Investment Banking puzzles, solved step by step

Puzzles: 100
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004A company trades at 20x earnings and 10x EBITDA. Equity value is 200, interest expense is 20 at a 5% interest rate, and depreciation and amortisation is 20. What is the tax rate?Valuation riddlesHardEvercoreMenlo Park · 2025EvercoreSan Francisco · 2026
Try it first
Which two numbers do you need to find before the tax rate falls out?
Show the worked solution
The tax rate is 50%. P/E of 20 on equity of 200 gives net income of 10. Interest of 20 at 5% means debt of 400, so enterprise value is 600 and EBITDA is 60. Take off depreciation of 20 and interest of 20 to reach pre-tax profit of 20. Tax is 20 minus 10, which is 10, so the rate is 50%.
Where do you start when every number looks equally useful?
Start from what the answer is made of. A tax rate is tax divided by pre-tax profit, so the job is to find the two profit lines either side of the tax charge. Everything in the question is a road to one of those two numbers. Say that out loud first: it tells the interviewer you have a plan before you touch the arithmetic.
Net income is one step: equity value over the P/E, 200 over 20, is 10. Pre-tax profit takes longer. The enterprise valueThe value of the whole business to all its funders: equity value plus debt, less cash. multiple is on EBITDA, so you need EV, which needs debt, which the interest line hands you: 20 of interest at 5% means 400 of debt.
Equity of 200 at 20x earnings gives net income of 10; interest of 20 at 5% gives debt of 400, so enterprise value is 600 and EBITDA at 10x is 60. EBITDA of 60 less depreciation of 20 and interest of 20 leaves pre-tax profit of 20, and a tax charge of 10 on that is a 50% rate.
What assumption are you making, and should you say it?
You are assuming the company holds no cash, so enterprise value is simply equity plus debt. Say the assumption before you use it, because with any cash on the balance sheet EV falls, EBITDA falls, and the tax rate changes. A 50% rate is also high for most countries, which is worth one sentence: the interviewer built round numbers, not a real company.
Where candidates lose it
Candidates grab EBITDA first and then stall, because EV/EBITDA looks like the obvious lever but EV is not given. The debt is hiding inside the interest line, and people who do not think of interest divided by rate never find it.
The second loss is silent assumptions. Saying no cash out loud costs two seconds and turns a lucky answer into a reasoned one.
What the interviewer asks next
Now the company holds 100 of cash. What is the tax rate?
What does a 50% effective tax rate tell you about this business?
If the P/E rises to 25x with everything else fixed, what happens to the implied tax rate?
Asked at Evercore, Investment Banking, Menlo Park, 2025 (Wall Street Oasis): P/E of 20x, Ev/EBITDa of 10x, 20 million interest expense, 5% IR, D/A of 20 million, EQ of 200 million -> find tax rate
Asked at Evercore, Mergers and Acquisitions, San Francisco, 2026 (Wall Street Oasis): Given the following, calculate the Tax Rate
020Company A trades at 12x earnings and 9x EV/EBITDA. Its closest peer B trades at 15x earnings and 8x EV/EBITDA. Build one set of numbers that explains how A is cheaper on P/E yet dearer on EV/EBITDA.Valuation riddlesHardBarclaysNew York · 2026
Try it first
Which difference does most of the work in a good answer?
Show the worked solution
Give B heavy depreciation and more debt, so much less of its EBITDA reaches shareholders. Both earn EBITDA of Rs 100 crore. A has D&A of 2 and interest of 18, keeps net income of 60, and at equity value 720 trades on 12x; its EV of 900 is 9x EBITDA. B has D&A of 25 and interest of 35, keeps only 30, and at equity value 450 trades on 15x, though its EV of 800 is 8x.
How can one company be cheaper on one multiple and dearer on another?
Two shops each take in Rs 1 lakh a month before equipment and loan costs. One works from a simple counter with a small loan; the other leases costly machines it must keep replacing and carries a large loan. Priced on takings before those costs, they look alike; priced on what the owner keeps, they do not. EV/EBITDA values the business before depreciation and before financing, while P/E values what is left for shareholders after both, so a company with heavy D&A or heavy interest can look cheap on one and dear on the other.
Both companies earn EBITDA of Rs 100 crore, but B's heavier D&A and interest leave it net income of Rs 30 crore against A's Rs 60 crore, so B looks cheaper on EV/EBITDA at 8.0x against 9.0x and dearer on P/E at 15.0x against 12.0x.
Which line explains how much of the gap?
Work down from EBITDA. B carries Rs 23 crore more D&A and Rs 17 crore more interest than A, which after 25% tax is Rs 17.25 crore and Rs 12.75 crore of net income: together exactly the Rs 30 crore gap between A's 60 and B's 30. Here depreciation does more of the work than debt, which you can show with EV/EBIT, a multiple that charges D&A but ignores financing: A is 9.2x and B 10.7x, so A is cheaper there too.
Rs crore A B
Enterprise value 900 800
Net debt, at 10% interest 180 350
Equity value 720 450
EBITDA 100 100
D&A 2 25
Interest 18 35
Net income, 25% tax 60 30
EV / EBITDA 9.0x 8.0x
EV / EBIT 9.2x 10.7x
P/E 12.0x 15.0x
Two invented companies built to match the multiples in the question.
So which company is actually cheaper?
It depends on whether B's D&A is a real cost. If it reflects equipment that must be replaced, which capital expenditureCash spent on buying or replacing long-term assets such as machines and buildings. running close to depreciation would confirm, then EBITDA flatters B and A is better value on every measure that counts the cost of staying in business. If B's D&A is mostly amortisation of acquired intangibles with no cash replacement, EBITDA is the fairer lens and B may be the cheap one. The answer the interviewer wants is the diagnosis: check capex against D&A and check leverage before trusting either multiple.
Where candidates lose it
The common slip is to say the market simply disagrees or that one multiple must be wrong. Both are correct; they measure different things, and the job is to name what sits between EBITDA and net income.
The second slip is naming only debt. Leverage can push P/E either way depending on the cost of debt, while heavy D&A reliably depresses earnings relative to EBITDA. Give both, with numbers, and say which one does more of the work.
What the interviewer asks next
How would you check whether B's D&A is a real economic cost?
If B refinanced its debt at 5% instead of 10%, what would its P/E become?
Which multiple would you use to compare a capital-heavy telecoms operator with a software company, and why?
Asked at Barclays, Investment Banking, New York, 2026 (Wall Street Oasis): A company is trading at a lower P/E but a higher EV/EBITDA than peers
049A bank trades at 2.0x book value, earns a 16% return on equity and has a 12% cost of equity. Using a dividend discount model, what long-run growth rate is the market pricing in?Valuation riddlesHardRothschild & CoRiyadh · 2026
Try it first
What growth is priced in?
Show the worked solution
About 8% a year, forever. In a dividend discount model, a bank that earns its ROE on book and grows at g must retain g over ROE of its earnings and pays out the rest, which makes its fair price to book (ROE minus g) over (cost of equity minus g). Setting (16% minus g) over (12% minus g) equal to 2.0 gives g of 8%, with 50% of earnings paid out. Whether 8% forever is believable is the real question.
Where does the price-to-book formula come from?
Think of a fruit tree. The fruit you eat is the dividend; the fruit you replant grows the orchard. A bank earning its ROE on book value can grow only by retaining earnings, so growth equals ROE times the share retained, and the rest is paid out as dividends. Put that dividend into the dividend discount modelA valuation that prices a share as the present value of its future dividends; with constant growth it becomes the next dividend divided by the cost of equity minus growth. and divide by book value, and price to book becomes (ROE minus g) over (r minus g).
With a 16% return on equity and a 12% cost of equity, the justified price to book rises from 1.33x at zero growth to 2.0x at 8% growth and 3.0x at 10%, so a 2.0x price implies the market expects 8% growth forever.
How do you solve for the growth?
Set the formula equal to the price. (16% minus g) over (12% minus g) equals 2.0, so 16 minus g equals 24 minus 2g, and g is 8%. Check it with rupees: on book of Rs 100 the bank earns Rs 16, keeps Rs 8 to grow book by 8%, and pays Rs 8. Rs 8 over (12% minus 8%) is Rs 200, twice book.
The relationship
$\frac{P}{B} = \frac{ROE - g}{r - g} \;\Rightarrow\; 2.0 = \frac{0.16 - g}{0.12 - g} \;\Rightarrow\; g = 8\%$
ROE return on equity, 16%
r cost of equity, 12%
g the growth rate the price implies
What it says in wordsThe justified price to book depends on how far ROE and the cost of equity each sit above growth.
Is 8% growth forever reasonable?
That is the judgement the interviewer is waiting for. Look at the curve: near 8% the price is very sensitive to growth, so a small shortfall, 7% instead of 8%, would justify only 1.8x book. Compare 8% with the nominal growth the economy could sustain over decades, a figure to look up rather than assume. And note what the formula says about returns: if the bank earned only its 12% cost of equity, it would be worth exactly book value, whatever its growth.
Where candidates lose it
Candidates reach for P/B equals ROE over cost of equity, get 1.33x, and cannot reconcile it with 2.0x. That shortcut is the no-growth case; the gap between 1.33x and 2.0x is exactly the growth the market is paying for.
The second loss is solving for 8% and stopping. The point of an implied number is to judge it: say whether 8% a year forever is plausible for a bank, and what happens to the price if it is not.
What the interviewer asks next
What payout ratio is consistent with this valuation?
If the market cut the bank to 1.5x book, what growth would that imply?
Why is a bank earning exactly its cost of equity worth book value whatever its growth?
Asked at Rothschild & Co, Generalist, Riyadh, 2026 (Wall Street Oasis): DDM model mechanics and A/P & A/R increase
087A DCF sets its terminal value at 10x final-year EBITDA. Free cash flow runs at half of EBITDA and the WACC is 10%. What perpetual growth rate is that exit multiple quietly assuming?Valuation riddlesHardElite boutique IBBulge bracket IB
Try it first
Roughly what growth rate does 10x EBITDA imply here?
Show the worked solution
About 4.8% a year, forever. Write the terminal value both ways. The multiple says 10 x EBITDA. The Gordon formula says 0.5 x EBITDA x (1 + g) / (0.10 - g). Setting them equal, EBITDA cancels: 1.0 - 10g = 0.5 + 0.5g, so g = 0.5 / 10.5 = 4.76%. Every exit multiple implies a growth rate, and the check is whether that rate is believable for this business.
Why does a multiple contain a growth rate at all?
If a buyer wants a 5% return and pays 25 times this year's rent for a flat, they are counting on the rent rising: with flat rent, a 5% return supports a price of only 20 times. A multiple is a shorthand price for a stream of future cash flows, so every multiple implies some combination of growth and discount rate. In a DCF the exit multiple sits beside an explicit WACC, which means the growth assumption can be solved for exactly.
Set the two terminal values equal. The multiple gives 10 x EBITDA. The perpetuity growth method gives next year's free cash flow over (r minus g), and next year's cash flow is 0.5 x EBITDA x (1 + g). EBITDA cancels from both sides, leaving one equation in g, which solves to 4.76%.
With free cash flow at half of EBITDA and a 10% WACC, a 10x exit multiple implies perpetual growth of 4.8%, and implied growth rises with the multiple while bending toward the 10% WACC, which it can never reach.
The relationship
$m = \frac{f(1+g)}{r-g} \;\Rightarrow\; g = \frac{mr - f}{m + f} = \frac{1.0 - 0.5}{10 + 0.5} \approx 4.76\%$
m exit multiple of EBITDA, 10x
f free cash flow as a share of EBITDA, 0.5
r WACC, 10%
g implied perpetual growth
What it says in wordsRearranging the perpetuity formula turns any exit multiple into the growth rate it assumes.
How do you judge whether 4.8% is believable?
Perpetual growth means growth forever, so it should not exceed the long-run nominal growth of the economy whose currency the cash flows are in, or the business eventually becomes larger than that economy. Whether 4.8% passes depends on that currency's inflation and real growth, so state the benchmark you are using rather than a number from memory. The useful habit is running the check both ways: an exit multiple that implies an unbelievable growth rate is a sign the multiple came from a peer set that does not match this business.
Say the convention too. If you apply the multiple to final-year EBITDA but take next year's cash flow without the (1 + g) uplift, the answer becomes 10% minus 0.5/10, which is 5.0%. A small difference, but naming the convention shows you know where it comes from.
Where candidates lose it
The common answer is that an exit multiple has no growth assumption, which is exactly backwards. Candidates who have only used the multiple method never think to translate it, and the interviewer is testing that translation.
The second loss is solving the algebra but not commenting on the result. Say the number, then say what you would compare it against and why.
What the interviewer asks next
What exit multiple implies zero growth here?
If free cash flow conversion rises to 70% of EBITDA, what growth does 10x imply?
Which method would you lead with in a fairness opinion, and why show both?

Rs crore	A	B
Enterprise value	900	800
Net debt, at 10% interest	180	350
Equity value	720	450
EBITDA	100	100
D&A	2	25
Interest	18	35
Net income, 25% tax	60	30
EV / EBITDA	9.0x	8.0x
EV / EBIT	9.2x	10.7x
P/E	12.0x	15.0x

ROE	return on equity, 16%
r	cost of equity, 12%
g	the growth rate the price implies

m	exit multiple of EBITDA, 10x
f	free cash flow as a share of EBITDA, 0.5
r	WACC, 10%
g	implied perpetual growth

Try it first

Where do you start when every number looks equally useful?

What assumption are you making, and should you say it?

Where candidates lose it

What the interviewer asks next

Try it first

How can one company be cheaper on one multiple and dearer on another?

Which line explains how much of the gap?

So which company is actually cheaper?

Where candidates lose it

What the interviewer asks next

Try it first

Where does the price-to-book formula come from?

How do you solve for the growth?

Is 8% growth forever reasonable?

Where candidates lose it

What the interviewer asks next

Try it first

Why does a multiple contain a growth rate at all?

How do you judge whether 4.8% is believable?

Where candidates lose it

What the interviewer asks next