Private Equity puzzles, solved step by step

360, or 90% of the sponsor's cost. Debt at 2x EBITDA of 120 is 240; at 5x it is 600. The 360 of new borrowing is paid out as a dividend, and 360 over the original 400 is 90%. The sponsor gets most of its money back in year three while keeping full ownership, and the business now carries two and a half times the debt it did.

What exactly is a dividend recap doing?

Think of a family that has paid down most of its home loan, then takes a fresh, larger loan against the same house and spends the cash. Nothing about the house changed; the family simply took equity out and left a bigger loan behind. A dividend recapA recapitalisation in which a company borrows new debt and pays the proceeds to its shareholders as a dividend. borrows against the company's current earnings and hands the cash to the owner, so the only new money is the increase in debt. Here that increase is 600 less 240, which is 360.

Has the sponsor made money, or just moved it?

Mostly moved it. Suppose the business is worth 10x EBITDA, which is 1,200. Before the recap the sponsor's equity is worth 1,200 less 240 of debt, which is 960. After it, the equity is worth 1,200 less 600, which is 600, plus 360 already in the bank: the same 960. A recap does not create value; it changes when the sponsor is paid and how much risk sits on the remaining equity. The returns arithmetic still improves, because cash returned in year three lifts the IRR and takes 90% of the cost off the table, which is why sponsors use the tool when credit markets are open.

Who carries the risk afterwards?

The company and its lenders. At an assumed 8% interest rate, interest rises from 19.2 to 48 a year, so EBITDA covers interest 2.5 times instead of 6.25. A 20% fall in EBITDA, to 96, now takes cover down to exactly 2x, a level at which many lenders start to worry, while the sponsor's money is already mostly home. Say the limitation too: lenders price this risk, covenants may cap the payout, and a board must be satisfied the company stays solvent after paying the dividend.

EBITDA must grow by 33.3% just to hand the sponsor its money back. Take EBITDA of 100: the business costs 1,200, funded by 600 of debt and 600 of equity. With no paydown the debt is still 600 at exit, so equity is 600 only if the business is again worth 1,200. At 9x that needs EBITDA of 133.3, which is 12 over 9 minus 1.

Why is the answer not 25%?

Picture a flat bought for 12 years of rent and sold later when buyers will pay only 9 years of rent. To get your price back, the rent has to rise until 9 years of it equals the old 12. That is a rise of 12 over 9, a third, not a quarter. A 25% cut in the multiple needs a 33.3% rise in earnings to undo it, because the recovery is measured on the smaller multiple. It is the same base effect that makes a 10% fall need an 11.1% rise.

Why does the 50% debt not change the answer?

Write the equity at exit as exit value less debt. The debt was 600 going in and, with no repayment, is 600 coming out. Equity is returned in full only when the exit value equals the entry value, so the break-even growth is the same at 30% debt, 50% debt or no debt at all. Leverage changes how fast equity grows once you pass break-even, and how fast it is wiped out below it. It does not move the break-even itself when nothing is repaid.

What does this do to a real growth plan?

Say the plan grows EBITDA 50% to 150. Sold at 12x, the business would be worth 1,800 and the equity 1,200, double the cheque. Sold at 9x it is worth 1,350 and the equity 750, only 1.25x. The first third of the growth plan is spent refilling the hole left by the lower multiple. This is why sponsors who pay high multiples underwrite their exit at the entry multiple or lower, and why the interviewer asks the question.

The company must repay 320, which is the whole equity cheque. The business costs 800, funded by 480 of debt and 320 of equity. With flat EBITDA and the same multiple it is still worth 800 at exit, so equity of 640 needs debt to fall to 160. That is 80 a year for four years, equal to all of EBITDA, so in practice this deal cannot reach 2x without growth.

Why does the repayment equal the equity cheque?

Think of a house bought for Rs 80 lakh with a Rs 48 lakh loan and Rs 32 lakh of your own money. If the house is still worth Rs 80 lakh years later, the only way your share doubles to Rs 64 lakh is for the loan to shrink to Rs 16 lakh. When the enterprise value does not move, equity gains exactly what the debt loses, so doubling the equity means repaying an amount equal to the original equity. That holds at any leverage, which makes it a fast check.

Is that repayment realistic?

Now test the answer against the business. Repaying 80 a year means every rupee of EBITDA goes to the lenders as principal. Interest on 480 of debt at 8% is about 38 in year one alone, before any tax or capex, so the cash to repay 80 a year simply does not exist. The honest conclusion is the useful one: a flat business bought at 10x with 6x debt cannot reach 2x from paydown. It needs EBITDA growth, a higher exit multiple, or a lower entry price.

At 10x they tie: each adds 100, taking equity from 600 to 700. One turn adds one year of EBITDA, 100. Ten per cent more EBITDA adds 10% of EBITDA times the multiple, 10 x 10, also 100. Above 10x the EBITDA growth adds more, below 10x the turn does. The crossover is one divided by the growth rate.

Why do the two levers tie at exactly 10x?

Enterprise value is EBITDA times the multiple, a product of two numbers, like a shop's takings being customers times average spend. A 10% rise in either factor lifts the product by 10%. One extra turn is a 10% rise in the multiple only when the multiple is 10, which is why the two levers tie there and nowhere else. At 8x, one turn is a 12.5% rise in the multiple; at 14x it is about 7%.

What does the equity holder actually feel, and which lever would you underwrite?

Net debt does not change in either case, so the full 100 lands on the equity: 600 to 700, a 16.7% lift. The arithmetic ties, but the two levers are not equally bankable: EBITDA growth is something the sponsor can plan and track, while the exit multiple is set by the market on the day of sale. That is why most buyout cases hold the exit multiple at or below entry and earn the return from EBITDA and debt paydown.

Give the general rule after the number: at 8x, 10% more EBITDA adds only 80 against 100 for a turn; at 14x it adds 140. One limitation is worth a sentence: higher EBITDA usually also brings extra cash that pays down debt, so in a full model growth gets a small bonus the turn does not.

About 1.38x, an IRR of roughly 11%. Year one cash flow is EBITDA 100 less interest 40, tax 10 and capex 15: 35. Interest falls as debt is repaid, so cash rises to about 38 and 40, and debt ends near 287. Exit at 700 leaves equity of about 413 on 300. Holding cash flat at 35 gives a quick 1.35x.

What do you set up before the years?

Entry price 700, debt 400, so equity 300. Then one year of cash. Think of a rented flat bought with a loan: the rent does not rise and the flat's price does not move, so the only way your stake grows is that rent left over after costs pays the loan down. With flat EBITDA and the same exit multiple, enterprise value is the same 700 at exit, so every rupee of equity gain is a rupee of debt repaid.

Why does the cash grow each year when EBITDA is flat?

Interest is charged on a shrinking balance. Every rupee repaid saves 10 paise of interest next year, 7.5 paise after tax, so cash flow climbs from 35 to 37.6 to 40.4 with no change in the business. That compounding is small over three years, which is why the quick answer of 35 a year, 105 in total and 1.35x, is close enough to say first. Then refine it if asked.

Close with the judgement the interviewer wants. A 11% IRR from debt paydown alone is below most buyout targets, so this deal needs EBITDA growth or a cheaper entry to work. That sentence turns the arithmetic into a view.