# 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%.

year Investment payout Deposits Payouts Cash flow Discount factors 0 1 2 3 4th 5 -1000 1000 1000 1000 1000 1000 600 600 600 600 600 -1000 400 400 400 400 400 0.909 0.826 0.751 0.683 0.621 364 331 301 273 248 516
 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, .

## 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, .
3. ^ Gabriel Preinreich: Valuation and Amortization . In: The Accounting Review , Vol. 12 (1937), Issue 3, pp. 209-226, .
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, .