Investment Banking puzzles, solved step by step

Puzzles: 100
Traced to a firm: 36
Topics: 12
Hard: 29

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056A company is funded 60% by equity at a 14% cost and 40% by debt at 10% before tax, and the tax rate is 25%. What is its weighted average cost of capital?DCF and cost of capitalWarm upBulge bracket IBMiddle market IB
Try it first
Pick the WACC.
Show the worked solution
The WACC is 11.4%. Equity is 60% of the funding at 14%, contributing 8.4 points. Debt is 40% at 10% before tax, but interest is tax-deductible, so its after-tax cost is 10% x (1 minus 25%) = 7.5%, contributing 3.0 points. Add them: 8.4 + 3.0 = 11.4%. Leaving out the tax shield gives 12.4%.
Why is it a weighted average and not a plain one?
A family buying a flat with 60% from savings and 40% from a home loan pays a blended cost that leans towards the bigger source. Each source of money counts in proportion to how much of the funding it provides. Here equity provides 60 rupees in every 100, so its 14% carries more weight than debt's rate. A plain average of 14% and 10% would be 12.0%, which pretends the two sources are the same size.
Equity is 60% of funding at 14% and contributes 8.4 points; debt is 40% at 7.5% after tax and contributes 3.0 points, so the WACC is 11.4%. Using the 10% pre-tax rate adds a wrong extra point and gives 12.4%.
Why does debt go in after tax?
Interest is deducted before profit is taxed, so every 10 rupees of interest cuts the tax bill by 2.50 rupees at a 25% rate. The tax saved pays a quarter of the interest, so the company's true cost of debt is 7.5%, not 10%. That saving is the tax shieldThe tax a company avoids because interest is deducted from profit before tax is worked out.. Equity has no such shield, because dividends are paid out of profit after tax, which is why the adjustment sits on the debt term only.
The relationship
$\text{WACC} = \tfrac{E}{V}\,r_e + \tfrac{D}{V}\,r_d(1-t) = 0.6(14\%) + 0.4(10\%)(0.75) = 11.4\%$
E/V equity's share of total funding, 60%
D/V debt's share of total funding, 40%
r_e cost of equity, 14%
r_d pre-tax cost of debt, 10%
t tax rate, 25%
What it says in wordsWeight each source's cost by its share of the funding, and cut the cost of debt by the tax it saves.
One more sentence wins the point. The weights should be market values, not the book values on the balance sheet, because investors demand a return on what their stake is worth today. Then say what the number is for: 11.4% is the rate at which you would discount the company's unlevered free cash flows in a DCF.
Where candidates lose it
The usual miss is forgetting the tax shield and answering 12.4%. It is the most common one-point error in a first-round cost of capital question, and interviewers ask it because it is so easy to skip.
The second is applying the tax adjustment to equity as well, or to the whole WACC. The shield belongs to interest alone, so say which term it sits on.
What the interviewer asks next
The company moves to 60% debt at the same rates. What happens to WACC, and why would the rates not stay the same?
Why is the cost of equity higher than the cost of debt?
If the company makes losses and pays no tax, what is its WACC?
071Final-year free cash flow is Rs 100 crore, long-run growth is 4% and WACC is 10%. By what percentage does the terminal value fall if WACC rises to 11%?DCF and cost of capitalCoreBulge bracket IBMiddle market IB
Try it first
Before you compute: how much does the terminal value fall?
Show the worked solution
It falls by about 14.3%, one seventh. Terminal value is next year's cash flow over the spread between WACC and growth. At 10% the spread is 6 points, so the value is Rs 104 crore over 0.06, about Rs 1,733 crore. At 11% the spread is 7 points, giving Rs 1,486 crore. The new value is 6 over 7 of the old, a fall of 14.3%. The spread, not the WACC, is what the value is sensitive to.
Why does a one point move in WACC cost a seventh of the value?
Think of a shop whose rent is Rs 10 a month and whose takings are Rs 16: the owner keeps Rs 6. If rent goes up by one rupee the owner's margin shrinks by a sixth, far more than the rent went up in percentage terms. The terminal value divides by the gap between WACC and growth, and when that gap is thin, a small move in either rate is a large move in the gap. Here the gap goes from 6 points to 7, so the value falls to six sevenths of itself.
With growth fixed at 4%, the terminal value of Rs 100 crore of final-year cash flow falls from Rs 1,733 crore at a 10% WACC to Rs 1,486 crore at 11%, a drop of 14.3%, because the spread in the denominator grows from 6 points to 7.
What is the shortcut, and when does it break?
You do not need either terminal value to answer. The ratio of new to old value is the ratio of the old spread to the new spread, 6 over 7, so the percentage change is the spread change divided by the new spread. The same shortcut runs the other way: a fall in WACC to 9% narrows the spread to 5 points, and the value rises by 6 over 5 minus 1, 20%. Notice the asymmetry. A one point fall lifts value 20% while a one point rise cuts it 14.3%, which is why DCF sensitivity tables look lopsided.
The relationship
$TV = \frac{FCF \times (1+g)}{WACC - g} \qquad \frac{TV_{11\%}}{TV_{10\%}} = \frac{0.10 - 0.04}{0.11 - 0.04} = \frac{6}{7} \Rightarrow -14.3\%$
FCF final-year free cash flow, Rs 100 crore
g long-run growth, 4%
WACC the discount rate, 10% rising to 11%
What it says in wordsThe cash flow cancels; the terminal value changes by the ratio of the old spread to the new spread.
WACC Spread over 4% growth Terminal value, Rs crore Change from the 10% case
8% 4 points 2,600 +50.0%
9% 5 points 2,080 +20.0%
10% 6 points 1,733 +0.0%
11% 7 points 1,486 -14.3%
12% 8 points 1,300 -25.0%
Each one point step in WACC moves the terminal value by a different percentage, larger on the way down in WACC than on the way up, because the value is a cash flow divided by a thin and changing spread.
Why does this matter beyond the arithmetic?
The terminal value is usually most of a DCF, often two thirds or more of the enterprise value. If a one point move in WACC moves the terminal value by a seventh, the whole valuation is a judgement about two inputs that nobody knows to within a point. That is why a banker shows a sensitivity table rather than a single number, and why a growth assumption of 4% against a WACC of 10% should be cross-checked against the implied exit multiple before anyone trusts it.
Where candidates lose it
The common loss is answering 10%, from the WACC moving from 10 to 11, or 1%, from the one point move itself. Both forget that the value divides by the spread, not by the WACC.
The other loss is computing both terminal values with the Rs 104 crore numerator and losing a minute. Say the spreads, 6 and 7, and the answer is one division.
What the interviewer asks next
What happens to the terminal value if growth rises from 4% to 5% with WACC fixed at 10%?
What exit EV/EBITDA multiple does a Rs 1,733 crore terminal value imply if final-year EBITDA is Rs 150 crore?
Why would a WACC of 10% with 4% growth be hard to defend for a mature company?
099A company with a 10% WACC and an equity beta of 1.0 is considering an all-equity project in a riskier business, where pure-play peers have an asset beta of 1.5. The risk-free rate is 7% and the equity risk premium is 5%. What discount rate should the project use?DCF and cost of capitalCoreBulge bracket IBMiddle market IB
Try it first
Which rate belongs in the project's DCF?
Show the worked solution
14.5%, from the project's own beta, not the company's 10% WACC. The project is all-equity, so its discount rate is its cost of equity at the peers' asset beta: 7% + 1.5 x 5% = 14.5%. Using 10% would accept projects that cannot earn their own risk. A project returning 12% looks like a winner at 10% but destroys value at 14.5%.
Why is the company's own WACC the wrong rate?
A bank with a low cost of funds still charges a risky borrower a high interest rate, because the rate should reflect the loan, not the bank. Projects work the same way. The discount rate is the return investors demand for the risk of these particular cash flows, so it belongs to the project, not to the company that happens to own it. The company's 10% WACC is the right rate for a project as risky as the company's average business, and this one is riskier.
Where does the risk come from? Pure-play peers, companies that do only this riskier business, have an asset betaThe beta of a business with its debt stripped out: how much its operating cash flows move with the market, regardless of how it is financed. of 1.5. Because the project is all-equity, its equity beta equals that asset beta. Plug it into CAPM: 7% + 1.5 x 5% = 14.5%. For comparison, the parent's own cost of equity at a beta of 1.0 is 12%, and its WACC is lower still because it blends in cheaper debt.
The project's own rate is 14.5% from an asset beta of 1.5, against the company's 10% WACC, so a project earning 12% shows an NPV of +20 at the wrong rate and -17.2 at the right one.
What does the wrong rate cost in practice?
The relationship
$r_{\text{project}} = r_f + \beta_{\text{asset}} \times \text{ERP} = 7\% + 1.5 \times 5\% = 14.5\%$
r_f risk-free rate, 7%
beta_asset the pure-play peers' asset beta, 1.5
ERP equity risk premium, 5%
What it says in wordsAn all-equity project's rate is the risk-free rate plus its own beta times the market premium.
Take a project costing 100 that pays 12 a year forever, an IRR of 12%. At 10% it is worth 120, an NPV of +20. At 14.5% it is worth 82.8, an NPV of -17.2. A company that discounts everything at its own WACC systematically accepts risky projects that destroy value and rejects safe ones that would create it. The limitation: peer betas are estimated with error, so use several peers and a median, and if the project were part-funded with debt you would relever the asset beta for that financing.
Where candidates lose it
The standard slip is answering 10% because the question opens with the company's WACC. That number is there as bait: the interviewer wants to hear that the discount rate follows the risk of the cash flows.
The second loss is relevering the 1.5 with the parent's debt when the question says the project is all-equity. Read the financing assumption before adjusting the beta.
What the interviewer asks next
If the project were funded 30% with debt at 9% pre-tax and a 25% tax rate, what rate would you use?
How would you find the asset beta if the peers carry debt?
Why might a conglomerate's divisions argue for a single company-wide hurdle rate, and what goes wrong?

E/V	equity's share of total funding, 60%
D/V	debt's share of total funding, 40%
r_e	cost of equity, 14%
r_d	pre-tax cost of debt, 10%
t	tax rate, 25%

FCF	final-year free cash flow, Rs 100 crore
g	long-run growth, 4%
WACC	the discount rate, 10% rising to 11%

WACC	Spread over 4% growth	Terminal value, Rs crore	Change from the 10% case
8%	4 points	2,600	+50.0%
9%	5 points	2,080	+20.0%
10%	6 points	1,733	+0.0%
11%	7 points	1,486	-14.3%
12%	8 points	1,300	-25.0%

r_f	risk-free rate, 7%
beta_asset	the pure-play peers' asset beta, 1.5
ERP	equity risk premium, 5%

Try it first

Why is it a weighted average and not a plain one?

Why does debt go in after tax?

Where candidates lose it

What the interviewer asks next

Try it first

Why does a one point move in WACC cost a seventh of the value?

What is the shortcut, and when does it break?

Why does this matter beyond the arithmetic?

Where candidates lose it

What the interviewer asks next

Try it first

Why is the company's own WACC the wrong rate?

What does the wrong rate cost in practice?

Where candidates lose it

What the interviewer asks next