Replication Portfolio in Binomial Model

In this post I a want to give a short derivation of the replication portfolio and the risk neutral probabilities in the binomial model from Cox-Ross-Rubinstein. Let V be the value of the underlying asset in t_0. In a project or investment this might be the present value of the project’s contribution (market related) cash flows. The positive development of V at time t_1, V^+, occurs with probability p, the negative development with value V^- in t_1 with probability 1-p. The twin security of the underlying in the open market takes a similar notation S, S^+, S^-,p. We consider an option with option value E in t_0 that leads to an option value of E^+ in the upper state V^+ and to an option value of E^- in the lower state V^-. As result we are searching the option value E at time t=0.

Next we replicate the option value in t_1 by a portfolio of n shares of twin security S partly financed by borrowings of amount B at the risk-free rate r. The values of the upper and lower state in t_1 are E^+=nS^+-\left(1+r\right)B and E^-=nS^--\left(1+r\right)B.

In efficient markets there exist no profitable arbitarge opportunities. Therefore the outcome of the option value E in t=1 must be the same in the upper and in the lower state. Setting E^+-nS^+=E^--nS^- we get:


Replacing n in the previous equations we obtain the value borrowed at the risk-free rate r:


The law of one price tells us that the value of assets that lead to the same cash flows must be the same. That means that the value of the option at time t_0 must be the same as the value of the portfolio at time t_0. Therefore we can set E=nS-B. With that we calculate the value of the option in t_0:


We create a new variable p^\prime to simplify the previous expression.


Hence we obtain:

    \[E=\frac{p\prime E^++\left(1-p\prime\right)E^-}{1+r}\]

p^\prime can be interpreted as probability for E^+, 1-p^\prime for E^-. p^\prime and 1-p^\prime are known as risk-neutral probabilities. Note that the value of the option does not explicitly involve the actual probabilities p and 1-p of the underlying. Instead, it is expressed in terms of risk-neutral probabilities. They allow to discount the expected future values at the risk-free rate.

Discounting at the risk-free rate is the main difference between decision tree analysis (DTA) and contingent claim analysis (CCA) or real options analysis (ROA). DTA does not take into account that the risk of the cash flow streams changes when you consider options and opportunities. ROA implements this issue correctly.

WACC, Return Rates & Betas with Debt

Joachim Kuczynski, 02 April 2021

In this post I want to summarize some interesting results concerning equity return rates , betas and WACC of a levered company. Regarding the market value balance sheet of a firm we can state that the value of the unlevered firm VU plus the present value of the tax shield VTS must be the same as the sum of levered equity E and debt D:


Further the rates of return on each side of the balance sheet are the weighted average of the component rates of return:


Substituting VU in the rate of return expression we get a general form of the equity return rate:


Consequently the general form of CAPM beta is given by:


The WACC is defined as the weighted average of equity and debt return rates including tax shield at corporate income tax rate T_C. If the tax shield savings are proportional to the taxes paid (see WACC with Tax Shield), the WACC is given by:


Substituting the equity return rate we get a general form of the WACC:


r_A, r_D, r_{TS} are the return rates of the unlevered asset, debt and tax shield. V and VTS are the values of the levered firm and the tax shield. D is the amount of debt, V is the value of the levered firm, namely the sum of equity and debt.

Modigliani and Miller: Constant debt value
If the firm keeps its dept value D constant, there are no specific market risks concerning the tax shield. Therefore we can set the tax shield discount rate r_{TS} equal to the debt discount rate, r_D. The tax shield present value with constant debt D is:


Hence we get simplified expressions for equity return, equity beta and WACC:




Assuming that debt interest rate does not depend on the market return rate (CAPM) we can set \beta_D=0. Hence we get the well-known Hamada equation for levered beta:


It is important to realize that Hamada’s equation is only valid if the value of debt is kept constant over time.

Harris and Pringle: Constant leverage ratio
Constant leverage ratio means that debt value is proportional to the value of the unlevered firm. According to Harris and Pringle that results in r_{TS}=r_A.




But we have to take care. Miles and Ezzell, Arzac and Glosten have shown that you have a tax shield discount rate of r_D in the first period, and of r_A in the following periods to have a constant leverage ratio over time. The premise of r_{TS}=r_A does not hold.

Miles and Ezzell
With a perpetuity growing rate g of debt and discounting in the first period with r_D instead of r_A we obtain:


Harris and Pringle
Taking the formula of Miles and Ezzell and setting r_TS equal to r_A in the first period, we get the a simplified expression for VTS:


General debt ratio
If the amount of leverage is flexible and not constant or growing with a constant growth rate over time, the previous formulas do not work. In this case you have to use the APV method, in which you calculate the tax shield in each time period seperately.

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.

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 of the asset, V_{asset}, is the sum of its cash flows’ (present) values, namely revenue V_{rev}, variable costs V_{var} and fixed costs, V_{fix}. Present values are linear, for the value of the asset we can write:

    \[V_{asset} = V_{rev} + V_{fix} + V_{var}\]

Rearranging leads us to:

    \[V_{rev} = V_{asset} - V_{var} - V_{fix}\]

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 the revenues is a weighted average of the betas of its component parts:


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_{fix} = 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_{rev} for \beta_{var}.


Setting V_{rev} + V_{var} = V_{asset} - V_{fix} we obtain:

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

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:

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

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)

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.

Pitfalls of Discounted Cash Flow Analysis

Correctly appraising capital projects with DCF analysis methods requires knowledge, practice and acute awareness of potentially serious pitfalls. I want to point out some important errors in project appraisal and suggest ways to avoid them. For many people DCF analysis seems to be quite easy, but it can be very difficult for complex projects. Here are some crucial issues from my point of view:

  • Decision focus: The calculation is focused on making the right decision concerning a project or an investment. That can be different from a calculation including all expenditures of the project or investment, e.g. sunk costs. For further comments concering this topic see Incremental Free Cash Flows.
  • Point of view: It has to be defined clearly from which perspective you are doing the decision and calculation. For example, the calculation can be different from the view of a business area and from the view of the overall company. The right perspective to the decision problem determines the relevant incremental cash flows.
  • Investment: Define clearly what you mean when talking about “investment”. Avoid the balance sheet view, look at investment as initial expenditures required for later contribution cash flows. In my point of view the term “investment” is best defined as commitments of resources made in the hope of realizing benefits that are expected to occur over a reasonably long period of time in the future.
  • Cash flows: A clear view of cash flow is important, avoid views from accounting and cost accounting, e.g. depreciation. And take into account tax effects.
  • Incremental cash flows: The correct definition of incremental cash flow is crucial. It is the difference between the relevant expected after-tax cash flows associated with two mutually exclusive scenarios: (1) the project goes ahead, and (2) the project does not go ahead (zero scenario). Sunk costs must not considered. For further comments see Incremental Free Cash Flows.
  • Comparing scenarios: Alway be aware of having a relative sight between the cash flow scenarios. Sometimes it is not so easy to define what would happen in the future without the project (zero scenario).
  • Risk-adjusted discount rates: Risk adjustment of discount rates has to be done for all (!) cash flows of the investment project that have significant risk differences: Fixed costs, investment expenses, one time expenses and payments, expenses for working capital, leasing, tax shields and contribution cash flows (turnover and variable costs) in various markets. For more infos concerning risk adjusted discount rates see Component Cash Flow Procedure.
  • Key figures: The only key figure that is valid for all types of projects and investment decision is the famous NPV. All other well-known figures like IRR, Baldwin rate, … are leading to false decisions in some cases. NPV also allows to build the bridge to financial calculation approaches like option valuation. Payback and liquidity requirements have to be considered carefully additionally to NPV.
  • Expected versus most likely cash flows: Quite often analysts take most likely cash flows. The right way is to consider the expected value (mathematical definition) of the cash flows.
  • Limited capacity: Do not forget internal capacity limitation when regarding market figures. Limited capacity has also to be considered when constructing the event tree in real options analysis. Besides that the temporal project value development with contribution cash flow’s discount rate has to be ensured in the binomial tree.
  • Hurlde rates: Avoid hurlde rates for project decisions, because the can also lead to false decisions. Especially when you take one hurdle rate for different projects.
  • Cash flow forecasting: Forecasts are often untruthful. Try to verify and countermeasure cash flows from different sources.
  • Inflation: Be careful considering inflation. In multinational project it might influence the foreign currency location’s required return. You can also consider a relationship between inflation rate and expected future exchange rates according to the purchasing power parity (PPP).
  • Real and nominal discount rates and cash flows: The procedure should be consistent for cash flows and discount rates. Usually we take nominal values for the calculation.
  • Real options: A DCF analysis should always be linked to a real options analysis. The more flexibility is in the project the more important is a real options analysis. Risk adds value to real options.
  • Precise cash flow timing: The influence of timing intervals can be significant. You can choose smaller time intervals in crucial time periods to increase accuracy.

Option to Wait

This is a simple example of an option to wait. We consider a 15 year project which requires an investment of 105 M€, that can be done anytime. Arbitrage Pricing Theory provides a yearly risk-adjusted capital discount rate (WACC) of 15%. Investment and internal risk cash flows are discounted by the risk-free rate. We assume for all years equal free net cash flow present values of 100/15 M€. Classical incremental cash flow analysis provides a present value of the market-related net cash flows of 100 M€. That means that the classic NPV of the project is -5 M€. Because of the negative NPV management should reject the project.

But management has an option to wait. It can wait with the decision and invest only if the market development is profitable. For sure the company loses revenues because of the delayed investment, but on the other hand management gets more information about market development. The question is: What is the value of this option to wait and how long should management wait with that investment decision? Can the project become profitable?

Monte-Carlo-Simulation of the project provides a project volatility of 30%. The risk-free rate is 5%. Next we are performing a real option analysis (ROA) of the waiting option with the binomial approach regarding 15 time steps, one for each year.

Real option analysis provides a project value of 21 M€. That means that the value added by the waiting option is 26 M€. Because the project value with waiting option is positive we should not reject the project any more. Management should go on with the project. Including the option in the project valuation leads to the opposite management decision. And besides the waiting option there might be additional options like the option to abandon or the option to expand/contract. They would bring additional value to the project.

Real option analysis also provides the information that there should be no investment done before the second year. Dependent from the market development management can decide when to invest according to the time value of the expected free cash flows.

In this example we assumed yearly cash flows that results in a decrease of the expected future cash flows. This corresponds to paying dividends at financial securities. Considering options in the lifetime of a project requires binomial valuation with leakage. If you assume relative leakage you get a recombining tree, with absolute leakage values you get a non-recombining binomial tree.

Sequential Compound Options

This is a simple example of a sequential compound option, which is typical in projects where the investment can be done in sequential steps. The option is valued by the binomial approach.

The project is divided into three sequential phases: (1) Land acquisition and permitting, (2) design and engineering and (3) construction. Each phase must be completed before the next phase can start. The company wants to bring the product to market in no more than seven years.
The construction will take two years to complete, and hence the company has a maximum of five years to decide whether to invest in the construction. The design and engineering phase will take two years to complete. Design and engineering has to be finished sucessfully before starting construction. Hence the company has a maximum of three years to decide whether to invest in the design and engineering phase. The land acquisition and permitting process will take two years to complete, and since it must be completed before the design phase can begin, the company has a maximum of one year from today to decide on the first phase.
Investments: Permitting is expected to cost 30 million Euro, design 90 million Euro, and construction another 210 million Euro.
Discounted cash flow analysis using an appropriate risk-adjusted discount rate values the plant, if it existed today, at 250 million Euro. The annual volatility of the logarithmic returns for the future cash flows for the plant is evaluated by a Monte-Carlo-Simulation to be 30%. The continuous annual risk-free interest rate over the next five years is 6%.
Static NPV approach: The private risk discount rate for investment is 9%. With that we get a NPV without any flexibility and option analysis of minus 2 million Euro. Because of its negative NPV we would reject the project neglecting any option flexibility.
ROA: Considering the real options mentioned above we calculate a positive project value of 41 million Euro. That means that the compound options give an additional real option value (ROV) of 43 million Euro. Thus we should implement the project.

Binomial valuation tree of a sequential compound option

The real option analysis additionally provides the information when and under which market development to invest in each phase. The investment for the first phase should be done in year 1, for the second phase in year 3 and for the third phase in year 5. The option valuation tree tells management what to do in which market development.
The valuation can be done in smaller time steps to increase accuracy. But the purpose of this example is to illustrate the principle of a sequential compound option valuation.
Details are specified in Kodukula (2006), p. 146 – 156.

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