Practice: Valuing a Stream of Dated Payments

1. The recommendation on A and B. At the stated cost of capital of $10\mathrm{\%}$:

B should be approved. It leaves the firm $213.03$ better off per $1000$ committed. The internal rates are correctly computed and they establish one thing each: A breaks even at $20.00\mathrm{\%}$ and B at $15.9962\mathrm{\%}$. Neither figure takes the firm's cost of capital as an input, so neither can express which project is preferable at $10\mathrm{\%}$. The note's phrase "the higher return" treats a break-even rate as though it were a measure of value, and those are different quantities. 2. The crossover. Net present value is a function of the discount rate; an internal rate is a fixed property of a stream. So the two NPV curves can cross even though the two internal rates cannot move. The mechanism is the timing. B's single receipt arrives at year 5, so it is divided by $(1+r{)}^5$ and is punished hard as $r$ rises; A's arrives at year 1 and is divided by $(1+r)$ only. At low rates B's larger total dominates; as $r$ increases, B's value falls faster, and at $15.0163\mathrm{\%}$ the two are equal. Above that A is worth more. At $15\mathrm{\%}$ the values are $43.4783$ and $44.0711$, with B still ahead; at $16\mathrm{\%}$ they are $34.4828$ and $-0.1627$, with A ahead. Note that A's internal rate of $20.00\mathrm{\%}$ is simply where its own curve reaches zero, which is a different question from where the two curves meet. 3. The restoration project. At $25\mathrm{\%}$:

Positive, so the project should be accepted, and the note's recommendation to reject is wrong. The reasoning fails because the rule "accept if the internal rate exceeds the cost of capital" assumes a stream whose net present value falls as the discount rate rises, true for a conventional investment, false here. This stream has NPV $-50.00$ at $0\mathrm{\%}$, zero at $10\mathrm{\%}$, $+24.00$ at $25\mathrm{\%}$, zero again at $50\mathrm{\%}$, and $-19.53$ at $60\mathrm{\%}$. It is worth doing only for rates strictly between the two roots. 4. Sign changes. The stream is out, in, out: two sign changes. Substituting $x=(1+r{)}^{-1}$ turns $\mathrm{NPV}(r)=0$ into a quadratic in $x$, and by Descartes' rule two sign changes permit up to two positive roots. Here there are exactly two, at $10\mathrm{\%}$ and $50\mathrm{\%}$. What should have been checked before quoting a rate: count the sign changes in the stream. One or none means an internal rate is unique or absent and can be quoted safely. More than one means the figure a spreadsheet returns is whichever root its search reached from its starting guess, with nothing in the output indicating that another exists. Tabulating or plotting NPV across a range of rates would have revealed both crossings immediately. 5. What I would put in the note. For the proposals: net present value of each at the firm's $10\mathrm{\%}$ cost of capital, $90.91$ for A and $303.93$ for B, with the recommendation to approve B. I would state the crossover rate of $15.0163\mathrm{\%}$ and note that the ranking would reverse if the applicable rate rose above it, since that is a genuine sensitivity the board should see. I would report the internal rates only as break-even rates, labelled as such, and would not use them to rank. For the restoration project: net present value at the applicable $25\mathrm{\%}$, namely $+24.00$, with the recommendation to accept. I would state explicitly that the stream changes sign twice and has two internal rates, $10\mathrm{\%}$ and $50\mathrm{\%}$, that neither should be compared against a hurdle, and that the project is worth doing only while the applicable rate lies between them, which makes the appraisal sensitive to the rate in an unusual direction and is worth flagging to the board rather than burying. For both: the discount rate used, stated on the face of the note, since a ranking without a rate is incomplete.

(a) The five level payments at $6\mathrm{\%}$.

Sum: $842.473$.

By the annuity formula, since the flows are level and consecutive:

The two agree exactly. They must: the formula is the closed form of the geometric series the column computes, not an approximation of it.

(b) The whole contract. The year-8 payment is a single flow:

Note the timing convention: every flow arrives at the end of its year, so nothing is undiscounted. A flow at $t=0$ would enter at face value.

(c) At $10\mathrm{\%}$.

The value fell by $245.221$, about $16.7\mathrm{\%}$. Within that, the annuity fell by $84.316$ ($10.0\mathrm{\%}$) and the single year-8 flow by $160.905$ ($25.6\mathrm{\%}$).

Why. The discount factor is $(1+r{)}^{-t}$, so the later the flow the larger the exponent and the more sensitive its value to the rate. A stream's sensitivity is therefore a statement about when its money arrives, which is why two contracts with equal nominal totals can rank differently at different rates.