Risk Management puzzles, solved step by step

No, not at 95% confidence, though it is close. With 500 obligors at 1%, you expect 5 defaults. The exact binomial chance of 9 or more is about 6.7%, above the 5% cut-off; the smallest count that would reject is 10. A normal approximation without a continuity correction gives z = 1.80 and wrongly rejects. One year of data rarely proves a grade wrong.

How surprising are nine defaults if the 1% is right?

If a coin lands heads 7 times in 10, you would not call it biased; 70 in 100 you would. Small samples swing. With 500 obligors at 1%, the expected count is only 5, and counts that small are lumpy. The question is not whether 9 is bigger than 5, but how often a correct 1% grade produces 9 or more by chance, and the exact answer is about one year in fifteen.

Why does the normal approximation get it wrong here?

The standard deviation of the count is the square root of 500 x 0.01 x 0.99, about 2.22. Nine is 1.80 standard deviations above five, past 1.645, so a quick z-test rejects. But the binomial is skewed and discrete at small counts; with a continuity correction z falls to 1.57 and the conclusion flips back. For counts under about ten, use the exact binomial.

Two further limitations make the case for caution stronger. Defaults are correlated through the economy, so a single bad year moves many obligors together and the true spread of counts is wider than the binomial says. And a validation team would look at several years and at neighbouring grades before recalibrating. The right answer is to flag the grade for watching, not to declare it broken.

About 42%. The property falls from Rs 120 crore to Rs 72 crore. A forced sale costs 10%, leaving Rs 64.8 crore. Received a year later and discounted at 12%, that is worth Rs 57.9 crore today. Against a Rs 100 crore loan the bank loses Rs 42.1 crore, a loss given default of about 42.1%, on a loan that started with 120% collateral cover.

Why does 120% cover not protect the lender?

A car bought for Rs 10 lakh with a Rs 8 lakh loan looks safe on day one. If the owner defaults three years later after a crash, the lender recovers a damaged, older car sold in a hurry, and pays a recovery agent and a lawyer while waiting. Collateral is worth what it fetches when the borrower defaults, not what it was worth when the loan was made, and defaults cluster in exactly the downturns that push collateral prices down. That link is why lenders model LGD under stressed values.

Which of the three haircuts do candidates forget?

The discounting. The sale cost is at least mentioned in the question; the time value is easy to skip because nothing appears to be lost. A year's wait at 12% removes about Rs 6.9 crore, as much as the sale cost, simply because money received later is worth less today. In jurisdictions where enforcing security takes several years, this line can become the largest of the three, which is why workout time sits at the centre of LGD models.

Say the limitations. The loan balance may have grown with unpaid interest by the time of sale, which raises the loss; legal costs would add to the sale cost; and the 40% fall is a scenario, not a forecast. The structure of the answer, value then cost then time, carries over to any secured loan.

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.

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.

About Rs 12.65 a day, a 1.26% move, which is roughly 20% annual volatility. A delta-hedged long gamma book earns half of gamma times the move squared each day and pays theta. Setting 250 times the move squared equal to Rs 40,000 gives a move of the square root of 160. Times the square root of 250 trading days, 1.26% a day is about 20% a year.

What does a long gamma book earn, and what does it pay?

Picture a shopkeeper who pays a fixed daily rent for a stall that earns money only when a crowd passes, and earns much more from a big crowd than a small one. A quiet day loses the rent; a busy day covers it many times. A delta-hedged long option book pays theta every day and earns from gamma, which pays in proportion to the square of the stock's move, so small moves lose and large moves win. Rebalancing the hedge after each move locks in that gamma profit: buying low after a fall and selling high after a rise.

Why does the breakeven turn into a volatility?

Because that is how the market priced the options. A Rs 12.65 move on a Rs 1,000 stock is 1.26% a day, and 1.26% times the square root of 250 is 20.0% a year. For a delta-hedged option book, the breakeven daily move is the implied volatility in disguise: the book profits if realised volatility beats about 20% and loses if it falls short. That is the one-line view a market risk manager wants: this book is long volatility at about 20%.

State the limits. Gamma and theta change as the stock moves and time passes, so the breakeven drifts; the relation holds day by day, not for a month in one go. And the average of squared moves is what counts, so one large day can pay for several quiet ones, which is why realised volatility, not the number of up days, decides the result.

About Rs 24.19. A game that ends on flip n pays 2 to the n with probability one half to the n, so each flip count adds exactly Rs 1. Uncapped, that sum is infinite. The client can pay only Rs 1 crore, which 2 to the 23 stays below, so 23 flip counts add Rs 23 and every longer game pays Rs 1 crore, adding about Rs 1.19.

Why is the uncapped game worth an infinite amount?

Picture a prize that doubles every time you survive another round, while the chance of surviving halves. Each round's prize times its chance is always the same Rs 1. In this game every possible length contributes exactly Rs 1 to the expected value, one Rs 1 for each flip count, forever, so the sum never stops. This is the St Petersburg paradox: the arithmetic says pay anything, yet nobody would pay even Rs 100.

How does the cap turn infinity into Rs 24?

Find where the cap bites: 2 to the 23 is Rs 83,88,608 and 2 to the 24 is over Rs 1 crore, so the first 23 flip counts pay in full. Those contribute Rs 23; every longer game pays the capped Rs 1 crore, and the chance of lasting past 23 flips is one half to the 23, so the tail adds Rs 1 crore divided by 2 to the 23, about Rs 1.19. The fair price is about Rs 24.19.

Why is this a counterparty risk question?

Because the value came entirely from payouts the client could never make. A promised payoff is worth only what the payer can actually pay, and most of this game's theoretical value sat in states where the payer would default. Raising the cap a thousandfold to Rs 1,000 crore only lifts the value to about Rs 34.16, because each doubling of capacity adds just one rupee. The same logic prices protection bought from a seller who could not survive the event it insures.

About 0.87%. Oversampling inflates every score by the same factor on the odds. The sample's odds of bad are 50 to 50; the population's are 2 to 98, one forty-ninth as high. The applicant's sample odds are 30 to 70, 0.429; divide by 49 to get 0.00875, which is a probability of 0.87%. The model ranks correctly but must be recalibrated.

Why does oversampling change the score?

Suppose a doctor learns to spot a rare illness from a teaching ward where half the patients have it. In a village clinic, where only one patient in fifty has it, the same symptoms mean far less. A model trained on a 50% bad sample has learned a base rate 25 times too high, so every score it produces is inflated, even though the ordering from safest to riskiest is still right. Oversampling is done on purpose, to give the model enough bad accounts to learn from, so the correction is a routine step, not a sign of a broken model.

How do you sanity-check the answer?

Take an applicant the model scores at exactly 50%, the sample average. After correction that applicant should sit at the population average, and the formula gives exactly 2%, which confirms the factor. Then note that 30% is below the sample average, so the corrected figure should be below 2%, and 0.87% is. The shortcut of scaling the probability by 2% over 50% gives 1.2%, which is close for low scores but breaks down badly for high ones: a sample score of 90% would scale to 3.6%, while the correct answer is about 15.5%.

Say what a validator would do next. The correction assumes the good and bad accounts were each sampled at random within their class. If the bad accounts were drawn from a different period or channel, the model's ranking may also be off, and the fix is to check calibration on a recent, unsampled holdout, comparing predicted and actual bad rates by score band.

A distance to default of about 1.50, implying a one-year default probability of about 6.7%. In the Merton model the firm defaults if assets end below the debt. The log of 100 over 70 is 0.357; add growth less half the variance, 0.019; divide by 25% volatility to get 1.50 standard deviations. The normal tail beyond that is 6.66%.

What is the model saying in plain words?

A homeowner with a Rs 70 lakh loan on a Rs 1 crore house is in trouble at repayment only if the house is then worth less than Rs 70 lakh. How likely that is depends on the cushion, Rs 30 lakh, and on how much house prices swing. The Merton model treats a firm the same way: default happens if the value of its assets at the debt's maturity falls below what it owes, so the default probability is the share of possible asset values that land below the debt. Distance to default is that cushion measured in standard deviations.

Why not just divide the cushion by volatility?

The simple version, Rs 30 crore of cushion over Rs 25 crore of one-year volatility, gives 1.2 standard deviations and a default probability of about 11.5%. It ignores two things that both favour the lender: assets are expected to grow 5%, and a lognormal asset value cannot fall as easily in rupees as it can rise, so the log cushion is wider than the simple ratio suggests. The simple form is still a useful first number in the room, as long as you say which way it errs.

Then give the model's limits. Asset value and asset volatility are not observed; in practice they are backed out from the equity price and equity volatility. The model assumes default happens only at maturity, a single debt payment, and normal log returns, and it tends to produce very low default probabilities for safe firms over short horizons. Commercial versions map distance to default onto historical default rates rather than trusting the normal tail.