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Internal rate of return

 
Investment Dictionary: Internal Rate Of Return - IRR
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The discount rate often used in capital budgeting that makes the net present value of all cash flows from a particular project equal to zero. Generally speaking, the higher a project's internal rate of return, the more desirable it is to undertake the project. As such, IRR can be used to rank several prospective projects a firm is considering. Assuming all other factors are equal among the various projects, the project with the highest IRR would probably be considered the best and undertaken first.

IRR is sometimes referred to as "economic rate of return (ERR)".

Investopedia Says:
You can think of IRR as the rate of growth a project is expected to generate. While the actual rate of return that a given project ends up generating will often differ from its estimated IRR rate, a project with a substantially higher IRR value than other available options would still provide a much better chance of strong growth.

IRRs can also be compared against prevailing rates of return in the securities market. If a firm can't find any projects with IRRs greater than the returns that can be generated in the financial markets, it may simply choose to invest its retained earnings into the market.


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Insurance Dictionary: Internal Rate of Return
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Method used to determine the Policyholder's return on premiums paid into a life insurance policy. This method is illustrated in two ways:

1. Surrender of Policy Approach-calculation of the interest rate required for the accumulated value of the total premiums paid (minus any Dividends) into the policy at a given time to equal the Cash Surrender Value of the policy at that time;

2. Death Benefit Paid Approach-calculation of the interest rate required for the accumulated value of the total premiums paid (minus any dividends) into the policy at a given time to equal the death benefit of the policy at that time.

Real Estate Dictionary: Internal Rate of Return (IRR)
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The true annual rate of earnings on an investment. Equates the value of cash returns with cash invested. Considers the application of Compound Interest factors. Requires a trial-and-error method for solution. The formula is

n periodic cash flow

∑ _______________ = investment amount

t - 1 (1 + i)t

where i = internal rate of return

t = each time interval

n = total time intervals

∑ = summation
Example: Abel sells for $200,000 land that he bought 4 years earlier for $100,000. There were no Carrying Charges or Transaction Costs. The internal rate of return was about 19%. That is the annual rate at which compound interest must be paid for $100,000 to become $200,000 in 4 years.
Example: Baker received $3,000 per year for 5 years on a $10,000 investment. The internal rate of return was about 15%.

Accounting Dictionary: Internal Rate of Return (IRR)
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Rate earned on a proposal. It is the rate of interest that equates the initial investment (I) with the present value (PV) of future cash inflows. That is, at IRR, I = PV, or NPV (net present value) = 0. Under the internal rate of return method, the decision rule is: accept the project if IRR exceeds the cost of capital; otherwise, reject the proposal.

For example, consider the following data:

Initial investment $16,200

Estimated life 10 years

Annual cash inflows $ 3000

Cost of capital (minimum required of return) 10%

Set up the following equality (I = PV):

$16,200 = $3000 x PV

Then PV = $16,200/$3000 = 5.400, which stands somewhere between 12% and 14% in the 10-year line of table 4 in the back of the book. Because the investment's IRR (13.15%) is greater than the cost of capital (10%), the investment should be accepted.

The IRR method is easy to use as long as cash inflows are even from year to year. Where cash flows are uneven, the IRR must be determined by trial and error. Assume, for example, that a company is considering an investment project that promises cash inflows of $400,000, $600,000, and $1,000,000 for each of the next three years for a given investment of $1,490,000. The IRR is found by selecting a rate and discounting the cash inflows. If the PV is greater than I, select a higher rate until one is found that equates the PV of the cash inflows with I. In this example, the IRR is approximately 14%, determined as follows:

An advantage of the IRR method is that it considers the Time Value of Money and is therefore more exact and realistic than Accounting Rate of Return (ARR). Disadvantages are: (1) it fails to recognize the varying size of investment in competing projects and their respective dollar profitabilities, and (2) in limited cases, where there are multiple reversals in the cash-flow streams, the project could yield more than one internal rate of return.

Wikipedia: Internal rate of return
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The internal rate of return (IRR) is a rate of return used in capital budgeting to measure and compare the profitability of investments. It is also called the discounted cash flow rate of return (DCFROR) or simply the rate of return (ROR).[1] In the context of savings and loans the IRR is also called the effective interest rate. The term internal refers to the fact that its calculation does not incorporate environmental factors (e.g., the interest rate or inflation).

Contents

Definition

Showing the position of the IRR on the graph of NPV(r) (r is labelled 'i' in the graph)

The internal rate of return on an investment or potential investment is the annualized effective compounded return rate that can be earned on the invested capital.

In more familiar terms, the IRR of an investment is the interest rate at which the costs of the investment lead to the benefits of the investment. This means that all gains from the investment are inherent to the time value of money and that the investment has a zero net present value at this interest rate.

Uses

Because the internal rate of return is a rate quantity, it is an indicator of the efficiency, quality, or yield of an investment. This is in contrast with the net present value, which is an indicator of the value or magnitude of an investment.

An investment is considered acceptable if its internal rate of return is greater than an established minimum acceptable rate of return. In a scenario where an investment is considered by a firm that has equity holders, this minimum rate is the cost of capital of the investment (which may be determined by the risk-adjusted cost of capital of alternative investments). This ensures that the investment is supported by equity holders since, in general, an investment whose IRR exceeds its cost of capital adds value for the company (i.e., it is profitable).

Calculation

Given a collection of pairs (time, cash flow) involved in a project, the internal rate of return follows from the net present value as a function of the rate of return. A rate of return for which this function is zero is an internal rate of return.

Given the (period, cash flow) pairs (n, Cn) where n is a positive integer, the total number of periods N, and the net present value NPV, the internal rate of return is given by r in:

\mbox{NPV} = \sum_{n=0}^{N} \frac{C_n}{(1+r)^{n}} = 0

Note that the period is usually given in years, but the calculation may be made simpler if r is calculated using the period in which the majority of the problem is defined (e.g., using months if most of the cash flows occur at monthly intervals) and converted to a yearly period thereafter.

Note that 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.

In the case that the cash flows are random variables, such as in the case of a life annuity, the expected values are put into the above formula.

Often, the value of r cannot be found analytically. In this case, numerical methods or graphical methods must be used.

Example

If an investment may be given by the sequence of cash flows

Year (n) Cash Flow (Cn)
0 -4000
1 1200
2 1410
3 1875
4 1050


then the IRR r is given by

NPV=-4000+\frac{1200}{(1+r)^1} + \frac{1410}{(1+r)^2} + \frac{1875}{(1+r)^3} + \frac{1050}{(1+r)^4} = 0.

In this case, the answer is 14.3%.

Numerical Solution

Since the above is a manifestation of the general problem of finding the roots of the equation NPV(r), there are many numerical methods that can be used to estimate r. For example, using the secant method, r is given by

r_{n+1} = r_n-\mbox{NPV}_n\left(\frac{(r_n-r_{n-1})}{\mbox{NPV}_n-\mbox{NPV}_{n-1}}\right).

where rn is considered the nth approximation of the IRR.

This formula initially requires two unique pairs of estimations of the IRR and NPV (r0,NPV0) and (r1,NPV1), and produces a sequence of

(r_0, \mbox{NPV}_0), (r_1, \mbox{NPV}_1), \dots, (r_{n-1}, \mbox{NPV}_{n-1}), (r_{n}, \mbox{NPV}_{n}), (r_{n+1}, \mbox{NPV}_{n+1})

that may converge to \scriptstyle (r, 0) as \scriptstyle n\to\infty. If the sequence converges, then iterations of the formula can continue indefinitely so that r can be found to an arbitrary degree of accuracy.

The convergence behaviour of the sequence is governed by the following:

  • If the function NPV(i) has a single real root r, then the sequence will converge reproducibly towards r.
  • If the function NPV(i) has n real roots \scriptstyle r_1,r_2,\dots,r_n, then the sequence will converge to one of the roots and changing the values of the initial pairs may change the root to which it converges.
  • If function NPV(i) has no real roots, then the sequence will tend towards infinity.

Having \scriptstyle{r_1 > r_0} when NPV or \scriptstyle{r_1 < r_0} when NPV0 < 0 may speed up convergence of rn to r.

Problems with using internal rate of return

As an investment decision tool, the calculated IRR should not be used to rate mutually exclusive projects, but only to decide whether a single project is worth investing in.

NPV vs discount rate comparison for two mutually exclusive projects. Project 'A' has a higher NPV (for certain discount rates), even though its IRR (=x-axis intercept) is lower than for project 'B' (click to enlarge)

In cases where one project has a higher initial investment than a second mutually exclusive project, the first project may have a lower IRR (expected return), but a higher NPV (increase in shareholders' wealth) and should thus be accepted over the second project (assuming no capital constraints).

IRR assumes consumption of positive cash flows during the project. If positive cash flows can be reinvested back into the project, then a suitable reinvestment rate is required in order to calculate the reinvestment cash flow and hence the IRR with cash flows reinvested.

When the calculated IRR is different from the true reinvestment rate for interim cash flows, the measure will accurately reflect the annual equivalent return from the project. The company may have additional projects, with equally attractive prospects, in which to invest the interim cash flows. [2]

This makes IRR a suitable (and popular) choice for analyzing venture capital and other private equity investments, as these strategies usually require several cash investments throughout the project, but only see one cash outflow at the end of the project (e.g., via IPO or M&A).

Since IRR does not consider cost of capital, it should not be used to compare projects of different duration. Modified Internal Rate of Return (MIRR) does consider cost of capital and provides a better indication of a project's efficiency in contributing to the firm's discounted cash flow.

In the case of positive cash flows followed by negative ones (+ + - - -) the IRR may have multiple values. In this case a discount rate may be used for the borrowing cash flow and the IRR calculated for the investment cash flow. This applies for example when a customer makes a deposit before a specific machine is built.

In a series of cash flows like (-10, 21, -11), one initially invests money, so a high rate of return is best, but then receives more than one possesses, so then one owes money, so now a low rate of return is best. In this case it is not even clear whether a high or a low IRR is better. There may even be multiple IRRs for a single project, like in the example 0% as well as 10%. Examples of this type of project are strip mines and nuclear power plants, where there is usually a large cash outflow at the end of the project.

In general, the IRR can be calculated by solving a polynomial equation. Sturm's theorem can be used to determine if that equation has a unique real solution. In general the IRR equation cannot be solved analytically but only iteratively.

When a project has multiple IRRs it may be more convenient to compute the IRR of the project with the benefits reinvestmented.[2] Accordingly, MIRR is used, which has an assumed reinvestment rate, usually equal to the project's cost of capital.

Despite a strong academic preference for NPV, surveys indicate that executives prefer IRR over NPV [3]. Apparently, managers find it easier to compare investments of different sizes in terms of percentage rates of return than by dollars of NPV. However, NPV remains the "more accurate" reflection of value to the business. IRR, as a measure of investment efficiency may give better insights in capital constrained situations. However, when comparing mutually exclusive projects, NPV is the appropriate measure.

Mathematics

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 net present value of zero (or equivalently, a rate of return that results in the correct value of zero after the last cash flow).

Thus internal rate(s) of return follow from the net present value as a function of the rate of return. This function is continuous. Towards a rate of return of -100% the net present value approaches infinity with the sign of the last cash flow, and towards a rate of return of positive infinity the net present value approaches the first cash flow (the one at the present). Therefore, if the first and last cash flow have a different sign there exists an internal rate of return. Examples of time series without an IRR:

  • Only negative cash flows - the NPV is negative for every rate of return.
  • (-1, 1, -1), rather small positive cash flow between two negative cash flows; the NPV is a quadratic function of 1/(1+r), where r is the rate of return, or put differently, a quadratic function of the discount rate r/(1+r); the highest NPV is -0.75, for r = 100%.

In the case of a series of exclusively negative cash flows followed by a series of exclusively positive ones, consider the total value of the cash flows converted to a time between the negative and the positive ones. The resulting function of the rate of return is continuous and monotonically decreasing from positive infinity to negative infinity, so there is a unique rate of return for which it is zero. Hence the IRR is also unique (and equal). Although the NPV-function itself is not necessarily monotonically decreasing on its whole domain, it is at the IRR.

Similarly, in the case of a series of exclusively positive cash flows followed by a series of exclusively negative ones the IRR is also unique.

  • Extended Internal Rate of Return: The Internal rate of return calculates the rate at which the investment made will generate cash flows. This method is convenient if the project has a short duration, but for projects which has an outlay of many years this method is not practical as IRR ignores the Time Value of Money. To take into consideration the Time Value of Money Extended Internal Rate of Return was introduced where all the future cash flows are first discounted at a discount rate and then the IRR is calculated. This method of calculation of IRR is called Extended Internal Rate of Return or XIRR.

See also


References

  1. ^ Project Economics and Decision Analysis, Volume I: Deterministic Models, M.A.Main, Page 269
  2. ^ a b Internal Rate of Return: A Cautionary Tale
  3. ^ Pogue, M.(2004). Investment Appraisal: A New Approach. Managerial Auditing Journal.Vol. 19 No. 4, 2004. pp. 565-570

Further reading

  • Bruce J. Feibel. Investment Performance Measurement. New York: Wiley, 2003. ISBN 0471268496

External links


This entry is from Wikipedia, the leading user-contributed encyclopedia. It may not have been reviewed by professional editors (see full disclaimer)

 
 

 

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