Risk Management puzzles, solved step by step

Puzzles: 100
Traced to a firm: 17
Topics: 13
Hard: 30

Showing 11–20 of 39 · filtered from 100Clear filters

024A desk's daily P&L has a standard deviation of Rs 2 crore. The head of the desk wants a loss number that is exceeded about one day a month. Which confidence level is that, and what is the VaR?VaR and expected shortfallCoreBank market riskTreasury and ALM
Try it first
Which confidence level matches one bad day a month?
Show the worked solution
About 95% confidence, and a VaR of about Rs 3.3 crore. A month has about 21 trading days, so one breach a month is a tail of 1 in 21, about 4.8%, a confidence level of about 95.2%. For a normal P&L that cut sits 1.67 standard deviations out, so the VaR is 1.67 times Rs 2 crore, about Rs 3.34 crore. At a round 95% it is Rs 3.29 crore.
How do you turn a frequency into a confidence level?
Think of a weather forecast that says a heavy rain day comes about once a month. That is the same statement as saying about 1 day in 30 is that wet. A VaR confidence level is just a frequency of bad days written as a percentage: one breach in 21 trading days is a 4.8% tail, which is a 95.2% confidence level. Counting trading days matters; using 30 calendar days would make the tail 3.3% and the level about 96.7%.
For daily P&L with a Rs 2 crore standard deviation, one loss day in 21 sits beyond Rs 3.34 crore, 1.67 standard deviations out, which is close to the Rs 3.29 crore 95% VaR.
The relationship
$\text{VaR} = z_{c}\,\sigma = 1.668 \times 2 \approx 3.34 \text{ crore} \qquad z_{95\%} = 1.645 \Rightarrow 3.29 \text{ crore}$
z_c the number of standard deviations that leaves the chosen tail below it
sigma the daily P&L standard deviation, Rs 2 crore
What it says in wordsMultiply the daily standard deviation by the number of standard deviations that matches the chosen frequency of bad days.
Why is this translation useful on a desk?
Because a desk head can act on a frequency and cannot act on a percentage. Saying we expect to lose more than Rs 3.3 crore about once a month, roughly 12 days a year, tells the head of the desk what to expect and when to worry. It also sets up the backtest: if losses beyond Rs 3.3 crore start happening three or four times a month, the model is understating risk, and that is visible within a quarter rather than a year.
Add the limits. The VaR says nothing about how bad the bad day is; a loss of Rs 3.4 crore and one of Rs 10 crore both count as one breach. The normal assumption also understates fat tails, so the real one-in-21 loss may sit further out than Rs 3.3 crore. And the Rs 2 crore standard deviation is itself an estimate that moves with market conditions.
Where candidates lose it
The trap is picking 99% because it is the regulatory number people remember. A 99% VaR is breached about once in 100 trading days, closer to once every five months than once a month.
The second miss is counting 30 calendar days. Losses happen on trading days, so the tail is 1 in 21, not 1 in 30.
What the interviewer asks next
What loss would the desk expect to exceed once a year?
The desk had five breaches last month. What do you conclude, and what would you check first?
How would the answer change if the P&L had fat tails with the same standard deviation?
026A Rs 100 crore loan earns Rs 3 crore of net interest income a year, costs Rs 0.8 crore to run, carries an expected loss of Rs 1 crore and needs Rs 8 crore of economic capital. What is its risk-adjusted return on capital, and does it clear a 14% hurdle?Capital and leverageCoreBank credit riskRisk GCC
Try it first
Where does the Rs 1 crore of expected loss go in the calculation?
Show the worked solution
RAROC is 15.0%, so the loan clears a 14% hurdle, but only just. Income of Rs 3 crore less Rs 0.8 crore of cost and Rs 1 crore of expected loss leaves Rs 1.2 crore of risk-adjusted profit. Divided by Rs 8 crore of economic capital that is 15.0%. After a 14% charge on the capital, Rs 1.12 crore, the loan adds only Rs 0.08 crore a year.
Why does expected loss sit in the numerator and unexpected loss in the denominator?
Think of a restaurant that knows about two plates in a hundred get dropped. That breakage is priced into the menu; it is a running cost. What the owner keeps cash aside for is the night a whole shelf comes down. Expected loss is the average, so it is a cost charged against income; unexpected loss is the bad year above the average, so it is covered by capital. RAROCRisk-adjusted return on capital: profit after costs and expected loss, divided by the capital held against unexpected loss. puts each in its proper place, which is why a bank can compare a thin-margin safe loan with a fat-margin risky one on the same scale.
Net interest income of Rs 3.0 crore, less Rs 0.8 crore of cost and Rs 1.0 crore of expected loss, leaves Rs 1.2 crore of risk-adjusted profit; over Rs 8 crore of economic capital that is a 15.0% RAROC, one point above the 14% hurdle.
The relationship
$\text{RAROC} = \frac{\text{income} - \text{cost} - \text{EL}}{\text{economic capital}} = \frac{3.0 - 0.8 - 1.0}{8} = 15.0\%$
EL expected loss, the average annual credit loss, Rs 1 crore
economic capital the buffer held against losses worse than average, Rs 8 crore
What it says in wordsTake the average loss off the income, then ask how hard the remaining profit works the capital held against a bad year.
How much room is there before the loan fails the hurdle?
Turn the hurdle into rupees. A 14% return on Rs 8 crore of capital is Rs 1.12 crore a year, and the loan earns Rs 1.2 crore, so it clears by Rs 0.08 crore. A margin that thin means one small change flips the answer. If expected loss rises from Rs 1.0 crore to Rs 1.08 crore, or economic capital rises from Rs 8 crore to Rs 8.57 crore, RAROC falls to exactly 14%. Saying those two breakeven numbers is what turns a ratio into a credit judgement.
State the limits too. This is a pre-tax, single-year figure; many banks compute RAROC after tax and add the return earned on the capital itself, which would lift the number. And the Rs 8 crore depends on the bank's own capital model, so the answer is only as good as that model's view of this borrower.
Where candidates lose it
The common slip is to subtract expected loss and then also add it to the capital, or to leave it out of the numerator on the grounds that capital covers losses. Either way the ratio comes out wrong by several points, and the interviewer hears that you do not know what capital is for.
The second slip is stopping at 15% and saying yes. A one-point margin deserves the breakeven sentence: how much worse can the loss or the capital get before the loan stops paying for itself.
What the interviewer asks next
The bank prices the loan 20 basis points lower to win the deal. Does it still clear the hurdle?
How would you estimate the Rs 8 crore of economic capital in the first place?
Why might a bank still make a loan with a RAROC below its hurdle?
029A fully collateralised netting set's value moves with a daily volatility of Rs 2 crore. Assuming a 10-day margin period of risk and normally distributed moves, what is the 99% potential future exposure?Counterparty exposure and collateralCoreCounterparty riskBank market risk
Try it first
Which number do you multiply the daily Rs 2 crore by to reach ten days?
Show the worked solution
About Rs 14.7 crore. Collateral covers the value up to the last margin call, so exposure is the move during the 10 days it takes to notice the default and close out. Ten days of Rs 2 crore daily volatility is 2 x root 10, Rs 6.32 crore. At 99%, 2.33 standard deviations, potential future exposure is Rs 14.71 crore.
If the trades are fully collateralised, where does any exposure come from?
A landlord holding a month's deposit is covered for last month's rent. If the tenant stops paying and takes three months to evict, the deposit covers one of them. Collateral protects you up to the last margin you actually received; the margin period of riskThe time from the last successful collateral exchange with a defaulting counterparty until its trades are closed out or re-hedged. is the gap after that, while the defaulter pays nothing and the market keeps moving. Ten days covers the missed call, the dispute, the formal default and the close-out.
From the last margin received, the 99% edge of the exposure fan opens with the square root of time, reaching Rs 4.65 crore after one day and Rs 14.7 crore after the 10-day margin period of risk, which is the exposure the collateral never covered.
Why the square root of ten and not ten?
Each day's move is independent, so some days undo others. Variances add across independent days, which makes volatility grow with the square root of time. Ten days is 2 x 3.16, Rs 6.32 crore, and 99% is 2.33 of those. Scaling linearly would give Rs 46.5 crore, more than three times too big, and a counterparty limit set on it would turn away business the bank could safely do.
The relationship
$\text{PFE}_{99\%} = z_{0.99}\,\sigma_{1d}\sqrt{\text{MPOR}} = 2.33 \times 2 \times \sqrt{10} \approx 14.7$
z_{0.99} the 99% one-sided normal quantile, 2.33
\sigma_{1d} daily volatility of the netting set value, Rs 2 crore
MPOR margin period of risk, 10 days
What it says in wordsExposure at a confidence level is that many standard deviations of the move over the days you are unprotected.
Name what the simple version leaves out. Only moves in your favour are exposure, so the lower half of the fan does not count. Collateral thresholds or a minimum transfer amount add a fixed layer on top. And a counterparty that defaults in a crisis defaults when volatility is high, so Rs 2 crore a day is likely too calm. Double the margin period to 20 days, as can happen with disputes or illiquid trades, and the figure rises to Rs 20.8 crore, not Rs 29.4 crore.
Where candidates lose it
Two ways to lose it. Some candidates say zero, because the set is fully collateralised, and miss that collateral only covers values already agreed. Others multiply by 10 instead of the square root of 10 and land on Rs 46.5 crore.
Say the timeline before the formula: last margin in, default, close-out. Once the interviewer hears that you know why the ten days exist, the arithmetic is a formality.
What the interviewer asks next
The counterparty has a Rs 5 crore threshold before it posts collateral. What is the PFE now?
Why might a regulator require a longer margin period of risk for some netting sets?
What is wrong-way risk, and how would it change this number?
030A distressed one-year bond costs 80. It pays 100 at maturity with probability 85%, and if the issuer defaults, holders recover 40. What is the expected return, and at what default probability does the trade break even?Credit risk arithmeticCoreBank credit riskAsset manager risk
Try it first
Before you calculate: at what default probability does paying 80 stop making sense?
Show the worked solution
The expected return is 13.75%, and the trade breaks even at a default probability of 33.3%. The expected payoff is 0.85 x 100 plus 0.15 x 40, which is 91, against a price of 80. Breakeven solves 80 = 100 minus 60p: p is 20 over 60, one third. The real question is whether default odds could be that high.
Why is the breakeven more useful than the expected return?
The 15% is somebody's estimate, and estimates of default for a stressed issuer are soft. Expected return inherits every error in the 15%; the breakeven tells you how wrong the estimate can be before you lose money. It is like buying a second-hand car that needs repairs: rather than guess the repair bill, you ask how large the bill could be before the deal stops being worth it. Here default odds could more than double, from 15% to one in three, before the trade loses on average.
Paying 80 for a bond that returns 100 with 85% probability and 40 on default gives an expected payoff of 91, a 13.75% expected return, and the trade only loses on average if the default probability exceeds 33.3%.
The relationship
$80 = (1-p)\,100 + p\,40 \;\Rightarrow\; p^{*} = \frac{100 - 80}{100 - 40} = \frac{1}{3}$
p one-year default probability
100 - 80 the gain if the bond pays at par
100 - 40 the gap between par and recovery
What it says in wordsBreakeven default probability is the discount to par divided by the loss given default measured from par.
What would a credit risk manager add before approving the trade?
Three things. First, the recovery is also a guess, and it moves the breakeven: if holders recover 20 rather than 40, breakeven default odds fall from one third to 25%, because each default now costs 60 from the purchase price instead of 40. Second, money has a time value. If you require an assumed 7% return from cash for the year, breakeven falls from 33.3% to 24.0%, because the bond must beat cash, not zero. Third, the payoff is lopsided, +25% or -50%, so position size matters more than the average: a book of such bonds survives, a single large position may not.
Where candidates lose it
Candidates compute 13.75% and stop, as if the 15% were a fact. The interviewer's follow-up is always what if the probability is wrong, and the candidate who has the one-third breakeven ready answers it before it is asked.
The second trap is using the 20-point discount to par as the breakeven default rate. The loss on default is 40 from where you bought, not 20, and mixing the two gives nonsense.
What the interviewer asks next
What default probability does the market price imply if investors demand a 7% return?
Recovery is uncertain, between 25 and 55. How does that change your view?
Why do distressed investors often care more about recovery analysis than default probability?
034Large fraud losses at a bank arrive at random at an average rate of 0.2 a year. What is the probability of no large loss in one year, and in five years?Operational loss and fraudCoreOperational riskQuant risk
Try it first
One large loss every five years on average. What is the chance of getting through five years with none?
Show the worked solution
About 81.9% for one year and 36.8% for five years. Random arrivals at a steady average rate follow a Poisson process, where the chance of none in t years is e to the minus rate times t. One year gives e to the minus 0.2; five years gives e to the minus 1. So a loss is more likely than not over five years, 63.2%, yet most single years are quiet.
Why is one every five years not a schedule?
Buses that arrive at random, on average every twenty minutes, do not arrive every twenty minutes. Sometimes two come together, sometimes you wait an hour. A rate of 0.2 a year says how often losses happen on average, not when; each year is an independent draw, and most draws come up empty. Losses that arrive independently at a steady average rate follow a Poisson processA model for events arriving at random and independently at a constant average rate, such as large fraud losses or defaults in a big pool., and the chance of seeing none falls away exponentially with time.
The relationship
$P(\text{no loss in } t \text{ years}) = e^{-\lambda t}, \qquad e^{-0.2} = 0.819, \quad e^{-1} = 0.368$
\lambda the average number of large losses a year, 0.2
t the number of years you are looking across
What it says in wordsThe chance of a clean run shrinks by the same factor every year, e to the minus 0.2, about 82%.
At 0.2 large losses a year the chance of none falls from 81.9% over one year to 36.8% over five and 13.5% over ten, so a single quiet year is the normal outcome and says little about whether the risk is still there.
What should an operational risk manager take from a quiet year?
Very little. A quiet year happens 82% of the time even when the risk is fully present, and three quiet years in a row still happen 55% of the time. A business that points to a clean record and asks for lighter controls is reading noise as evidence. The flip side is equally useful: when a loss arrives after a long gap, it is not proof that controls suddenly failed; the average gap here is 5 years, and gaps of ten years happen 13.5% of the time.
The limit to say out loud: the Poisson model assumes losses arrive independently at a constant rate. Fraud often clusters, one weak control exploited several times, and rates change as the business changes. The model is a baseline for reading the record, not a forecast.
Where candidates lose it
The common error is treating the average as a timetable: one loss is due by year five, so the chance of none is near zero. That misreads a rate as a schedule and is the same slip as expecting a coin to come up heads because it has shown tails four times.
The other miss is computing 0.8 to the power 5, 32.8%, which treats each year as a yes or no event with at most one loss. It is close, but say why e to the minus 1 is the right form: more than one loss can arrive in a year.
What the interviewer asks next
What is the probability of two or more large losses in five years?
A unit has had no large loss in eight years. What does that tell you about its true rate?
How would clustering of fraud losses change the capital you hold?
036A bond portfolio holds 50 names, each with a 2% one-year default probability, and defaults are independent. What is the expected number of defaults in a year, and what is the chance of four or more?Probability and base ratesCoreBank credit risk
Try it first
Expected defaults are 1. Roughly how likely is a year with four or more?
Show the worked solution
One default expected, and about a 1.8% chance of four or more. The expected count is 50 x 2% = 1. The chance of none is 0.98 to the power 50, 36.4%; of exactly one, 37.2%; two, 18.6%; three, 6.1%. Those sum to 98.2%, so four or more is about 1.78%, roughly one year in 56.
Why is the expected count not enough to size the risk?
A school with 50 pupils, each with a 2% chance of being off sick on a given day, expects one absence. Most days it gets none, one or two; some days it gets four, and the class still has to run. An average of one default tells you what a normal year costs; it says nothing about how bad the bad year is, and capital exists for the bad year. Here, with Rs 10 crore in each name and a 60% loss on default, the expected loss is Rs 6 crore, but a four-default year costs Rs 24 crore.
Across 50 independent names at 2% each, one default is the most likely outcome at 37.2%, yet four or more defaults still happen 1.78% of the time, a tail that turns a Rs 6 crore expected loss into a Rs 24 crore bad year.
How do you get 1.8% without a calculator?
Use the Poisson approximationFor many independent rare events, the count is close to a Poisson distribution with the same mean, so P(k) is about e to the minus mean times mean to the k over k factorial.. With a mean of 1, the chances of 0, 1, 2 and 3 are about 0.368, 0.368, 0.184 and 0.061. They add to about 0.981, so four or more is about 1.9%, within a whisker of the exact binomial 1.78%. Saying you are using the approximation, and why it works here, earns as much credit as the exact figure.
The relationship
$P(X \ge 4) = 1 - \sum_{k=0}^{3} \binom{50}{k}(0.02)^k(0.98)^{50-k} \approx 1 - 0.982 = 0.018$
X the number of defaults in the year
\binom{50}{k} the number of ways to pick which k names default
What it says in wordsFour or more is one minus the chance of zero, one, two or three.
Then say the assumption that matters most. Independence is the weak link: names in the same sector or region default together in a downturn. With correlation, the expected count stays at one, but years with no defaults and years with many both become more common, and four or more can be several times more likely than 1.8%. That is why credit portfolio models spend most of their effort on correlation, not on the individual default probabilities.
Where candidates lose it
Candidates give the expected count and stop, or say four defaults is basically impossible because the average is one. The interviewer asked for the tail precisely because averages do not size capital.
The second trap is overconfidence in the 1.8%. Offer the independence caveat before being asked; it shows you know which assumption the answer is most sensitive to.
What the interviewer asks next
If defaults are correlated, what happens to the chance of zero defaults?
How many names would you need for the chance of four or more to exceed 10%?
Each name has a different default probability. How does that change your method?
039A retiree holds Rs 1 crore and withdraws Rs 10 lakh at the end of each year. Compare two years of minus 20% then plus 20% with plus 20% then minus 20%. Where does each order leave the retiree?Compounding and drawdownsCoreAsset manager risk
Try it first
Without withdrawals, both orders end at Rs 96 lakh. With the Rs 10 lakh withdrawals, which order ends better?
Show the worked solution
Up first ends at Rs 78 lakh; down first ends at Rs 74 lakh. Without withdrawals both orders end at Rs 96 lakh, because returns multiply. With withdrawals, the Rs 10 lakh taken out after year one sits out year two: it escapes a 20% fall in one order and misses a 20% gain in the other, a gap of 10 x 0.4, Rs 4 lakh.
Why does order matter only once money is withdrawn?
Multiplication does not care about order: 0.8 x 1.2 is the same as 1.2 x 0.8, 0.96 either way. A withdrawal breaks the chain, because rupees taken out before a return do not experience it. Think of a farmer who sells part of the harvest each year. A good year followed by a bad one lets him sell from a big crop first; a bad year first forces him to sell from a small one, and the portion sold never gets the chance to recover. This is sequence riskThe risk that the order of returns, not just their average, changes the outcome, which happens whenever money is being added or withdrawn..
With Rs 10 lakh withdrawn each year, a fall of 20% followed by a rise of 20% leaves Rs 74 lakh, while the same returns in the opposite order leave Rs 78 lakh; without withdrawals both orders would end at Rs 96 lakh.
Where exactly does the Rs 4 lakh gap come from?
Track the first withdrawal. In both orders the retiree takes Rs 10 lakh at the end of year one. That Rs 10 lakh then skips year two: in the up-first order it skips a 20% fall and saves Rs 2 lakh, in the down-first order it skips a 20% rise and loses Rs 2 lakh. The difference is 10 x (1.2 minus 0.8), Rs 4 lakh. The second withdrawal is taken at the very end and does not depend on the order at all.
The relationship
$W_2 = (W_0(1+r_1) - D)(1+r_2) - D \;\Rightarrow\; W_2^{\text{up first}} - W_2^{\text{down first}} = D\,(r_{\text{up}} - r_{\text{down}}) = 10 \times 0.4 = 4$
W_0 starting wealth, Rs 100 lakh
D the yearly withdrawal, Rs 10 lakh
r_1, r_2 the two years' returns
What it says in wordsThe starting wealth sees both returns in any order; only the withdrawn money depends on which return it misses.
Scale it up and the effect grows. Over twenty years of withdrawals, a bad run in the first few years forces sales at low prices from a pot that then has less left to recover, and two retirees with the same average return can end decades apart. The limit of this puzzle is its size: two years and one gap of Rs 4 lakh undersell how much the order of returns matters over a long retirement.
Where candidates lose it
The instinct is to say order cannot matter because multiplication is commutative. That is true only without cash flows, and the question gave you a withdrawal precisely to break it.
The second miss is getting 74 and 78 but being unable to say why. The single sentence about the first withdrawal skipping year two is what the interviewer wants to hear.
What the interviewer asks next
What if the retiree added Rs 10 lakh each year instead of withdrawing it?
How would you reduce sequence risk for a new retiree without changing the expected return?
Over thirty years, why does a bad first five years matter more than a bad last five?
042You want a curve steepener that is neutral to parallel moves: buy the 2-year bond and short the 10-year. The 10-year has a DV01 of 0.09 per 100 of face and the 2-year has 0.019. How much 2-year do you buy for each Rs 100 crore of 10-year you short?Duration and ratesCoreTreasury and ALMBank market risk
Try it first
Roughly how much 2-year face value balances Rs 100 crore of 10-year?
Show the worked solution
About Rs 474 crore of 2-year for each Rs 100 crore of 10-year. A parallel-neutral trade matches rupees per basis point, not face value. Rs 100 crore of 10-year moves Rs 9 lakh per basis point. The 2-year moves 0.019 per 100, so matching Rs 9 lakh needs 100 x 0.09 / 0.019, Rs 473.7 crore. A parallel shift then leaves the book flat; only a change in the gap between the two yields makes or loses money.
Why size by DV01 rather than by face value?
Balancing a see-saw is about weight times distance from the pivot, not the number of children on each end. DV01 is each leg's weight: the rupees it gains or loses when its yield moves one basis point. A 10-year bond is about 4.7 times as sensitive per rupee of face as a 2-year, so a parallel-neutral trade needs about 4.7 times as much 2-year face. Equal face value would leave the book mostly a bet on the 10-year, and a parallel move would swamp the curve view.
Rs 473.7 crore of 2-year bonds bought and Rs 100 crore of 10-year bonds sold each carry Rs 9 lakh of DV01, so the trade is level under a parallel move even though the 2-year face is almost five times larger.
The relationship
$N_2 = N_{10} \times \frac{DV01_{10}}{DV01_{2}} = 100 \times \frac{0.09}{0.019} = 473.7$
N face value of each leg, Rs crore
DV01 price change per 100 of face for a one basis point yield move
What it says in wordsSet the two legs' rupees per basis point equal, then solve for the face value of the hedge leg.
What does the trade make, and what can still go wrong?
If the 10-year yield rises 10 basis points while the 2-year stays put, the curve steepens and the short 10-year gains Rs 90 lakh. If both yields rise 10 basis points together, the 2-year loses Rs 90 lakh and the 10-year short gains Rs 90 lakh: flat, which is the design. Had you used equal face, that same parallel rise would have made about Rs 71 lakh, a large outright bet you did not mean to place.
Name the limits. DV01 is a local measure and drifts as yields move and time passes, so the ratio must be rebalanced. CarryThe income a position earns or pays while it is held unchanged: coupons received less the cost of funding and of shorting. differs across the two legs and can quietly dominate a slow trade. And short and long yields rarely move by the same amount; some desks weight the 2-year leg by its historical beta to the 10-year instead of one for one, which gives a different ratio.
Where candidates lose it
The fast wrong answer is equal face value, Rs 100 crore against Rs 100 crore. It leaves the trade roughly four-fifths an outright short of the 10-year, and the first parallel sell-off or rally decides the P&L, not the curve.
Candidates also invert the ratio and answer about Rs 21 crore, dividing 0.019 by 0.09. Check the direction: the less sensitive bond always needs the bigger face.
What the interviewer asks next
The trade should also be neutral to a 1-for-0.8 move between 2-year and 10-year yields. How does the ratio change?
How much does the position make if the curve steepens by 15 basis points?
Why might a treasury desk prefer futures to cash bonds for this trade?
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.Logic, estimation and brainteasersCoreTreasury 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 relationship
$40\text{ lakh} \times 0.75 \times 0.30 \times \text{Rs } 3{,}000 \div 5 = \text{Rs } 54 \text{ crore}$
0.75 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?
049A fund has an expected annual return of 12% and annual volatility of 20%, on Rs 100 crore. What is its one-year 95% VaR with and without the expected return, and when does the mean matter?VaR and expected shortfallCoreAsset manager risk
Try it first
Including the 12% expected return, what is the one-year 95% VaR?
Show the worked solution
Rs 32.9 crore ignoring the mean and Rs 20.9 crore including it. The 95% cut is 1.645 standard deviations below the mean: 1.645 x 20% is 32.9%, and a 12% expected return lifts the cut to minus 20.9%. At one year the mean cuts VaR by more than a third. At one day it barely matters: Rs 2.07 crore against Rs 2.02 crore.
Why does the mean matter at one year but not at one day?
On a short walk the path you take wanders more than your average direction moves you; on a long journey the direction wins. Expected return grows in proportion to time while volatility grows with its square root, so over short horizons the mean is noise and over long ones it is material. At one day the mean is 12% / 252, about 0.048%, against a daily volatility of 1.26%; the VaR figures differ by 2.3%. At one year the gap is 36%.
Over one year a 12% expected return shifts the whole distribution right, moving the 5% cut from minus 32.9% to minus 20.9%, whereas over one day the same mean moves VaR from Rs 2.07 crore only to Rs 2.02 crore.
The relationship
$\text{VaR}_{95\%} = (1.645\,\sigma - \mu) \times W = (32.9\% - 12\%) \times 100 = 20.9$
\sigma annual volatility, 20%
\mu expected annual return, 12%
W portfolio value, Rs 100 crore
What it says in wordsVaR is how far the 5% worst outcome sits below zero: the volatility term pulls it down, the mean pushes it back up.
Which figure should a risk report quote?
Say which you are quoting, because both are used. Relative VaRVaR measured from the expected outcome rather than from zero, so it captures only the uncertainty and ignores the expected gain. ignores the mean and measures pure uncertainty; absolute VaR includes it and measures the loss from today's value. For a one-year horizon, many risk teams quote the zero-mean figure deliberately: a 12% expected return is an assumption, and counting it as a cushion lets an optimistic forecast shrink the risk number. Daily trading VaR usually ignores the mean for the simpler reason that it makes no difference.
Limits to name: over a year, compounding and fat tails matter, so a normal model with a constant 20% volatility understates the chance of a large loss, and volatility itself changes over the year. The one-year figure is a rough guide to the size of a bad year, not a promise about it.
Where candidates lose it
The most common slip is adding the mean to the loss, giving 44.9, which moves the cut the wrong way. The second is ignoring the question's own hint and giving only 32.9.
The quieter miss is not answering the when part. The square-root rule for volatility against the linear growth of the mean is the one sentence the interviewer is waiting for.
What the interviewer asks next
Over what horizon does the mean cut VaR by half?
Why might a regulator prefer the zero-mean figure?
How would you compute one-year VaR if returns were lognormal?

z_c	the number of standard deviations that leaves the chosen tail below it
sigma	the daily P&L standard deviation, Rs 2 crore

EL	expected loss, the average annual credit loss, Rs 1 crore
economic capital	the buffer held against losses worse than average, Rs 8 crore

z_{0.99}	the 99% one-sided normal quantile, 2.33
\sigma_{1d}	daily volatility of the netting set value, Rs 2 crore
MPOR	margin period of risk, 10 days

p	one-year default probability
100 - 80	the gain if the bond pays at par
100 - 40	the gap between par and recovery

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