Exam FM — Bonds Practice Flashcards

**Answer: (C).** This is the basic price formula $P = Fr\,a_{\overline{n|}} + C v^{n}$ with $Fr = 0.09(1000) = 90$, $C = 1000$, $i = 0.07$, $n = 12$. $a_{\overline{12|}} = \dfrac{1 - 1.07^{-12}}{0.07} \approx 7.942686$ and $1.07^{-12} \approx 0.444012$. $P = 90(7.942686) + 1000(0.444012) \approx 714.842 + 444.012 = \$1{,}158.85$. Since the $9\%$ coupon exceeds the $7\%$ yield, the bond sells at a premium ($P > C$). (Discounting the coupons as an annuity-*due* gives the distractor $\$1{,}208.89$; pricing the coupons at $9\%$ instead of the yield gives $\$1{,}000$.)

**Answer: (A).** Convert everything to the semiannual period: coupon $Fr = \dfrac{0.06}{2}(1000) = 30$, yield $i = \dfrac{0.08}{2} = 0.04$, periods $n = 7 \times 2 = 14$. $a_{\overline{14|}} = \dfrac{1 - 1.04^{-14}}{0.04} \approx 10.563123$ and $1.04^{-14} \approx 0.577475$. $P = 30(10.563123) + 1000(0.577475) \approx 316.894 + 577.475 = \$894.37$. The $3\%$ periodic coupon is below the $4\%$ periodic yield, so the bond is at a discount. (Forgetting to convert to semiannual and using $i=0.08$, $n=7$, $Fr=60$ gives the wrong-convention distractor near $\$1{,}000$.)

**Answer: (C).** Makeham: $P = K + \dfrac{g}{i}(C - K)$, where $K = C v^{n}$ and $g = \dfrac{Fr}{C}$. Here $C = 1000$, $g = \dfrac{80}{1000} = 0.08$, $i = 0.06$, $n = 20$. $K = 1000(1.06)^{-20} = 1000(0.311805) \approx 311.805$. $P = 311.805 + \dfrac{0.08}{0.06}(1000 - 311.805) = 311.805 + 1.\overline{3}(688.195) \approx 311.805 + 917.593 = \$1{,}229.40$. A direct check $80\,a_{\overline{20|}} + 1000 v^{20}$ gives the same $\$1{,}229.40$.

**Answer: (B).** Coupons are computed on the face: $Fr = 0.07(1000) = 70$. The redemption uses $C = 1100$, not the face. $a_{\overline{15|}} = \dfrac{1 - 1.07^{-15}}{0.07} \approx 9.107914$ and $1.07^{-15} \approx 0.362446$. $P = 70(9.107914) + 1100(0.362446) \approx 637.554 + 398.691 = \$1{,}036.24$. Even though the coupon rate equals the yield rate, the price exceeds par because $C > F$. (Using $C = 1000$ instead of $1100$ in the redemption term gives exactly $\$1{,}000$.)

**Answer: (B).** The correct premium/discount test uses the standard notation $F$ (face), $C$ (redemption), $Fr$ (coupon), and $i$ (yield): when $C \neq F$, compare the coupon $Fr$ to $Ci$, **not** the coupon rate to the yield rate. Equivalently, compare the modified coupon rate $g = \dfrac{Fr}{C}$ to $i$. Here $Fr = 55$ and $Ci = 1050(0.05) = 52.50$ (and $g = \dfrac{55}{1050} \approx 5.24\% > i = 5\%$). Since $Fr = 55 > Ci = 52.50$, the bond sells at a **premium** relative to its redemption value $C$ (i.e., $P > C = 1050$). Choice (A) reaches the right conclusion by the wrong test — comparing the $5.5\%$ face-coupon rate to $5\%$ ignores that coupons are on $F = 1000$ but the test rate applies to $C = 1050$. Choice (C) incorrectly uses $Ci = 1050(0.055)$.

**Answer: (C).** With no coupons, $620 = 1000(1+i)^{-8}$, so $(1+i)^{8} = \dfrac{1000}{620} = 1.612903$. $1 + i = 1.612903^{1/8}$. Since $\ln 1.612903 \approx 0.477934$, dividing by $8$ gives $0.059742$, and $e^{0.059742} \approx 1.061563$. $i \approx 6.16\%$. Using simple interest, $\dfrac{1000/620 - 1}{8} \approx 7.66\%$ (distractor D); inverting the ratio gives the negative distractor (A).

**Answer: (C).** Use the prospective formula on the $n - t = 10 - 5 = 5$ remaining coupons at the original yield $i = 6\%$: $B_{5} = Fr\,a_{\overline{5|}} + C v^{5} = 80\,a_{\overline{5|}} + 1000 v^{5}$. $a_{\overline{5|}} = \dfrac{1 - 1.06^{-5}}{0.06} \approx 4.212364$ and $1.06^{-5} \approx 0.747258$. $B_{5} = 80(4.212364) + 1000(0.747258) \approx 336.989 + 747.258 = \$1{,}084.25$. As a premium bond, $B_{5}$ lies between the purchase price $\$1{,}147.20$ and the redemption $\$1{,}000$, amortizing toward $C$.

**Answer: (B).** First find the price: $P = 100\,a_{\overline{8|}} + 1000 v^{8}$ at $6\%$. With $a_{\overline{8|}} \approx 6.209794$ and $1.06^{-8} \approx 0.627412$, $P \approx 620.979 + 627.412 = 1248.39 = B_{0}$. Interest earned in coupon 1 $= i B_{0} = 0.06(1248.39) \approx 74.90$. Write-down (principal adjustment) $= Fr - iB_{0} = 100 - 74.90 = \$25.10$. Check with the direct formula $(Fr - Ci)v^{\,n} = (100 - 60)(1.06)^{-8} = 40(0.627412) \approx \$25.10$. (The $\$74.90$ distractor is the interest earned, not the write-down.)