Optimal service life

The optimum useful life of a system is reached at a point in time at which its falling annual marginal profit corresponds to the imputed interest rate on the capital value of the replacement investment .

General

The optimal useful life is a form developed in business administration to determine the useful life of items of property, plant and equipment . It is based on the premise that there is an infinitely elastic capital market and that the system and its replacement investment produce a constant output . The decision about a replacement investment also depends on the knowledge of the optimal service life of the system to be replaced.

The higher the imputed interest rate, the longer the optimal useful life, and higher depreciation shortens it.

detection

As part of the investment calculation, the economically optimal useful life of an asset can be determined. There are three variants: ${\displaystyle n}$

• one-time investment or
• one-time repeated investment or
• investment repeated infinitely.

In all three cases, a fixed and known discount rate and known cash flows E _t -A _t = Z _{t are} assumed. In addition, the liquidation proceeds L _t to be achieved in each case are known for all periods . ${\displaystyle i}$ ${\displaystyle t}$

One-time investment

There are two methods for determining the optimal useful life of a one-time investment:

Determination of the period with the highest net present value

For each period or term , the net present value is calculated that is reached when the use of the asset ends at the end of this period and the asset is sold. The calculation is made using the formula: ${\displaystyle T}$

${\displaystyle K(T,i)=\sum _{t=0}^{T}Z_{t}\cdot \left(1+i\right)^{-t}+L\cdot \left(1+i\right)^{-T}}$.

The period for which the net present value is highest represents the optimal useful life.

Determination of the last period with a positive marginal contribution to the net present value

For each period, it is determined which marginal contribution the balance of income and costs and depreciation of the period makes to the net present value. The following are taken into account:

• the payout in the period ( A _t ),
• the depreciation of the asset ( L _t-1 -L _t ),
• the interest on the capital tied up at the beginning of the period or the lost interest on unrealized liquidation proceeds ( i * L _t-1 ).

The marginal contribution of the period to the net present value of the investment is positive if:

${\displaystyle (E_{t}-A_{t})-(L_{t}-L_{t-1})-(i\cdot L_{t-1})>0}$.

Only then will the investment be continued in the period . The last period with a positive marginal contribution determines the optimal useful life. ${\displaystyle t}$

One-time repeated investment

An investment is repeated once. Again it is assumed that all relevant parameters are known with the exception of the useful life to be determined.

First of all, the optimal useful life T ₂ for the second investment is determined using one of the methods described for the one-time investment. The net present value C _0.2 that the second investment achieves when viewed in isolation must also be calculated. (Note: Strictly speaking, the net present value of the second investment is not to be related to time 0, but to time T ₁ , the start of the second investment project.)

Then the optimal useful life T _{1 of} the first investment is determined. For this purpose, it is checked up to which period a continuation of the investment produces a positive marginal contribution to the net present value of the entire investment project. In addition to the incoming and _outgoing payments E _T1 , A _T1 , the loss in value L _T1-1 -L _T1 and the lost interest income i * L _T1-1 , the opportunity costs that arise from continuing the first investment must also be taken into account the net present value of the second investment is not realized until a period later.

The opportunity costs of continuing the first investment are: i * C _T1.2 . This results in the test variable:

${\displaystyle (E_{T1,1}-A_{T1,1})-(L_{T1-1,1}-L_{T1,1})-(i\cdot L_{T1-1,1})-(i\cdot C_{T1,2})>0}$.

(The first index shows the point in time, the second numbers the investment project.)

The first investment is only continued, i.e. H. T ₁ shifted further back as long as this condition is met. Otherwise, the start of investment 2 is more worthwhile.

Infinitely repeated investment

The aim is to determine the optimal useful life of an individual investment, assuming that this investment is repeated indefinitely. It is assumed that the individual investments are identical in terms of their framework conditions and parameters.

Under these conditions, there is no final investment that can be used as a starting point for determining the optimal useful life. Instead, the annuity method is used, which means that the duration of use is sought that offers the maximum withdrawal options.

The procedure for determining the optimal useful life then includes three steps:

• The useful life-dependent capital values ​​of an individual investment are converted into annuities. This is done using

${\displaystyle K\cdot ANF_{n,i}={\frac {i\cdot (1+i)^{n}}{(1+i)^{n}-1}}\ \cdot K}$.

• The service life that corresponds to the highest annuity is selected.
• The net present value of the entire investment chain then corresponds to the present value of a perpetuity . With a given annuity and discount rate, this is : ${\displaystyle a}$ ${\displaystyle i}$

${\displaystyle K=a\cdot {\frac {1}{i}}}$.

literature

• Lutz Kruschwitz: Investment calculation. Oldenbourg Wissenschaftsverlag, 13th edition, Munich, ISBN 978-3-4867-0531-7 .
• Günter Wöhe / Ulrich Döring : Introduction to general business administration. Vahlen, 25th edition, Munich 2013, ISBN 978-3-8006-4687-6 .

Individual evidence

1. ↑ Wolfgang Breuer / Thilo Schweizer / Claudia Breuer (eds.), Gabler Lexikon Corporate Finance , 2003, p. 162
2. ^ Klaus Reiche, Calculation of the useful life , in: Wolfgang Lück (Ed.), Lexikon der Betriebswirtschaft, 2004, p. 481