Risk Management puzzles, solved step by step

About 0.8% get through if the checkers are truly independent. Each misses 20% of what reaches them, so 100 errors become 20, then 4, then 0.8. The real figure is likely higher because the checkers are not independent: later checkers relax because someone already looked. If checkers two and three catch only 40%, 7.2% get through, nine times as many.

Why do independent checks multiply?

Think of three sieves stacked on top of each other. If each lets one grain in five through, the first passes 20 of 100 grains, the second 4 of those 20, and the third 0.8 of the 4. When checks are independent, the chance of an error getting past all of them is the product of each one's miss rate. That is why a three-check process looks almost watertight on paper: 0.2 cubed is 0.008.

Why is the real leakage higher than 0.8%?

Because people who know others are checking check less carefully. The second checker assumes the first caught the obvious problems, and the third signs off because two colleagues already did. Checks that rely on each other are not independent, and once they are correlated the multiplication rule overstates how much they catch. There is also a common cause: an error that looks like a normal payment, a correct-looking amount to a familiar name, fools all three checkers for the same reason.

What would an operational risk team do about it?

Measure each check on its own, not the chain. Seed known errors into the flow and count what each checker catches; rotate who checks first; give each checker a different thing to look for rather than the same whole payment. The goal is to make the checks genuinely independent, because three checks that behave like one are more dangerous than one check everyone knows is the only one.

About Rs 40. The typical one-year move is 20% of Rs 1,000, Rs 200. Over three months it is Rs 200 times the square root of 0.25, Rs 100. The call is worth about 0.4 of that typical move, Rs 40. A full Black-Scholes calculation at zero rates gives Rs 39.88, so the rule is within a rupee.

Why is the premium a slice of the typical move?

Imagine insuring a shop's daily takings against a bad day. The premium depends on how much the takings usually swing, not on how large the takings are. An at-the-money option pays out on the upside half of the stock's moves, so its value is proportional to the size of a typical move over the option's life, not to the stock price itself. The price enters only through converting volatility into rupees.

Where does 0.4 come from, and when does the rule fail?

If the stock's move is roughly normal with standard deviation of one typical move, the average payoff from the upside half is that deviation divided by the square root of 2 pi, 0.399. The 0.4 is not a fudge factor; it is the average size of the positive half of a normal move. The rule is built for at-the-money options with short expiries and low rates. It fails for options far in or out of the money, where the payoff is mostly intrinsic value or mostly zero, and over long horizons, where interest rates and the skew of returns start to matter.

A risk manager uses this to sanity-check a trader's mark in ten seconds. If a three-month at-the-money call on a Rs 1,000 stock is marked at Rs 70, the mark implies volatility of about 35%, which is a question worth asking.

About 27.2%. One manager beats the index five years running with probability one half to the fifth, 1 in 32. The chance that none of the ten does it is 31/32 to the power 10, about 0.728. So at least one streak appears 27.2% of the time with no skill anywhere. With 100 managers it is 95.8%.

Why is a streak among many managers weak evidence?

Ask a hall of 300 people to toss a coin five times, and about nine will throw five heads. Nobody would call them skilled tossers. The chance that one named person has a streak is small, but the chance that someone in a crowd has one is large, and a fund manager with a five-year record is usually picked from a crowd. Which question you are answering decides everything: the probability for manager G, or the probability for whoever turned out to have the streak.

How do you get there fast without a calculator?

Use the expected count first: ten managers times 1 in 32 is about 0.31 streaks. When the expected count is small, the chance of at least one is a little below it, so 31% is an upper bound and the exact answer is about 27%. The gap comes from the chance of two or more streaks, which the simple sum counts twice. Saying the bound and then the exact figure shows the interviewer you can check your own work.

What does an asset manager's risk team do with this?

It sets the bar for evidence. A five-year record of beating the index is common among hundreds of funds even if none has skill, so the team looks at how the returns were earned: the size of the edge against its volatility, whether it comes from one bet or many, and whether the process explains the outcome. Survivorship matters too: the managers whose streaks broke often closed their funds, so the crowd you see is already filtered toward winners.

Not precisely enough to claim skill. The standard error of the mean is the daily volatility over the square root of the number of days: 1% over the square root of 250, about 0.063%. The estimate of 0.05% is only 0.79 standard errors from zero, and a two standard error band runs from -0.076% to 0.176%. You would need about 1,600 days, 6.4 years, to clear zero.

Why is a year of daily data not enough?

Weigh yourself on a bathroom scale that jumps by two kilos each time you step on it. If you lost 100 grams last month, a week of readings will not show it; the jumps drown the signal. The precision of an average improves only with the square root of the number of observations, so a small edge buried in large daily noise takes years to show. Here the daily noise is 1% and the edge is 0.05%, twenty times smaller.

How long would it take, and what does that mean for judging desks?

Set the t-statistic to 2 and solve for n: n equals (2 times 1% over 0.05%) squared, which is 1,600 days. An annual Sharpe ratio of about 0.79 needs more than six years of data before it is statistically distinguishable from zero. That is longer than most desks keep the same strategy, so a risk team cannot rely on the P&L record alone; it looks at whether the edge has a reason, whether it survives out of sample, and how much the result depends on a few days.

State the assumptions. The calculation treats daily returns as independent with constant volatility. Fat tails and volatility clustering make the true uncertainty larger, so six years is a floor, not a promise.

Bank B, with about 6.05% of assets in equity against 3.5% for bank A. Equity to assets is the CET1 ratio times the share of assets that count as risk-weighted: 14% times 25% for A, 11% times 55% for B. A's higher ratio rests on low risk weights, so it is levered about 29 times against B's 16.5 times.

Why can a higher capital ratio mean less capital?

Two people each say they save 20% of their income. One counts only the salary left after rent; the other counts the whole salary. The first sounds equally thrifty but saves far fewer rupees. A CET1 ratio divides equity by risk-weighted assets, not total assets, so a bank that assigns low risk weights to its loans can show a high ratio on a thin equity base. The share of assets that ends up risk-weighted is called {term('RWA density', 'Risk-weighted assets divided by total assets; a low figure means the bank judges most of its assets to be low risk.')}, and it is the number that reconciles the two views.

Which number should a credit analyst trust?

Both, because they answer different questions. The CET1 ratio is right if the risk weights are right. Equity to assets makes no judgement about risk, so it is the check on whether the risk weights are doing too much of the work. Test A's number: if its true risk density were 40% rather than 25%, perhaps because its internal models are optimistic, its CET1 ratio would be 3.5% over 40%, only 8.75%. That is why regulators set a leverage ratio floor alongside risk-based ratios; the current minimum is set by the regulator, so confirm it before quoting it.

What would make bank A's low density legitimate?

A book of home loans with low loan-to-value ratios, or large holdings of government bonds, genuinely carries low risk weights, and a 25% density can be honest. The question is whether the weights are earned. An analyst asks what the assets are, whether the density has fallen while the book stayed the same, and how the bank's model-based weights compare with the standardised ones.

25%. Drawdown is measured from the highest value reached so far. The fund peaks at 120 and falls to 90, a 25% drop. It then peaks at 130 and falls to 100, a 23.1% drop. The larger of the two, 25%, is the maximum drawdown. Measured from the start the worst point looks like only 10%, which understates the pain.

Why measure from the running peak?

If your savings climbed to Rs 1.2 lakh and then fell to Rs 90,000, you would not console yourself that you started with Rs 1 lakh. You lost Rs 30,000 of money you had. Drawdown measures the fall from the highest value an investor has held so far, because that is the loss an investor who bought at the top actually suffers. The running peak resets upward each time the fund makes a new high, and each drawdown is measured against it.

What does the number not tell you?

Two things worth saying. Maximum drawdown is a single worst episode, so it depends heavily on the sample: a longer history can only make it larger, never smaller. It also ignores time. The fall from 120 to 90 took one period and the recovery took one more; a fund that takes three years to climb out of a 25% hole is a different experience from one that recovers in a quarter. A risk team reports duration of the drawdown and time to recovery alongside the depth, and remembers that a 25% fall needs a 33.3% gain to get back.

About 14.3% in B, giving a minimum volatility of about 8.78%. The minimum variance weight in B is A's variance less the covariance, over the sum of the variances less twice the covariance: 160 over 1,120. The portfolio is less risky than A alone for any weight in B up to 28.6%, even though B is three times as volatile.

How can a riskier asset lower total risk?

An umbrella seller and an ice cream seller at the same beach each have bad days, but rarely the same bad days. A shopkeeper who sells mostly ice cream and a few umbrellas has steadier takings than one who sells only ice cream, even if umbrella sales swing wildly. For a small weight, the risk an asset adds depends on how it moves with what you already hold, and a negatively correlated asset offsets part of the existing risk. B's own variance enters with the square of its weight, so at small weights it barely registers, while the offset enters in proportion to the weight.

How do you check the minimum volatility?

Plug the weight back in: 0.857 squared times 100, plus 0.143 squared times 900, plus twice 0.857 times 0.143 times minus 60. That is 73.5 plus 18.4 less 14.7, about 77.1, whose square root is 8.78%. A shortcut confirms it: the minimum variance for two assets is the product of the variances times one minus rho squared, over the same denominator, 90,000 times 0.96 over 1,120, which is 77.1. Two routes agreeing is the check the interviewer is listening for.

Then give the limit. The answer rests on a correlation estimate, and correlations drift. If the correlation moved to zero, the best weight would fall to 10% and the minimum would rise to about 9.5%; if it turned positive in a stress, the benefit would shrink further. A risk manager treats the minimum variance weight as a range, not a point.

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.

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.

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.

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.

Rs 63 crore. The monthly losses double: 1, 2, 4, 8, 16 and 32 crore, and they add to 2 to the power 6, less 1, which is 63. Each month's loss is one more than everything before it combined, so the hole grows as fast as the bets. For a control function, the pattern is the signal: position size growing geometrically with no matching approved limit.

Why does doubling down grow the hole so fast?

A gambler at a roulette table who doubles his bet after every loss plans to recover everything with one win. After six losses in a row he has lost 63 times his first stake and must bet 64 times it to break even, if the table allows it. Doubling means each new bet equals all past losses plus one, so the cumulative loss is always just under the next bet, and a short losing run produces an enormous number. The strategy fails when the losing run is longer than the capital or the limit.

What should a control function see before month six?

Several things, and none needs the trader's honesty. A position growing geometrically, a P&L that is flat or smooth while the gross position explodes, and large trades that are cancelled, amended or booked to unusual accounts are the classic signs of concealed losses. Gross notional limits catch the growth even when net risk looks small. Reconciling traded volumes with confirmations from counterparties catches fictitious offsetting trades. A mandatory two-week leave rule catches positions that need the trader to keep rolling them.

The operational risk lesson is that the fraud is the smaller part of the loss. A Rs 1 crore trading loss, reported on day one, is a normal cost of business; the controls failed on the other Rs 62 crore. That is why control reviews measure time to detection, not only whether a loss was eventually found.