Portfolio Management puzzles, solved step by step

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  1. 077Money doubles every six years. How long does it take to grow to eight times, and roughly what annual return does that imply?Compounding and fee dragWarm upWealth managementAsset management

    Try it first

    How many years to reach eight times?

    Show the worked solution

    Eighteen years, at about 12.2% a year. Eight is two times two times two, so reaching eight times takes three doubling periods of six years each. The return that doubles money in six years is 2 to the power one sixth, less one, which is 12.2%; the rule of 72 gives 12% as a quick check.

    Why count doublings instead of dividing?

    A rumour passed on by each listener to two new people reaches 2, then 4, then 8. Nobody counts that as adding 2 each round; it doubles each round. A compounding balance works the same way, so any multiple that is a power of two is just a count of doubling periods. Eight is two cubed: three doublings, eighteen years. Sixteen times would be four doublings, twenty four years.

    Eight times is three doublings of six years each1x2x4x8xYear 0Year 62xYear 124xYear 188xx 2x 2x 2Doubling every 6 years means2^(1/6) - 1 = 12.2% a yearRule of 72 check: 72 / 6 = 12%
    Money that doubles every six years is worth 2 times at year 6, 4 times at year 12 and 8 times at year 18, and the smooth path between those points is a steady 12.2% compounded each year.

    How do you get the annual return without a calculator?

    Use the rule of 72: the doubling time multiplied by the rate is roughly 72, so 72 divided by 6 is 12%. The exact figure is 12.25%, and the rule of 72 is within a quarter point anywhere from about 6% to 12%. Give 12% first, then say it is a shade over, which shows you know the rule is an approximation.

    The relationship
    (1+r)6=2  ⇒  r=21/6−1≈12.2%(1+r)^6 = 2 \;\Rightarrow\; r = 2^{1/6} - 1 \approx 12.2\%
    rthe annual return
    6years to double
    What it says in wordsThe annual return is the sixth root of two, less one.

    Portfolio managers ask this because clients think in multiples and managers think in annual rates. Being able to move between the two in your head is the whole skill.

    Where candidates lose it

    The fast wrong answer is 48 years, eight times six, which treats the multiple as a count of periods. It is the same straight-line habit that makes people underestimate what compounding does over a career.

    The quieter loss is saying 12% and stopping. Give 12% from the rule of 72, then correct it to about 12.2%, so the interviewer sees you know where the shortcut comes from.

    What the interviewer asks next

    • How long to reach 10 times at the same rate?
    • If fees take 1.5% a year off that return, how long does one doubling take?
    • Why does the rule of 72 get worse at very high rates?