Private Equity puzzles, solved step by step

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  1. 049EBITDA grows from 100 to 250 over 6 years. What is the compound annual growth rate?Compounding and time valueWarm upMid-market buyout fundIndian mid-market PE

    Try it first

    Pick the CAGR.

    Show the worked solution

    About 16.5% a year. CAGR is the ratio of end to start, 2.5, raised to one over the number of years, less one. 1.15 to the sixth is about 2.31 and 1.17 to the sixth about 2.57, so the rate sits about three quarters of the way between them, near 16.5%. Dividing the 150% gain by 6 gives 25%, which would compound to 381.

    Why is 25% wrong?

    A savings account that pays interest on interest grows faster each year, so it needs a lower rate than you might think to reach a target. CAGR is the single rate that, compounded every year, turns the start into the end, so it is the sixth root of 2.5, not 150% divided by six. 25% compounded for six years would reach 381, far past 250.

    The right rate compounds to 250; 25% a year compounds to 381100200300400Yr 0Yr 1Yr 2Yr 3Yr 4Yr 5Yr 625% a year: 38125016.5% a yearBracket it1.15^6 = 2.311.17^6 = 2.572.50 is 0.74 ofthe way upso about 16.5%
    Compounding at 16.5% a year takes EBITDA from 100 to 250 in six years, while the 25% from dividing 150% by six would compound to 381; bracketing between 1.15 and 1.17 to the sixth finds the rate.
    The relationship
    CAGR=(250100)1/6−1≈16.5%CAGR = \left(\frac{250}{100}\right)^{1/6} - 1 \approx 16.5\%
    250 / 100the ratio of end value to start value, 2.5
    1/6one over the number of years
    What it says in wordsThe compound growth rate is the ratio of end to start, rooted by the number of years, less one.

    How do you find a sixth root in your head?

    Bracket it with rates whose sixth powers you can build. 1.15 squared is 1.3225, cubed that is about 2.31. 1.17 squared is 1.3689, cubed that is about 2.57. 2.5 sits about 0.74 of the way from 2.31 to 2.57, so the rate is about 15% plus 0.74 of 2 points, near 16.5%. A cross-check with the rule of 72: at 16.5% money doubles in about 4.4 years, and 2.5x in 6 years is a little more than one doubling, which fits.

    Say what the number hides. A CAGR smooths the path: a business could have been flat for four years and then jumped, and the CAGR would be the same. A buyout investor asks for the yearly figures before trusting the rate, and checks whether the 250 includes acquisitions.

    Where candidates lose it

    The common loss is dividing the total growth by the years and saying 25%. That is the average simple growth, and it overstates the compound rate badly over six years.

    The second loss is freezing on the sixth root. You do not need logarithms; bracket the rate between two you can compute and slide.

    What the interviewer asks next

    • What CAGR turns 100 into 300 over 5 years?
    • If EBITDA grew 40% in year one and was flat after, what is the CAGR over six years?
    • How would you strip acquired EBITDA out of the growth rate?
  2. 082Money compounds at 9% a year. Roughly how long does it take to double by the rule of 72, and how close is that to the exact answer?Compounding and time valueWarm upMid-market buyout fund

    Try it first

    Answer inside five seconds.

    Show the worked solution

    About 8 years by the rule of 72, and 8.04 years exactly. Divide 72 by the rate in per cent: 72 over 9 is 8. The exact answer is the log of 2 over the log of 1.09. The rule is near exact around 8% a year and drifts at very low or very high rates, where 69 or 70 works better.

    Why does dividing 72 by the rate work?

    Doubling needs the growth factor to reach 2, and the log of 2 is about 0.693. For small rates, the log of 1 plus r is close to r, so doubling time is about 69.3 divided by the rate in per cent. 72 is used instead of 69 because it divides cleanly by 2, 3, 4, 6, 8, 9 and 12, and because it corrects for the approximation at the rates people use most. It is a calculator in your head, like knowing that a dozen eggs at Rs 6 each is Rs 72.

    At 9% money doubles in just over 8 years; the rule of 72 says 82x1xexact: 8.04 yearsrule of 72: 72 / 9 = 80246810years at 9%RateRule of 72ExactGap4%18.017.67+0.338%9.09.01-0.019%8.08.04-0.0412%6.06.12-0.1224%3.03.22-0.22Closest around 8%; drifts at the extremes.
    At 9% a year money crosses twice its starting value at 8.04 years against the rule of 72's 8, and across rates the rule stays within about a tenth of a year from 8% to 12% but drifts to 0.33 years at 4% and 0.22 years at 24%.
    The relationship
    t=ln⁡2ln⁡(1.09)=0.6930.0862≈8.04rule: 729=8t = \frac{\ln 2}{\ln(1.09)} = \frac{0.693}{0.0862} \approx 8.04 \qquad \text{rule: } \frac{72}{9} = 8
    tyears to double
    ln 2natural log of 2, about 0.693
    ln(1.09)natural log of the growth factor, about 0.0862
    What it says in wordsExact doubling time is log 2 over log of one plus the rate; the rule of 72 approximates it with simple division.

    Where does a buyout interviewer use this?

    Everywhere returns are quoted. A deal that doubles the money in about four years has an IRR near 18%, and one that doubles in about three years is near 24%, because 72 over 4 is 18 and 72 over 3 is 24. That lets you check a quoted IRR against a quoted money multiple in seconds. At 24% the rule says 3 years and the truth is about 3.2, so for high-return deals you shade the rule slightly.

    Where candidates lose it

    The common slip is using simple interest and saying about 11 years, 100 divided by 9. Compounding means each year's interest itself earns interest, so doubling comes sooner.

    The other is giving 8 and stopping when asked how exact it is. Know that the rule is closest around 8% and that the true figure here is just over 8 years.

    What the interviewer asks next

    • How long does it take money to triple at 9%?
    • A deal returns 2x in 3 years. Roughly what IRR is that?
    • Why does the rule of 72 overstate doubling time at low rates and understate it at high rates?