Risk Management puzzles, solved step by step

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

Showing 1–10 of 100

001A bank's CET1 ratio is 13% and its minimum requirement plus buffers is 10.5%. Its risk-weighted assets are half its total assets. What loss, as a share of total assets, can it absorb before it breaches?Capital and leverageCoreBank credit risk
Try it first
Pick the loss, as a share of total assets, before you work it.
Show the worked solution
About 1.25% of total assets. The cushion is 13% minus 10.5%, which is 2.5 points of risk-weighted assets. RWA are half of total assets, so the cushion is 1.25% of assets. On Rs 1,00,000 crore of assets that is Rs 1,250 crore of loss before the bank hits the floor, assuming RWA do not move.
Why is the gap not simply 2.5%?
Think of a bus pass priced per kilometre and a taxi fare priced per minute. Both are numbers, but you cannot compare them until you put them on the same unit. A capital ratio is measured against risk-weighted assets, while a loss is measured against the assets that went bad, so the two sit on different bases. Put numbers on it. A bank with Rs 1,00,000 crore of assets and Rs 50,000 crore of RWA holds 13% of 50,000, which is Rs 6,500 crore of CET1. The floor is 10.5% of 50,000, Rs 5,250 crore. The spare capital is Rs 1,250 crore.
Measured against risk-weighted assets the bank has 2.5 points of headroom, but measured against total assets the same Rs 1,250 crore is only 1.25 points, because risk-weighted assets are half the balance sheet.
The relationship
$\text{loss capacity} = (13\% - 10.5\%) \times \frac{\text{RWA}}{\text{total assets}} = 2.5\% \times 0.5 = 1.25\%$
13% - 10.5% spare capital in ratio points of RWA
RWA / total assets the average risk weight, here 0.5
What it says in wordsConvert the spare ratio points into asset terms by multiplying by the average risk weight.
What refinement shows you understand the ratio?
The first answer holds RWA fixed. If the loss comes from writing off loans that carried a 100% risk weight, those loans leave the RWA too, so the denominator shrinks with the numerator and the bank can absorb slightly more. Solving 6,500 minus L over 50,000 minus L equal to 10.5% gives L of about Rs 1,397 crore, around 1.40% of assets. Say the base answer first, then offer this as the second-order effect.
Two limits are worth one sentence each. The 10.5% figure is the question's assumption; actual minimums and buffers vary by bank and by regulator, so confirm the current numbers for any real institution. And capital is only one constraint: a bank can hit its leverage ratio, a liquidity limit or a large exposure limit before it runs out of CET1.
Where candidates lose it
The fast wrong answer is 2.5%. It treats a percentage of risk-weighted assets as if it were a percentage of the balance sheet, and it overstates the cushion by a factor of two for this bank.
The quieter loss is stopping at 1.25% without saying you held RWA fixed. Name the assumption; it invites the refinement and shows you know what moves in the denominator.
What the interviewer asks next
The bank's RWA density rises to 70% of assets. How much loss can it absorb now?
How much capital must it raise to rebuild a 2.5 point buffer after a 1% loss on assets?
Why might the leverage ratio bind before the CET1 ratio for a bank holding mostly government bonds?
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?
004Your exposure to a counterparty is Rs 50 crore. The collateral agreement has a Rs 20 crore threshold and a Rs 2 crore minimum transfer amount, and you already hold Rs 25 crore of collateral. How much collateral do you call?Counterparty exposure and collateralCoreCounterparty risk
Try it first
How much do you call?
Show the worked solution
Call Rs 5 crore. The agreement requires collateral on exposure above the threshold: Rs 50 crore minus Rs 20 crore is Rs 30 crore. You hold Rs 25 crore, so the shortfall is Rs 5 crore. That is above the Rs 2 crore minimum transfer amount, so the call goes out. Even afterwards, Rs 20 crore of exposure stays unsecured.
What does each term in the agreement do?
Think of a shopkeeper who lets a regular customer run a tab of up to Rs 2,000 before asking for anything, and who does not bother collecting amounts under Rs 200. The thresholdThe level of exposure below which no collateral is required under the collateral agreement. is credit you have chosen to extend without security, and the minimum transfer amount stops tiny calls that cost more to process than they protect. The collateral you are owed is exposure minus threshold. The call is that amount minus what you already hold, sent only if it clears the minimum.
Of Rs 50 crore of exposure, the first Rs 20 crore is the agreed threshold, Rs 25 crore is covered by collateral already held, and the last Rs 5 crore is the call, which clears the Rs 2 crore minimum transfer amount.
Why does the answer not reach Rs 25 crore?
Because you signed away the first Rs 20 crore. A threshold is unsecured exposure that the credit team approved when the agreement was negotiated, usually because the counterparty was strong, and it stays unsecured until the agreement is renegotiated. After the call lands you hold Rs 30 crore against Rs 50 crore of exposure, and the remaining Rs 20 crore is exactly the size of the threshold. A good answer says that residual number, because it is what the credit limit has to cover.
Add one practical point. The exposure number is itself a valuation that the counterparty may dispute, and collateral usually carries a haircut, so Rs 25 crore of bonds may count as less than Rs 25 crore. Rounding conventions can also change the call by a small amount. None of that changes the method: required equals exposure minus threshold, call equals required minus held, sent only above the minimum.
Where candidates lose it
The common wrong answer is Rs 25 crore: exposure minus collateral held, with the threshold forgotten. It asks the counterparty for more than the agreement allows, and in a real call it starts a dispute you will lose.
The opposite miss is saying nothing is due because Rs 25 crore is already more than the Rs 20 crore threshold. The threshold is subtracted from the exposure, not compared with the collateral.
What the interviewer asks next
The exposure drops to Rs 46 crore the next day. What happens to the collateral you hold?
Your counterparty is downgraded and the threshold falls to zero. What is the call now?
Why might a bank accept a high threshold from one counterparty but not another?
005A rating grade shows a 2% cumulative probability of default after one year and 5% after two years. What is the probability of default in year two for a borrower that survived year one?Credit risk arithmeticCoreBank credit riskQuant risk
Try it first
What is the year-two default probability for a survivor?
Show the worked solution
About 3.06%. Start with 10,000 borrowers. 200 default in year one, leaving 9,800. By the end of year two 500 have defaulted, so 300 did so in year two. For a borrower who reached the start of year two, the chance is 300 out of 9,800, which is 3.06%, a little above the 3% you get by subtracting.
Why divide by the survivors?
Think of a school where 2 of every 100 students leave in class nine and 5 in total have left by the end of class ten. If you are a class ten student today, your chance of leaving this year is measured against the 98 who are still in the room, not the 100 who started. A conditional probability of default is always measured against the borrowers who survived to the start of the period. The unconditional slice, 3% of the original pool, is the right number only if you are standing at the start of year one.
Of 10,000 borrowers, 200 default in year one and 300 in year two, so a borrower who survives year one faces 300 defaults out of 9,800 survivors, 3.06%, not 300 out of 10,000.
The relationship
$PD_{2|1} = \frac{C_2 - C_1}{1 - C_1} = \frac{0.05 - 0.02}{0.98} \approx 3.06\%$
C_1, C_2 cumulative default probabilities at one and two years
1 - C_1 the share still alive at the start of year two
What it says in wordsTake the extra defaults in year two and divide by the share of borrowers still alive to default.
Where does this matter on a credit desk?
Whenever you price or provision a loan over several years. Expected loss in year two uses the marginal default of the original pool, while a hazard rateThe probability of default in a short period for a borrower that has survived to the start of it. used to model a surviving borrower uses the conditional figure. Mixing them up is a small error at 2% and 5%, 3.06% against 3.00%, but at high-yield default rates the gap widens: 20% and 35% cumulative gives 18.75% conditional against 15% by subtraction.
Say the limitation too. Cumulative default tables are averages across many cohorts and economic cycles, so a borrower in a downturn year may face a higher rate than the table shows. The arithmetic is exact; the inputs are estimates.
Where candidates lose it
The trap is answering 3% by subtracting. It feels complete because the numbers are clean, but it answers a different question: what share of the original pool defaults in year two, not what a surviving borrower faces.
Give 3.06%, then say why it differs from 3%. The interviewer is listening for the word survivors.
What the interviewer asks next
If the year-two conditional default rate is the same as year one's 2%, what is the two-year cumulative rate?
Convert the 2% one-year figure into a constant hazard rate.
Why do cumulative default curves for high-yield grades often flatten in later years?
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.
007A bank holds liquid assets equal to 12% of its deposits. In a run, depositors withdraw 5% of the remaining deposits every day. On which day does the bank run out of liquid assets?Liquidity and balance sheetCoreTreasury and ALMBank credit risk
Try it first
When is the liquid buffer exhausted?
Show the worked solution
On day 3. Out of every Rs 100 of deposits, Rs 5 leaves on day 1 and Rs 4.75 on day 2, 9.75 in total, still inside the Rs 12 buffer. Day 3 takes another Rs 4.51, and the cumulative outflow passes Rs 12 part way through the day, about 2.5 days into the run. A buffer that sounds comfortable lasts under three days.
Why do shrinking withdrawals not save the bank?
Picture a water tank with a leak that loses 5% of what is left each hour. The leak slows as the tank empties, but in the first few hours it is losing almost 5 litres an hour from a 100 litre tank. A percentage outflow on a large base shrinks only slowly, so for the first days it behaves almost like a fixed outflow of 5 a day. Twelve divided by five says about two and a half days, and the exact answer is only slightly longer.
Daily withdrawals of 5.00, 4.75 and 4.51 per 100 of deposits push cumulative outflows to 9.75 by the end of day 2 and 14.26 by the end of day 3, crossing the 12 of liquid assets about 2.5 days into the run.
The relationship
$1 - 0.95^{n} = 0.12 \;\Rightarrow\; n = \frac{\ln 0.88}{\ln 0.95} \approx 2.49 \text{ days}$
0.95^n the share of deposits still in the bank after n days
0.12 the liquid assets, as a share of the original deposits
What it says in wordsThe buffer is gone when cumulative withdrawals, one minus what remains, reach the liquid assets.
What would a treasury risk manager add?
That the puzzle is a survival horizonHow long a bank can meet outflows under stress from its own liquid assets, before it must sell less liquid assets or borrow. calculation, and that the real answer depends on what the bank can do on day 3. A bank does not fail the moment the liquid buffer is empty; it fails when it can no longer turn other assets into cash fast enough. It can pledge loans to the central bank, sell securities at a discount or borrow, and each has a cost and a limit. The puzzle also assumes a constant 5% a day; real runs usually accelerate once they become news.
Close with the lesson. Liquidity measured as a percentage of deposits sounds like a lot, but outflows in a run are also measured in percentages of deposits, per day. That mismatch in time units is why regulators express liquidity buffers against stressed outflows over a set horizon, rather than as a plain share of the balance sheet.
Where candidates lose it
The trap is dividing 12 by 5 and answering day 2, or, worse, reasoning that a shrinking outflow never exhausts the buffer. The first stops before the buffer is actually gone; the second confuses a slowing leak with a stopped one.
Give day 3, then the exact 2.49 days from the log formula, then say what the bank would do next.
What the interviewer asks next
What liquid buffer would keep the bank solvent for 30 days at 5% a day?
Withdrawals start at 5% a day and rise by one point every day. When does the buffer run out now?
Which deposits run first, and how would you weight them in a stress test?
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?
009A bank has 20 small operational losses a year averaging Rs 5 lakh each, and one large loss of about Rs 5 crore every five years. What is the expected annual loss, and what share of it comes from the rare event?Operational loss and fraudCoreOperational risk
Try it first
What share of the expected annual loss comes from the rare event?
Show the worked solution
Expected annual loss is about Rs 2 crore, and the rare event is half of it. Twenty small losses at Rs 5 lakh cost Rs 1 crore a year. A Rs 5 crore loss every five years averages Rs 1 crore a year. So an event that is about 1% of the count is about 50% of the expected loss, and in the year it strikes the total is about Rs 6 crore.
Why does counting events mislead you?
Think of a household budget. Daily tea and snacks happen hundreds of times a year and feel like the big drain, but one hospital bill every few years can cost as much as all of them together. Expected loss is frequency times severity, and a rare event with a large severity can match or exceed a flood of small ones. Here twenty small losses a year produce Rs 1 crore, and one-fifth of a Rs 5 crore event a year also produces Rs 1 crore.
Small losses are about 99% of all operational loss events but only 50% of the expected annual loss; the Rs 5 crore event that happens once in five years is about 1% of events and the other half of the Rs 2 crore expected loss.
The relationship
$E[L] = \sum_i \lambda_i \, \mu_i = 20 \times 0.05 + 0.2 \times 5 = 1 + 1 = 2 \text{ crore}$
lambda_i how many events of type i occur a year on average
mu_i the average loss per event, in Rs crore
What it says in wordsFor each kind of loss, multiply how often it happens by how much it costs, then add the kinds together.
Why is the expected loss not the number a risk manager plans capital around?
Because no single year looks like the average. In four years out of five the bank loses about Rs 1 crore, and in the fifth it loses about Rs 6 crore; capital has to cover the bad year, not the average one. This is the gap between expected and unexpected lossThe loss above the average that a bank must be able to absorb in a bad year, usually covered by capital rather than by pricing or provisions.. Small, frequent losses are usually budgeted in the cost base; the rare, severe loss is what operational risk capital exists for.
Say the estimation problem too. The small-loss average rests on hundreds of data points; the rare-loss figure rests on very few, perhaps one event in the bank's own history. That is why operational risk teams add external loss data and scenario workshops: the half of expected loss that matters most is also the half measured least precisely.
Where candidates lose it
The trap is reading the count and saying the rare event barely matters. It is one event in about a hundred, but it carries half the expected loss, and all of the year-to-year volatility.
The second loss is stopping at Rs 2 crore. Say what the bad year looks like, Rs 6 crore, because that is the number a capital discussion starts from.
What the interviewer asks next
If the rare event is really Rs 10 crore once every ten years, does expected loss change? What does change?
How would insurance with a Rs 2 crore deductible change the expected loss to the bank?
Why might the bank's own loss history understate the rare event?
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?

13% - 10.5%	spare capital in ratio points of RWA
RWA / total assets	the average risk weight, here 0.5

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_1, C_2	cumulative default probabilities at one and two years
1 - C_1	the share still alive at the start of year two

Try it first

Why is the gap not simply 2.5%?

What refinement shows you understand the ratio?

Where candidates lose it

What the interviewer asks next

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

Try it first

Why does the cash count as zero?

What does beta leave out?

Where candidates lose it

What the interviewer asks next

Try it first

What does each term in the agreement do?

Why does the answer not reach Rs 25 crore?

Where candidates lose it

What the interviewer asks next

Try it first

Why divide by the survivors?

Where does this matter on a credit desk?

Where candidates lose it

What the interviewer asks next

Try it first

Why is NPV a dot product at all?

What does the interviewer want to hear beyond the number?

Where candidates lose it

What the interviewer asks next

Try it first

Why do shrinking withdrawals not save the bank?

What would a treasury risk manager add?

Where candidates lose it

What the interviewer asks next

Try it first

Why is zero the wrong answer?

Why would a risk interviewer ask a clock question?

Where candidates lose it

What the interviewer asks next

Try it first

Why does counting events mislead you?

Why is the expected loss not the number a risk manager plans capital around?

Where candidates lose it

What the interviewer asks next

Try it first

How do you turn a Greek into rupees?

When is the Rs 20 lakh estimate wrong?

Where candidates lose it

What the interviewer asks next