Exam FM — Cash-Flow Analysis (NPV / IRR / Yield) Practice Flashcards

**Answer: (A).** The four returns form a level annuity-immediate. With $i=0.07$, $a_{\overline{4|}}=\dfrac{1-1.07^{-4}}{0.07}=\dfrac{1-0.762895}{0.07}\approx 3.387211.$ Present value of inflows: $9000\times 3.387211\approx 30{,}484.90.$ $\mathrm{NPV}=30{,}484.90-25{,}000\approx \mathbf{\$5{,}485}.$ (Option (E) forgets to subtract the cost; (D) is the undiscounted total $36{,}000-25{,}000$.)

**Answer: (C).** Solve $12{,}000(1+i)^{3}=15{,}972$, so $(1+i)^{3}=\dfrac{15{,}972}{12{,}000}=1.331.$ $1+i=1.331^{1/3}=1.10000$, giving $i=\mathbf{10.0\%}.$ (Option (E) is the total three-year return $33.1\%$ left un-annualized; (D) divides that total by 3 instead of taking the cube root.)

**Answer: (C).** Let $v=(1+i)^{-1}$. The equation of value is $5000=2500v+3300v^{2}$, i.e. $3300v^{2}+2500v-5000=0.$ $v=\dfrac{-2500+\sqrt{2500^{2}+4(3300)(5000)}}{2(3300)}=\dfrac{-2500+\sqrt{6{,}250{,}000+66{,}000{,}000}}{6600}=\dfrac{-2500+\sqrt{72{,}250{,}000}}{6600}.$ $\sqrt{72{,}250{,}000}=8500$, so $v=\dfrac{6000}{6600}=0.909091$ and $1+i=\dfrac{1}{0.909091}=1.10.$ Thus $i=\mathbf{10.0\%}.$ Check: $2500/1.1+3300/1.21=2272.73+2727.27=5000.$ (Option (E) comes from solving the quadratic with a sign error on the linear term.)

**Answer: (D).** Linear interpolation for the rate at which NPV crosses zero: $i^{*}\approx 6\%+(9\%-6\%)\cdot\dfrac{420}{420-(-180)}=6\%+3\%\cdot\dfrac{420}{600}.$ $\dfrac{420}{600}=0.70$, so $i^{*}\approx 6\%+3\%(0.70)=6\%+2.1\%=\mathbf{8.1\%}.$ (Option (A) uses the wrong fraction $180/600$; the IRR sits closer to $9\%$ because the NPV magnitude at $6\%$ is the larger of the two.)

**Answer: (C).** At $i=0.10$: $\mathrm{NPV}=-8000-\dfrac{3000}{1.10}+\dfrac{13000}{1.10^{2}}=-8000-2727.27+\dfrac{13000}{1.21}.$ $\dfrac{13000}{1.21}=10743.80,$ so $\mathrm{NPV}=-8000-2727.27+10743.80\approx \mathbf{\$16.53}\approx \$17.$ The cash flows have a single sign change ($-,-,+$), so the IRR is unique and NPV is a decreasing function of $i$. Because $\mathrm{NPV}(10\%)>0$, the curve crosses zero at a rate **above** $10\%$. Hence the IRR is just above $10\%$. (Option (B) gets the NPV right but the IRR direction backward — positive NPV at $10\%$ implies the yield exceeds $10\%$.)

**Answer: (C).** IRRs: Project X gives $\dfrac{11{,}600}{10{,}000}-1=16.0\%.$ Project Y gives $\left(\dfrac{13{,}300}{10{,}000}\right)^{1/2}-1=\sqrt{1.33}-1\approx 15.33\%.$ So **IRR ranks X above Y**. NPVs at $8\%$: $\mathrm{NPV}_X=\dfrac{11{,}600}{1.08}-10{,}000=10{,}740.74-10{,}000=740.74.$ $\mathrm{NPV}_Y=\dfrac{13{,}300}{1.08^{2}}-10{,}000=\dfrac{13{,}300}{1.1664}-10{,}000=11{,}402.61-10{,}000=1{,}402.61.$ So **NPV ranks Y above X**. The rules conflict because IRR ignores the longer horizon and the scale of value created. At an $8\%$ hurdle the correct choice is the higher-NPV project, Y. Thus NPV picks Y while IRR picks X.

**Answer: (A).** With $v=(1+i)^{-1}$, set NPV to zero: $-1000+2350v-1365v^{2}=0,$ i.e. $1365v^{2}-2350v+1000=0.$ $v=\dfrac{2350\pm\sqrt{2350^{2}-4(1365)(1000)}}{2(1365)}=\dfrac{2350\pm\sqrt{5{,}522{,}500-5{,}460{,}000}}{2730}=\dfrac{2350\pm\sqrt{62{,}500}}{2730}.$ $\sqrt{62{,}500}=250$, so $v=\dfrac{2350\pm 250}{2730}.$ $v=\dfrac{2600}{2730}=0.952381\Rightarrow 1+i=1.05\Rightarrow i=5\%.$ $v=\dfrac{2100}{2730}=0.769231\Rightarrow 1+i=1.30\Rightarrow i=30\%.$ The two sign changes produce two legitimate yields, $\mathbf{5\%}$ and $\mathbf{30\%}$, so the IRR alone is ambiguous here.

**Answer: (C).** The sufficient condition for a unique positive IRR is exactly one sign change in the ordered cash-flow sequence (all outflows before all inflows, or vice versa). - (A) $-,+,-$: two sign changes. - (B) $+,-,+$: two sign changes. - (C) $-,-,+$: **one** sign change — guaranteed unique IRR. - (D) $-,+,-,+$: three sign changes. - (E) $+,-,+,+$: two sign changes. Only sequence (C) has a single sign change, so it is the only one guaranteed a unique positive yield.