Valuing a Stream of Dated Payments

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.

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.

That every project has an internal rate of return, and exactly one. The internal rate is a root of the present-value polynomial in the discount factor, so the usual facts about roots apply. A stream whose cash flows are all the same sign has no root at all and no internal rate. A stream that changes sign more than once may have several: the flows $-1000$, $+2600$, $-1650$ have net present value exactly zero at both $10\mathrm{\%}$ and $50\mathrm{\%}$, positive strictly between them and negative outside, so there are two internal rates and neither is "the" one. Software returning a single figure has found one root, usually the first encountered from its starting guess, and reports it without indicating that others exist.

That between two projects the one with the higher internal rate of return is the better investment. The internal rate takes no discount rate as input, so it cannot express a preference that depends on the rate the decision-maker actually faces, and rankings frequently do depend on it. A project paying $1200$ at the end of one period on an outlay of $1000$ has an internal rate of $20.00\mathrm{\%}$; a project paying $2100$ after five periods on the same outlay has $15.9962\mathrm{\%}$. The internal rate prefers the first at every discount rate. Net present value prefers the second below $15.0163\mathrm{\%}$ and the first above it: at $10\mathrm{\%}$ the values are $90.91$ against $303.93$, and at $18\mathrm{\%}$ they are $16.95$ against $-82.07$. The internal rate is also insensitive to scale, so a small project with a high rate outranks a large one that creates far more value.