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. 052A company carries debt of 600 against EBITDA of 100, so it is levered 6.0x. It repays 100 of debt every year and EBITDA grows 10% a year. What is leverage after three years?Leverage and capital structureCoreMid-market buyout fundIndian mid-market PE

    Try it first

    Pick the closest figure for leverage at the end of year 3.

    Show the worked solution

    About 2.25x. Three repayments take debt from 600 to 300, and three years of 10% growth take EBITDA from 100 to 133.1. Leverage is 300 over 133.1, or 2.25x, down from 6.0x. Halving the debt alone would have left 3.0x; growth removes most of another turn, and that matters to a lender reading the covenant.

    Why does leverage fall faster than the debt?

    Think of a family loan measured against the family's income. Paying down the loan helps, and so does a pay rise, because the bank looks at how many years of income the loan represents. Leverage is debt divided by EBITDA, so it falls when the top shrinks and when the bottom grows, and here both are happening at once. Track both lines year by year rather than jumping to the end.

    Debt shrinks from the top while EBITDA grows from the bottom600100Year 06.0x500110Year 14.5x400121Year 23.3x300133.1Year 32.25xDebtEBITDALeverage after 3 years if...only debt repaid3.00xonly EBITDA grows4.51xboth happen2.25xDebt halves: divide by 2.000EBITDA x 1.331: divide by 1.3316.0 / 2.000 / 1.331 = 2.25xRepayment does 71%, growth 29%
    Debt falls from 600 to 300 while EBITDA rises from 100 to 133.1, so leverage drops from 6.0x to 2.25x; repayment alone would leave 3.0x and growth alone 4.51x.
    YearDebtEBITDADebt / EBITDA
    0600100.06.00x
    1500110.04.55x
    2400121.03.31x
    3300133.12.25x
    Each year debt falls by a flat 100 and EBITDA rises by 10%, so the ratio drops from 6.00x to 4.55x, 3.31x and finally 2.25x.

    How much of the drop came from growth?

    Split the move multiplicatively, which is exact for a ratio. Halving the debt divides leverage by 2.0; growing EBITDA by 33.1% divides it by 1.331. Six divided by 2.0 and then by 1.331 is 2.25x. Measured that way, repayment does about 71% of the work and growth about 29%, so growth is worth roughly three quarters of a turn here. Say the limit too: the 100 a year of repayment is an assumption, and in a real deal the cash for it comes from the same EBITDA that is growing.

    Where candidates lose it

    The common slip is answering 3.0x: halving the debt and stopping. It treats leverage as a debt number when it is a ratio, and it misses the 33% rise in EBITDA that does a large share of the work.

    The other slip is compounding the debt reduction, as if debt fell 100 then a percentage. The repayment is a flat 100 each year; only EBITDA compounds. Keep the two lines apart and the ratio falls out.

    What the interviewer asks next

    • What leverage would a lender see at year 3 if EBITDA had fallen 10% a year instead?
    • The covenant steps down to 2.5x at year 3. Is there headroom?
    • Where does the 100 a year of repayment actually come from, and is it realistic at 6.0x?
  2. 088A business has EBITDA of 100 and debt of 5x EBITDA at 10%. Capex is 20 and tax is ignored. What is interest cover, and how far can EBITDA fall before free cash flow after interest reaches zero?Leverage and capital structureCorePrivate creditMid-market buyout fund

    Try it first

    How far can EBITDA fall before free cash flow hits zero?

    Show the worked solution

    Interest cover is 2.0x, and free cash flow reaches zero at EBITDA of 70, a 30% fall. Debt of 500 at 10% costs 50. EBITDA of 100 less capex of 20 and interest of 50 leaves 30 of free cash. That 30 is the cushion. At EBITDA of 70 the cover ratio still reads 1.4x, which looks fine just as the cash runs out.

    Why is cover of 2.0x not the cushion it looks like?

    A household earning Rs 1 lakh a month with a Rs 50,000 loan payment has income twice its instalment. But if it also spends Rs 20,000 on things it cannot skip, school fees and rent, only Rs 30,000 is truly spare. Interest cover ignores capex, so the real cushion is free cash flow after interest, 30 here, not the gap between EBITDA and interest. Cover would suggest EBITDA can halve; the cash says it can fall 30%.

    The cushion is the 30 of free cash, not the 2.0x coverEBITDA 100cover 2.0xcapex 20interest 50free cash 30EBITDA 70cover 1.4xcapex 20interest 50free cash 0breakeven at 70: a 30% fallCover still reads 1.4x at the point free cash flow hits zero
    EBITDA of 100 pays capex of 20 and interest of 50 and leaves 30 of free cash; at EBITDA of 70, a 30% fall, free cash is zero while interest cover still reads 1.4x.
    The relationship
    cover=10050=2.0×breakeven EBITDA=20+50=70\text{cover} = \frac{100}{50} = 2.0\times \qquad \text{breakeven EBITDA} = 20 + 50 = 70
    100EBITDA
    50interest, 10% on debt of 500
    20capex the business must spend
    What it says in wordsCover compares EBITDA with interest; the breakeven adds the capex that must also be paid.

    What does a private credit lender do with this?

    A lender tests the downside in cash, not in ratios: how far EBITDA can fall before the company must borrow more, cut capex, or miss a payment. Here the answer is 30%, and a lender would compare it with how far EBITDA fell in the sector's last downturn. If capex could be cut to 10 in a crisis, the cushion widens to a 40% fall, which is why lenders ask how much of capex is maintenance and how much is growth.

    One limitation: with tax ignored the picture is generous. Tax would take a slice of the 30, and working capital swings in a downturn usually absorb cash too.

    Where candidates lose it

    Candidates say EBITDA can halve because cover is 2.0x. That forgets capex, which the business must pay whether or not EBITDA falls.

    The second miss is giving the breakeven as an EBITDA level only. Say the percentage fall too, 30%, because that is the number a credit committee compares with history.

    What the interviewer asks next

    • Add tax at 25% on EBITDA less D&A of 20 less interest. Where is the breakeven now?
    • If the rate rises to 12%, how much cushion is left?
    • Which covenant would you set for this loan, and at what level?