{ "deckName": "Exam FM — Cash-Flow Analysis (NPV / IRR / Yield)", "examCode": "Exam FM", "cards": [ { "front": "Define the **net present value (NPV)** of a series of cash flows $C_{t}$ valued at interest rate $i$.", "back": "$\\mathrm{NPV}(i)=\\sum_{t} C_{t}\\,v^{t}=\\sum_{t} \\frac{C_{t}}{(1+i)^{t}}$, where $v=(1+i)^{-1}$ and inflows are positive, outflows negative.\nA project is acceptable on a stand-alone basis when $\\mathrm{NPV}(i)\\geq 0$ at the investor's required rate of return $i$.", "tag": "NPV & IRR" }, { "front": "What is the **internal rate of return (IRR)** of a cash-flow stream, and how is it characterized?", "back": "The IRR (or **yield rate**) $i^{*}$ is the rate that makes the net present value zero: $\\mathrm{NPV}(i^{*})=\\sum_{t} C_{t}\\,(1+i^{*})^{-t}=0$.\nEquivalently it is the rate at which the present value of inflows equals the present value of outflows — the solution of the equation of value.", "tag": "NPV & IRR" }, { "front": "How does $\\mathrm{NPV}(i)$ typically behave as the interest rate $i$ rises, for a normal investment (outflow first, inflows later)?", "back": "$\\mathrm{NPV}(i)$ is a **decreasing** function of $i$: discounting the later inflows more heavily lowers their present value.\nThe IRR is where this downward-sloping curve crosses zero. Hence if your required rate is **below** the IRR the NPV is positive (accept); above the IRR it is negative (reject).", "tag": "NPV & IRR" }, { "front": "A project costs $\\$10{,}000$ today and returns $\\$4{,}000$ at the end of year 1, $\\$4{,}000$ at end of year 2, and $\\$5{,}000$ at end of year 3. Find the NPV at an annual effective rate of $8\\%$.", "back": "Discount each inflow at $i=0.08$:\n$4000v=4000/1.08\\approx 3703.70$.\n$4000v^{2}=4000/1.08^{2}\\approx 3429.36$.\n$5000v^{3}=5000/1.08^{3}\\approx 3969.16$.\nSum of inflows $\\approx 11{,}102.22$. Subtract the cost: $\\mathrm{NPV}=11{,}102.22-10{,}000\\approx \\$1{,}102.22$.\nNPV $>0$, so the project clears an $8\\%$ hurdle.", "tag": "NPV & IRR" }, { "front": "You invest $\\$1{,}000$ now and receive $\\$1{,}300$ in exactly two years (no intermediate flows). Find the annual effective yield (IRR).", "back": "Solve $1000(1+i)^{2}=1300$, i.e. $(1+i)^{2}=1.3$.\n$1+i=\\sqrt{1.3}\\approx 1.140175$.\n$i\\approx 0.140175$, so the yield is about $\\mathbf{14.02\\%}$ annual effective.", "tag": "NPV & IRR" }, { "front": "An investor pays $\\$5{,}000$ today and receives $\\$2{,}000$ at the end of year 1 and $\\$3{,}500$ at the end of year 2. Set up the equation of value for the yield $i$ and verify that $i\\approx 9.5\\%$ is close.", "back": "Equation of value (PV of inflows = outflow):\n$5000=\\frac{2000}{1+i}+\\frac{3500}{(1+i)^{2}}$.\nAt $i=0.095$: $2000/1.095\\approx 1826.48$ and $3500/1.095^{2}\\approx 2919.04$; sum $\\approx 4745.52<5000$, so the true yield is a bit lower.\nAt $i=0.06$: $1886.79+3114.99\\approx 5001.78\\approx 5000$, so the IRR $\\approx \\mathbf{6.0\\%}$. (The $9.5\\%$ guess overshoots.)", "tag": "NPV & IRR" }, { "front": "State the **sign-change (Descartes-type) condition** that guarantees a *unique* positive IRR for a cash-flow stream.", "back": "If the sequence of cash flows $C_{0},C_{1},\\ldots,C_{n}$ has **exactly one sign change** (all outflows precede all inflows, or vice versa), there is a unique IRR $i^{*}>-1$.\nMultiple sign changes *may* produce multiple yield rates or none in the feasible range; a single sign change rules that out.", "tag": "Uniqueness of IRR" }, { "front": "Why can a project with cash flows $-1000,\\ +2300,\\ -1320$ (two sign changes) have **more than one** IRR?", "back": "Two sign changes allow the NPV polynomial in $v$ to have two positive roots. Solving $-1000+2300v-1320v^{2}=0$ gives $v=\\frac{2300\\pm\\sqrt{2300^{2}-4\\cdot1320\\cdot1000}}{2\\cdot1320}$.\nDiscriminant $=5{,}290{,}000-5{,}280{,}000=10{,}000$, so $v=\\frac{2300\\pm100}{2640}$, giving $v\\approx0.9091$ or $v\\approx0.8333$, i.e. $i\\approx10\\%$ **and** $i\\approx20\\%$. Both are legitimate IRRs, so IRR alone is ambiguous here.", "tag": "Uniqueness of IRR" }, { "front": "When the IRR is not unique (or doesn't exist), what is the standard remedy for ranking or evaluating a project?", "back": "Use the **net present value at a specified rate** instead of the IRR. NPV gives a single, unambiguous accept/reject signal once the required rate is fixed.\nA related fix is to assume an explicit reinvestment rate and compute a single accumulated value (a modified yield), removing the ambiguity caused by multiple sign changes.", "tag": "Uniqueness of IRR" }, { "front": "When comparing two mutually exclusive projects, why can the IRR rule disagree with the NPV rule, and which should you trust?", "back": "IRR implicitly assumes intermediate cash flows are reinvested at the IRR itself, which differs across projects and may be unrealistic; it also ignores project **scale**. A small project can have a higher IRR yet a smaller NPV.\nFor a fixed required rate, **NPV** correctly ranks value created, so prefer NPV when the two rules conflict.", "tag": "Uniqueness of IRR" }, { "front": "Distinguish the **dollar-weighted** (money-weighted) rate of return from the **time-weighted** rate of return.", "back": "The **dollar-weighted** yield solves the fund's equation of value and is sensitive to the *amount and timing* of deposits and withdrawals — it measures the investor's actual experience.\nThe **time-weighted** yield chain-links the returns of each sub-period between cash flows and is *insensitive* to contribution timing — it measures the manager's investment performance.", "tag": "Dollar vs time-weighted" }, { "front": "Give the simple-interest approximation for the **dollar-weighted** rate of return over one year.", "back": "$i\\approx \\dfrac{I}{A+\\sum_{t} C_{t}\\,(1-t)}$, where $A$ is the beginning balance, $C_{t}$ is the net deposit at time $t$ (a fraction of the year), and $I$ is the interest earned over the year.\nInterest is $I=B-A-\\sum C_{t}$, with $B$ the ending balance. Each deposit is weighted by the fraction of the year $(1-t)$ it was invested.", "tag": "Dollar vs time-weighted" }, { "front": "A fund starts the year at $\\$100{,}000$. A deposit of $\\$24{,}000$ is made at $t=\\tfrac{1}{3}$ (end of month 4) and $\\$18{,}000$ is withdrawn at $t=\\tfrac{3}{4}$ (end of month 9). The year-end balance is $\\$112{,}000$. Find the dollar-weighted yield (simple-interest method).", "back": "Interest earned: $I=112{,}000-100{,}000-(24{,}000-18{,}000)=6{,}000$.\nExposure denominator: $A+\\sum C_{t}(1-t)=100{,}000+24{,}000(1-\\tfrac13)+(-18{,}000)(1-\\tfrac34)$\n$=100{,}000+24{,}000(0.6667)-18{,}000(0.25)=100{,}000+16{,}000-4{,}500=111{,}500$.\n$i\\approx \\dfrac{6{,}000}{111{,}500}\\approx 0.05381=\\mathbf{5.38\\%}$.", "tag": "Dollar vs time-weighted" }, { "front": "How do you compute a **time-weighted** rate of return when there are cash flows during the year?", "back": "Break the year at each cash-flow date. For each sub-period $k$ compute the growth factor $1+r_{k}=\\dfrac{\\text{value just before the flow}}{\\text{value just after the previous flow}}$ using values **immediately before** each new deposit/withdrawal.\nThen $1+i_{TW}=\\prod_{k}(1+r_{k})$, and $i_{TW}$ is the chain-linked product minus 1.", "tag": "Dollar vs time-weighted" }, { "front": "A fund is $\\$50{,}000$ on Jan 1. On Jul 1 it has grown to $\\$55{,}000$, then the investor deposits $\\$20{,}000$ (balance $\\$75{,}000$). On Dec 31 the fund is worth $\\$78{,}000$. Find the time-weighted rate of return.", "back": "Sub-period 1 (Jan–Jul): $1+r_{1}=\\dfrac{55{,}000}{50{,}000}=1.10$.\nSub-period 2 (Jul–Dec), value just after deposit is $75{,}000$: $1+r_{2}=\\dfrac{78{,}000}{75{,}000}=1.04$.\n$1+i_{TW}=1.10\\times1.04=1.144$, so $i_{TW}=\\mathbf{14.4\\%}$.\n(The mid-year deposit is excluded from the return because it sits just after the measurement point.)", "tag": "Dollar vs time-weighted" }, { "front": "If a large deposit is made just **before** a strong sub-period, how will the dollar-weighted return compare to the time-weighted return, and why?", "back": "The **dollar-weighted** return will be *higher* than the time-weighted return, because more money is exposed to the strong period and the dollar-weighted measure rewards favorable timing of contributions.\nThe **time-weighted** return is unchanged by the deposit timing — it isolates the per-dollar investment performance and ignores how much was invested when.", "tag": "Dollar vs time-weighted" }, { "front": "A portfolio returns $+25\\%$ in year 1 and $-20\\%$ in year 2 (no cash flows). Find the time-weighted (geometric) rate of return per year.", "back": "$1+i_{TW}=(1.25)(0.80)=1.00$, so the two-year growth factor is exactly $1$.\nThe cumulative return is $0\\%$; per year $(1+i)^{2}=1.00\\Rightarrow i=0\\%$.\nNote the *arithmetic* average $(+25\\%-20\\%)/2=+2.5\\%$ overstates the truth — geometric chain-linking is the correct time-weighted measure.", "tag": "Dollar vs time-weighted" }, { "front": "What is a **reinvestment-rate** problem, and why can the realized yield differ from the quoted rate?", "back": "When interest or coupon payments are received before the horizon, they must be **reinvested**, often at a rate $j$ different from the original yield $i$. The realized (overall) yield depends on $j$.\nIf $ji$ it exceeds $i$. Reinvestment risk is the uncertainty in the rate at which interim cash is redeployed.", "tag": "Reinvestment" }, { "front": "$\\$10{,}000$ is invested for 5 years at an annual effective rate of $6\\%$, with interest paid annually and reinvested at $4\\%$. Find the total accumulated value at $t=5$ and the realized annual yield.", "back": "Annual interest is $0.06\\times10{,}000=600$, paid at the end of years 1–5 and reinvested at $4\\%$: accumulated value of the interest stream is $600\\,s_{\\overline{5|}\\,4\\%}=600\\cdot\\dfrac{1.04^{5}-1}{0.04}\\approx600\\times5.41632\\approx3{,}249.79$.\nAdd the returned principal $10{,}000$: total $\\approx \\$13{,}249.79$.\nRealized yield: $(13{,}249.79/10{,}000)^{1/5}-1\\approx 1.05789-1\\approx \\mathbf{5.79\\%}$ — below $6\\%$ because reinvestment was only $4\\%$.\n($s_{\\overline{n|}}$ denotes the accumulated value of an annuity-immediate of 1.)", "tag": "Reinvestment" }, { "front": "Deposits of $\\$1{,}000$ are made at the end of each year for 10 years into an account earning $5\\%$ effective. Find the accumulated value at $t=10$.", "back": "This is an annuity-immediate accumulation: $\\text{AV}=1000\\,s_{\\overline{10|}\\,5\\%}=1000\\cdot\\dfrac{1.05^{10}-1}{0.05}$.\n$1.05^{10}\\approx1.628895$, so $s_{\\overline{10|}}\\approx\\dfrac{0.628895}{0.05}\\approx12.57789$.\n$\\text{AV}\\approx1000\\times12.57789\\approx \\$12{,}577.89$.", "tag": "Reinvestment" }, { "front": "$\\$8{,}000$ is lent for 4 years. The borrower pays only the annual interest of $7\\%$ to the lender each year, and the lender reinvests those payments at $5\\%$; the $\\$8{,}000$ principal is returned at $t=4$. Find the lender's overall annual yield.", "back": "Annual interest received $=0.07\\times8000=560$, reinvested at $5\\%$: $560\\,s_{\\overline{4|}\\,5\\%}=560\\cdot\\dfrac{1.05^{4}-1}{0.05}\\approx560\\times4.310125\\approx2{,}413.67$.\nTotal at $t=4$: $8{,}000+2{,}413.67=10{,}413.67$.\nYield: $(10{,}413.67/8{,}000)^{1/4}-1=(1.301709)^{1/4}-1\\approx1.06811-1\\approx\\mathbf{6.81\\%}$, just below the $7\\%$ coupon because reinvestment earned only $5\\%$.", "tag": "Reinvestment" }, { "front": "Under the reinvestment framework, write the accumulated value at time $n$ of a single deposit of $1$ earning $i$ with interest reinvested at rate $j$ (interest payable annually).", "back": "$\\text{AV}=1+i\\,s_{\\overline{n|}\\,j}=1+i\\cdot\\dfrac{(1+j)^{n}-1}{j}$.\nThe principal of $1$ is returned at $t=n$, and the $n$ interest payments of $i$ accumulate at the reinvestment rate $j$. When $j=i$ this collapses to $(1+i)^{n}$.", "tag": "Reinvestment" }, { "front": "Contrast the **portfolio method** and the **investment-year method** for crediting interest on funds.", "back": "The **portfolio method** credits *all* funds — old and new — the same average (portfolio) rate each year, regardless of when they entered.\nThe **investment-year method (new-money method)** credits new deposits a rate that depends on their *year of entry* for an initial period, after which they roll into the portfolio rate. It better reflects the rates available when money was actually invested.", "tag": "Portfolio & investment-year" }, { "front": "How do you read an **investment-year-method rate table** $i_{y}^{t}$, and when does a deposit switch to the portfolio rate?", "back": "Rows are indexed by calendar year of investment $y$; columns $t=1,2,\\ldots,m$ give the rate in the $t$-th year *since* that deposit was made. So $i_{y}^{t}$ applies to year-$y$ money during its $t$-th year.\nAfter the $m$-year select period, the deposit earns the **portfolio rate** $i^{y+m-1+s}$ for all subsequent calendar years (read down the portfolio column).", "tag": "Portfolio & investment-year" }, { "front": "A deposit is made in year $z$. The investment-year rates for its first three years are $i_{z}^{1}=7\\%$, $i_{z}^{2}=6.5\\%$, $i_{z}^{3}=6\\%$. Accumulate $\\$1{,}000$ over those three years.", "back": "Apply each year's rate multiplicatively:\n$\\text{AV}=1000(1.07)(1.065)(1.06)$.\n$1.07\\times1.065=1.13955$; $\\times1.06\\approx1.207923$.\n$\\text{AV}\\approx1000\\times1.207923\\approx \\$1{,}207.92$.", "tag": "Portfolio & investment-year" }, { "front": "Money invested in year $y$ earns investment-year rates $9\\%$ then $8\\%$ for two years, after which it earns the portfolio rate, which is $6\\%$ in year $y+2$. Accumulate $\\$5{,}000$ to the end of year $y+2$ (3 years total).", "back": "Year 1 (investment-year): $\\times1.09$. Year 2 (investment-year): $\\times1.08$. Year 3 (now portfolio): $\\times1.06$.\n$\\text{AV}=5000(1.09)(1.08)(1.06)$.\n$1.09\\times1.08=1.1772$; $\\times1.06\\approx1.247832$.\n$\\text{AV}\\approx5000\\times1.247832\\approx \\$6{,}239.16$.", "tag": "Portfolio & investment-year" }, { "front": "Define the **spot rate** $s_{t}$ and write the present value of a single cash flow $C_{t}$ using the spot-rate curve.", "back": "The spot rate $s_{t}$ is the annual effective yield on a **zero-coupon** bond maturing at time $t$ — the rate that applies to a single payment $t$ periods away.\nPresent value: $\\mathrm{PV}=\\dfrac{C_{t}}{(1+s_{t})^{t}}$. A stream is valued by discounting *each* flow at its own spot rate: $\\mathrm{PV}=\\sum_{t}\\dfrac{C_{t}}{(1+s_{t})^{t}}$.", "tag": "Spot & forward rates" }, { "front": "Define the **forward rate** and give the no-arbitrage relationship between forward rates and spot rates.", "back": "The forward rate $f_{[t,\\,t+k]}$ is the rate, agreed today, that applies to a loan/investment running from time $t$ to time $t+k$.\nNo-arbitrage: $(1+s_{t+k})^{t+k}=(1+s_{t})^{t}\\,(1+f)^{k}$.\nFor a one-year forward starting at $t$: $(1+s_{t+1})^{t+1}=(1+s_{t})^{t}(1+f_{t})$.", "tag": "Spot & forward rates" }, { "front": "The 1-year spot rate is $s_{1}=4\\%$ and the 2-year spot rate is $s_{2}=5\\%$. Find the 1-year forward rate $f_{1}$ that applies from $t=1$ to $t=2$.", "back": "Use $(1+s_{2})^{2}=(1+s_{1})(1+f_{1})$.\n$1.05^{2}=1.1025$ and $1+s_{1}=1.04$.\n$1+f_{1}=\\dfrac{1.1025}{1.04}\\approx1.060096$, so $f_{1}\\approx\\mathbf{6.01\\%}$.\nThe forward exceeds both spots because the curve is upward-sloping.", "tag": "Spot & forward rates" }, { "front": "Given spot rates $s_{1}=3\\%$, $s_{2}=3.6\\%$, $s_{3}=4\\%$, find the 2-year forward rate beginning at $t=1$ (covering years 2 and 3).", "back": "Use $(1+s_{3})^{3}=(1+s_{1})\\,(1+f)^{2}$ where $f$ is the annual forward for $t=1$ to $t=3$.\n$(1.04)^{3}=1.124864$; $1+s_{1}=1.03$.\n$(1+f)^{2}=\\dfrac{1.124864}{1.03}\\approx1.092101$, so $1+f=\\sqrt{1.092101}\\approx1.045037$.\n$f\\approx\\mathbf{4.50\\%}$ per year over years 2–3.", "tag": "Spot & forward rates" }, { "front": "Price a 3-year annuity-immediate of $\\$1{,}000$ per year off the spot curve $s_{1}=4\\%$, $s_{2}=4.5\\%$, $s_{3}=5\\%$.", "back": "Discount each payment at its own spot rate:\n$\\dfrac{1000}{1.04}\\approx961.54$; $\\dfrac{1000}{1.045^{2}}=\\dfrac{1000}{1.092025}\\approx915.73$; $\\dfrac{1000}{1.05^{3}}=\\dfrac{1000}{1.157625}\\approx863.84$.\nPrice $\\approx961.54+915.73+863.84\\approx \\$2{,}741.11$.\nUsing a single yield would misprice it — each maturity carries a different spot rate.", "tag": "Spot & forward rates" }, { "front": "A 2-year bond pays a $\\$60$ coupon at $t=1$ and $\\$1{,}060$ at $t=2$. Spot rates are $s_{1}=5\\%$, $s_{2}=6\\%$. Find the bond's price and its (single) yield to maturity, then compare.", "back": "Price off spots: $\\dfrac{60}{1.05}+\\dfrac{1060}{1.06^{2}}=57.14+\\dfrac{1060}{1.1236}\\approx57.14+943.40\\approx\\$1{,}000.54$.\nThe YTM $y$ solves $1000.54=\\dfrac{60}{1+y}+\\dfrac{1060}{(1+y)^{2}}$; since price $\\approx$ par, $y\\approx5.97\\%$, a PV-weighted blend of the $5\\%$ and $6\\%$ spots.\nThe YTM lies between the two spot rates, closer to $s_{2}$ because most of the value is in the $t=2$ flow.", "tag": "Spot & forward rates" }, { "front": "What does the **shape of the term structure** (yield curve) tell you, and what are the three standard shapes?", "back": "The term structure plots spot rates against maturity. **Upward-sloping (normal):** longer maturities yield more; forward rates exceed spot rates. **Flat:** spot rates equal across maturities; forwards equal spots. **Inverted (downward):** short rates exceed long rates; forwards fall below spots.\nForward rates always lie *above* the spot curve when it rises and *below* it when it falls.", "tag": "Spot & forward rates" }, { "front": "Given the 1-year spot $s_{1}=5\\%$ and the 1-year forward rates $f_{1}=6\\%$ (years 1→2) and $f_{2}=7\\%$ (years 2→3), build the 3-year spot rate $s_{3}$.", "back": "Chain the one-period factors: $(1+s_{3})^{3}=(1.05)(1.06)(1.07)$.\n$1.05\\times1.06=1.113$; $\\times1.07\\approx1.19091$.\n$1+s_{3}=1.19091^{1/3}\\approx1.059956$, so $s_{3}\\approx\\mathbf{6.00\\%}$.\nThe 3-year spot is the geometric average of the embedded one-year forwards.", "tag": "Spot & forward rates" }, { "front": "What does it mean to **replicate** a cash-flow stream, and what no-arbitrage principle links a stream to its replicating portfolio?", "back": "To replicate a target stream you assemble a portfolio of traded instruments (e.g. zero-coupon bonds) whose combined cash flows exactly match the target at every date.\n**Law of one price / no-arbitrage:** two portfolios with identical future cash flows must have the same price today. So the target's fair value equals the cost of its replicating portfolio.", "tag": "Replicating cash flows" }, { "front": "You must fund liabilities of $\\$1{,}000$ at $t=1$ and $\\$1{,}000$ at $t=2$. Zero-coupon bonds (face $\\$1{,}000$) are priced at $\\$952.38$ for 1-year and $\\$889.00$ for 2-year. Find the cost to exactly replicate (cash-flow match) the liabilities.", "back": "Buy one 1-year zero ($\\$1{,}000$ at $t=1$) and one 2-year zero ($\\$1{,}000$ at $t=2$); their cash flows exactly equal the liabilities.\nCost $=952.38+889.00=\\$1{,}841.38$.\nBy no-arbitrage this is the unique fair price of the liability stream; the implied spots are $s_{1}=1000/952.38-1\\approx5\\%$ and $s_{2}=(1000/889)^{1/2}-1\\approx6\\%$.", "tag": "Replicating cash flows" }, { "front": "Replicate a 2-year bond paying a $\\$50$ coupon at $t=1$ and $\\$1{,}050$ at $t=2$ using zero-coupon bonds. How many of each (per $\\$1$ face) do you buy, and how does this price the bond?", "back": "Hold $50$ units of the 1-year zero (face $\\$1$) and $1{,}050$ units of the 2-year zero (face $\\$1$); their payoffs replicate the coupon bond exactly.\nIf the 1-year and 2-year zero prices (per $\\$1$ face) are $P_{1}$ and $P_{2}$, the bond price is $50P_{1}+1050P_{2}=50/(1+s_{1})+1050/(1+s_{2})^{2}$ — the spot-rate valuation, enforced by no-arbitrage.", "tag": "Replicating cash flows" }, { "front": "Using spot rates $s_{1}=4\\%$ and $s_{2}=5\\%$, replicate a guaranteed loan that pays you $\\$300$ at $t=1$ and $\\$300$ at $t=2$. What is its arbitrage-free price?", "back": "Buy $\\$300$ face of the 1-year zero and $\\$300$ face of the 2-year zero.\n$\\mathrm{PV}=\\dfrac{300}{1.04}+\\dfrac{300}{1.05^{2}}=288.46+\\dfrac{300}{1.1025}=288.46+272.11=\\$560.57$.\nAny market price different from $\\$560.57$ would allow a riskless arbitrage between the loan and the zero-coupon portfolio.", "tag": "Replicating cash flows" }, { "front": "A dealer can borrow at the 1-year spot $s_{1}=4\\%$ and the 2-year spot $s_{2}=6\\%$. A customer wants to lock a 1-year loan starting in one year. What rate should the dealer quote, and how is it hedged?", "back": "Quote the implied 1-year forward $f_{1}$: $(1+s_{2})^{2}=(1+s_{1})(1+f_{1})\\Rightarrow1+f_{1}=\\dfrac{1.06^{2}}{1.04}=\\dfrac{1.1236}{1.04}\\approx1.080385$, so $f_{1}\\approx\\mathbf{8.04\\%}$.\nThe dealer replicates by borrowing 2-year and lending 1-year today, locking the forward and earning a riskless spread if the customer is charged above $f_{1}$.", "tag": "Replicating cash flows" }, { "front": "A project requires outflows of $\\$1{,}000$ at $t=0$ and $\\$500$ at $t=1$, and returns $\\$1{,}800$ at $t=2$. Compute the NPV at $10\\%$ and state whether the IRR exceeds or falls short of $10\\%$.", "back": "At $i=0.10$: $\\mathrm{NPV}=-1000-\\dfrac{500}{1.10}+\\dfrac{1800}{1.10^{2}}=-1000-454.55+\\dfrac{1800}{1.21}$.\n$1800/1.21\\approx1487.60$, so $\\mathrm{NPV}\\approx-1000-454.55+1487.60\\approx \\$33.06$.\nSince $\\mathrm{NPV}(0.10)>0$ and NPV decreases in $i$, the **IRR exceeds $10\\%$** (the curve crosses zero at a higher rate). The single sign change in the outflow-then-inflow pattern guarantees that IRR is unique.", "tag": "NPV & IRR" } ] }