Private Equity puzzles, solved step by step

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All topicsCredit and PIK maths8Returns maths10Mental paper LBOs8Operating levers and margin maths8Valuation riddles10Fund economics numeracy9Market sizing and estimation9Compounding and time value7Mental maths8Probability and expected value in deals8Leverage and capital structure9Logic and brainteasers6
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  1. 003In year three of a buyout, EBITDA is 120 and debt sits at 2x EBITDA. The sponsor re-levers to 5x and pays all the new debt out as a dividend. The sponsor's original equity cheque was 400. What share of its cost does it get back?Mental paper LBOsCoreLarge-cap buyout fundPrivate credit

    Try it first

    Pick the share of the 400 that comes back.

    Show the worked solution

    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.

    Re-levering from 2x to 5x: 360 of new debt goes straight to the sponsorBeforedebt 240 (2.0x)After240+360 new debt= 600 (5.0x)Dividend to the sponsor: 360= 90% of the 400 it investedInterest at an assumed 8%Before: 19.2 a year, EBITDA covers it 6.25xAfter: 48.0 a year, EBITDA covers it 2.50xAt 10x EBITDA the business is 1,200.Equity 960 before; 600 + 360 cash after.Same 960 in total, more risk on it.
    Re-levering EBITDA of 120 from 2x to 5x lifts debt from 240 to 600, and the 360 difference is paid to the sponsor as 90% of its original 400, while interest cover at an assumed 8% rate falls from 6.25x to 2.50x.

    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.

    Where candidates lose it

    The common error is answering 600 over 400, or 150%, by treating the whole new debt level as the payout. The 240 already on the balance sheet was borrowed at the start of the deal and spent on the purchase price; only the increase is new cash.

    The second miss is calling the recap free money. Say in one sentence that total sponsor value is unchanged at the same valuation and that the risk has moved onto the balance sheet.

    What the interviewer asks next

    • What is the sponsor's DPI after the recap, and what does it mean for its IRR?
    • Why might lenders agree to fund a recap at 5x?
    • If EBITDA falls to 90 the year after, what is leverage and what is interest cover?
  2. 046In your head: you buy a business with EBITDA of 50 at 8x, funded 50% with debt. EBITDA grows to 75 over 5 years, cumulative free cash flow of 100 repays debt, and you exit at 8x. What are the money multiple and the IRR?Mental paper LBOsCoreNeuberger BermanNew York · 2022

    Try it first

    What is the money multiple on the equity?

    Show the worked solution

    2.5x and an IRR of about 20%. Entry EV is 50 x 8 = 400, half debt, so equity is 200. Exit EV is 75 x 8 = 600, and debt has fallen from 200 to 100, so exit equity is 500. 500 over 200 is 2.5x. Over five years that is about 20% a year, since 1.2 to the fifth is about 2.49.

    What is the fastest way to do this in your head?

    Buy a flat for Rs 40 lakh with a Rs 20 lakh loan. Five years later it is worth Rs 60 lakh and you have paid the loan down to Rs 10 lakh. Your stake went from Rs 20 lakh to Rs 50 lakh. Every paper LBO is two balance sheets, entry and exit, and the answer is exit equity over entry equity, where equity is EV less debt each time. Write the four numbers, subtract twice, divide once.

    Two balance sheets and a subtractionEV 4008x EBITDA 50Debt 200Equity 200Entry, year 0EV 6008x EBITDA 75Debt 100Equity 500Exit, year 5500 / 2002.5xIRR about 20%Where the 300 gain came fromgrowth 200debt 10025 more EBITDA x 8 = 200cash flow repays 100 of debtmultiple unchanged, adds 0
    Entry EV of 400 is funded with 200 of debt and 200 of equity; at exit EV is 600 and debt 100, so equity is 500, a 2.5x return made up of 200 from EBITDA growth and 100 from debt paydown.

    How do you get the IRR without a calculator?

    Carry three anchors for a five-year hold. 2x in five years is about 15% a year, 2.5x is about 20%, and 3x is about 25%. 1.2 to the fifth is 1.2 x 1.2 = 1.44, then 1.73, 2.07 and 2.49, so 2.5x is a touch over 20%: 20.1%. Saying the anchor and then the check sounds far more assured than reaching for a formula.

    The relationship
    MOIC=E5 m−D5E0 m−D0=75×8−10050×8−200=500200=2.5xMOIC = \frac{E_5 \, m - D_5}{E_0 \, m - D_0} = \frac{75 \times 8 - 100}{50 \times 8 - 200} = \frac{500}{200} = 2.5x
    E_0, E_5EBITDA at entry and exit, 50 and 75
    mthe multiple at both ends, 8x
    D_0, D_5debt at entry and exit, 200 and 100
    What it says in wordsThe money multiple is exit equity over entry equity, each one EV less debt.

    Then split the gain, because the interviewer will ask where the return came from. EBITDA rose 25 at a constant 8x, adding 200 of value; cash flow repaid 100 of debt; the multiple did not move, so it added nothing. The limitation: the sums ignore fees at entry and any cash left on the balance sheet at exit, both of which a real model would carry.

    Where candidates lose it

    The common loss is computing the EV return, 600 over 400, and calling it 1.5x. The question is about the equity, and leverage plus paydown makes the equity grow much faster than the EV.

    The second loss is forgetting the debt paydown and using 200 of debt at exit, which gives 400 of equity and 2.0x. Track the debt at both ends.

    What the interviewer asks next

    • Exit at 7x instead of 8x. What are the multiple and IRR now?
    • What would the return have been with no debt at all?
    • How much of the return would you attribute to the sponsor's work, and how much to leverage?

    Asked at Neuberger Berman, Private Equity, New York, 2022 (Wall Street Oasis): The most difficult was the more advanced industry-specific technicals and paperback LBOs

  3. 055A sponsor buys a business with EBITDA of 80 at 10x, funded 60% with debt. EBITDA stays flat for four years and the exit is also at 10x. How much debt must be repaid for the sponsor to make 2x its money?Mental paper LBOsCoreMid-market buyout fundIndian mid-market PE

    Try it first

    How much debt has to be repaid over the four years?

    Show the worked solution

    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.

    No growth, no multiple change: 2x has to come entirely out of the debtDebt 480Equity 320Entry: EV 800Debt 160Equity 640Exit: EV 800Repay320Each year, Rs croreEBITDA available80Needed: repayment + year 1 interest at 8%8038320 / 4 years = 80 a year= 100% of EBITDA, beforeinterest, tax or capexNot achievable: growth is needed
    With enterprise value stuck at 800, equity doubles from 320 to 640 only if debt falls from 480 to 160; that repayment of 320 is 80 a year, equal to all of EBITDA before year 1 interest of about 38.

    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.

    The relationship
    Repayment=(2−1)×E0=320IRR=21/4−1≈18.9%\text{Repayment} = (2 - 1) \times E_0 = 320 \qquad \text{IRR} = 2^{1/4} - 1 \approx 18.9\%
    E_0the sponsor's equity at entry, 320
    2the target money multiple
    1/4four years of hold
    What it says in wordsWith no change in enterprise value, the debt repaid must equal the gain you want on the equity.

    Where candidates lose it

    Candidates get 320 and stop, which wins half the point. The interviewer is waiting for the second sentence: 80 a year is all of EBITDA, so the plan does not work. Answering the arithmetic without testing it against the cash flow misses why the question was asked.

    The other slip is answering 160, the debt left at exit, instead of the 320 repaid. Say both numbers so there is no doubt which you mean.

    What the interviewer asks next

    • How much EBITDA growth gets the same deal to 2x with no debt repaid at all?
    • If the exit multiple rises to 11x, how much repayment do you still need?
    • 2x in four years is about 18.9% a year. Would a fund accept that on this risk?