Exam FM — Annuities (Varying & Continuous) Practice Flashcards

**Answer: (B).** The payments are $4,8,12,\ldots,32 = 4\times(1,2,\ldots,8)$, so the present value is $4\,(Ia)_{\overline{8|}}$. At $i=0.06$: $v^{8}=1.06^{-8}\approx 0.627412$, $a_{\overline{8|}}=\dfrac{1-v^{8}}{0.06}\approx 6.209794$, and $\ddot{a}_{\overline{8|}}=a_{\overline{8|}}(1.06)\approx 6.582382$. $(Ia)_{\overline{8|}}=\dfrac{\ddot{a}_{\overline{8|}}-8v^{8}}{i}=\dfrac{6.582382-8(0.627412)}{0.06}=\dfrac{6.582382-5.019296}{0.06}\approx 26.0514$. PV $=4(26.0514)\approx \mathbf{\$104.21}$.

**Answer: (D).** The payments are $200,190,\ldots,10 = 10\times(20,19,\ldots,1)$, so PV $=10\,(Da)_{\overline{20|}}$. At $i=0.07$: $a_{\overline{20|}}=\dfrac{1-1.07^{-20}}{0.07}\approx 10.594014$. $(Da)_{\overline{20|}}=\dfrac{n-a_{\overline{n|}}}{i}=\dfrac{20-10.594014}{0.07}=\dfrac{9.405986}{0.07}\approx 134.3712$. PV $=10(134.3712)\approx \mathbf{\$1{,}343.71}$. (Distractor (A) comes from mistakenly using $(Ia)_{\overline{20|}}$ instead of $(Da)_{\overline{20|}}$.)

**Answer: (D).** The payments are $5\times(1,2,3,\ldots)$, so PV $=5\,(Ia)_{\overline{\infty|}}$. For an increasing perpetuity-immediate, $(Ia)_{\overline{\infty|}}=\dfrac{1}{i}+\dfrac{1}{i^{2}}=\dfrac{1}{0.05}+\dfrac{1}{0.05^{2}}=20+400=420$. PV $=5(420)=\mathbf{\$2{,}100}$. (Distractor (A) is the level part $5/i$ only; (C) is $5/i^{2}$ only; (E) multiplies by $(1+i)$ as if it were a perpetuity-due.)

**Answer: (D).** Use the general arithmetic formula with $P=1000$, $Q=50$, $n=15$, $i=0.06$: $\operatorname{PV}=P\,a_{\overline{n|}}+Q\,\dfrac{a_{\overline{n|}}-n\,v^{n}}{i}.$ At $6\%$: $a_{\overline{15|}}\approx 9.712249$, $v^{15}=1.06^{-15}\approx 0.417265$. $\dfrac{a_{\overline{15|}}-15v^{15}}{i}=\dfrac{9.712249-15(0.417265)}{0.06}=\dfrac{9.712249-6.258978}{0.06}\approx 57.5545$. PV $=1000(9.712249)+50(57.5545)=9712.25+2877.73\approx \mathbf{\$12{,}590}$. (Distractor (A) is the level $\$1000$ piece only; (E) wrongly multiplies the full $(Ia)$ by $Q$ instead of the increasing-only piece.)

**Answer: (C).** Geometric annuity with $P=5000$, $g=0.04$, $i=0.08$, $n=25$: $\operatorname{PV}=\dfrac{P}{i-g}\left[1-\left(\dfrac{1+g}{1+i}\right)^{n}\right]=\dfrac{5000}{0.04}\left[1-\left(\dfrac{1.04}{1.08}\right)^{25}\right].$ $\left(\dfrac{1.04}{1.08}\right)^{25}=(0.962963)^{25}\approx 0.389260$. PV $=125000\,(1-0.389260)=125000(0.610740)\approx \mathbf{\$76{,}343}$. (Distractor (E) is $\dfrac{P}{i-g}$ treated as a perpetuity; (A) uses $i+g$ in the denominator.)

**Answer: (B).** The force of interest is $\delta=\ln(1.06)\approx 0.0582689$. $v^{20}=1.06^{-20}\approx 0.311805$, so $\bar{a}_{\overline{20|}}=\dfrac{1-v^{20}}{\delta}=\dfrac{1-0.311805}{0.0582689}=\dfrac{0.688195}{0.0582689}\approx \mathbf{11.81}.$ Check: $a_{\overline{20|}}\approx 11.469921$ and $\bar{a}_{\overline{20|}}=a_{\overline{20|}}\cdot\dfrac{i}{\delta}=11.469921\times\dfrac{0.06}{0.0582689}\approx 11.81$. ✓ (Distractor (A) is the discrete annuity-immediate $a_{\overline{20|}}$.)

**Answer: (D).** This is the accumulated increasing annuity $(Is)_{\overline{12|}}=(Ia)_{\overline{12|}}(1+i)^{12}$. At $5\%$: $v^{12}=1.05^{-12}\approx 0.556837$, $a_{\overline{12|}}\approx 8.863252$, $\ddot{a}_{\overline{12|}}=a_{\overline{12|}}(1.05)\approx 9.306414$. $(Ia)_{\overline{12|}}=\dfrac{\ddot{a}_{\overline{12|}}-12v^{12}}{i}=\dfrac{9.306414-12(0.556837)}{0.05}=\dfrac{9.306414-6.682049}{0.05}\approx 52.4873$. $(Is)_{\overline{12|}}=52.4873\,(1.05)^{12}=52.4873(1.795856)\approx \mathbf{94.26}$. (Distractor (A) is $(Ia)_{\overline{12|}}$ — the present value, not accumulated.)

**Answer: (C).** The payments are $50,48,\ldots,2 = 2\times(25,24,\ldots,1)$, so PV $=2\,(Da)_{\overline{25|}}$. At $i=0.05$: $a_{\overline{25|}}=\dfrac{1-1.05^{-25}}{0.05}\approx 14.093945$. $(Da)_{\overline{25|}}=\dfrac{n-a_{\overline{n|}}}{i}=\dfrac{25-14.093945}{0.05}=\dfrac{10.906055}{0.05}\approx 218.1211$. PV $=2(218.1211)\approx \mathbf{\$436.24}$.