Preinreich Lücke Theorem

The Preinreich Lücke Theorem tells us that the present value of the residual incomes is equal to the present value of the corresponding cash flows. This might be important, because it is the link of yearly reported figures like economic value added (EVA) to the value of a complete future cash flow stream. In this post I provide the proof of the theorem. Furthermore I want to discuss the premises and consequences critically.

Proof of Preinreich Lücke Theorem

The residual income is income minus capital costs. Capital means in this view all expenditures that are amortized and do not affect income directly. The residual income in period t is defined as:

    \[I_t^{res}:=f_t+c_t-c_{t-1}-c_{t-1}i_t\]

f_t is the cash flow in period t, c_t the fixed capital in period t and i_t the discount rate in period t. c_t - c_{t-1} is just the depreciation in period t. The present value of the residual incomes is equal to the present value of the corresponding cash flows, if the difference of them is zero. The difference is:

    \[\sum_{t=0}^{n}(f_t-I_t^{res})\rho_t=\sum_{t=0}^{n}(c_{t-1}(1+i_t)-c_t)\rho_t\]

\rho_t is the discount factor in period t and decreases in period t by the factor 1+i_t. That means \rho_t = \rho_{t-1} / (1+i_t). With that we obtain:

    \[\sum_{t=0}^{n}(f_t-I_t^{res})\rho_t=\sum_{t=0}^{n}(c_{t-1}\rho_{t-1} - c_t \rho_t)\]

Within that sum all terms in the middle cancel out. Only the first and the last term remain. Assuming that there is no fixed capital before t=0 we can set c_{-1} = 0. And if all fixed asset is depreciated in the considered n periods, we can set c_n=0. With these two premises we realize that all terms of the sum become zero.

    \[\sum_{t=0}^{n}(f_t-I_t^{res}) \rho_t = c_{-1} \rho_{-1} - c_n \rho_n=0\]

That means that there is no difference of discounting cash flows or discounting residual incomes. This is exactly what we wanted to proof.

Discussion

At first I want to point out that the Preinreich Lücke Theorem requires the same discount factor in one period it for all cash flows. They can differ from one period to another, but within the same period all cash flows and residual incomes are discounted with the same factor. In reality each cash flow can have its own risks (risk premiums) and its own financing structure. That means that each cash flow can require its own specific appropriate discount factors. But with different discount factors the Preinreich Lücke Theorem does not work any more.

Secondly, the fixed capital must be amortized completely in the considered n periods. If there is a residual book value in the last period n, the Preinreich Lücke Theorem is not valid any more.

As a third point I want to mention that you get residual values after having calculated the cash flows. The calculation with residual incomes is an additional calculation loop with no real benefit.

As a last point I want to mention that figures like EVA are used widely because they can be calculated in addition to an income statement easily. But this is not the same as it is done in the Preinreich Lücke Theorem. There we have a calculation of one investment / project in many periods and not only one.

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