Preinreich-Lücke theorem
The Preinreich-Lücke theorem states that the net present value of profits (change in assets + (income - expenses)) corresponds to the net present value of excess payments (incoming payments - outgoing payments). This only applies if the differences are based on different periods of payments or payments compared to costs and services. The theorem was proposed by Wolfgang Lücke in 1955 as an alternative to the cash flow- based investment calculation and thus found its way into German economics . In the US, on the other hand, this correspondence was described earlier by Gabriel Preinreich in 1937 .
Assumptions
The following conditions must be met so that the net present value calculated from the income and expenditure of a project matches the net present value determined from the income and expense parameters and thus leads to the same optimal investment decision:
First, the sum of the payment surpluses of all periods must be equal to the sum of all period profits:
- Period profit at the time
- Excess cash at the time
- Deposits at the time
- Payouts at the time
- running time
- respective period
This corresponds to the congruence principle of Schmalenbach's dynamic balancing, which states that the sum of the successes in the section must be congruent with the total success.
Second, the profit for the period determined as the difference between income and expenses must be corrected by imputed interest on the capital commitment of the previous period :
With
- Capital stock at the time
- Capital stock of the previous period
- respective period that has expired up to the point in time .
proof
The capital commitment, i.e. the difference between the cumulative profits up to the point in time and the cumulative payment surpluses, is determined at the beginning of each period as follows:
- Cost of capital
If both of the above The following equation results for the net present value at the time of the decision:
- Residual profit at the time
example
The interest rate assumed for the following examples is 10%.
year | 0 | 1 | 2 | 3 | 4th | 5 |
---|---|---|---|---|---|---|
Investment payout | -1000 | |||||
Deposits | 1000 | 1000 | 1000 | 1000 | 1000 | |
Payouts | 600 | 600 | 600 | 600 | 600 | |
Cash flow | -1000 | 400 | 400 | 400 | 400 | 400 |
Discount factors | 0.909 | 0.826 | 0.751 | 0.683 | 0.621 | |
Present value | 364 | 331 | 301 | 273 | 248 | |
NPV | 516 |
year | 0 | 1 | 2 | 3 | 4th | 5 |
revenues | 1000 | 1000 | 1000 | 1000 | 1000 | |
costs | 600 | 600 | 600 | 600 | 600 | |
Depreciation | 200 | 200 | 200 | 200 | 200 | |
Operating result (before calculated interest) | 200 | 200 | 200 | 200 | 200 | |
lime. Interest (based on assets) | 100 | 80 | 60 | 40 | 20th | |
Residual result (BE after calculating interest) | 100 | 120 | 140 | 160 | 180 | |
Discount factors | 0.909 | 0.826 | 0.751 | 0.683 | 0.621 | |
Present value | 91 | 99 | 105 | 109 | 112 | |
NPV | 516 |
Table 3: Calculation of net present value based on costs according to the Preinreich-Lücke theorem with a modified depreciation method
year | 0 | 1 | 2 | 3 | 4th | 5 |
revenues | 1000 | 1000 | 1000 | 1000 | 1000 | |
costs | 600 | 600 | 600 | 600 | 600 | |
Depreciation | 400 | 275 | 175 | 100 | 50 | |
Operating profit | 0 | 125 | 225 | 300 | 350 | |
lime. interest | 100 | 60 | 32.5 | 15th | 5 | |
Residual result | -100 | 65 | 192.5 | 285 | 345 | |
Discount factors | 0.909 | 0.826 | 0.751 | 0.683 | 0.621 | |
Present value | -91 | 54 | 145 | 195 | 214 | |
NPV | 516 |
From Table 3 it becomes clear that under the conditions of the Preinreich-Lücke theorem, a change in the depreciation method has no influence on the calculation of the net present value in the case of a net present value calculation based on profit or loss. The tabular form used is the basis for the method of the complete financial plan , which allows the Lücke theorem to be easily understood.
literature
- Wolfgang Lücke: Investment calculation based on expenses or costs ?. In: Zeitschrift für Handelswissenschaftliche Forschung / NF , vol. 7 (1955), pp. 310-324, ISSN 1619-6147 .
Individual evidence
- ^ Lutz Kruschwitz: Investment calculation . 11th edition Oldenbourg Wissenschaftsverlag, Munich 2007, p. 197ff., ISBN 978-3-486-58306-9 .
- ↑ Wolfgang Lücke: Investment calculation based on expenses or costs? In: Zeitschrift für Handelswissenschaftliche Forschung / NF , vol. 7 (1955), pp. 310-324, ISSN 1619-6147 .
- ^ Gabriel Preinreich: Valuation and Amortization . In: The Accounting Review , Vol. 12 (1937), Issue 3, pp. 209-226, ISSN 0001-4826 .
- ^ Josef Kloock : Multi-period investment calculations on the basis of imputed and commercial income statements . In: Schmalenbach's magazine for business research , Vol. 34 (1981), pp. 873-890, ISSN 0341-2687 .