Risk Management puzzles, solved step by step

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

Showing 1–10 of 20 · filtered from 100Clear filters

002A credit card balance carries interest of 3.5% a month, compounded monthly. What is the effective annual rate?Compounding and drawdownsWarm upNBFC credit riskBank credit risk
Try it first
Answer inside ten seconds: roughly what is the effective annual rate?
Show the worked solution
About 51.1% a year. Rs 100 left unpaid grows by 3.5% each month on a balance that already includes last month's interest, so after twelve months it is 100 times 1.035 to the 12th, which is Rs 151.1. The simple rate of 12 times 3.5% is 42%, so compounding adds about 9.1 points.
Why is 42% the wrong answer?
Picture a jar of rice where a helper adds 3.5% of whatever is in the jar every month. In month two the helper adds 3.5% of a bigger jar than in month one. Monthly compounding charges interest on the interest already added, so the annual rate is always above twelve times the monthly rate. The 42% figure is what you would pay only if the lender added interest to a separate pile that never itself earned interest.
Rs 100 left on a card at 3.5% a month climbs in monthly steps to Rs 151.1 after a year, while adding a flat Rs 3.50 a month reaches only Rs 142, so compounding adds about 9.1 points to the annual rate.
The relationship
$\text{EAR} = (1 + m)^{12} - 1 = 1.035^{12} - 1 \approx 51.1\%$
m the monthly rate, 3.5%
12 the number of compounding periods in a year
What it says in wordsGrow one rupee for twelve months at the monthly rate and subtract the rupee you started with.
How do you get 1.035 to the 12th without a calculator?
Square it in steps. 1.035 squared is about 1.0712. Square again for four months, about 1.1475. Cube that for twelve months: 1.1475 squared is about 1.3168, and times 1.1475 again is about 1.511. Three multiplications you can do out loud get you to within a tenth of a point. A faster check is the log approximation: twelve times 3.44%, the log of 1.035, is 41.3%, and e to the 0.413 is about 1.51.
Then say why a credit risk team cares. The effective rate is what a borrower who rolls the balance actually pays, and a borrower paying above 50% a year is a borrower whose debt can outgrow their income quickly. The stated monthly figure is how the product is sold; the effective annual rate is the number that belongs in a comparison with other loans.
Where candidates lose it
The trap is answering 42% because the question sounds like a multiplication. It misses that the lender adds interest to the balance every month, and it understates the cost by about 9 points.
The second loss is freezing on the arithmetic. Say the formula, then square in steps: 1.035 squared, squared again, then cubed. Reaching 1.51 out loud is worth more than a silent calculator answer.
What the interviewer asks next
What monthly rate gives an effective annual rate of exactly 36%?
If the card compounds daily at the same annual simple rate, is the effective rate higher or lower, and by how much?
A borrower pays only the minimum of 5% of the balance each month. How long until the balance halves?
003A portfolio holds 60% in a stock with a beta of 1.2 and 40% in cash. The index falls 10%. What move do you expect in the portfolio from market exposure alone?Correlation and diversificationWarm upAsset manager risk
Try it first
What is the expected move in the portfolio?
Show the worked solution
About 7.2% down. Portfolio beta is the weighted average of the holdings' betas: 0.6 times 1.2 for the stock plus 0.4 times zero for the cash, which is 0.72. Multiply by the index move of minus 10% and the expected move is minus 7.2%. The stock itself is expected to fall 12%, but the cash dilutes it.
Why does the cash count as zero?
Imagine a household where one earner's pay swings with the economy and the other's savings sit in a bank account. When the economy dips, only the first half of the income moves. Beta measures how much a holding moves for each 1% move in the index, and cash does not move with the index at all, so its beta is zero. The portfolio beta is then the weighted sum: 0.6 times 1.2, which is 0.72, plus 0.4 times zero.
The stock is 60% of the portfolio with a beta of 1.2 and contributes 0.72; the cash contributes nothing, so a 10% fall in the index maps to an expected 12% fall in the stock but only 7.2% in the portfolio.
The relationship
$\beta_p = \sum_i w_i \beta_i = 0.6 \times 1.2 + 0.4 \times 0 = 0.72 \qquad \Delta_p = 0.72 \times (-10\%) = -7.2\%$
w_i each holding's share of the portfolio
beta_i each holding's sensitivity to the index
What it says in wordsWeight each holding's beta by its share of the money, add them up, and scale the index move by the result.
What does beta leave out?
Everything that is not the market. Beta gives the expected move from market exposure; the stock's own news adds a separate, unpredictable move on top. On Rs 10 lakh the expected loss is Rs 0.72 lakh, about Rs 72,000, but the actual loss could be larger or smaller depending on what happens to that one company. A single stock position carries a lot of this idiosyncratic riskRisk specific to one company, such as a product failure or a management change, which does not move with the market as a whole., which is why a risk manager quotes beta as an expectation, not a forecast.
Also say that beta is estimated from past returns and drifts over time. A stock measured at 1.2 over the last three years can behave like 1.5 in a sell-off, because correlations tend to rise when markets fall. The 7.2% is the right answer to the question as posed; the conversation that follows is about how much to trust the 1.2.
Where candidates lose it
Candidates answer 12%, which is the stock's expected move, and forget that 40% of the money is in cash. The question is about the portfolio, and the weights are the point.
The second miss is presenting 7.2% as what will happen. Call it the expected move from market exposure, and name the stock-specific risk that sits on top.
What the interviewer asks next
How much of the stock would you sell to bring the portfolio beta to 0.5?
The cash is replaced with a bond fund with a beta of 0.1. What is the new portfolio beta?
How would you hedge the market exposure with index futures, and what risk would remain?
006Write NPV as the product of two vectors. The cash flows are minus 100, 30, 40, 50 and 20 in years 0 to 4, and the discount rate is 10%. What is the NPV?Duration and ratesWarm upMoody'sNew York · 2018
Try it first
Which operation turns the two vectors into the NPV?
Show the worked solution
NPV is the dot product of the cash flow vector and the discount factor vector, and here it is about 11.56. The discount factors at 10% are 1, 0.909, 0.826, 0.751 and 0.683. Multiplying term by term gives minus 100, 27.27, 33.06, 37.57 and 13.66, which sum to 11.56. A positive NPV means the project earns more than 10%.
Why is NPV a dot product at all?
Think of a grocery bill. One list holds the quantity of each item, another holds each item's price, and the bill is quantity times price for each line, added up. NPV has the same shape: one vector holds the cash flows, the other holds what one rupee in each year is worth today, and the NPV is the sum of their products. Writing it this way separates the project, which is the cash flow vector, from the market, which is the discount factor vector. Change the rate and only the second vector changes.
The relationship
$\text{NPV} = \mathbf{c} \cdot \mathbf{d} = \sum_{t=0}^{4} c_t\, d_t, \qquad d_t = \frac{1}{(1+r)^t}$
c the cash flow vector, year 0 to year 4
d the discount factor vector, one entry per year
r the discount rate, 10%
What it says in wordsMultiply each cash flow by the value today of one rupee in that year, and add the results.
Multiplying the cash flows by the discount factors year by year gives present values of minus 100, 27.27, 33.06, 37.57 and 13.66, and adding them from minus 100 upward reaches an NPV of 11.56.
What does the interviewer want to hear beyond the number?
The thought process was part of the question, so say it in order. First build the discount factor vector from the rate, then take the dot product, then sanity check the sign and size. The undiscounted inflows are 140 against 100 out, so a positive but much smaller NPV is expected once four years of 10% are taken out. And in a spreadsheet the same idea is one SUMPRODUCT of two ranges, which is why the vector form is how a model is usually built.
Then give the extension that shows range. With a term structure of rates, only the discount factor vector changes: each entry uses its own year's rate. With several scenarios, stack the cash flow vectors into a matrix and one matrix multiplication gives every scenario's NPV at once. The limitation is that the vector form assumes the cash flows are known; uncertain cash flows need expected values or scenarios first.
Where candidates lose it
Candidates reach for the NPV formula and start adding fractions, which gets the number but misses the question. The interviewer asked for two vectors precisely to see whether you can separate what the project pays from what time is worth.
The other slip is discounting year 0. The first discount factor is 1; the minus 100 is already in today's money.
What the interviewer asks next
How would you write the IRR condition using the same two vectors?
The rate for year 1 is 8% and for later years 10%. What changes in the vector form?
How would you compute the NPV for 1,000 cash flow scenarios in one operation?
Asked at Moody's, Analytics, New York, 2018 (Wall Street Oasis): Construct an NPV formula using 2 vectors and show me your thought process.
008What is the angle between the hour hand and the minute hand of a clock at 3:15?Logic, estimation and brainteasersWarm upBank market riskRisk GCC
Try it first
Answer inside five seconds.
Show the worked solution
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.
At 3:15 the minute hand points at the 3, 90 degrees from 12, while the hour hand has moved a quarter of the way towards the 4, to 97.5 degrees, leaving a 7.5 degree gap between them.
The relationship
$\theta = \left| 30H + 0.5M - 6M \right| = |90 + 7.5 - 90| = 7.5^{\circ}$
H the hour, here 3
M the minutes past the hour, here 15
30H + 0.5M the hour hand's angle from 12
6M the minute hand's angle from 12
What it says in wordsWork out each hand's angle from 12, then take the difference.
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.
Where candidates lose it
The trap is answering zero, fast. The question is short, the picture feels obvious, and the hour hand's drift is exactly the detail the static picture hides.
If you catch yourself, say so: both hands look as if they point at the 3, but the hour hand has moved a quarter of the way on. Correcting out loud is almost as good as getting it right first time.
What the interviewer asks next
What is the angle at 9:45?
How many times a day do the hands overlap, and when is the first overlap after 12:00?
At what time between 3 and 4 are the hands exactly opposite each other?
010An options book has a vega of Rs 5 lakh per volatility point. Implied volatility drops from 22% to 18% overnight. What is the P&L?Options and GreeksWarm upBank market risk
Try it first
What is the P&L on the book?
Show the worked solution
A loss of about Rs 20 lakh. Positive vega means the book is long volatility and gains Rs 5 lakh for each point implied volatility rises. Volatility fell from 22% to 18%, four points, so the book loses four times Rs 5 lakh. That is a first-order estimate: vega itself changes as volatility and time move.
How do you turn a Greek into rupees?
Think of an electricity tariff quoted per unit. The bill is the rate times the units used, and you need to know which unit the rate refers to. VegaThe change in an option position value for a one point change in implied volatility. is a rate quoted per volatility point, so the P&L is vega times the number of points volatility moved, with the sign. Here the book has positive vega, so it is long options, and volatility moved minus 4 points: Rs 5 lakh times minus 4 is minus Rs 20 lakh.
Implied volatility falls four points from 22% to 18%, and at a vega of Rs 5 lakh per point the book loses Rs 5 lakh on each point, a loss of Rs 20 lakh in total.
The relationship
$\Delta V \approx \nu \times \Delta\sigma = 5 \text{ lakh} \times (18 - 22) = -20 \text{ lakh}$
nu vega, Rs lakh per volatility point
Delta sigma the change in implied volatility, in points
What it says in wordsMultiply the book's sensitivity per point by how many points volatility moved.
When is the Rs 20 lakh estimate wrong?
In three ways worth naming. Vega is a local slope, so a four point move is large enough for vega itself to change, and the true loss can differ from the straight-line estimate. Second, a single vega number assumes every strike and maturity moved by the same four points; in practice short-dated volatility often moves more than long-dated, and the skew can twist. Third, overnight the book also loses or gains time decay and any delta and gamma from the underlying move, so the full P&L explain has more than one line.
On a risk desk, say how you would check it. Compare the vega estimate with a full revaluation at 18%, and bucket vega by maturity so a twist in the volatility surface is visible. If the explained P&L and the actual P&L differ by much, the gap is what the risk team investigates next.
Where candidates lose it
The trap is treating the move as a percentage change: 4 over 22 is about 18%, and some candidates multiply vega by that. Vega is quoted per point of volatility, so the move is four points, not 18%.
The second slip is the sign. Positive vega loses when volatility falls; say the direction before the number.
What the interviewer asks next
What position would have made money on this move, and what would its vega be?
Short-dated volatility fell 6 points and long-dated only 2. How would you estimate the P&L now?
How would you hedge the book's vega without changing its delta?
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?
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?
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?

m	the monthly rate, 3.5%
12	the number of compounding periods in a year

w_i	each holding's share of the portfolio
beta_i	each holding's sensitivity to the index

c	the cash flow vector, year 0 to year 4
d	the discount factor vector, one entry per year
r	the discount rate, 10%

H	the hour, here 3
M	the minutes past the hour, here 15
30H + 0.5M	the hour hand's angle from 12
6M	the minute hand's angle from 12

Try it first

Why is 42% the wrong answer?

How do you get 1.035 to the 12th without a calculator?

Where candidates lose it

What the interviewer asks next

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Why does the cash count as zero?

What does beta leave out?

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Why is NPV a dot product at all?

What does the interviewer want to hear beyond the number?

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Why is zero the wrong answer?

Why would a risk interviewer ask a clock question?

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How do you turn a Greek into rupees?

When is the Rs 20 lakh estimate wrong?

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Why does maturity not drive a floater's rate risk?

How do you check the number quickly?

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Why does the funding cost drop out of the answer?

Is 3.2 points the true gap?

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

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

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What does each leg of the collar do?

How do you check the two ends quickly?

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Why does the same VaR hide such different risk?

What does the ratio of expected shortfall to VaR tell you?

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