Exam FM — Bonds Flashcards

$F$ = **face (par) value**, the amount on which coupons are computed. $r$ = **coupon rate per period**; the periodic coupon is $Fr$. $C$ = **redemption value** paid at maturity (often $C=F$, but not always). $i$ = **yield rate per period** used to discount. $n$ = **number of coupons** remaining. $g$ = **modified coupon rate** $=\frac{Fr}{C}$, the coupon as a fraction of the redemption value.

**Face value** = **par value** $F$ — the printed amount that coupons are calculated on ($Fr$). **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 ($C<F$). Coupons always use $F$; the maturity payment and the premium/discount test use $C$.

$P = Fr\,a_{\overline{n|}} + C v^{n}$, where $a_{\overline{n|}}=\frac{1-v^{n}}{i}$ is the annuity-immediate factor and $v=(1+i)^{-1}$. The price is the present value of the coupon stream $Fr$ plus the present value of the redemption $C$.

Here $Fr=80$, $C=1000$, $i=0.06$, $n=10$. $a_{\overline{10|}}=\frac{1-1.06^{-10}}{0.06}\approx 7.36009$ and $1.06^{-10}\approx 0.558395$. $P = 80(7.36009) + 1000(0.558395) \approx 588.807 + 558.395 = \$1{,}147.20$. Since $P>C$ the bond sells at a **premium** (coupon rate $>$ yield).

$P = C + (Fr - Ci)\,a_{\overline{n|}}$. The 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.

Compare the coupon to the yield (relative to the redemption value): **Premium** when $Fr>Ci$, i.e. the coupon rate exceeds the yield. Then $P>C$ and the book value **amortizes down** to $C$. **Discount** when $Fr<Ci$, i.e. the coupon rate is below the yield. Then $P<C$ and the book value **accretes up** to $C$. When $C=F$ this is simply coupon rate vs yield rate.

**Premium** $= P - C$ (used when $P>C$): the excess paid above redemption, recovered by writing book value down. **Discount** $= C - P$ (used when $P<C$): the shortfall below redemption, recovered by writing book value up. Either way, over the bond's life the total write-up/write-down equals $|P-C|$, bringing book value to exactly $C$ at maturity.

$Fr=50$, $C=1000$, $i=0.07$, $n=8$. $a_{\overline{8|}}=\frac{1-1.07^{-8}}{0.07}\approx 5.971299$ and $1.07^{-8}\approx 0.582009$. $P = 50(5.971299) + 1000(0.582009)\approx 298.565 + 582.009 = \$880.57$. Since $Fr=50 < Ci=70$, it is a **discount** bond; discount $=1000-880.57=\$119.43$.

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. $a_{\overline{20|}}=\frac{1-1.04^{-20}}{0.04}\approx 13.590326$ and $1.04^{-20}\approx 0.456387$. $P = 35(13.590326)+1000(0.456387)\approx 475.661 + 456.387 = \$932.05$. It is a **discount** bond since the $3.5\%$ periodic coupon is below the $4\%$ periodic yield.

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 $n$. Exam 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.

$P = K + \frac{g}{i}\,(C - K)$, where $K = C v^{n}$ is the present value of the redemption payment and $g=\frac{Fr}{C}$ is the modified coupon rate. The 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 $i$.

$C=1000$, $g=\frac{70}{1000}=0.07$, $i=0.05$, $n=15$. $K = 1000(1.05)^{-15} = 1000(0.481017)\approx 481.017$. $P = K + \frac{g}{i}(C-K) = 481.017 + \frac{0.07}{0.05}(1000-481.017)$ $= 481.017 + 1.4(518.983)\approx 481.017 + 726.576 = \$1{,}207.59$.

Coupons use the face: $Fr=60$. The redemption uses $C=1050$. $a_{\overline{12|}}=\frac{1-1.06^{-12}}{0.06}\approx 8.383844$ and $1.06^{-12}\approx 0.496969$. $P = 60(8.383844) + 1050(0.496969)\approx 503.031 + 521.817 = \$1{,}024.85$. Note the bond is above par even though coupon rate $=$ yield, because $C>F$.

Use $Ci$, not the coupon rate on face. The bond is at a premium when $Fr>Ci$ and a discount when $Fr<Ci$. Example: $F=1000$, 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.

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. Prospectively $B_{t}=Fr\,a_{\overline{n-t|}} + C v^{n-t}$, 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 $C$.

Recursion: $B_{t}=B_{t-1}(1+i) - Fr$. Interest earned in the period $= i\,B_{t-1}$. Principal 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).

Write-down in coupon $t$ $= (Fr - Ci)\,v^{\,n-t+1}$. It 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 $C$.

$B_{0}=1140.47$, $i=0.07$, $Fr=90$. Interest earned $= iB_{0}=0.07(1140.47)\approx \$79.83$. Write-down (principal adjustment) $= Fr - iB_{0}=90 - 79.83 = \$10.17$. Check via the formula $(Fr-Ci)v^{\,n}=(90-70)(1.07)^{-10}=20(0.508349)\approx \$10.17$. New book value $B_{1}=1140.47-10.17=\$1{,}130.30$.

Use $(Fr-Ci)\,v^{\,n-t+1}$ with $Fr-Ci=90-70=20$, $i=0.07$, $n=10$, $t=3$, so the exponent is $10-3+1=8$. $1.07^{-8}\approx 0.582009$. Write-down $=20(0.582009)\approx \$11.64$. It exceeds the first coupon's $\$10.17$ write-down, confirming write-downs grow by $(1+i)$ each period.

Write-up in coupon $t$ $= (Ci - Fr)\,v^{\,n-t+1}$. It 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.

$B_{0}=875.80$, $i=0.06$, $Fr=40$. Interest earned $= iB_{0}=0.06(875.80)\approx \$52.55$. The coupon paid is only $40$, so the shortfall is added to book value: write-up $=52.55-40=\$12.55$. Check: $(Ci-Fr)v^{\,n}=(60-40)(1.06)^{-8}=20(0.627412)\approx \$12.55$. New book value $B_{1}=875.80+12.55=\$888.35$.

Use the prospective formula on the remaining $n-t=5$ coupons: $B_{5}=Fr\,a_{\overline{5|}} + C v^{5}$ at $i=7\%$. $a_{\overline{5|}}=\frac{1-1.07^{-5}}{0.07}\approx 4.100197$, $1.07^{-5}\approx 0.712986$. $B_{5}=90(4.100197)+1000(0.712986)\approx 369.018 + 712.986 = \$1{,}082.00$. Being a premium bond, $B_{5}$ sits between the price $1140.47$ and the redemption $1000$.

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 $P = Fr\,a_{\overline{n|}} + C v^{n}$. It assumes the bond is held to maturity and that all coupons are reinvested at that same rate $i$.

With no coupons, $600 = 1000(1+i)^{-10}$, so $(1+i)^{10}=\frac{1000}{600}=1.\overline{6}$. $1+i = 1.6667^{1/10}$. Since $\ln 1.6667 \approx 0.510826$, divide by $10$ to get $0.051083$, and $e^{0.051083}\approx 1.05241$. $i \approx 5.241\%$ annual effective.

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. Remember to interpret $I/Y$ as the per-coupon-period rate when coupons are not annual.

Solve $1050 = Fr\,a_{\overline{10|}} + 1000\,v^{10}$ at $i=6\%$. $a_{\overline{10|}}\approx 7.360087$, $1.06^{-10}\approx 0.558395$. $Fr = \frac{1050 - 1000(0.558395)}{7.360087} = \frac{1050 - 558.395}{7.360087}=\frac{491.605}{7.360087}\approx 66.79$. Coupon rate $=\frac{66.79}{1000}\approx 6.68\%$. (Coupon $>$ yield, consistent with the premium price.)

Solve $1000 = 60\,a_{\overline{10|}} + C v^{10}$ at $i=6.5\%$. $a_{\overline{10|}}=\frac{1-1.065^{-10}}{0.065}\approx 7.188830$, $1.065^{-10}\approx 0.532726$. $C = \frac{1000 - 60(7.188830)}{0.532726}=\frac{1000 - 431.330}{0.532726}=\frac{568.670}{0.532726}\approx \$1{,}067.47$.

**Coupon rate** $r$ is fixed by the contract and sets the cash coupon $Fr$. **Yield rate** $i$ (YTM) is the market discount rate that equates price to discounted cash flows; it moves with price. **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).

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.

Use the **earliest** possible call date. At 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.

Use the **latest** possible date (usually final maturity). At 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.

Coupon $>$ yield, so the bond is at a **premium** → worst case is the **earliest** call (year $8$). Price at $n=8$: $a_{\overline{8|}}=\frac{1-1.06^{-8}}{0.06}\approx 6.209794$, $1.06^{-8}\approx 0.627412$. $P = 80(6.209794)+1000(0.627412)\approx 496.784 + 627.412 = \$1{,}124.20$. This is the minimum across $n=8,9,10$, so paying $\$1{,}124.20$ guarantees at least a $6\%$ yield.

Coupon $<$ yield, so the bond is at a **discount** → worst case is the **latest** date (year $10$). $a_{\overline{10|}}\approx 7.360087$, $1.06^{-10}\approx 0.558395$. $P = 40(7.360087)+1000(0.558395)\approx 294.403 + 558.395 = \$852.80$. This is the minimum across the call dates, so $\$852.80$ guarantees at least a $6\%$ yield.

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.

Because the redemption value differs, price both scenarios: Call at year $5$ ($C=1050$): $70\,a_{\overline{5|}} + 1050\,v^{5} = 70(4.212364)+1050(0.747258)\approx 294.865 + 784.621 = 1079.49$. Maturity year $10$ ($C=1000$): $70\,a_{\overline{10|}} + 1000\,v^{10}=70(7.360087)+1000(0.558395)\approx 515.206 + 558.395 = 1073.60$. Price to worst (minimum) $= \$1{,}073.60$.

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. The **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.

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). It is added to the seller's proceeds because the buyer will collect the whole next coupon $Fr$.

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 $\approx \$999.41$. Accrued interest $=t\cdot Fr = 0.5(35)=\$17.50$. Clean $=999.41 - 17.50 = \$981.91$.

**Theoretical:** $\text{dirty}=B_{0}(1+i)^{t}$ — compound the prior book value over the fraction $t$. **Practical (semi-theoretical):** $\text{dirty}=B_{0}(1+ti)$ — simple-interest accumulation over $t$. Both 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 $\$981.91$.

A semiannual coupon period is $6$ months, so $t=\frac{2}{6}=\frac{1}{3}$ of the period has elapsed. $\text{AI}=t\cdot Fr = \frac{1}{3}(40)\approx \$13.33$. The 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.

The day-count fixes $t$, the fraction of the coupon period elapsed, which drives **accrued interest** $\text{AI}=t\cdot Fr$. **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. **Actual/actual:** uses the true calendar days, $t=\frac{\text{actual days elapsed}}{\text{actual days in period}}$. Standard for U.S. Treasuries. The 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. Example — 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 $\text{AI}_{\text{act/act}}\approx 0.497238(40)\approx \$19.89$.

Per semiannual period: coupon $Fr=\frac{0.08}{2}(1000)=40$ and yield $i=\frac{0.06}{2}=0.03$. Under 30/360, $3$ months $=90$ of the period's $180$ days, so $t=\frac{90}{180}=0.5$. **Accrued interest:** $\text{AI}=t\cdot Fr = 0.5(40)=\$20.00$. **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 $\approx \$1{,}165.87$. **Clean (market) price:** $\text{clean}=\text{dirty}-\text{AI}=1165.87-20.00=\$1{,}145.87$. The 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.

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 $C$. The 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.

Accumulate coupons at $j=4\%$: $s_{\overline{5|}}=\frac{1.04^{5}-1}{0.04}\approx 5.416323$, so coupon FV $=50(5.416323)\approx 270.82$. Total at maturity $=270.82 + 1000 = \$1{,}270.82$. Realized 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 $i^{*}\approx 5.99\%$.