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.

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.

Operating Leverage

Operating leverage is the sensitivity of an asset’s value on the market development caused by the operational cost structure, fixed and variable costs. The asset can be a company, a project or another economic unit. A production facility with high fixed costs is said to have high operating leverage. High operating leverage means a high asset beta caused by high fixed costs. The cash flows of an asset mainly consists of revenues, fixed and variable expenses:

cash flow = revenues – fixed expenses – variable expenses

Costs are variable if they depend on the output rate. Fixed costs do not depend on the output rate. The present value (PV) of the asset is the present value of its cash flows. Present values are linear, we obtain for the asset’s PV:

PV(asset) = PV(revenues) – PV(fixed expenses) – PV(variable expenses)

Rearranging leads us to:

PV(revenues) = PV(fixed expenses) + PV(variable expenses) + PV(asset)

Those who receive the fixed expenses are like debtholders in the project. They get fixed payments. Those who receive the net cash flows of the asset are like shareholders. They get whatever is left after payment of the fixed expenses. Now we analyze how the beta of the asset is related to the betas of revenues and expenses. The beta of PV(revenue) is a weighted average of the betas of its component parts:

    \[\beta_{revenue}=\beta_{\text{fixed exp.}}\frac{\text{PV(fixed exp.)}}{\text{PV(revenue)}}+\]

    \[+\beta_{\text{var. exp.}}\frac{\text{PV(var. exp.)}}{\text{PV(revenue)}}+\beta_{\text{asset}}\frac{\text{PV(asset)}}{\text{PV(revenue)}}\]

The fixed expense beta is close to zero, because the fixed expenses do not depend on the market development. The receivers of the fixed expenses get a fixed stream of cash flows however the market develops. That means \beta_{\text{fixed exp.}} = 0. The betas of revenues and variable expenses are more or less the same, because they are both related to the output. Therefore we can substitute \beta_{\text{revenue}} for \beta_\text{var.exp.}.


Setting PV(revenue) – PV(var.exp.) = PV(asset) + PV(fixed exp.) we obtain:

    \[\beta_{\text{asset}}=\beta_{\text{revenue}}\left[ 1 + \frac{\text{PV(fixed exp.)}}{\text{PV(asset)}}\right]\]

This is the relationship of asset beta to the beta of turnover. The asset beta increases with increasing fixed costs. As an accounting measure we define the degree of operating leverage (DOL) as:

    \[\text{DOL}= 1 + \frac{\text{fixed exp.}}{\text{profits}}\]

The degree of operating leverage measures the change in profits when revenues change.

Valuing the equity beta is a standard issue in DCF analysis. In many cases you take an industry segment beta and adjust it to your company or project. The adjustment of the industry beta also includes the adjustment of operating leverage. We assume that \beta_{\text{revenue}} is the same for all companies in the industry segment. \beta_{\text{revenue}} is the beta of the segment without operating leverage. The \beta_{\text{asset}}^{\text{ind. segm.}} is the average asset beta of the industry segment, which has an average ratio of fixed expenses to profits. \beta_{\text{asset}}^{\text{ind. segm.}} is provided by public databases.

For detailed information see: Brealey/Myers/Allen: Principles of Corporate Finance, 13th edition, p. 238, McGraw Hill Education, 2020)

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.

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.

Component Cash Flow Analysis

Project and component cash flow procedure

In this post I want to give some remarks on the component cash flow procedure. I am using this approach in my DCF analysis. It is the only correct way to discount the cash flows of complex investments, projects or diversified firms. In investment valuation many analysts take the WACC of the company, the business unit or the investment project to discount the cash flows. Taking only one single discount rate in a valuation is called project cash flow procedure (PCFP)

But most projects consist of a mixture of cash flows with different risks. To take this into account the present value of each cash flow should be calculated using a discount rate appropriate for its risk. The present values of all of the cash flow components should then be summed up. Projects should hence be selected using the present value criterion applied to its total present values. This approach is called component cash flow procedure (CCFP).

Disadvantages of using a single discount rate

In general, it is possible to find a composite discount rate for a project that gives the same NPV for the project as the NPV derived from the component cash flow procedure. The net cash flows can be determined that gives the same present value as the sum of the present values of its components. But there are at least three problems in applying one discount rate to the sum of the cash flows (Bierman, Smidt / Advanced Capital Budgeting, 2007).

  • Any change in assumptions about the project will tend to lead to a change in the composite discount rate. Changing the life of the project or the proportion of any of its cash flow components would likely require a different composite discount rate for the total cash flows.
  • If the correct composite discount rate is applied to the net cash flows of a project, then although the NPV of the project will be correct, the present values assigned to the cash flow components using this rate will be inaccurate. For example, the present value of the depreciation tax shields will usually be underestimated. In addition, the present value of the total cash flows in a particular year or a particular period will usually be inaccurate. This may lead to errors in decisions, such as estimating the value of the project at various future dates.
  • If the cash flow mixture is changed, the present value calculated using the previous composite discount rate would not produce accurate present values. This is particularly important in making choice between mutually exclusive alternatives that frequently involve a change in the mixture of cash flows, for example, the substitution of capital for labor.

If the project life is finite and greater than one year, then finding the composite discount rate of a project requires finding an IRR. There may be projects for which an IRR does not exist, or is not unique. For those projects, there may be no composite rate, or the composite rate may not be unique.

My proposal for appropriate discount rates

I propose the following way to discount the cash flows in a DCF analysis:

  • Market related (contribution) cash flows that depend on the overall market development should be discounted with their appropriate risk adjusted WACC, corrected by extracting Operating Leverage. For this I take the CAPM (including additional country risk premiums for non-diversifiable country specific risks) and the APT. These WACC should include appropriate default free interest rates, market and country risk premiums, the appropriate debt to equity ratios, equity betas and marginal tax rates. Revenues, variable costs, taxes and expenses for working capital are typically part of these market contribution cash flows. If you have a project with revenues in various markets with different currencies and risks, you have to discount each cash flow with its appropriate discount rate.
  • Capex / investment cash flows should be discounted with the reinvestment rate of the corresponding currency.
  • One time payments and expenses should be discounted with the risk-free rate of the corresponding currency.
  • Fixed costs should be discounted with the reinvestment rate of the corresponding currency. In some real options, e.g. switch options, savings in fixed costs are part of the option value. Option values are discounted with the risk-free rate, which is close to the WACC with a CAPM beta of 0. In this way DCF and ROA concepts match.
  • Leasing revenues and expenses should be discounted with the risk-free rate of the corresponding currency. The buy versus lease decision illustrates well the desirability of using different discount rates for cash flows with different characteristics. The use of different discount rates for different cash flow components is a widely accepted practice in analyzing buy vs lease problems.
  • Interests and tax shield savings depend on the leverage strategy of the company. If the amount of debt is fixed over time, the discount rate should be the risk-free rate of the corresponding currency. If the share of debt is kept constant over time, the discount rate should be the specific risk-adjusted market WACC.

Binomial Model vs. Black Scholes

Joachim Kuczynski, 06 December 2022

There are two basic ways to evaluate real options. The first way is the evaluation with exact analytical approaches like the famous Black Scholes Merton model. They have their origin in financial option pricing and deliver exact results. But they have certain underlying conditions that do not match reality of real options many times. The second basic approach to evaluate real options are discrete approximative models like the Binomial Model from Cox, Ross and Rubinstein. They are approximations of the exact analytical solutions when they have the same underlying conditions. In my real option analysis I prefer the approximative approach because of many reasons:

  • Volatility: The volatility, or risk respectively, defines the variance of the binomial tree branches (ups and downs). Volatility can change over time because of many reasons. But in the Black Scholes Merton model volatility is fixed for the considered time interval. Taking the binomial approach it is up to you to change volatility whenever you want. For sure, the binomial tree can become more complicated, e.g. if you have to change from a recombining to a non-recombining binomial tree. But with computer support also a more complicated event tree is no problem.
  • Exercise price: The exercise price at a node is the time value of the corresponding future cash flows which do not depend on the market development and are not diversiviable (mainly investments, fixed expenses and fixed earnings). Each of these components can change in time. That means that the exercise price of the real option can be differerent in each period. In the Black Scholes Merton model the exercise price is the same over time. In case of changing exercise prices you have to use approximative (binomial) approaches.
  • Discount rates: The discount rate includes many parameters like risk free rate, non-diversifiable market risks of the cash flow, the investor’s capital structure, the investor’s opportunity portfolio and tax shields. All these parameters can vary over time, and hence change the discount rate. If the discount rate changes, you might get a non-recombining binomial tree. Take care that the discount rate in each period has to match the expected values of the binomial tree branches in each period.
  • Decision tree: The decision tree does not have to be a complete binomial tree as required at the approximation of the Black Scholes Merton approach. Some branches might not exist or there can be more than one possibility in a node of the binomial tree (tri- or multinomial trees). Sometimes these exceptions represent reality more accurately and can be calculated in a discrete model. But these cases are not an approximation of Black Scholes Merton model any more. The Black Scholes Merton model cannot handle such tasks.
  • Time steps: Times between nodes of an event tree can be different at each link. You can adapt the times to your specific problem if required. The nodes of the event tree can for example be determined by the decision time of the investment project. At the Black Scholes approach the temporal development is fixed by the input parameters. There is no possibility to adapt it anyway. In the binomial approach you can adapt time steps as required by the corresponding problem.

In general, the (binomial) approximative approaches are much more flexible and can be adjusted to the specific problem. With specific input parameters the binomial model is an exact approximation of the Black Scholes Merton model. But many times the input parameters describing reality are different. Black Scholes Merton comes from financial option markets, where situations are less complex as at real options many times. The binomial approach is much more suitable for real option analysis. Because of the inaccuracy of many input variables at real options, the approximative character of the binomial model does not distort the result mostly.

Baldwin Rate of Return | MIRR

Joachim Kuczynski, 05 March 2021

Baldwin rate definition

The modified internal rate of return (MIRR), or Baldwin rate of return respectively, is an advancement of the internal rate of return (IRR). But also the MIRR can be misleading and can generate false investment decisions. The MIRR is defined as:

    \[MIRR=\sqrt[n]{\frac{\text{FV(contribution cash flows,WACC)}}{\text{PV(invest cash flows, financing rate)}}}-1\]

FV means the final value at the last considered period, PV stands for the present value.

Pitfalls of the Badwin rate

These points have to be considered carefully when applying the MIRR:

  • Cash flows in different countries, with different currencies, equity betas, tax rates, capital structure, etc. should be evaluated with specific risk-adjusted discount rates. In MIRR the project is profitable, if the rate of return is higher than the required WACC. But which WACC do we mean in projects with various differing cash flows ? No diversification of cash flows can be taken into account in MIRR. But that is crucial in evaluating international projects.
  • You need premises about the reinvestment rate of the contribution cash flows that affect the profitability of the project. These premises are not required in the (e)NPV concept. Hence you add an additional element of uncertainty in your calculation when using the MIRR, without any need.
  • Reinvesting contribution cash flows (numerator of the root) with the risk adjusted WACC means that the return of the project increases when the project risks and WACC increases. That cannot be true. In pinciple you should not use key figures that require assumptions about reinvestment return rates. You are evaluating a certain project and not of other unknown investment sources. In general you can discount all cash flows with its appropriate discount rate and capitalize it to the last period.
  • Does capitalizing (or rediscounting) a cash flow with a risk-adjusted discount rate to a future period make sense in general? I do not think so. To rediscount cash flows with a risk adjusted discount rate including a risk premium means that you are increasing the risk of the project. The project does not remain the same, because its risk increases. Only taking a riskless discount rate for reinvestments would not increase the risk of the project. The MIRR comes from a classical perspective with no risk adjustment of the discount rates. If you are using risk-adjusted discount rates, you are mixing two concepts that do not fit.
  • An additional positive cash flow must improve the profitability of the project. If you add an additional, small cash flow in an additional period n+1, the Baldwin rate can decrease. This is because the number of periods increases and the value of the root decreases. Cases that lead to wrong results are not acceptable for decision key figure.
  • If you have e.g. an after sales market with small positive cash flows, the Baldwin rate decreases by considering these cash flows in your calculation. This is because the number of periods increases and the n-th root decreases. Thus an after sales market cannot be implemented in your calculation.
  • You have to define clearly, which cash flow is in the numerator and which is in the denominator of the root. There is no clear and logic distinction. Thus you can find different definitions in literature. Anyway avoid to take balance sheet definitions of “investment”. Note that besides investments also fixed costs and leasing payments have to be discounted with a default free discount rate in general.
  • You cannot compare mutual exclusive investment projects, if the investments or the project periods are different.
  • You can also not evaluate investment projects with negative value contribution to the firm. But anyway such projects exist and have to be decided.
  • All cash flows should be considered as expected value of a probability distributions. The expected value of the Baldwin rate is not the Baldwin rate of the expected values of the cash flows.


The (e)NPV concept is much better than the MIRR or Baldwin Rate of Return. The (e)NPV does not have all the pitfalls mentioned above. Further you can also evaluate and compare value-loosing investment alternatives and do not need any premises about reinvestment rates. There are only disadvantage of the MIRR / Baldwin rate compared to the (e)NPV, try to avoid the application of MIRR / Baldwin rate.

After-Tax Discount Rate

In this post I want do derive the after-tax discount rate from the before-tax discount rate. “Before tax” means that the tax shield is not considered in the discount rate. It does not mean that the tax expenses (without tax shield) are not considered in the free cash flow. The tax expenses (without tax shield) are a part of the free cash flow in the before-tax and in the after-tax discount rate. For further information have a look at my other post WACC with Tax Shield. Abbreviations:

r … before-tax discount rate
r^{*} … after-tax discount rate
L … rate of debt to sum of equity E and debt D, L=D/(E+D)
r_D … debt interest rate
r_E … equity interest rate
t … marginal corporate tax rate

We assume that the values of r_D, r_E and L are known. Then the before-tax discount rate is:

    \[r=\left( 1-L \right)r_{E}+Lr_D\]

Rearranging the above to solve for r_{E} we have:


The after-tax discount rate at a constant leverage rate is:

    \[r^{*}=\left( 1-L \right)r_{E}+L\left( 1-t \right)r_D\]

This is the famous equation most financial analysts might know. The factor “-t” comes from the tax shield and decreases the discount rate. Hence the discount rate after taxes is lower than the return rate before taxes. But you have to take care. This after-tax formula is only valid if the leverage rate L remains constant. Additionally it assumes that the total amount of tax expenses can be deducted by tax shield. If these two premises are not true, the previous formula does not work and you have to an analyze the topic with the adjusted present value (APV) approach. For a general view see this post. By substituting r_{E} we get:

    \[r^{*}= r -Ltr_D\]

This formula can be useful, because you do not have to know the equity return rate to calculate the after-tax return rate. But have in mind that this is only valid, if the leverage ratio is constant and the total tax shield amount can really be deducted from the tax expenses.

Return Rate Aggregation

Joachim Kuczynski, 09 February 2023

In many books you can read that the return rate of a set of several assets is the weighted average of the single asset’s return rates. But up to now I did not found any proof for this statement. In this post I provide a derivation of that relationship. An additional benefit of that calculation is to understand the conditions under which that relationship is valid basically.

Let us start with an asset value at time t, C(t), which is the sum of different assets values C_i(t):


At time t=0 the asset values are C(0) and C_i(0) with C(0)=\sum_{i}^{}C_i(0). The asset value C_i is developing in time t with its specific return rate r_i, that means:

    \[C_i (t)=C_i(0)exp(r_i t)\]

Now we are searching an aggregated return rate r, that describes the development of the aggregated asset value C. Setting C(t)=C(0)exp(rt) we obtain:

    \[r=\frac{1}{t}ln\frac{C(t)}{C(0)}=\frac{1}{t}ln\left( \sum_{i}^{} \left \frac{C_i(0)}{C(0)} exp \left( r_i t \right) \right \right)\]

This is the exact relationship between the aggregated return rate r and the differential return rates r_i. This expression cannot be simplified any more. Now we develop the exponential and logarithmic functions using Taylor series and take the polynomial approximation only up to its first oder. That means \text{exp}\left( x \right)\simeq 1+x and \text{ln}\left( x \right)\simeq x-1. Hence we get a first order approximation of r:

    \[r\simeq \frac{1}{t}\left( \sum_{i}^{} \left( \frac{C_i(0)}{C(0)} \left( 1+ r_i t \right) \right) -1 \right)\]

This simplyfies to:

    \[r\simeq \sum_{i}^{} \frac{C_i(0)}{C(0)} r_i\]

This is the result, that many authors present and use in their books. Also the calculation of the WACC, or aggregated return / discount rate respectively, is told to be the weighted average of debt D return rate r_D and equity E return rate r_E:


But take care, that all is only an approximation. And in some cases is can be an inaccurate approximation. With increasing differential return rates r_i and increasing time t the approximation becomes more and more inaccurate. If you require an exact calculation, take the formula presented above.

Note that it does not matter whether you take C_i (t)=C_i(0)exp(r_i t) or C_i (t)=C_i(0)(1+r_i^*)^t. With a substitution of r=ln(1+r^*), you can transform these two return rates into each other.

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.

Lease or Buy / Make

Joachim Kuczynski, 03 April 2023

A rental agreement that extends for a year or more by a series of fixed payments is called a lease. Firms lease as an alternative to buying capital equipment. Cars, aircraft, ships, farm equipment and trucks are leased many times. In principal every kind of asset can be leased. In this post I want to describe the valuation of a lease contract and how you can analyse whether to prefer buying / making or leasing an asset from the financial point of view.

The correct way

At first you have to figure out all free cash flows that are different between buying an asset in comparison to leasing it. Usually this concerns cash flows from purchasing, tax shield because of EBIT reduction caused by depreciation or leasing payments, tax shields because of debt interest deduction, maintenance costs and salvage value of the asset. After that you have to discount all cash flows with the appropriate risk adjusted discount factor (\gamma_{b,i} for the buying and \gamma_{l,j} for the leasing scenario). Adding up all present values you get a net present value (NPV) of the buying case and a NPV of the leasing case.



The case with the higher NPV is the better one from financial point of view. That means leasing is better than buying, if NPV_l > NPV_b.

Many firms do not value this way because they are not familiar with risk appropriate discounting of cash flows. Therefore they use simplified formulas described in many corporate finance books. But in general that can lead to false results if the premises of the simplifications do not fit reality. So take care and do not use the simplified formulas without checking the concrete situation.

Simplified ways

The short ways are characterized by unifying discount factors for cash flows. This assumes that the cash flows have the same risk adjustment. In general that is not the case obviously. Each cash flow has its own risk in principle. In my calculation I consider each cash flow and calculate its specific risk. That is not much additional work. But I can be sure that I get the correct result.

In some cases you can assume the same discount factor. Tax shield because of EBIT reduction, caused by depreciation and leasing payments, and debt interest deduction can be considered as fixed cash flows in most cases. They have no market dependency. Hence you can discount their cash flows with the company’s debt interest rate. Depending on the debt interest deductibility of the cash flows you have to take the before or after interest tax rate (operating and financial lease).

If some cash flows have market dependency (e.g. salvage value) or are realised in different currencies, the discount factors might not be the same. Additionally, the changing capital structure and the dependency of tax deductibility on the market development can lead to different discount rates. In all these cases you cannot use the simplifies formulas. Then you have to apply the APV method and calculate all NPV contributions seperately. This is the procedure of the previous section.

Debt-equivalent cash flows

The authors of many books about corporate finance use the term debt-equivalent cash flows. That are the additional cash flows that occur when financing a free cash flow stream by an equivalent loan. At leasing vs buying the free cash flow stream is leasing FCF minus buying FCF. After that you calculate interests of the loan and the interest tax shields with APV or with simplified adjusted discount rates (as described before). But take care! This simplified version is only valid, if leasing and buying cash flows have the same discount factor (risks) in each period. Otherwise the calculation with debt-equivalent cash flows provides false results!

Options in leasing contracts

Many leasing contracts include options like buying the assets at the end of the leasing time or cancelling the leasing contract before expiration. Any option can be valued with Real Options Analysis. The analysis is problem-specific. But in general each option in a leasing contract has a specific, well defined value.


The analysis of leasing contracts (and their comparion to buying / making) is completely the standard analysis of asset valuation. But you should do that in the basic accurate way. That means DCF analysis with appropriate risk-adjusted discount rates (Component Cash Flow Procedure) including the adjusted present value (APV) approach. It is not much additional work to do. But you can be sure, that your results are right.

Binomialmodell Wahrscheinlichkeiten

Joachim Kuczynski, 24. Oktober 2023


In diesem Artikel möchte ich die Abhängigkeit des Optionswerts von der Eintrittswahrscheinlichkeit der realen (binomialen) Zustände innerhalb des Modells von Cox, Ross and Rubinstein darstellen. Auch in der Literatur ist mancherorts zu lesen, dass die risikoneutralen Wahrscheinlichkeiten und damit die Optionswerte nicht von den Wahrscheinlichkeiten der Realzustände abhängen. Die Optionswerte hängen aber implizit sehr wohl von ihnen ab. Dies möchte ich in diesem Post kurz darstellen und herleiten.

Binomialmodel von Cox, Ross und Rubinstein

Optionen können mit dem Binomialmodell von Ross, Cox and Rubinstein bewertet werden. Der Optionswert C_0 zum Zeitpunkt t=0 ist:

    \[C_0=\frac{\alpha C_{u,t_1}+(1-\alpha )C_{d,t_1}}{(1+r)^T }\]

C_{u,t_1} and C_{d,t_1} sind die Optionswerte der up und down Entwicklungen zum Zeitpunkt t_1. r ist der risikofreie (bzw. zuschlagsfreie) Zinssatz und T die Zeit zwischen t_0 und t_1. \alpha ist hierin die risikoneutrale Wahrscheinlichkeit der up Bewegung in t_1, 1-\alpha ist die risikoneutrale Wahrscheinlichkeit der down Bewegung in t_1. Das Binomialmodel stellt uns folgende Beziehung zur Verfügung:


Einsetzen von \alpha liefert uns diese Darstellung für C_0:

    \[C_0=\frac{\frac{(1+r)^T-d}{u-d} C_{u,t_1}+(1-\frac{(1+r)^T-d}{u-d} )C_{d,t_1}}{(1+r)^T }\]

Nach einer kleinen Umformung erhalten wir:

    \[C_0=\frac{( (1+r)^T-d ) C_{u,t_1}+(u-(1+r)^T )C_{d,t_1}}{(1+r)^T (u-d)}\]

u und d sind definiert als die Verhältnisse von up und down Entwicklung zum Erwartungswert des Zustands in t_0, EV(S_{t_0}):

    \[u= \frac{EV(S_{t_0})}{S_{u,t_1}}\]

    \[d= \frac{EV(S_{t_0})}{S_{d,t_1}}\]

Bis jetzt sind die Wahrscheinlichkeiten von Zustand up S_{u,{t_1}} und Zustand down S_{d,{t_1}} nicht vorgekommen. Vielfach wird nun argumentiert, dass die Wahrscheinlichkeiten der beiden Realzustände den Optionswert nicht beeinflussen. Dies stimmt nicht. Der Erwartungwert von Zustand S_{t1} und damit von S_{t0} hängen von diesen Wahrscheinlichkeiten ab. Der Erwartungswert des Zustands in t_0 ist der diskontierte Wert des Zustands in t_1. Mit D als jährlichen kostanten Diskontierungsrate erhalten wir:


Für u und d erhalten wir:



Das Endergebnis lautet nun:

    \[C_0=\frac{( (1+r)^T-\frac{pS_{u,{t_1}}+(1-p)S_{d,{t_1}}}{S_{d,t_1}(1+D)^{T}}) C_{u,t_1}}{(1+r)^T ( \frac{{pS_{u,{t_1}}+(1-p)S_{d,{t_1}}}}{{S_{u,t_1}(1+D)^{T}}}-\frac{{pS_{u,{t_1}}+(1-p)S_{d,{t_1}}}}{{S_{d,t_1}(1+D)^{T}}})}+\]

    \[+\frac{(\frac{{pS_{u,{t_1}}+(1-p)S_{d,{t_1}}}}{{S_{u,t_1}(1+D)^{T}}}-(1+r)^T )C_{d,t_1}}{(1+r)^T ( \frac{{pS_{u,{t_1}}+(1-p)S_{d,{t_1}}}}{{S_{u,t_1}(1+D)^{T}}}-\frac{{pS_{u,{t_1}}+(1-p)S_{d,{t_1}}}}{{S_{d,t_1}(1+D)^{T}}})}\]

Dies ist die grundlegende Beziehung zwischen Optionswert zum Zeitpunkt t_0 und expliziten problemspezifischen Variablen.


Wir sehen, dass der Optionswert C_0 explizit von den Wahrscheinlichkeiten p und 1-p der realen up S_{u,t_1} und down Zustände S_{d,t_1} abhängt. Das wollten wir zeigen. q.e.d.

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