Risk Management puzzles, solved step by step
- Puzzles
- 100
- Traced to a firm
- 17
- Topics
- 13
- Hard
- 30
019Three traders, Asha, Bilal and Chen, are asked who breached a limit. Exactly one of them did, and exactly one of them tells the truth. Asha says Bilal did it, Bilal says he did not, and Chen says he did not. Who breached the limit?Rating agencyRisk GCC
Try it first
Who breached the limit?
Show the worked solution
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.
Assuming Asha is the culprit gives two true statements and assuming Bilal gives two, both breaking the rule, while assuming Chen gives exactly one true statement, Bilal's, so Chen breached the limit. 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.
Where candidates lose it
The trap is trusting the first statement and following it: Asha accuses Bilal, so candidates test Bilal first and then get tangled. Starting from an accusation lets the story steer you.
The other loss is solving it silently. Say each assumption and its count of true statements; the reasoning is what the interviewer scores.
What the interviewer asks next
- Change the rule to exactly one liar. Who breached now?
- Add a fourth trader, Dev, who says Asha is lying. Does the answer change?
- How would you set up a truth table for this in a spreadsheet?
044Estimate the cash a city's ATMs must dispense each weekday, given a population of 40 lakh, 75% of them adults, 30% of adults withdrawing once a week, an average withdrawal of Rs 3,000, and withdrawals spread evenly over five weekdays.Treasury and ALMRisk GCC
Try it first
Which order of magnitude is right for cash dispensed per weekday?
Show the worked solution
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.
Multiplying 40 lakh people by 75% adults, 30% weekly users and Rs 3,000 a withdrawal gives Rs 270 crore a week, about Rs 54 crore a weekday, and the answer stays between Rs 36 and 72 crore while the weekly share sits between 20% and 40%. 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.
The relationship0.75 the share of the population who are adults 0.30 the share of adults who withdraw once a week 5 weekdays the withdrawals are spread across What it says in wordsMultiply the chain of assumptions, then divide by the days the total is spread over.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.
Where candidates lose it
The common loss is racing to a number without stating the chain, then being unable to defend it when the interviewer challenges one input. The interviewer cares more about the structure than whether you land on 54.
The other slip is forgetting the last step and quoting the weekly Rs 270 crore as the daily figure. Say the units at every link: people, withdrawals a week, rupees a week, rupees a day.
What the interviewer asks next
- How would you size the peak day, around the first of the month?
- If 10% of the city's ATMs are down at any time, how does the cash plan change?
- How would you estimate the number of ATMs the city needs?
071Assume Rs 100 lakh crore of government bonds are outstanding with an average modified duration of 7. Roughly how much does their total market value change for a 1 basis point rise in yields?Treasury and ALMBank market risk
Try it first
Order of magnitude first: what does one basis point cost the holders?
Show the worked solution
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.
Rs 100 lakh crore of bonds times a modified duration of 7 times one basis point is 0.07 lakh crore, which is Rs 7,000 crore of value per basis point. Scaled to a 100 basis point move the change is Rs 7 lakh crore. 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.
The relationshipD_mod modified duration, 7 V market value of the bonds, 100 lakh crore, assumed \Delta y the change in yield, one basis point or 0.0001 What it says in wordsValue change is duration times value times the yield change, with a minus sign because prices fall when yields rise.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.
Where candidates lose it
The trap is the units. Candidates multiply correctly to 0.07 and then say Rs 0.07 crore or Rs 70 crore because they lose track of lakh crore. Carry the unit through every step and convert once.
The second slip is forgetting that duration is in per cent per per cent, then dividing by 100 twice. One basis point is 0.0001 as a decimal; multiply by it once.
What the interviewer asks next
- If banks hold a third of the stock, what is the banking system's DV01?
- Why is the loss for a 100 basis point rise slightly less than 100 times the DV01?
- How would a bank hedge part of this exposure?
083You have eight identical-looking balls, one of them slightly heavier, and a two-pan balance. What is the fewest number of weighings that guarantees you find the heavy ball?Bank market riskRisk GCC
Try it first
What is the minimum number of weighings that always works?
Show the worked solution
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.
Weighing 1, 2 and 3 against 4, 5 and 6 sends every case down one of three branches, and a second weighing within each branch names the heavy ball, so eight balls are always solved in two weighings. 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.
The relationshipw number of weighings n number of balls, 8 3 outcomes of one weighing What it says in wordsYou need enough weighings that three to the power of that number covers every ball.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.
Where candidates lose it
Most candidates split four against four, then two against two, then one against one, and answer three. It works, but it is not the minimum, because a balance never happens in the first two steps.
The other loss is finding two weighings without saying why one is impossible. Give the counting bound: one weighing, three outcomes, fewer than eight balls.
What the interviewer asks next
- What is the most balls you can handle in three weighings?
- Now you do not know whether the odd ball is heavier or lighter. How many weighings for twelve balls?
- What if the scale shows the weight difference rather than just which side drops?