NPV

The NPV ( English value present , PV , present value or present value ) is a financial ratio of the dynamic investment calculation . The net present value is the sum of the future success of an investment discounted to the present. By discounting future successes to the present, the time value of money is taken into account: the sooner one can dispose of money, the more value it has. By discounting successes to a uniform point in time, payments that occur at different points in time can also be compared.

Delimitations

Success factors in determining capital values

With the help of capital values, the current values of property, plant and equipment (land, buildings, machines, etc.), financial assets (stocks, bonds, participations, etc.) or even of entire companies are regularly determined. Due to different calculations and delimitations of future successes , there are many variants for calculating capital values. The capitalization ( discounting ) of surplus payments ( cash flows ) is internationally common today when evaluating investments . Cash flows often reflect more realistically than profits the added value of the assets under consideration. The calculated / reported profits are regularly distorted by margins in the accounting (e.g. when calculating depreciation , measuring provisions, etc.). In valuation practice, it has become established to evaluate so-called operating profits or cash flows that are not overlaid by financial activities. These are performance indicators before interest payments (more precisely before financial results). The consideration of the operational level has the advantage that future successes can be conclusively predicted from an analysis of the market, company and competition. The financing transactions and interest payments associated with the financing of the assets are not included in the operational performance indicators. Due to the unpredictable developments in the financial market , future interest payments and financial flows are usually difficult to forecast.

The selection of the suitable discount rate corresponds to the definition of the performance variable. Operational results or operational cash flows create value for all capital providers (owners and lenders) and must therefore be discounted with the return demand of all capital providers . This compound rate of interest is also known as the Weighted Average Cost of Chapter ( WACC ). If, on the other hand, success variables are considered after interest payments (the claims of the lenders are then already settled), the future successes only need to be discounted with the return demand of the owners. The determination of the return requirements of the equity provider is usually based on capital market theoretical models, mostly on the Capital Asset Pricing Model (CAPM).

To assess whether an investment in the above-mentioned assets is worthwhile, the net present value must be compared with the value of the investment payment required at the beginning. The future cash flows and the current value of the investment payoff of an investment is the difference between the net present value / present value even as a net present value ( english value net present , NPV , net present value). The net present value of an investment then corresponds to the sum of the present values of all payments associated with the investment (deposits and withdrawals). The investment disbursements are often current market prices. In this respect, the net present value regularly compares the values with the prices of investments. An investment is always worthwhile if the net present value is greater than zero or the (capital) value of the investment exceeds its price.

Examples: The net present value of companies results from the discounted / capitalized free cash flows of the company. Buying the whole company is worthwhile if the capital value exceeds the current market price ( enterprise value ). The value of a bond is calculated from the net present value of all payments expected in the future (i.e. coupon payments and par value repayment). Investing in a bond makes sense if the net present value exceeds the currently required price on the stock exchange. The value of real estate is calculated from the discounted future income from the real estate. An investment in the property would be worthwhile if a purchase price can be negotiated that is below the net present value of the property.

The net present value method (also NPV method or net present value method or NGW method ) allows companies to assess expansion investments and determine the optimal replacement time .

calculation

Calculation of (net) capital values

The net present value of an investment is formally calculated as follows:

{\ displaystyle C_ {0} (i) = - I + \ sum _ {t = 1} ^ {T} {\ frac {Z_ {t}} {\ left (1 + i \ right) ^ {t}}} + L \ cdot \ left (1 + i \ right) ^ {- T} = \ sum _ {t = 0} ^ {T} \ left (1 + i \ right) ^ {- t} \ cdot Z'_ {t}}

${\ displaystyle C_ {0}}$ : Net present value based on the point in time ${\ displaystyle t = 0}$
${\ displaystyle i}$ : Discount rate
${\ displaystyle Z_ {t}}$ : Cash flow in period , where (Incoming payments - Outgoing payments in period ) represents, or in general stands for a payment vector. ${\ displaystyle t}$ ${\ displaystyle Z_ {t} = E_ {t} -A_ {t}}$ ${\ displaystyle t}$ ${\ displaystyle Z '_ {t}}$
${\ displaystyle I}$ : Investment payment at the time (can also be understood as) ${\ displaystyle t = 0}$ ${\ displaystyle Z_ {0} = - I}$
${\ displaystyle L}$ : Liquidation proceeds / residual proceeds at the time (it applies ) ${\ displaystyle t = T}$ ${\ displaystyle Z '_ {T} = Z_ {T} + L}$
${\ displaystyle T}$ : Duration / observation period (in periods)

Note: In the case of liquidation proceeds, it is not the book value (depreciation) but the expected sales proceeds that are used for the calculation.

This calculation with a uniform discount rate assumes that the debit and credit interest for all future payments in and out is identical. This can only be justified under the assumption of a perfect capital market . In addition, the uniform discount rate assumes that the interest rate is the same in all future periods and for all terms. This is only given with a flat interest structure . ${\ displaystyle i}$ ${\ displaystyle i}$ ${\ displaystyle t = 1, \ ldots, T}$

With different interest factors in the different periods, the net present value is calculated as follows:

{\ displaystyle C_ {0} = - I + {\ frac {Z_ {1}} {q_ {1}}} + {\ frac {Z_ {2}} {q_ {1} \ cdot q_ {2}}} + {\ frac {Z_ {3}} {q_ {1} \ cdot q_ {2} \ cdot q_ {3}}} + \ ldots + {\ frac {Z_ {T}} {q_ {1} \ cdot q_ { 2} \ cdot q_ {3} \ cdot \ ldots \ cdot q_ {n}}} = - I + \ sum \ limits _ {t = 1} ^ {T} \ cdot {\ frac {Z_ {t}} {\ prod \ limits _ {t = 1} ^ {n} q_ {t}}}}

${\ displaystyle Z_ {t}}$ : Cash flow in period ${\ displaystyle t}$
${\ displaystyle q_ {t}}$ : Interest factor of the period with ${\ displaystyle t}$ ${\ displaystyle q_ {t} = 1 + i_ {t}}$

Simplification for the same payments over a limited period T

If the same payments are always made during the period of use, the net present value can also be easily determined using the pension present value formula :

{\ displaystyle C_ {0} = - I + Z \ cdot {\ frac {\ left (1 + i \ right) ^ {T} -1} {\ left (1 + i \ right) ^ {T} \ cdot i}} + L \ cdot \ left (1 + i \ right) ^ {- T}}

${\ displaystyle Z}$ : Constant payment in every period

Constant payments are often found with fixed-rate debt . The calculation of a capital value with the help of the pension present value formula is therefore particularly taken into account when evaluating debt capital. This also includes listed bonds .

Simplification for the same payments over an infinite term T → ∞

In the case of an infinite term, the net present value factor converges against or the capital value converges against a perpetuity . In this case, the net present value can also be calculated simply by: ${\ displaystyle 1 / i}$

{\ displaystyle C_ {0} = - I + {\ frac {Z} {i}}}

Please note that there can be no liquidation proceeds for an infinite term. An infinite term is regularly assumed when valuing companies and stocks. The dividend discount model for stock valuation is based on the application of the formula for perpetuity.

Simplification for evenly growing payments over an infinite term T → ∞

The assumption of constant payments proves to be unrealistic in many applications. Many variables in the economy, including the cash flows of companies, grow over time. Assuming constant growth (annual growth rate ) of the cash flows, the net present value can also be determined as follows: ${\ displaystyle g}$

{\ displaystyle C_ {0} = - I + {\ frac {Z_ {0} (1 + g)} {ig}}}

An infinite term with constantly growing cash flows is regularly assumed when valuing companies and stocks. The discounted cash flow model for company valuation is based on the application of this present value formula.

Considerations in constant time

In abstract, economic considerations, capital values are often also analyzed over time. The discrete-time formula of net present value

{\ displaystyle \ mathrm {C_ {0}} (i) = \ sum _ {t = 0} ^ {N} {\ frac {Z_ {t}} {(1 + i) ^ {t}}}}

can also be represented in a continuous form

{\ displaystyle \ mathrm {C_ {0}} (i) = \ int _ {t = 0} ^ {\ infty} (1 + i) ^ {- t} \ cdot z (t) \, \ mathrm {d } t}

.

Here is the flow rate of deposits and withdrawals in money per time, with cash flow when the investment is finished. ${\ displaystyle z (t)}$ ${\ displaystyle z (t) = 0}$

The net present value can also be viewed as the Laplace or z- transformed of the cash flow with the integral operator including the complex number (corresponds roughly to the interest rate or more precisely ) from the real number space. This results in known simplifications from cybernetics , system theory and control engineering . Imaginary parts of the complex number s describe the tendency to oscillate (compare with the pig cycle and the phase shift of price and supply as well as the explanatory cobweb theorem ), real the compound interest effect (compare with damping ). ${\ displaystyle s}$ ${\ displaystyle i}$ ${\ displaystyle s = \ ln (1 + i)}$

{\ displaystyle F (s) = {\ mathcal {L}} \ left \ {f (t) \ right \} = \ int _ {0} ^ {\ infty} e ^ {- st} f (t) \ , \ mathrm {d} t}

.

interpretation

An investment is absolutely beneficial if its net present value is greater than zero. In this case, the net present value of the investment exceeds the value of the investment payoff.

Net capital value = 0: The investor receives his invested capital back and interest on the outstanding amounts in the amount of the discount rate. The investment has no advantage over investing in the capital market at the same (risk-equivalent) interest rate. This is where the internal rate of return is located .

Net capital value> 0: The investor receives his invested capital back and interest on the outstanding amounts that exceed the discount rate.

Net capital value <0: The investment cannot guarantee a return on the capital employed at the calculated interest rate.

If several mutually exclusive investment alternatives are compared, the one with the largest net present value is the relatively most advantageous. Furthermore, it is possible to add up the capital values of various non-mutually exclusive investments with different calculation interest rates, since this is an additive method.

criticism

advantages

It is a computationally simple procedure that enables easy interpretation, since the capital value is expressed in monetary units (absolute result). It is still possible to carry out calculations that conform to the interest structure, as the discount rate can be adjusted in every period. In addition, the net present value method has the advantages of dynamic calculation (observing the timing of payments) compared to static calculation.

disadvantage

Problems with the use of the net present value method, like all other discounted cash flow methods, are the assumption of the perfect capital market, in particular the assumption of the equality of debit and credit interest rates, the discount rate based on subjective assumptions and the amount of future cash flows. Due to the simple calculation and interpretability, there is a risk of using the results without comment. It is therefore important that the assumptions made, above all about the level of the risk premium, the discount rate and the future cash flows, are named and justified.

If the possibility of making an investment several times is overlooked, this can lead to wrong decisions. The annuity method provides a remedy here . It is assumed that the income will be reinvested at the capital market rate.

literature

Louis Perridon , Manfred Steiner : Finance of the company. 14th revised and expanded edition. Franz Vahlen Verlag, Munich 2007, ISBN 978-3-8006-3359-3 , 732 S. ( Vahlens Handbooks of Economics and Social Sciences ).
Jean-Paul Thommen : Management-oriented business administration . 7th revised and expanded edition. Versus Verlag, Zurich 2004, ISBN 3-03909-000-3 , 960 pp.
Rudolf Volkart: Corporate Finance. Fundamentals of finance and investment. 3rd revised and expanded edition. Versus Verlag, Zurich 2007, ISBN 978-3-03909-091-4 , 1343 pp.

Web links

A simulation for calculating capital values can be found here: [1]

Individual evidence

^ Lutz Kruschwitz, Sven Husmann: Financing and Investment. 2012, Oldenbourg Wissenschaftsverlag, ISBN 978-3486702590 , p. 6

↑ Horst Hanusch: Benefit - Costs - Analysis ( Memento of the original from May 23, 2005 in the Internet Archive ) Info: The @1@ 2 archive link was automatically inserted and not yet checked. Please check the original and archive link according to the instructions and then remove this notice. . 2nd edition. Faculty of Economics at the University of Augsburg, 2004. pp. 23, 27 ff. (PDF file; 141 kB)

↑ Robert W. Grubbström: On the Application of the Laplace Transform to Certain Economic problem . In: Management Science . 13, 1967, pp. 558-567. doi : 10.1287 / mnsc.13.7.558 .

↑ Steven Buser: LaPlace Transforms as Present Value Rules: A Note (PDF; 156 kB), The Journal of Finance, Vol. 41, No. 1, March, 1986, pp. 243-247.