Risk Management puzzles, solved step by step

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100
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All topicsCapital and leverage6Compounding and drawdowns8Correlation and diversification8Counterparty exposure and collateral7Credit risk arithmetic10Duration and rates7Liquidity and balance sheet7Logic, estimation and brainteasers7Operational loss and fraud7Options and Greeks7Probability and base rates8Statistics and estimation10VaR and expected shortfall8
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  1. 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.

    A loss shrinks both sides of the ratio, but the numerator falls much fasterCET1, Rs crore100Before80After loss95Needed at 10%raise 15-20% on the numeratorRisk-weighted assets, Rs crore1,000Before950After write-off-50-5% on the denominator80 / 9508.42%
    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
    raise=0.10×(1,000−50)−(100−20)=95−80=15\text{raise} = 0.10 \times (1{,}000 - 50) - (100 - 20) = 95 - 80 = 15
    0.10the target CET1 ratio
    1,000 - 50risk-weighted assets after the write-off
    100 - 20CET1 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?
  2. 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
    EA=ERWA×RWAAA:14%×25%=3.5%B:11%×55%=6.05%\frac{E}{A} = \frac{E}{RWA} \times \frac{RWA}{A} \qquad A: 14\% \times 25\% = 3.5\% \qquad B: 11\% \times 55\% = 6.05\%
    ECET1 equity
    RWArisk-weighted assets
    Atotal assets
    What it says in wordsEquity per rupee of assets is the CET1 ratio multiplied by the RWA density.
    Divide by assets instead of risk-weighted assets and the ranking flips14.00%Bank Ahigher11%Bank BCET1 ratio (equity / RWA)3.50%Bank A6.05%Bank BhigherEquity / total assetsRisk weightsA: RWA = 25% of assetsB: RWA = 55% of assets14% x 25% = 3.5%
    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?