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: ${\displaystyle Z_{t}=EZ_{t}-AZ_{t}}$ ${\displaystyle G_{t}}$

${\displaystyle \sum _{t=0}^{T}G_{t}=\sum _{t=0}^{T}Z_{t}}$

• ${\displaystyle G_{t}}$ Period profit at the time ${\displaystyle t}$
• ${\displaystyle Z_{t}}$ Excess cash at the time ${\displaystyle t}$
• ${\displaystyle EZ_{t}}$ Deposits at the time ${\displaystyle t}$
• ${\displaystyle AZ_{t}}$ Payouts at the time ${\displaystyle t}$
• ${\displaystyle T}$ running time
• ${\displaystyle t}$ 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 : ${\displaystyle G_{t}}$ ${\displaystyle KB_{t-1}}$

${\displaystyle \sum _{s=0}^{t-1}G_{s}=KB_{t-1}+\sum _{s=0}^{t-1}Z_{s}}$

With ${\displaystyle KB_{0}=KB_{T}=0}$

• ${\displaystyle KB_{t}}$ Capital stock at the time ${\displaystyle t}$
• ${\displaystyle KB_{t-1}}$ Capital stock of the previous period
• ${\displaystyle s}$respective period that has expired up to the point in time . ${\displaystyle t-1}$

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: ${\displaystyle t}$

${\displaystyle KB_{t-1}=\sum _{s=0}^{t-1}G_{s}-\sum _{s=0}^{t-1}(EZ_{s}-AZ_{s})}$

• ${\displaystyle i}$ Cost of capital

If both of the above The following equation results for the net present value at the time of the decision:

${\displaystyle K_{0}=\sum _{t=0}^{T}{\frac {EZ_{t}-AZ_{t}}{(1+i)^{t}}}=\sum _{t=0}^{T}{\frac {G_{t}-i\cdot KB_{t-1}}{(1+i)^{t}}}=\sum _{t=0}^{T}{\frac {G_{t}^{*}}{(1+i)^{t}}}}$

• ${\displaystyle G_{t}^{*}}$ Residual profit at the time ${\displaystyle t}$

example

The interest rate assumed for the following examples is 10%.

Table 1: Investment calculation based on payments with an investment amount of 1000

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

Table 2: Calculation of net present value based on costs according to the Preinreich-Lücke theorem

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

1. ^ Lutz Kruschwitz: Investment calculation . 11th edition Oldenbourg Wissenschaftsverlag, Munich 2007, p. 197ff., ISBN 978-3-486-58306-9 .
2. ↑ Wolfgang Lücke: Investment calculation based on expenses or costs? In: Zeitschrift für Handelswissenschaftliche Forschung / NF , vol. 7 (1955), pp. 310-324, ISSN  1619-6147 .
3. ^ Gabriel Preinreich: Valuation and Amortization . In: The Accounting Review , Vol. 12 (1937), Issue 3, pp. 209-226, ISSN  0001-4826 .
4. ^ 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 .

Web links

• http://hans-markus.de/finance/86/hauptstudium_drei/luecke_theorem/