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?
Show the worked solution
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?
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
Try it first
Which number do you multiply the daily Rs 2 crore by to reach ten days?
Show the worked solution
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?
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?
Show the worked solution
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?

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	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

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 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