Internal rate of return

An internal rate of return (short: IZF ; also: internal interest rate ; English : Internal Rate of Return , IRR ) of an investment is a discount rate , the use of which results in a net present value of zero. Interpreted differently, an internal rate of return is the discount factor , when used, the discounted future payments correspond to the current price or the initial investment. If this interest rate is greater than the calculated interest rate (i.e. the return is greater than the capital interest plus risk premium ), the investment is economical over the entire term.

The internal interest rate method ( internal interest rate method , method of the internal interest rate / rate , IZM ) is a method of dynamic investment calculation within investment theory . It enables a (theoretical) average annual return to be calculated for an investment or capital investment that generates irregular and fluctuating returns .

The internal rate of return method was originally developed to increase the profitability of investment decisions in companies. The aim of the calculations was to determine the investment decision that would have the most beneficial effect on the company as a whole.

Action

The interest rate i is sought at which the net present value

${\ displaystyle KW = -I + \ sum _ {t = 1} ^ {T} {\ frac {C_ {t}} {(1 + i) ^ {t}}} \; {\ overset {\ mathrm {! }} {=}} \; 0}$

of the given project is zero . The investment is compared to the sum of all discounted cash flows (payments) at points in time . ${\ displaystyle I}$ ${\ displaystyle C}$ ${\ displaystyle t}$

To solve the equation, i. H. To determine the rate of interest i , an interpolation method is usually used :

One chooses a first estimated rate of interest and uses it to calculate the capital value of the investment object.

{\ displaystyle i_ {1}}

{\ displaystyle KW_ {1}}

If ( ), one chooses a rate of return ( ) and calculates with it , so that ( ).

{\ displaystyle KW_ {1}> 0}

{\ displaystyle KW_ {1} <0}

{\ displaystyle i_ {2}> i_ {1}}

{\ displaystyle i_ {2} <i_ {1}}

{\ displaystyle KW_ {2}}

{\ displaystyle KW_ {2} <0}

{\ displaystyle KW_ {2}> 0}

From the values , and , the intersection point with the x-axis is determined using the straight line equation
and thus an approximate value for the actual interest rate : ${\ displaystyle i_ {1}}$ $i_1$ ${\ displaystyle i_ {2}}$ $i_ {2}$ ${\ displaystyle KW_ {1}}$ $KW_ {1}$ ${\ displaystyle KW_ {2}}$ $KW_ {2}$ ${\ displaystyle i ^ {*}}$ $i ^ {*}$ ${\ displaystyle i}$ $i$
- ${\ displaystyle i ^ {*} = i_ {1} - {\ frac {KW_ {1}} {KW_ {2} -KW_ {1}}} \ cdot \ left (i_ {2} -i_ {1} \ right)}$ .

With the newly calculated interest rate one determines the new capital value . If this is not close enough to zero, the procedure is repeated - the previous positive (negative) net present value and its interest are replaced by the newly calculated positive (negative) net present value and the newly gained interest. This process is repeated until sufficient accuracy is achieved.

{\ displaystyle KW_ {3}}

With regard to the trial interest rates ( , ), the following applies: The closer the trial interest rates are together, the smaller the interpolation error. ${\ displaystyle i_ {1}}$ ${\ displaystyle i_ {2}}$

In practice, in addition to the mathematical solution method presented above for geometric series based on Regula falsi , the Newton method is also used. Modern spreadsheet programs such as Microsoft Excel contain add-ons that support the calculation of the zeros ( Solver - in German : "Target value search"). The regular falsi formula can be displayed very easily in OpenOffice.org Calc as well as in MS Excel with the IKV function (internal interest rate function).

The problem, however, is that geometric series with more than one change of sign lead to the fact that, mathematically, there may be several zeros. Example: An investment of 10 leads to a payback of 21 after one year and dismantling costs of 11 after the second year. The correct solutions are 0% and 10%.

Critical assessment

Granting or borrowing credit

If the following two projects are compared, the internal rate of return method does not help:

Project	${\ displaystyle I}$	${\ displaystyle C_ {1}}$	IZF	KW at 10%
A.	−2,000	+3,000	+ 50%	+727
B.	+2,000	−3,000	+ 50%	−727

Both projects have the same internal rate of return ( and ) and are therefore equally attractive according to this method. However, when looking at KW (or in this case: looking closely) it becomes clear that 50% of the money is initially loaned in project A and borrowed in project B. When borrowing money, a lower interest rate is desired, that is, the IZF should be lower than the opportunity cost , not higher. ${\ displaystyle -2,000 + {\ tfrac {3,000} {1 {,} 50}} = 0}$ ${\ displaystyle +2,000 - {\ tfrac {3,000} {1 {,} 50}} = 0}$

Multiple internal rates of return

In most countries, taxes are paid in the following year; This means that the profit and the tax burden do not arise in the same period. The following example is a project that requires an investment of € 2,000,000 and an additional profit of € 600,000 per year during its (here five-year) term. a. brings in. The tax rate is 50% and is paid in the following period:

	${\ displaystyle C_ {0}}$	${\ displaystyle C_ {1}}$	${\ displaystyle C_ {2}}$	${\ displaystyle C_ {3}}$	${\ displaystyle C_ {4}}$	${\ displaystyle C_ {5}}$	${\ displaystyle C_ {6}}$
Gross cash flow (before taxes)	−2,000	+600	+600	+600	+600	+600
Taxes		+1,000	−300	−300	−300	−300	−300
Net cash flow	−2,000	+1,600	+300	+300	+300	+300	−300

(Note: The investment of € 2 million in reduces the tax burden for this period by € 1,000,000, which will be added in.) ${\ displaystyle C_ {0}}$ ${\ displaystyle C_ {1}}$

The calculations of the IZF and KW result in the following:

IZF	KW at 10%
−50% and 15.2%	149.71 or 149,710 €

The condition is met for both interest rates . The reason for this is that it is not a normal investment (a maximum of one sign change in the series of payments): According to Descartes' sign rule , a polynomial equation can have as many positive zeros as there are sign changes. In the example, this double change in sign means that the (mathematically correct) result is not unique (which internal rate of return is correct?). ${\ displaystyle KW = 0}$

In practice, such series do not only come about due to the delay in tax payments, but can also arise from maintenance costs during the term of the project or the scrapping of a system at the end of the term.

One way of avoiding a final (second) change in sign is to calculate a modified IZF: The cash flow in the 6th year is calculated and added to this in the 5th and the IZF is calculated again. However, this leads to the result being falsified - the net present value of the original series of payments is still positive even above the interest rate determined in this way.

Mutually exclusive projects

In order to fulfill a particular assignment, companies often have a choice between mutually exclusive projects. Here, too, the IZF method can be misleading:

Project	${\ displaystyle C_ {0}}$	${\ displaystyle C_ {1}}$	IZF	KW at 10%
C.	−20,000	+40,000	+ 100%	+16,363
D.	−40,000	+70,000	+ 75%	+23,636

Both projects are lucrative and, according to the IZF decision rule, project C would have to be carried out, but as KW shows, D is preferable to C because it has the higher monetary value. Nevertheless, the IZF method can also be used here: When considering the incremental cash flows (the difference between the two projects), the internal rate of return leads to the same result as the net present value method (the incremental IZF is 50%, i.e. if the incremental IZF is greater than the discount rate, should the "larger" project - here: Example D - be carried out).

Neglecting the yield structure

The IZF method is based on the assumption that the short-term and long-term interest rates are identical (see formula, only one interest rate). That is seldom the case in reality. The interest rates differ significantly in terms of maturity. Short money , d. H. Loan with a relatively short term, has a lower interest rate, so it is cheaper than long money , i. H. Longer Term Loans. Inverse interest rate structures were observed in the early 1990s, for example. This is not a problem with the net present value method, since the payment flows can simply be discounted with different interest rates:

${\ displaystyle KW = C_ {0} + {\ frac {C_ {1}} {(1 + i_ {1})}} + {\ frac {C_ {2}} {(1 + i_ {2}) ^ {2}}} + \ ldots}$

An alternative is to calculate with the weighted average of the interest over the term, but critics of this variant object that it unnecessarily increases the complexity of the calculation if a simple solution is available.

In practice, the interest structure problem and thus the question of which interest rate the IZF should be compared with ( , or ), is mostly neglected. The discount rate used in the investment calculation is also never just the financing interest, but rather a required minimum interest rate. This can be adjusted to the interest rate structure based on the term. Expected changes in the interest rate level can be taken into account by modifying the interest rates: ${\ displaystyle i_ {1}}$ ${\ displaystyle i_ {2}}$ ${\ displaystyle i_ {5}}$

${\ displaystyle KW = C_ {0} + {\ frac {C_ {1}} {(1 + i_ {1})}} + {\ frac {C_ {2}} {(1 + i_ {1}) ( 1 + i_ {2})}} + \ ldots}$

Conclusion

The internal rate of return method is not suitable for comparing several investment projects of different size, duration and investment timing. It is quite possible that an investment with a higher internal rate of return will have a lower net present value than another investment with a lower IZF.

The informative value of the calculated value is limited depending on the investment object. For financial investments, the internal rate of return corresponds to the effective interest rate. In the case of real investments, on the other hand, the internal rate of return is only a theoretical marginal interest rate up to which an investment would be economical.

Furthermore, this method assumes that all capital returns are reinvested at the internal interest rate ( reinvestment premises ) and not at the market interest rate (capital value method). In practice, the reinvestment premise is largely classified as unrealistic.

The examples given show that it is entirely possible to modify the IZF method in such a way that it delivers useful results. The question arises, however, whether this is necessary in view of the reliability and mathematical simplicity of the net present value method.

The internal interest rate method is well suited in practice for assessing individual investments in incompletely defined scenarios. Measure size is a desired minimum return. If the interest rate exceeds this minimum return, the investment makes sense in and of itself.

The shown options for making the internal rate of return method practically usable result in an application of the capital value method: The specific investment is compared with a reference interest rate indirectly (IZF method) or directly (capital value method).

Variants of the internal rate of return method

In practice, there are different variants of the internal rate of return method, depending on whether linear or exponential interest rates are used. This is shown below.

Practical variants of the internal rate of return method
	ISMA	SIA	Treasury	Moosmüller
Conversion of a period return to an annual return	exponentially	linear during the year	linear during the year	exponentially
Discounting of the first full coupon period	exponentially	exponentially	linear	linear
Calculation of the return for remaining terms of less than one year	exponentially	exponentially	linear	exponentially

Explanation:

ISMA - International Securities Market Association (now: ICMA - International Capital Market Association )
SIA - Securities Industries Association
Treasury

literature

Lutz Kruschwitz: Investment calculation . 11th edition Oldenbourg Wissenschaftsverlag, Munich 2007, ISBN 978-3-486-58306-9
Louis Perridon, Manfred Steiner: Finance of the company . 14th edition Franz Vahlen, Munich 2007, ISBN 3-8006-3359-0
Gerhard Mensch: Investment . 1st edition. Oldenbourg Wissenschaftsverlag, Munich 2002. ISBN 978-3-486-25946-9 . P. 87 ff.
Richard A. Brealey, Stewart Clay Myers: Principles of Corporate Finance . 7th edition McGraw-Hill, London 2002/2003, ISBN 978-0-07-294043-5

Individual evidence

↑ Gerhard Mensch: Investment . Oldenbourg Wissenschaftsverlag, Munich 2002, ISBN 978-3-486-25946-9 . P. 87.
^ Richard A. Brealey, Stewart Clay Myers: Principles of Corporate Finance . 7th edition McGraw-Hill, London 2002/2003, ISBN 978-0-07-294043-5 , p. 105.

[1] Gerhard Mensch: Investment . Oldenbourg Wissenschaftsverlag, Munich 2002, ISBN 978-3-486-25946-9 . P. 87.

[2] Richard A. Brealey, Stewart Clay Myers: Principles of Corporate Finance . 7th edition McGraw-Hill, London 2002/2003, ISBN 978-0-07-294043-5 , p. 105.