Financial Analysis puzzles, solved step by step

Roughly 21,800 to 26,200. Scale the 16% annual volatility by the square root of the time, not the time itself: four months is a third of a year, the square root of a third is about 0.58, and 16% times 0.58 is about 9.2%. That is about 2,217 points either side of 24,000, a band that holds the close about two times in three.

Why can you not just say where the index will close?

You cannot, and saying so is part of a good answer. The honest reply to a where-will-it-close question is a central point and a range, with the range doing most of the work. Today's level is the natural centre over a few months, since the expected drift is small next to the spread. The interviewer reported here asked a sales and trading candidate exactly this, and what they are testing is whether you can turn a volatility number into a range in your head.

Why does volatility scale with the square root of time?

Picture someone walking along a lane, taking each step forward or back at random. After four steps they are not usually four steps away; good and bad steps partly cancel, and the typical distance is about two steps, the square root of four. Price moves behave the same way: variance adds up with time, so the standard deviation grows with the square root of time. A third of a year is not a third of the annual volatility; it is the square root of a third, about 58% of it.

How do you check it, and what are you leaving out?

Check by another route. Monthly volatility is 16% over the square root of 12, about 4.6%, and four months is the square root of 4, which is 2, times that: about 9.2%. Same answer. Two standard deviations, about 19,600 to 28,400, covers roughly 95% of outcomes, and naming that wider band shows you know a one standard deviation range will be wrong a third of the time. Say the limits too: prices compound, so the upper side is slightly wider, about 21,900 to 26,300 on a log basis; the drift is ignored; and real index returns have fatter tails than a bell curve.

It under-forecasts by Rs 20 lakh every month. On a straight-line trend, the average of the last three months equals the middle month, so the average lags the latest month by one month. The forecast is for the following month, one more month ahead. Two months of Rs 10 lakh growth is Rs 20 lakh. For a window of n months, the miss is the monthly growth times (n + 1) / 2.

Why does an average lag behind a trend?

Think of judging how fast a child is growing by averaging her height over the last three birthdays. The average describes her at the middle birthday, not today, and certainly not next year. A moving average describes the middle of its window, so on a rising trend it always sits below the latest value and further below the next one. With months 4, 5 and 6 at 130, 140 and 150, the average is 140, the month 5 figure, while month 7 will be 160.

What does that mean for the window you choose?

A longer window smooths noise better but lags more: a 12-month average on the same trend would miss by 10 x 6.5 = Rs 65 lakh. Moving averages trade noise against lag, so they suit flat, noisy series and fail on trending ones. The fix on a trend is to model the trend itself: add the slope back, or use a method that tracks level and slope separately, such as Holt's linear exponential smoothing. Say the limit both ways: if sales are seasonal, a three-month average also mixes high and low months, and if the trend turns, any trend-adjusted method overshoots for a while.

Not on this evidence: most of the drop is regression to the mean. A branch that tops the table usually had skill plus a good year. The good year does not repeat, so the same branches fall back toward the average whatever the manager does. Here only 2 points of a 20 point lead survived, so most of last year's lead was luck. Judge the head against similar branches that kept their managers, not against their own lucky year.

Why do the best performers fall back with no cause at all?

Think of a class test. The student who topped it probably knows the subject and also had a good day: the questions suited her and two guesses landed. In the next test the knowledge stays but the luck is drawn again, so her score is likely to be lower while staying above average. Any result that is part skill and part luck will, when it is extreme, usually be followed by a less extreme one. Francis Galton called this regression toward the mean when he compared the heights of parents and their children.

How much of the 30% was luck?

Measure the lead, not the level. The top five were 20 points above average last year and 2 points above this year. Keeping 2 points of a 20 point lead means about a tenth of last year's outperformance was persistent, and nine tenths was noise that happened to land on those five branches. Look at the bottom of the chart as well: the weakest five rose from -10.0% to 8.3% with no new manager, which is the same drift running the other way.

How would you test whether the new head made a difference?

Build a comparison. Take branches that were about as strong last year but kept their managers, and see how they grew this year. If they also fell to around 12%, the new head is doing exactly what chance predicts. If they held near 20%, the head has a real question to answer. The fair benchmark for a group picked for being extreme is a similar group picked the same way, never its own best year. The same trap sits in sales incentives, fund selection and bonus pools, where praising last year's winners and punishing last year's losers both appear to work.