DCF Analysis - Strategic Investments

26/07/202106/10/2025

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 interest rate.
One time payments and expenses should be discounted with the interest rate.
Fixed costs should be discounted with the interest rate. 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 interest 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 interest 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.

02/04/202124/10/2023

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:

$VU+VTS=E+D$

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

$r_A\frac{VU}{VU+VTS}+r_{TS}\frac{VTS}{VU+VTS}=r_E\frac{E}{E+D}+r_D\frac{D}{E+D}$

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

$r_E=r_A+\left(r_A-r_D\right)\frac{D}{E}-\left(r_A-r_{TS}\right)\frac{VTS}{E}$

Consequently the general form of CAPM beta is given by:

$\beta_E=\beta_A+\left(\beta_A-\beta_D\right)\frac{D}{E}-\left(\beta_A-\beta_{TS}\right)\frac{VTS}{E}$

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:

$WACC=r_E\frac{E}{E+D}+r_D\left(1-T_C\right)\frac{D}{E+D}$

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

$WACC=r_A\left(1-\frac{VTS}{V}\right)-r_DT_C\frac{D}{V}+r_{TS}\frac{VTS}{V}$

$r_A$ , $r_D$ , $r_{TS}$ 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:

$VTS=\sum_{j=1}^{\infty}{DT_C\left(\frac{1}{1+r_D}\right)}^j=\frac{DT_C}{r_D}$

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

$r_E=r_A+\left(r_A-r_D\right)\frac{D}{E}\left(1-T_C\right)$

$\beta_E=\beta_A+\left(\beta_A-\beta_D\right)\frac{D}{E}\left(1-T_C\right)$

$WACC=r_A\left(1-T_C\frac{D}{E+D}\right)$

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:

$\beta_E=\beta_A\left(1+\frac{D}{E}\left(1-T_C\right)\right)$

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 $r_{TS}=r_A$ .

$r_E=r_A+\left(r_A-r_D\right)\frac{D}{E}$

$\beta_E=\beta_A+\left(\beta_A-\beta_D\right)\frac{D}{E}$

$WACC=r_A-r_DT_C\frac{D}{E+D}$

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:

${VTS}^{ME}=\frac{Dr_DT_C\left(1+r_A\right)}{\left(r_A-g\right)\left(1+r_D\right)}$

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 $VTS$ :

${VTS}^{HP}=\frac{Dr_DT_C}{\left(r_A-g\right)}$

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.

05/03/202125/10/2023

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.

Conclusion

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.

03/03/202120/12/2023

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:

$\beta_{rev}=\beta_{asset}\frac{V_{asset}}{V_{rev}}-\beta_{var}\frac{V_{var}}{V_{rev}}-\beta_{fix}\frac{V_{fix}}{V_{rev}}$

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 $\beta_{var}$ .

$\beta_{asset}=\beta_{rev}\frac{V_{rev}+V_{var}}{V_{asset}}$

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)

20/02/202124/10/2023

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:

$r_{E}=\frac{r-Lr_D}{1-L}$

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.

09/02/202117/11/2023

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.