Question

Every student loves the Nobel-prize-winning theory of capital structure irrelevance in a world of

perfect competition without taxes or private information (and no transactions costs), formulated by
Modigliani and Miller, or M&M for short. Suppose there are no personal or corporate taxes and no
transactions costs nor private information. In this perfect world, LGB and FJB are crypto issuers, Bitcoin
competitors, not yet bankrupt like FTX, identical in all respects, except that LGB is unlevered and FJB has
$10,000,000 of 10% bonds outstanding. Assume that EBIT is $3,000,000 for both, and that the cost of
equity to LGB is 15% and the cost of equity to FJB is 20%. Next assume that LGB = $22,000,000 and FJB =
$20,000,000. Using the M&M propositions, demonstrate how to make a riskless profit from LGB and FJB
without investing a penny of one’s own wealth and yes, LGB is the famous “Let’s Go Brandon” crypto
currency whose first LGB coin was issued in 2021.


b. Now in the same world without personal or corporate taxes and no transactions costs, assume
FJB’s debt is risk-free, so that the cost of debt RD equals the risk-free rate Rf. Define βA as FJB’s asset beta
– that is, the systematic risk of the firm’s assets. Define βE to be the beta of FJB’s equity. Use the CAPM
along with M&M’s Proposition II to find the relationship between FJB’s βA and βE.

Answer 1

Using the CAPM and M&M's Proposition II, the relationship between FJB's asset beta (βA) and equity beta (βE) can be determined. βE is equal to βA divided by the ratio of equity to the total value of the firm (E/V).

Step-by-step explanation:

The relationship between FJB's asset beta (βA) and equity beta (βE) can be determined using the Capital Asset Pricing Model (CAPM) and Modigliani and Miller's Proposition II.

The CAPM formula is:

RE = Rf + βE(RM - Rf)

Where:

• RE is the expected return on equity
• Rf is the risk-free rate
• βE is the equity beta
• RM is the expected return on the market

M&M's Proposition II states that the cost of equity (RE) is equal to the cost of unlevered equity (R0) plus the leverage premium (RP):

RE = R0 + RP

If FJB's debt is risk-free, the cost of debt (RD) is equal to the risk-free rate (Rf). Therefore, the asset beta (βA) is equal to the equity beta (βE) times the ratio of equity to the total value of the firm (E/V):

βA = βE * (E/V)

Combining these equations, we can find the relationship between βA and βE as follows:

βE = βA / (E/V)