Risk Management puzzles, solved step by step

Chen breached the limit, and Bilal is the one telling the truth. Assume each trader in turn is the culprit and count the true statements. Asha as culprit makes two statements true, Bilal's and Chen's. Bilal as culprit also makes two true, Asha's and Chen's. Only Chen as culprit leaves exactly one true statement, Bilal's denial, which matches the rules.

Where do you start when every statement might be a lie?

Think of a lost house key and three rooms. Rather than arguing about who saw what, you check each room in turn and stop when one fits. With three possible culprits, the fastest route is to assume each one did it, work out which statements are then true, and keep the case that matches the rule of exactly one truth. That turns an argument about who is lying into three short checks.

Is there a shortcut that finds it faster?

Yes: look for two statements that cannot both be false. Asha says Bilal did it and Bilal says he did not, so exactly one of those two is true whoever the culprit is. That uses up the single true statement, which means Chen's statement must be false. Chen saying he did not do it is false, so Chen did it. Saying this pairing out loud shows the interviewer you look for structure before grinding through cases.

Then connect it to the job, in one line. A limit breach investigation often starts with conflicting accounts, and the same discipline applies: list the possibilities, check each against the hard evidence, and discard the ones that contradict it. The trade records play the part of the rule here; the accounts are only as good as what they agree with.

About 21%, and volatility would need to fall to about 6.2%. A loss means a return below zero, which is 8 points, or 0.8 standard deviations, under the mean. About 21.2% of a normal distribution lies below that, roughly one year in five. For a 10% chance, zero must sit 1.28 standard deviations below the mean, so volatility must be 8 divided by 1.28, about 6.2%.

How do a return target and a volatility target fix the chance of loss?

Think of a commute that takes 40 minutes on average but varies from day to day. Whether you are ever late for a 50 minute deadline depends on how much it varies, not only on the average. The chance of a losing year depends on how many standard deviations the expected return sits above zero: here 8 divided by 10, which is 0.8. Look up 0.8 in the normal table and about 21.2% of years fall below zero. The investor who hears 8% and thinks losses are rare is wrong one year in five.

What would you actually set as targets, and what is wrong with this model?

Set the targets as a pair, and state the trade-off. If the investor cannot tolerate losing more than one year in ten, then either volatility must come down to about 6.2%, which usually lowers the expected return too, or the loss tolerance must be stated over a longer horizon. Over five years the mean grows five times but the volatility only by the square root of five, so the chance of a losing five-year stretch is much lower. Asking about the horizon is the question a good risk manager raises first.

Then name the model's limits. Real returns have fatter left tails than a normal curve, so the chance of a large loss is understated; returns are not independent from year to year; and the 8% expected return is an assumption, not a promise. A drawdown limit, such as no more than a 15% fall from peak, is often more useful to an institution than a probability of a losing year.

About 95% confidence, and a VaR of about Rs 3.3 crore. A month has about 21 trading days, so one breach a month is a tail of 1 in 21, about 4.8%, a confidence level of about 95.2%. For a normal P&L that cut sits 1.67 standard deviations out, so the VaR is 1.67 times Rs 2 crore, about Rs 3.34 crore. At a round 95% it is Rs 3.29 crore.

How do you turn a frequency into a confidence level?

Think of a weather forecast that says a heavy rain day comes about once a month. That is the same statement as saying about 1 day in 30 is that wet. A VaR confidence level is just a frequency of bad days written as a percentage: one breach in 21 trading days is a 4.8% tail, which is a 95.2% confidence level. Counting trading days matters; using 30 calendar days would make the tail 3.3% and the level about 96.7%.

Why is this translation useful on a desk?

Because a desk head can act on a frequency and cannot act on a percentage. Saying we expect to lose more than Rs 3.3 crore about once a month, roughly 12 days a year, tells the head of the desk what to expect and when to worry. It also sets up the backtest: if losses beyond Rs 3.3 crore start happening three or four times a month, the model is understating risk, and that is visible within a quarter rather than a year.

Add the limits. The VaR says nothing about how bad the bad day is; a loss of Rs 3.4 crore and one of Rs 10 crore both count as one breach. The normal assumption also understates fat tails, so the real one-in-21 loss may sit further out than Rs 3.3 crore. And the Rs 2 crore standard deviation is itself an estimate that moves with market conditions.

RAROC is 15.0%, so the loan clears a 14% hurdle, but only just. Income of Rs 3 crore less Rs 0.8 crore of cost and Rs 1 crore of expected loss leaves Rs 1.2 crore of risk-adjusted profit. Divided by Rs 8 crore of economic capital that is 15.0%. After a 14% charge on the capital, Rs 1.12 crore, the loan adds only Rs 0.08 crore a year.

Why does expected loss sit in the numerator and unexpected loss in the denominator?

Think of a restaurant that knows about two plates in a hundred get dropped. That breakage is priced into the menu; it is a running cost. What the owner keeps cash aside for is the night a whole shelf comes down. Expected loss is the average, so it is a cost charged against income; unexpected loss is the bad year above the average, so it is covered by capital. RAROCRisk-adjusted return on capital: profit after costs and expected loss, divided by the capital held against unexpected loss. puts each in its proper place, which is why a bank can compare a thin-margin safe loan with a fat-margin risky one on the same scale.

How much room is there before the loan fails the hurdle?

Turn the hurdle into rupees. A 14% return on Rs 8 crore of capital is Rs 1.12 crore a year, and the loan earns Rs 1.2 crore, so it clears by Rs 0.08 crore. A margin that thin means one small change flips the answer. If expected loss rises from Rs 1.0 crore to Rs 1.08 crore, or economic capital rises from Rs 8 crore to Rs 8.57 crore, RAROC falls to exactly 14%. Saying those two breakeven numbers is what turns a ratio into a credit judgement.

State the limits too. This is a pre-tax, single-year figure; many banks compute RAROC after tax and add the return earned on the capital itself, which would lift the number. And the Rs 8 crore depends on the bank's own capital model, so the answer is only as good as that model's view of this borrower.

About 5.73%. Tracking error is the volatility of the fund's return minus the benchmark's. Its variance is 18 squared plus 16 squared minus 2 x 0.95 x 18 x 16, which is 324 plus 256 minus 547.2, or 32.8. The square root is 5.73%, nearly three times the 2-point gap in volatilities.

What exactly is tracking error measuring?

Two friends walk to the same office. How far apart they are at any moment depends less on how fast each walks than on whether they take the same streets. Tracking error is the volatility of the return difference, fund minus benchmark, so it depends on how much the two move apart, not on how much each moves. The fund and index can both swing wildly and still track closely if they swing together. That is why the formula needs the correlationA number from minus 1 to 1 describing how closely two returns move together; 1 means perfect lockstep., not just the two volatilities.

Why does the correlation matter more than the volatilities?

Look at the table in the figure. Keeping 18 and 16 fixed, moving correlation from 0.99 to 0.90 takes tracking error from 3.12% to 7.85%, more than doubling it. At high correlations each hundredth of correlation moves tracking error a lot, because the large shared term 2 x rho x 18 x 16 almost cancels the two variances. Now hold correlation at 0.95 and give both sides 16% volatility: tracking error is still 5.06%. The volatility gap contributes a little; the imperfect correlation contributes most.

Close with the limit. The formula uses a correlation estimated from history, and correlations drift, often falling in stressed markets. A fund reporting 5.7% tracking error in calm years can run well above it in a sell-off, so a risk team watches realised tracking error alongside the model figure.

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 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.

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.

Forecaster A, with a mean squared error of 5 against 6.25. Mean squared error splits into bias squared plus variance. A pays 1 squared for its bias and 2 squared for its spread, 5 in all. B pays nothing for bias but 2.5 squared, 6.25, for its spread. A's typical error, the square root, is 2.24 against 2.50.

How can a biased forecaster beat an unbiased one?

Two archers. One groups every arrow tightly but slightly left of centre; the other is centred on average but scatters arrows all over the target. Ask which one lands closer to the bullseye on a typical shot, and the tight grouping wins. Mean squared error charges for two things, how far off you are on average and how much you scatter, and a small, steady bias can cost far less than a large scatter. Being unbiased only removes the first charge.

When would your answer flip, and what would you do with A?

Solve for the tie: A matches B when bias squared plus 4 equals 6.25, so a bias of 1.50. Below that, A wins. More useful still, a bias that is stable can be measured and subtracted: correct A by one point and its MSE falls to 4, better than either original. That is the practical lesson for a risk team: a model that is consistently off in one direction is fixable, while a noisy model is not. The trade-off is also why risk teams use shrinkagePulling a noisy estimate towards a simpler, steadier target, accepting a little bias in return for much lower variance. on covariance matrices built from short histories.

The limit: MSE punishes large errors heavily because it squares them, and it treats over-forecasts and under-forecasts alike. A risk manager forecasting losses may care more about under-forecasting than over-forecasting, in which case a symmetric score is the wrong yardstick and the ranking could change.