Internal Rate of Return (IRR)

The internal rate of return (IRR) is the discount rate that makes the net present value of all of an investment's cash flows equal to zero. Put differently, it is the annual compound return the investment is expected to earn on the money that remains tied up in it. Corporate finance teams compare a project's IRR with a required return, or hurdle rate; private equity, venture capital and real-estate funds report IRR as their headline performance measure.

IRR and NPV use the same cash flows and both rest on the time value of money, but they answer different questions. NPV says how much value a project adds in money; IRR says what percentage return it earns. IRR is popular because a percentage is easy to compare with a borrowing cost or a target, yet it has well-known traps that can mislead when projects differ in size, length or cash-flow pattern.

Suppose you lend a relative some money and they pay you back in uneven instalments over a few years. You want to know what interest rate that deal was really worth to you, as if it had been a savings account. The IRR is that rate: the single steady rate that would turn your original outlay into exactly the payments you received.

If that rate beats what a bank or another investment would have paid, the deal was a good use of your money. If it falls short, you would have done better elsewhere. IRR squeezes a messy pattern of payments into one familiar-looking percentage, which is both its appeal and its weakness.

The IRR is the rate $r^{*}$ that solves:

• $C_t$ is the net cash flow in period $t$, with the initial outlay at $t = 0$ entered as a negative number.
• $r^{*}$ is the internal rate of return per period.
• $n$ is the number of periods.

There is usually no neat algebraic answer, so the rate is found by trial and error or by spreadsheet functions such as IRR and XIRR (the second handles irregular dates). Worked example: an investment costs 1,000 and returns 600 at the end of year 1 and 600 at the end of year 2. At 13% the inflows are worth $531 + 470 = 1{,}001$; at 13.1% they are worth about 999. The IRR is therefore about 13.07%. If the investor's required return is 10%, the project passes; its NPV at 10% is positive, at about 41.

Private funds draw investors' money in stages and return it in stages, so a time-weighted return is hard to apply and IRR has become the standard reporting measure. Funds also quote multiples, such as total value to paid-in capital, because IRR alone can flatter results. A fund that returns cash quickly can show a high IRR on a modest profit, and managers can raise reported IRR by using short-term borrowing facilities to delay calling investors' capital, a practice industry bodies such as the Institutional Limited Partners Association have asked to be disclosed. Investors comparing funds therefore look at IRR, multiples and public-market-equivalent measures together, alongside a risk-adjusted return view of the strategy.

• Scale. A small project returning 40% may add far less value than a large one returning 15%. When only one project can be chosen, NPV gives the right ranking and IRR may not.
• Multiple or no solutions. If cash flows change sign more than once, for example a mine that needs a large clean-up payment at the end, the equation can have two IRRs or none at all.
• Reinvestment assumption. Comparing projects by IRR implicitly treats interim cash as if it could be reinvested at the IRR itself, which is unrealistic for high-return projects. The modified IRR (MIRR) assumes reinvestment at the cost of capital instead.
• Timing and duration. IRR says nothing about how long the money is at work; a 25% return for one year and a 25% annual return for ten years produce very different wealth.
• Changing rates. Where the appropriate discount rate differs over time, as the yield curve often implies, a single hurdle rate is a simplification.

For a bond, the IRR of buying at today's price and holding to the end is the yield to maturity. For an investment held over several years with no interim cash flows, IRR equals the compound annual growth rate. IRR differs from simple return on investment, which ignores timing, and from the general rate of return reported by mutual funds, which is usually time-weighted so that investors' own deposits and withdrawals do not distort the manager's record. Because IRR does not adjust for risk, analysts compare it with a hurdle rate built from the risk-free rate plus a risk premium, often derived from the Capital Asset Pricing Model (CAPM).

• HM Treasury: The Green Book, appraisal and evaluation in central government
• Aswath Damodaran, NYU Stern School of Business: valuation data and teaching materials