Risk Management puzzles, solved step by step

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

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004Your exposure to a counterparty is Rs 50 crore. The collateral agreement has a Rs 20 crore threshold and a Rs 2 crore minimum transfer amount, and you already hold Rs 25 crore of collateral. How much collateral do you call?Counterparty exposure and collateralCoreCounterparty risk
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How much do you call?
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Call Rs 5 crore. The agreement requires collateral on exposure above the threshold: Rs 50 crore minus Rs 20 crore is Rs 30 crore. You hold Rs 25 crore, so the shortfall is Rs 5 crore. That is above the Rs 2 crore minimum transfer amount, so the call goes out. Even afterwards, Rs 20 crore of exposure stays unsecured.
What does each term in the agreement do?
Think of a shopkeeper who lets a regular customer run a tab of up to Rs 2,000 before asking for anything, and who does not bother collecting amounts under Rs 200. The thresholdThe level of exposure below which no collateral is required under the collateral agreement. is credit you have chosen to extend without security, and the minimum transfer amount stops tiny calls that cost more to process than they protect. The collateral you are owed is exposure minus threshold. The call is that amount minus what you already hold, sent only if it clears the minimum.
Of Rs 50 crore of exposure, the first Rs 20 crore is the agreed threshold, Rs 25 crore is covered by collateral already held, and the last Rs 5 crore is the call, which clears the Rs 2 crore minimum transfer amount.
Why does the answer not reach Rs 25 crore?
Because you signed away the first Rs 20 crore. A threshold is unsecured exposure that the credit team approved when the agreement was negotiated, usually because the counterparty was strong, and it stays unsecured until the agreement is renegotiated. After the call lands you hold Rs 30 crore against Rs 50 crore of exposure, and the remaining Rs 20 crore is exactly the size of the threshold. A good answer says that residual number, because it is what the credit limit has to cover.
Add one practical point. The exposure number is itself a valuation that the counterparty may dispute, and collateral usually carries a haircut, so Rs 25 crore of bonds may count as less than Rs 25 crore. Rounding conventions can also change the call by a small amount. None of that changes the method: required equals exposure minus threshold, call equals required minus held, sent only above the minimum.
Where candidates lose it
The common wrong answer is Rs 25 crore: exposure minus collateral held, with the threshold forgotten. It asks the counterparty for more than the agreement allows, and in a real call it starts a dispute you will lose.
The opposite miss is saying nothing is due because Rs 25 crore is already more than the Rs 20 crore threshold. The threshold is subtracted from the exposure, not compared with the collateral.
What the interviewer asks next
The exposure drops to Rs 46 crore the next day. What happens to the collateral you hold?
Your counterparty is downgraded and the threshold falls to zero. What is the call now?
Why might a bank accept a high threshold from one counterparty but not another?
016A swap has an expected positive exposure of about Rs 5 crore across its five-year life. The counterparty's credit spread is 200 basis points. Roughly what is the credit valuation adjustment?Counterparty exposure and collateralHardCounterparty riskQuant risk
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Which rough estimate is closest?
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Roughly Rs 45 lakh. The credit spread is what the market charges each year for bearing the counterparty's default risk. Rs 5 crore of expected exposure at 2% is Rs 10 lakh a year. Over five years that is Rs 50 lakh, and discounting at an assumed 4% brings it to about Rs 44.5 lakh. That is the value you give up for trading with a counterparty that can default.
Why does a spread times an exposure give you CVA?
Think of a friend who owes you money and a moneylender who would charge that friend 2% more a year than a safe borrower. That 2% is the market's price for the chance your friend does not pay. The credit valuation adjustmentThe reduction in the value of a derivative to reflect the chance that the counterparty defaults while owing you money. is roughly the counterparty's credit spread charged on the amount it is expected to owe you, year by year, discounted to today. The spread already folds together the chance of default and the loss if it happens.
Rs 5 crore of expected exposure charged at a 2% credit spread costs Rs 10 lakh a year, and discounting each year at 4% gives Rs 9.62, 9.25, 8.89, 8.55 and 8.22 lakh, a rough CVA of about Rs 44.5 lakh.
The relationship
$\text{CVA} \approx s \sum_{t=1}^{5} \text{EPE}_t \, D_t = 0.02 \times 5 \times \sum_{t=1}^{5} 1.04^{-t} \approx 0.445 \text{ crore}$
s the counterparty's credit spread, 2% a year
EPE_t expected positive exposure in year t, Rs 5 crore
D_t the discount factor for year t, at an assumed 4%
What it says in wordsCharge the spread on each year's expected exposure, discount it to today, and add the years.
What does the rough version leave out?
Three things, each worth a sentence. First, the rough formula ignores that a counterparty that defaults in year two cannot default again in year three, so it slightly overstates the charge. Modelled explicitly, with a 60% loss given default and a constant default rate implied by the spread, the CVA comes to about Rs 41 lakh. Second, exposure on a swap is rarely flat; it usually rises and then falls as payments are made. Third, if the counterparty's credit worsens exactly when your exposure grows, the charge is larger, which is wrong-way risk.
The discount rate here is an assumption for the illustration, and the market spread would come from the counterparty's bonds or credit default swaps where they trade. The structure is what the interviewer wants: exposure profile, times default cost, discounted and summed.
Where candidates lose it
The trap is answering Rs 10 lakh, one year of the spread, or treating the spread as a one-off probability of losing the whole exposure. The spread is an annual rate, and the exposure lasts five years.
The second loss is not naming the assumptions. Say that you used a flat exposure, a 4% discount rate and the spread as a stand-in for default probability times loss given default.
What the interviewer asks next
How does a collateral agreement with a zero threshold change this CVA?
The exposure rises from Rs 2 crore in year one to Rs 8 crore in year five, with the same average. Is CVA higher or lower?
What is debit valuation adjustment, and why do some people dislike booking it as profit?
029A fully collateralised netting set's value moves with a daily volatility of Rs 2 crore. Assuming a 10-day margin period of risk and normally distributed moves, what is the 99% potential future exposure?Counterparty exposure and collateralCoreCounterparty riskBank market risk
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Which number do you multiply the daily Rs 2 crore by to reach ten days?
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About Rs 14.7 crore. Collateral covers the value up to the last margin call, so exposure is the move during the 10 days it takes to notice the default and close out. Ten days of Rs 2 crore daily volatility is 2 x root 10, Rs 6.32 crore. At 99%, 2.33 standard deviations, potential future exposure is Rs 14.71 crore.
If the trades are fully collateralised, where does any exposure come from?
A landlord holding a month's deposit is covered for last month's rent. If the tenant stops paying and takes three months to evict, the deposit covers one of them. Collateral protects you up to the last margin you actually received; the margin period of riskThe time from the last successful collateral exchange with a defaulting counterparty until its trades are closed out or re-hedged. is the gap after that, while the defaulter pays nothing and the market keeps moving. Ten days covers the missed call, the dispute, the formal default and the close-out.
From the last margin received, the 99% edge of the exposure fan opens with the square root of time, reaching Rs 4.65 crore after one day and Rs 14.7 crore after the 10-day margin period of risk, which is the exposure the collateral never covered.
Why the square root of ten and not ten?
Each day's move is independent, so some days undo others. Variances add across independent days, which makes volatility grow with the square root of time. Ten days is 2 x 3.16, Rs 6.32 crore, and 99% is 2.33 of those. Scaling linearly would give Rs 46.5 crore, more than three times too big, and a counterparty limit set on it would turn away business the bank could safely do.
The relationship
$\text{PFE}_{99\%} = z_{0.99}\,\sigma_{1d}\sqrt{\text{MPOR}} = 2.33 \times 2 \times \sqrt{10} \approx 14.7$
z_{0.99} the 99% one-sided normal quantile, 2.33
\sigma_{1d} daily volatility of the netting set value, Rs 2 crore
MPOR margin period of risk, 10 days
What it says in wordsExposure at a confidence level is that many standard deviations of the move over the days you are unprotected.
Name what the simple version leaves out. Only moves in your favour are exposure, so the lower half of the fan does not count. Collateral thresholds or a minimum transfer amount add a fixed layer on top. And a counterparty that defaults in a crisis defaults when volatility is high, so Rs 2 crore a day is likely too calm. Double the margin period to 20 days, as can happen with disputes or illiquid trades, and the figure rises to Rs 20.8 crore, not Rs 29.4 crore.
Where candidates lose it
Two ways to lose it. Some candidates say zero, because the set is fully collateralised, and miss that collateral only covers values already agreed. Others multiply by 10 instead of the square root of 10 and land on Rs 46.5 crore.
Say the timeline before the formula: last margin in, default, close-out. Once the interviewer hears that you know why the ten days exist, the arithmetic is a formality.
What the interviewer asks next
The counterparty has a Rs 5 crore threshold before it posts collateral. What is the PFE now?
Why might a regulator require a longer margin period of risk for some netting sets?
What is wrong-way risk, and how would it change this number?
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
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Without netting, what does the bank stand to lose if the counterparty defaults today?
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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 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
$E_{\text{gross}} = \sum_i \max(V_i,0) = 45 \qquad E_{\text{net}} = \max\Big(\sum_i V_i,0\Big) = 15$
V_i the 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?
067A forward contract's value to the bank at the one-year horizon is normally distributed with mean zero and standard deviation Rs 10 crore. What is the expected positive exposure to the counterparty at that date?Counterparty exposure and collateralCoreCounterparty riskQuant risk
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The expected value of the forward is zero. What is the expected exposure?
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About Rs 4.0 crore, or 0.4 standard deviations. Exposure is the forward's value when it is positive and zero when it is negative. For a normal distribution with mean zero, the expected value of the positive part is sigma divided by the square root of 2 pi, about 0.399 x Rs 10 crore. Half the outcomes contribute nothing; the other half average about Rs 8 crore.
Why is the exposure positive when the expected value is zero?
Lending a friend your bicycle is a one-sided risk: if the friend loses it, you are out a bicycle; if the friend buys you a better one, you gain nothing from the loan itself. Counterparty exposure has the same shape. When the contract is in your favour, a default costs you its value; when it is against you, you still owe the full amount, so negative outcomes count as zero, not as a gain. Cutting off the left half leaves a positive average.
The forward's value in one year is centred on zero with a standard deviation of Rs 10 crore, but only the shaded right half is exposure. Counting the left half as zero gives an expected exposure of about Rs 4.0 crore, 0.4 standard deviations, while the 97.5% potential future exposure is about Rs 19.6 crore.
Where does 0.4 come from, and what is it called?
Integrating v times the normal density from zero to infinity gives sigma times the density at zero, which is 1 over the square root of 2 pi, about 0.399. At a single future date desks usually call this number the expected exposure; averaged over the life of the trade it becomes expected positive exposure, the input to credit valuation adjustment. The tail measure, potential future exposureA high percentile of the exposure distribution at a future date, used to size limits on how much a counterparty could owe. at 97.5%, is 1.96 sigma, about Rs 19.6 crore.
The relationship
$EE = E[\max(V,0)] = \int_0^{\infty} v\,\phi(v)\,dv = \frac{\sigma}{\sqrt{2\pi}} \approx 0.399 \times 10 = 3.99$
V the forward's value to the bank at the horizon
\phi the normal density with mean zero and standard deviation sigma
\sigma Rs 10 crore
What it says in wordsExpected exposure averages only the positive outcomes, which for a zero-mean normal is about 0.4 standard deviations.
The limitation: this assumes no collateral and no netting with other trades. A daily margin agreement would cut the exposure to the move over the few days it takes to call and receive margin, which is far smaller than a year's worth of drift.
Where candidates lose it
The trap is answering zero because the forward is fair and its expected value is zero. Exposure is an option-like quantity: it keeps the upside of the value and discards the downside, so it is always at least zero on average.
The second slip is quoting Rs 8 crore, the average of the positive half alone. That forgets the half of outcomes where exposure is zero. Average across all outcomes: about Rs 4 crore.
What the interviewer asks next
If the forward's expected value were Rs 5 crore instead of zero, would the expected exposure rise by more or less than Rs 5 crore?
How does daily collateral change this number?
Why does exposure on an interest rate swap rise and then fall over its life?
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
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The bonds fall 10%. Are you still covered?
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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.
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
$h \approx z_{99\%} \cdot \sigma_{daily} \cdot \sqrt{t} = 2.33 \times 1\% \times \sqrt{10} \approx 7.4\%$
h the haircut
z 2.33, the one-sided 99% point of a normal distribution
t days 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?
092A counterparty has a 2% probability of default. Your exposure to it is Rs 10 crore in normal states, but in the states where it defaults the exposure is Rs 40 crore. With 60% loss given default, compare the expected loss with and without that link.Counterparty exposure and collateralHardCounterparty riskQuant risk
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How much larger is the expected loss once exposure and default are linked?
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Rs 12 lakh if independent, Rs 48 lakh when linked: four times as much. Independent: 2% times 60% times Rs 10 crore. Linked: the defaults happen exactly when exposure is Rs 40 crore, so 2% times 60% times Rs 40 crore. That link is wrong-way risk. Averaging exposure first gives only about Rs 12.7 lakh and misses it.
Why does the link between exposure and default matter so much?
Insuring a house against fire with an insurer whose only office is in the same building is poor protection: the insurer is least able to pay in exactly the event you insured against. Expected loss is probability of default times loss given default times the exposure at the moment of default, and wrong-way risk is when that exposure is largest in the states where the counterparty fails. Here the default probability and the recovery are unchanged; only the exposure in the bad states moved, and the expected loss quadrupled.
With the same 2% default probability and 60% loss given default, exposure of Rs 10 crore in every state gives an expected loss of Rs 12 lakh, but exposure that jumps to Rs 40 crore in the default states gives Rs 48 lakh, four times as much.
The relationship
$EL = PD \times LGD \times E[\text{exposure} \mid \text{default}] \qquad 0.02 \times 0.6 \times 40 = 0.48$
PD probability of default, 2%
LGD loss given default, 60%
E[exposure | default] the expected exposure in the states where default happens, Rs 40 crore
What it says in wordsUse the exposure conditional on default, not the average exposure, because only the default states produce a loss.
Where does this happen in practice?
Whenever the trade and the counterparty share a driver. A bank that sells a put on a country's currency to a bank in that country gains exposure just as that bank weakens. A lender taking a company's own shares as collateral sees the collateral fall as the company fails. The general signal is a trade whose value to you rises in the same scenario that damages the counterparty. A limit system that measures average exposure, about Rs 10.6 crore here, reports the position as a Rs 10 crore line and misses where the loss sits.
Say how desks handle it. Specific wrong-way trades are flagged and either priced with the conditional exposure, collateralised more heavily or refused. General wrong-way risk, from shared exposure to a market factor, is harder to see and needs a stress scenario in which the counterparty and the market move together.
Where candidates lose it
The common error is averaging first: 98% of 10 plus 2% of 40 is Rs 10.6 crore, and 2% times 60% of that is about Rs 12.7 lakh, barely above the independent case. It treats the link as a small adjustment when it is the whole story.
The other miss is changing the default probability instead of the exposure. The link is in the exposure; the default rate is the same in both trees.
What the interviewer asks next
Give an example of right-way risk, where exposure falls as the counterparty weakens.
How would collateral posted by the counterparty change the Rs 48 lakh?
How do banks reflect wrong-way risk in CVA?

s	the counterparty's credit spread, 2% a year
EPE_t	expected positive exposure in year t, Rs 5 crore
D_t	the discount factor for year t, at an assumed 4%

z_{0.99}	the 99% one-sided normal quantile, 2.33
\sigma_{1d}	daily volatility of the netting set value, Rs 2 crore
MPOR	margin period of risk, 10 days

V_i	the current value of trade i from the bank's side
\max(\cdot,0)	only amounts owed to the bank count as exposure

V	the forward's value to the bank at the horizon
\phi	the normal density with mean zero and standard deviation sigma
\sigma	Rs 10 crore

Try it first

What does each term in the agreement do?

Why does the answer not reach Rs 25 crore?

Where candidates lose it

What the interviewer asks next

Try it first

Why does a spread times an exposure give you CVA?

What does the rough version leave out?

Where candidates lose it

What the interviewer asks next

Try it first

If the trades are fully collateralised, where does any exposure come from?

Why the square root of ten and not ten?

Where candidates lose it

What the interviewer asks next

Try it first

Why does netting cut the exposure by two thirds?

What happens to the negative trades without netting?

Where candidates lose it

What the interviewer asks next

Try it first

Why is the exposure positive when the expected value is zero?

Where does 0.4 come from, and what is it called?

Where candidates lose it

What the interviewer asks next

Try it first

What is a haircut, in plain terms?

How would a risk team have chosen 8%?

Where candidates lose it

What the interviewer asks next

Try it first

Why does the link between exposure and default matter so much?

Where does this happen in practice?

Where candidates lose it

What the interviewer asks next