Risk Management puzzles, solved step by step

Puzzles
100
Traced to a firm
17
Topics
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. 007A bank holds liquid assets equal to 12% of its deposits. In a run, depositors withdraw 5% of the remaining deposits every day. On which day does the bank run out of liquid assets?Liquidity and balance sheetCoreTreasury and ALMBank credit risk

    Try it first

    When is the liquid buffer exhausted?

    Show the worked solution

    On day 3. Out of every Rs 100 of deposits, Rs 5 leaves on day 1 and Rs 4.75 on day 2, 9.75 in total, still inside the Rs 12 buffer. Day 3 takes another Rs 4.51, and the cumulative outflow passes Rs 12 part way through the day, about 2.5 days into the run. A buffer that sounds comfortable lasts under three days.

    Why do shrinking withdrawals not save the bank?

    Picture a water tank with a leak that loses 5% of what is left each hour. The leak slows as the tank empties, but in the first few hours it is losing almost 5 litres an hour from a 100 litre tank. A percentage outflow on a large base shrinks only slowly, so for the first days it behaves almost like a fixed outflow of 5 a day. Twelve divided by five says about two and a half days, and the exact answer is only slightly longer.

    Rs 100 of deposits, Rs 12 of liquid assets: the buffer is gone on day 306121824Liquid assets: 125.00Day 14.75Day 24.51Day 34.29Day 44.07Day 59.7514.2618.5522.62empty at 2.49 daysBars: that day's withdrawals. Line: cumulative withdrawals.
    Daily withdrawals of 5.00, 4.75 and 4.51 per 100 of deposits push cumulative outflows to 9.75 by the end of day 2 and 14.26 by the end of day 3, crossing the 12 of liquid assets about 2.5 days into the run.
    The relationship
    1−0.95n=0.12  ⇒  n=ln⁡0.88ln⁡0.95≈2.49 days1 - 0.95^{n} = 0.12 \;\Rightarrow\; n = \frac{\ln 0.88}{\ln 0.95} \approx 2.49 \text{ days}
    0.95^nthe share of deposits still in the bank after n days
    0.12the liquid assets, as a share of the original deposits
    What it says in wordsThe buffer is gone when cumulative withdrawals, one minus what remains, reach the liquid assets.

    What would a treasury risk manager add?

    That the puzzle is a survival horizonHow long a bank can meet outflows under stress from its own liquid assets, before it must sell less liquid assets or borrow. calculation, and that the real answer depends on what the bank can do on day 3. A bank does not fail the moment the liquid buffer is empty; it fails when it can no longer turn other assets into cash fast enough. It can pledge loans to the central bank, sell securities at a discount or borrow, and each has a cost and a limit. The puzzle also assumes a constant 5% a day; real runs usually accelerate once they become news.

    Close with the lesson. Liquidity measured as a percentage of deposits sounds like a lot, but outflows in a run are also measured in percentages of deposits, per day. That mismatch in time units is why regulators express liquidity buffers against stressed outflows over a set horizon, rather than as a plain share of the balance sheet.

    Where candidates lose it

    The trap is dividing 12 by 5 and answering day 2, or, worse, reasoning that a shrinking outflow never exhausts the buffer. The first stops before the buffer is actually gone; the second confuses a slowing leak with a stopped one.

    Give day 3, then the exact 2.49 days from the log formula, then say what the bank would do next.

    What the interviewer asks next

    • What liquid buffer would keep the bank solvent for 30 days at 5% a day?
    • Withdrawals start at 5% a day and rise by one point every day. When does the buffer run out now?
    • Which deposits run first, and how would you weight them in a stress test?
  2. 082A Rs 200 crore bond position has a one-day 99% VaR of Rs 3 crore. The bid-ask spread on the bond is 40 basis points. What is the liquidity-adjusted VaR?Liquidity and balance sheetCoreBank market riskTreasury and ALM

    Try it first

    How much does the spread add to the Rs 3 crore VaR?

    Show the worked solution

    About Rs 3.4 crore: the Rs 3 crore VaR plus Rs 0.4 crore to get out. The position is marked at mid, but selling means taking the bid, half the spread below mid. Half of 40 basis points is 0.20%, and 0.20% of Rs 200 crore is Rs 0.4 crore. The exit cost adds 13% to the risk number, and more if spreads widen in a stress.

    Why is ordinary VaR missing a cost?

    A second-hand car dealer will quote you two prices for the same car: what he pays and what he sells for. Your car is worth the middle on paper, but if you need cash today you get the lower one. VaR is computed on mid prices, so it measures how far the value might move, not what it costs to actually leave the position. For a liquid government bond the difference is small; for a corporate bond the spread can be a meaningful share of the risk.

    Getting out costs half the spread, even on a day prices do not moveask 100.20mid 100.00bid 99.80you sell hereOne bond quote, price per 100Half spread = 0.40% / 2 = 0.20%0.20% x Rs 200 cr = Rs 0.4 cr3.03.0Market VaR3.03.4Normal spread3.0+1.24.2Stressed spread+0.4Rs crore, one day, 99%
    Selling at the bid costs half the 40 basis point spread, 0.20% of Rs 200 crore or Rs 0.4 crore, which lifts a Rs 3.0 crore market VaR to a liquidity-adjusted Rs 3.4 crore, and to Rs 4.2 crore if the spread widens to 120 basis points.
    The relationship
    LVaR=VaR+12×s×P=3.0+0.5×0.0040×200=3.4\text{LVaR} = \text{VaR} + \tfrac{1}{2} \times s \times P = 3.0 + 0.5 \times 0.0040 \times 200 = 3.4
    VaRmarket VaR on mid prices, Rs 3 crore
    sbid-ask spread as a fraction of price, 0.40%
    Pposition value, Rs 200 crore
    What it says in wordsAdd the cost of crossing half the spread to the market VaR, because that cost is paid even if prices do not move.

    What makes the adjustment larger than it looks?

    Two things, and both arrive in a stress. Spreads widen exactly when you need to sell, and a large position moves the price against you as you sell it. If the spread triples to 120 basis points, the exit cost is Rs 1.2 crore and the adjusted figure is Rs 4.2 crore, 40% above plain VaR. A risk team would also ask how many days it takes to exit Rs 200 crore without moving the market; if the answer is five days rather than one, the market VaR itself should be scaled to that horizon.

    Say the limit of the simple version: it treats the spread as a fixed number. A fuller treatment uses the spread's own volatility, adding a multiple of its standard deviation, so the adjustment reflects how bad the spread gets in a bad week, not how it looks on an average one.

    Where candidates lose it

    The common mistake is adding the full 40 basis points, Rs 0.8 crore. Positions are marked at mid, so selling costs only the distance from mid to bid, half the spread.

    The second is saying VaR already includes liquidity because it uses market prices. It uses mid prices and a one-day horizon, and assumes you could exit at mid; the question is testing whether you see that gap.

    What the interviewer asks next

    • The bond takes five days to sell without moving the price. How would you change the VaR?
    • Why might a desk argue against a liquidity add-on for government bonds?
    • How would you estimate a bid-ask spread for a bond that rarely trades?