Risk Management puzzles, solved step by step

7.5 degrees. The minute hand at 15 minutes points straight at the 3, 90 degrees from 12. The hour hand moves 30 degrees an hour, so half a degree a minute; at 3:15 it sits at 90 plus 7.5, which is 97.5 degrees. The gap is 7.5 degrees, not zero.

Why is zero the wrong answer?

Picture a train that leaves at 3 o'clock and a car that starts after it. If you only check where the train was at 3 o'clock, you will think the car has caught up when it reaches that station. The hour hand never waits at a number; it moves half a degree every minute, so at a quarter past it has already moved a quarter of the way to the next number. Most people see the static clock face of a child's drawing, where the hour hand points exactly at the 3.

Why would a risk interviewer ask a clock question?

Because it tests one habit that matters on a risk desk. The question is designed so that the static picture gives a confident wrong answer, and the interviewer is watching whether you check what moves. A risk number is full of the same trap: a position that looks hedged at the close can drift out of balance intraday, a limit measured at month end can be breached in between. Getting 7.5 is fine; saying why the answer is not zero is what earns the point.

Have the general formula ready, because the follow-up usually asks for another time. At 9:45, the minute hand is at 270 degrees and the hour hand at 270 plus 22.5, a gap of 22.5 degrees. The hands overlap eleven times in twelve hours, roughly every 65.45 minutes, and working that out is the usual second question.

Chen breached the limit, and Bilal is the one telling the truth. Assume each trader in turn is the culprit and count the true statements. Asha as culprit makes two statements true, Bilal's and Chen's. Bilal as culprit also makes two true, Asha's and Chen's. Only Chen as culprit leaves exactly one true statement, Bilal's denial, which matches the rules.

Where do you start when every statement might be a lie?

Think of a lost house key and three rooms. Rather than arguing about who saw what, you check each room in turn and stop when one fits. With three possible culprits, the fastest route is to assume each one did it, work out which statements are then true, and keep the case that matches the rule of exactly one truth. That turns an argument about who is lying into three short checks.

Is there a shortcut that finds it faster?

Yes: look for two statements that cannot both be false. Asha says Bilal did it and Bilal says he did not, so exactly one of those two is true whoever the culprit is. That uses up the single true statement, which means Chen's statement must be false. Chen saying he did not do it is false, so Chen did it. Saying this pairing out loud shows the interviewer you look for structure before grinding through cases.

Then connect it to the job, in one line. A limit breach investigation often starts with conflicting accounts, and the same discipline applies: list the possibilities, check each against the hard evidence, and discard the ones that contradict it. The trade records play the part of the rule here; the accounts are only as good as what they agree with.

The price ends 4% lower, and the NPA ratio rose by 1 percentage point, which is a 50% increase. 100 up 20% is 120, and 20% off 120 is 24, leaving 96. The ratio moved from 2% to 3%: the gap is 1 percentage point, and 1 over the starting 2 is 50%. Saying it rose 1% would be wrong on both counts.

Why does up 20% then down 20% lose money?

A shopkeeper marks a shirt up 20% from Rs 100 to Rs 120, then runs a 20% off sale. The discount is taken on Rs 120, so it is Rs 24, and the shirt sells for Rs 96. Each percentage is measured on whatever base exists at the time, and the fall happens on a bigger base than the rise. The two moves multiply rather than add: 1.2 x 0.8 is 0.96. In general, up x then down x leaves you down x squared, here 0.2 x 0.2, or 4%.

What is the difference between a percentage point and a percent?

When the quantity is itself a percentage, there are two honest ways to describe a change. Percentage points measure the gap between two rates by subtraction; percent measures that gap relative to where you started. From 2% to 3% is +1 point by subtraction and +50% relative to 2. A gross NPA ratioNon-performing assets, loans on which the borrower has stopped paying for a set period, as a share of total loans before provisions. is a rate, so a risk report must say which one it means, and the two carry different messages: one point sounds mild; half as many bad loans again sounds serious.

In a risk committee, both descriptions are used and both can mislead. A desk that wants to play down deterioration quotes points; one that wants attention quotes percent. The disciplined habit is to quote the level and the change in points together, 3% from 2%, and let the reader see the relative move for themselves.

About Rs 54 crore a weekday. 40 lakh people x 75% adults is 30 lakh adults. 30% of them withdraw weekly, 9 lakh withdrawals. At Rs 3,000 each that is Rs 270 crore a week, and spread over five weekdays, Rs 54 crore a day. Moving the weekly share between 20% and 40% puts the answer between Rs 36 and 72 crore.

How do you structure the estimate so the interviewer can follow it?

Planning food for a wedding, you do not guess the total rice; you count guests, portions per guest and grams per portion. A sizing answer is a chain of multipliers, each one a stated assumption, so the interviewer can challenge any link without the whole answer collapsing. Say the chain before you multiply: people, adults, weekly users, ticket size, days. Then the arithmetic is almost an afterthought.

How do you sanity check Rs 54 crore?

Test it from a second angle. Rs 270 crore a week across 30 lakh adults is Rs 900 per adult per week, a plausible cash habit in a city where digital payments carry much of the load. It also means about 1.8 lakh transactions a weekday; if you assumed a number of ATMs, you could check whether each would handle a sensible number of withdrawals a day. Then name the weakest link: the 30% weekly share. Doubling it doubles the answer; the population and adult share are far less uncertain.

Finish with why a treasury desk cares. Cash is a liquidityThe ability to meet a payment obligation, here physical cash in machines, on time and without loss. obligation that cannot be deferred, and the average is not the peak. Salary days, festivals and long weekends bunch demand, so the cash plan is sized to a peak day plus a buffer, and an empty machine is an operational failure as much as a customer one. Rs 54 crore is the average the peaks are built on.

Probably not: the ratio improved only because the book grew. The bad loan ratio fell from 3.0% to 2.8%, but the bad loans themselves rose 16.7%, from Rs 120 crore to Rs 140 crore. The Rs 1,000 crore of new lending is too young to have defaulted. Set against last year's book, bad loans are 3.5% of the loans that could have gone bad.

How can a ratio fall while the problem grows?

A school with 40 failing students out of 1,000 has a 4% failure rate. Admit 500 new students in April, before any exams, and the rate drops to 2.7% without a single student improving. Any ratio can fall because its denominator grew, and a fast-growing loan book dilutes its bad loan ratio with loans that have not yet had time to fail. Here bad loans rose Rs 20 crore while the book rose Rs 1,000 crore.

What would you check before calling it either way?

Loans take time to go bad, a process lenders call seasoningThe time a loan needs before its true default rate shows, because few borrowers default in the first months after taking a loan.. The fair test compares bad loans with the book that was old enough to produce them, which is why risk teams track lagged ratios and default rates by the year a loan was written. A lagged ratio of 3.5% against 3.0% says the old book is getting worse, not better.

The limitation: the lagged ratio assumes the new loans added nothing to the Rs 140 crore. Some of the extra Rs 20 crore could come from new loans that failed fast, which would itself be a warning about how they were underwritten. Either way, the headline ratio is the weakest of the three readings.

About Rs 7,000 crore of value lost for a 1 basis point rise. A bond's price falls by roughly its modified duration times the change in yield. For the whole stock that is Rs 100 lakh crore x 7 x 0.0001, which is 0.07 lakh crore, or Rs 7,000 crore. The same arithmetic makes a 100 basis point rise cost about Rs 7 lakh crore.

Why does a hundredth of a per cent move so much money?

A one paisa rise in the price of petrol means nothing to one driver and a great deal to an oil company selling crores of litres. The move is small; the base is enormous. The DV01The change in value of a bond or portfolio for a one basis point change in yield, sometimes called PV01. of a holding is value times duration times one basis point, so a stock worth Rs 100 lakh crore with a duration of 7 carries Rs 7,000 crore of value per basis point.

How do you keep the units straight under pressure?

Convert once, at the end, and say the conversion aloud. One lakh crore is 1,00,000 crore, so 0.07 lakh crore is 7,000 crore; do the multiplication in the big unit and translate only the answer. A useful check: duration 7 means a 1% move changes value by 7%, so 1 basis point, a hundredth of that, changes it by 0.07%, and 0.07% of 100 lakh crore is 0.07 lakh crore.

State the limits. The stock and duration are assumptions for the puzzle, not current figures, which you would look up. The duration rule ignores convexity, harmless at one basis point but not at a hundred, where the true fall is a little smaller than Rs 7 lakh crore. And the loss lands on whoever holds the bonds, which is why a bank's treasury watches the DV01 of its government bond book daily.

Two weighings. Put three balls on each pan and leave two aside. If one pan drops, the heavy ball is among its three: weigh one against another, and a balance points to the third. If the pans balance, weigh the two set aside against each other. A balance has three outcomes, so two weighings separate up to nine balls.

Why is halving the wrong instinct?

A guessing game where someone answers only yes or no halves the possibilities with each question. A balance answers in three ways: left heavier, right heavier or level. Each weighing has three outcomes, so the best split is into three groups, and the balanced outcome is information, not a wasted turn. Four against four uses only two of the three outcomes, because the pans can never balance, and so it wastes a third of what the scale can tell you.

How do you prove two is the minimum and find the limit?

One weighing has three outcomes, and eight balls need eight different answers, so one weighing cannot be enough. Two weighings have 3 times 3, nine outcomes. With w weighings you can separate at most 3 to the power w balls, so nine balls also need only two, and a tenth ball needs a third weighing. Saying the bound, not only the procedure, is what turns a trick into reasoning.

Why would a risk interviewer ask this?

It tests whether you count what a test can tell you before you run it. The same habit applies to a control check: a review that can only return pass or fail tells you less than one that also flags items for a closer look. Name the three outcomes first and the answer follows.