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  1. 023At 9% a year, roughly how long does money take to double? Check the rule of 72 against the exact answer and say where the rule breaks down.Growth and compoundingWarm upMiddle market IBPrivate equity

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    Answer inside five seconds.

    Show the worked solution

    About 8 years: 72 divided by 9 is 8.0, and the exact answer is 8.04 years. The exact doubling time is the log of 2 divided by the log of 1.09. The rule is a shortcut built for moderate rates: it is almost exact around 8%, slightly long at low rates and increasingly short at high ones. At 40% it says 1.8 years against an exact 2.06.

    Why does 72 work at all?

    Think of a sapling that grows 9% taller each year; the question is how many of those steps multiply up to 2. The exact answer uses logarithms: years equal ln 2 divided by ln(1 + r). For small r, ln(1 + r) is close to r, and ln 2 is 0.693, so the exact rule is close to 69.3 divided by the rate in per cent. 72 replaces 69.3 because it divides cleanly by 2, 3, 4, 6, 8, 9 and 12, and because ln(1 + r) sits below r at the rates people actually meet, which pushes the true constant up. At 9%, ln 1.09 is 0.0862, and 0.693 / 0.0862 is 8.04.

    Rule of 72: almost exact near 8%, increasingly short at high rates2%10%20%30%40%0122436Annual rateYears to double9%: rule 8.0, exact 8.04exact72 / rate+5%-5%-10%-15%010%20%30%40%Annual rateRule's error against exact2%: +2.8%20%: -5.3%40%: -12.6%within 1%: 6% to 10%
    The rule of 72 and the exact doubling time almost coincide at moderate rates, 8.0 against 8.04 years at 9%, but the rule runs about 3% long at a 2% rate and 12.6% short at 40%.
    The relationship
    t=ln⁡2ln⁡(1.09)=0.69310.0862=8.04t72=729=8.0t = \frac{\ln 2}{\ln(1.09)} = \frac{0.6931}{0.0862} = 8.04 \qquad t_{72} = \frac{72}{9} = 8.0
    ln 2the natural log of 2, about 0.693, because the money must double
    ln(1.09)the log of one year's growth factor at 9%
    72 / 9the rule of 72 with the rate in per cent
    What it says in wordsThe exact doubling time is the log of 2 over the log of one year's growth; the rule of 72 approximates that ratio for moderate rates.

    Where does the rule break down?

    At high rates. ln(1 + r) falls further below r as r grows, so the true doubling time is longer than 72 divided by r: at 20% the rule says 3.6 years against 3.80, and at 40% it says 1.8 against 2.06, an error of 12.6%. At very low rates it errs the other way: 36 years against 35.0 at 2%. The rule is within about 1% of the exact answer only between roughly 6% and 10%, and should be adjusted outside that band.

    A common adjustment for high rates adds one to the 72 for every three points of rate above 8%. At 20% that gives 76 divided by 20, 3.80 years, and at 40% about 82.7 divided by 40, 2.07 years, both within a hundredth or two of the exact figures. For deal work, where target returns of 20% to 30% are common, that adjustment is worth knowing.

    Where candidates lose it

    Candidates either answer 8 and stop, or try to compute logarithms in their head and stall. Give 8 at once, then say the exact figure is a touch above, about 8.04, because the rule is tuned for rates near 8%.

    The loss that costs more is not knowing where the rule fails, when the question asks. Say that at high rates it understates the time, give the 40% example, and offer the adjustment of one extra point on the 72 for every three points above 8.

    What the interviewer asks next

    • How long does money take to triple at 9%?
    • Why is 69.3 the exact constant under continuous compounding?
    • An investment doubles in 5 years. What annual return is that, roughly and exactly?
  2. 046Fund A returns +30% and then -10%. Fund B returns +10% and then +10%. Starting with Rs 100 in each, which ends higher?Growth and compoundingWarm upMiddle market IBPrivate equity

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    Which ends higher?

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    Fund B ends higher, at Rs 121 against Rs 117. Fund A goes to 130 and then loses 10% of 130, ending at 117. Fund B compounds 10% twice to 121. Both average 10% a year, but A's compound growth rate is only about 8.2% because its returns swing. Volatility drags on compound returns: roughly half the variance comes off the average each year.

    Why does the same average give different endings?

    Think of a salary that rises 30% one year and is cut 10% the next, against one that rises 10% twice. The cut lands on the higher salary, so it takes away more rupees than the same percentage would have earlier. Returns multiply rather than add, so a loss after a gain is taken from a bigger base, and an uneven path ends below a smooth one with the same average. Rs 100 grows to 130 and then gives back Rs 13; the steady fund never gives anything back.

    Same average return, different ending: volatility drags on compounding100110120130StartYear 1Year 2A: +30% to 130B: +10% to 110A: -10% to 117B: +10% to 121Average v compoundABAverage10%10%Compound8.2%10%Ends at117121Drag estimate for A:half of 0.2 squared = 2 points10% - 2% = 8%, exact 8.2%
    Fund A rises to 130 and falls to 117 while Fund B climbs to 110 and then 121; both average 10% a year, but A compounds at only 8.2% because of its swing, so B ends Rs 4 higher.

    How big is the drag, and can you estimate it in your head?

    The growth rate that matters is the geometric averageThe constant yearly return that would turn the starting amount into the ending amount over the same period.. Fund A's is the square root of 1.17 minus 1, about 8.2%, against B's 10%, even though both arithmetic averages are 10%. A quick estimate: subtract half the variance. A's returns sit 20 points either side of 10%, so half of 0.2 squared is 2 points, and 10% minus 2% is about 8%, close to the exact 8.2%.

    The relationship
    gA=1.30×0.90−1≈8.2%gA≈rˉ−σ22=10%−2%=8%g_A = \sqrt{1.30 \times 0.90} - 1 \approx 8.2\% \qquad g_A \approx \bar r - \tfrac{\sigma^2}{2} = 10\% - 2\% = 8\%
    r-barthe arithmetic average return, 10%
    sigmahow far returns swing around the average, 20 points for Fund A
    What it says in wordsCompound growth is roughly the average return minus half the variance.

    Why would an interviewer care?

    Because fund reports often quote average returns, and investors live on compound ones. Two funds with the same average return can leave an investor with very different money, and the more volatile one leaves less. The same arithmetic explains why a fund that rises 50% and then falls 50% is down 25%, and why leverage that doubles volatility can lower long-run growth even while it raises the average.

    Where candidates lose it

    The trap is answering that they end level, because both funds average 10% a year. Averaging percentages assumes they add, and returns multiply.

    The second loss is getting 117 and 121 without saying why. Name volatility drag and give the half-the-variance estimate; that turns a calculation into an insight.

    What the interviewer asks next

    • A fund rises 50% and then falls 50%. Where does it end?
    • What steady yearly return matches Fund A over the two years?
    • Fund C returns +40% and then -20%. How does it compare with A and B?
  3. 082A company's revenue grew from Rs 100 crore to Rs 250 crore over five years. What was its compound annual growth rate?Growth and compoundingWarm upMiddle market IBPrivate equity

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    Answer before you calculate.

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    About 20.1% a year. Revenue multiplied by 2.5 over five years, so the annual rate is 2.5 to the power one fifth, minus one. A quick check: 1.2 to the fifth power is about 2.49, so the rate is just above 20%. Dividing the 150% total growth by five gives 30%, which overstates it, because 30% compounded for five years would take 100 to about 371.

    Why is 30% wrong when 150% over five years looks like 30% a year?

    A child's height or pocket money grows on what is already there. If your allowance rises 10% a year, year three's rise is 10% of a bigger allowance than year one's. Compound growth earns on its own past growth, so a constant rate produces bigger rupee steps every year, and dividing the total by the years overstates the rate. Thirty per cent a year would take 100 to 130, 169, 220, 286 and finally 371.

    Divide total growth by years and you overshootTempting shortcutTotal growth 150% / 5 years= 30% a yearCheck: 100 x 1.30^5 = 371, not 250Compound annual rate(250 / 100)^(1/5) - 1= 20.1% a yearCheck: 100 x 1.201^5 = 250100250Yr 0Yr 1Yr 2Yr 3Yr 4Yr 530%: 37120.1%: 250Revenue, Rs crore
    Dividing 150% total growth by five years gives 30%, but 30% compounded reaches 371; the rate that actually takes revenue from 100 to 250 in five years is 20.1%.

    How do you get 20.1% without a calculator?

    The relationship
    CAGR=(endstart)1/n−1=2.51/5−1≈20.1%\text{CAGR} = \Big(\frac{\text{end}}{\text{start}}\Big)^{1/n} - 1 = 2.5^{1/5} - 1 \approx 20.1\%
    end / startthe total multiple, 250 / 100 = 2.5
    nthe number of years, 5
    What it says in wordsFind the single yearly multiplier that, applied n times, turns start into end.

    Guess and check from a round rate. Try 20%: 1.2 squared is 1.44, cubed is 1.73, then 2.07, then 2.49, just shy of 2.5, so the answer is a hair above 20%. A second route is the rule of 72: at 20% money doubles in about 3.6 years, and 2.5x in five years is a little more than one doubling plus a bit, consistent with roughly 20%.

    Say the limitation. CAGR describes only the start and end points. Revenue could have jumped to 250 in year one and sat flat, or fallen and recovered; the CAGR is the same 20.1%. If the path matters, ask for the yearly figures.

    Where candidates lose it

    Saying 30% is the whole trap. It sounds reasonable and the interviewer asks it fast precisely to see whether you notice that growth compounds.

    The second loss is knowing the formula but freezing on the fifth root. Guess 20%, multiply up five times out loud, and adjust. Interviewers prefer that to silence.

    What the interviewer asks next

    • What CAGR doubles revenue in five years?
    • If revenue then grows 10% a year for five more years, what is the ten-year CAGR?
    • Why might a company quote the simple average growth rate instead of the CAGR?