Risk Management puzzles, solved step by step

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.

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.

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.

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.

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.

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.

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.

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.

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 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.

About Rs 30,000 crore, 1.5 times book. For a bank growing steadily, price to book equals ROE less growth, over cost of equity less growth: (15 - 6) / (12 - 6) = 1.5. Book of Rs 20,000 crore times 1.5 gives Rs 30,000 crore. A cross-check: earnings of Rs 3,000 crore at a P/E of 10 gives the same figure.

Why does ROE against cost of equity decide the answer?

Imagine a fixed deposit that pays exactly the return you demand. Nobody would pay more than its face value, and nobody would sell it for less. A deposit paying more than you demand is worth a premium; one paying less sells at a discount. A bank's book equity is that deposit. A bank that earns exactly its cost of equity is worth its book, and every point of ROE above that cost is priced as a premium to book. Here the bank earns 15% on capital that shareholders price at 12%, so you already know the answer is above Rs 20,000 crore before touching the arithmetic.

Where does the formula come from, so you can rebuild it under pressure?

Start from a dividend growth model. To grow book at 6% while earning 15%, the bank must keep 6 over 15, or 40%, of its profit, and it can pay out the other 60%. That means the dividend is book times ROE less growth, and dividing by cost of equity less growth gives the price. Earnings are Rs 3,000 crore, the payout is Rs 1,800 crore, and Rs 1,800 crore divided by 12% less 6% is Rs 30,000 crore.

What would a risk interviewer want you to add?

Say how fragile the number is. The denominator is only 6 points wide, so a one point rise in the cost of equity to 13% cuts price to book from 1.5 to 1.29, a fall of 14% in market value. The word sustainable is also doing work: a 15% ROE earned in a benign credit year, before loan losses normalise, is not the same as 15% through a cycle. A stress-testing team cares because a bank trading below book is telling you the market doubts its ROE or its asset values.

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.

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.