Risk Management puzzles, solved step by step

About 24% to touch, against about 12% to finish below: roughly double. Twenty days at 2% a day is 8.94% of volatility. A 10% fall is ln 0.9, or 10.5% in log terms, which is 1.18 standard deviations, so the chance of ending below is 11.9%. The reflection principle doubles it for a touch: 23.9%.

Why is touching twice as likely as finishing below?

Picture a drunk walker on a pavement with a kerb one step to the left. By the end of the evening he may be well to the right, yet at some point he almost certainly stepped off the kerb. Paths wander. Once a driftless path has touched the level, it is equally likely to finish above it or below it, so for every path that finishes below there is a twin that touched and came back. This is the reflection principleFor a random walk with no drift, the path after it first hits a level is equally likely to be its mirror image about that level.: flip the path after the first touch and you swap an above-finisher for a below-finisher.

How do you get the numbers in the room?

Scale volatility by the square root of time: 2% times the square root of 20 is 8.94%. Work in log returns, because a 10% fall is a log move of 10.5%, not 10%. That is 1.18 standard deviations, and the normal table gives 11.9% below it. Double for the touch. If you skip the log step and use 10% flat you get 26.4%, close enough to earn credit if you say it is an approximation.

Where does the doubling rule stop being exact?

Two places. With drift the mirror twins are no longer equally likely, and a positive drift pulls the touch probability below double. And if the stop only triggers on daily closes, a path can dip through during the day and recover by the close, so fewer touches count; a standard correction for daily monitoring gives about 19%. The practical reading is that a stop-loss level is hit far more often than a finishing-price distribution suggests, which is why stops set from end-of-period VaR fire more than the desk expects.

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.

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.

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.

The expected return is 13.75%, and the trade breaks even at a default probability of 33.3%. The expected payoff is 0.85 x 100 plus 0.15 x 40, which is 91, against a price of 80. Breakeven solves 80 = 100 minus 60p: p is 20 over 60, one third. The real question is whether default odds could be that high.

Why is the breakeven more useful than the expected return?

The 15% is somebody's estimate, and estimates of default for a stressed issuer are soft. Expected return inherits every error in the 15%; the breakeven tells you how wrong the estimate can be before you lose money. It is like buying a second-hand car that needs repairs: rather than guess the repair bill, you ask how large the bill could be before the deal stops being worth it. Here default odds could more than double, from 15% to one in three, before the trade loses on average.

What would a credit risk manager add before approving the trade?

Three things. First, the recovery is also a guess, and it moves the breakeven: if holders recover 20 rather than 40, breakeven default odds fall from one third to 25%, because each default now costs 60 from the purchase price instead of 40. Second, money has a time value. If you require an assumed 7% return from cash for the year, breakeven falls from 33.3% to 24.0%, because the bond must beat cash, not zero. Third, the payoff is lopsided, +25% or -50%, so position size matters more than the average: a book of such bonds survives, a single large position may not.

About 0.25 years at most, and on average about an eighth of a year. Every coupon after the next one resets to the market rate, so a change in rates today is passed straight into future coupons and leaves their value unchanged. Only the next coupon is fixed, so the note behaves like a three-month bill: modified duration of about 0.25 just after a reset, falling towards zero before the next one.

Why does maturity not drive a floater's rate risk?

Think of renting a flat with the rent reset to the market every quarter. If rents jump tomorrow, your lease is not a bargain or a burden for more than a few months, because the next reset catches up. A bond loses value when rates rise only because its cash flows are fixed below the new market rate; a floater's cash flows are not fixed beyond the next reset. At each reset date the note is worth par again, provided the issuer's credit has not changed, so for rate purposes it is a claim to par plus one known coupon, three months away.

How do you check the number quickly?

Right after a reset, the note is worth the present value of par plus one coupon, received in a quarter. That is a single cash flow a quarter away, so Macaulay duration is 0.25 years and modified duration is 0.25 divided by 1.02, about 0.25. A day before the next reset it is almost zero. Compare a 7-year fixed bond paying 8% quarterly at par: its modified duration is 5.32, so a one point rate rise costs it about 5.3 per 100 against about 0.25 for the floater.

Now say where the 7 years still matter. The coupon is the market rate plus a fixed quoted marginThe fixed spread over the reference rate that a floating rate note pays, set at issue and unchanged for the life of the note.. If the issuer's credit worsens and investors demand a wider spread, that fixed margin is too low for all 28 remaining quarters, so the note's spread duration is close to a fixed bond's, several years. A treasury desk that files floaters under no rate risk and forgets spread risk has the right answer to the wrong question.

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%.

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.

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.

The price ends 4% lower, and the NPA ratio rose by 1 percentage point, which is a 50% increase. 100 up 20% is 120, and 20% off 120 is 24, leaving 96. The ratio moved from 2% to 3%: the gap is 1 percentage point, and 1 over the starting 2 is 50%. Saying it rose 1% would be wrong on both counts.

Why does up 20% then down 20% lose money?

A shopkeeper marks a shirt up 20% from Rs 100 to Rs 120, then runs a 20% off sale. The discount is taken on Rs 120, so it is Rs 24, and the shirt sells for Rs 96. Each percentage is measured on whatever base exists at the time, and the fall happens on a bigger base than the rise. The two moves multiply rather than add: 1.2 x 0.8 is 0.96. In general, up x then down x leaves you down x squared, here 0.2 x 0.2, or 4%.

What is the difference between a percentage point and a percent?

When the quantity is itself a percentage, there are two honest ways to describe a change. Percentage points measure the gap between two rates by subtraction; percent measures that gap relative to where you started. From 2% to 3% is +1 point by subtraction and +50% relative to 2. A gross NPA ratioNon-performing assets, loans on which the borrower has stopped paying for a set period, as a share of total loans before provisions. is a rate, so a risk report must say which one it means, and the two carry different messages: one point sounds mild; half as many bad loans again sounds serious.

In a risk committee, both descriptions are used and both can mislead. A desk that wants to play down deterioration quotes points; one that wants attention quotes percent. The disciplined habit is to quote the level and the change in points together, 3% from 2%, and let the reader see the relative move for themselves.

About 81.9% for one year and 36.8% for five years. Random arrivals at a steady average rate follow a Poisson process, where the chance of none in t years is e to the minus rate times t. One year gives e to the minus 0.2; five years gives e to the minus 1. So a loss is more likely than not over five years, 63.2%, yet most single years are quiet.

Why is one every five years not a schedule?

Buses that arrive at random, on average every twenty minutes, do not arrive every twenty minutes. Sometimes two come together, sometimes you wait an hour. A rate of 0.2 a year says how often losses happen on average, not when; each year is an independent draw, and most draws come up empty. Losses that arrive independently at a steady average rate follow a Poisson processA model for events arriving at random and independently at a constant average rate, such as large fraud losses or defaults in a big pool., and the chance of seeing none falls away exponentially with time.

What should an operational risk manager take from a quiet year?

Very little. A quiet year happens 82% of the time even when the risk is fully present, and three quiet years in a row still happen 55% of the time. A business that points to a clean record and asks for lighter controls is reading noise as evidence. The flip side is equally useful: when a loss arrives after a long gap, it is not proof that controls suddenly failed; the average gap here is 5 years, and gaps of ten years happen 13.5% of the time.

The limit to say out loud: the Poisson model assumes losses arrive independently at a constant rate. Fraud often clusters, one weak control exploited several times, and rates change as the business changes. The model is a baseline for reading the record, not a forecast.

Between a loss of Rs 52 and a gain of Rs 58 a share. The put guarantees a sale at 450 at worst; the sold call hands over anything above 560. Between the strikes the investor simply holds the stock. The pair costs Rs 12 minus Rs 10, a net Rs 2, so the outcome runs from 450 less 500 less 2, minus 52, to 560 less 500 less 2, plus 58.

What does each leg of the collar do?

A farmer worried about a price crash agrees with a trader: if prices fall below a floor, the trader pays the floor; in return, if prices soar above a ceiling, the farmer sells at the ceiling. The farmer gives up the dream harvest to remove the nightmare one. The bought put is the floor, the sold call is the ceiling, and the premium from the call pays for most of the put. That structure is a collarA position that holds a stock, buys a put below the current price and sells a call above it, locking the outcome between two strikes.: the investor still owns the stock between 450 and 560 and nothing outside it.

How do you check the two ends quickly?

Take one price in each region and walk through it. At 400, the stock is worth 400 and the put pays 50, so the holding is worth 450; the call expires worthless. At 620, the stock is worth 620 and the call costs 60, so the holding is worth 560. Whatever happens, the holding ends between 450 and 560, and subtracting the Rs 500 cost and the Rs 2 net premium gives the range of minus 52 to plus 58. Breakeven is Rs 502, the starting price plus the net premium.

Say what the collar does not do. It removes the tails; it does nothing for moves inside the band. And the cheap net premium is not free protection: the investor paid by selling every rupee of gain above 560. Whether that trade is sensible depends on what the investor needs, a floor for a known liability, say, rather than on the Rs 2.

One default expected, and about a 1.8% chance of four or more. The expected count is 50 x 2% = 1. The chance of none is 0.98 to the power 50, 36.4%; of exactly one, 37.2%; two, 18.6%; three, 6.1%. Those sum to 98.2%, so four or more is about 1.78%, roughly one year in 56.

Why is the expected count not enough to size the risk?

A school with 50 pupils, each with a 2% chance of being off sick on a given day, expects one absence. Most days it gets none, one or two; some days it gets four, and the class still has to run. An average of one default tells you what a normal year costs; it says nothing about how bad the bad year is, and capital exists for the bad year. Here, with Rs 10 crore in each name and a 60% loss on default, the expected loss is Rs 6 crore, but a four-default year costs Rs 24 crore.

How do you get 1.8% without a calculator?

Use the Poisson approximationFor many independent rare events, the count is close to a Poisson distribution with the same mean, so P(k) is about e to the minus mean times mean to the k over k factorial.. With a mean of 1, the chances of 0, 1, 2 and 3 are about 0.368, 0.368, 0.184 and 0.061. They add to about 0.981, so four or more is about 1.9%, within a whisker of the exact binomial 1.78%. Saying you are using the approximation, and why it works here, earns as much credit as the exact figure.

Then say the assumption that matters most. Independence is the weak link: names in the same sector or region default together in a downturn. With correlation, the expected count stays at one, but years with no defaults and years with many both become more common, and four or more can be several times more likely than 1.8%. That is why credit portfolio models spend most of their effort on correlation, not on the individual default probabilities.

Desk A's tail average is Rs 6.5 crore and desk B's is Rs 16.0 crore, so desk B is far riskier. Both desks have a 99% VaR of Rs 5 crore, which only says where the worst 1% of days begins. Averaging the losses beyond it, the expected shortfall, shows A's bad days cluster just past the line while B's run to Rs 30 crore.

Why does the same VaR hide such different risk?

Two rivers both have a flood mark at five metres that is crossed one year in a hundred. On one river, those floods reach six or seven metres; on the other, one of them reached thirty and washed the town away. The flood mark is the same; the town planner should care about the second river. VaR is the threshold of the bad days, not their size; two desks can share a threshold and have tails of completely different weight. Expected shortfallThe average loss on the days worse than VaR, so it measures the size of the tail rather than just where it starts. answers the question VaR leaves open: when it goes wrong, how wrong on average?

What does the ratio of expected shortfall to VaR tell you?

Divide one by the other. A's ratio is 1.3 and B's is 3.2. For normally distributed returns at 99%, expected shortfall is only about 15% above VaR, so a ratio of 3.2 is a flag that something is behaving far from normal: an option position losing faster as the market moves, a concentrated name gapping, or a liquidity cliff. Desk A looks close to normal; desk B needs its positions read line by line.

State the limit honestly. Three observations make a fragile average: one more bad day for desk A, or one fewer outlier for desk B, would move the numbers a lot. Expected shortfall is also harder to backtest than VaR, because you are checking an average of rare events rather than a count of breaches. It is still the better answer to the question asked.