Risk Management puzzles, solved step by step

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  1. 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.

    The same Rs 1,250 crore of headroom, measured on two different basesCET1 / risk-weighted assetsWhat the ratio is quoted on0%2%4%6%8%10%12%14%need 10.5%have 13%2.5CET1 / total assets0%2%4%6%8%10%12%14%need 5.25%have 6.5%RWA are half the assets,so every mark halvesLoss it can absorb1.25%of total assetsBank with Rs 1,00,000 crore of assets, Rs 50,000 crore of RWA and Rs 6,500 crore of CET1; the floor is Rs 5,250 crore.Headroom: Rs 1,250 crore, which is 2.5% of RWA but only 1.25% of assets.
    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
    loss capacity=(13%−10.5%)×RWAtotal assets=2.5%×0.5=1.25%\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 assetsthe 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?
  2. 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.

    RAROC: expected loss is a cost on top, unexpected loss is capital underneath3.0Net interestincome-0.8Operatingcost-1.0Expectedloss1.2Risk-adjustedprofitRs crore a year on a Rs 100 crore loanDivide by economic capitalheld against unexpected loss1.2 / 8 = 15.0%RAROC15.0%Hurdle14.0%Profit above the capital charge:1.2 - 14% x 8 = Rs 0.08 croreClears the hurdle, by a thin margin
    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
    RAROC=income−cost−ELeconomic capital=3.0−0.8−1.08=15.0%\text{RAROC} = \frac{\text{income} - \text{cost} - \text{EL}}{\text{economic capital}} = \frac{3.0 - 0.8 - 1.0}{8} = 15.0\%
    ELexpected loss, the average annual credit loss, Rs 1 crore
    economic capitalthe 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?
  3. 076A bank has book equity of Rs 20,000 crore, a sustainable return on equity of 15%, a cost of equity of 12% and long-run growth of 6%. Estimate its market capitalisation.Capital and leverageCoreScotiabankToronto · 2025

    Try it first

    Before any formula: is this bank worth more or less than its Rs 20,000 crore book?

    Show the worked solution

    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.

    A bank is worth more than its book only when ROE beats the cost of equity0.5x1.0x1.5x2.0x2.5x0ROE 12% = cost of equity: 1.0xROE 15%: 1.5x bookEarns less than it costs:trades below bookEarns more than it costs:trades above book6%9%12%15%18%21%Return on equityPrice to book(ROE - g) / (ke - g)= (15 - 6) / (12 - 6)= 1.5x bookBook Rs 20,000 crx 1.5Rs 30,000 crmarket cap
    With a 12% cost of equity and 6% growth, price to book crosses 1.0 at an ROE of 12% and reaches 1.5 at 15%, so the bank's Rs 20,000 crore of book equity is worth about Rs 30,000 crore.

    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.

    The relationship
    PB=ROE−gke−g=0.15−0.060.12−0.06=1.5\frac{P}{B} = \frac{ROE - g}{k_e - g} = \frac{0.15 - 0.06}{0.12 - 0.06} = 1.5
    P/Bmarket capitalisation over book equity
    ROEsustainable return on book equity, 15%
    k_ecost of equity, 12%
    glong-run growth in book and dividends, 6%
    What it says in wordsPrice to book is the excess of ROE over growth, divided by the excess of the cost of equity over growth.

    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.

    Where candidates lose it

    The common miss is answering Rs 20,000 crore, treating book value as the value of a bank. Book is only the starting point; the premium or discount comes entirely from ROE against the cost of equity.

    The second is plugging 15% and 12% into a P/E formula without growth and getting lost. Say the price to book shortcut out loud, then back it up with earnings of Rs 3,000 crore at a P/E of 10.

    What the interviewer asks next

    • Loan losses rise and sustainable ROE falls to 10%. What happens to the market capitalisation?
    • What does a bank trading at 0.6 times book tell you about how the market views its loan book?
    • Why might a bank with a high CET1 ratio still earn a low ROE?

    Asked at Scotiabank, Risk, Toronto, 2025 (Wall Street Oasis): The market cap of the bank