Risk Management puzzles, solved step by step

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

Showing 61–70 of 100

061A loan book is 60% retail loans with a 3% default rate and 40% corporate loans with a 1% default rate. A loan picked at random has defaulted. What is the probability it was a retail loan?Probability and base ratesCoreBank credit riskRisk GCC
Try it first
Which number is closest?
Show the worked solution
About 81.8%. Retail defaults make up 60% x 3% = 1.8% of the book and corporate defaults 40% x 1% = 0.4%, so 2.2% of all loans default. Given that a loan is among the 2.2%, the chance it is retail is 1.8 divided by 2.2, which is 9 in 11, or about 82%.
Why is the answer not simply 60%?
Suppose 60% of the cars on a road are small hatchbacks and 40% are trucks, but trucks are far more likely to have a broken tail light. If a policeman stops a car for a broken tail light, it is more likely to be a truck than the 40% share suggests. The stop tells you something. Learning that a loan defaulted is evidence, and it shifts the odds towards the segment that defaults more often.
Retail loans are 60% of the book and default 3% of the time, contributing 1.8% of all loans as defaults; corporate loans contribute 0.4%. Of the 2.2% that default, 1.8 is retail, so a defaulted loan is retail with probability 81.8%.
What is the quickest way to do this in your head?
Use 1,000 loans. There are 600 retail loans, of which 18 default, and 400 corporate loans, of which 4 default. Out of 22 defaults, 18 are retail, so the answer is 18 over 22, and natural numbers make the base rate impossible to forget. This is Bayes' rule without the notation, and it is the fastest way to be right out loud.
The relationship
$P(R\mid D) = \frac{P(D\mid R)P(R)}{P(D\mid R)P(R)+P(D\mid C)P(C)} = \frac{0.03 \times 0.6}{0.018+0.004} = \frac{9}{11}$
P(R) the retail share of the book, 60%
P(D|R) the retail default rate, 3%
P(D|C) the corporate default rate, 1%
What it says in wordsThe chance a default came from retail is retail's share of all defaults.
For a risk team this is not an exam trick. The mix of a defaulted pool drives which collections team gets the work, which recovery rate applies and which model needs recalibrating. A 60% share of the book becomes an 82% share of the defaults.
Where candidates lose it
The trap is answering 60%, the share of the book, and ignoring the fact that the loan defaulted. That is base rate thinking run backwards: the prior is treated as the answer.
The other slip is answering 75% by comparing the default rates alone, 3 against 1. That forgets the book is not split evenly. Weight both: 1.8 against 0.4.
What the interviewer asks next
If the book were 50:50, what would the answer be?
Retail LGD is 60% and corporate LGD is 30%. What share of the loss comes from retail?
Two loans default. What is the chance both are retail?
062Your prior estimate of a hidden fair price is 10 with variance 4. A noisy measurement comes in at 12 with measurement variance 1. After one Kalman filter update, what is your new estimate and its variance?Statistics and estimationCoreUBSLondon · 2022
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Where does the new estimate land?
Show the worked solution
The new estimate is 11.6 with variance 0.8. The Kalman gain is the prior variance over the total, 4 over 5, or 0.8. The estimate moves 80% of the way from 10 towards 12, landing at 11.6. The variance becomes (1 - 0.8) x 4 = 0.8, smaller than either the prior's 4 or the measurement's 1.
How does the filter decide how far to move?
Two friends guess your commute time. One has ridden with you a hundred times, the other once. You would average their guesses, but lean heavily on the first. A Kalman filterA method that updates an estimate of something you cannot see directly each time a noisy measurement arrives, weighting old estimate and new data by how precise each is. does exactly that: it weights the prior and the measurement by their precision, one over variance. Precision 0.25 against 1 gives the measurement 80% of the weight.
The prior centred at 10 is wide, with variance 4, and the measurement at 12 is narrow, with variance 1. The update lands at 11.6, four fifths of the way to the measurement, and its variance of 0.8 makes it narrower than either source.
Why is the new variance smaller than both inputs?
Because two independent pieces of evidence together know more than either alone. Precisions add: 1 over 4 plus 1 over 1 is 1.25, and one over 1.25 is a variance of 0.8. That is the part candidates skip. The filter does not just move the estimate; it becomes more confident with every measurement, until new data carry little weight and the estimate settles.
The relationship
$K = \frac{P}{P+R} = 0.8 \qquad \hat{x} = 10 + 0.8(12-10) = 11.6 \qquad P' = (1-K)P = 0.8$
P the prior variance, 4
R the measurement variance, 1
K the Kalman gain, the weight on the new measurement
\hat{x} the updated estimate
What it says in wordsMove from the prior towards the measurement by the gain, and shrink the variance by the same share.
Say the limitation: this single step assumes both errors are normal and the hidden price did not move between the prior and the measurement. A full filter adds a prediction step that lets the price drift and widens the variance before each update. Risk teams use the idea to track hidden quantities such as a hedge ratio that changes over time.
Where candidates lose it
The trap is averaging the two numbers and answering 11. That treats the measurement and the prior as equally trustworthy, which the variances say they are not.
The second trap is getting 11.6 and then saying the variance is somewhere between 1 and 4. Combining evidence always reduces uncertainty, so the new variance must be below both: 0.8.
What the interviewer asks next
A second measurement of 11 arrives with variance 1. What is the estimate now?
What happens to the gain as the number of measurements grows?
How would you use a Kalman filter to estimate a time-varying hedge ratio?
Asked at UBS, Risk, London, 2022 (Wall Street Oasis): Explain what kalman filter is.
063From 250 days of daily P&L, the five worst losses are Rs 18, 14, 11, 9.5 and 8 crore. What is the 99% one-day historical VaR, and what is the expected shortfall beyond it?VaR and expected shortfallCoreUBSRemote · 2020
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How many of the worst days sit beyond the 99% line in a 250-day sample?
Show the worked solution
99% VaR is about Rs 12.5 crore and expected shortfall about Rs 15 crore. One per cent of 250 days is 2.5 observations, so VaR sits between the second worst loss, 14, and the third, 11: Rs 12.5 crore by interpolation, or Rs 11 crore if the desk takes the third worst. Expected shortfall averages the worst 2.5 days: (18 + 14 + half of 11) / 2.5 = 15.
Why is historical VaR a position in a list rather than a formula?
If you rank a class of 250 students by exam score, the student at the 99th percentile is not computed, they are counted: roughly the second or third from the top. Historical VaR works the same way with losses. Rank the daily P&L from worst to best and count 1% of the sample in from the bad end; the loss you land on is the 99% VaR. With 250 days you land 2.5 places in, between two real days.
In a 250-day sample the 99% line sits 2.5 observations from the worst end, between losses of Rs 14 crore and Rs 11 crore, so historical VaR is Rs 12.5 crore by interpolation or Rs 11 crore by the third-worst convention. Expected shortfall, the average of the worst 2.5 observations, is Rs 15 crore.
Which convention is right, and does it matter?
Neither is wrong; both are used, and the gap here is Rs 1.5 crore on a Rs 12.5 crore number. What is wrong is not saying which one you used, because two desks can report different VaR from identical data. Expected shortfall moves with the convention too: averaging only the two losses beyond Rs 11 crore gives Rs 16 crore rather than Rs 15 crore. Name the rule, then give the number.
The relationship
$ES_{99\%} = \frac{18 + 14 + 0.5 \times 11}{2.5} = \frac{37.5}{2.5} = 15$
2.5 the number of observations in the worst 1% of 250 days
0.5 x 11 half of the third worst loss, the fraction of it inside the tail
What it says in wordsExpected shortfall is the average loss across the worst 1% of days, counting the third worst at half weight.
The limitation is size. A 99% number from 250 days rests on two or three losses, so one bad day entering or leaving the window can move it by several crore. That is why regulators ask for backtesting and why expected shortfall, which uses the whole tail, is preferred where the tail is thin.
Where candidates lose it
The first trap is picking the worst loss, Rs 18 crore, as the 99% VaR. That is closer to a 99.6% number. Count 1% of the sample, 2.5 days, then read off the list.
The second is giving a single number without the convention. Say 2.5 observations, say which rule you use, and show that expected shortfall is larger than VaR because it averages what lies beyond.
What the interviewer asks next
What would the 97.5% expected shortfall be from the same list?
The worst day drops out of the window tomorrow. What happens to VaR and ES?
Why might a regulator prefer expected shortfall to VaR for setting capital?
Asked at UBS, Risk Management, Remote, 2020 (Wall Street Oasis): Calculate VaR
064A bank has CET1 of Rs 100 crore on risk-weighted assets of Rs 1,000 crore, a 10% ratio. A Rs 20 crore loan loss also removes Rs 50 crore of risk-weighted assets as the defaulted loans are written off. How much capital must it raise to get back to 10%?Capital and leverageHardBank credit riskTreasury and ALM
Try it first
How much new equity does the bank need?
Show the worked solution
Rs 15 crore. The loss takes CET1 from Rs 100 crore to Rs 80 crore, and writing off the defaulted loans takes risk-weighted assets from Rs 1,000 crore to Rs 950 crore, a ratio of 8.42%. To be back at 10% the bank needs 10% of Rs 950 crore, which is Rs 95 crore, so it must raise Rs 15 crore of new equity, assuming the new cash carries no risk weight.
Why is the answer not simply the Rs 20 crore lost?
A cricket team that needs a run rate of 6 an over has lost two overs to rain. It needs fewer runs than before, not the same total, because the target is a rate. A capital ratio is a rate too. The loss cuts the numerator by 20%, but the write-off also cuts the denominator by 5%, and the target is 10% of the new, smaller denominator. That makes the gap Rs 15 crore.
CET1 falls 20%, from Rs 100 crore to Rs 80 crore, while risk-weighted assets fall only 5%, from Rs 1,000 crore to Rs 950 crore, so the ratio drops to 8.42%. Ten per cent of Rs 950 crore is Rs 95 crore, which leaves a Rs 15 crore capital gap.
Where does the Rs 50 crore of RWA come from, and is it realistic?
Think of Rs 50 crore of defaulted loans at a 100% risk weight. The bank loses Rs 20 crore on them, a 40% loss given default, and recovers Rs 30 crore in cash, which carries no risk weight. Both sides of the ratio move on the same event, but a loss always hits capital at full size while the RWA falls only by the risk weight of what left. A write-off never rescues the ratio: 80 over 950 is still well below 10%.
The relationship
$\text{raise} = 0.10 \times (1{,}000 - 50) - (100 - 20) = 95 - 80 = 15$
0.10 the target CET1 ratio
1,000 - 50 risk-weighted assets after the write-off
100 - 20 CET1 after the loss
What it says in wordsThe raise is the target ratio times the new risk-weighted assets, less the capital left after the loss.
The limitations are worth one line each. The new equity is assumed to sit in cash or government bonds at a zero risk weight; if the bank lends it out, RWA grows and it needs more. And a real bank must also rebuild any buffer above the minimum, whose size depends on current rules you would check rather than assume.
Where candidates lose it
The common answer is Rs 20 crore, replacing the loss. It forgets the denominator moved. The opposite slip is to credit the RWA fall at full size and answer Rs 5 crore, as if removing Rs 50 crore of assets were worth Rs 50 crore of capital.
Write the ratio after the loss, 80 over 950, then solve for the numerator that gives 10%. Two lines of arithmetic, said out loud, and the answer defends itself.
What the interviewer asks next
What if the recovered Rs 30 crore is relent at a 100% risk weight?
How much RWA would the bank have to shed instead of raising capital?
Why might a bank prefer to shrink RWA rather than raise equity in a downturn, and what does that do to lending?
065You can bet repeatedly at even odds with a 60% chance of winning each bet. Compare the long-run growth of your capital if you bet 20% of it each time with betting 50% each time.Compounding and drawdownsHardQuant riskAsset manager risk
Try it first
Betting 50% a time on a 60% edge: what happens over many bets?
Show the worked solution
Betting 20% grows capital about 2.0% a bet; betting 50% shrinks it about 3.3% a bet. Growth per bet is the average log return: 0.6 ln(1 + f) + 0.4 ln(1 - f). At 20%, the Kelly fraction 2p - 1, that is +2.01%. At 50% it is -3.40%. After 100 bets the typical 20% bettor has about 7.5 times the start; the 50% bettor about 0.03 times.
How can a bet with a positive edge lose money?
A shopkeeper who stakes half his stock on every festival season, and wins more seasons than he loses, can still go under: one bad season halves him, and a good one only adds half back. Losses on a shrunken base are expensive to recover. The typical outcome of repeated betting is set by the average log return, not the average return, and at 50% the losses dominate the logs.
With a 60% chance of winning at even odds, growth per bet peaks at the 20% Kelly fraction, about +2.0% a bet, crosses zero near 39% and is -3.4% a bet at 50%, so oversizing a winning bet turns steady growth into steady decline.
Where does the 20% come from?
Set the slope of the growth curve to zero. For even odds the Kelly fractionThe fraction of capital to bet each time that maximises the long-run compound growth rate of the capital. is the probability of winning minus the probability of losing: 0.6 minus 0.4, or 20%. Betting less than Kelly gives up some growth for a much smoother ride; betting more than about twice Kelly gives up growth altogether. Here the curve crosses zero near 39%, just under twice Kelly.
The relationship
$g(f) = p\ln(1+f) + (1-p)\ln(1-f) \qquad f^* = 2p - 1 = 0.2$
f the fraction of capital bet each time
p the probability of winning a bet, 0.6
g(f) the expected log growth of capital per bet
What it says in wordsLong-run growth is the probability-weighted average of the log of what each outcome does to your capital.
For a risk manager the lesson is position sizing. A desk with a genuine edge can still destroy capital by running too much leverage, and the average P&L will look fine right up until the drawdowns compound. Real edges are also uncertain, which is why practitioners bet a fraction of Kelly.
Where candidates lose it
The trap is answering that 50% grows faster because each bet has a positive expected value and bigger bets earn more of it. That is true of the arithmetic average and false for the capital you actually end up with, which compounds.
The second slip is knowing Kelly but not being able to show why 50% loses. Have the one-line check ready: six wins and four losses at 50% multiply capital by about 0.71.
What the interviewer asks next
What is the Kelly fraction if you win 55% of the time at even odds?
What if the payoff is 2 to 1 with a 40% chance of winning?
Why do traders often bet half Kelly rather than full Kelly?
066A long-short pair holds equal rupee amounts of two stocks, long one and short the other, each with 2% daily volatility. What happens to the pair's daily volatility when the correlation between the stocks falls from 0.9 to 0.6?Correlation and diversificationHardBank market riskAsset manager risk
Try it first
The correlation drops by a third. What does the pair's volatility do?
Show the worked solution
It doubles, from about 0.89% to 1.79% a day, as a share of one leg. For equal and opposite legs with the same volatility, pair variance is 2 x sigma squared x (1 - rho). The unhedged part, 1 - rho, goes from 0.1 to 0.4, four times as large, so volatility doubles. On Rs 100 crore a leg, daily volatility goes from about Rs 0.89 crore to Rs 1.79 crore.
Why does a modest fall in correlation hit a hedged book so hard?
Two boats tied close together on a river rise and fall almost as one; the rope between them barely strains. Loosen the tie and every wave that hits one boat and not the other shows up as strain on the rope. A long-short pair is the rope. It only carries the part of the two stocks that does not move together, and that part is 1 minus rho, which grows fast as correlation slips from a high level.
With equal legs of 2% daily volatility, the pair's volatility is 0.89% at a correlation of 0.9 and 1.79% at 0.6, double, because the uncorrelated part 1 minus rho quadruples from 0.1 to 0.4.
What does this mean for the desk's risk numbers?
VaR scales with volatility, so the pair's VaR doubles with no change in position. On Rs 100 crore a leg the 99% one-day VaR moves from about Rs 2.1 crore to Rs 4.2 crore, entirely from a correlation the desk does not control. Correlations between related stocks do fall in exactly the markets where one company's news breaks from its peer's, which is when the pair was supposed to be safe.
The relationship
$\sigma_{\text{pair}} = \sqrt{\sigma^2 + \sigma^2 - 2\rho\sigma^2} = \sigma\sqrt{2(1-\rho)}$
\sigma each leg's daily volatility, 2%
\rho the correlation between the two stocks, 0.9 then 0.6
What it says in wordsThe pair's volatility is each leg's volatility times the square root of twice the uncorrelated share.
Say the limitation. The formula assumes equal volatilities and equal notionals; a real pair is often beta-weighted, which changes the coefficients but not the lesson. A risk limit that assumes the historical correlation of 0.9 is a limit on a number that can quietly double.
Where candidates lose it
The trap is reasoning linearly: correlation fell by a third, so risk rises by a third. The risk depends on 1 minus rho, which went up fourfold, and the square root turns that into a doubling.
The second trap is treating the pair as market neutral and therefore low risk. Neutral to the market is not neutral to correlation, and a good answer names correlation as the position's real exposure.
What the interviewer asks next
At what correlation does the pair's volatility equal a single leg's 2%?
How would you hedge the correlation risk itself?
If the legs are beta-weighted instead of equal, how does the formula change?
067A forward contract's value to the bank at the one-year horizon is normally distributed with mean zero and standard deviation Rs 10 crore. What is the expected positive exposure to the counterparty at that date?Counterparty exposure and collateralCoreCounterparty riskQuant risk
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The expected value of the forward is zero. What is the expected exposure?
Show the worked solution
About Rs 4.0 crore, or 0.4 standard deviations. Exposure is the forward's value when it is positive and zero when it is negative. For a normal distribution with mean zero, the expected value of the positive part is sigma divided by the square root of 2 pi, about 0.399 x Rs 10 crore. Half the outcomes contribute nothing; the other half average about Rs 8 crore.
Why is the exposure positive when the expected value is zero?
Lending a friend your bicycle is a one-sided risk: if the friend loses it, you are out a bicycle; if the friend buys you a better one, you gain nothing from the loan itself. Counterparty exposure has the same shape. When the contract is in your favour, a default costs you its value; when it is against you, you still owe the full amount, so negative outcomes count as zero, not as a gain. Cutting off the left half leaves a positive average.
The forward's value in one year is centred on zero with a standard deviation of Rs 10 crore, but only the shaded right half is exposure. Counting the left half as zero gives an expected exposure of about Rs 4.0 crore, 0.4 standard deviations, while the 97.5% potential future exposure is about Rs 19.6 crore.
Where does 0.4 come from, and what is it called?
Integrating v times the normal density from zero to infinity gives sigma times the density at zero, which is 1 over the square root of 2 pi, about 0.399. At a single future date desks usually call this number the expected exposure; averaged over the life of the trade it becomes expected positive exposure, the input to credit valuation adjustment. The tail measure, potential future exposureA high percentile of the exposure distribution at a future date, used to size limits on how much a counterparty could owe. at 97.5%, is 1.96 sigma, about Rs 19.6 crore.
The relationship
$EE = E[\max(V,0)] = \int_0^{\infty} v\,\phi(v)\,dv = \frac{\sigma}{\sqrt{2\pi}} \approx 0.399 \times 10 = 3.99$
V the forward's value to the bank at the horizon
\phi the normal density with mean zero and standard deviation sigma
\sigma Rs 10 crore
What it says in wordsExpected exposure averages only the positive outcomes, which for a zero-mean normal is about 0.4 standard deviations.
The limitation: this assumes no collateral and no netting with other trades. A daily margin agreement would cut the exposure to the move over the few days it takes to call and receive margin, which is far smaller than a year's worth of drift.
Where candidates lose it
The trap is answering zero because the forward is fair and its expected value is zero. Exposure is an option-like quantity: it keeps the upside of the value and discards the downside, so it is always at least zero on average.
The second slip is quoting Rs 8 crore, the average of the positive half alone. That forgets the half of outcomes where exposure is zero. Average across all outcomes: about Rs 4 crore.
What the interviewer asks next
If the forward's expected value were Rs 5 crore instead of zero, would the expected exposure rise by more or less than Rs 5 crore?
How does daily collateral change this number?
Why does exposure on an interest rate swap rise and then fall over its life?
068A Rs 100 crore loan has a 2% probability of default and a 50% loss given default. What is the expected loss, and what is the standard deviation of the loss that capital has to cover?Credit risk arithmeticCoreBank credit risk
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How does the standard deviation of loss compare with the expected loss?
Show the worked solution
Expected loss is Rs 1 crore; the standard deviation of loss is Rs 7 crore. If the loan defaults, the bank loses 50% of Rs 100 crore, Rs 50 crore; otherwise it loses nothing. The average is 2% x 50 = Rs 1 crore, which the spread should cover. The standard deviation is 50 x the square root of 0.02 x 0.98, which is Rs 7 crore, the unexpected loss capital exists for.
Why is the average loss the wrong thing to hold capital against?
A shopkeeper insuring against a one-in-fifty chance of a Rs 5 lakh fire can budget Rs 10,000 a year for the premium, the average. But in the year the fire happens, the budget is useless; what saves the shop is savings. Expected loss is a cost of doing business and belongs in the price; unexpected lossThe spread of possible losses around the expected loss, usually measured by a standard deviation or a high percentile, which capital is held to absorb. is the surprise, and it is what capital is for.
The loan either loses nothing, 98% of the time, or loses Rs 50 crore, 2% of the time. The expected loss is Rs 1 crore and the standard deviation is Rs 7 crore, so the surprise the bank must be able to absorb is seven times the average it prices in.
How do you get Rs 7 crore quickly?
Treat default as a coin that lands bad with probability p. Its standard deviation is the square root of p times (1 - p): the square root of 0.0196, which is 0.14. Multiply by the loss if it lands bad, Rs 50 crore, and you have Rs 7 crore. The ratio of seven to one is typical for a low-default loan, and it grows as the default probability falls: rarer defaults mean a mean closer to zero and a relatively wider spread.
The relationship
$EL = PD \times LGD \times EAD = 1 \qquad UL = LGD \times EAD \times \sqrt{PD(1-PD)} = 50 \times 0.14 = 7$
PD probability of default, 2%
LGD loss given default, 50%
EAD exposure at default, Rs 100 crore
What it says in wordsThe average loss is the default chance times the loss; the spread is the loss times the square root of p times one minus p.
Say the limitation. For one loan, the standard deviation is a poor summary: the real outcome is 0 or 50, never 7. In a large, diversified portfolio losses spread out and the standard deviation becomes meaningful, and capital models set it at a high percentile of the portfolio loss rather than one standard deviation of a single loan.
Where candidates lose it
The trap is stopping at the expected loss of Rs 1 crore and treating it as the risk. It is the one number the bank should never be surprised by; the risk is the spread.
The second slip is using the variance formula for a continuous variable, or forgetting the square root and quoting Rs 0.98 crore. Say the Bernoulli formula aloud, p times one minus p, then take the root.
What the interviewer asks next
What happens to UL if the default probability falls to 0.5%?
The bank holds 100 such loans, independent of each other. What is the portfolio UL?
Why do defaults in a real portfolio not behave independently?
069Portfolio A holds only 5-year zero-coupon bonds. Portfolio B holds 2-year and 8-year zeros, weighted so its duration is also 5. Which portfolio does better after a large parallel move in yields, and which one loses if the curve steepens?Duration and ratesHardTreasury and ALMBank market risk
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After a 200 basis point parallel move in either direction, which portfolio is ahead?
Show the worked solution
The barbell wins a large parallel move either way; it loses when the curve steepens. Both portfolios have duration 5, but the barbell's convexity is 34 against the bullet's 25. For a 200 basis point rise the barbell loses 9.35% against 9.52%, and for a fall it gains 10.72% against 10.52%. If the curve steepens 10 basis points per year around 5 years, the bullet is flat and the barbell loses 0.88%.
If the durations match, why do the portfolios behave differently?
Two families each spend an average of Rs 50,000 a month. One spends it evenly; the other spends Rs 20,000 some months and Rs 80,000 others. A small price rise hits them alike, but a big shock hits the second family's heavy months much harder. Averages match; the spread does not. Duration is the average maturity, so it only describes small, parallel moves; the barbell spreads its money further from 5 years, which gives it more convexity and exposes it to changes in the curve's shape.
The bullet sits at 5 years and the barbell splits evenly between 2 and 8 years, both at duration 5. The barbell does better in a 200 basis point parallel move either way because its convexity is 34 against 25, and it loses 0.88% when the curve steepens around the 5-year point while the bullet is unaffected.
Why does the steepener hurt only the barbell?
In this steepener the 5-year yield does not move, so the bullet does not move. The 2-year yield falls 30 basis points and the 8-year yield rises 30. The barbell's long leg carries four times the duration of its short leg, so the loss on the 8-year bond swamps the gain on the 2-year. Roughly: half the money times 8 years times 0.30% is a 1.2% loss, and half times 2 years times 0.30% is a 0.3% gain, a net loss of about 0.9%.
The relationship
$w\cdot 2 + (1-w)\cdot 8 = 5 \Rightarrow w = 0.5 \qquad C_{\text{barbell}} = 0.5(2^2) + 0.5(8^2) = 34 > 25 = 5^2$
w the share of value in the 2-year zero
C convexity; for a zero with continuous compounding it is maturity squared
What it says in wordsMatching duration fixes the weights at half and half, and the barbell ends up with more convexity.
The limitation: the gains from convexity are small for small moves and are usually priced, since a barbell typically yields less than a bullet of the same duration. In a quiet market that yield give-up can outweigh the convexity benefit, so the choice is a view on volatility and curve shape, not a free lunch.
Where candidates lose it
The trap is saying the two are the same risk because duration matches. Duration is one number describing one kind of move, and the interviewer built the question to see if you know the two other risks it misses: convexity and curve shape.
The second slip is getting convexity right and missing the steepener. Give both halves: the barbell wins a big parallel move and loses when the long end sells off relative to the short end.
What the interviewer asks next
Which portfolio is hurt by a flattening instead?
How would you hedge the barbell's steepener exposure?
Why does a barbell usually yield less than a bullet of the same duration?
070A fund holds Rs 1,000 crore of bonds financed by repo, putting up Rs 20 crore of its own capital against a 2% haircut. The lender raises the haircut to 10%. With its capital unchanged, how much must the fund sell?Liquidity and balance sheetHardTreasury and ALMAsset manager risk
Try it first
How much of the Rs 1,000 crore position has to go?
Show the worked solution
It must sell Rs 800 crore, four fifths of the position. The haircut is the share of the position the fund funds itself. At 2%, Rs 20 crore supports Rs 1,000 crore of bonds, 50 times leverage. At 10%, the same Rs 20 crore supports only Rs 200 crore, 10 times leverage. Keeping the full position would need Rs 100 crore of capital, Rs 80 crore more than it has.
Why does a small change in haircut force such a big sale?
Buying a flat with a 5% down payment lets Rs 10 lakh of savings control a Rs 2 crore flat. If the bank suddenly demands 20% down, the same Rs 10 lakh controls only a Rs 50 lakh flat. The haircutThe share of a security value a lender will not lend against, so the borrower must fund that share with its own capital. sets leverage directly, at one over the haircut, so moving it from 2% to 10% cuts the position the capital can support from 50 times to 10 times.
At a 2% haircut, Rs 20 crore of capital and Rs 980 crore of repo carry Rs 1,000 crore of bonds. At 10%, the same Rs 20 crore carries only Rs 200 crore with Rs 180 crore of repo, so Rs 800 crore of bonds must be sold.
What happens next in a real market?
The sale itself moves prices. If many leveraged holders face the same haircut increase, they all sell the same bonds, prices fall, their capital shrinks, and the lower capital supports an even smaller position. That feedback loop between haircuts, prices and forced selling is the liquidity spiral seen in past funding crises. A fund at 50 times leverage loses its entire Rs 20 crore of capital on a 2% price fall.
The relationship
$\text{max position} = \frac{\text{capital}}{\text{haircut}} = \frac{20}{0.10} = 200 \qquad \text{sell} = 1{,}000 - 200 = 800$
capital the fund's own money in the position, Rs 20 crore
haircut the share the lender will not finance, now 10%
What it says in wordsThe largest position the fund can hold is its capital divided by the haircut.
The limitation: the answer assumes the fund cannot raise capital or find another lender at the old haircut. In practice it would try both, and a risk manager's job is to know in advance how many days that would take and how much the forced sale would cost at stressed prices.
Where candidates lose it
The trap is answering Rs 80 crore: 8% more haircut on Rs 1,000 crore. That is the extra capital needed to keep the position, not the amount the fund must sell when it has no extra capital. The question asks for the sale.
The second miss is stopping at the arithmetic. Say what the sale does to prices and to the fund's remaining capital; the interviewer wants to hear the word spiral.
What the interviewer asks next
What price fall on the remaining position would wipe out the fund's capital at 10 times leverage?
The fund can raise Rs 30 crore more. How much must it sell now?
Why do lenders raise haircuts precisely when markets are falling?

P(R)	the retail share of the book, 60%
P(D\|R)	the retail default rate, 3%
P(D\|C)	the corporate default rate, 1%

2.5	the number of observations in the worst 1% of 250 days
0.5 x 11	half of the third worst loss, the fraction of it inside the tail

0.10	the target CET1 ratio
1,000 - 50	risk-weighted assets after the write-off
100 - 20	CET1 after the loss

\sigma	each leg's daily volatility, 2%
\rho	the correlation between the two stocks, 0.9 then 0.6

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Why is the answer not simply 60%?

What is the quickest way to do this in your head?

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How does the filter decide how far to move?

Why is the new variance smaller than both inputs?

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Why is historical VaR a position in a list rather than a formula?

Which convention is right, and does it matter?

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Why is the answer not simply the Rs 20 crore lost?

Where does the Rs 50 crore of RWA come from, and is it realistic?

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How can a bet with a positive edge lose money?

Where does the 20% come from?

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Why does a modest fall in correlation hit a hedged book so hard?

What does this mean for the desk's risk numbers?

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Why is the exposure positive when the expected value is zero?

Where does 0.4 come from, and what is it called?

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Why is the average loss the wrong thing to hold capital against?

How do you get Rs 7 crore quickly?

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If the durations match, why do the portfolios behave differently?

Why does the steepener hurt only the barbell?

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Why does a small change in haircut force such a big sale?

What happens next in a real market?

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