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.

Anywhere from about -0.75 to 0.95. Read each correlation as the cosine of an angle between two arrows. X is 78.5 degrees from Y, and Z is 60 degrees from Y. Z can swing to the same side as X, leaving them 18.5 degrees apart, or to the other side, 138.5 degrees apart. The cosines of those angles are 0.9485 and -0.7485.

Why can a correlation be drawn as an angle?

Think of three people walking away from the same lamp post. Knowing how far apart the first two point, and how far apart the second and third point, limits how far apart the first and third can point, but only loosely. If you standardise each variable, its correlation with another is the cosine of the angle between them, so correlations must behave like angles in space. A correlation of 0.2 is an angle of 78.5 degrees; 0.5 is 60 degrees.

Where does the formula come from, and when does it bite?

The three-by-three correlation matrix must not imply a negative variance for any mix of X, Y and Z, which means its determinant cannot go below zero. Solving that condition gives the formula above, the cosine rule for the difference and sum of two angles. The constraint is loose when the known correlations are weak and tight when they are strong. With 0.2 and 0.5, almost the whole scale stays open. With 0.9 and 0.9, X and Z must correlate at least 0.62: two things each closely tied to Y cannot drift far from each other. Say the practical use: a risk model that fills in missing correlations by hand can break this rule and produce a matrix that is not valid.

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.

Both hold when the two stocks share a slow driver that changes month to month, while their day-to-day moves push in opposite directions. On any one day the shared drift is a small part of each move and the opposing noise wins, so the daily correlation is negative. Over a month the drift adds up 21 times while the noise partly cancels, so the shared part dominates and the correlation turns positive.

How can two stocks move apart by the day but together by the month?

Take two neighbours' electricity meters. On a given day one runs high because guests are staying while the other family is away, so daily readings look opposed. But both bills climb every summer when the air conditioners come on. Correlation is not a fixed property of two assets; it belongs to a horizon, because different forces drive short moves and long moves. Daily returns are dominated by trading noise and stock-specific news; monthly returns by sector and macro trends.

Why does the shared driver win over a month?

Write each daily return as a shared drift plus the stock's own noise. Over 21 days the drift adds up in a straight line, so a month's drift is 21 times a day's and its variance is 21 x 21 = 441 times larger. Noise does not line up day after day, so its variance grows only 21 times. Whatever the two stocks share pulls harder the longer you hold them, because shared moves stack while unshared moves wash out. With daily noise variance of 1, a noise correlation of minus 0.3 and a drift variance of 0.107, the monthly correlation comes out at exactly plus 0.6.

The model also shows how sensitive the result is. Halve the drift variance and the monthly correlation drops from 0.6 to 0.39. Read across the whole year, daily returns would show a correlation of about -0.17: still negative, because on any single day the drift is too small to matter.

What else could cause it, and what would you check?

Two other mechanisms produce the same pattern. A lead-lag, where one stock reacts to shared news a day later, hides co-movement in daily data that monthly data captures. And short-term trading that pushes money from one stock into the other, such as a pairs or index rebalancing flow, creates opposing daily moves inside a shared trend. Then question the evidence. Twelve monthly points give a correlation of 0.6 a standard error of about 0.19, so the true figure could plausibly be anywhere from about 0.2 to near 1. Check other years before building a hedge on it, because a hedge ratio estimated on daily data can have the wrong sign for a position held for months.

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.

A's overall rate is 9.2%, B's is 3.9%, and A is still the better underwriter in each segment. A: 2 defaults among 100 salaried plus 90 among 900 self-employed, 92 in 1,000. B: 27 plus 12, 39 in 1,000. An overall rate is a weighted average of segment rates, and the weights are the mix of the book. A's book is 90% self-employed, where everyone's rate is high, so its average is pulled up by whom it lends to, not by how well it lends.

How can a branch be better at everything and worse in total?

Two surgeons. One takes the hardest cases in the hospital and loses fewer of them than anyone else would; the other takes mostly routine cases. Count deaths per patient and the first surgeon looks worse, because her patients were sicker to begin with. An overall rate is a weighted average of segment rates, and the weights are the mix of the book, so a branch that lends mostly to the riskier segment carries a higher average even when it is better in every segment. A's rate is 0.1 x 2% + 0.9 x 10% = 9.2%; B's is 0.9 x 3% + 0.1 x 12% = 3.9%.

How do you compare the two branches fairly?

Give them the same customers. Apply each branch's segment rates to one common mix and the mix stops voting. At B's mix, 90% salaried, A's rates give 0.9 x 2% + 0.1 x 10% = 2.8% against B's actual 3.9%. At A's mix, B's rates give 0.1 x 3% + 0.9 x 12% = 11.1% against A's actual 9.2%. At a 50:50 mix, A is 6.0% and B is 7.5%. On any common mix A is lower, so A underwrites better; its higher total is the price of the book it chose to build. This reversal, where a comparison flips when groups are combined, is known as Simpson's paradox.

What should the analyst say beyond the arithmetic?

Three things. First, both statements are true at once: A underwrites better, and A's book loses more money, because capital is charged on actual losses and A chose a riskier book. Whether that was a good choice depends on the pricing: if A earns enough extra yield on self-employed loans to cover the extra 8 points of defaults, its strategy can be sound. Second, ask why the mixes differ, since location, product and targets all shape who walks in. Before judging two rates, ask whether the two populations are the same; if they are not, standardise to one mix or compare segment by segment.

Third, say the limitation. Each segment is itself a mix: of loan sizes, vintages and cities. A's salaried borrowers might be larger-ticket government employees and B's might be gig workers on salary slips, and a finer split could flip the comparison back. Standardising to the segments you can see removes the mix you know about, never the mix you do not. The honest close is that A looks the better underwriter on this cut, and you would want the segment rates by vintage before signing that view.