Risk Management puzzles, solved step by step
- Puzzles
- 100
- Traced to a firm
- 17
- Topics
- 13
- Hard
- 30
041Bank A holds one Rs 100 crore loan. Bank B holds one hundred loans of Rs 1 crore each. Every loan has a 2% default probability, total loss on default and independent defaults. Both expect to lose Rs 2 crore. What is each bank's chance of losing more than Rs 10 crore in a year?Bank credit riskRisk GCC
Try it first
Bank A has a 2% chance of losing more than Rs 10 crore. Bank B's chance is:
Show the worked solution
Bank A: 2%. Bank B: about 5.6 in a million, roughly 1 in 177,116. Bank A loses Rs 100 crore whenever its single loan defaults. Bank B loses more than Rs 10 crore only if 11 or more of its 100 loans default when 2 are expected, which independent defaults almost never produce. Expected loss is Rs 2 crore for both; the tail differs by a factor of several thousand.
Why does splitting the same money into small loans change the tail?
A shopkeeper who sells to one wholesale buyer is either paid in full or not at all; one who sells to a hundred households loses a few payments every month but never the whole book. Granularity leaves expected loss unchanged but pulls the loss distribution in around its average, because independent small defaults rarely pile up. Bank A's outcome is all or nothing: 98% nothing, 2% the full Rs 100 crore. Bank B's losses cluster between Rs 0 and 6 crore.
Both banks expect to lose Rs 2 crore, but bank A faces a 2% chance of losing its whole Rs 100 crore while bank B's losses cluster around Rs 2 crore and exceed Rs 10 crore only about 5.6 times in a million. How do you estimate bank B's number in the room?
The defaults are a binomial with mean 2 and standard deviation about 1.40. Eleven defaults is more than six standard deviations above the mean, so the answer is tiny; the exact binomial tail is 5.6 in a million. Compare the standard deviations of loss directly: bank A's is Rs 14.0 crore, bank B's is Rs 1.4 crore, ten times smaller, because splitting one exposure into a hundred independent ones divides the spread by the square root of 100.
The relationship\sigma standard deviation of the annual loss, Rs crore 0.02 default probability of each loan What it says in wordsSplitting one exposure into n independent pieces cuts the spread of losses by the square root of n.Now the caveat that matters in practice. The 99% loss quantile is Rs 100 crore for A and Rs 6 crore for B, which is why concentration riskExtra risk from large single exposures or from many exposures that fail together, which average-based measures do not show. gets its own capital charge. But granularity only helps if defaults are independent. A hundred small loans to one industry in one city will default together in a local downturn, and bank B's tail then fattens towards bank A's. Diversification across names is not diversification across causes.
Where candidates lose it
The first trap is saying both banks are equally risky because both expect to lose Rs 2 crore. Expected loss is exactly the number that cannot tell these two banks apart.
The second is assuming more loans means more risk because more loans can default. The count of defaults rises, but each is small, and the chance of many at once collapses. Close by naming the independence assumption as the thing that could undo it.
What the interviewer asks next
- What is each bank's 99% VaR?
- If bank B's loans all go to one sector with default correlation, what happens to its tail?
- How would you set a single-name concentration limit using this logic?
075A Rs 100 crore loan is secured by property worth Rs 120 crore. In a downturn the property falls 40% in value, a forced sale costs 10% of the sale value, and recovery takes one year, discounted at 12%. What is the loss given default?Bank credit riskNBFC credit risk
Try it first
With 120% collateral cover at the start, where does LGD land?
Show the worked solution
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.
Rs 120 crore of collateral falls to Rs 72 crore in the downturn, to Rs 64.8 crore after a 10% forced sale cost and to Rs 57.9 crore after a year of discounting at 12%. Set against the Rs 100 crore loan, that is a loss of Rs 42.1 crore, a loss given default of 42.1%. 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.
The relationship120 collateral value at origination, Rs crore 0.40 the downturn fall in property value 0.10 forced sale cost as a share of sale value 1.12 one year of discounting at 12% What it says in wordsLoss given default is one minus the present value of what the collateral fetches, as a share of the loan.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.
Where candidates lose it
The trap is answering zero because the loan is over-collateralised. The interviewer is testing whether you see that collateral values fall exactly when defaults happen, which is the whole reason downturn LGD exists.
The second trap is taking only the 40% fall and answering 28%. Apply all three steps in order, value, sale cost and time, and give the number with its assumptions.
What the interviewer asks next
- How long can recovery take before LGD reaches 50%?
- What collateral cover at origination would give zero loss in this scenario?
- Why do regulators ask banks to estimate LGD for a downturn rather than on average?
100A firm's assets are worth Rs 100 crore with 25% volatility and 5% expected growth; Rs 70 crore of debt is due in one year. What is the distance to default, and what default probability does it imply?Quant riskBank credit risk
Try it first
Roughly what one-year default probability does the Merton model give?
Show the worked solution
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.
With assets of Rs 100 crore, 25% volatility and 5% growth, the one-year asset value centres near Rs 101.9 crore and only the tail below the Rs 70 crore of debt, 6.66% of outcomes, ends in default, a distance to default of 1.50. The relationshipV asset value today, Rs 100 crore D debt due, Rs 70 crore mu expected asset growth, 5% sigma asset volatility, 25% N the standard normal distribution What it says in wordsDistance to default is the log cushion plus expected drift, in units of asset volatility; the default probability is the normal tail beyond it.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.
Where candidates lose it
The common error is using the simple ratio, 1.2 standard deviations, and quoting 11.5% as the Merton answer. It is a quick estimate, not the model; show the log form or say that you are approximating.
The second is forgetting the minus half sigma squared term. It is small here, 0.031, but leaving it out, or adding it, moves the distance to default and shows the interviewer the formula is memorised rather than understood.
What the interviewer asks next
- Asset volatility rises to 35%. What happens to the default probability?
- How would you estimate asset value and asset volatility from the share price?
- Why does the Merton model tend to understate short-term default risk?