Private Equity puzzles, solved step by step

Puzzles: 100
Traced to a firm: 22
Topics: 12
Hard: 30

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029A sponsor buys a platform at 11x EBITDA of 100, then adds three bolt-ons with EBITDA of 10, 15 and 25, bought at 6x, 7x and 5x. What is the blended entry multiple for the whole group?Valuation riddlesCoreMid-market buyout fundIndian mid-market PE
Try it first
Which is closest to the blended multiple?
Show the worked solution
About 9.27x: total price of 1,390 over total EBITDA of 150. The platform costs 1,100. The bolt-ons cost 60, 105 and 125, which is 290 for 50 of EBITDA, a 5.8x average of their own. Add prices, add EBITDA, divide. The simple average of the four multiples, 7.25x, is wrong because it weights a 10 of EBITDA deal the same as a 100.
Why can you not average the multiples?
Buy 10 kg of rice at Rs 60 a kg and 1 kg of saffron rice at Rs 600 a kg. The average price per kg is not Rs 330; it is Rs 1,200 over 11 kg, about Rs 109. A multiple is a price per unit of EBITDA, so blending multiples means total price over total EBITDA, which weights each deal by its size. The platform holds two thirds of the group's EBITDA, so it pulls the blend towards 11x.
Drawn with width equal to EBITDA and height equal to the multiple, each deal's area is its price; the four blocks hold 1,390 of price over 150 of EBITDA, a blended multiple of 9.27x, well above the 7.25x simple average.
What does the blend tell you about the buy-and-build story?
The sponsor has bought 150 of EBITDA for 9.27x. If the larger group is worth the platform's 11x at exit, the same EBITDA is worth 1,650, against 1,390 paid. That gap of 260 is the multiple arbitrage that buy-and-build plans are built on, and it exists only if the market values the combined group at the platform's multiple.
Block EBITDA Multiple Price
Platform 100 11x 1,100
Bolt-on 1 10 6x 60
Bolt-on 2 15 7x 105
Bolt-on 3 25 5x 125
Group 150 9.27x 1,390
Adding prices and EBITDA separately gives 1,390 over 150, a blended 9.27x for the group.
Say the limitation. Bolt-on EBITDA is often bought on the seller's figures, before integration costs, lost customers and the management time each deal eats. Small businesses sell for lower multiples partly because they are riskier. The blend is a fair measure of what was paid; whether the group deserves 11x at exit is a separate judgement.
Where candidates lose it
The average of the four multiples, 7.25x, is the trap, and it is tempting because the question lists four multiples side by side. The interviewer is checking whether you treat a multiple as a price per unit and weight it by size.
The second loss is stopping at the number. A buyout interviewer wants the next sentence: the blend is below the platform multiple, and that gap is the paper value the strategy is counting on.
What the interviewer asks next
What exit multiple on the group would leave the sponsor no better off than the entry blend?
Why do smaller businesses trade at lower multiples, and is that gap sure to close once they are inside a platform?
How would integration costs change the blended multiple you report to the investment committee?
037You buy 30% of a company from its existing shareholders for 300. The company has net debt of 500 and EBITDA of 160. What EV/EBITDA multiple did you pay?Valuation riddlesCoreGrowth equityMid-market buyout fund
Try it first
Which multiple did you pay?
Show the worked solution
About 9.4x EV/EBITDA. Paying 300 for 30% values all the equity at 300 / 0.3 = 1,000. Enterprise value is equity plus net debt, 1,000 + 500 = 1,500. Divided by EBITDA of 160, that is 9.38x. Two steps turn a minority cheque into a whole-company multiple: gross up to 100%, then add back the debt.
Why can you not divide the cheque by EBITDA?
Buying a 30% share of a flat for Rs 30 lakh tells you the owners think the whole flat's equity is worth Rs 1 crore. If the flat also carries a Rs 50 lakh home loan, the flat itself is worth Rs 1.5 crore. EBITDA is earned by the whole company for all its funders, so it has to be compared with the whole company's value, which is all the equity plus the net debt. The cheque is a fraction of only one of those two pieces.
Paying 300 for 30% implies equity of 1,000, and adding net debt of 500 gives an enterprise value of 1,500, which is 9.38x EBITDA of 160; had the 300 gone into the company as new money, the multiple would have been 7.5x.
The relationship
$\frac{EV}{EBITDA} = \frac{\text{Cheque}/\text{stake} + \text{Net debt}}{EBITDA} = \frac{300/0.3 + 500}{160} = 9.375$
Cheque / stake the value of 100% of the equity implied by the price paid
Net debt borrowings less cash, 500
EBITDA 160
What it says in wordsGross the cheque up to the whole equity, add net debt, and divide by EBITDA.
Does it matter whether the money goes to the sellers or into the company?
Yes, and growth investors are asked this often. If the 300 is new money paid into the company, the company's cash rises by 300, net debt falls to 200, and the same price buys a business valued at only 1,200, or 7.5x. Buying existing shares sends the cash to the sellers, so the company's debt is unchanged and the multiple is 9.38x. Ask which it is before you answer.
Say the limitation. A minority price can carry a discount for lack of control, or a premium if the investor gets preference shares with downside protection. So the implied multiple is a starting point, not a clean comparison with what a buyer of 100% would pay.
Where candidates lose it
The common loss is forgetting the debt: grossing the cheque up to 1,000 and reporting 6.25x. That mixes an equity value with an enterprise-level profit.
The second loss is not asking whether the investment is primary or secondary. New money changes net debt, and the multiple moves from 9.4x to 7.5x on the same cheque.
What the interviewer asks next
What P/E did you pay if net income is 80?
The 300 is new money. What is the post-money equity value and your stake?
Why might a minority investor accept a higher implied multiple than a buyer of the whole company?
054A company has an enterprise value of 500 and holds 300 of net cash, so its equity is worth 800. A buyer offers a 20% premium on the equity value. By what percentage does the implied enterprise value rise?Valuation riddlesCoreLarge-cap buyout fund
Try it first
The equity premium is 20%. What is the premium on the enterprise value?
Show the worked solution
The implied enterprise value rises 32%, from 500 to 660. A 20% premium on equity of 800 is 160, taking the offer to 960. The 300 of cash is worth 300 to everyone, so the whole 160 is a premium on the operating business. With net cash, the premium on EV is always larger than the headline premium on equity; with net debt, it is smaller.
Why can the premium not land on the cash?
Picture buying a shop that has Rs 3 lakh sitting in its till. You may pay extra for the shop's location and customers, but nobody pays Rs 3.6 lakh for Rs 3 lakh of cash: the cash is worth its face value to every buyer. The equity price is cash plus the business, and since the cash is fixed at 300, the full 160 of premium has to sit on the business. That turns a 20% premium into a 32% premium on the part the buyer is really valuing.
Equity rises 20% from 800 to 960, but the 300 of cash does not move, so the enterprise value rises from 500 to 660, a 32% premium; with no cash the same offer would be 20% on both.
The relationship
$\Delta EV\% = \frac{p \times E}{E - C} = \frac{0.20 \times 800}{800 - 300} = 32\%$
p the premium on equity, 20%
E equity value before the offer, 800
C net cash, 300
What it says in wordsThe premium on the business is the rupee premium divided by the business value, which is equity less cash.
Why would a buyout investor care which premium you quote?
Because the sponsor underwrites the business, not the cash. If the business earned EBITDA of 50, the market price would be 10x and the offer is 13.2x. A 20% headline premium sounds modest, but on a cash-rich target it can mean paying a third more for the operations. The limit runs the other way too: on a company with large net debt, a 20% equity premium is a much smaller premium on EV, so headline premiums are never comparable across capital structures without this step.
Where candidates lose it
The common answer is 20%: candidates assume the premium spreads evenly across equity and enterprise value. It only does when there is no cash and no debt.
The second slip is subtracting the cash from the wrong side and getting a smaller number, as though cash dilutes the premium. Cash is the one part of the company with no premium on it, which is why the rest carries more.
What the interviewer asks next
Now the company has 300 of net debt instead of cash. What is the premium on EV?
If the business earns EBITDA of 50, what multiple does the buyer pay? (13.2x)
Why might a target with large cash still demand a premium on the cash, and how would you respond?
056Two companies each have EBITDA of 100 and an enterprise value of 900. One spends 10 a year on capex, the other 50. If you could value a business on one metric only, levered free cash flow, EBIT or EBITDA, which would you choose, and which of these two is cheaper?Valuation riddlesCoreWarburg PincusSan Francisco · 2014
Try it first
Both trade at 9.0x EBITDA. What are they on EBITDA less capex?
Show the worked solution
Choose levered free cash flow, and Company A is the cheaper one. Both cost 9.0x EBITDA, but A keeps 90 after capex and B only 50, so the same 900 buys cash at 10.0x in A and 18.0x in B. After 25% tax the gap holds at 13.3x against 24.0x. Levered free cash flow is the one number that has already paid for capex, working capital, interest and tax: it is what an owner can actually take out.
What does EBITDA leave out that matters here?
Picture two taxis, each bringing in Rs 1 lakh a month after fuel and the driver. One is new and needs little upkeep; the other is old and eats half its takings in repairs. Before repairs they look identical, and nobody would pay the same for both. EBITDA is earnings before the business pays for the assets it uses up, so two companies with equal EBITDA can leave their owners very different amounts of cash. Company B spends half its EBITDA just to keep running.
Both companies cost 900 for EBITDA of 100, 9.0x, but after capex Company A keeps 90 and Company B keeps 50, so the same price is 10.0x A's cash and 18.0x B's; after tax the gap is 13.3x against 24.0x.
Why levered free cash flow rather than EBIT?
EBIT does subtract depreciation, which here happens to equal capex, so EBIT would rank the two correctly. But depreciation is an accounting estimate of past spending, not the cash going out this year, and it ignores working capital, interest and tax. Levered free cash flow is the only one of the three that is cash in the owner's hand after every claim ahead of the owner has been paid. Assume no debt and depreciation equal to capex: A's free cash flow is 100 less tax of 22.5 less capex of 10, which is 67.5; B's is 37.5.
What is the limit of your answer?
Say two things. First, one year of free cash flow can be lumpy: a big capex year or a working capital swing can make a good business look expensive. With a single period of data, levered free cash flow is the most honest number, but it is also the noisiest. Second, levered cash flow belongs to the equity, so you compare it with equity value, not enterprise value. In this example there is no debt, so the two are the same; with debt you would divide by equity value instead.
Where candidates lose it
Candidates pick EBITDA because buyout people talk in EBITDA multiples all day. The question is built to see whether you know what EBITDA leaves out: on these two companies it gives the same answer for a business that keeps 90 and one that keeps 50.
The second loss is choosing levered free cash flow and then dividing it by enterprise value. Levered cash is the equity's cash; set it against the equity value, or say clearly that with no debt the two coincide.
What the interviewer asks next
Company B's capex is growth capex for a new plant. Does that change your view?
Which of the three metrics would you pick for a bank, and why?
How would you adjust one year of free cash flow that includes a large working capital release?
Asked at Warburg Pincus, Private Equity, San Francisco, 2014 (Wall Street Oasis): If you knew nothing about a company or industry and were able to use one metric as a means of valuation, what would you choose?
094A sponsor needs a 25% IRR over 5 years. It expects to exit at an enterprise value of 1,500 with net debt of 300. What is the most equity it can put in, and with half the entry price funded by debt, the most it can pay for the business?Valuation riddlesCoreLarge-cap buyout fundMid-market buyout fund
Try it first
Roughly what is the maximum equity cheque?
Show the worked solution
About 393 of equity, and an entry enterprise value of about 786. Exit equity is 1,500 less 300 of net debt, 1,200. A 25% IRR for five years means multiplying the money by 1.25 to the fifth, about 3.05x, so the most the sponsor can invest is 1,200 divided by 3.05. If equity is half the price, the business is worth at most twice that.
Why work backwards from the exit?
A house buyer who wants to double their money by the time they sell, and expects to sell for Rs 1 crore after clearing the loan, knows they cannot put in more than Rs 50 lakh today. A target return plus an expected exit turns into a ceiling on the price, which is how sponsors set their bids. The exit equity is fixed by the assumptions; the only free number is what you pay today.
Exit enterprise value of 1,500 less 300 of net debt leaves exit equity of 1,200, which divided by 3.05, 1.25 to the fifth, caps entry equity at about 393; with half the price in debt, the most the sponsor can pay is about 786.
The relationship
$E_0 = \frac{1{,}500 - 300}{1.25^5} = \frac{1{,}200}{3.052} \approx 393 \qquad EV_0 = \frac{393}{0.5} \approx 786$
E_0 maximum equity at entry
1.25^5 five years of compounding at 25%, about 3.05
0.5 equity's share of the entry price
What it says in wordsDivide exit equity by the required money multiple to get the maximum cheque, then gross up by equity's share of the price.
How do you get 1.25 to the fifth without a calculator?
Square twice and multiply once. 1.25 squared is 1.5625, about 1.56; squared again is about 2.44; times 1.25 is about 3.05. A 25% IRR over five years is roughly a 3x money multiple, worth knowing by heart because buyout targets are often quoted that way. Then check consistency: entry debt of about 393 falling to 300 at exit means about 93 repaid over five years, which you would test against the business's cash flow.
Where candidates lose it
The usual slip is simple interest: 25% times 5 is 125%, so 2.25x, which gives a cap of about 533 and overpays. Returns compound.
The other is dividing the exit enterprise value, not the exit equity, by the multiple. The sponsor only owns what is left after the 300 of net debt.
What the interviewer asks next
The target IRR falls to 20%. How much more can the sponsor pay?
If exit EBITDA is 150, what entry multiple does the 786 imply against the exit multiple?
Why might a sponsor bid above this ceiling anyway?

Block	EBITDA	Multiple	Price
Platform	100	11x	1,100
Bolt-on 1	10	6x	60
Bolt-on 2	15	7x	105
Bolt-on 3	25	5x	125
Group	150	9.27x	1,390

Cheque / stake	the value of 100% of the equity implied by the price paid
Net debt	borrowings less cash, 500
EBITDA	160

p	the premium on equity, 20%
E	equity value before the offer, 800
C	net cash, 300

E_0	maximum equity at entry
1.25^5	five years of compounding at 25%, about 3.05
0.5	equity's share of the entry price

Try it first

Why can you not average the multiples?

What does the blend tell you about the buy-and-build story?

Where candidates lose it

What the interviewer asks next

Try it first

Why can you not divide the cheque by EBITDA?

Does it matter whether the money goes to the sellers or into the company?

Where candidates lose it

What the interviewer asks next

Try it first

Why can the premium not land on the cash?

Why would a buyout investor care which premium you quote?

Where candidates lose it

What the interviewer asks next

Try it first

What does EBITDA leave out that matters here?

Why levered free cash flow rather than EBIT?

What is the limit of your answer?

Where candidates lose it

What the interviewer asks next

Try it first

Why work backwards from the exit?

How do you get 1.25 to the fifth without a calculator?

Where candidates lose it

What the interviewer asks next