Where did this formula come from. Derivation.
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Just as we can express the beta of any investment portfolio as a market-value weighted average of the betas of the investments in that portfolio, we can express
the systematic risk of each of the sources of a company’s capital in a similar manner
using the Hamada equation . In other words, we can represent
the systematic risk of the assets of the entire company as a weighted average of the
systematic risk of the company’s debt and equity:
Beta asset = D/V(total value)* beta d + equity/V (total value)*Beta equity
According to Modigliani and Miller, the company’s cost of capital does not depend
on its capital structure but rather is determined by the business risk of the company.
As the level of debt rises, however, the risk of the company defaulting on its debt
increases. These costs are borne by the equityholders. So, as the proportionate use
of debt rises, the equity’s beta, βe, also rises. By reordering the formula of βa to solve
for βe, we get –
beta equity = beta asset + (beta asset – beta debt)D/E
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