Risk Management puzzles, solved step by step

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

Showing 31–40 of 100

031A floating rate note pays a coupon that resets every three months to the market rate, and it matures in 7 years. Roughly what is its interest rate duration?Duration and ratesWarm upTreasury and ALM
Try it first
Pick the duration before you reason it out.
Show the worked solution
About 0.25 years at most, and on average about an eighth of a year. Every coupon after the next one resets to the market rate, so a change in rates today is passed straight into future coupons and leaves their value unchanged. Only the next coupon is fixed, so the note behaves like a three-month bill: modified duration of about 0.25 just after a reset, falling towards zero before the next one.
Why does maturity not drive a floater's rate risk?
Think of renting a flat with the rent reset to the market every quarter. If rents jump tomorrow, your lease is not a bargain or a burden for more than a few months, because the next reset catches up. A bond loses value when rates rise only because its cash flows are fixed below the new market rate; a floater's cash flows are not fixed beyond the next reset. At each reset date the note is worth par again, provided the issuer's credit has not changed, so for rate purposes it is a claim to par plus one known coupon, three months away.
A 7-year quarterly floater has only its next coupon fixed, so its rate duration is about 0.25 years just after a reset, while a 7-year fixed 8% bond has a modified duration of about 5.32 years and loses about twenty times as much for a one point rise in rates.
How do you check the number quickly?
Right after a reset, the note is worth the present value of par plus one coupon, received in a quarter. That is a single cash flow a quarter away, so Macaulay duration is 0.25 years and modified duration is 0.25 divided by 1.02, about 0.25. A day before the next reset it is almost zero. Compare a 7-year fixed bond paying 8% quarterly at par: its modified duration is 5.32, so a one point rate rise costs it about 5.3 per 100 against about 0.25 for the floater.
Now say where the 7 years still matter. The coupon is the market rate plus a fixed quoted marginThe fixed spread over the reference rate that a floating rate note pays, set at issue and unchanged for the life of the note.. If the issuer's credit worsens and investors demand a wider spread, that fixed margin is too low for all 28 remaining quarters, so the note's spread duration is close to a fixed bond's, several years. A treasury desk that files floaters under no rate risk and forgets spread risk has the right answer to the wrong question.
Where candidates lose it
The instinct is to answer 7 years because the note matures in 7 years, or about 5 because that is what a 7-year bond usually carries. Both confuse maturity with the length of time cash flows are fixed.
The overcorrection is saying zero. Until the next reset one coupon is locked, and the credit spread is locked for the whole life. Give the rate answer, then name spread duration unprompted.
What the interviewer asks next
What is the floater's spread duration, roughly, and why?
An inverse floater pays 16% minus the market rate. What is its duration?
How would you hedge a fixed-rate loan book funded with floating-rate deposits?
032A bank holds Rs 1,000 crore of liquid government bonds yielding 6.8% instead of lending the money at 10%, and it funds the whole amount at 6%. What does carrying this liquidity buffer cost the bank each year?Liquidity and balance sheetWarm upTreasury and ALM
Try it first
Which comparison gives the cost of the buffer?
Show the worked solution
About Rs 32 crore a year, before adjusting for credit losses on the loans. The money is funded at 6% either way, so funding drops out. The cost of the buffer is the income it gives up: lending would earn 10% and the bonds earn 6.8%, a 3.2 point gap on Rs 1,000 crore. The buffer still earns Rs 8 crore over funding; it just earns Rs 32 crore less than loans would.
Why does the funding cost drop out of the answer?
A family that keeps Rs 5 lakh in a savings account instead of prepaying a home loan pays the same salary-funded EMI either way. The cost of the emergency fund is the gap between the loan rate saved and the savings rate earned. The cost of any buffer is an opportunity cost: what the same rupees would have earned in their next-best use, with everything common to both uses cancelling out. Funding at 6% is common to both uses here, so the only number that matters is 10% minus 6.8%.
Lending earns 10.0% and liquid bonds earn 6.8% on money funded at 6.0%, so holding Rs 1,000 crore of bonds instead of loans gives up 3.2 points, Rs 32 crore a year, even though the bonds still earn Rs 8 crore over funding.
Is 3.2 points the true gap?
Not quite, and saying why is the part interviewers listen for. A 10% loan yield is before credit losses and before the capital loans consume; government bonds need neither. If the loans carry an assumed expected loss of 1.2% a year, the like-for-like gap narrows to 2.0 points, and the buffer costs about Rs 20 crore rather than Rs 32 crore. Capital would narrow it further. The headline number is an upper bound; the risk-adjusted number is what the treasurer should defend.
The relationship
$\text{cost} = \text{size} \times (y_{\text{loan}} - EL - y_{\text{liquid}}) = 1{,}000 \times (10.0\% - 1.2\% - 6.8\%) = 20$
y_loan yield on the loans that could have been made, 10%
EL an assumed annual expected credit loss on those loans, 1.2%
y_liquid yield on the liquid bonds, 6.8%
What it says in wordsCompare what the money earns in each use after the costs that differ between them.
Close by naming what the premium buys. The buffer is insurance: in a deposit run it can be sold or pledged within days, while loans cannot. Liquidity coverageA regulatory measure comparing high-quality liquid assets with the net cash a bank could lose in a 30-day stress. rules make a floor of it compulsory; confirm the current requirement with the regulator rather than from memory. Above that floor, a bank is choosing how much insurance to buy, and Rs 20 to 32 crore a year is the price tag to weigh against the run it protects against.
Where candidates lose it
The common wrong answer is zero, because the bonds earn 6.8% against a 6% funding cost and so look profitable. Positive carry is not the same as no cost; the bank has given up a better use of the money.
The second miss is quoting Rs 32 crore as if loans were risk-free. Mention expected loss and capital in one sentence and you show you compare like with like.
What the interviewer asks next
Rates on liquid bonds rise to 7.5% with everything else fixed. What happens to the cost?
Why might a bank hold more liquidity than the regulatory minimum?
How would you allocate this cost to the business lines that create the liquidity need?
033A price rises 20% and then falls 20%. Separately, a bank's gross NPA ratio moves from 2% to 3%. What is the net price change, and how would you describe the NPA move, both in percentage points and in percent?Logic, estimation and brainteasersWarm upRisk GCCAsset manager risk
Try it first
The gross NPA ratio went from 2% to 3%. Which description is wrong?
Show the worked solution
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%.
A price that rises 20% from 100 to 120 and then falls 20% ends at 96, a 4% loss; a gross NPA ratio that moves from 2% to 3% has risen by 1 percentage point, which is a 50% increase on its starting level.
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.
Where candidates lose it
On the price, the trap is answering zero because plus 20 and minus 20 seem to cancel. They cancel only when percentages are added, and returns multiply.
On the ratio, the trap is saying it rose by 1%. That phrase means 2.02%, a rounding-level change, and a risk manager who uses it in a committee has understated a 50% jump in bad loans.
What the interviewer asks next
A price falls 20% and then rises 20%. Where does it end?
A 10% default rate rises to 12%. Describe the change both ways.
Why do regulators and banks prefer basis points when quoting changes in rates?
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?
035A stock trades at Rs 500. An investor who owns it buys a 450 put for Rs 12 and sells a 560 call for Rs 10, both expiring on the same date. What is the range of outcomes at expiry?Options and GreeksWarm upAsset manager risk
Try it first
What is the worst loss per share at expiry, including the premiums?
Show the worked solution
Between a loss of Rs 52 and a gain of Rs 58 a share. The put guarantees a sale at 450 at worst; the sold call hands over anything above 560. Between the strikes the investor simply holds the stock. The pair costs Rs 12 minus Rs 10, a net Rs 2, so the outcome runs from 450 less 500 less 2, minus 52, to 560 less 500 less 2, plus 58.
What does each leg of the collar do?
A farmer worried about a price crash agrees with a trader: if prices fall below a floor, the trader pays the floor; in return, if prices soar above a ceiling, the farmer sells at the ceiling. The farmer gives up the dream harvest to remove the nightmare one. The bought put is the floor, the sold call is the ceiling, and the premium from the call pays for most of the put. That structure is a collarA position that holds a stock, buys a put below the current price and sells a call above it, locking the outcome between two strikes.: the investor still owns the stock between 450 and 560 and nothing outside it.
The collar follows the stock between the 450 and 560 strikes and is flat outside them, so the outcome runs from a loss of Rs 52 to a gain of Rs 58 a share, with breakeven at Rs 502 after the net premium of Rs 2.
How do you check the two ends quickly?
Take one price in each region and walk through it. At 400, the stock is worth 400 and the put pays 50, so the holding is worth 450; the call expires worthless. At 620, the stock is worth 620 and the call costs 60, so the holding is worth 560. Whatever happens, the holding ends between 450 and 560, and subtracting the Rs 500 cost and the Rs 2 net premium gives the range of minus 52 to plus 58. Breakeven is Rs 502, the starting price plus the net premium.
Say what the collar does not do. It removes the tails; it does nothing for moves inside the band. And the cheap net premium is not free protection: the investor paid by selling every rupee of gain above 560. Whether that trade is sensible depends on what the investor needs, a floor for a known liability, say, rather than on the Rs 2.
Where candidates lose it
The frequent slip is to forget the premiums and quote minus 50 to plus 60. The interviewer gave you two premium numbers for a reason; the net Rs 2 moves both ends.
The opposite slip is reading the sold call as an unlimited risk. The investor owns the shares, so the call is covered: its cost is the lost upside above 560, not an open-ended loss.
What the interviewer asks next
Which strikes would make the collar cost exactly zero, and what do you give up?
The stock is at 440 a month before expiry. How has the collar's delta changed?
Why might a promoter holding a large stake use a collar rather than simply selling shares?
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?
037Forecaster A has a bias of 1 point and a forecast error standard deviation of 2. Forecaster B is unbiased with a standard deviation of 2.5. Using the mean squared error decomposition, which forecaster is better?Statistics and estimationCoreBlackRockNew York · 2026
Try it first
Which forecaster has the lower mean squared error?
Show the worked solution
Forecaster A, with a mean squared error of 5 against 6.25. Mean squared error splits into bias squared plus variance. A pays 1 squared for its bias and 2 squared for its spread, 5 in all. B pays nothing for bias but 2.5 squared, 6.25, for its spread. A's typical error, the square root, is 2.24 against 2.50.
How can a biased forecaster beat an unbiased one?
Two archers. One groups every arrow tightly but slightly left of centre; the other is centred on average but scatters arrows all over the target. Ask which one lands closer to the bullseye on a typical shot, and the tight grouping wins. Mean squared error charges for two things, how far off you are on average and how much you scatter, and a small, steady bias can cost far less than a large scatter. Being unbiased only removes the first charge.
The relationship
$\text{MSE} = \text{bias}^2 + \text{variance}: \quad A = 1^2 + 2^2 = 5, \qquad B = 0^2 + 2.5^2 = 6.25$
bias the average forecast error, forecast minus actual
variance the spread of the errors around their own average, the standard deviation squared
What it says in wordsSquared error on average equals the squared average error plus the spread of errors around it.
Forecaster A's mean squared error is 1 of bias squared plus 4 of variance, 5 in total, while unbiased Forecaster B carries 6.25 of pure variance, so A's small bias buys a larger cut in variance and gives the lower error.
When would your answer flip, and what would you do with A?
Solve for the tie: A matches B when bias squared plus 4 equals 6.25, so a bias of 1.50. Below that, A wins. More useful still, a bias that is stable can be measured and subtracted: correct A by one point and its MSE falls to 4, better than either original. That is the practical lesson for a risk team: a model that is consistently off in one direction is fixable, while a noisy model is not. The trade-off is also why risk teams use shrinkagePulling a noisy estimate towards a simpler, steadier target, accepting a little bias in return for much lower variance. on covariance matrices built from short histories.
The limit: MSE punishes large errors heavily because it squares them, and it treats over-forecasts and under-forecasts alike. A risk manager forecasting losses may care more about under-forecasting than over-forecasting, in which case a symmetric score is the wrong yardstick and the ranking could change.
Where candidates lose it
Candidates pick B on reflex because unbiased sounds like correct. The question is built to see whether you know that MSE has two parts and can do the two-line arithmetic.
The quieter miss is stopping at 5 against 6.25. Add that A's bias can be corrected, taking its MSE to 4, and you have turned a statistics answer into a model-risk judgement.
What the interviewer asks next
What bias would make the two forecasters exactly equal?
Why might a regulator prefer the unbiased forecaster even with a higher MSE?
How would you test whether A's bias is stable over time?
Asked at BlackRock, Restructuring, New York, 2026 (Wall Street Oasis): Which equities have duration ? multiple stocks vs value stocks MSE Forecasting equation
038Two trading desks both report a 99% one-day VaR of Rs 5 crore. On their worst 1% of days, desk A lost Rs 6, 6.5 and 7 crore and desk B lost Rs 6, 12 and 30 crore. Compute the average tail loss for each desk and say which is riskier.VaR and expected shortfallWarm upBank market risk
Try it first
Before you average: which desk's tail average is larger, and by roughly how much?
Show the worked solution
Desk A's tail average is Rs 6.5 crore and desk B's is Rs 16.0 crore, so desk B is far riskier. Both desks have a 99% VaR of Rs 5 crore, which only says where the worst 1% of days begins. Averaging the losses beyond it, the expected shortfall, shows A's bad days cluster just past the line while B's run to Rs 30 crore.
Why does the same VaR hide such different risk?
Two rivers both have a flood mark at five metres that is crossed one year in a hundred. On one river, those floods reach six or seven metres; on the other, one of them reached thirty and washed the town away. The flood mark is the same; the town planner should care about the second river. VaR is the threshold of the bad days, not their size; two desks can share a threshold and have tails of completely different weight. Expected shortfallThe average loss on the days worse than VaR, so it measures the size of the tail rather than just where it starts. answers the question VaR leaves open: when it goes wrong, how wrong on average?
Desk A and desk B share a 99% VaR of Rs 5 crore, but desk A's worst days average Rs 6.5 crore while desk B's average Rs 16.0 crore, because one of B's tail days lost Rs 30 crore.
What does the ratio of expected shortfall to VaR tell you?
Divide one by the other. A's ratio is 1.3 and B's is 3.2. For normally distributed returns at 99%, expected shortfall is only about 15% above VaR, so a ratio of 3.2 is a flag that something is behaving far from normal: an option position losing faster as the market moves, a concentrated name gapping, or a liquidity cliff. Desk A looks close to normal; desk B needs its positions read line by line.
The relationship
$ES_A = \frac{6 + 6.5 + 7}{3} = 6.5, \qquad ES_B = \frac{6 + 12 + 30}{3} = 16$
ES expected shortfall, the average of losses beyond VaR
3 the number of days in the worst 1%, which implies about 300 days of history
What it says in wordsAverage the losses on the days worse than VaR to see how heavy the tail is.
State the limit honestly. Three observations make a fragile average: one more bad day for desk A, or one fewer outlier for desk B, would move the numbers a lot. Expected shortfall is also harder to backtest than VaR, because you are checking an average of rare events rather than a count of breaches. It is still the better answer to the question asked.
Where candidates lose it
The trap is saying the desks are equally risky because their VaR is equal, or ranking them on the single worst day without computing anything. The question hands you the tail precisely so you use it.
The quieter miss is not naming the measure. Call the tail average expected shortfall and say one sentence about why regulators moved towards it: VaR is blind past its own line.
What the interviewer asks next
Desk B's Rs 30 crore day came from one option position. What would you ask the desk?
Why is expected shortfall harder to backtest than VaR?
If you combined the two desks, could the combined VaR exceed the sum of the two? Could the expected shortfall?
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?
040A portfolio holds Rs 60 crore of asset A with 20% volatility and Rs 40 crore of asset B with 30% volatility, and their correlation is 0.5. Compute the portfolio's volatility and each asset's contribution to it.Correlation and diversificationHardBank market riskQuant risk
Try it first
Asset A is 60% of the capital. What share of the portfolio's risk does it contribute?
Show the worked solution
Portfolio volatility is about Rs 20.78 crore, 20.8% of the Rs 100 crore, and each asset contributes exactly half of it. A carries 60 x 20% = Rs 12 crore of standalone volatility and B carries 40 x 30% = Rs 12 crore. The variance is 144 + 144 + 2 x 0.5 x 12 x 12 = 432, so volatility is 20.78. With equal standalone risk, each contributes Rs 10.39 crore: 60% of the capital, 50% of the risk.
Why convert to rupee volatility first?
A household spends 60% of its budget on rent and 40% on a car loan. If the rent is fixed and the car loan's rate floats, most of the budget's uncertainty comes from the smaller item. Risk depends on size times volatility, so the first step is to express each position's volatility in rupees: that puts the two assets on one scale. Here the smaller, more volatile position carries exactly as much standalone risk as the larger, calmer one: Rs 12 crore each.
The relationship
$\sigma_P^2 = 12^2 + 12^2 + 2(0.5)(12)(12) = 432, \qquad \sigma_P = 20.78$
12 rupee volatility of each position, Rs crore
0.5 correlation between A and B
What it says in wordsPortfolio variance is each position's variance plus twice the shared part.
How do you split the portfolio's risk between the two assets?
Use the Euler allocationSplitting total risk so each position gets its own variance plus its share of every covariance, divided by total volatility; the pieces add up exactly to the total.: each asset's contribution is its own variance plus its covariance with the other, divided by total volatility. For A that is (144 + 0.5 x 12 x 12) over 20.78, which is 216 over 20.78, Rs 10.39 crore. B's contribution is the same Rs 10.39 crore, and the two add back to exactly Rs 20.78 crore, so the split is 50/50 while the capital split is 60/40.
Asset A holds 60% of the capital but contributes only 50% of the risk, Rs 10.39 crore of the Rs 20.78 crore portfolio volatility, because asset B's higher volatility makes its Rs 40 crore as risky as A's Rs 60 crore.
Two points to add. First, the diversification benefitThe amount by which portfolio risk falls short of the sum of the standalone risks, because the positions do not move in perfect lockstep.: standalone risks sum to Rs 24 crore, the portfolio carries Rs 20.78 crore, and Rs 3.22 crore disappears because correlation is below 1. Euler allocation spreads that benefit across both assets rather than handing it to whoever joined last. Second, the limit: contributions depend on a correlation estimate, and correlations rise in stress. At a correlation of 1 there is no benefit left to share.
Where candidates lose it
The common error is assuming risk shares match capital shares, 60/40, because A is the bigger position. The whole point of the puzzle is that the smaller, more volatile asset can carry as much risk as the larger one.
The second error is adding standalone volatilities, Rs 24 crore, as the portfolio risk. That ignores correlation and overstates risk by more than Rs 3 crore; always go through variance.
What the interviewer asks next
How much of B would you sell to make the risk split 60/40?
What happens to both contributions if the correlation rises to 0.9?
Why do risk-parity funds size positions by risk contribution instead of capital?

y_loan	yield on the loans that could have been made, 10%
EL	an assumed annual expected credit loss on those loans, 1.2%
y_liquid	yield on the liquid bonds, 6.8%

\lambda	the average number of large losses a year, 0.2
t	the number of years you are looking across

X	the number of defaults in the year
\binom{50}{k}	the number of ways to pick which k names default

bias	the average forecast error, forecast minus actual
variance	the spread of the errors around their own average, the standard deviation squared

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