Risk Management puzzles, solved step by step

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100
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17
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13
Hard
30
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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. 043A bank holds Rs 5,000 crore of bonds at market value, of which Rs 3,200 crore are pledged in repo and at the clearing house. Bond prices fall 10%, and the pledged value must be restored to Rs 3,200 crore. How much free collateral is left?Liquidity and balance sheetHardTreasury and ALMBank credit risk

    Try it first

    Free collateral started at Rs 1,800 crore. By roughly how much does it fall?

    Show the worked solution

    Rs 1,300 crore, down 27.8% from Rs 1,800 crore. The holding falls 10% to Rs 4,500 crore. The pledged bonds are now worth Rs 2,880 crore, so Rs 320 crore of free bonds must be posted to restore Rs 3,200 crore. Free bonds, themselves down to Rs 1,620 crore, lose that Rs 320 crore too, leaving Rs 1,300 crore.

    Why does free collateral fall faster than prices?

    Someone who has pawned most of their jewellery owes the lender a fixed amount of value. When gold prices fall, the pawnbroker asks for more pieces to keep the loan covered, and those pieces come from the small pile still at home. The pledged claim is fixed in rupees, so the entire fall in value across all Rs 5,000 crore lands on the free slice. That makes free collateral a leveraged position on bond prices, with leverage of 5,000 over 1,800, about 2.78 times.

    Prices fall 10%; free collateral falls 28%pledged 3,200free 1,8005,000Beforepledged 2,880+320 top-upfree 1,3004,500After a 10% fallFree collateral1,800 to 1,300down 27.8%= 10% x 2.78(5,000 / 1,800)All free bonds gone aftera 36% fallRs crore, market value
    After a 10% price fall, Rs 320 crore of free bonds must be posted to keep the pledged value at Rs 3,200 crore, so free collateral drops from Rs 1,800 crore to Rs 1,300 crore, a 27.8% fall.
    The relationship
    free1=total (1−d)−pledged=5,000×0.9−3,200=1,300\text{free}_1 = \text{total}\,(1 - d) - \text{pledged} = 5{,}000 \times 0.9 - 3{,}200 = 1{,}300
    dthe fall in bond prices, 10%
    pledgedthe market value that must stay posted, Rs 3,200 crore
    What it says in wordsFree collateral is whatever is left of the whole holding after the fixed pledged value is carved out.

    At what point does the bank run out, and what would you add?

    Solve for zero: 5,000 x (1 minus d) equals 3,200 when d is 36%. A 36% price fall exhausts every free bond, and any further fall produces a margin call the bank cannot meet from securities. Real life is worse than this sum: lenders raise haircutsThe discount a lender applies to collateral value; a 5% haircut means Rs 100 of bonds secures only Rs 95 of borrowing. in a sell-off, which increases the pledged amount needed at exactly the moment prices fall, and the same stress usually brings deposit outflows that draw on the same free bonds.

    That is why a liquidity risk team tracks the encumbrance ratio, here 64% before the fall and 71% after, and stresses free collateral against price falls and haircut rises together. A buffer reported at Rs 1,800 crore is really a buffer of Rs 1,800 crore only on the day prices stand still.

    Where candidates lose it

    The instinct is to take 10% off the free Rs 1,800 crore and answer Rs 1,620 crore. That misses the top-up: the pledged bonds also lost value and must be replenished from the free pile.

    The overcorrection is to subtract the top-up but forget that the free bonds fell too. Do it in one line from the total: 4,500 less 3,200 is 1,300, and the leverage of 5,000 over 1,800 explains the 27.8% in one sentence.

    What the interviewer asks next

    • The repo lender also raises its haircut from 2% to 5%. How much free collateral is left?
    • Which bonds would you pledge first, and which would you keep free?
    • How would a 10% fall combined with a Rs 400 crore deposit outflow change your answer?
  2. 070A fund holds Rs 1,000 crore of bonds financed by repo, putting up Rs 20 crore of its own capital against a 2% haircut. The lender raises the haircut to 10%. With its capital unchanged, how much must the fund sell?Liquidity and balance sheetHardTreasury and ALMAsset manager risk

    Try it first

    How much of the Rs 1,000 crore position has to go?

    Show the worked solution

    It must sell Rs 800 crore, four fifths of the position. The haircut is the share of the position the fund funds itself. At 2%, Rs 20 crore supports Rs 1,000 crore of bonds, 50 times leverage. At 10%, the same Rs 20 crore supports only Rs 200 crore, 10 times leverage. Keeping the full position would need Rs 100 crore of capital, Rs 80 crore more than it has.

    Why does a small change in haircut force such a big sale?

    Buying a flat with a 5% down payment lets Rs 10 lakh of savings control a Rs 2 crore flat. If the bank suddenly demands 20% down, the same Rs 10 lakh controls only a Rs 50 lakh flat. The haircutThe share of a security value a lender will not lend against, so the borrower must fund that share with its own capital. sets leverage directly, at one over the haircut, so moving it from 2% to 10% cuts the position the capital can support from 50 times to 10 times.

    A haircut change is a forced deleveraging: capital / haircut = the most you can holdBefore: 2%50x leverageRepo 980Bonds 1,000capital 20After: 10%10x leverageRepo 180Bonds 200, of which capital 20Must sell 800Maximum position = capital / haircut20 / 2% = 1,000 then 20 / 10% = 200
    At a 2% haircut, Rs 20 crore of capital and Rs 980 crore of repo carry Rs 1,000 crore of bonds. At 10%, the same Rs 20 crore carries only Rs 200 crore with Rs 180 crore of repo, so Rs 800 crore of bonds must be sold.

    What happens next in a real market?

    The sale itself moves prices. If many leveraged holders face the same haircut increase, they all sell the same bonds, prices fall, their capital shrinks, and the lower capital supports an even smaller position. That feedback loop between haircuts, prices and forced selling is the liquidity spiral seen in past funding crises. A fund at 50 times leverage loses its entire Rs 20 crore of capital on a 2% price fall.

    The relationship
    max position=capitalhaircut=200.10=200sell=1,000−200=800\text{max position} = \frac{\text{capital}}{\text{haircut}} = \frac{20}{0.10} = 200 \qquad \text{sell} = 1{,}000 - 200 = 800
    capitalthe fund's own money in the position, Rs 20 crore
    haircutthe share the lender will not finance, now 10%
    What it says in wordsThe largest position the fund can hold is its capital divided by the haircut.

    The limitation: the answer assumes the fund cannot raise capital or find another lender at the old haircut. In practice it would try both, and a risk manager's job is to know in advance how many days that would take and how much the forced sale would cost at stressed prices.

    Where candidates lose it

    The trap is answering Rs 80 crore: 8% more haircut on Rs 1,000 crore. That is the extra capital needed to keep the position, not the amount the fund must sell when it has no extra capital. The question asks for the sale.

    The second miss is stopping at the arithmetic. Say what the sale does to prices and to the fund's remaining capital; the interviewer wants to hear the word spiral.

    What the interviewer asks next

    • What price fall on the remaining position would wipe out the fund's capital at 10 times leverage?
    • The fund can raise Rs 30 crore more. How much must it sell now?
    • Why do lenders raise haircuts precisely when markets are falling?
  3. 094A bank has assets of Rs 1,000 crore with a duration of 4 and liabilities of Rs 900 crore with a duration of 1. What is the duration of its equity, and what happens to its equity if rates rise 1%?Liquidity and balance sheetHardTreasury and ALMBank market risk

    Try it first

    If rates rise 1%, roughly how much of the bank's Rs 100 crore of equity is lost?

    Show the worked solution

    The duration of equity is about 31, so a 1% rate rise costs about Rs 31 crore, 31% of equity. Assets fall 4% of Rs 1,000 crore, Rs 40 crore. Liabilities fall 1% of Rs 900 crore, Rs 9 crore. Equity absorbs the difference, Rs 31 crore on a base of Rs 100 crore. Leverage of ten to one turns a modest duration gap into a large equity sensitivity.

    Why is equity so much more sensitive than either side?

    A family buys a Rs 1 crore flat with Rs 10 lakh down and a Rs 90 lakh loan. If the flat's value falls 4%, they lose Rs 4 lakh, 40% of their stake, even though the flat barely moved. Equity is the thin difference between two large numbers, so any gap in how assets and liabilities respond to rates is magnified by the leverage. A bank that funds four-year assets with one-year deposits has that gap built in.

    A thin slice of equity absorbs the whole duration gapBeforeAssets1,000Equity 100Liabilities900After rates rise 1%Assets960Equity 69Liabilities891duration 4duration 1Assets: -4% x 1,000-40Liabilities: -1% x 900-9Equity-31, or -31%Duration of equity(4,000 - 900) / 100 = 31
    A 1% rate rise cuts assets of Rs 1,000 crore by Rs 40 crore and liabilities of Rs 900 crore by Rs 9 crore, so the Rs 100 crore equity slice falls Rs 31 crore to Rs 69 crore, a duration of equity of 31.
    The relationship
    DE=A⋅DA−L⋅DLE=1000×4−900×1100=31D_E = \frac{A \cdot D_A - L \cdot D_L}{E} = \frac{1000 \times 4 - 900 \times 1}{100} = 31
    A, L, Eassets 1,000, liabilities 900, equity 100, in Rs crore
    D_A, D_Lasset duration 4 and liability duration 1
    D_Eduration of equity
    What it says in wordsEquity's duration is the rupee duration of assets less that of liabilities, spread over the small equity base.

    How do you say this in the language a treasury uses?

    Two equivalent ways. The leverage-adjusted duration gap is 4 less 0.9 times 1, which is 3.1; multiply by the 1% move and by total assets, Rs 1,000 crore, and you get the same Rs 31 crore. Either way the answer is an economic value of equity loss, the number supervisors ask banks to measure for interest rate risk in the banking book. Say too that the income view is different: in the first year, deposits reprice faster than loans, so net interest income is squeezed as well.

    Name the assumptions. The calculation treats the rate rise as a parallel shift, uses first-order duration without convexity, and takes the deposit duration of 1 at face value. Deposits that customers leave in place for years behave as longer-dated funding; deposits that can leave overnight behave as shorter. That behavioural assumption can move the answer more than the arithmetic.

    Where candidates lose it

    The common error is answering 3%, the simple gap between 4 and 1, and applying it to equity. The gap has to be weighted by the balance sheet sizes and then divided by equity, which multiplies it roughly ten times.

    The second is forgetting that liabilities fall in value too. Subtract the Rs 9 crore gain on liabilities, or the answer comes out at 40% instead of 31%.

    What the interviewer asks next

    • What liability duration would make the equity immune to a parallel rate move?
    • How would you use interest rate swaps to shorten the asset duration?
    • Why might the economic value of equity fall while net interest income rises?