemption is worst: it pushes the redemption gain furthest out and lowers the price the most. Pricing to the latest date guarantees the target yield even if the issuer never calls early.",
      "tag": "Callable bonds"
    },
    {
      "front": "A $\\$1{,}000$ par bond, $8\\%$ annual coupons, callable at par at the end of years $8$, $9$, or $10$, is to yield at least $6\\%$. Find the purchase price.",
      "back": "Coupon $>$ yield, so the bond is at a **premium** → worst case is the **earliest** call (year $8$).\nPrice at $n=8$: $a_{\\overline{8|}}=\\frac{1-1.06^{-8}}{0.06}\\approx 6.209794$, $1.06^{-8}\\approx 0.627412$.\n$P = 80(6.209794)+1000(0.627412)\\approx 496.784 + 627.412 = \\$1{,}124.20$.\nThis is the minimum across $n=8,9,10$, so paying $\\$1{,}124.20$ guarantees at least a $6\\%$ yield.",
      "tag": "Callable bonds"
    },
    {
      "front": "A $\\$1{,}000$ par bond, $4\\%$ annual coupons, redeemable at par, callable at the end of years $8$, $9$, or $10$, to yield at least $6\\%$. Find the price.",
      "back": "Coupon $<$ yield, so the bond is at a **discount** → worst case is the **latest** date (year $10$).\n$a_{\\overline{10|}}\\approx 7.360087$, $1.06^{-10}\\approx 0.558395$.\n$P = 40(7.360087)+1000(0.558395)\\approx 294.403 + 558.395 = \\$852.80$.\nThis is the minimum across the call dates, so $\\$852.80$ guarantees at least a $6\\%$ yield.",
      "tag": "Callable bonds"
    },
    {
      "front": "Why is the safest approach to callable-bond pricing simply to compute the price at **every** candidate date and take the minimum?",
      "back": "The premium/discount shortcut (earliest call if premium, latest if discount) can break when there is a **call premium** that varies by date — the redemption value differs across call dates, so the bond can be a premium at one date and a discount at another. Computing the price at each date and taking the **minimum** always yields the worst case directly, with no ambiguity, and guarantees the target yield.",
      "tag": "Callable bonds"
    },
    {
      "front": "A bond pays $\\$70$ annual coupons (par $\\$1{,}000$) and is callable at the end of year $5$ for $\\$1{,}050$ or held to maturity at year $10$ for $\\$1{,}000$. At a $6\\%$ yield, find the price guaranteeing $6\\%$.",
      "back": "Because the redemption value differs, price both scenarios:\nCall at year $5$ ($C=1050$): $70\\,a_{\\overline{5|}} + 1050\\,v^{5} = 70(4.212364)+1050(0.747258)\\approx 294.865 + 784.621 = 1079.49$.\nMaturity year $10$ ($C=1000$): $70\\,a_{\\overline{10|}} + 1000\\,v^{10}=70(7.360087)+1000(0.558395)\\approx 515.206 + 558.395 = 1073.60$.\nPrice to worst (minimum) $= \\$1{,}073.60$.",
      "tag": "Callable bonds"
    },
    {
      "front": "What are the **dirty (full)** price and the **clean (market)** price of a bond between coupon dates?",
      "back": "The **dirty price** (a.k.a. full or invoice price) is the total amount the buyer actually pays — the value of the bond including the portion of the next coupon that has accrued.\nThe **clean price** is the dirty price minus **accrued interest**: $\\text{clean} = \\text{dirty} - \\text{AI}$. Quoted/market prices are clean; settlement is on the dirty price.",
      "tag": "Between coupon dates"
    },
    {
      "front": "Define **accrued interest** between coupon dates (practical method) and give the formula.",
      "back": "Accrued interest is the share of the upcoming coupon earned by the seller for holding the bond part-way through the period: $\\text{AI} = t \\cdot Fr$, where $t$ is the fraction of the coupon period elapsed since the last coupon (computed on the relevant day-count basis).\nIt is added to the seller's proceeds because the buyer will collect the whole next coupon $Fr$.",
      "tag": "Between coupon dates"
    },
    {
      "front": "A bond's book value just after its last coupon is $\\$980.00$; the semiannual yield is $4\\%$ and each coupon is $\\$35$. Three months ($t=0.5$ of the period) later, find the **dirty** and **clean** prices (theoretical method).",
      "back": "Theoretical dirty price compounds the book value: $\\text{dirty}=B_{0}(1+i)^{t}=980.00(1.04)^{0.5}$. Since $1.04^{0.5}\\approx 1.019804$, dirty $\\approx \\$999.41$.\nAccrued interest $=t\\cdot Fr = 0.5(35)=\\$17.50$.\nClean $=999.41 - 17.50 = \\$981.91$.",
      "tag": "Between coupon dates"
    },
    {
      "front": "Contrast the **theoretical** and **practical (semi-theoretical)** methods for the dirty price between coupon dates.",
      "back": "**Theoretical:** $\\text{dirty}=B_{0}(1+i)^{t}$ — compound the prior book value over the fraction $t$.\n**Practical (semi-theoretical):** $\\text{dirty}=B_{0}(1+ti)$ — simple-interest accumulation over $t$.\nBoth subtract the same linear accrued interest $t\\cdot Fr$ to get the clean price. For the $\\$980$, $i=4\\%$, $Fr=35$, $t=0.5$ example: practical dirty $=980(1+0.5(0.04))=\\$999.60$, clean $=999.60-17.50=\\$982.10$, slightly above the theoretical $\\$981.91$.",
      "tag": "Between coupon dates"
    },
    {
      "front": "A bond pays semiannual coupons of $\\$40$. Two months after the last coupon, how much **accrued interest** does the buyer pay the seller?",
      "back": "A semiannual coupon period is $6$ months, so $t=\\frac{2}{6}=\\frac{1}{3}$ of the period has elapsed.\n$\\text{AI}=t\\cdot Fr = \\frac{1}{3}(40)\\approx \\$13.33$.\nThe buyer reimburses the seller this $\\$13.33$ (added on top of the clean price) because the buyer will receive the full $\\$40$ coupon at the next coupon date.",
      "tag": "Between coupon dates"
    },
    {
      "front": "Contrast the **30/360** and **actual/actual** day-count conventions, and explain how the chosen day-count feeds the **clean (market)** vs **dirty (full)** price relationship.",
      "back": "The day-count fixes $t$, the fraction of the coupon period elapsed, which drives **accrued interest** $\\text{AI}=t\\cdot Fr$.\n**30/360:** every month counts as $30$ days and a year as $360$, so a semiannual period is $180$ days; $t=\\frac{\\text{30/360 days elapsed}}{180}$. Common for U.S. corporate/municipal bonds.\n**Actual/actual:** uses the true calendar days, $t=\\frac{\\text{actual days elapsed}}{\\text{actual days in period}}$. Standard for U.S. Treasuries.\nThe relationship is always $\\text{clean}=\\text{dirty}-\\text{AI}$: quoted (market) prices are **clean**, but the buyer settles on the **dirty** (full/invoice) price $=$ clean $+$ AI.\nExample — coupon dates Jan 1 / Jul 1, settling Apr 1 (90 actual days elapsed, $181$-day period). Under **30/360**, $t=\\frac{90}{180}=0.5$. Under **actual/actual**, $t=\\frac{90}{181}\\approx 0.497238$. With a semiannual coupon $Fr=40$: $\\text{AI}_{30/360}=0.5(40)=\\$20.00$ versus $\\text{AI}_{\\text{act/act}}\\approx 0.497238(40)\\approx \\$19.89$.",
      "tag": "Between coupon dates"
    },
    {
      "front": "A $\\$1{,}000$ par bond with $8\\%$ annual coupons paid semiannually is priced to yield $6\\%$ convertible semiannually with $10$ years ($20$ periods) left. The book value just after the last coupon is $\\$1{,}148.77$. Using **30/360**, settlement falls $3$ months ($t=0.5$ of the period) later. Find the accrued interest and the **dirty** and **clean** prices.",
      "back": "Per semiannual period: coupon $Fr=\\frac{0.08}{2}(1000)=40$ and yield $i=\\frac{0.06}{2}=0.03$.\nUnder 30/360, $3$ months $=90$ of the period's $180$ days, so $t=\\frac{90}{180}=0.5$.\n**Accrued interest:** $\\text{AI}=t\\cdot Fr = 0.5(40)=\\$20.00$.\n**Dirty (full) price** (theoretical method, compounding the book value): $\\text{dirty}=B_{0}(1+i)^{t}=1148.77(1.03)^{0.5}$. Since $1.03^{0.5}\\approx 1.014889$, dirty $\\approx \\$1{,}165.87$.\n**Clean (market) price:** $\\text{clean}=\\text{dirty}-\\text{AI}=1165.87-20.00=\\$1{,}145.87$.\nThe buyer pays the dirty $\\$1{,}165.87$ at settlement; the bond is quoted at the clean $\\$1{,}145.87$, and the seller's $\\$20.00$ accrued interest compensates for the coupon earned so far.",
      "tag": "Between coupon dates"
    },
    {
      "front": "Define the **realized (reinvestment) yield** when coupons are reinvested at a rate $j$ that differs from the bond's yield.",
      "back": "The total accumulated value at maturity is $Fr\\,s_{\\overline{n|}}^{\\,j} + C$, where $s_{\\overline{n|}}^{\\,j}=\\frac{(1+j)^{n}-1}{j}$ accumulates the coupons at the **reinvestment rate** $j$, plus the redemption $C$.\nThe realized annual yield $i^{*}$ over the holding period solves $P(1+i^{*})^{n} = Fr\\,s_{\\overline{n|}}^{\\,j} + C$. If $j$ is below the YTM, the realized yield falls short of the YTM.",
      "tag": "Yield to maturity"
    },
    {
      "front": "A $\\$1{,}000$ par bond pays $\\$50$ annual coupons for $5$ years, redeems at par, and is bought for $\\$950$. If coupons are reinvested at $4\\%$, find the realized annual yield.",
      "back": "Accumulate coupons at $j=4\\%$: $s_{\\overline{5|}}=\\frac{1.04^{5}-1}{0.04}\\approx 5.416323$, so coupon FV $=50(5.416323)\\approx 270.82$.\nTotal at maturity $=270.82 + 1000 = \\$1{,}270.82$.\nRealized yield: $(1+i^{*})^{5}=\\frac{1270.82}{950}=1.337705$, so $1+i^{*}=1.337705^{1/5}$. Since $\\ln 1.337705\\approx 0.290957$, divide by $5$ to get $0.058191$ and $e^{0.058191}\\approx 1.059917$. Thus $i^{*}\\approx 5.99\\%$.",
      "tag": "Yield to maturity"
    }
  ]
}