Risk Management puzzles, solved step by step

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100
Traced to a firm
17
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13
Hard
30
Topic
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. 054A bank has four derivative trades with one counterparty, currently valued at plus 30, minus 20, plus 15 and minus 10 crore from the bank's side. What is the bank's exposure if the counterparty defaults, with and without an enforceable netting agreement?Counterparty exposure and collateralWarm upCounterparty riskBank credit risk

    Try it first

    Without netting, what does the bank stand to lose if the counterparty defaults today?

    Show the worked solution

    Rs 45 crore without netting and Rs 15 crore with it. Without netting, each trade stands alone, so the bank is exposed to every trade in its favour, 30 plus 15, and must still pay the 30 it owes. With an enforceable netting agreement all four collapse into one net claim: 30 minus 20 plus 15 minus 10, which is Rs 15 crore.

    Why does netting cut the exposure by two thirds?

    Two flatmates keep a running tab: one owes the other Rs 3,000 for rent, the other owes Rs 2,000 for groceries. If they settle as one tab, Rs 1,000 changes hands. If one of them walks out, the other would want the tab settled as one, not to pay the grocery bill in full while chasing the rent. A netting agreementA legal contract under which all trades between two parties are combined into a single net amount if one of them defaults. turns many trades into one claim, so money you owe the defaulter is set against money it owes you.

    Four trades, one counterparty: gross exposure against the netted amount0+30Trade 1-20Trade 2+15Trade 3-10Trade 4Green: they owe you. Red: you owe them.No netting45Exposure = 30 + 15 on defaultand you still pay the 30 you oweEnforceable netting1530 - 20 + 15 - 10 = 15one claim, one number
    Four trades worth plus 30, minus 20, plus 15 and minus 10 crore give an exposure of Rs 45 crore if each is treated alone, because only positive values are at risk. An enforceable netting agreement collapses them into a single claim of Rs 15 crore.

    What happens to the negative trades without netting?

    They still get paid, by the bank. The administrator of a failed counterparty will collect every trade where the bank owes money and join the queue of creditors for every trade where the bank is owed. That asymmetry is called cherry-picking, and it is why exposure without netting is the sum of the positive values, never the net. Here the bank pays Rs 30 crore out and recovers only whatever the estate pays on Rs 45 crore.

    The relationship
    Egross=∑imax⁡(Vi,0)=45Enet=max⁡(∑iVi,0)=15E_{\text{gross}} = \sum_i \max(V_i,0) = 45 \qquad E_{\text{net}} = \max\Big(\sum_i V_i,0\Big) = 15
    V_ithe current value of trade i from the bank's side
    \max(\cdot,0)only amounts owed to the bank count as exposure
    What it says in wordsWithout netting take the positive part of each trade; with netting take the positive part of the total.

    The limitation to say out loud: netting only helps where it is enforceable in the counterparty's jurisdiction, which is why banks obtain legal opinions before counting it. Where that is uncertain, the risk system should fall back to the gross number.

    Where candidates lose it

    The most common error is answering Rs 15 crore for both cases, because the trades feel as if they offset. Without a legal right to set them off, they do not.

    The other slip is adding all four absolute values to get Rs 75 crore. Money the bank owes is not exposure; it is an obligation it pays in full. Exposure counts only what the counterparty owes you.

    What the interviewer asks next

    • The counterparty posts Rs 10 crore of collateral under the netting agreement. What is the exposure now?
    • How does netting change the potential future exposure, not just today's?
    • Why would a regulator want a legal opinion before a bank counts netting?
  2. 079You lend Rs 92 crore against bonds worth Rs 100 crore, an 8% haircut. How far can the bonds fall before the loan is uncovered, and what does that tell you about how the haircut was set?Counterparty exposure and collateralWarm upCounterparty riskBank credit risk

    Try it first

    The bonds fall 10%. Are you still covered?

    Show the worked solution

    The bonds can fall 8%, to Rs 92 crore, before the loan is uncovered. The haircut is exactly that cushion: collateral of 100 less a loan of 92. A haircut is set to cover the largest price fall likely while you seize and sell the bonds; with 1% daily volatility and ten days to sell, a 99% move is about 7.4%, so 8% covers it with little to spare.

    What is a haircut, in plain terms?

    A pawnbroker lends Rs 8,000 against a gold chain worth Rs 10,000. The Rs 2,000 gap is there because gold prices move and because the chain has to be sold if the loan is not repaid. A haircut is the price fall the lender can absorb before the collateral is worth less than the loan. Here the gap is Rs 8 crore on Rs 100 crore, so the bonds can lose 8% before the lender is exposed.

    The 8% haircut is the fall the bonds can take before the loan is uncovered80859095100100Today95Bonds -5%92Bonds -8%88Bonds -12%-4 shortthe loan, Rs 92 crore8% cushionCollateral value, Rs crore (axis starts at 80)Why about 8%?daily volatility 1%x 2.33 for 99%x sqrt(10 days to sell)= 7.4% move8% covers it witha thin buffer
    Rs 100 crore of bonds cover a Rs 92 crore loan after falls of 5% and 8%, but a 12% fall leaves them at Rs 88 crore, Rs 4 crore short, and the 8% cushion sits just above a 7.4% ten-day 99% move for a bond with 1% daily volatility.

    How would a risk team have chosen 8%?

    Ask two questions: how volatile is the collateral, and how long would it take to get out? The second one is called the margin period of riskThe time between the last good margin call and the moment the lender has sold the collateral after a default.. A haircut is roughly the collateral's daily volatility, scaled to the days needed to sell it, at a high confidence level. With 1% daily volatility, ten days and a 99% level, the move is 2.33 times 1% times the square root of 10, about 7.4%. Round up for the bid-ask cost of a forced sale and you reach about 8%.

    The relationship
    h≈z99%⋅σdaily⋅t=2.33×1%×10≈7.4%h \approx z_{99\%} \cdot \sigma_{daily} \cdot \sqrt{t} = 2.33 \times 1\% \times \sqrt{10} \approx 7.4\%
    hthe haircut
    z2.33, the one-sided 99% point of a normal distribution
    tdays to liquidate the collateral
    What it says in wordsThe haircut covers the price fall that would be exceeded only one time in a hundred over the time it takes to sell.

    Then name what breaks it. The haircut assumes the bonds keep their normal volatility and can be sold in ten days; in a stress both assumptions fail together. If the bond issuer is linked to the borrower, the collateral falls just as the borrower defaults, and no haircut sized on normal days is enough.

    Where candidates lose it

    The common slip is saying the bonds can fall 8.7%, dividing 8 by 92. The cushion is measured on the collateral's value, so it is 8 over 100.

    The bigger miss is stopping at the number. The interviewer asked what the haircut says: it is a volatility times a liquidation period, and naming both shows you know why haircuts widen in a crisis.

    What the interviewer asks next

    • The bonds are less liquid and take twenty days to sell. What haircut would you set?
    • The collateral is shares of the borrower's parent. What changes?
    • Why do haircuts rise across the market during a stress, and what does that do to borrowers?