Reading: Estimating the Cost of Capital

Before, we have argued that Amazon's investors require an average return of 8.53% per year. But who exactly are these investors?

Because we have estimated Amazon's beta based on the firm's historical equity returns, the resulting risk measure applies to the firm's equity! Consequently, the return of 8.53% per year is the average return the shareholders expect to earn. It is the so-called cost of equity. If we are interested in the firm's overall cost of capital, we also have to incorporate the return expectations of other providers of capital, most notably the debtholders.

Most often, we are interested in the valuation of the whole firm or project, not only their equity component. Consequently, we need to find a discount rate that is representative of the average return required by the overall providers capital, not only the shareholders. The following generic balance sheet at market values summarizes the situation. Let us briefly explain the terminology. For simplicity, we use the term "firm." That could obviously be a firm with a single project.

• The green element to the left depicts the total market value of the firm.
     
  • The firm's business activities are not risk-free.
     
  • Hence, the firm needs to earn a certain return on its assets (at market value).
     
  • This required return on the assets is what we call the overall cost of capital or cost of assets, k_A.
     
    
• The assets are typically financed with two sources of capital, debt (red element to the right) and equity (blue element to the right).
   
  • The debtholders require a certain return for the systematic risk associated with their investment. That's the cost of debt, k_D.
     
    
  • Similarly, the shareholder want to be compensated for the systematic risk of their investment. That's the cost of equity, k_E. 

So far, we have learned how to compute the cost of equity, k_E, using the CAPM. However, as we have mentioned above, we are generally interested in the overall cost of capital, k_A. How can we now get from k_E to k_A?

Here is what we need to do:

• Remember that the above graph represents a balance sheet. If there is one thing that we know about balance sheets, it is that they should be balanced: Assets = Debt + Equity.
   
• Put differently, the firm can only distribute to its debt and equity holders what it generates on its assets. The total (dollar) return on the assets must therefore correspond to the total (dollar) return on the debt and equity:
   
  \( k_A \times A = k_D \times D + k_E \times E \)
   
• When we solve the above expression for the overall cost of capital, k_A, we get (remember that A = D + E):
   
  \( k_A = k_D \times \frac{D}{D+E} + k_E \times \frac{E}{D+E} \)
   
• This is the expression that we have been looking for! To obtain the overall cost of capital, we need to compute the weighted average of the firm's cost of debt and cost of equity.
    

Example

To practice the calculation of the overall cost of assets, let us go back to the case of Amazon. We make a few additional assumptions:

• Debt financing: 10% of firm value are financed with debt. Consequently, equity accounts for 90% of the balance sheet at market value:  \(\frac{D}{D+E} = 0.1 \implies \frac{E}{D+E} = 0.9\)
    
• Cost of debt (k_D): Let's assume that Amazon can borrow at a rate of 3.20%. In February 2019, this was the annual yield that investors could expect to earn on Amazon's bonds. Consequently, we assume that \(k_D = 0.032 = 3.2\%\).

Together with the estimated cost of equity \(k_E\) of 8.53% from before, we can now compute Amazon's overall cost of capital:

\(k_{A,Amazon}=k_D \times \frac{D}{D+E} + k_E \times \frac{E}{D+E} \) \(= 0.032 \times 0.1 + 0.0853 \times 0.9 = 0.0797 = 8\% \)

The result is an overall cost of capital (\(k_A\)) of approximately 8%. Since the firm is mostly equity financed, it is not surprising that this overall cost of capital is close to the estimated cost of equity.

This overall cost of capital of approximately 8% is the appropriate discount rate to value a project with the typical risk of Amazon. 

It is also the discount rate that we could use to value a project of a company that is not publicly traded (so we have no data to estimate its equity beta) but has similar risk as Amazon. We will discuss such a case in the next course section.

Before doing so, we have to address one final key issue in the estimation of the cost of capital: Taxes.