IB3H7: Mergers and Acquisitions

Professor Nickolay Gantchev

Problem Set #1 Solution

Part 1 (Cost of Capital)

GS Inc. has two distinct but interrelated divisions, both of which serve institutional investors. One
division develops post-trade processing software; the other builds and operates specialized data
centres.

Each of the firm’s two divisions has two (equally good) pure-play competitors (“Software” A and B
and “Data centre” A and B, respectively). The table below contains betas and debt to equity ratios for
the comparable firms.

Firm Asset beta Equity beta Debt beta D/E ratio

Software A ? 2.03 0.08 0.12
Software B ? 2.26 0.11 0.16
Data centre A 0.66 0.90 0.20 0.78
Data centre B 1.02 1.35 0.85 2.89

All firms face a 35% tax rate. The risk-free rate is 2.96% and the risk premium of the S&P 500 index
over the Treasury bill rate is 7.48%.

Question 1: Given the data above, what is your best estimate of the current unlevered cost of capital
(rA) for each of GS Inc.’s divisions (software and data centres)?

Software A’s asset beta is

D
β EA = β AA + (1− tc ) ( β AA − β DA ) ⇔ 2.03 = β AA + ( 0.12 ) ( 0.65) ( β AA − 0.08) ⇒ β AA = 1.89
E

Software B’s asset beta is

D
β EB = β AB + (1− tc ) ( β AB − β DB ) ⇔ 2.26 = β AB + ( 0.16 ) ( 0.65) ( β AB − 0.11) ⇒ β AB = 2.06
E

The simple average of the two asset betas is 1.98, which is our estimate for the asset beta of
GS Inc.’s software division.

The unlevered cost of capital (rA) of the software division is

rASoftware = rf + β A ( rM − rf ) = 2.96% +1.98 ( 7.48%) = 17.77%.

The simple average of the asset betas of Data center A and Data center B is 0.84, which is our
estimate for the asset beta of GS Inc.’s data center division.

The unlevered cost of capital (rA) of the data center division is

rAData = rf + β A ( rM − rf ) = 2.96% + 0.84 ( 7.48%) = 9.24%.

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 IB3H7: Mergers and Acquisitions
Professor Nickolay Gantchev

The software division will be operated with a constant D/E ratio of 0.10 and has a debt beta of 0.05.
The data centre division will be operated with a constant D/E ratio of 1.50, resulting in a debt beta of
0.45 for that division.

Question 2: Given your answer to the previous question, estimate the weighted average cost of
capital of each division.

Software
The cost of debt of the software division is rD = 2.96% + 0.05 ( 7.48%) = 3.33%

Its D/V ratio is 0.10/1.10 = 0.0909. As a result,

D
(1− tc ) (rASoftware − rDSoftware ) = 0.1777 + (0.10)(0.65)(0.1777 − 0.0333) = 0.1871
rESoftware = rASoftware +
E
Software E D
⇒ rWACC = rESoftware + (1− tc ) rDSoftware = 0.9091(0.1871) + 0.0909(0.65)(0.0333) = 0.1721.
V V
Data
The cost of debt of the data center division is rD = 2.96% + 0.45 ( 7.48%) = 6.33%

and its D/V ratio is 1.50/2.50 = 0.60. As a result,

rEData = 0.0924 + (1.50 ) ( 0.65) ( 0.0924 − 0.0633) = 0.1208

Data
⇒ rWACC = 0.40 ( 0.1208) + 0.60(0.65) ( 0.0633) = 0.073

75% of GS Inc.’s total unlevered free cash flow of $35 million is expected to come from the software
division and the rest from the data centre division. The unlevered FCF from software is expected to
grow at 3.5% in perpetuity while the unlevered FCF from data centres is expected to grow at 0.65%.

Question 3: Find the value of each division and the total value of GS Inc.

The value of the software division is $35M*0.75/(0.1721-0.035) = $191.47 million and the value of
the data centre division is $35M*0.25/(0.073-0.0065) = $131.58 million. The total value of the firm is
$191.47 + $131.58 = $323.05.

Question 4: Indicate whether each of the following statements is correct or not.

1. According to the CAPM, a firm that increases its debt-to-equity ratio (D/E) will decrease its asset
beta.

No. The asset beta of a firm is not affected by the firm’s financial leverage.

2. When calculating a free cash flow to the firm, one should not adjust the cash flow to account for the
tax shield on debt.

Yes. WACC adjusts the discount rate so no adjustment is needed for the cash flow.

3. Firms A and B have the same tax rate and the same equity beta (assume that their debt betas are
zero) but Firm A’s D/E ratio is higher than Firm B’s D/E ratio. This implies that Firm A’s asset beta
(beta of unlevered equity) is smaller than Firm’s B’s asset beta.

Yes. To have the same equity risk but less leverage than Firm A, Firm B must have risker operations.

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 IB3H7: Mergers and Acquisitions
Professor Nickolay Gantchev

Question 5: The CAPM implies that the expected excess return (r-rf) investors require to be willing to
own an asset should only depend on that asset’s beta and the market risk premium (rm-rf). Does the
CAPM imply that this should also be true for an entrepreneur who has invested all her wealth in the
firm and owns 100% of it? Explain briefly why this is or is not the case.

No. A crucial premise of the CAPM is that all assets are held in well-diversified portfolios, which
does not seem to be the case for the entrepreneur and her firm.

Part 2 (APV)

B&G Inc. is considering a new project. The project requires an investment of $5,000 today.
Ignoring financing costs, the project is expected to generate a $500 after-tax cash flow at
year- end for the first 5 years and $750 every year after that. The firm is going to use $2,500
of debt to finance the investment. The debt will be paid back over 5 years and the debt
repayment schedule is based on equal installments of principal. There is no other debt
associated with this project. The interest rate on the debt is equal to the firm’s market cost of
debt. The firm’s equity beta (obtained from stock-market data of recent years) is 0.875,
generating a cost of equity of 15%. The firm’s debt-to-equity ratio (D/E) in recent years has
been 50%, its tax rate is 50%, and its debt beta is 0.125. The market risk premium is 8%.
Assume that there are no personal taxes, and the risk-free rate on equity is equal to the risk-
free rate on debt.

Question 6: Calculate the APV of the project for the equity holders of B&G.

First, calculate the unlevered cost of capital (rA). B&G’s asset beta 0.725.

𝐷
𝛽! = 𝛽" + (1 − 𝑡# )(𝛽" − 𝛽$ ) ⟺ 0.875 = 𝛽" + (1 − 0.5) ∗ 0.5 ∗ (𝛽" − 0.125) ⟹ 𝛽" = 0.725
𝐸

Using the CAPM, 𝑟! = 15% = 𝑟% + 0.875 ∗ 0.08 ⟹ 𝑟% = 0.15 − 0.875 ∗ 0.08 = 0.08.

Then, the unlevered cost of capital is 𝑟" = 0.08 + 0.725 ∗ 0.08 = 13.8%.

The present value of unlevered cash flows - $500 each year for 5 years and $750 a year thereafter – is
calculated by discounting them at the unlevered cost of capital of 13.8%:
&'' &'' &'' &'' &'' ,&' (
𝐹𝐶𝐹 = (.(*+ + (.(*+! +(.(*+" + (.(*+# + (.(*+$ + '.(*+ :(.(*+$; = 4572.37.

Using the CAPM, the cost of debt is 𝑟$ = 0.08 + 0.125 ∗ 0.08 = 9%.

The interest payments in the first 5 years are $225=0.09*2500, $180=0.09*2000, $135, $90, and $45.
Discounting the interest savings at 9%, the cost of debt, gives
--& (+' (*& .' /&
𝐼𝑛𝑡𝑒𝑟𝑒𝑠𝑡 = (.'. + (.'.! +(.'." + (.'.# + (.'.$ = 555.17.

The discounted tax shield equals 0.5*$555.17 = 277.59.

As a result, the NPV of the project is $4572.37+277.59 – 5000 = -150.04.

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 IB3H7: Mergers and Acquisitions
Professor Nickolay Gantchev

Question 7: Now suppose that the loan is given to B&G at a zero-interest rate, but you are allowed to
subtract principal repayments for tax purposes (similar to interest). Everything else remains the same
as in question 6 (including the market cost of debt that you calculated). Calculate the new NPV of the
project for the equity holders of B&G.

The present value of the unlevered cash flows remains $4572.37.

Since the interest rate is zero, there is no tax shield on the debt. The subsidized loan does, however,
have a positive NPV. The debt payments are $500 at the end of every year during the first five years
and zero after that. Discounted at 9%, the cost of debt, their present value is $1944.83.

As a result, the NPV of the loan is $2500 – $1944.83 = $555.17.

The project’s NPV is $4,572.37 + $555.17 - $5,000 = $127.54.

Question 8: Discuss the differences in value to shareholders between the loan with and without interest
and make a recommendation regarding which alternative is better.

Even if the entire value of tax shields is lost with a zero-interest rate, the value of the subsidy more than
offsets that loss. At a zero-interest rate, the NPV of financing is large enough to offset the negative NPV
of the stand-alone project. Therefore, B&G should undertake the project if it can get zero-interest
financing.

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