Problem

Calculating the WACC In the previous problem, suppose the company’s stock has a beta of 1....

Calculating the WACC In the previous problem, suppose the company’s stock has a beta of 1.2. The risk-free rate is 5.2 percent, and the market risk premium is 7 percent. Assume that the overall cost of debt is the weighted average implied by the two outstanding debt issues. Both bonds make semiannual payments. The tax rate is 35 percent. What is the company’s WACC?

Step-by-Step Solution

Solution 1

\(\begin{array}{ll}\text { Number of Shares outstanding } & =7,500,000 \\ \text { Current Share price } & =\$ 49 \\ \text { Book value per Share } & =\$ 4\end{array}\)

First issue:

\(\begin{array}{ll}\text { Bond face value } & =\$ 60,000,000 \\ \text { Coupon rate } & =7 \% \\ \text { Selling price } & =93 \% \text { of Par } \\ \text { Maturity } & =10 \text { Years }\end{array}\)

Second issue:

\(\begin{array}{ll}\text { Bond face value } & =\$ 50,000,000 \\ \text { Coupon rate } & =6.5 \% \\ \text { Selling price } & =96.5 \% \text { of Par } \\ \text { Maturity } & =6 \text { Years } \\ \text { Beta } & =1.2 \\ \text { Risk-free rate } & =5.2 \% \\ \text { Market risk premium } & =7 \% \\ \text { Tax rate } & =35 \%\end{array}\)

of equity, cost of debt is very important. Therefore, first we will calculate cost of equity, and then we can compute the cost of debt. We can use CAPM method to find out the cost of equity.

Step 1:

$$ \begin{aligned} &\text { Cost of equity = Risk-free rate }+\text { Beta (Risk premium) } \\ &\begin{aligned} R_{E} &=5.2 \%+1.2(7 \%) \\ &=5.2 \%+8.4 \% \\ &=13.6 \% \end{aligned} \end{aligned} $$

Step 2: From the Given information, the face value of first bond is at \(93 \%\) that is \(\$ 1,000 \times 93 \%=\$ 930\). Interest is at \(7 \%\) and paid in semi annual basis.

$$ \begin{aligned} \text { Interest } &=7 \% \times \$ 1,000 \\ &=\$ 70 \\ &=\$ 70 / 2 \\ &=\$ 35 \end{aligned} $$

Since it is on semi annual the number of years will be multiplied by 2 that is 10 years \(\times 2=20\) years . As per Table A-2 we can find out the value of PVIFA for \(4 \%\) for 20 Years that is PVIFA @ \(4 \%\) for 20 years which is 13.5903. As per Table A-1 we can find out the value of PVIF @ \(4 \%\) for 20 years which is \(0.4564 .\)

$$ \begin{aligned} \text { Price of bond } &=\left[\begin{array}{c} {[\text { Interest } \times \text { PVIFA } @ 4 \% \text { for } 20 \text { years }]} \\ + \\ {[\text { Face value } \times \text { PVIF } @ 4 \% \text { for } 20 \text { years }]} \end{array}\right] \\ &=[\$ 35 \times 13.5903]+[\$ 1,000 \times 0.4564] \\ &=\$ 475.6605+\$ 456.4 \\ &=\$ 932.0605 \end{aligned} $$

Step 3: Since it is on semi annual the number of years will be multiplied by 2 that is 10 years \(\times 2=20\) years. Interest is as per Step 2 and other variables are as per given information. As per Table A-2 we can find out the value of PVIFA for \(5 \%\) for 20 Years that is PVIFA@ \(5 \%\) for 20 years which is \(12.4622\). As per Table A-1 we can find out the value of PVIF (a) \(5 \%\) for 20 years which is \(0.3769\).

$$ \begin{aligned} \text { Price of bond } &=\left[\begin{array}{l} {[\text { Interest } \times \text { PVIFA } @ 5 \% \text { for } 20 \text { years }]} \\ {[\text { Face value } \times \text { PVIF } @ 5 \% \text { for } 20 \text { years }]} \end{array}\right] \\ &=[\$ 35 \times 12.4622]+[\$ 1,000 \times 0.3769] \\ &=\$ 436.177+\$ 376.9 \\ &=\$ 813.077 \end{aligned} $$

Step 4: Price of bond with \(5 \%\) is as per Step 3, Price of bond with \(4 \%\) is as per Step 2 and other variables are as per given information.

$$ \begin{aligned} &=\left[4 \%+\left(\frac{\$ 932.0605-\$ 930}{\$ 932.0605-\$ 813.077}\right) \times 1 \%\right] \times 2 \% \\ &=\left[4 \%+\left(\frac{\$ 2.0605}{\$ 118.9835}\right) \times 1 \%\right] \times 2 \% \\ &=[4 \%+(0.0173 \times 1 \%)] \times 2 \% \\ &=4.0173 \% \times 2 \% \\ &=8.03 \% \end{aligned} $$

Step 5: From the Given information, the face value of Second bond is at \(96.5 \%\) that is

$$ \begin{aligned} \$ 1,000 \times & 96.5 \%=\$ 965 \\ \text { Interest } &=6.5 \% \times \$ 1,000 \\ &=\$ 65 \\ &=\frac{\$ 65}{2} \\ &=\$ 32.5 \end{aligned} $$

Since it is on semi annual the number of years will be multiplied by 2 that is 6 years \(\times 2=12\) years. As per Table A-2 we can find out the value of PVIFA for \(4 \%\) for 12 Years that is PVIFA @ \(4 \%\) for 12 years which is \(9.3851\). As per Table A-1 we can find out the value of PVIF @ \(4 \%\) for 12 years which is \(0.6246\)

$$ \begin{aligned} \text { Price of bond } &=\left[\begin{array}{l} {[\text { Interest } \times \text { PVIFA@ } 4 \% \text { for } 12 \text { years }]} \\ + \\ {[\text { Face value } \times \text { PVIF@ } @ \% \text { for } 12 \text { years }]} \end{array}\right] \\ &=[\$ 32.5 \times 9.3851]+[\$ 1,000 \times 0.6246] \\ &=\$ 305.01575+\$ 624.6 \\ &=\$ 929.61575 \end{aligned} $$

Step 6: Since it is on semi annual the number of years will be multiplied by 2 that is 6 years \(\times 2=12\) years. Interest is as per Step 5 and other variables are as per given information. As per Table A-2 we can find out the value of PVIFA for \(3 \%\) for 12 Years that is PVIFA @ \(3 \%\) for 12 years which is \(9.9540\). As per Table A-1 we can find out the value of PVIF @ \(3 \%\) for 12 years which is \(0.7014\).

$$ \begin{aligned} \text { Price of bond } &=\left[\begin{array}{l} {[\text { Interest } \times \text { PVIFA@ } 3 \% \text { for } 12 \text { years }]} \\ + \\ {[\text { Face value } \times \text { PVIF@ } @ \% \text { for } 12 \text { years }]} \end{array}\right] \\ &=[\$ 32.5 \times 9.9540]+[\$ 1,000 \times 0.7014] \\ &=\$ 323.505+\$ 701.4 \\ &=\$ 1,024.905 \end{aligned} $$

Step 7: Price of bond with \(4 \%\) is as per Step 5, Price of bond with \(3 \%\) is as per Step 6 and other variables are as per given information.

$$ \begin{aligned} &=\left[3 \%+\left(\frac{\$ 1,024.905-\$ 965}{\$ 1,024.905-\$ 929.61575}\right) \times 1 \%\right] \times 2 \% \\ &=\left[3 \%+\left(\frac{\$ 59.905}{\$ 95.28925}\right) \times 1 \%\right] \times 2 \% \\ &=[3 \%+(0.62866 \times 1 \%)] \times 2 \% \\ &=3.62866 \% \times 2 \% \\ &=7.26 \% \end{aligned} $$

Step 8: Cost of first bond is as per Step 4, cost of second bond is as per Step 7 and other variables are as per given information.

\(\begin{aligned} \text { Total value of bond } &=\text { Value of first bond }+\text { Value of second bond } \\ &=(0.93 \times \$ 60,000,000)+(0.965 \times \$ 50,000,000) \\ &=\$ 104,050,000 \\ \text { Weight of first bond } &=\frac{\$ 55,800,000}{\$ 104,050,000} \\ &=0.536 \end{aligned}\)

\(\begin{aligned} \text { Weight of second bond } &=\frac{\$ 48,250,000}{\$ 104,050,000} \\ &=0.464 \end{aligned}\)

Step 9: Calculation of total cost of debt after tax: Cost of first bond is as per Step 4, cost of second bond is as per Step 7, and weights of the bonds are as per Step 8 and other variables are as per given information.

$$ \left(\begin{array}{c} \text { Total after tax } \\ \text { cost of debt } \end{array}\right)=(1-\operatorname{tax} \text { rate }) \times\left[\begin{array}{c} \text { (Cost of first bond } \times \text { Weight of first bond) } \\ + \\ (\text { Cost of Second bond } \times \text { Weight of Second bond }) \end{array}\right] $$

On substituting the variables in the above equation:

$$ \begin{aligned} &=(1-0.35) \times[(8.03 \% \times 0.536)+(7.26 \% \times 0.464)] \\ &=0.65 \times[4.304 \%+3.369 \%] \\ &=4.98 \% \end{aligned} $$

Step 10: Value of Debt is as per Step 8, Cost of equity is as per Step 1, cost of debt is as per Step 9 and other variables are as per given information.

$$ \begin{aligned} \text { Value of Equity } &=\text { Number of shares } \times \text { Price per share } \\ &=7,500,000 \times \$ 49 \\ &=\$ 367,500,000 \end{aligned} $$

$$ \begin{aligned} \text { Total value of firm } &=\text { Value of debt }+\text { Value of equity } \\ &=\$ 104,050,000+\$ 367,500,000 \\ &=\$ 471,550,000 \\ \text { WACC } &=[\text { Cost of equity } \times \text { Weight of equity }]+[\text { Cost of debt } \times \text { weight of debt }] \\ &=\left[13.60 \% \times\left(\frac{\$ 367,500,000}{\$ 471,550,000}\right)\right]+\left[4.98 \% \times\left(\frac{\$ 104,050,000}{\$ 471,550,000}\right)\right] \\ &=[13.60 \% \times 0.7793]+[4.98 \% \times 0.2207] \\ &=10.60 \%+1.1 \% \\ =11.70 \% & \end{aligned} $$

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