Internal rate of return
The term internal refers to the fact that the calculation excludes external factors, such as the risk-free rate, inflation, the cost of capital, or financial risk.
[2] The IRR of an investment or project is the "annualized effective compounded return rate" or rate of return that sets the net present value (NPV) of all cash flows (both positive and negative) from the investment equal to zero.
This is in contrast with the NPV, which is an indicator of the net value or magnitude added by making an investment.
This discrepancy leads to overestimation of the rate of return which might be an incorrect representation of the value of the project.
Given a collection of pairs (time, cash flow) representing a project, the NPV is a function of the rate of return.
Any fixed time can be used in place of the present (e.g., the end of one interval of an annuity); the value obtained is zero if and only if the NPV is zero.
For example, Microsoft Excel and Google Sheets have built-in functions to calculate IRR for both fixed and variable time-intervals; "=IRR(...)" and "=XIRR(...)".
Of particular interest is the case where the stream of payments consists of a single outflow, followed by multiple inflows occurring at equal periods.
In the above notation, this corresponds to: In this case the NPV of the payment stream is a convex, strictly decreasing function of interest rate.
) has been shown to be almost 10 times more accurate than the secant formula for a wide range of interest rates and initial guesses.
If applied iteratively, either the secant method or the improved formula always converges to the correct solution.
When the objective is to maximize total value, the calculated IRR should not be used to choose between mutually exclusive projects.
When the objective is to maximize total value, IRR should not be used to compare projects of different duration.
For example, the NPV added by a project with longer duration but lower IRR could be greater than that of a project of similar size, in terms of total net cash flows, but with shorter duration and higher IRR.
Max Value is also happy, because she has filled her capital budget straight away, and decides she can take the rest of the year off investing.
With multiple internal rates of return, the IRR approach can still be interpreted in a way that is consistent with the present value approach if the underlying investment stream is correctly identified as net investment or net borrowing.
In the context of survivorship bias which makes the high IRR of large private equity firms a poor representation of the average, according to Ludovic Phalippou, "...a headline figure that is often shown prominently as a rate of return in presentations and documents is, in fact, an IRR.
These early winners have set up those firms' since-inception IRR at an artificially sticky and high level.
In passing, this generates some stark injustice because it is easier to game IRRs on LBOs in Western countries than in any other PE investments.
That means that the rest of the PE industry (e.g. emerging market growth capital) is sentenced to look relatively bad forever, for no reason other than the use of a game-able performance metric.
In most cases, they post information on their past performance in PE, but nothing that enables any meaningful benchmarking.
As IRR is often misleading and can never be aggregated or compared to stock-market returns, such information is basically useless for gauging performance.
This applies in real life for example when a customer makes a deposit before a specific machine is built.
[16] Mathematically, the value of the investment is assumed to undergo exponential growth or decay according to some rate of return (any value greater than −100%), with discontinuities for cash flows, and the IRR of a series of cash flows is defined as any rate of return that results in a NPV of zero (or equivalently, a rate of return that results in the correct value of zero after the last cash flow).
[18][19][20][21][22][23] To understand the source of this confusion let's consider an example with a 3-year bond of $1000 face value and coupon rate of 5% (or $50).
To understand why, we need to calculate the present value (PV) of our future cash flows, effectively reproducing IRR calculations manually: The IRR can be used to measure the money-weighted performance of financial investments such as an individual investor's brokerage account.
For this scenario, an equivalent,[24] more intuitive definition of the IRR is, "The IRR is the annual interest rate of the fixed rate account (like a somewhat idealized savings account) which, when subjected to the same deposits and withdrawals as the actual investment, has the same ending balance as the actual investment."
There are examples where the replicating fixed rate account encounters negative balances despite the fact that the actual investment did not.
It has been shown that this way of charging interest is the root cause of the IRR's multiple solutions problem.
[25][26] If the model is modified so that, as is the case in real life, an externally supplied cost of borrowing (possibly varying over time) is charged on negative balances, the multiple solutions issue disappears.
