Valuing a Stream of Dated Payments

A promise of 1000 next year is not worth 1000 today, because 1000 held today can be lent. Discounting is the conversion between the two, and it is exact: a flow $C$ arriving $t$ periods out is worth $C(1+r{)}^{-t}$ now, where $r$ is the rate available.

Everything else in this unit follows from applying that consistently. A level stream of payments is a geometric progression in the discount factor, so its value has a closed form; letting the stream run forever gives another, simpler one. Neither formula needs memorising once the series is recognised.

The interesting part is what to do with the resulting number.

Net present value takes the rate as an input, which means it can answer a question whose answer depends on the rate. The internal rate of return takes no rate at all, it is the rate at which the stream breaks even, and that independence is exactly why it cannot rank projects when the ranking depends on what else the money could earn.

On the pair worked later, one project has an internal rate of $20.00\mathrm{\%}$ and the other $15.9962\mathrm{\%}$. The first wins on that criterion at every discount rate. By net present value the second is worth more below $15.0163\mathrm{\%}$ and the first above it, and which of those regions a firm is in is a fact about the firm rather than about the projects.

The unit closes with a stream whose internal rate is not a number at all, because the equation defining it has two roots.

The canonical statements above give the discount factor, the two closed forms and the definition of the internal rate. What follows is the conditions attached to each.

The rate must match the period. $(1+r{)}^{-t}$ prices a flow $t$ periods out at the per-period rate $r$. Applying an annual rate to monthly flows requires the per-month equivalent $(1+r{)}^{1/12}-1$, not $r/12$: at $r=0.12$ these are $0.009489$ and $0.01$, and the difference compounds over a long stream.

The annuity formula is a partial sum, so it stops where the stream stops.

prices exactly $n$ payments, the first one period from now. A stream beginning immediately is worth $(1+r)$ times as much, and mismatching the convention is the most common error in applying it.

The perpetuity is its limit, and the limit needs . As $n$ grows, $(1+r{)}^{-n}\to 0$ and the bracket tends to 1, giving $C/r$. At $r=0$ the sum diverges, which is the arithmetic correctly reporting that an infinite undiscounted stream has no finite value.

The internal rate is a root of a polynomial. Substituting $x=(1+r{)}^{-1}$ turns $\mathrm{NPV}(r)=0$ into

an ordinary polynomial equation. Three consequences follow directly, none of them special to finance:

Descartes' rule of signs supplies the bound. It bounds rather than determines: two sign changes permit two positive roots and do not guarantee them.

Net present value is additive; an internal rate is not. The value of two independent projects taken together is the sum of their values, which is what lets a firm evaluate a portfolio piecewise. Internal rates cannot be added or averaged. The combined stream has its own root, unrelated to the two separate ones.

And net present value depends on $r$ while an internal rate does not. That difference is the whole of the ranking problem: a criterion that ignores the discount rate cannot express a preference that changes with it.

Where the value of a long stream actually sits. At $5\mathrm{\%}$, a perpetuity of 1000 a year is worth $\mathrm{20,000}$. The first twenty payments alone are worth $\mathrm{12,462.21}$, $62.31\mathrm{\%}$ of the total, and everything from year twenty-one to infinity accounts for the remaining $\mathrm{7,537.79}$.

This matters when a valuation depends on assumptions about the distant future. The distant future contributes, and it contributes less than its unboundedness suggests, because each payment is divided by a factor growing geometrically.

The two criteria differ in what they take as input. Net present value is a function of the discount rate: give it a rate, it returns a number. The internal rate of return is the rate at which that function crosses zero, so it consumes no rate at all.

That independence sounds like an advantage and is the source of the problem. Consider a project paying $1200$ a year from now on an outlay of $1000$, against one paying $2100$ in five years on the same outlay.

The internal rate prefers the short project in both columns, because it never learns what the money could otherwise earn. Net present value prefers the long one at $10\mathrm{\%}$ and the short one at $18\mathrm{\%}$, and the switch happens at $15.0163\mathrm{\%}$.

Neither answer is wrong about its own question. The internal rate correctly reports that the short project breaks even at a higher rate. The decision, though, is which project to fund, and that depends on the firm's actual cost of capital, which is precisely the input the internal rate does not take.

Scale is the second casualty. A project turning 10 into 12 has an internal rate of $20\mathrm{\%}$; one turning 10,000 into 11,500 has $15\mathrm{\%}$. The rate prefers the first and the firm is better off by 1,500 with the second. A ratio cannot express magnitude.

And the root need not be unique. Once a stream changes sign more than once, the polynomial defining the internal rate can cross zero more than once. The stream $-1000$, $+2600$, $-1650$ has net present value exactly zero at $10\mathrm{\%}$ and again at $50\mathrm{\%}$: positive between them, negative outside. Asking a spreadsheet for "the" internal rate returns whichever root its search happens to find first, with nothing in the output to say there was another.

A bond paying a fixed coupon to maturity. A level stream with a lump sum at the end: the annuity formula prices the coupons and one discount factor prices the redemption. The stream changes sign once, from the purchase outflow to the receipts, so its internal rate exists and is unique. This is the redemption yield, and quoting it is standard precisely because uniqueness is guaranteed here.

A mortgage. The same arithmetic run backwards. The payment is chosen so the present value of $n$ payments equals the sum advanced, which is the annuity formula solved for $C$ rather than for the value. The quoted rate and the payment are two views of one equation.

Two mutually exclusive capital projects. The case where the criterion matters. If one is short and one is long, their internal rates can rank them opposite to net present value, and the ranking depends on the firm's cost of capital. Reporting net present value across a range of rates answers the question; reporting two internal rates does not.

A mine with restoration costs. Money out to open it, money in while it produces, money out to close the site. Two sign changes, so the present-value polynomial can have two roots, and the worked example shows a stream where it does. Here the internal rate is not merely misleading for ranking. It is not a well-defined number.

A perpetual endowment. A fund intended to pay a fixed sum forever is priced at $C/r$, and the formula makes the dependence on the rate stark: at $5\mathrm{\%}$ a payment of 1000 needs $\mathrm{20,000}$, at $2.5\mathrm{\%}$ it needs $\mathrm{40,000}$. Halving the assumed rate doubles the capital required, which is why endowment arithmetic is so sensitive to the long-run return assumed.

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What decides the criterion in each case. The first two ask for a rate and the stream's single sign change makes that rate well defined. The third asks which project to fund, which depends on a rate the criterion must accept as input. The fourth has no single rate to report. The fifth is a pricing question with no ranking at all. The pattern is that the internal rate answers at what rate does this break even and nothing else; where the question is different, so is the tool.

To value a dated stream.

1. Lay out the flows against their dates, including the sign of each. A cost is negative; the discipline of writing signs explicitly is what later reveals how many times the stream changes direction.
2. Fix the period and match the rate to it. Monthly flows take a monthly rate, obtained from an annual one as $(1+r{)}^{1/12}-1$ rather than $r/12$.
3. Discount each flow by $(1+r{)}^{-t}$, with $t$ counted in periods from the valuation date.
4. Sum. The result is the net present value at that rate, and it is a number attached to a rate rather than a property of the project.

To use the closed forms. For a level stream of $n$ payments beginning one period from now, $\mathrm{PV}=C\frac{1-(1+r{)}^{-n}}{r}$. If the first payment is immediate, multiply by $(1+r)$. For an endless level stream with , $\mathrm{PV}=C/r$.

To compare projects.

1. Establish the rate the decision-maker faces, and its plausible range if it is uncertain.
2. Compute net present value for each project across that range, not at a single point.
3. If the ranking is constant across the range, report it and the range.
4. If it reverses, locate the crossover rate and report where the decision-maker sits relative to it. A reversal is a finding about the projects rather than an inconvenience.
5. Check scale. Two projects of very different size can both be worth doing; ranking matters only when they are mutually exclusive or capital is rationed.

To handle an internal rate of return.

1. Count the sign changes in the stream. Zero means there is no internal rate. One means at most one positive root. More than one means several are possible.
2. If you compute one, plot or tabulate net present value across a range of rates and look for other crossings. A single number from a spreadsheet function is one root found from one starting guess.
3. Do not rank mutually exclusive projects by it. Use net present value at the relevant rate.
4. Report it, if at all, as a break-even rate: the rate at which this stream is worth nothing. That is what it means, and it is a legitimate thing to know.

Checks. Confirm that net present value at the reported internal rate is zero, which catches arithmetic errors in both. Confirm the annuity formula against a direct sum for a short stream, which catches the beginning-of-period convention. And before quoting a ranking, state the rate it was computed at. A ranking without a rate is incomplete rather than merely terse.

(a) An annuity, and how much of a perpetuity it captures. A stream of $1000$ at the end of each year, discounted at $5\mathrm{\%}$.

Twenty payments:

The same payment forever:

So the tail from year 21 to infinity is worth $\mathrm{20,000}-\mathrm{12,462.21}=\mathrm{7,537.789657}$, and the first twenty years hold $62.31\mathrm{\%}$ of the whole infinite stream's value. Discounting compresses the far future hard: an unbounded number of payments contributes barely more than a third.

(b) Two projects whose ranking depends on the rate. Both cost $1000$ at time zero.

• A returns $1200$ at the end of year 1.
• B returns $2100$ at the end of year 5.

Internal rates. For A, $1200/(1+r)=1000$ gives $r=20.00\mathrm{\%}$ exactly. For B, $2100/(1+r{)}^5=1000$ gives $r=15.9962\mathrm{\%}$.

By that criterion A wins, and it wins whatever the firm's cost of capital, because the criterion never sees it.

Net present values.

The ranking reverses at $15.0163\mathrm{\%}$, between the $15\mathrm{\%}$ and $16\mathrm{\%}$ rows. At $20\mathrm{\%}$ project A's net present value is exactly zero, which is what its internal rate of $20\mathrm{\%}$ means.

At $10\mathrm{\%}$ the long project is worth more than three times the short one, $303.93$ against $90.91$, while having the lower internal rate. A firm borrowing at $10\mathrm{\%}$ that chose A on the strength of its $20\mathrm{\%}$ internal rate would forgo $213$ of value per $1000$ committed.

(c) A stream with two internal rates. Cash flows $-1000$, $+2600$, $-1650$: out, in, out.

Net present value is zero at $10\mathrm{\%}$ and at $50\mathrm{\%}$, positive strictly between, negative outside. Both are internal rates of return.

The reason is visible in the signs: the stream changes sign twice, so by Descartes' rule the present-value polynomial may have up to two positive roots, and here it has exactly two. Asking for "the" internal rate presupposes a uniqueness the arithmetic does not provide, and a spreadsheet function will return one of them, determined by its starting guess, without indicating that the other exists.

What to do instead. Compute net present value at the rate the decision-maker actually faces. For this stream at $25\mathrm{\%}$ it is $24.00$ and the project is worth taking; at $5\mathrm{\%}$ it is $-20.41$ and it is not. That is a complete answer, and it required no root at all.

Net present value against internal rate, on the same two projects.

The internal rate ranks A above B at every discount rate, because it consumes none. Net present value ranks B above A below $15.0163\mathrm{\%}$ and A above B beyond it. A firm borrowing at $10\mathrm{\%}$ that followed the internal rate would give up $213.03$ per $1000$ committed.

Twenty payments against infinitely many.

Everything from year 21 to the end of time is worth $\mathrm{7,537.79}$, less than the first two decades. The difference between a long stream and an endless one is smaller than the difference between their descriptions.

One sign change against two.

A stream that goes out once and comes in thereafter has at most one positive root, so its internal rate is unique and quoting it is safe. A stream that goes out, comes in and goes out again may have two: $-1000$, $+2600$, $-1650$ is zero at $10\mathrm{\%}$ and at $50\mathrm{\%}$. The difference is not the size of the flows but the number of direction changes, which is why counting signs is the first step rather than an afterthought.

Annual rate divided by twelve against the twelfth root.

At a quoted $12\mathrm{\%}$, the monthly equivalents are $1\mathrm{\%}$ by division and $0.9489\mathrm{\%}$ by $(1.12{)}^{1/12}-1$. The first overstates the monthly rate, and over a long stream the error accumulates in the same direction throughout. Both appear in practice; they price different contracts, and which is correct depends on how the quoted rate was defined rather than on preference.

A ratio against an amount.

A project turning $10$ into $12$ has an internal rate of $20\mathrm{\%}$; one turning $\mathrm{10,000}$ into $\mathrm{11,500}$ has $15\mathrm{\%}$. The rate prefers the first and the second leaves the firm $\mathrm{1,500}$ better off. A ratio cannot express magnitude, which is a limitation of the quantity rather than a mistake in computing it.

Ranking mutually exclusive projects by internal rate. The quantity takes no discount rate, so it cannot express a preference that depends on one, and rankings routinely do. On the worked pair the internal rate prefers the short project always, while net present value prefers the long one below $15.0163\mathrm{\%}$. Following the rate at a cost of capital of $10\mathrm{\%}$ forgoes $213.03$ per $1000$.

Quoting "the" internal rate for a stream that changes sign more than once. The flows $-1000$, $+2600$, $-1650$ break even at $10\mathrm{\%}$ and again at $50\mathrm{\%}$. A spreadsheet function returns whichever root its search reaches first, with nothing in the output to indicate the other. Count the sign changes before asking for a rate.

Dividing an annual rate by twelve. At $12\mathrm{\%}$ annually the monthly equivalent is $(1.12{)}^{1/12}-1=0.9489\mathrm{\%}$, not $1\mathrm{\%}$. The error is small per period and always in the same direction, so it compounds across a long stream.

Mismatching the annuity convention. The formula prices $n$ payments with the first one period out. A stream beginning immediately is worth $(1+r)$ times as much. Nothing in the result looks wrong when the convention is mismatched, which is why checking a short stream against a direct sum is worth the minute it takes.

Treating a high rate as a large gain. A ratio carries no magnitude. Ten becoming twelve beats ten thousand becoming eleven and a half thousand on the rate, and loses by $\mathrm{1,498}$ on the amount.

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Two errors about the discount rate itself.

Using a rate the decision-maker does not face. Net present value is only as meaningful as its input. A rate taken from a textbook, or from another firm's disclosures, prices the stream for somebody else. Where the true rate is uncertain, the report should give net present value across a range, and the range is part of the finding.

Reading a perpetuity's value as robust. $C/r$ is acutely sensitive at low rates: halving the assumed rate doubles the value. An endowment priced at $5\mathrm{\%}$ needs $\mathrm{20,000}$ per $1000$ of annual payment and at $2.5\mathrm{\%}$ needs $\mathrm{40,000}$. The formula is exact and its input is an assumption about the indefinite future.

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What the internal rate is legitimately for. It reports the rate at which a stream breaks even, which is a real and useful fact. It is the redemption yield of a bond, and it summarises a single-sign-change investment compactly. The failures above all arise from asking it to do something else: rank alternatives, or describe a stream that crosses zero more than once.

Capital budgeting under a constraint. When a firm can fund only some of the projects available, the ranking is the decision. Finance functions that rank by internal rate systematically favour small, quick projects, because the rate rewards speed and ignores scale. The standard correction is to rank by net present value at the firm's own cost of capital, and where capital is rationed, by value per unit of capital consumed.

Infrastructure appraisal. Public projects have costs now and benefits spread over decades, so the discount rate chosen dominates the answer. Treasury guidance in several countries specifies the rate to use precisely to stop the choice being made project by project, and published appraisals report sensitivity across a range rather than a single figure.

Bond pricing. The redemption yield is an internal rate, and quoting it is safe here because a bond's stream changes sign once: an outflow to buy, inflows thereafter. The uniqueness that justifies the convention is a property of the sign pattern rather than of bonds.

Decommissioning and restoration. Mining, nuclear and offshore projects pay out at the end, giving a stream that goes out, comes in, and goes out again. Two sign changes, so the internal rate may not be unique, and appraisal in these industries is conducted on net present value for that reason.

Pension and endowment funding. A commitment to pay a sum indefinitely is priced at $C/r$, and the sensitivity is severe: at $5\mathrm{\%}$ each $1000$ of annual payment requires $\mathrm{20,000}$ of capital, at $2.5\mathrm{\%}$ it requires $\mathrm{40,000}$. Long-term funding disputes are usually disputes about $r$ rather than about arithmetic, which is why the assumed return is disclosed and contested.

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The recurring shape. In every case the discounting is routine and the decision turns on two things the arithmetic cannot supply: which rate the decision-maker actually faces, and whether the question being asked is one a single break-even rate can answer. Stating both explicitly is what makes an appraisal auditable rather than merely arithmetic.