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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.)