Operating Leverage in Practice

Joachim Kuczynski, 15 December 2023


A key issue in each asset valuation is the calculation of an asset’s equity beta. The preferred procedure is to take the appropriate industry segment equity beta and adjust it to your specific asset. This adjustment mainly consists of financial leverage and operating leverage. In this article I explain the procedure of operating leverage adjustment. In the following discussion we name the asset, that we want to value, our project. In general, the operating leverage of all assets can be analyzed this way. Especially it is also valid for single contribution cash flows, if you want to use the approach in the component cash flow procedure, which I am preferring in my valuations because of its accuracy.


Most valuations base on average industry segment equity betas, which are adapted to the specific asset. We receive equity industry segment betas \beta^{ind}_{asset} from databases in literature or in the web. In my articles concerning operating leverage and operating leverage in CCFP you can see, that the average industry equity beta for revenues and variable costs can be calculated in this way:

    \[\beta^{ind}_{rev}=\beta ^{ind}_{asset} \left( 1-\frac{V^{ind}_{fix}}{V^{ind}_{asset}} \right)^{-1}\]

The equity beta for revenues and variable costs are the same for the industry segment and for the considered project. That means \beta^{ind}_{rev} = \beta^{proj}_{rev}. Hence we obtain:

    \[\beta^{proj}_{rev}=\beta ^{ind}_{asset} \left( 1-\frac{V^{ind}_{fix}}{V^{ind}_{asset}} \right)^{-1}\]

This is an important result for the component cash flow procedure. It provides the equity beta for revenues and variable cost cash flows. Using the CAPM you derive the discount rates for revenues and variable cost cash flows.


We can calculate the asset equity beta of the project as well:

    \[\beta^{proj}_{asset}=\beta ^{proj}_{rev} \left( 1-\frac{V^{proj}_{fix}}{V^{proj}_{asset}} \right)=\]

    \[= \beta ^{ind}_{asset} \frac{1-\frac{V^{proj}_{fix}}{V^{proj}_{asset}}}{1-\frac{V^{ind}_{fix}}{V^{ind}_{asset}}}\]

This is the appropriate equity beta of the considered project, if you use the project cash flow procedure. Making use of the CAPM you get the discount rates for the project’s cash flows in general with the corrected operating leverage.


We want to value a simple project with infinite lifetime, a turnover of EUR 120 and costs of EUR 50 per year. The yearly profit is EUR 70. The riskless rate is 5%, the expected market return rate is 10%.

Case 1: The project has only variable costs and no fixed costs. The project has a beta of 1. The discount rate of the project (CAPM) is r=5%+1.0 (10%-5%)=10%. With infinite project lifetime we get a project value of EUR 70 / 10% = EUR 700.

Case 2: Let us assume now the EUR 50 cost per year to be fixed instead of variable. At first we discount the cash flows seperately acc. to the component cash flow procedure. Turnover still has a beta of 1 and a discount rate of 10%. We obtain a turnover value of EUR 120 / 10% = EUR 1200. The fixed costs have to be discounted with the riskless rate of 5%. Hence we get a value of EUR 50 / 5% = EUR 1000. The project value is EUR 1200 – EUR 1000 = EUR 200.
Next we apply the project cash flow procedure, which discounts all cash flows with a unique project discount rate. The beta of the project’s revenues does not depend on the fixed costs share, it is still 1. With formula described above we can calculate the project beta with the new operating leverage:

    \[\beta^{project}_{asset}=1.0 \left( 1+ \frac {1000}{200} \right)=6\]

A second way to calculate the project beta is by taking the weighted average of its components’ betas:

    \[\beta_{P}=\frac{R}{R-C}\beta_{R} - \frac{C}{R-C}\beta_{C}=\]

    \[=\frac{1200}{1200-1000}1 - \frac{1000}{1200-1000}0=6\]

Both approaches provide a project beta of 6. According to CAPM we get a project discout rate of r=5%+6(10%-5%)=35%. Hence we obtain a project value of EUR 70 / 35% = EUR 200, the same result as calculated with the component cash flow procedure.

But the key message of this example is that the value of the project falls from EUR 700 to EUR 200 because of higher operating leverage. This illustrated that an adaption of betas and discount rates because of operating leverage is crucial in many cases. Only in this way we can get valid results.

In more complex projects with many different kind of cash flows I prefer the component cash flow analysis. A change in one cash flow changes only the value of this cash flow. If you use the project cash flow procedure, you always have to calculate a new project discount rate at any change of one parameter.

Approximation with P&L statement

The (present) values of fixed costs V_{fix}, variable costs V_{var} and revenues V_{rev} cannot be found easily. But we can try to approximate them with P&L or balance sheet figures, which are available more eaysily. Let us consider eternal yearly revenues R, yearly fixed costs \alpha R and yearly variable costs \beta R. The risk free rate is r_f and the market return rate r_m. With the expressions above we get the beta of the asset:

    \[\beta _{asset} = \beta_{rev} \left( 1 - \frac{V_{fix}}{V_{asset}} \right) \]

The values of fixed costs and asset are:

    \[V_{fix}=- \frac{\alpha R}{r_f}\]

    \[V_{asset}= \frac{R - \alpha R - \beta R}{r_m}\]

With that we obtain for \beta_{asset}:

    \[\beta _{asset} = \beta_{rev} \left( 1 + \frac{\frac{\alpha R}{r_f}}{\frac{R - \alpha R - \beta R}{r_m}} \right) = \beta_{rev} \left( 1 + \frac{r_m}{r_f}\frac{\alpha}{1-\alpha-\beta} \right) \]

r_m \simeq 10  \% and r_f \simeq 5 \% approximately, so \frac{r_m}{r_f} \simeq 2, \frac{\alpha}{1-\alpha-\beta} is the ratio of annual fixed costs to annual profit. That means that you have to multiply that factor with the 2, when applying P&L figures in approximation and not present values.

Binomial Model Probabilities

Joachim Kuczynski, 24 October 2023


In this article I want to derive the explicit relationship between an option value and the probability of occurrence of its event states in the binomial model of Cox, Ross and Rubinstein. In many cases I read that the risk neutral probabilities and therefore the option value do not depend on the probabilities of the real state values. But the options values depend on them implicitly. That is what I will derive in this post.

Binomial Model by Cox, Ross and Rubinstein

Options can be valued with the binomial model from Ross, Cox and Rubinstein. The value C_0 of an option at time t_0 is given by:

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

C_{u,t_1} and C_{d,t_1} are the option values of the up and down development at time t=1. r is the risk free rate and T is the time between t_0 and t_1, T=t_1-t_0. \alpha is the risk neutral probability of the up movement in t_1, 1-\alpha is the risk neutral probability of the down movement in t_1. The binomial model provides the following relationship:


Including \alpha provides this expression for C_0

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

Hence we obtain:

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

u and d are defined as ratio of up and down movement in relation to the expected value in t_0, EV(S_{t_0}):

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

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

Up to now the probabilities of up state S_{u,{t_1}} and down state S_{d,{t_1}} have not occured. Many times that leads to the argument that these probabilities do not influence the option value. But that is not true. The expected value of the state S_{t1} and therefore S_{t0} depends on the probabilities. The expected value of the event state in t_0 is the discounted value of event state in t_1. With D as yearly constant discount rate we get:


For u and d we get the following:



As final result we obtain:

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

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

This is the basic relationship between the value of an option at a time t_0 and explicit problem specific variables.


We realize that the option value C_0 expicitely depends on the probability p of the up state S_{u,t_1}, and 1-p of the down state S_{d,t_1} respectively. That is what we wanted to prove. The argument that this dependency does not exist, does not take into account that the value of state S_0 depends on the state probabilities in t_1. Hence there is no disappearance mystery of real life or real states probabilities in options valuation. q.e.d.

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.

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.

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 value of the asset (project or investment), V_{asset}, is the sum of revenues’ value, V_{rev}, the value of variable cost cash flows, V_{var}, and value of the fixed cost cash flows, V_{fix}:


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

    \[\beta_{asset}=\frac{\frac{\partial }{\partial r_m}V_{asset}}{V_{asset}}=\frac{\frac{\partial }{\partial r_m} V_{rev} + \frac{\partial }{\partial r_m} V_{var}+ \frac{\partial }{\partial r_m} V_{fix}}{V_{asset}}.\]

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

    \[\beta_{asset}=\frac{\frac{\partial }{\partial r_m} V_{rev} + \frac{\partial }{\partial r_m} V_{var}}{V_{asset}}=\frac{V_{rev}\beta_{rev} + V_{var}\beta_{var}}{V_{asset}}\]

The betas of revenues and variable costs are the same, \beta_{rev}=\beta_{var}. We get:

    \[\beta_{asset}=\beta_{rev} \frac{V_{rev} + V_{var}}{V_{asset}}=\beta_{rev} \left( 1-\frac{V_{fix}}{V_{asset}} \right)\]

Note that the value of the fixed cash flows is negative, V_{fix} \le 0, and the value of the asset is positive, V_{asset} \ge 0, in ordinary cases. The expression in the bracket becomes \ge 1. That means that the beta of the asset is higher than the beta of the revenues, \beta_{asset} \ge \beta_{rev}. Obviously that is true, because adding fixed costs increases the beta of the asset. This effect 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_{rev} and \beta_{var}. V_{fix} and V_{asset} 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_{rev} 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:

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

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

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

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.

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.

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 NPV Timeline

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.

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.

Matching Volatility in Binomial Model

Let us consider one step of a CRR binomial model with time interval \Delta t, initial state S_0 in t=0, an up-state of S^+=S_0u in t = \Delta t and a down state of S^-=S_0d in t= \Delta t. The probability of an up movement is assumed to be p, the probability of a down movement 1-p respectively, with 0<p<1. The expected value after the first time step \Delta t is S_0 exp( \mu \Delta t) and has to be the same as the expected value of the two binomial states in \Delta t:

    \[p S_0 u + (1-p) S_0 d=e^{\mu \Delta t}\]

Let E(X) be the expected value of a random variable X. Then the variance var(X) of X equals to var(X)=E(X^2)-[E(X)]^2. With that the variance of the two states of the binomial tree in \Delta t is:

    \[pS_0^2 u^2+(1-p) S_0^2 d^2-(puS_0+(1-p)dS_0 )^2 \]

A stock price, or a project value (for real options analysis) respectively, follows a Geometric Brownian motion (stochastic Wiener process). Let \sigma be the expected annual volatility of the process. \sigma is defined as standard deviation of the normal distribution \Phi of the annual relative returns:

    \[\frac{ \Delta S}{S} \thickapprox \Phi ( \mu \Delta t, \sigma \sqrt{\Delta t})\]

The variance of such a Geometric Brownian motion is:

    \[var(S)=S^2 \sigma ^2 \Delta t\]

Our binomial model should match the parameters of the continuous model. Therefore the variance of the binomial model and the variance of the Geometric Brownian motion have to be the same. Hence we get the following equation:

    \[p u^2 + (1-p) d^2 - ( p u+ (1-p) d)^2 = \sigma ^2 \Delta t\]

In my point of view this is a very important expression, because it determines the correlation of u and d to match volatility. When terms in\Delta t^2 and higher powers of \Delta t are ignored, one solution of this equation is:

    \[u=e^ {( \sigma \sqrt {\Delta t})}\]

    \[d=e^ {( - \sigma \sqrt {\Delta t})}\]

These are the values of u and d proposed by Cox, Ross and Rubinstein in their Binomial Model for matching u and d. But in principle there are infinite possible solutions of this equations. If you define an up or down movement, you can calculate the other value approximative numerically, e.g. with the goal seek function in Excel, even though there might be no closed solution.

Additionally it can be proofed that the variance does not depend on the expected return \mu, when \Delta t tends to zero. That means that the volatility is independent from the expected return. This is known as Girsanov’s theorem. When we move from a world with one set of risk preferences to a world with another set of risk preferences, the expected growth rates in variables change, but their volatilities remain the same.

Switch Options to Shut Down Example

Problem Framing

In this post I want to give a simple example of a switch option to shut down operations. Project management can decide each year whether to continue production or shut down operations in this year. If operations is shut down, management can generate cost savings by reducing fixed costs. If considered production continues, the project can generate additional revenues and contribution cash flows.
The project lifetime is 10 years, the WACC of the market contribution cash flows is 10.0% and constant over time. The contribution cash flows are shown in the graph below. The default (or risk) free rate is 5.0%. The switches between the mode of operation to the mode of shut down require costs in both directions. Switch costs, contribution cash flows and savings are shown in the table. Savings and switch costs increase with an inflation rate of 1.5% per year.

Swith Option to Shut Down Valuation

We want to give answers to the following questions: What is the value of this option to switch between the two modes? When do we have to switch between operation and shut down to get the maximum value added to the project?

DCF analysis provides a present value of the market contribution cash flows of 461 million EUR. A Monte-Carlo-Simulation provides a project volatility of 0.30. We want to analyze this switch option with the binomial approach of Cox-Ross-Rubinstein. As time periods we take one time step per year, so that you can read the figures in the lattice properly. For higher accuracy we could take smaller time periods anytime.

Below you can have look at the two binomial lattices. The first value at each node is the value of the underlying (market contribution cash flows). The second value is the value of the underlying including the switch option value. In the third line you can see whether you have to take the option and switch to the other mode or not. “Sw” means to switch to the other mode, “go” means to stay in the mode.

Binomial tree of the project’s active operation mode
Binomial tree of the project’s shut down mode

Beginning the project with the mode of operation (production), the value of the project including the option is 513 MEUR. That is 52 MEUR higher than the value of the project without any option (461 MEUR). Hence the option adds a value of 52 MEUR to the project, the ROV (real option value) is 52 MEUR. That is also the maximum that project management should invest in having the option. If you start the project with the shut down mode, the value added is only 33 million EUR.

The value added of 52 million EUR means that the NPV of the classical DCF analysis may convert from negative to positive because of this additional value. That could lead to a reconsidering of the project decision. The ENPV (expanded NPV) is the NPV of the project plus the value of the switch option. The savings of the shut down mode must be separate outgoing cash flows of the DCF analysis. They have to be discounted with the default free discount rate of 5.0%, because the savings are part of the project’s fixed costs. That is consistent to the component cash flow procedure (CCFP) approach. Investment and fixed costs cash flows have to be discounted with the default (or risk) free discount rate and not with the WACC of the market contribution cash flows.

In the binomial lattices you can also see what you have to do in which situation in the 10 years. Depending on the economic development, you should stay in the mode or switch to the other mode. This is a practical guideline for project management.

Additional Remarks

I constructed the event tree of the project’s contribution cash flows out of a Monte-Carlo-Simulation. Of course you can also create the nodes of the event trees by calculating the time values of the outstanding contribution cash flows explicitly. Discussion with management and marketing can provide the required cash flows and probabilities of the event tree. Analytical valuation methods like Black-Scholes-Merton cannot provide any solution for option types like this.

Option to Choose Example

Problem Framing

In this post I want to give a simple example of a choose option valuation. The considered project lasts 10 years, the WACC of the contribution (market) cash flows is 10.0% and constant over time. The contribution cash flows are shown in the graph below. Let the risk free rate be 5.0%. The project provides 3 options: 1) Option to expand with an investment of 100 kEUR (increasing with an inflation rate of 2% per year). The contribution cash flows would be increased by 20%, if managements invests in the expansion. 2) Option to abandon the project with a salvage value of 100 kEUR (increasing with an inflation rate of 2% per year) and 3) Option to contract with a savings of 200 kEUR (increasing with an inflation rate of 2% per year) and a contraction factor of 0.9. DCF analysis provides a present value of the market contribution cash flows of 1 million EUR. A Monte-Carlo-Simulation shows a project volatility of 0.30. We analyze this choose option with the binomial approach of Cox-Ross-Rubinstein. As time periods we take two time steps per year, so that you can read the figures in the lattice properly. For higher accuracy we could take smaller time periods anytime.

Choose Option Valuation

We want to give answers to the following questions: What is the value of this option to choose? When do we have to take which option to get the maximum value added for the project?

We take the binomial approach from Cox-Ross-Rubinstein to solve that issue. Following you can have look at the binomial lattice. The first value of each node is the value of the underlying (market contribution cash flows). The sevond value is the value of the options at this node. In the third line you can see whether you have to take an option and which option you have to take. “exp” means to invest in the expansion of the project, “go” means to take no option at this node and “con” means to invest in the contraction of the project.

The total value of the three options is 137 kEUR. That is the maximum investment that should be done for the three options in sum. The option to abandon is not taken in any node. That means that you should not invest in this abandonment option. In the first year you should not take any option, let the project evolve. In the second year there might be the first opportunities to take the expansion option. In the following years the expansion option is a good opportunity in case of positive cash flow development. Your project controlling should have a look at the cash flow development and go the right path through the lattice over time. In the first six years the contraction option provides no addition value to the project. But in the last three years of the project the contraction option becomes more important.

Note that this choose option adds a value of 137 kEUR to the project. That means that the NPV of the classical DCF analysis can convert from negative to positive because of the additional value. That could lead to a reconsidering of the project decision. The ENPV (expandedNPV) is the NPV of the project plus the value of the choose option.

Additional Remarks

I constructed the event tree from the project’s contribution cash flow volatility out of a Monte-Carlo-Simulations. Of course you can also create the nodes of the event trees by calculating the time values of the outstanding contribution cash flows explicitly. Discussion with management and marketing can provide the required cash flows and probabilities of the event tree.

This option to choose combines the three options of expansion, contraction and abandonment. This example also shows that the value of various options is not the sum of its individual options. Although the option to abandon has an option value by itself, it contributes no additional value to the option to choose, because it is not required in any node. Analytical valuation methods like Black-Scholes-Merton cannot provide exact solutions for such interdependent options.

Option to Switch Example

Problem Framing

In this post I want to give a simple example of a swich option valuation. The considered project lasts 10 years, the WACC of the contribution (market) cash flows is 10.0% and constant over time. The contribution cash flows are shown in the graph below. Let the risk free rate be 5.0%. The project provides 2 operation modes (possibilities), one with technology A and the other with technology B. A Monte-Carlo-Simulation of technology A shows a project’s contribution cash flow volatility of 0.30. The corresponding Monte-Carlo-Simuation of technology B provides a project’s contribution cash flow volatility of 0.15. We analyze this switch option with a binomial approach. As time periods we choose one time step per year, so that you can read the figures in the lattices. For higher accuracy we could take smaller time periods anytime. The present value of the project market contribution cash flows is 1 million USD. Switch costs from technology A to B start with 145 kUSD, from technology B to A start with 80 KUSD. These costs increase with an inflation rate of 2.0%.

Switch Option Valuation

We want to give answers to the following questions: What is the value of this option to switch between technology A and technology B? How much can we invest in this flexibility keeping a positive added value to the project? And when do I have to switch between the 2 technologies to get the maximum value added?

We take the binomial approach from Cox-Ross-Rubinstein to solve that issue. Following you can have look at the binomial lattices for both technologies. The first value of each node is the value of the underlying (market contribution cash flows), the sevond value is the value of the switch option at the corresponding node. In the third line you can see whether you have to switch to the other technology or not (“sw” means to switch, “go” means to stay in the technology).

Binomial tree of switch option for technology A
Binomial tree of switch option for technology B

The value of the switch option starting with technology A is 21 kUSD. Starting with technology B the switch option value is 157 kUSD. If the investment for the switch option flexibility is below these option values, you should do the investment. If you can choose the starting technology, you should start with B. This results in a higher value added, the corresponding investments not taken into account. In the first year you have to make no switch in any case. In the second year it depends on the cash flow development, if you start with technology A. Switch in the bad case to technology B and stay in technology A in case of positive development. Starting with technology B you should stay there and make no switch, independent of the cash flow development. In the third year you should switch to technology A in the best cash flow development. Analyzing the event/option tree through time in this way provides a guideline for what to do in which year, depending on the cash flow development.

Note that this switch option adds a value of 157 kEUR to the project. That means that the NPV of the classical DCF analysis can convert from negative to positive because of this additional value. That could lead to a reconsidering of the project decision. The ENPV (expanded NPV) is the NPV of the project plus the value of the switch option.

Additional Remarks

I constructed the event tree from the project’s contribution cash flow volatilities out of Monte-Carlo-Simulations. Of course you can also create the nodes of the event trees by calculating the time values of the outstanding contribution cash flows explicitly. Discussion with management and marketing can provide the required cash flows and probabilities of the event tree.

This switch option valuation approach can be applied to many problems with different operation possibilities. Switch options are compound options with path-dependency. They are a good examples that real options can be more complicated than financial options. Analytical valuation methods like Black-Scholes-Merton cannot give you a solutions for problems like that.

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