Risk Management puzzles, solved step by step

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

Showing 1–5 of 5 · 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?
026A Rs 100 crore loan earns Rs 3 crore of net interest income a year, costs Rs 0.8 crore to run, carries an expected loss of Rs 1 crore and needs Rs 8 crore of economic capital. What is its risk-adjusted return on capital, and does it clear a 14% hurdle?Capital and leverageCoreBank credit riskRisk GCC
Try it first
Where does the Rs 1 crore of expected loss go in the calculation?
Show the worked solution
RAROC is 15.0%, so the loan clears a 14% hurdle, but only just. Income of Rs 3 crore less Rs 0.8 crore of cost and Rs 1 crore of expected loss leaves Rs 1.2 crore of risk-adjusted profit. Divided by Rs 8 crore of economic capital that is 15.0%. After a 14% charge on the capital, Rs 1.12 crore, the loan adds only Rs 0.08 crore a year.
Why does expected loss sit in the numerator and unexpected loss in the denominator?
Think of a restaurant that knows about two plates in a hundred get dropped. That breakage is priced into the menu; it is a running cost. What the owner keeps cash aside for is the night a whole shelf comes down. Expected loss is the average, so it is a cost charged against income; unexpected loss is the bad year above the average, so it is covered by capital. RAROCRisk-adjusted return on capital: profit after costs and expected loss, divided by the capital held against unexpected loss. puts each in its proper place, which is why a bank can compare a thin-margin safe loan with a fat-margin risky one on the same scale.
Net interest income of Rs 3.0 crore, less Rs 0.8 crore of cost and Rs 1.0 crore of expected loss, leaves Rs 1.2 crore of risk-adjusted profit; over Rs 8 crore of economic capital that is a 15.0% RAROC, one point above the 14% hurdle.
The relationship
$\text{RAROC} = \frac{\text{income} - \text{cost} - \text{EL}}{\text{economic capital}} = \frac{3.0 - 0.8 - 1.0}{8} = 15.0\%$
EL expected loss, the average annual credit loss, Rs 1 crore
economic capital the buffer held against losses worse than average, Rs 8 crore
What it says in wordsTake the average loss off the income, then ask how hard the remaining profit works the capital held against a bad year.
How much room is there before the loan fails the hurdle?
Turn the hurdle into rupees. A 14% return on Rs 8 crore of capital is Rs 1.12 crore a year, and the loan earns Rs 1.2 crore, so it clears by Rs 0.08 crore. A margin that thin means one small change flips the answer. If expected loss rises from Rs 1.0 crore to Rs 1.08 crore, or economic capital rises from Rs 8 crore to Rs 8.57 crore, RAROC falls to exactly 14%. Saying those two breakeven numbers is what turns a ratio into a credit judgement.
State the limits too. This is a pre-tax, single-year figure; many banks compute RAROC after tax and add the return earned on the capital itself, which would lift the number. And the Rs 8 crore depends on the bank's own capital model, so the answer is only as good as that model's view of this borrower.
Where candidates lose it
The common slip is to subtract expected loss and then also add it to the capital, or to leave it out of the numerator on the grounds that capital covers losses. Either way the ratio comes out wrong by several points, and the interviewer hears that you do not know what capital is for.
The second slip is stopping at 15% and saying yes. A one-point margin deserves the breakeven sentence: how much worse can the loss or the capital get before the loan stops paying for itself.
What the interviewer asks next
The bank prices the loan 20 basis points lower to win the deal. Does it still clear the hurdle?
How would you estimate the Rs 8 crore of economic capital in the first place?
Why might a bank still make a loan with a RAROC below its hurdle?
051A bank has CET1 capital of Rs 900 crore and risk-weighted assets of Rs 8,000 crore. It takes a Rs 200 crore loss. What happens to its CET1 ratio?Capital and leverageWarm upBank credit riskRisk GCC
Try it first
Before you divide anything: roughly where does the ratio land?
Show the worked solution
The CET1 ratio falls from 11.25% to 8.75%, a drop of 2.5 percentage points. The Rs 200 crore loss comes straight out of CET1, taking it from Rs 900 crore to Rs 700 crore, while risk-weighted assets stay close to Rs 8,000 crore. A loss equal to 2.5% of risk-weighted assets wipes out 22% of the capital.
Why does a loss that looks small against the balance sheet hurt so much?
Think of a family with a Rs 80 lakh home loan and Rs 9 lakh in savings. A Rs 2 lakh medical bill is a rounding error against the loan and a painful bite out of the savings, because the savings are what pay for it. A bank works the same way. The CET1Common equity tier 1: the shareholders money and retained profits a bank can use to absorb losses while it keeps operating. ratio is capital over risk-weighted assetsThe bank assets, each scaled by a weight that reflects how risky it is, so a safe government bond counts for little and an unsecured loan counts in full.. Losses are paid from the numerator rupee for rupee, while the denominator moves only a little, so a ratio of 11% can lose a quarter of its height in one hit.
CET1 of Rs 900 crore over risk-weighted assets of Rs 8,000 crore is 11.25%. A Rs 200 crore loss takes CET1 to Rs 700 crore over the same Rs 8,000 crore, which is 8.75%, so the ratio loses 2.5 points.
Is there a shortcut for the size of the drop?
Yes. With the denominator fixed, the ratio falls by the loss divided by risk-weighted assets: 200 over 8,000 is 2.5 points. Every Rs 80 crore of loss costs this bank one full point of CET1 ratio. That is the number a risk manager keeps in their head, because it turns a loss estimate from a stress test straight into a capital headline.
The relationship
$\Delta\,\text{ratio} = -\frac{\text{loss}}{\text{RWA}} = -\frac{200}{8{,}000} = -2.5\text{ pts}$
loss the post-tax loss that reduces CET1, here Rs 200 crore
RWA risk-weighted assets, held at Rs 8,000 crore
What it says in wordsWith the denominator unchanged, the ratio drops by the loss as a share of risk-weighted assets.
Say the limitation in one line. The written-off loans do leave the balance sheet, so risk-weighted assets fall a little too, and a tax credit on the loss can soften the hit. Both effects are small next to the Rs 200 crore coming out of capital. Where the result lands against the bank's minimum depends on the current regulatory figure and any buffers, which you would confirm rather than quote from memory.
Where candidates lose it
The common slip is to compare the loss with the balance sheet, call it 2.5%, and then shave 2.5% off the ratio to get about 11%. That confuses a fall of 2.5 percentage points with a fall of 2.5 per cent of the ratio, and it misses that the loss lands entirely on the capital line.
Say 11.25% to 8.75%, then add the shortcut: loss over risk-weighted assets gives the drop in points. It shows you can run a stress number in your head.
What the interviewer asks next
How big a loss takes this bank to an 8% CET1 ratio?
The written-off loans carried Rs 300 crore of risk-weighted assets. What is the ratio now?
Why do regulators use a risk-weighted denominator rather than total assets?
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?
089Bank A reports a CET1 ratio of 14%, with risk-weighted assets equal to 25% of its total assets. Bank B reports 11%, with risk-weighted assets at 55% of total assets. Which bank has more equity per rupee of assets?Capital and leverageHardBank credit riskRating agency
Try it first
Which bank holds more equity for every rupee of assets?
Show the worked solution
Bank B, with about 6.05% of assets in equity against 3.5% for bank A. Equity to assets is the CET1 ratio times the share of assets that count as risk-weighted: 14% times 25% for A, 11% times 55% for B. A's higher ratio rests on low risk weights, so it is levered about 29 times against B's 16.5 times.
Why can a higher capital ratio mean less capital?
Two people each say they save 20% of their income. One counts only the salary left after rent; the other counts the whole salary. The first sounds equally thrifty but saves far fewer rupees. A CET1 ratio divides equity by risk-weighted assets, not total assets, so a bank that assigns low risk weights to its loans can show a high ratio on a thin equity base. The share of assets that ends up risk-weighted is called {term('RWA density', 'Risk-weighted assets divided by total assets; a low figure means the bank judges most of its assets to be low risk.')}, and it is the number that reconciles the two views.
The relationship
$\frac{E}{A} = \frac{E}{RWA} \times \frac{RWA}{A} \qquad A: 14\% \times 25\% = 3.5\% \qquad B: 11\% \times 55\% = 6.05\%$
E CET1 equity
RWA risk-weighted assets
A total assets
What it says in wordsEquity per rupee of assets is the CET1 ratio multiplied by the RWA density.
Bank A's CET1 ratio of 14% beats bank B's 11%, but because A's risk-weighted assets are only 25% of its assets against B's 55%, A holds 3.5% of assets in equity and B holds 6.05%, and the ranking flips.
Which number should a credit analyst trust?
Both, because they answer different questions. The CET1 ratio is right if the risk weights are right. Equity to assets makes no judgement about risk, so it is the check on whether the risk weights are doing too much of the work. Test A's number: if its true risk density were 40% rather than 25%, perhaps because its internal models are optimistic, its CET1 ratio would be 3.5% over 40%, only 8.75%. That is why regulators set a leverage ratio floor alongside risk-based ratios; the current minimum is set by the regulator, so confirm it before quoting it.
What would make bank A's low density legitimate?
A book of home loans with low loan-to-value ratios, or large holdings of government bonds, genuinely carries low risk weights, and a 25% density can be honest. The question is whether the weights are earned. An analyst asks what the assets are, whether the density has fallen while the book stayed the same, and how the bank's model-based weights compare with the standardised ones.
Where candidates lose it
Most candidates answer bank A, reading the higher ratio as more capital. The question is built to see whether you ask what the denominator is.
The second miss is getting B and then calling A unsafe. A low density can be legitimate; say what you would check before concluding the weights are too low.
What the interviewer asks next
What happens to bank A's CET1 ratio if its risk weights rise to the level of bank B's?
Why do regulators set a leverage ratio at all if risk-weighted ratios exist?
Which kinds of assets carry low risk weights, and which of them surprised people in past crises?

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

EL	expected loss, the average annual credit loss, Rs 1 crore
economic capital	the buffer held against losses worse than average, Rs 8 crore

loss	the post-tax loss that reduces CET1, here Rs 200 crore
RWA	risk-weighted assets, held at Rs 8,000 crore

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

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 does expected loss sit in the numerator and unexpected loss in the denominator?

How much room is there before the loan fails the hurdle?

Where candidates lose it

What the interviewer asks next

Try it first

Why does a loss that looks small against the balance sheet hurt so much?

Is there a shortcut for the size of the drop?

Where candidates lose it

What the interviewer asks next

Try it first

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?

Where candidates lose it

What the interviewer asks next

Try it first

Why can a higher capital ratio mean less capital?

Which number should a credit analyst trust?

What would make bank A's low density legitimate?

Where candidates lose it

What the interviewer asks next