Internal interest rate

the internal interest rate method, also internal interest rate method (English ERR: Internal rate OF Return), is a procedure of the dynamic investment calculation.

procedure

it looks for itself the interest rate, with which the present value (NPV = Net Present VALUE) of the project (investment plus the sum all abgezinsten cash-flow (payments)) zero are alike:

[itex] NPV = - I+ \ sum_ {t=1} ^T \ frac {C_t} {(1+i) ^t} =0< /math>

But one avails oneself mostly an interpolation procedure:

1. One selects an interest foot [itex] i_1< /math> and compute thereby the present value of the investment object

2. Is [itex] KW_1 > 0< /math> ( [itex] KW_1 < 0< /math> ), then one selects an interest rate [itex] i_2 > i_1< /math> ( [itex] i_2 < i_1< /math> ) and compute thereby [itex] KW_2< /math>

3. As approximate value for [itex] r< /math> one determines [itex] r*< /math> assistance of a suitable interpolation formula

Concerning the attempt interest rates (i1, i2) it should be mentioned that during financial investments only a difference of up to 0,5% is meaningful. With expenditures on capital assets against it larger differences are possible (up to 5%)

general apply: The more near the attempt interest rates together lie, all the smaller are that Interpolation error.

The following interpolation formula is suitable:

[itex] r^*=i_1 - \ frac {KW_1} {KW_2 - KW_1} \ cdot \ left (i_2 - i_1 \ right) [/itex]

In practice for it frequently a computer uses a mathematical solution procedure for geometrical rows like the Newton procedure or Regula falsi. Modern spread-sheet programs such as MS Excel contain ADD in, which supports the zero computation (Solver).

Problematic it is however that geometrical rows with frequent sign changes lead to the fact that computationally possibly several zeros exist. Such rows must be settled only. For this an initial value for the interest rate can be indicated.

interpretation

is larger this interest rate than the calculation interest rate, is absolutely economical the investment.

critical estimate

granting of credit or raising of credit

one compares the following two projects does not help with one another the internal interest rate method not:

 Project [itex] C_0< /math> [itex] C_1< /math> IZF NPV with 10% A -2,000 +3,000 +50% +1,000 B +2,000 -3,000 +50% -1,000

as one sees both projects the same internal interest rate (- 1,000 + 1,500/1.50 = 0 and +1,000 - 1,500/1.50 = 0) having, is thus equivalent attractive according to this method. However becomes with view of the NPV (or in this case: the exact Hinsehen) clearly that with project A initially money is lent to 50% and with project B is borrowed. If money is borrowed is a low interest rate desired, i.e. the IZF should be lower than the Opportunitätskosten and not more highly.

several internal interest rates

in most countries are paid the taxes in the subsequent year, i.e. that the profit and fiscal charges in the same period do not result. One imagines a project, an investment at a value of 2.000.000€ required and during its (here five-year) running time an additional profit at a value of 600.000€ per annum. brings in. The control item amounts to 50% and in the subsequent period is paid:

 [itex] C_0< /math> [itex] C_1< /math> [itex] C_2< /math> [itex] C_3< /math> [itex] C_4< /math> [itex] C_5< /math> [itex] C_6< /math> 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 i.H.v. 2 millions€ in [itex] C_0< /math> reduces fiscal charges for this period by 1.000.000€, which we in [itex] C_1< /math> add.)

the computations of the IZF and NPV result in the following:

 IZF NPV with 10% -50% and 15.2% 149.71 or 149,710€

with both interest rates is fulfilled the condition NPV=0. The reason for this lies in the twice sign change in the payment row: After Descartes a polynomial equation can have as many mathematically valid solutions like sign changes. In the example this twice sign change leads to the fact that (mathematically correct) the result is not economically interpretable (which internal interest rate is correct?).

In practice such rows come off not only by the delay of the payments of taxes, but can result also from maintenance costs at run time the project or the scrapping of a plant at the end of the running time.

A possibility in the evasion of a locking (second) sign change consists of computing a modified IZF: The cash flow in 6. Year becomes in 5. computed and to this added and the IZF computes again.

which are mutually exclusive projects

around a certain order to fulfill itself have companies often the choice between which are mutually exclusive projects. The IZF method can in errs also here to lead:

 Project [itex] C_0< /math> [itex] C_1< /math> IZF NPV with 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 would have we project C to accomplish, but like the NPV, is D shows in relation to C to be preferred, since it has the higher monetary value. Nevertheless the IZF method can be used also here: During view of the incremental payment stream (the difference of both projects) the internal interest rate leads to the same result as the present value method (the incremental IZF are 50%, i.e. if the incremental IZF is larger than the calculation interest rate should the “larger” project be accomplished.).

neglect of the interest structure

the IZF method proceeds from the assumption that short and long-term interest rates are identical (see formula, only one interest rate). This applies in the reality rarely and the interest rates differs regarding the Fristigkeit substantially (“short money”, i.e. Credits with a relatively short running time exhibit a lower interest rate, are thus more cheaply, than “long money”, i.e. Credits with longer running time. Inverse interest structures were observed for example at the beginning of the 1990er years.). With the present value method this no problem represents, since simply the payment stream with different interest rates can be abgezinst:

[itex] NPV = C_0+ \ frac {C_1} {(1+i_1)}+ \ frac {C_2} {(1+i_2) ^2} +…< /math>

An alternative consists of counting on the weighted average of the interest over the running time however critics of this variant object that she increases the complexity of the calculation unnecessarily when being present a simple solution.

In practice the interest structure problem, and with it the question with which interest rate of the IZF will be compared is ([itex] i_1< /math>, [itex] i_2< /math> or [itex] i_5< /math>), usually neglects.

result

• the method of the internal interest rate is not suitable for it, several investment projects of different height to compare duration and investment times with one another. It is well possible that an investment with a higher internal interest rate has a smaller present value than another investment with lower IZF.
• Further this method assumes all refluxes of capital are again put on to the internal interest rate (reinvestment premise) and not to the market interest rate (present value method). The reinvestment premise is arranged in practice predominantly as unrealistic.
• The examples mentioned show that it is quite possible to modify the IZF method in such a way that them supply useful results. However the question arises whether this is necessary, if one considers, how the present value method functions reliably and mathematically simply.
• The method of the internal interest rate is suitable in practice well for the evaluation of single investments in incompletely defined scenarios. Measure size is a desired minimum net yield. If the interest rate exceeds this minimum net yield, then the investment for itself is taken meaningful.
• The shown possibilities of making the internal interest rate method practically usable come down in the result to an application of the present value method: The concrete investment becomes over detours (IZF method) or directly (present value method) with a reference interest rate compared.

literature

• Hans Meyer: To the general theory of the investment calculation. In: Series of publications the enterprise, Duesseldorf 1977