New Overall Required Rate of Return for Arizona Rock

C

b(old, firm) = 1.25.

k(old, firm) = 0.07 + (14 - 7)1.25 = 15.75%.

b(new, firm) = 0.9(1.25) + 0.1(1.1) = 1.235.

k(new, firm) = 0.07 + 1.235(0.07) = 15.645%.

k(new, assets) = 0.07 + 1.1(0.07) = 14.7%.

To calculate the new overall required rate of return for Arizona Rock, we need to use the formula for the weighted average cost of capital (WACC). The WACC is the average rate of return required by all of the firm's investors, taking into account both the cost of equity and the cost of debt (if applicable). However, in this case, it is mentioned that Arizona Rock is an all-equity firm, so we can focus on the cost of equity.

The formula for the cost of equity using the capital asset pricing model (CAPM) is: k(E) = k(RF) + β(E) * [k(M) - k(RF)]

Where: k(E) is the cost of equity k(RF) is the risk-free rate of return β(E) is the beta of the equity k(M) is the expected market rate of return

Given that Arizona Rock's current beta is 1.25, k(RF) is 7%, and k(M) is 14%, we can calculate the current cost of equity:

k(E) = 0.07 + 1.25 * (0.14 - 0.07) = 0.07 + 1.25 * 0.07 = 0.07 + 0.0875 = 0.1575 or 15.75%

Next, let's calculate the new overall beta of the firm after selling 10% of its assets and purchasing new assets. To do this, we need to find the weighted average beta using the proportion of assets before and after the transaction.

Before the transaction: Proportion of old assets = 1 (100%) Beta of old assets = 1.25

After the transaction: Proportion of new assets = 0.1 (10%) Beta of new assets = 1.1

Weighted average beta = (Proportion of old assets * Beta of old assets) + (Proportion of new assets * Beta of new assets) = (1 * 1.25) + (0.1 * 1.1) = 1.25 + 0.11 = 1.36

Now, we can calculate the new overall required rate of return using the updated beta:

k(E) = k(RF) + β(E) * [k(M) - k(RF)] = 0.07 + 1.36 * (0.14 - 0.07) = 0.07 + 1.36 * 0.07 = 0.07 + 0.0952 = 0.1652 or 16.52%

Therefore, the new overall required rate of return for Arizona Rock is 16.52%.

To find the rate of return that the new assets must produce in order to leave the stock price unchanged, we can use the concept of the capital asset pricing model (CAPM). The stock price remains unchanged when the expected return on the stock matches the required rate of return.

Using the CAPM formula, we can calculate the rate of return required for the new assets:

k(E) = k(RF) + β(E) * [k(M) - k(RF)]

Since the stock price is unchanged, the cost of equity for the new assets must be the same as the new overall required rate of return, which is 16.52%. Therefore:

16.52% = 0.07 + β(E) * (0.14 - 0.07)

Solving for β(E):