1. Consider the following information on expected and historical returns for the S&P 500 index, a 3-month treasury bill and Ateb Corporation:

Note - the return on Ateb in 1999 is now -5%

Year  R(Ateb)    R(S&P500)    T-bill
2001     5%             20%             3%
2000   15%             18%             3%
1999    -5%             23%             3%
1998   50%             20%             3%
1997     5%             10%             3%

- What is the beta for Ateb Corporation based on its historical data?

Beta is calculated as the slope of a regression line of the returns on Ateb on the returns on the market portfolio. The definition of the slope coefficient is the covariance between the returns on Ateb and the market, divided by the variance of returns on the the market.

From the above table we have the following:

E(R_Ateb) = 14%

E(R_S&P500) = 18.2%

E(R_f) = R_f =3%

COV(R_Ateb,R_S&P500)=0.0031 (Calculated as the sum over the 5 years of [(R_Ateb - E(R_Ateb)] * [(R_S&P500 - E(R_S&P500)]

VAR(R_S&P500)=0.00968 (Calculated as the sum over the 5 years of [(R_S&P500 - E(R_S&P500)]²

^{Beta = 0.0031 / 0.00968 = 0.320248}

- What is the expected return for Ateb Corporation?

According to the CAPM: R_Ateb = 3% + 0.320248 * [18.2% - 3%] = 7.87%

- What discount rate should Ateb use if it uses all-equity financing for projects with a similar risk profile as the whole firm?

If Ateb is 100% equity financed, the R_E = R₀ = R_WACC, and hence, the firm should use 7.87% as the discount rate.R₀ is the return on a 100% euity firm.

- What discount rate should Ateb use if it decides to move to a debt-to-equity ratio of 0.5 for projects with similar risk as the whole firm? Assume the beta of debt to be equal to zero.

With no corporate taxes and according to Miller and Modigliani Proposition I, we have that V_U = V_L, and hence that R_WACC is independent of leverage. Hence, the firm should still use a 7.87% discount rate.

We can verify this statement by looking at the new weighted average cost of capital. The firm has R_D=3% (risk-free), and R₀=7.87%. From MM Proposition II, we have R_E = R₀ + (D/E) * (R₀ - R_D)=10.30%. The capital structure weights D/V and E/V are respectively, 1/3 and 2/3, hence R_WACC = 1/3 * 3% + 2/3 * 10.30% = 7.87%.

- How would your answer change given a 30% corporate tax rate?

MM Proposition II with Corporate Tax: R_E = R₀ + (D/E) * (1 - T_c) * (R₀-R_D)=9.57%. The capital structure weights D/V and E/V are respectively, 1/3 and 2/3, hence R_WACC = 1/3 * 0.7 * 3% + 2/3 * 9.57% = 7.08%. The decrease in the weighted average cost of capital reflects the increase in the firm value (MM Proposition I with Corporate Taxes: V_U + Present Value of the Interest Tax Shields = V_L) because of the value of the tax shields.

- If after Ateb reaches its target debt-to-equity ratio, they pay 16% on newly issued debt (before-tax),  what is the weighted-average cost of capital for Ateb (Rwacc)? What does this imply for the after-tax beta of debt?

With Debt this risky, we can't use the MM propositions, because it is assumed that debt is less risky than the unlevered equity. While some of the firm's debt might be this risky - it seems strange if the overall cost of debt is this high to even use any leverage. If debt has for example a more reasonable yield-to-maturity of 5%, we would have to recalculate the return on equity using Proposition II and then recalculate the weighted-average cost of capital. To find the beta on debt, we could solve for the CAPM equation, where we have R_D = R_f + Beta * (R_M - R_F).