Risk Management puzzles, solved step by step

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

Showing 1–10 of 50 · filtered from 100Clear filters

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?
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?
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?
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?
011Two loans each have a 5% one-year probability of default and a 2% chance of defaulting together. What is the probability that at least one defaults, and how does it compare with the 9.75% you would get if they were independent?Probability and base ratesCoreQuant riskBank credit risk
Try it first
What is the probability that at least one of the two loans defaults?
Show the worked solution
8%, lower than the 9.75% for independent loans. The chance of at least one default is 5% plus 5% minus the 2% where both default, which would otherwise be counted twice. Independent loans default together only 0.25% of the time, so they give 9.75%. Correlation makes any single default slightly less likely but a double default eight times more likely.
Why subtract the joint probability?
Think of a class where 5 students play cricket and 5 play football, and 2 play both. If you ask how many play at least one game, adding 5 and 5 counts the two all-rounders twice, so the answer is 8. The chance of at least one default is the sum of the single chances minus the chance of both, because the joint case sits inside each single case. Only when the joint case is tiny does adding the two chances get close.
Out of 400 equally likely outcomes, correlated loans put 8 cells in both-default and 32 cells in any-default, 8%, while independent loans put only 1 cell in both-default and spread defaults over 39 cells, 9.75%.
The relationship
$P(A \cup B) = P(A) + P(B) - P(A \cap B) = 5\% + 5\% - 2\% = 8\%$
P(A), P(B) each loan's probability of default, 5%
P(A and B) the chance both default together, 2%
What it says in wordsAdd the chances of each loan defaulting, then take away the overlap you counted twice.
Why does correlation cut the chance of any default but raise the chance of both?
Because the defaults are bunched into the same outcomes. When defaults tend to happen together, the bad outcomes overlap, so fewer outcomes contain a default at all, but the ones that do are worse. Independent loans default together only 5% of 5%, 0.25% of the time. Here it is 2%, eight times more often, and the implied default correlationThe correlation between two yes-or-no default outcomes, computed from the joint and single default probabilities. is about 0.37.
That is the lesson a credit portfolio manager takes from the puzzle. The expected number of defaults is 0.1 in both cases, because expectations add regardless of correlation. What correlation changes is the shape of losses: fewer mild years, more years where everything goes wrong at once. Capital is held for those years, which is why correlated books need more of it even when the expected loss is identical.
Where candidates lose it
The first trap is answering 10%, adding the two 5% figures and forgetting the overlap. The second is using the independent formula, 9.75%, when the question has handed you a joint probability that is not 0.25%.
The quieter miss is stopping at 8% and not saying what it means. The interviewer wants to hear that correlation shifts risk from single defaults to joint defaults.
What the interviewer asks next
What is the probability that exactly one loan defaults?
What joint default probability would make the two loans perfectly correlated?
With 100 such loans, how does correlation change the distribution of the number of defaults?
014A stock goes from 100 to 150 and back to 100. Compare the average of the simple returns with the average of the log returns.Compounding and drawdownsCoreAsset manager riskQuant risk
Try it first
What is the average simple return over the two periods?
Show the worked solution
The simple returns average +8.3%, the log returns average 0%. The simple returns are +50% and -33.3%, and their average suggests a gain although the price is back at 100. The log returns are +40.5% and -40.5%, which add to zero and match what happened. Log returns add over time; simple returns do not.
Why does the simple average show a gain that never happened?
Imagine a shop that raises a price from Rs 100 to Rs 150 and then cuts it back to Rs 100. The rise is 50 on a base of 100, and the cut is 50 on a base of 150. Simple returns are each measured against a different starting price, so averaging them mixes percentages of different bases and overstates growth whenever prices bounce around. The arithmetic mean of +50% and -33.3% is +8.3%, but the investor has exactly the money they started with.
For a price that goes 100, 150, 100, simple returns of +50% and -33.3% average +8.3% although the price made nothing, while log returns of +40.5% and -40.5% add to zero and average zero.
The relationship
$r_t = \ln\frac{P_t}{P_{t-1}}, \qquad r_1 + r_2 = \ln\frac{150}{100} + \ln\frac{100}{150} = \ln\frac{100}{100} = 0$
r_t the log return in period t
P_t the price at the end of period t
What it says in wordsLog returns add up across periods to the log of the total change, so a round trip sums to zero.
Which one should a risk manager use?
It depends on what you are adding up. Log returns add across time, so they are the natural choice for compounding a return over many days and for most statistical models of a single asset. Simple returns add across assets, so a portfolio's return for one period is the weighted average of its holdings' simple returns, which log returns do not give you. The gap between the two averages is the volatility dragThe shortfall of compound growth below the arithmetic average return, which grows with the variance of returns.: roughly half the variance, and here the swings are wild enough to make it 8 points.
Close with the practical warning. A fund that reports the arithmetic average of its yearly returns will look better than the growth its investors actually got, and the more volatile the fund, the larger the flattering gap. The honest single number for growth over time is the geometric average, which here is zero.
Where candidates lose it
The trap is quoting +8.3% as the average return and calling it performance. It is a correct average of the wrong thing: percentages taken on different bases.
The second miss is saying log returns are simply better. They add over time but not across assets, so a risk manager needs both and should say when each applies.
What the interviewer asks next
What is the geometric average return here, and how does it relate to the log returns?
A fund returns +20% and -20% in alternate years. What is its compound growth rate?
Why do most VaR models use log returns for single assets but simple returns to aggregate a portfolio?
015A 60/40 portfolio holds equities with 15% volatility and bonds with 6% volatility, and the correlation between them is minus 0.2. What is the portfolio volatility?Correlation and diversificationCoreAsset manager risk
Try it first
Pick the portfolio volatility.
Show the worked solution
About 8.84%. The equity term is 0.6 times 15, squared, which is 81. The bond term is 0.4 times 6, squared, 5.76. The cross term is 2 times 0.6 times 0.4 times minus 0.2 times 15 times 6, which is minus 8.64. The variance is 78.12 and its square root is 8.84%, well below the 11.4% weighted average.
Why is the answer not the weighted average of 15% and 6%?
Think of two friends walking a dog on separate leads. If they always pull the same way, the dog is dragged as far as their combined pull. If one sometimes pulls left while the other pulls right, the pulls partly cancel. Volatilities only add in a straight line when the correlation is exactly one; any lower correlation lets the swings offset, and a negative one subtracts risk outright. The weighted average of 15% and 6% is 11.4%, the answer for a correlation of one.
The relationship
$\sigma_p^2 = (w_1\sigma_1)^2 + (w_2\sigma_2)^2 + 2\,w_1 w_2\,\rho\,\sigma_1\sigma_2 = 81 + 5.76 - 8.64 = 78.12$
w_1, w_2 the weights, 0.6 in equities and 0.4 in bonds
sigma_1, sigma_2 the volatilities, 15% and 6%
rho the correlation, minus 0.2
What it says in wordsAdd each asset's own variance contribution, then add the cross term, which is negative when the correlation is negative.
Equities contribute 81 and bonds 5.76 squared percentage points of variance, and the minus 0.2 correlation subtracts 8.64, leaving 78.12, a volatility of 8.84% against 9.31% at zero correlation and 11.4% at a correlation of one.
What do you add after the number?
Two things. First, equities carry almost all the risk: 81 of the 86.76 squared points before the cross term, so a 60/40 portfolio is mostly an equity risk portfolio with a bond cushion. A risk contributionThe share of a portfolio total variance that comes from one holding, including its share of the cross terms. breakdown makes that visible and is usually the next question. Second, the minus 0.2 is an estimate from history, and correlations between equities and bonds have changed sign across decades. If it turned positive at plus 0.3, the volatility would rise to about 10%.
Say the limit too. Volatility treats upside and downside swings alike and assumes the correlation holds in a crisis. In a sharp sell-off correlations can move together, so the diversification shown here is the benefit in normal conditions, not a promise for the worst month.
Where candidates lose it
The trap is answering 11.4%, the weighted average of the volatilities, which ignores diversification entirely. The second is getting the sign of the cross term wrong and adding 8.64 instead of subtracting it.
Say the formula before the numbers, and square the weighted volatilities first: 9 squared and 2.4 squared are easier out loud than 0.36 times 225.
What the interviewer asks next
What bond weight minimises the portfolio volatility?
What is the portfolio volatility if the correlation is plus 0.3?
What share of the portfolio's risk comes from equities once the cross term is split between the two?
017A Rs 1,000 crore loan pool is tranched into equity from 0 to 5%, mezzanine from 5 to 15% and senior from 15 to 100%. The pool loses 12%. How much does each tranche lose as a share of its size, and what pool loss wipes out the mezzanine?Credit risk arithmeticCoreMoody'sNew York · 2024
Try it first
What share of the mezzanine tranche is lost when the pool loses 12%?
Show the worked solution
Equity loses 100%, mezzanine 70% and senior nothing; the mezzanine is wiped out at a 15% pool loss. The Rs 120 crore loss fills the tranches from the bottom. Equity absorbs its full Rs 50 crore. The remaining Rs 70 crore falls on the Rs 100 crore mezzanine. The senior tranche starts losing only once pool losses pass 15%, the point where the mezzanine is gone.
How do losses move through a tranche stack?
Picture a building flooding from the ground up. The ground floor is soaked before a drop reaches the first floor, and the top floors stay dry until the water climbs to them. Losses fill the tranches from the bottom: each tranche loses nothing until the pool loss passes its attachment pointThe level of pool loss at which a tranche starts to lose money., and everything once the loss passes its detachment point. The equity attaches at 0% and detaches at 5%; the mezzanine attaches at 5% and detaches at 15%.
A 12% loss on the Rs 1,000 crore pool wipes out the Rs 50 crore equity tranche, takes Rs 70 crore, or 70%, of the Rs 100 crore mezzanine, and leaves the senior tranche untouched until pool losses pass 15%.
The relationship
$\text{tranche loss share} = \frac{\min(L, D) - \min(L, A)}{D - A} = \frac{12\% - 5\%}{15\% - 5\%} = 70\%$
L the pool loss, 12%
A the attachment point, 5% for the mezzanine
D the detachment point, 15% for the mezzanine
What it says in wordsThe part of the pool loss that falls between a tranche's lower and upper edges, divided by the tranche's thickness.
Why does thickness decide how risky a tranche is?
Because a thin tranche goes from untouched to wiped out over a small range of pool losses. The mezzanine is only 10 points thick, so a pool loss moving from 5% to 15% takes it from zero to total loss, while the same move barely registers on the pool as a whole. That is the leverage inside structured finance: the mezzanine's loss share moved 7 times as far as the pool's 12% average suggests from 5% onwards. A rating analyst evaluating the deal asks how likely the pool loss is to cross each attachment point, which depends heavily on how correlated the loans are.
Name the risks the structure does not remove. Correlation among the loans decides whether pool losses cluster at a few percent or occasionally jump past 15%. The collateral data may be weak. And the waterfall rules in the documents, such as when cash is diverted to protect senior holders, can shift losses between tranches in ways this simple loss-only picture does not show.
Where candidates lose it
The trap is answering 12% for every tranche, as if losses were shared in proportion. The whole point of tranching is that they are not.
The second miss is saying the mezzanine loses 7%, the points above its attachment, and forgetting to divide by its 10 point thickness. Loss share is always relative to the tranche's own size.
What the interviewer asks next
What pool loss would cost the senior tranche 10% of its value?
How does rising correlation among the loans change the risk of the equity versus the senior tranche?
Why might a mezzanine tranche be rated well below the pool's average credit quality?
Asked at Moody's, Credit Risk, New York, 2024 (Wall Street Oasis): What is structured finance, how would you evaluate it, and what are the credit risks?
018A bond callable at 102 trades at 101. If yields fall 100 basis points, does its price rise as much as an otherwise identical non-callable bond, and what happens to its duration?Duration and ratesCoreTreasury and ALMBank market risk
Try it first
When yields fall 100 basis points, roughly how does the callable bond move?
Show the worked solution
No. The callable bond rises only about 1.0 point, to about 102.0, against about 7.2 points for the straight bond, and its duration collapses. The issuer will call the bond once refinancing is cheaper, so investors will not pay much above 102. In this stylised example its effective duration is about 3.4 against 6.8 for the straight bond, and it shrinks further as yields fall.
Why can the callable bond not keep rising?
Think of a home loan with no prepayment penalty. When rates fall, the borrower refinances, and the lender who was enjoying a high rate gets the money back. A call option lets the issuer do the same: when yields fall enough, it buys the bond back at 102, so no investor will pay much more than 102 for it. The price is effectively capped at the call price, while a straight bond with the same coupon and maturity keeps gaining as yields fall.
In this stylised 10 year 8% bond, a 100 basis point fall in yield lifts the straight bond from 101.7 to 108.9 but the callable bond only from 101.0 to 102.0, because its price flattens under the 102 call price.
What happens to duration and convexity?
Duration is how much the price moves for a yield change, and near the cap the callable bond hardly moves. As yields fall towards the level where the call is exercised, the callable bond's effective duration shrinks, and its price curve bends the wrong way: negative convexityWhen a bond gains less from a fall in yields than it loses from an equal rise, because its price curve bends downward.. Here a 100 basis point rise costs both bonds about the same, roughly 6 points, while a 100 basis point fall gives the callable bond only about 1. That lopsided payoff is the price of the call the investor has sold to the issuer, and it is paid for through a higher yield.
Be clear about the model. The curve here is a stylised cap chosen for illustration, not a full option pricing model; a desk would use an interest rate model to value the call and compute effective duration by bumping the whole yield curve. The shape is what matters for the interview: gains capped, losses intact, duration that shortens exactly when you would want it long.
Where candidates lose it
The trap is applying the straight bond's duration and predicting a 7 point gain. Standard duration assumes the cash flows are fixed, and a callable bond's cash flows change when the call is exercised.
The second miss is saying duration rises because the price is near par. Effective duration falls as the call becomes more likely, because the bond starts to behave like a short bond ending at the call date.
What the interviewer asks next
Why do mortgage-backed securities show the same pattern?
How would you hedge a portfolio of callable bonds against falling yields?
What would a putable bond's price curve look like against the same straight bond?

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

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

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

lambda_i	how many events of type i occur a year on average
mu_i	the average loss per event, in Rs crore

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

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

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Why divide by the survivors?

Where does this matter on a credit desk?

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Why do shrinking withdrawals not save the bank?

What would a treasury risk manager add?

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Why does counting events mislead you?

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

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Why subtract the joint probability?

Why does correlation cut the chance of any default but raise the chance of both?

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Why does the simple average show a gain that never happened?

Which one should a risk manager use?

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Why is the answer not the weighted average of 15% and 6%?

What do you add after the number?

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How do losses move through a tranche stack?

Why does thickness decide how risky a tranche is?

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Why can the callable bond not keep rising?

What happens to duration and convexity?

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