Now suppose the unlevered beta for the other division is 0.9 instead of 0.6, and the risk-free rate is 5.3% instead of 3%. What is your answer to the previous question in this case?

Write your answer in percentage point with one digit after the decimal. For example, if your answer is 6.7%, just write 6.7

Question

Now suppose the unlevered beta for the other division is 0.9 instead of 0.6, and the risk-free rate is 5.3% instead of 3%. What is your answer to the previous question in this case?

Write your answer in percentage point with one digit after the decimal. For example, if your answer is 6.7%, just write 6.7

BlackTom AI · Accepted Answer

To approach the adjusted scenario, start by identifying which quantities are affected and which inputs influence them.

First, note that the question changes two inputs: the division’s unlevered beta (Beta_U) rises from 0.6 to 0.9, and the risk-free rate (Rf) rises from 3% to 5.3%. The central task is to determine the impact on the required return or the percentage output expected in the previous setup, which typically relies on a levered beta and the cost of capital formula.

Second, recall the core relationships often used in these problems. In many corporate finance and CAPM-based exercises, the levered beta Beta_L reflects the financial structure (debt and equity) via Beta_L = Beta_U × [1 + (1 − T) × (D/E)], where T is the tax rate and D/E is the debt-to-equity ratio. While the exact debt mix may not be specified here, the key takeaway is that increasing Beta_U raises Beta_L, all else equal, which in turn raises the equity expected return via CAPM: Re = Rf + Beta_L × ERP, where ERP is the equity market risk premium (the extra return over the risk-free rate that investors require for taking on market risk).

Third, consider the impact of a higher risk-free rate. Even if Beta_L remained the same, boosting Rf directly increases Re (Cost of equity) by the amount of the Rf increase, since Re = Rf + Beta_L × ERP. Therefore, with Rf rising from 3% to 5.3%, you would expect Re to be higher, all else equal.

Now, step through the two main effects separately and then consider them together:
- Effect of increasing Beta_U to 0.9: A higher unlevered beta implies a higher levered beta after accounting for any leverage. Since levered beta scales with Beta_U, you would anticipate a higher Beta_L, which raises Re through the Beta_L × ERP term. This pushes the required return upward, assuming a positive market risk premium.
- Effect of increasing Rf to 5.3%: This directly raises Re by (new Rf − old Rf), independent of Beta_L changes, reinforcing the upward pressure on the required return.

Fourth, when combining both effects, the net change in the required return depends on the magnitude of ERP and the leverage adjustment. If the levered beta increase is substantial (due to a larger Beta_U and any present leverage), the product Beta_L × ERP would increase notably, compounding the increase from a higher Rf. Conversely, if the debt/equity structure or tax shield altered Beta_L only modestly, the dominant change could still be the higher risk-free rate.

Fifth, to perform a precise numeric update, you would typically:
- Recompute Beta_L using Beta_U = 0.9 and the given or assumed leverage parameters (D/E and tax rate T).
- Recalculate the cost of equity via Re = Rf + Beta_L × ERP, with Rf = 5.3% and ERP taken from the market data used in the original problem.
- If the original question involved a weighted average cost of capital (WACC) or a project-specific hurdle rate, apply the updated Re (and any revised after-tax cost of debt if your framework uses WACC) to obtain the new required rate in percentage points.

Sixth, given that the original numeric answer in the prior setup was 8.8 percentage points and you now have both a higher Beta_U (0.9) and a higher Rf (5.3%), the trend would be up (i.e., the required rate should increase rather than decrease). The exact new value hinges on ERP and the leverage structure, but the direction is clearly upward due to both inputs moving higher.

In short, with Beta_U rising to 0.9 and Rf increasing to 5.3%, you should expect the calculated required return to be higher than in the original scenario, reflecting both a larger systematic risk and a higher baseline risk-free return. To finalize the exact numerical result, substitute the updated Beta_L into the cost of equity formula Re = 5.3% + Beta_L × ERP, using the ERP from your prior context and the project’s capital structure.

SP2025.B62.FIN.534 Quiz 4 (Lecture 4)

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