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. 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?