Private Equity puzzles, solved step by step

Puzzles
100
Traced to a firm
22
Topics
12
Hard
30
Topic
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
Level
AnyWarm upCoreHard
Source
AnyReported at a firmStandard
Showing 1–1 of 1 · filtered from 100Clear filters
  1. 067You make 8x your money in 6 years. Estimate the IRR using the rule of 72, then check it exactly with nothing more than a basic calculator.Returns mathsCoreWarburg PincusNew York · 2012

    Try it first

    What does the rule of 72 give, and is the true IRR higher or lower?

    Show the worked solution

    The rule of 72 gives 36%; the exact IRR is about 41.4%. Eight times is three doublings, so the money doubles every two years, and 72 divided by 2 is 36. The exact rate is 8 to the power one sixth, which is the square root of 2, so the IRR is 1.414 minus 1, 41.4%. The rule runs low because it is tuned for rates near 8%.

    How does the rule of 72 apply to a multiple bigger than 2?

    Think of folding a sheet of paper: each fold doubles the thickness, so eight layers is three folds. Any money multiple that is a power of 2 is a count of doublings, and once you know the years per doubling, the rule of 72 gives the rate. Eight is 2 x 2 x 2, so six years hold three doublings of two years each. The rule says a doubling every two years needs 72 over 2, which is 36% a year.

    Count the doublings first, then correct the rule of 72 upwardsYear 0Year 1Year 2Year 3Year 4Year 5Year 61x2x4x8xdouble in 2 yearsdouble in 2 yearsdouble in 2 yearsRate from the rule of 72 against the exact rateRule of 72: 72 / 236%, gives only 6.33xExact: 8^(1/6) - 1 = root 2 - 141.4%, gives 8.00xThe rule of 72 suits rates near 8%; for a 2-year doubling the right numerator is about 83
    Eight times in six years is three doublings of two years each, which the rule of 72 turns into 36% a year; the exact rate is 41.4%, and 36% compounded for six years only reaches 6.33x.

    How do you get the exact answer with a basic calculator?

    You need the sixth root of 8. Split it: the sixth root is the square root of the cube root, and the cube root of 8 is 2, so the answer is the square root of 2. One press of the square root key gives 1.414, an IRR of 41.4%. Without a square root key, use trial and error: 1.4 squared is 1.96, and 1.96 cubed is about 7.53, a little short of 8, while 1.42 to the sixth is about 8.20, a little over. The answer sits between 40% and 42%.

    The relationship
    (1+r)6=8  ⇒  1+r=81/6=83 1/2=2=1.414(1+r)^6 = 8 \;\Rightarrow\; 1 + r = 8^{1/6} = \sqrt[3]{8}^{\,1/2} = \sqrt{2} = 1.414
    rthe IRR
    8the money multiple
    6years held
    What it says in wordsThe IRR is the sixth root of the multiple, less one; here that is the square root of two, less one.

    Then say why the rule ran low. The rule of 72 is built around rates near 8%; the higher the rate, the bigger the number you should divide into. For a two-year doubling the exact rate implies a numerator of about 83, not 72. Checking the shortfall out loud, 36% for six years gives only 6.33x, shows the interviewer you know the rule's limit.

    Where candidates lose it

    The most common slip is dividing 72 by the six years and answering 12%, as if 8x were a single doubling. Count the doublings first: three.

    The second loss is giving 36% as the final answer. The interviewer who says use the rule of 72 is often waiting to hear that the rule understates at high rates, and that the exact answer is the square root of 2 minus 1.

    What the interviewer asks next

    • What IRR does 3x in 5 years give, by rule of thumb and exactly?
    • 4x in 6 years: rule of 72 and exact?
    • Why does the rule of 72 understate at high rates and overstate at very low ones?

    Asked at Warburg Pincus, Private Equity, New York, 2012 (Wall Street Oasis): If I make 8 times my money in 6 years, what's my IRR? You have to use the rule of 72 to figure this out.