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

$\mathrm{NPV}(i)=\sum_{t} C_{t}\,v^{t}=\sum_{t} \frac{C_{t}}{(1+i)^{t}}$, where $v=(1+i)^{-1}$ and inflows are positive, outflows negative. A project is acceptable on a stand-alone basis when $\mathrm{NPV}(i)\geq 0$ at the investor's required rate of return $i$.

The IRR (or **yield rate**) $i^{*}$ is the rate that makes the net present value zero: $\mathrm{NPV}(i^{*})=\sum_{t} C_{t}\,(1+i^{*})^{-t}=0$. Equivalently it is the rate at which the present value of inflows equals the present value of outflows — the solution of the equation of value.

$\mathrm{NPV}(i)$ is a **decreasing** function of $i$: discounting the later inflows more heavily lowers their present value. The 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).

Discount each inflow at $i=0.08$: $4000v=4000/1.08\approx 3703.70$. $4000v^{2}=4000/1.08^{2}\approx 3429.36$. $5000v^{3}=5000/1.08^{3}\approx 3969.16$. Sum of inflows $\approx 11{,}102.22$. Subtract the cost: $\mathrm{NPV}=11{,}102.22-10{,}000\approx \$1{,}102.22$. NPV $>0$, so the project clears an $8\%$ hurdle.

Solve $1000(1+i)^{2}=1300$, i.e. $(1+i)^{2}=1.3$. $1+i=\sqrt{1.3}\approx 1.140175$. $i\approx 0.140175$, so the yield is about $\mathbf{14.02\%}$ annual effective.

Equation of value (PV of inflows = outflow): $5000=\frac{2000}{1+i}+\frac{3500}{(1+i)^{2}}$. At $i=0.095$: $2000/1.095\approx 1826.48$ and $3500/1.095^{2}\approx 2919.04$; sum $\approx 4745.52<5000$, so the true yield is a bit lower. At $i=0.06$: $1886.79+3114.99\approx 5001.78\approx 5000$, so the IRR $\approx \mathbf{6.0\%}$. (The $9.5\%$ guess overshoots.)

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 $i^{*}>-1$. Multiple sign changes *may* produce multiple yield rates or none in the feasible range; a single sign change rules that out.

Two sign changes allow the NPV polynomial in $v$ to have two positive roots. Solving $-1000+2300v-1320v^{2}=0$ gives $v=\frac{2300\pm\sqrt{2300^{2}-4\cdot1320\cdot1000}}{2\cdot1320}$. Discriminant $=5{,}290{,}000-5{,}280{,}000=10{,}000$, 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.

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. A 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.

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. For a fixed required rate, **NPV** correctly ranks value created, so prefer NPV when the two rules conflict.

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. The **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.

$i\approx \dfrac{I}{A+\sum_{t} C_{t}\,(1-t)}$, 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. Interest 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.

Interest earned: $I=112{,}000-100{,}000-(24{,}000-18{,}000)=6{,}000$. Exposure denominator: $A+\sum C_{t}(1-t)=100{,}000+24{,}000(1-\tfrac13)+(-18{,}000)(1-\tfrac34)$ $=100{,}000+24{,}000(0.6667)-18{,}000(0.25)=100{,}000+16{,}000-4{,}500=111{,}500$. $i\approx \dfrac{6{,}000}{111{,}500}\approx 0.05381=\mathbf{5.38\%}$.

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. Then $1+i_{TW}=\prod_{k}(1+r_{k})$, and $i_{TW}$ is the chain-linked product minus 1.

Sub-period 1 (Jan–Jul): $1+r_{1}=\dfrac{55{,}000}{50{,}000}=1.10$. Sub-period 2 (Jul–Dec), value just after deposit is $75{,}000$: $1+r_{2}=\dfrac{78{,}000}{75{,}000}=1.04$. $1+i_{TW}=1.10\times1.04=1.144$, so $i_{TW}=\mathbf{14.4\%}$. (The mid-year deposit is excluded from the return because it sits just after the measurement point.)

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. The **time-weighted** return is unchanged by the deposit timing — it isolates the per-dollar investment performance and ignores how much was invested when.

$1+i_{TW}=(1.25)(0.80)=1.00$, so the two-year growth factor is exactly $1$. The cumulative return is $0\%$; per year $(1+i)^{2}=1.00\Rightarrow i=0\%$. Note the *arithmetic* average $(+25\%-20\%)/2=+2.5\%$ overstates the truth — geometric chain-linking is the correct time-weighted measure.

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 $j$. If $j<i$ the realized yield is below $i$; if $j>i$ it exceeds $i$. Reinvestment risk is the uncertainty in the rate at which interim cash is redeployed.

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 $600\,s_{\overline{5|}\,4\%}=600\cdot\dfrac{1.04^{5}-1}{0.04}\approx600\times5.41632\approx3{,}249.79$. Add the returned principal $10{,}000$: total $\approx \$13{,}249.79$. Realized yield: $(13{,}249.79/10{,}000)^{1/5}-1\approx 1.05789-1\approx \mathbf{5.79\%}$ — below $6\%$ because reinvestment was only $4\%$. ($s_{\overline{n|}}$ denotes the accumulated value of an annuity-immediate of 1.)

This is an annuity-immediate accumulation: $\text{AV}=1000\,s_{\overline{10|}\,5\%}=1000\cdot\dfrac{1.05^{10}-1}{0.05}$. $1.05^{10}\approx1.628895$, so $s_{\overline{10|}}\approx\dfrac{0.628895}{0.05}\approx12.57789$. $\text{AV}\approx1000\times12.57789\approx \$12{,}577.89$.

Annual interest received $=0.07\times8000=560$, reinvested at $5\%$: $560\,s_{\overline{4|}\,5\%}=560\cdot\dfrac{1.05^{4}-1}{0.05}\approx560\times4.310125\approx2{,}413.67$. Total at $t=4$: $8{,}000+2{,}413.67=10{,}413.67$. Yield: $(10{,}413.67/8{,}000)^{1/4}-1=(1.301709)^{1/4}-1\approx1.06811-1\approx\mathbf{6.81\%}$, just below the $7\%$ coupon because reinvestment earned only $5\%$.

$\text{AV}=1+i\,s_{\overline{n|}\,j}=1+i\cdot\dfrac{(1+j)^{n}-1}{j}$. The 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 $(1+i)^{n}$.

The **portfolio method** credits *all* funds — old and new — the same average (portfolio) rate each year, regardless of when they entered. The **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.

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. After 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).

Apply each year's rate multiplicatively: $\text{AV}=1000(1.07)(1.065)(1.06)$. $1.07\times1.065=1.13955$; $\times1.06\approx1.207923$. $\text{AV}\approx1000\times1.207923\approx \$1{,}207.92$.

Year 1 (investment-year): $\times1.09$. Year 2 (investment-year): $\times1.08$. Year 3 (now portfolio): $\times1.06$. $\text{AV}=5000(1.09)(1.08)(1.06)$. $1.09\times1.08=1.1772$; $\times1.06\approx1.247832$. $\text{AV}\approx5000\times1.247832\approx \$6{,}239.16$.

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. Present value: $\mathrm{PV}=\dfrac{C_{t}}{(1+s_{t})^{t}}$. A stream is valued by discounting *each* flow at its own spot rate: $\mathrm{PV}=\sum_{t}\dfrac{C_{t}}{(1+s_{t})^{t}}$.

The forward rate $f_{[t,\,t+k]}$ is the rate, agreed today, that applies to a loan/investment running from time $t$ to time $t+k$. No-arbitrage: $(1+s_{t+k})^{t+k}=(1+s_{t})^{t}\,(1+f)^{k}$. For a one-year forward starting at $t$: $(1+s_{t+1})^{t+1}=(1+s_{t})^{t}(1+f_{t})$.

Use $(1+s_{2})^{2}=(1+s_{1})(1+f_{1})$. $1.05^{2}=1.1025$ and $1+s_{1}=1.04$. $1+f_{1}=\dfrac{1.1025}{1.04}\approx1.060096$, so $f_{1}\approx\mathbf{6.01\%}$. The forward exceeds both spots because the curve is upward-sloping.

Use $(1+s_{3})^{3}=(1+s_{1})\,(1+f)^{2}$ where $f$ is the annual forward for $t=1$ to $t=3$. $(1.04)^{3}=1.124864$; $1+s_{1}=1.03$. $(1+f)^{2}=\dfrac{1.124864}{1.03}\approx1.092101$, so $1+f=\sqrt{1.092101}\approx1.045037$. $f\approx\mathbf{4.50\%}$ per year over years 2–3.

Discount each payment at its own spot rate: $\dfrac{1000}{1.04}\approx961.54$; $\dfrac{1000}{1.045^{2}}=\dfrac{1000}{1.092025}\approx915.73$; $\dfrac{1000}{1.05^{3}}=\dfrac{1000}{1.157625}\approx863.84$. Price $\approx961.54+915.73+863.84\approx \$2{,}741.11$. Using a single yield would misprice it — each maturity carries a different spot rate.

Price off spots: $\dfrac{60}{1.05}+\dfrac{1060}{1.06^{2}}=57.14+\dfrac{1060}{1.1236}\approx57.14+943.40\approx\$1{,}000.54$. The 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. The YTM lies between the two spot rates, closer to $s_{2}$ because most of the value is in the $t=2$ flow.

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. Forward rates always lie *above* the spot curve when it rises and *below* it when it falls.

Chain the one-period factors: $(1+s_{3})^{3}=(1.05)(1.06)(1.07)$. $1.05\times1.06=1.113$; $\times1.07\approx1.19091$. $1+s_{3}=1.19091^{1/3}\approx1.059956$, so $s_{3}\approx\mathbf{6.00\%}$. The 3-year spot is the geometric average of the embedded one-year forwards.

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. **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.

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. Cost $=952.38+889.00=\$1{,}841.38$. By no-arbitrage this is the unique fair price of the liability stream; the implied spots are $s_{1}=1000/952.38-1\approx5\%$ and $s_{2}=(1000/889)^{1/2}-1\approx6\%$.

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. If 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.

Buy $\$300$ face of the 1-year zero and $\$300$ face of the 2-year zero. $\mathrm{PV}=\dfrac{300}{1.04}+\dfrac{300}{1.05^{2}}=288.46+\dfrac{300}{1.1025}=288.46+272.11=\$560.57$. Any market price different from $\$560.57$ would allow a riskless arbitrage between the loan and the zero-coupon portfolio.

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 $f_{1}\approx\mathbf{8.04\%}$. The 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 $f_{1}$.

At $i=0.10$: $\mathrm{NPV}=-1000-\dfrac{500}{1.10}+\dfrac{1800}{1.10^{2}}=-1000-454.55+\dfrac{1800}{1.21}$. $1800/1.21\approx1487.60$, so $\mathrm{NPV}\approx-1000-454.55+1487.60\approx \$33.06$. Since $\mathrm{NPV}(0.10)>0$ 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.