Comprehensive Guide to the Net Present Value Method (Kapitalwertmethode)

Financial Mathematical Foundations of the Net Present Value Method

On the capital market, money that is invested earns interest for the investor. For instance, if $1.000,00\,\text{€}$ is invested today, this capital will earn interest over the course of one year, after which the investor receives the original capital plus the accumulated interest. The interest per year can be calculated using basic percentage math. When determining the current value of a payment occurring one year in the future, that payment must be "cleared" of interest. This process identifies the Present Value (Barwert). To find the Present Value, the future Euro amount is divided by $(1+p)^n$. In this context, $p$ represents the internal discount rate (Kalkulationszinsfuß), and $n$ represents the number of years or periods, where $t = 0, 1, \dots, n$.

Because values further in the future are subject to more compounding interest, they possess a lower Present Value than values closer to the present day. This occurs because the more distant values must be adjusted or "cleared" of a higher total interest amount. The components of this calculation include the time ($t$), the discount rate ($p$), and the number of utilization periods ($n$). The Discount Factor (Abzinsungsfaktor) is defined as \n\frac{1}{(1 + \frac{p}{100})^n}. The Present Value (BW) is calculated by multiplying the future Time Value ($ZW_n$) by this factor: $BW = ZW_n \times \frac{1}{(1 + \frac{p}{100})^n}$.

Methodology and Decision Rules of the Net Present Value (NPV) Method

The Net Present Value Method (Kapitalwertmethode) is utilized to decide whether a specific investment should be undertaken. This method considers the entire useful life of the object. The decision rule is as follows: if the Net Present Value (KW) is greater than or equal to 0 ($KW \geq 0$), the investment should be performed. If the Net Present Value is less than 0 ($KW < 0$), the investment should not be performed. When $KW = 0$, the project's internal interest rate matches the market interest rate.

Financial inputs for this method include Cash Inflows (Einzahlungen), calculated as Price $\times$ Quantity ($p \times x$), and Cash Outflows (Auszahlungen), calculated as the sum of other fixed costs and variable unit costs multiplied by quantity ($\text{sonst. } K_{fix} + (k_{var} \times x)$). The resulting Surpluses (Überschüsse) are found by subtracting Outflows from Inflows: $(p \times x) - (\text{sonst. } K_{fix} + k_{var} \times x)$. The Net Present Value itself is the sum of these discounted surpluses minus the initial acquisition costs (Anschaffungskosten).

Several specific rules apply to these calculations. First, Liquidationserlös (liquidation proceeds) is added to the inflows in the final year of the useful life if the machine is sold at that time. Second, while inflows and outflows refer to cash receipts and expenditures (Einnahmen and Ausgaben), cost categories such as imputed interest (kalkulatorische Zinsen) and imputed depreciation (kalkulatorische Abschreibungen) are not included. These are excluded because they are not cash-effective (ausgabewirksam); they do not represent an actual physical outflow of money. Furthermore, the outflow in Year 0 (acquisition cost) is never discounted because it is already expressed in terms of today's value. Finally, to maximize interest earnings, companies aim to receive inflows as early as possible and postpone outflows as late as possible.

Criticisms of the Net Present Value Method

Despite its mathematical precision, the method faces several critical limitations. Primarily, the projected cash inflows and outflows are mere estimates, which may not reflect future reality. Additionally, the company has the autonomy to choose the discount rate ($p$) itself, which allows them to manipulate how advantageous an investment appears. There is also a significant problem regarding the exact allocation of earnings to a specific investment project, as returns are often the result of multiple interacting assets.

Case Study: Investment Without Liquidation Proceeds (FELGE AG)

FELGE AG plans to purchase a new machine next year with the following data: Acquisition costs of $720.000,00\,\text{€}$, a maximum capacity of $5.000\,\text{pieces/year}$, and other fixed costs totaling $72.000,00\,\text{€/year}$. Of these fixed costs, only $75\%$ ($54.000,00\,\text{€}$) are cash-effective. Imputed interest is stated as $21.600,00\,\text{€}$ but is ignored in the calculation as it is not cash-effective. The variable unit costs are $130,00\,\text{€}$, the unit revenue is $215,00\,\text{€}$, and capacity utilization is projected to be constant at $80\%$ ($4.000\,\text{units}$). The machine remains in the company at the end of its 3-year term (no liquidation proceeds). The discount rate is $6\%$.

The annual Cash Inflows are $215,00 \times 4.000 = 860.000,00\,\text{€}$. The annual Cash Outflows are the cash-effective fixed costs plus variable costs: $54.000,00 + (130,00 \times 4.000) = 574.000,00\,\text{€}$. This results in a yearly Surplus of $286.000,00\,\text{€}$. To calculate the Net Present Value, each surplus is discounted:

• Year 1 Present Value: $286.000,00 \times \frac{1}{(1 + \frac{6}{100})^1} = 269.811,32\,\text{€}$
• Year 2 Present Value: $286.000,00 \times \frac{1}{(1 + \frac{6}{100})^2} = 254.538,98\,\text{€}$
• Year 3 Present Value: $286.000,00 \times \frac{1}{(1 + \frac{6}{100})^3} = 240.131,11\,\text{€}$

The Net Present Value ($KW$) is $-720.000,00 + 269.811,32 + 254.538,98 + 240.131,11 = 44.481,41\,\text{€}$. Because the Net Present Value is positive, the investment in the machine should be carried out.

Case Study: Required Liquidation Proceeds (SOUNDON AG)

SOUNDON AG produces speaker boxes and is considering a system with acquisition costs of $1.020.000,00\,\text{€}$. Full capacity is $60\,\text{units/month}$ ($720\,\text{units/year}$). Variable costs are $465,00\,\text{€/unit}$, and other fixed costs are $95.000,00\,\text{€}$. The selling price is $815,00\,\text{€/unit}$. The plant must be sold after 5 years, and the discount rate is $5\%$. The company requires a specific Net Present Value of $86.749,60\,\text{€}$. The goal is to determine what the liquidation proceeds must be to achieve this NPV.

Annual Inflows total $815,00 \times 720 = 586.800,00\,\text{€}$. Annual Outflows total $95.000,00 + (465,00 \times 720) = 429.800,00\,\text{€}$, resulting in a recurring annual surplus of $157.000,00\,\text{€}$. The present values for the first four years are calculated as follows:

• Year 1: $149.523,81\,\text{€}$
• Year 2: $142.403,63\,\text{€}$
• Year 3: $135.622,50\,\text{€}$
• Year 4: $129.164,29\,\text{€}$

The total target for discounted values from Year 5 is found by taking the required NPV ($86.749,60\,\text{€}$), adding back the initial investment ($1.020.000,00\,\text{€}$), and subtracting the sum of the present values for Years 1 through 4 ($556.714,23\,\text{€}$). This leaves a required Year 5 Present Value of $550.035,37\,\text{€}$. To find the actual Year 5 Surplus, this present value is compounded: $550.035,37 \times (1 + \frac{5}{100})^5 = 702.000,00\,\text{€}$. Finally, subtracting the standard operating surplus ($157.000,00\,\text{€}$) from this total Year 5 surplus yields a required Liquidation Proceeds of $545.000,00\,\text{€}$. Through backward calculation using the formula: $KW = -AK + BW_1 + BW_2 + BW_3 + BW_4 + \frac{Überschuss_5 + L}{(1+p)^5}$, the necessary liquidation value is confirmed.