Operating Leverage

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

cash flow = revenues – fixed expenses – variable expenses

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

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

Rearranging leads us to:

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

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

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

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

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


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

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

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

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

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

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

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

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 DCF 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 common errors in project appraisal and suggest ways of avoiding them. For many people DCF analysis seems to be quite easy but it is not 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 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 reqired 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 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 WACC 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 Option

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.

CRR Binomial Method

Das Binomialmodell von Cox-Ross-Rubinstein aus dem Jahr 1979 ist der Grundstein für die klassische Optionsbewertung von Finanztiteln. Aber auch für die Bewertung von Realoptionen ist es das zentrale Modell, mit welchem flexibel unterschiedliche Optionsarten simultan analysiert werden können.


Consent Management Platform by Real Cookie Banner