Private Equity puzzles, solved step by step

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  1. 006A sponsor invests 300 of equity in a buyout. Four years later it sells, and the equity is worth 750. What are the MOIC and the IRR?Mental paper LBOsWarm upWarburg PincusSan Francisco · 2014

    Try it first

    Which IRR is closest?

    Show the worked solution

    The MOIC is 2.5x and the IRR is about 25.7%. MOIC is money out over money in: 750 divided by 300. The IRR is the yearly rate that compounds 300 into 750 over four years, the fourth root of 2.5. Take the square root twice: 2.5 becomes 1.581, then 1.257. Check: 1.25 to the fourth is 2.44, just short of 2.5, so the rate sits a little above 25%.

    Why work out the multiple before the rate?

    Picture a plant that grew from 30 centimetres to 75 in four years. Asking how many times bigger it got is one division; asking how fast it grew each year needs a root. Getting the multiple first strips the problem down to one number, 2.5, so the only hard step left is a fourth root. In a paper LBO the multiple is also what the interviewer will ask for first, because it is the number a sponsor quotes when it talks about a deal.

    300 to 750 in four years: the same x1.257 step every year300Year 0377Year 1x1.257474Year 2x1.257596Year 3x1.257750Year 4x1.257dashed: a straight line, which is nothow compounding growsStep 1: the multiple750 / 300 = 2.5xStep 2: the fourth rootsquare root of 2.5 = 1.581square root of 1.581 = 1.257IRR = 25.7%
    Compounding at 25.7% a year takes 300 to 377, 474, 596 and then 750, the same x1.257 step each year, which is why a 2.5x multiple over four years is a 25.7% IRR and not the 37.5% a straight line suggests.

    How do you take a fourth root in your head?

    A fourth root is a square root of a square root. The square root of 2.5 is about 1.58, because 1.58 squared is about 2.5, and the square root of 1.58 is about 1.257, because 1.26 squared is 1.59. Then check from the other side with a round rate you can compound: 1.25 squared is 1.5625, and squared again is 2.44. That is a little short of 2.5, so the IRR is a little above 25%, which matches 25.7%.

    The relationship
    MOIC=750300=2.5IRR=2.51/4−1≈25.7%\text{MOIC} = \frac{750}{300} = 2.5 \qquad \text{IRR} = 2.5^{1/4} - 1 \approx 25.7\%
    MOICmultiple on invested capital: equity out divided by equity in
    1/4the fourth root, because the money is invested for four years
    What it says in wordsThe multiple tells you how much the money grew; the root of the multiple tells you how fast it grew each year.

    What would change the IRR without changing the multiple?

    Time and interim cash. The same 2.5x earned over five years is about 20%, and over three years about 36%, so a sponsor always quotes both numbers together. Say also that this calculation assumes one cash flow in and one out. A dividend paid in year two, or a second equity cheque in year one, means the IRR has to be solved over several cash flows, and the shortcut gives only an approximation.

    Where candidates lose it

    The fast wrong answer is 37.5%: a 150% gain split evenly over four years. It is the most common error in paper LBOs because the multiple and the rate get mixed up under time pressure.

    The other loss is a correct number with no check. A fourth root guessed at 26% sounds lucky; 1.25 to the fourth equals 2.44, so slightly above 25% sounds reasoned.

    What the interviewer asks next

    • What IRR does 2.5x give over five years?
    • The sponsor also took a 100 dividend in year two. Is the IRR higher or lower, and roughly by how much?
    • What exit equity value gives a 30% IRR over four years?

    Asked at Warburg Pincus, Private Equity, San Francisco, 2014 (Wall Street Oasis): The interviews consisted of case studies and technical questions (paper LBO, basic returns, IRR, etc.)

  2. 044Work backwards: EBITDA grows from 100 to 140 over 5 years and the business exits at 9x. Entry debt is 5x EBITDA and 200 of it is repaid by exit. What is the highest entry multiple that still returns 3x the equity?Mental paper LBOsHardNeuberger BermanNew York · 2022

    Try it first

    Where does the maximum entry multiple land?

    Show the worked solution

    About 8.2x. Exit EV is 140 x 9 = 1,260. Debt falls from 500 to 300, so exit equity is 960. A 3x return allows entry equity of 960 / 3 = 320. Add the 500 of entry debt and the most you can pay is an EV of 820, which is 8.2x entry EBITDA. Pay more and the 3x target is missed; fees and costs would lower the ceiling further.

    Why work backwards instead of guessing an entry price?

    If you want to arrive at a wedding by 7 and the drive takes two hours with half an hour of traffic, you leave at 4:30. You fix the end point and walk back. A target multiple fixes the exit equity you need relative to the cheque, so the maximum price is found by running the LBO in reverse: exit value, less exit debt, divided by the target, plus entry debt. Every number you need is in the question.

    Start at the exit and walk back to the most you can pay1,260Exit EV140 x 9300Exit debt500 - 200960Exit equity1,260 - 300320Entry equity960 / 3500Entry debt5 x 100820Max entry EV320 + 500read right to left820 / 100 = 8.2x
    Exit EV of 1,260 less 300 of remaining debt leaves exit equity of 960, which supports entry equity of 320 at a 3x target; adding 500 of entry debt sets the maximum entry EV at 820, or 8.2x EBITDA of 100.
    The relationship
    EV0max=E5×mexit−D5target+D0=1260−3003+500=820EV_{0}^{max} = \frac{E_5 \times m_{exit} - D_5}{\text{target}} + D_0 = \frac{1260 - 300}{3} + 500 = 820
    E_5 x m_exitexit EV, 140 x 9
    D_5debt left at exit, 300
    targetthe money multiple required, 3x
    D_0entry debt, 5 x 100 = 500
    What it says in wordsThe most you can pay is the exit equity divided by the target multiple, plus the debt you borrow at entry.

    What does 8.2x tell the investment committee?

    It is a ceiling, not an offer. Entering at 8.2x and exiting at 9x means the plan relies on a little multiple expansion as well as 40% EBITDA growth and 200 of debt paydown, so the committee will test each of those three. 3x over five years is an IRR of about 25%: 3 to the power one fifth is about 1.246.

    Say the limitations. Transaction fees and financing costs come out of the equity at entry, so the true ceiling on the headline price is lower. A real model would also hold some cash at exit and might pay interest in kind. And if the lenders will not provide 5x at an 8.2x price, the equity cheque grows and the ceiling falls.

    Where candidates lose it

    The common loss is forgetting to add the entry debt back: candidates divide exit equity by 3, get 320, and say 3.2x. That is the equity cheque over EBITDA, not the enterprise multiple.

    The second loss is using entry debt instead of exit debt when computing exit equity, which ignores the 200 repaid and makes the ceiling look lower than it is. Track debt at both ends.

    What the interviewer asks next

    • If the exit multiple is 8x instead of 9x, what is the new ceiling?
    • How much does the ceiling fall if 20 of fees are paid out of the equity at entry?
    • What IRR does 3x in five years imply, and what multiple would 25% need?

    Asked at Neuberger Berman, Private Equity, New York, 2022 (Wall Street Oasis): Interviews 4-5 were very technical again and also included multiple paperback LBOs and other, more advanced technicals.

  3. 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