Exam FM — Annuities (Level) Practice Flashcards

**Answer: (B).** The present value of an annuity-immediate is $800\,a_{\overline{12|}}$ at $i=0.06$. $a_{\overline{12|}}=\dfrac{1-(1.06)^{-12}}{0.06}=\dfrac{1-0.496969}{0.06}=\dfrac{0.503031}{0.06}=8.383844$. PV $=800\times 8.383844=\$6{,}707.08$. (C) uses the annuity-*due* value $800(1.06)a_{\overline{12|}}$. (E) just sums the undiscounted payments $800\times 12$. (A) discounts one extra year ($800\,v\,a_{\overline{12|}}$).

**Answer: (E).** For an annuity-due, $\ddot{a}_{\overline{8|}}=(1+i)\,a_{\overline{8|}}$ at $i=0.07$. $a_{\overline{8|}}=\dfrac{1-(1.07)^{-8}}{0.07}=\dfrac{1-0.582009}{0.07}=\dfrac{0.417991}{0.07}=5.971299$. $\ddot{a}_{\overline{8|}}=1.07\times 5.971299=6.389290$. PV $=1500\times 6.389290=\$9{,}583.94\approx\$9{,}584$. (C) drops the due adjustment (uses $a_{\overline{8|}}$): $1500\times 5.971299=\$8{,}957$. (B) uses a $7$-year annuity-due $\ddot{a}_{\overline{7|}}$ (one payment short). (A) uses the $7$-year annuity-*immediate* $a_{\overline{7|}}$.

**Answer: (C).** The accumulated value at time $15$ is $2000\,s_{\overline{15|}}$ at $i=0.05$. $s_{\overline{15|}}=\dfrac{(1.05)^{15}-1}{0.05}=\dfrac{2.078928-1}{0.05}=\dfrac{1.078928}{0.05}=21.578564$. AV $=2000\times 21.578564=\$43{,}157.13$. (D) applies the annuity-due accumulation $\ddot{s}_{\overline{15|}}=(1.05)s_{\overline{15|}}$ in error. (B) uses $s_{\overline{14|}}$ (off-by-one term). (A) just sums the deposits.

**Answer: (C).** Deposits at the start of each year accumulate as an annuity-due to time $10$: $1000\,\ddot{s}_{\overline{10|}}=1000(1+i)s_{\overline{10|}}$ at $i=0.08$. $s_{\overline{10|}}=\dfrac{(1.08)^{10}-1}{0.08}=\dfrac{2.158925-1}{0.08}=\dfrac{1.158925}{0.08}=14.486562$. $\ddot{s}_{\overline{10|}}=1.08\times 14.486562=15.645487$. AV $=1000\times 15.645487=\$15{,}645.49$. (B) forgets the extra year of interest (uses $s_{\overline{10|}}$ — the value at time $9$, not $10$): $1000\times 14.486562=\$14{,}487$. (A) uses $i=0.06$ by mistake (and the immediate accumulation): $1000\,s_{\overline{10|}}$ at $6\%=\$13{,}181$. (E) sums the deposits undiscounted as accumulation by another error.

**Answer: (D).** A perpetuity-immediate has present value $\dfrac{\text{payment}}{i}$. PV $=\dfrac{3500}{0.055}=\$63{,}636.36$. (E) uses the perpetuity-*due* value $\dfrac{3500}{d}$ with $d=\dfrac{0.055}{1.055}=0.052133$, giving $\dfrac{3500}{0.052133}=\$67{,}136$. (B) discounts the immediate value one extra year ($\dfrac{3500}{0.055}\cdot v$). (A) discounts it two extra years.

**Answer: (C).** A perpetuity-due has present value $\dfrac{\text{payment}}{d}$, where $d=\dfrac{i}{1+i}=\dfrac{0.06}{1.06}=0.056604$. PV $=\dfrac{2400}{0.056604}=\$42{,}400.00$. Equivalently $\dfrac{2400}{0.06}+2400=40{,}000+2400=\$42{,}400$. (B) is the perpetuity-*immediate* value $\dfrac{2400}{0.06}$ (drops the immediate first payment). (A) divides by $(1+i)$ in error.

**Answer: (A).** The first payment is at time $6$, so this is a $20$-year annuity-immediate deferred $5$ years: PV $=1000\,v^{5}a_{\overline{20|}}$ at $i=0.07$. $a_{\overline{20|}}=\dfrac{1-(1.07)^{-20}}{0.07}=\dfrac{1-0.258419}{0.07}=\dfrac{0.741581}{0.07}=10.594014$. $v^{5}=(1.07)^{-5}=0.712986$. PV $=1000\times 0.712986\times 10.594014=\$7{,}553.39$. (B) defers only $4$ years (treats the first payment as time $5$, i.e. $v^{4}$). (D) forgets to defer (uses $a_{\overline{20|}}$ alone). (E) defers but uses $a_{\overline{20|}}$ undiscounted plus an erroneous accumulation.

**Answer: (B).** Value the perpetuity one period before its first payment. The first payment is at time $9$, so value it at time $8$: $\dfrac{5000}{0.04}=125{,}000$. Then discount $8$ years to today. $v^{8}=(1.04)^{-8}=0.730690$. PV $=125{,}000\times 0.730690=\$91{,}336.28$. (A) discounts $9$ years instead of $8$ (values the perpetuity at time $9$, not time $8$). (C) discounts $7$ years. (E) forgets to discount the time-$8$ value.