Risk Management puzzles, solved step by step

Puzzles: 100
Traced to a firm: 17
Topics: 13
Hard: 30

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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
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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.
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
$1 - 0.95^{n} = 0.12 \;\Rightarrow\; n = \frac{\ln 0.88}{\ln 0.95} \approx 2.49 \text{ days}$
0.95^n the share of deposits still in the bank after n days
0.12 the 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?
032A bank holds Rs 1,000 crore of liquid government bonds yielding 6.8% instead of lending the money at 10%, and it funds the whole amount at 6%. What does carrying this liquidity buffer cost the bank each year?Liquidity and balance sheetWarm upTreasury and ALM
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Which comparison gives the cost of the buffer?
Show the worked solution
About Rs 32 crore a year, before adjusting for credit losses on the loans. The money is funded at 6% either way, so funding drops out. The cost of the buffer is the income it gives up: lending would earn 10% and the bonds earn 6.8%, a 3.2 point gap on Rs 1,000 crore. The buffer still earns Rs 8 crore over funding; it just earns Rs 32 crore less than loans would.
Why does the funding cost drop out of the answer?
A family that keeps Rs 5 lakh in a savings account instead of prepaying a home loan pays the same salary-funded EMI either way. The cost of the emergency fund is the gap between the loan rate saved and the savings rate earned. The cost of any buffer is an opportunity cost: what the same rupees would have earned in their next-best use, with everything common to both uses cancelling out. Funding at 6% is common to both uses here, so the only number that matters is 10% minus 6.8%.
Lending earns 10.0% and liquid bonds earn 6.8% on money funded at 6.0%, so holding Rs 1,000 crore of bonds instead of loans gives up 3.2 points, Rs 32 crore a year, even though the bonds still earn Rs 8 crore over funding.
Is 3.2 points the true gap?
Not quite, and saying why is the part interviewers listen for. A 10% loan yield is before credit losses and before the capital loans consume; government bonds need neither. If the loans carry an assumed expected loss of 1.2% a year, the like-for-like gap narrows to 2.0 points, and the buffer costs about Rs 20 crore rather than Rs 32 crore. Capital would narrow it further. The headline number is an upper bound; the risk-adjusted number is what the treasurer should defend.
The relationship
$\text{cost} = \text{size} \times (y_{\text{loan}} - EL - y_{\text{liquid}}) = 1{,}000 \times (10.0\% - 1.2\% - 6.8\%) = 20$
y_loan yield on the loans that could have been made, 10%
EL an assumed annual expected credit loss on those loans, 1.2%
y_liquid yield on the liquid bonds, 6.8%
What it says in wordsCompare what the money earns in each use after the costs that differ between them.
Close by naming what the premium buys. The buffer is insurance: in a deposit run it can be sold or pledged within days, while loans cannot. Liquidity coverageA regulatory measure comparing high-quality liquid assets with the net cash a bank could lose in a 30-day stress. rules make a floor of it compulsory; confirm the current requirement with the regulator rather than from memory. Above that floor, a bank is choosing how much insurance to buy, and Rs 20 to 32 crore a year is the price tag to weigh against the run it protects against.
Where candidates lose it
The common wrong answer is zero, because the bonds earn 6.8% against a 6% funding cost and so look profitable. Positive carry is not the same as no cost; the bank has given up a better use of the money.
The second miss is quoting Rs 32 crore as if loans were risk-free. Mention expected loss and capital in one sentence and you show you compare like with like.
What the interviewer asks next
Rates on liquid bonds rise to 7.5% with everything else fixed. What happens to the cost?
Why might a bank hold more liquidity than the regulatory minimum?
How would you allocate this cost to the business lines that create the liquidity need?
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
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Free collateral started at Rs 1,800 crore. By roughly how much does it fall?
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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.
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
$\text{free}_1 = \text{total}\,(1 - d) - \text{pledged} = 5{,}000 \times 0.9 - 3{,}200 = 1{,}300$
d the fall in bond prices, 10%
pledged the 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?
057A bank has loans of Rs 95 crore and deposits of Rs 100 crore. Deposits fall 10%. If the bank keeps its loan-to-deposit ratio at 95%, by how much must its loans shrink?Liquidity and balance sheetWarm upTreasury and ALMBank credit risk
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Quick answer: how much lending goes?
Show the worked solution
Loans must shrink by Rs 9.5 crore, from Rs 95 crore to Rs 85.5 crore. Deposits fall 10% to Rs 90 crore, and 95% of Rs 90 crore is Rs 85.5 crore. The Rs 10 crore that leaves is met by Rs 9.5 crore of loans running off and Rs 0.5 crore of liquid assets, so the bank's lending falls by the same 10% as its deposits.
Why does a deposit outflow become a lending cut?
A household that lives on its salary and lends a cousin money every month has to stop lending if the salary is cut. The cousin has done nothing wrong; the money simply is not there. A bank funding its loans from deposits is in the same position. Holding the loan-to-deposit ratio fixed means every rupee of deposit flight passes straight into less lending, scaled by the ratio.
Deposits fall from Rs 100 crore to Rs 90 crore. Keeping loans at 95% of deposits takes loans from Rs 95 crore to Rs 85.5 crore, so Rs 9.5 crore of lending runs off and liquid assets fall by Rs 0.5 crore to cover the rest of the outflow.
What makes this harder in practice than on paper?
Loans do not shrink on command. Term loans run off only as they repay, and calling them early harms the borrower and the bank's franchise. In the short run the bank has to meet the outflow from liquid assets or new funding, and a buffer of Rs 5 crore against a Rs 10 crore outflow is not enough. That gap is why liquidity rules ask banks to hold enough high quality liquid assets to survive a stressed outflow without selling loans.
The relationship
$\Delta L = \text{LDR} \times \Delta D = 0.95 \times (-10) = -9.5$
\Delta L the change in loans, Rs crore
\text{LDR} the loan-to-deposit ratio held fixed at 95%
\Delta D the change in deposits, a fall of Rs 10 crore
What it says in wordsWith the ratio fixed, loans change by the ratio times the change in deposits.
Say the limitation: a real bank also has equity and wholesale funding on the liability side, and it could replace lost deposits with borrowing at a higher cost. The puzzle shuts that door deliberately, to show how directly a deposit run reaches lending when no other funding is available.
Where candidates lose it
The quick wrong answer is Rs 10 crore, matching the deposit fall one for one. That ignores that the ratio is 95%, not 100%, so only 95 paise of lending goes for every rupee of deposits.
The more costly miss is stopping at the arithmetic. The interviewer wants to hear that loans cannot shrink overnight, so the outflow is met first from liquid assets, which is what a liquidity buffer is for.
What the interviewer asks next
If the bank instead keeps loans unchanged, what does its ratio become?
How much liquid asset buffer would it need to meet a 20% outflow without shrinking loans?
Why do regulators care about the speed at which different deposits can leave?
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
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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.
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
$\text{max position} = \frac{\text{capital}}{\text{haircut}} = \frac{20}{0.10} = 200 \qquad \text{sell} = 1{,}000 - 200 = 800$
capital the fund's own money in the position, Rs 20 crore
haircut the 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?
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
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How much does the spread add to the Rs 3 crore VaR?
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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.
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
$\text{LVaR} = \text{VaR} + \tfrac{1}{2} \times s \times P = 3.0 + 0.5 \times 0.0040 \times 200 = 3.4$
VaR market VaR on mid prices, Rs 3 crore
s bid-ask spread as a fraction of price, 0.40%
P position 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?
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 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
$D_E = \frac{A \cdot D_A - L \cdot D_L}{E} = \frac{1000 \times 4 - 900 \times 1}{100} = 31$
A, L, E assets 1,000, liabilities 900, equity 100, in Rs crore
D_A, D_L asset duration 4 and liability duration 1
D_E duration 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?

0.95^n	the share of deposits still in the bank after n days
0.12	the liquid assets, as a share of the original deposits

y_loan	yield on the loans that could have been made, 10%
EL	an assumed annual expected credit loss on those loans, 1.2%
y_liquid	yield on the liquid bonds, 6.8%

d	the fall in bond prices, 10%
pledged	the market value that must stay posted, Rs 3,200 crore

\Delta L	the change in loans, Rs crore
\text{LDR}	the loan-to-deposit ratio held fixed at 95%
\Delta D	the change in deposits, a fall of Rs 10 crore

Try it first

Why do shrinking withdrawals not save the bank?

What would a treasury risk manager add?

Where candidates lose it

What the interviewer asks next

Try it first

Why does the funding cost drop out of the answer?

Is 3.2 points the true gap?

Where candidates lose it

What the interviewer asks next

Try it first

Why does free collateral fall faster than prices?

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

Where candidates lose it

What the interviewer asks next

Try it first

Why does a deposit outflow become a lending cut?

What makes this harder in practice than on paper?

Where candidates lose it

What the interviewer asks next

Try it first

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

What happens next in a real market?

Where candidates lose it

What the interviewer asks next

Try it first

Why is ordinary VaR missing a cost?

What makes the adjustment larger than it looks?

Where candidates lose it

What the interviewer asks next

Try it first

Why is equity so much more sensitive than either side?

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

Where candidates lose it

What the interviewer asks next