Exam FMCash flow analysisFree to read

Internal rate of return

The rate that makes NPV zero. Unique for a conventional project, and possibly not unique when the sign of the cash flows changes more than once.

The formulas

Definition

$$NPV(i^{\ast })=\sum _{t\ge 0}{C}_t(1+i^{\ast }{)}^{-t}=0$$

Descartes' bound

$$\#\text{sign changes}\ge \#\text{positive roots}$$

Dollar-weighted (simple interest approximation)

$$i\approx \frac{I}{A+{\sum }_t{C}_t(1-t)}$$

Time-weighted

$$1+i=\prod _k\frac{B_k}{B_{k-1}+C_{k-1}}$$

Where it comes from

1. Setting the NPV to zero gives a polynomial in $v$; each sign change in the cash-flow sequence permits one positive root.
2. A conventional project (one outflow then inflows) has exactly one sign change and therefore exactly one IRR.
3. The dollar-weighted rate measures the FUND's performance including the timing of deposits; the time-weighted rate strips timing out and measures the MANAGER's.

Worked example

This answer is recomputed from the site’s own interest-theory and probability functions every time the test suite runs, so the page and the mathematics cannot drift apart.

Memory hooks

• One sign change, one IRR. More sign changes, check before trusting a single answer.
• Dollar-weighted judges the investor; time-weighted judges the manager.

Traps

• Ranking mutually exclusive projects by IRR — use NPV.
• Applying the simple-interest dollar-weighted approximation to a period other than one year.

Related

• Net present value
• Bond yield and callable bonds

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