Question

A firm has 5,000,000 shares of common stock outstanding, each with a market price of $10.00 per share. It has 55,000 bonds outstanding, each selling for $990 with a $1,000 face value. The bonds mature in 15 years, have a coupon rate of 8 percent, and pay coupons semiannually. The firm's equity has a beta of 2.0, and the expected market return is 15 percent. The tax rate is 21 percent and the WACC is 16 percent. Calculate the risk-free rate. Multiple Choice 1.79 percent 2.12 percent 3.56 percent 27.68 percent

Answer 1

Answer:
Debt:
Number of bonds outstanding = 55,000
Face Value = $1,000
Current Price = $990

Market Value of Debt = 55,000 * $990
Market Value of Debt = $54,450,000

Annual Coupon Rate = 8%
Semiannual Coupon Rate = 4%
Semiannual Coupon = 4% * $1,000 = $40

Time to Maturity = 15 years
Semiannual Period to Maturity = 30

Let semiannual YTM be i%

$990 = $40 * PVIFA(i%, 30) + $1,000 * PVIF(i%, 30)

Using financial calculator:
N = 30
PV = -990
PMT = 40
FV = 1000

I = 4.06%

Semiannual YTM = 4.06%
Annual YTM = 2 * 4.06%
Annual YTM = 8.12%

Before-tax Cost of Debt = 8.12%
After-tax Cost of Debt = 8.12% * (1 - 0.21)
After-tax Cost of Debt = 6.41%

Equity:

Number of shares outstanding = 5,000,000
Current Price = $10

Market Value of Common Stock = 5,000,000 * $10
Market Value of Common Stock = $50,000,000

Market Value of Firm = Market Value of Debt + Market Value of Common Stock
Value of Firm = $54,450,000 + $50,000,000
Value of Firm = $104,450,000

Weight of Debt = $54,450,000 / $104,450,000
Weight of Debt = 0.5213

Weight of Common Stock = $50,000,000 / $104,450,000
Weight of Common Stock = 0.4787

Let the Cost of Common Stock be “x%”

WACC = Weight of Debt*After-tax Cost of Debt + Weight of Common Stock*Cost of Common Stock
16% = (0.5213 * 6.41%) + (0.4787 * x%)
16% = 3.3415% + 0.4787x%
12.6585% = 0.4787x%
x% = 26.44%

Cost of Common Equity = Risk-free Rate + Beta * Market Risk Premium
26.44% = Risk Free Rate + 2.0 * (15% - Risk Free Rate)
26.44% = Risk Free Rate + 30% - 2.0 * Risk Free Rate
Risk Free Rate = 3.56%