Private Wealth Management puzzles, solved step by step

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100
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Hard
30
Topic
All topicsCompounding and doubling8Returns arithmetic9Fee and cost drag8Inflation and real return7Tax arithmetic7Probability and risk of loss9Retirement and withdrawal8Fixed income numeracy8Behavioural traps8Estimation and sizing8Options and structured products6Leverage and borrowing6Wealth business economics8
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  1. 053At 9% a year, roughly how long does Rs 10 lakh take to become Rs 80 lakh? Answer in your head, then check it.Compounding and doublingWarm upIndian wealth management

    Try it first

    Your first answer, inside ten seconds?

    Show the worked solution

    About 24 years. Rs 10 lakh to Rs 80 lakh is eight times the money, which is three doublings. At 9% the rule of 72 gives 72 / 9 = 8 years per doubling, so three doublings take about 24 years. The exact figure, the log of 8 over the log of 1.09, is 24.13 years, so the shortcut is off by under two months.

    Why count doublings instead of years?

    A staircase is easier to climb in your head than a ramp. Compound growth is a ramp, but every doubling takes the same number of years at a fixed rate, so you can turn it into equal steps. First ask how many times the money must double, then multiply by the years one doubling takes. Eight times is 2 x 2 x 2, three doublings, and at 9% each takes about 8 years.

    Count the doublings, not the years: Rs 10 lakh to Rs 80 lakh at 9%10 lakhYear 020 lakhYear 8x2 in 8 yrs40 lakhYear 16x2 in 8 yrs80 lakhYear 24x2 in 8 yrsexact: 24.1yearsRule of 72: 72 / 9 = 8 years per doubling. Three doublings = 24 years.
    At 9% a year Rs 10 lakh reaches Rs 20 lakh in about 8 years, Rs 40 lakh in about 16 and Rs 80 lakh in about 24, because every doubling takes the same time; the exact path reaches Rs 80 lakh in 24.1 years.

    How good is the rule of 72 at 9%?

    Very good. The exact doubling time at 9% is the log of 2 over the log of 1.09, which is 8.04 years, against 8.00 from the rule. The rule of 72 is most accurate for rates around 8%, which is why it serves so well for Indian savings and equity assumptions. Three doublings carry the small error three times, which is how 24 becomes 24.13.

    The relationship
    t=ln⁡8ln⁡1.09=2.0790.0862≈24.13 yearst = \frac{\ln 8}{\ln 1.09} = \frac{2.079}{0.0862} \approx 24.13\ \text{years}
    8the growth multiple needed, 80 lakh over 10 lakh
    1.09one year of growth at 9%
    tyears needed
    What it says in wordsThe years needed are the log of the multiple divided by the log of one year's growth.

    Then turn it into a client sentence. Rs 10 lakh at 9% needs a working lifetime, not a decade, to reach Rs 80 lakh, and the last doubling, from Rs 40 lakh to Rs 80 lakh, adds more rupees than the first two combined. That is the argument for starting early, said with numbers rather than slogans.

    Where candidates lose it

    Candidates divide 80 by 10, get 8, and then try to compound year by year in their head. They lose the room in arithmetic. The move is to see that 8 is 2 cubed before touching the rate.

    The second slip is quoting 8 years, which is the time for one doubling, because the rule of 72 is the first thing that comes to mind. Say how many doublings first, then the time for each.

    What the interviewer asks next

    • How long does the same money take to reach Rs 1 crore?
    • At 12% instead of 9%, how many years does the client save on the way to Rs 80 lakh?
    • Why is the rule of 72 less accurate at 20% a year?
  2. 077A client tells you his Rs 5 lakh investment became Rs 20 lakh in 12 years. Without a calculator, what annual growth rate is that?Compounding and doublingWarm upWealth management

    Try it first

    Your instinct, in five seconds.

    Show the worked solution

    About 12% a year. Rs 5 lakh to Rs 20 lakh is four times, which is two doublings. Two doublings in 12 years means one every 6 years, and the rule of 72 gives 72 divided by 6, which is 12%. The exact rate is 4 to the power one twelfth, less 1, which is 12.25%.

    Why turn the multiple into doublings first?

    If someone tells you a town's population went from 5,000 to 20,000, you naturally say it doubled and doubled again. Doublings are easy to count and hard to get wrong. A growth multiple that is a power of two converts straight into a number of doublings, and the rule of 72 turns years per doubling into a rate. Four times is two doublings; eight times would be three.

    Four times in 12 years is two doublings, six years eachRs 5 lakhyear 0Rs 10 lakhyear 6Rs 20 lakhyear 12x 2x 206 years12 yearsfirst doublingsecond doublingRule of 72: rate x years to double = 7272 / 6 = 12% a yearExact: 4 to the power 1/12, less 112.25% a year
    Rs 5 lakh doubles to Rs 10 lakh in six years and doubles again to Rs 20 lakh by year 12, so the rule of 72 gives 72 / 6 = 12% a year against an exact 12.25%.

    How good is the rule of 72 here, and when does it slip?

    The relationship
    r=41/12−1=12.25%rule of 72: 72/6=12%r = 4^{1/12} - 1 = 12.25\% \qquad \text{rule of 72: } 72/6 = 12\%
    4the multiple, Rs 20 lakh over Rs 5 lakh
    1/12one twelfth, because the growth happened over 12 years
    rthe compound annual growth rate
    What it says in wordsThe exact rate is the twelfth root of the multiple, less one; the rule of 72 gets within a quarter of a point.

    The rule of 72 is most accurate for rates around 8%, and it drifts at the edges: at 12% it undershoots slightly, and at 20% or more it undershoots by more. For interview purposes, 12% with the words "a shade over" is the answer that shows you know it is an approximation. If the multiple is not a clean power of two, say 5 times in 12 years, estimate the doublings: 5 is a bit over two doublings, about 2.3, so a doubling every 5.2 years and roughly 14%.

    Then turn it back to the client. A 12% compound rate over 12 years is a good result, but ask what it was in and what the fees and taxes were, because the client quoted a pre-tax figure from memory.

    Where candidates lose it

    The fast wrong answer is 25%: a 300% gain split evenly over 12 years. It treats the growth as a straight line and overstates the rate by more than double.

    The second slip is reaching for a calculator or saying "about 10%" without a method. Say two doublings, six years each, 72 over 6: the method is what the interviewer is listening for.

    What the interviewer asks next

    • The same Rs 5 lakh became Rs 40 lakh in 18 years. What rate is that?
    • At 12%, how long does it take Rs 20 lakh to reach Rs 1 crore?
    • The client says 12% beat the market. What do you ask him next?