Operating Leverage in CCFP

Joachim Kuczynski, 04 February 2023


The only correct way to discount cash flows and value an asset is the Component Cash Flow Procedure (CCFP). Hereby each cash flow is discounted with its appropriate risk-adjusted discount rate. All fixed cash flows, that have no market risk and are diversifiable for investors, have to be discounted without additional equity risk premiums. Fixed cash flows include fixed operative costs, R&D, investments and cash flows for fixed expenditures like customer and tooling expenses & payments. In contrast to that contribution cash flows depend on the market development. They must be discounted with rates that include equity risk premiums for the investors. Primarily, contribution cash flows consist of turnover, variable costs and working capital cash flows.

In most cases discount rates for contribution cash flow are based on bottom-up betas and not on historical data. These are averaged equity betas representing an average of similar companys in an industry segment in a certain country or market. But these bottom-up betas take into account all cash flows, contribution cash flows as well as fixed cash flows. Hence these bottom-up betas are not correct to discount contribution cash flows alone. But that is exactly what is required in the CCFP. The content of this post is the derivation of the contribution cash flow beta from the bottom-up industry segment beta.


Base for the derivation is the market balance sheet where the present value of the asset (project or investment), PV_{asset}, is the sum of present value of the contribution cash flows, PV_{contr}, and the present value of the fixed cash flows, PV_{fix}:


Now we derive in respect to the market portfolio return rate r_m and devide by PV_{asset}. The relative change of PV_{asset} in respect to r_m is just the beta of the asset, \beta_{asset}, by definition. Hence we get

    \[\frac{\frac{\partial }{\partial r_m}PV_{asset}}{PV_{asset}}=\beta_{asset}=\frac{\frac{\partial }{\partial r_m} PV_{contr} + \frac{\partial }{\partial r_m} PV_{fix}}{PV_{asset}}.\]

The fixed cash flows are independent of the market return rate. That means that the derivation \frac{\partial }{\partial r_m} PV_{fix}=0 and we get:

    \[{\frac{1}{\beta_{asset}}=\frac{PV_{asset}}{\frac{\partial }{\partial r_m} PV_{contr}}=\frac{PV_{contr}+PV_{fix}}{\frac{\partial }{\partial r_m} PV_{contr}}\]

The relative change of the conribution cash flow present value in respect to the market portfolio return rate is just the beta or the contribution cash flows. Taking that into account and rearranging the equation leads us to:

    \[\frac{1}{\beta_{asset}}=\frac{1}{\beta_{contr}}\left( 1+ \frac{PV_{fix}}{PV_{contr}} \right)\]

    \[\beta_{contr}=\beta_{asset}\left( 1+ \frac{PV_{fix}}{PV_{contr}} \right)\]

Note that the present value of the fixed cash flows is negative, PV_{fix} \le 0, and the present value of the contribution cash flows is positive, PV_{contr} \ge 0, in ordinary cases. That means that the expression in the bracket becomes \le 1. That means that the beta of the contribution cash flows is smaller than the asset’s beta, \beta_{contr} \le \beta_{asset}. Obviously that is true, because adding fixed costs increases the beta of the asset. this is known as operating leverage.

Application in CCFP

The bottom up beta from an industry segment includes fixed costs, it corresponds to \beta_{asset}. Applying the previous derivation you can calculate the beta of the contribution cash flows, \beta_{contr}. PV_{fix} and PV_{contr} are values from the market balance sheet of the averaged industry segment companies. If these values are more or less stable over time you can also take the corresponding figures from the averaged income statement of the industry segment. These data are available in several statistical sources. With \beta_{contr} you can calculate the equity risk premium of the contribution cash flow discount rate and hence the discount rate itself, for instance according to the capital asset pricing model (CAPM). With r_f as risk free rate and r_m as average return rate of the industry segment, the equity return rate is:

    \[r_E=r_f+\beta _{contr}\left( r_m - r_f \right)\]

The return rate for contribution cash flows r_{contr} considering debt D and equity E financing including tax shield effect with constant debt to equity ratio and marginal tax rate t is:

    \[r_{contr}=\frac{D}{E+D} \left( 1-t \right) r_D + \frac{E}{E+D} r_E\]

With that procedure we get the correct discount rate for the contribution cash flows in the CCFP approach.

If there are several contribution cash flows from different industry segments with specific risks, we can do that procedure with each kind of contribution cash flow. In this way we get the appropriate return rate for each kind of contribution cash flow.

Profitability Index Annuity

Joachim Kuczynski, 29 September 2022

Project ranking is an important issue in capital rationing, when the company has limited financial resources. A company can rank its projects with various measures: NPV, eNPV, PI, PIA, IRR, MIRR, Baldwin rate of return and many others. The profitability index annuity is a further development of the profitability index to consider the temporal cash flow distribution in the decision making process.

Profitability Index

The profitability index (PI) of an investment project is the present value of all cash flows without investment (let us name them contribution cash flows), PV_C, divided by the present value of all investment cash flows, PV_I. Hereby it is important how we define the term investment. You can have a look at my proposal in this post. Hence we get for the profitibility index:


The PI shows how much value is created per investment. All projects with a PI>1 are creating additional value to the investors, all projects with a PI<1 are destroying value.

Sometimes the profitability index is defined net present value of the project devided by the present value of all investment cash flows. Let us name this definition of the profitability index PI*:


No matter which definition we take, the ranking of projects remains the same. The difference is only an “offset” of 1.

Profitability Index Annuity

Most managers prefer projects in which the returns of the investment are in the beginning. They prefer early cash flow return. The profitability index cannot provide any information concerning the timing of the cash flows. The profitability index annuity (PIA) tries to solve that issue. It is defined as profitability index divided by the annuity factor A_{n,r}:


An annuity is a sequence of equal cash flows paid each period for a specified number of periods n. The sum of all the discount factors equals the annuity factor. With n as the project lifetime and r as annual discount rate, the annuity factor is defined as:

    \[A_{n,r}=\frac{1}{r}\left( 1-\frac{1}{\left( 1+r \right)^n} \right)\]

The previous equation is only valid, if all cash flows are discounted with the same discount rate in all periods. If you have different discount rates in the considered periods, the formula not valid any more but can be adapted easily. If you use the Component Cash Flow Procedure, there might be no solution or more than one possible solution for the PIA value. So take care when applying the PIA ranking!


The annuity factor considers how much the investment project is influenced by the discounting effect. An investor that prefers early cash flow returns wants to have small discounting effects. The smaller the discounting effect the smaller the annuity factor and the higher the profitability index annuity. Looking at two projects with the same PI the investor prefers the project with the higher PIA.

Project rankings based on the profitability index annuity (PIA) have some desirable properties. Projects with shorter lives tend to have higher ranking PIAs, and short life implies rapid cash generation. Choosing projects ranked by their PIA values can help managers to invest in projects combining three desirable characteristics: High PV, low capital requirement and rapid cash generation.

In my point of view PIA is an additional useful figure to evaluate investments and projects. But we should not use it as the only measure. It relies on the term investment that we have to consider carefully. Always think about what is the true financial limitation in the company and then refer to that.

Investment Term

In this post I would like to make some comments concerning the term INVESTMENT. It is a central notion in investment and project valuation. Many key figures are based on this term. Hence it is important that we really know what it is.

An asset valuation is always done from the fund / capital providers’ or investors’ sight. From that point of view an investment is the amount of cash that the investor has to provide to run the investment project. The cash flow between investor and investment project is the basis of the valuation. Thereby it is not important when the investment cash flow takes place. An investment cash flow does not has to be an initial cash flow in the beginning of the project. An investor can always shift the cash flows by using the capital markets. He can borrow money from a bank and pays it back later with additional interests.

From the investor’s point of view it is not important how the cash is handled in the income statement and balance sheet. The investor is only interested in cash flows concerning him / her. The notion investment does not depend on any classification in income statement and balance sheet. Further it is irrevelant whether the investment cash flow is depreciated / amortized or not.

The investment of a project are all fixed cash flows. That are all cash flows that have no dependency on the market devopment. These are for example cash flows for machines, land, buildings, product development, fixed costs for production, patents, fixed customer payments for various items and further more. It is not important whether these cash flows are incoming or outgoing. In a discounted cash flow analysis the investment cash flows must be discounted with the riskless rate that includes no risk premium. It is incorrect to discount investment cash flows with any discount rate including market risk premiums, for example the WACC. For that reason all figures valuing a project with a single rate (IRR, MIRR, …) are doubtful. They cannot value projects with cash flows having different risk and with it discount rates.

All cash flows that depend on the market development are not investment cash flows. That mainly includes turnover and variable costs. They must be discounted with the appropriate risk adjusted discount rate, because the investor requires a risk premium for taking that non diversifiable risks. Each risky cash flow has its specific own risk and requires an appropriate risk adjusted discount rate.

Preinreich Lücke Theorem

Joachim Kuczynski, 31 August 2022

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:


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:


\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.


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.

Internal Rate of Return (IRR)

The internal rate of return (IRR) is a widespread used figure to evaluate investment projects. But it is a very dangerous figure that can lead to wrong decisions easily. The figure is only valid if the investment project fulfills special conditions.

IRR Definition

I want to give you a clear derivation of the IRR definition to point out some very critical issues. The internal rate of return is defined to be the zero of a polynomial that calculates the net present value (NPV) of an investment project. The NPV is the sum of all discounted incremental cash flows to the firm after taxes, c_i. The discount factors \gamma_i can be different for each cach flow c_i.

    \[NPV=\sum_{i}^{}\gamma _i c_i\]

Let us assume that we can sum up the products in each period t, because all cash flows one period t have the same discount factor \gamma _t (uniformity). Let C_t be the sum of the cash flows in period t. Hence we can simplify the NPV calculation by summing over all time periods instead of over all cash flows:

    \[NPV=\sum_{t}^{}\gamma _t C_t\]

Further we assume that the discount factor \gamma_t has polynomial character in t (flatness). In this case we can rewrite the NPV as a polynomial:

    \[NPV=\sum_{t}^{}C_t\gamma ^t\]

Let i be the annual interest rate to discount cash flows. Setting \gamma=\left( 1+i \right)^{-1} we get the well-known formula:

    \[NPV=\sum_{t}^{}C_t\left( 1+i \right) ^{-t}\]

The internal rate of return i^{IRR} is now defined to be the values of i for which the NPV is zero:

    \[NPV=\sum_{t}^{}C_t\left( 1+i^{IRR} \right) ^{-t}=0\]

I want to point out that we require two restrictive premises to get an expression for the IRR, flatness and uniformity.

Flat and uniform discount rates

To give the NPV the form of a polynomial we required two important assumptions. At first the investment project must have a uniform discount rate. That means that the discount rate has to be the same for all cash flows in one period. All cash flows are assumed to have the same equity risk premium and the same risk free rate. In almost all investment projects this assumption does not hold. Secondly, the discount rate structure must be flat. That means that you have the same discount rate for each period. The risk free rate and the equity risk premiums are assumed to be constant over time. In most investment projects this is also not true.

We can state the the premises uniformity and flatness to let the NPV get a polynomial cannot be applied in almost all investment projects. Especially in complex projects with many different cash flows, currencies and risks these assumptions are not valid.


The IRR are the zero points of a polynomial. A polynomial of order n can have n different zeros. That means that you get n different IRR in general, the solution for possible IRR can be ambiguous. If you get more than one IRR, which one is the right one? Or you can even get no solution. But even when you get no solution for the IRR, there exists a contribution of the project to the company’s value. NPV (or eNPV) can evaluate all kinds of value contributions to the firm and must be preferred.

IRR decision rule

According to the decision rule of the IRR method an investment should be realized, if the IRR is bigger than the WACC. That is only true, if the the NPV has a negative derivative at the IRR. A zero point of a n-order polynomial can have a positive or negative first derivation at zero points. Accordingly you cannot say that the NPV decreases with increasing discount rate in general. That means that the IRR decision rule does not hold when the first derivative is positive.

Mutually exclusive investment projects

Firms often have to choose between several alternative ways of doing the same job or using the same facility. In other words, they need to choose between mutually exclusive projects. The IRR decision rule can be misleading in that case as well. An alternative can have a bigger IRR but a smaller NPV than another alternative at the same time. The investors of the project (who are the relevant deciders) are interested in a maximum increase of value of the firm. That is represented by the NPV. Hence comparing alternatives by their IRR can lead to false decisions.


My conclusion is the the IRR method with its decision rule should only be applied at very simple projects with one investment in the beginning and uniform / flat discount rates. It should not be applied as decision figure in a standardized investment valuation concept of a company. In all complex projects the NPV (or eNPV) is the much better concept. Additionally the NPV / eNPV gives the investors the information they demand, namely the contribution of the investment project to the value of the firm and hence the value of their shares.

Risk-Free Rate

Importance of risk-free rates

The risk-free rate of return r_f is the theoretical return rate of an investment with no risk. The risk-free rate represents the interest an investor would expect from an absolutely risk-free investment over a specified period of time. It plays a central role in financial valuation and is the planning base for return rates of risky assets.

Prospective cost of debt (return rate requirement) can be estimated by adding default spreads to the risk-free rate. This can be done by synthetic rating of the company with its interest coverage ratio and adding a default spread according to the synthetic rating of the company. Additionally, in an emerging market the country of the occuring cash flows can generate a further country default spread added to the risk free rate.

Equity return rates r_e are calculated by adding risk premiums to the risk-free rate. Considering the Capital Asset Pricing Model (CAPM) the expected value of the equity return rate is:

    \[E \left( r_e \right)= r_f+\beta \left( E\left( r_m \right) - r_f \right)\]

\beta is the sensitity of the risky cash flow return rate to the efficient market protfolio return rate r_m.

Risk-free rates are also very important in real options analysis. The time values of real options are weighted by risk-neutral probabilities and discounted with the risk-free rate.

Because of all these reasons it is very important to understand the risk-free rate in detail. Most books neglect that issue. In my point of view the best analysis is done by Aswath Damodaran in his book about investment valuation (2011).

Risk-free rate criteria

There are two criteria that a return rate of an asset can be considered as risk-free: 1) First the asset does not have any default risk. This excludes private entities, since even the largest and safest ones have some measure of default risk. The only securities that have a chance of being risk-free are government securities, not because governments are better than corporations, but because they usually control the printing of currency. 2) Secondly the asset must not have any reinvestment risk. The return rate has to be ensured fo the whole period of time.

Discounting Period

Government zero-coupon bonds and securities have different return rates for different time horizons. Usually the return rate increases with longer time horizons. Well-behaved term structures would include an upward-sloping yield curve, where long-term rates are at most 2 to 3 percent higher than short-term rates. For each maturity the investor gets a different guaranteed return on the investment. That means that each maturity has a specific discount rate.

Most DCF valuations use only one risk-free rate corresponding to a specific maturity. In most cases they take long term Treasury bonds, because cash flows mainly occur years away from present time (t=0) of the valuation. But be aware that this is an approximation and can be false in some cases.


The risk-free rate used to come up with expected returns should be measured conisistently with how the cash flows are measured. Thus, if cash flows are estimated in nominal U.S. dollar terms, the risk-free rate will be the U.S. Treasury bond rate. This also implies that it is not where a firm is domiciled that determines the choice of a risk-free rate, but the currency in which the cash flows of the firm are estimated. If we assume purchasing power parity, then differences in interest rates reflect differences in expected inflation. Both the cash flows and the discount rate are affected by expected inflation; thus, a low discount rate arising from a low risk-free rate will be exactly offset by a decline in expected nominal growth rates for cash flows, and the value will remain unchanged. If the difference in interest rates across two currencies does not adequately reflect the difference in expected inflation in these currencies, the values obtained using the different currencies can be different.

Default Spreads

The interest rates on bonds are determined by the default risk that investors perceive in the issuer of the bonds. This default risk is often measured with a bond rating, and the interest rate that corresponds to the rating is estimated by adding a default spread to the riskless rate. An Euro government bond from Greek or Italian government has a higher default spread than a bond emitted by German government. You always have to subtract the default spread from the effective bond interest rate to get the risk-less (or better default-free) rate.


In most books concerning capital budgeting and DCF analysis the risk-free rate is the interest rate of a long term Treasury government bond. But also maturity, currency and default risk of the bond emitter have to be considered to get the right interest rate of a default-free asset. The investment analyst has to be aware of that and also has to know the inaccuracy when making approximations.

Investment Return Requirement

Joachim Kuczynski, 02 July 2022

In this post I want to give a derivation of the return requirement of an additional investment opportunity for an investor having an existing investment / security portfolio. In my point of view this is the key point of portfolio theory to understand the discounting of cash flows in a DCF analysis.

Let us assume an investor which owns a portfolio of investments or securities with relative shares x_i having annual return rates R_i and standard deviations of the annual return rates \sigma\left( R_i \right). The variance of the portfolio return rate is given by:

    \[var\left( R_P \right)=\sum_{i}^{}x_i\text{cov}\left( R_i, R_P \right) \text{, or}\]

    \[ var\left( R_P \right) =\sum_{i}^{}x_i\sigma\left( R_i \right)\sigma\left( R_P \right)\text{corr}\left( R_i, R_P \right)\]

Dividing both sides by standard deviation \sigma\left( R_P \right) gives the standard deviation \sigma\left( R_P \right):

    \[\sigma\left( R_P \right)=\sum_{i}^{}x_i\sigma\left( R_i \right)\text{corr}\left( R_i, R_P \right)\]

That means that the incremental risk contribution of each investment to the risk of the portfolio ist \sigma\left( R_i \right)\text{corr}\left( R_i, R_P \right).

Instead of including a new investment into the portfolio the investor can also increase the return of the protfolio by increasing the risk of the portfolio. This reward-to-volatility ratio of the tangential portfolio is given by the Sharpe Ratio:

    \[\frac{E\left( R_P\right)-r_f}{\sigma \left( R_P \right)}\]

E\left( R_P\right) is the expected value of R_P and r_f is the risk-free or default-free rate. The investor wants to invest in the new opportunity, if the additional return rate of this investment is higher than an investment in the existing portfolio with the same risk changes. Hence we obtain the requirement to invest in the new investment opportunity:

    \[\text{E}\left( R_i \right)-r_f > \sigma\left( R_i \right)\text{corr}\left( R_i,R_P \right)\frac{E\left( R_P\right)-r_f}{\sigma\left( R_P \right)}\]

With that we can define the sensitivity \beta_i^P of the new investment to the existing portfolio:

    \[\beta_i^P=\frac{\sigma\left( R_i \right)\text{corr}\left( R_i,R_P \right)}{\sigma\left( R_P \right)}\]

Substituting with \beta_i^P the requirement for the new investment becomes the well-known equation:

    \[\text{E}\left( R_i \right) > r_f+\beta_i^P\left( E\left( R_P\right)-r_f \right)\]

With that we can define a minimal annual return rate of the investment r_i:

    \[ r_i = r_f+\beta_i^P\left( E\left( R_P\right)-r_f \right) \]

This is the right (leveraged) discount rate for cash flows financed by equity. It is the minimum rate at which an investor would decide to allocate the new investment opportunity in his portfolio, because the expected risk-adjusted return rate is higher that the risk-adjusted rate of the existing portfolio. It is easy to see that each cash flow has to be discounted with its specific risk-adjusted rate, when they have different risks. Because of the additivity of net present values the investor can discount each cash flow seperately and sum up the NPV of all cash flows. This is called component cash flow procedure, see this post.

If cash flows are financed by debt and equity, the discount rate is the weighted average of debt and equity return rate requirements (WACC).

Quite often the portfolio of a so called marginal investor is not known in detail. One possibility is to assume that his portfolio consists of all available securities in the market with its specific weighted shares. With the assumptions of the Capital Asset Pricing Model (CAPM) the efficient tangential portfolio is the market protfolio and the expected portfolio return rate is the expected return rate of the market. In most cases the S&P500 is taken as reference portfolio.

NPV Sign and Time Scale

In this post I want to discuss the sign of NPV when time scale is shifted by an amount of \Delta t. The sign of NPV indicates whether an asset generates value to the fund providers (debt and equity) or not. If only the amount, but not the sign of NPV changes by a time shift, the decision to allocate the project or not does not change. That means that the decision itself does not depend on the time scale. In many books I have read that argument. But is this really true?

Project Cash Flow Procedure

The project cash flow procedure (PCFP) takes the same annual discount rate r for all cash flows C_i. By substituting \alpha = ln (1+r) we can write e^{- \alpha t} instead of (1+r)^{-t}. The NPV of the project at the “present” time t=0 without time shift, let´s name it NPV_0, is:

    \[NPV_0=\sum_{i}^{}E(C_i) e^{-\alpha t_i}\]

E(C_i) is the expected value of the i-th cash flow component. When we shift time by \Delta t, we get a new NPV, let`s call it NPV_ {\Delta t}:

    \[NPV_ {\Delta t} =\sum_{i}^{}E(C_i) e^{-\alpha (t_i+ {\Delta t} )}\]

    \[ NPV_ {\Delta t} = e^{ -\alpha  \Delta t} \sum_{i}^{}E(C_i) e^{-\alpha (t_i )}= e^{ -\alpha  \Delta t} NPV_0\]

Because e^{ -\alpha  \Delta t}>0, the sign of NPV_0 and NPV_ {\Delta t} is the same for all \Delta t. That means the decision, when based on the NPV sign, remains the same: Invest in case of positive NPV and do not invest in case of negative NPV. In the PCFP a time shift does not affect the decision.

Component Cash Flow Procedure

The component cash flow procedure (CCFP) takes the specific appropriate risk adjusted discount rate for each cash flow. That means that you have a specific discount rate r_i, or \alpha _ i = ln (1+r_i) respectively, for cash flow C_i. The NPV of the project without time shift, NPV_0, is:

    \[NPV_0=\sum_{i}^{}E(C_i) e^{-\alpha_i t_i}\]

When we shift time by \Delta t, we get a new NPV_ {\Delta t}:

    \[NPV_ {\Delta t} =\sum_{i}^{}E(C_i) e^{-\alpha_i (t_i+ {\Delta t} )}\]

We cannot make further simplifications because each term has an individual discount rate \alpha_i. That means that the sign of NPV with time shift does not have to be the same as the NPV without time shift, the sign can change. Hence the decision whether to allocate the project or investment can also change. Furthermore that means that we have to pay attention to take the right time scale in the calculation to come to the right decision.


In this post I described the advantages of the CCFP over the PCFP. The PCFP is only a simplification of the CCFP, PCFP can lead to wrong decisions. Because CCFP is the preferred and valid procedure of calculation, we can state that the sign of NPV can change by a time shift in general. The decision based on the NPV sign is only valid for the “present” time, which is t=0 in the DCF calculation. If you want the calculation to be the basis for an investment decision, you have to ensure that t=0 in the calculation is the point in time of the decision!

Several companies set t=0 at the beginning of the project`s revenues or the first investment cash flows. But this neglects the uncertainty and riskiness of cash flows from the decision point of time to their starting points. That is false in general, because of the arguments above. t=0 has to be the point in time of the decision. Otherwise you can generate false corporate decisions.

Incremental Free Cash Flows

In this post I want to point out some important characteristics of incremental free cash flows in a discounted cash flow (DCF) analysis. Incremental cash flows after taxes are the basis of each DCF analysis and in this way of each investment valuation.

Relative View to Alternative Scenario

Incremental means that the cash flows are caused by a positive investment decision. Positive decision means that the investment project is decided to be realized. Incremental cash flows are the cash flows that are additional to the cash flows of a negative investment decision (zero scenario or better alternative scenario). But to get the additional cash flows to the alternative scenario, you have to know the alternative scenario! In many cases, especially in “green field” projects, it is easy. There are simply no cash flows concerning your company in the alternative scenario. But it can become more complicated, if the alternative scenario depends on other investment project decisions that are not decided yet. In this case the investment projects are interrelated. Investments projects can for example require the same investment resources. But the company has to purchase it only once. If resources are used by several investment projects, the investment planning and DCF calculation must be done in a comprehensive company view. Only in this way you can allocate the investments to the investment projects. But nevertheless this is not always clear, because this allocation to a certain project can be ambiguous. Many companies have a central investment planning department which ensures that interrelated investment projects are harmonized.

A company can have the choice between two exclusive scenarios, which are both unprofitable stand-alone. But the company has to choose on of these two alternatives. When making a DCF analysis of one of these scenarios you have to take into account that the other scenario is not zero cash flow but the second excluding alternative scenario. That can lead to the fact that the DCF analysis becomes profitable because the negative effects of the alternative scenario can be prevented. Can that be true? Yes, because a DCF analysis is always decision focused and the decision can be more profitable than the realization of the alternative scenario. It is very important to have in mind this relative characteristic of a DCF analysis.

Sunk Costs and Opportunity Costs

An additional crucial point is that the incremental free cash flows must be caused by the decision to realize an investment project (principle of cause and effect). Costs that are linked to the project but cannot be influenced by the investment decision itself are so called sunk costs. Sunk costs are not part of the incremental free cash flows. In this way your DCF calculation shows whether the decision to realize the investment project is profitable for the company or not. But generally it does not show whether the project itself is profitable or not. Management cannot avoid sunk costs, hence the calculation show straightforward the ability to influence the future development. But you also have to take into account opportunity costs. These are negative cash flows effects caused by the investment decision but actually part of other projects. For example, the launch of a new product can substitute the volume of another (just released) investment project. Another example is a price reduction on the product which volume is part of the contribution cash flows of another investment project. This relationship also has to be considered if you are doing a recalculation of an investment project.


You should always consider these two point in a DCF analysis: 1) The relative view compared to the alternative scenario and 2) the decision making focus ignoring sunk costs and including opportunity costs. Sometimes companies are making recalculations of their investment projects. Recalculations can only be compared with the original, decision focused DCF analysis when taking the same assumptions. In an interrelated investment world you cannot recalculate one investment project without knowing the relationship to the other investment projects.

WACC with Tax Shield

In this post I want to provide a derivation of the discount rate that includes savings because of interest tax shield. Further I can show a general expression for tax shields implementation, wherein this well known WACC formula is only a special case:

    \[WACC=\frac{D}{D+E}r_{D}\left( 1-t \right)+\frac{E}{D+E}r_{E}\]

The formula includes “-t” that comes from tax shield savings. D and E stand for debt and equity of the firm, r_D and r_E are the required return rates for debt and equity, t is the marginal tax rate.

General case

We consider one time period starting at t_0 and ending at time t_1. In t_1 we have a cash flow excluding tax shield of C_1 and an absolute tax shield value of T_1. r is the discount rate without tax shield. We are searching a discount rate r^* that allows us to discount the cash flow excluding tax shield but including the tax shield effect in the present value in t_0. Hence we have to adapt the discount rate. The discounted value of the cash flows in t_0 has to be the same for both discount rates:


That leads to a general relationship of r^{*} and r:


That expression allows us to include a cash flow T_1 into the discount rate and to discount the cash flows in t_1 excluding T_1 with the adapted discount rate r^*. It enables us to calculate the discount rate r^* from the cash flows C_1 and T_1 in t_1. You do not require any values from t_0. Especially there is no need to have knowledge about the capital structure of the company. But for sure the capital structure is required to calculate T_1 in most cases.

Including capital structure

Next we want to consider the capital structure in t_0. The discounted asset value consists of debt D_0 and equity E_0. The discounted value of the cash flows must be the sum of debt and equity. With \left( D_0 + E_0 \right) \left( 1+r \right) = C_1 + T_1 we obtain:


Setting r as weighted average return rates of debt and equity without tax shield we get:

    \[r^*=\frac{D_0}{D_0+E_0}r_{D}+\frac{E_0}{D_0+E_0}r_{E} -\frac{T_1}{D_0+E_0} \]

Famous WACC after taxes

In most cases the tax shield is the interests paid on D_0 times the marginal tax rate t. That means T_1=D_0 r_D t. Hence we get the well known expression for r^*:

    \[r^*=\frac{D_0}{D_0+E_0}r_{D}\left( 1-t \right)+\frac{E_0}{D_0+E_0}r_{E}\]

This is the discount rate or “WACC” after taxes which is quoted in most books. But take care! It is only valid, if really the complete amount of interests paid can be deducted from taxes. Sometimes the company does not have enough profit to deduct all interest payments. In other cases the amount of tax deduction is limited by some constraints. In these two cases the previous formula does not work any more. The equation also shows that the capital structure in t_0 is important and not the capital structure in t_1.

Maximum constraint of tax shield

If the company has for example a maximum for the tax shield T_1^{\text{max}}, maybe a maximum share of EBITDA in t_1, we obtain another expression:

    \[r^*=\frac{D_0}{D_0+E_0}r_{D}+\frac{E_0}{D_0+E_0}r_{E} -\frac{ \min \left( D_0 r_D t , T_1^{\text{max}} \right)}{D_0+E_0} \]

Or if the company has tax shield savings from other periods in t_1, the equation is not valid, too.

In my point of view, the Adjusted Present Value (APV) approach is much better than the WACC approach for the implementation of tax shield. Each time period has clear tax shield amounts. And in complex cases you do not have to adapt the WACC in each time period.

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