Joachim Kuczynski, 04 February 2023

## Introduction

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**.

## Derivation

Base for the derivation is the market balance sheet where the present value of the asset (project or investment), , is the sum of present value of the contribution cash flows, , and the present value of the fixed cash flows, :

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

The fixed cash flows are independent of the market return rate. That means that the derivation and we get:

The relative change of the conribution cash flow present value in respect to the market portfolio return rate is just the beta or the contribution cash flows. Taking that into account and rearranging the equation leads us to:

Note that the present value of the fixed cash flows is negative, , and the present value of the contribution cash flows is positive, , in ordinary cases. That means that the expression in the bracket becomes . That means that the beta of the contribution cash flows is smaller than the asset’s beta, . Obviously that is true, because adding fixed costs increases the beta of the asset. this is known as operating leverage.

## Application in CCFP

The bottom up beta from an industry segment includes fixed costs, it corresponds to . Applying the previous derivation you can calculate the beta of the contribution cash flows, . and 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 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 as risk free rate and as average return rate of the industry segment, the equity return rate is:

The return rate for contribution cash flows considering debt and equity financing including tax shield effect with constant debt to equity ratio and marginal tax rate is:

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.