Private Equity puzzles, solved step by step

It is worth 1,050, and growth of about 4.76% makes it worth exactly 20x. Next year's cash flow is 50 x 1.05, or 52.5, and dividing by the 10% discount rate less 5% growth gives 1,050. For 20x, value must be 1,000: solving 50(1 + g) / (0.10 - g) = 1,000 gives g = 50/1,050, about 4.76%. A quarter of a point of growth moves value by 50.

Why does the formula use next year's cash flow?

If you value a fruit tree today, this season's crop has already been picked and sold; what you are buying is next season's crop and every one after it. A perpetuity values the stream of future cash flows, and the first one in that stream arrives a year from now, already grown by 5%. So the numerator is 50 x 1.05, which is 52.5. The denominator is the discount rate less growth, 10% less 5%, which is 5%. 52.5 divided by 0.05 is 1,050, or 21x this year's cash flow.

How do you solve for the growth rate behind a 20x multiple?

Set the value to 1,000, which is 20 x 50, and solve. 50(1 + g) = 1,000 x (0.10 - g), so 50 + 50g = 100 - 1,000g, which gives 1,050g = 50 and g = 4.76%. Dropping growth by just 0.24 of a point, from 5% to 4.76%, takes value down by 50, from 1,050 to 1,000. That is the real lesson of the question, and it is worth saying as a sentence rather than leaving inside the algebra.

Why is a perpetuity so sensitive near the discount rate?

Because the growth rate sits in the denominator as a subtraction. Each point of growth shrinks the gap between the discount rate and growth, and value is one divided by that gap, so the closer growth gets to 10%, the faster value explodes. At 6% the business is worth 1,325; at 8%, 2,700. This is why a terminal value in a buyout model is usually cross-checked against an exit multiple: a small, unexaminable change in a perpetual growth rate can swing the answer by more than any operational assumption. Say also that growth above the long-run growth of the economy cannot last forever, so the formula breaks down when g is set high.

Reinvest 25% and the P/E is 15x; at a 10% return on capital it falls to 10x. To grow 5% a year on a 20% return, the business reinvests 5 over 20, a quarter of profit, and pays out 75 of every 100. 75 over 10% less 5% is 1,500, fifteen times profit. At a 10% return it must reinvest half, pays out 50, and 50 over 5% is 1,000, ten times profit, the same value it would have with no growth at all.

Why does growth cost anything in the first place?

Picture a tiffin service that wants 5% more customers next year. It needs more tiffin boxes and a bigger kitchen, and that comes out of this year's profit before the owner can take anything home. Growth has to be funded, and the amount of profit that must be ploughed back is the growth rate divided by the return the business earns on new capital. A business earning 20% on capital needs to reinvest 5 over 20, a quarter of profit, to add 5% a year. One earning 10% needs 5 over 10, half its profit, for the same growth. The owner of the second business gives up twice as much to get the same thing.

How does the reinvestment rate turn into a multiple?

Value is the cash the owner actually receives, growing at 5%, discounted at 10%. With 20% returns the owner receives 75 of every 100, so value is 75 over 5%, which is 1,500 and a P/E of 15x; with 10% returns the owner receives only 50, and 50 over 5% is 1,000, a P/E of 10x. Compare that with no growth at all: the owner receives the whole 100, value is 100 over 10%, which is also 1,000 and 10x. So growth at a 10% return on capital has moved the multiple from 10x to exactly 10x. The 5% growth consumed exactly as much as it created, because each rupee reinvested earned precisely the rate the owner demanded.

What is the lesson for a buyer, and where does the formula break?

That a growth story is only worth paying for when the growth is earned at a return above the cost of capital. Below 10% in this example, growth destroys value: the business would reinvest more than it earns on the reinvestment, and the multiple would fall below the no-growth 10x. That is why a buyout investor asks what return the incremental capital earns before asking how fast revenue grows. The limits are real: the formula assumes the growth and the return last forever, that profit is a fair proxy for cash, and that the business can keep finding projects at the same return, which it usually cannot as it grows.

The EBITDA margin is 20%. Both multiples share the same enterprise value on top. Divide EV/Sales by EV/EBITDA and EV cancels, leaving EBITDA divided by sales: 2 over 10, which is 20%. Check it with any EV you like: at 1,000, sales are 500 and EBITDA is 100, and 100 is 20% of 500.

Why does the enterprise value not matter?

A cricket bat costs as much as 10 balls, and as much as 2 sets of pads. How many balls is a set of pads worth? Five, and you never needed the price of the bat. When two ratios share the same top line, dividing one by the other cancels it and leaves the ratio of the two bottom lines. Here the shared top line is enterprise value, and the bottom lines are sales and EBITDA.

How do you check the direction of the division?

A margin must be smaller than 100%, and EBITDA is a slice of sales, so EBITDA has to be the smaller number. The higher multiple sits on the smaller number, so the margin is the low multiple over the high multiple, never the other way. Dividing 10 by 2 gives 5, which would be a 500% margin and is impossible. Dividing 2 by 10 gives 0.2.

A buyout investor uses this the other way round all the time. If comparable companies trade at 2x sales and the target earns a 10% margin, then 2x sales is 20x EBITDA for this target, which is expensive. Sales multiples hide margin differences; converting to EBITDA puts them back. The limitation: EBITDA multiples carry their own blind spots, such as heavy capital spending that EBITDA leaves out.

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.

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.

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.

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.

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.

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.

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.

EBITDA is 100. Net income of 40 at 15x gives equity value of 600. Add net debt of 200 to reach enterprise value of 800. The EV/EBITDA multiple of 8x then gives EBITDA of 800 divided by 8, which is 100. Two multiples and the bridge between equity and enterprise value pin down the missing line.

Why can you not go straight from net income to EBITDA?

Net income sits below interest, tax, depreciation and amortisation, and the question gives none of them. The two multiples work on different values, P/E on equity and EV/EBITDA on the whole business, so the route runs through value, not through the income statement. It is like knowing a flat's price per square foot and the loan on it: you get to the total value first and then back out what you need.

What follow-up can you get ahead of?

Interviewers often ask what the gap between EBITDA of 100 and net income of 40 is made of. Sixty of EBITDA goes on depreciation, interest and tax, and with net debt of 200 interest is only a modest slice, so depreciation or tax must be large. At an assumed 8% rate, interest would be 16, leaving 44 for depreciation and tax. Saying that shows you read the result, not only compute it.

State one assumption as you go: net debt here is all the claims that sit between equity and enterprise value. If there were minority interests or preference shares, they would be added too and EBITDA would come out higher.

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.

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.