Risk Management puzzles, solved step by step

About 3.06%. Start with 10,000 borrowers. 200 default in year one, leaving 9,800. By the end of year two 500 have defaulted, so 300 did so in year two. For a borrower who reached the start of year two, the chance is 300 out of 9,800, which is 3.06%, a little above the 3% you get by subtracting.

Why divide by the survivors?

Think of a school where 2 of every 100 students leave in class nine and 5 in total have left by the end of class ten. If you are a class ten student today, your chance of leaving this year is measured against the 98 who are still in the room, not the 100 who started. A conditional probability of default is always measured against the borrowers who survived to the start of the period. The unconditional slice, 3% of the original pool, is the right number only if you are standing at the start of year one.

Where does this matter on a credit desk?

Whenever you price or provision a loan over several years. Expected loss in year two uses the marginal default of the original pool, while a hazard rateThe probability of default in a short period for a borrower that has survived to the start of it. used to model a surviving borrower uses the conditional figure. Mixing them up is a small error at 2% and 5%, 3.06% against 3.00%, but at high-yield default rates the gap widens: 20% and 35% cumulative gives 18.75% conditional against 15% by subtraction.

Say the limitation too. Cumulative default tables are averages across many cohorts and economic cycles, so a borrower in a downturn year may face a higher rate than the table shows. The arithmetic is exact; the inputs are estimates.

Equity loses 100%, mezzanine 70% and senior nothing; the mezzanine is wiped out at a 15% pool loss. The Rs 120 crore loss fills the tranches from the bottom. Equity absorbs its full Rs 50 crore. The remaining Rs 70 crore falls on the Rs 100 crore mezzanine. The senior tranche starts losing only once pool losses pass 15%, the point where the mezzanine is gone.

How do losses move through a tranche stack?

Picture a building flooding from the ground up. The ground floor is soaked before a drop reaches the first floor, and the top floors stay dry until the water climbs to them. Losses fill the tranches from the bottom: each tranche loses nothing until the pool loss passes its attachment pointThe level of pool loss at which a tranche starts to lose money., and everything once the loss passes its detachment point. The equity attaches at 0% and detaches at 5%; the mezzanine attaches at 5% and detaches at 15%.

Why does thickness decide how risky a tranche is?

Because a thin tranche goes from untouched to wiped out over a small range of pool losses. The mezzanine is only 10 points thick, so a pool loss moving from 5% to 15% takes it from zero to total loss, while the same move barely registers on the pool as a whole. That is the leverage inside structured finance: the mezzanine's loss share moved 7 times as far as the pool's 12% average suggests from 5% onwards. A rating analyst evaluating the deal asks how likely the pool loss is to cross each attachment point, which depends heavily on how correlated the loans are.

Name the risks the structure does not remove. Correlation among the loans decides whether pool losses cluster at a few percent or occasionally jump past 15%. The collateral data may be weak. And the waterfall rules in the documents, such as when cash is diverted to protect senior holders, can shift losses between tranches in ways this simple loss-only picture does not show.

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.

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.

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.

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.

About Rs 57.6 lakh, or Rs 0.576 crore. The line is Rs 48 crore drawn and Rs 32 crore undrawn. Exposure at default adds half the undrawn part, Rs 16 crore, for Rs 64 crore. Expected loss is exposure times default probability times loss given default: 64 x 2% x 45% = Rs 0.576 crore.

Why is today's drawn balance the wrong exposure?

Think of a friend with a credit card who is quietly losing his job. In the months before he stops paying, the card balance climbs, because he is using the card to cover what his salary no longer does. Companies behave the same way with bank lines. Borrowers draw down their undrawn limits on the way to default, so exposure at default is the drawn balance plus a credit conversion factorThe share of an undrawn limit a bank assumes will be drawn by the time the borrower defaults. share of what is still undrawn.

How much does the conversion factor change the answer?

A lot. Using only the drawn balance gives Rs 43.2 lakh; using the full limit gives Rs 72 lakh. The 50% conversion factor moves the expected loss by a third over the drawn-only figure, with no change in the borrower's credit quality. That is why the conversion factor is estimated from the bank's own default history on committed lines, not guessed.

Say the limitation plainly. Expected loss is an average, the amount the bank should price into the loan's spread and provide for, not the loss it would face if this borrower actually defaulted, which would be 45% of Rs 64 crore, or Rs 28.8 crore.

Expected loss is Rs 1 crore; the standard deviation of loss is Rs 7 crore. If the loan defaults, the bank loses 50% of Rs 100 crore, Rs 50 crore; otherwise it loses nothing. The average is 2% x 50 = Rs 1 crore, which the spread should cover. The standard deviation is 50 x the square root of 0.02 x 0.98, which is Rs 7 crore, the unexpected loss capital exists for.

Why is the average loss the wrong thing to hold capital against?

A shopkeeper insuring against a one-in-fifty chance of a Rs 5 lakh fire can budget Rs 10,000 a year for the premium, the average. But in the year the fire happens, the budget is useless; what saves the shop is savings. Expected loss is a cost of doing business and belongs in the price; unexpected lossThe spread of possible losses around the expected loss, usually measured by a standard deviation or a high percentile, which capital is held to absorb. is the surprise, and it is what capital is for.

How do you get Rs 7 crore quickly?

Treat default as a coin that lands bad with probability p. Its standard deviation is the square root of p times (1 - p): the square root of 0.0196, which is 0.14. Multiply by the loss if it lands bad, Rs 50 crore, and you have Rs 7 crore. The ratio of seven to one is typical for a low-default loan, and it grows as the default probability falls: rarer defaults mean a mean closer to zero and a relatively wider spread.

Say the limitation. For one loan, the standard deviation is a poor summary: the real outcome is 0 or 50, never 7. In a large, diversified portfolio losses spread out and the standard deviation becomes meaningful, and capital models set it at a high percentile of the portfolio loss rather than one standard deviation of a single loan.

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.

About 5% a year. The spread roughly pays for expected loss, which is default probability times loss given default. So default probability is the spread divided by the loss: 300 basis points over 60% is 5%. That is an upper bound for the real-world rate, because part of every spread pays for risk and illiquidity, not expected loss.

Why is spread roughly default probability times loss?

Suppose you lend Rs 100 to each of 100 shopkeepers for a year. If 5 of them fail, and you get back only 40 paise in the rupee from each, you lose Rs 300 across the group. To break even you need to charge 3% more than a loan to the government. The spread is the extra yield that pays for expected loss, and expected loss is how often borrowers default times how much you lose when they do. This is sometimes called the credit triangle.

Why is 5% probably too high as a real forecast?

Because bond investors demand more than their expected loss. A spread also pays a risk premium for bearing uncertain losses and a liquidity premium for holding a bond that is hard to sell, so the implied probability is a risk-neutral figure that sits above the real-world default rate. For investment grade bonds, historical default rates are often a small fraction of what the spread implies. Say this as a limitation, and add that a rating agency would compare the implied 5% against the default history of similar ratings before drawing any conclusion.

Also check the recovery assumption. If recovery were 20% rather than 40%, LGD would be 80% and the implied default probability would fall to 3.75%; the answer moves a lot with a number that is itself a guess.

Net income falls Rs 7.5 crore, cash rises Rs 2.5 crore, and both sides of the balance sheet fall Rs 7.5 crore. Pre-tax profit drops 10, tax drops 2.5, so net income drops 7.5. The cash flow statement adds back the non-cash 10, leaving cash up 2.5. On the balance sheet, cash is up 2.5 and fixed assets down 10, total assets down 7.5, matched by retained earnings down 7.5.

How does a non-cash charge put cash in the bank?

Think of a shopkeeper who can deduct the wear on his delivery van from his taxable income. Writing the van down costs him nothing today, since he paid for it years ago, but it lowers the tax bill he pays this year. Depreciation moves no cash itself; the only cash effect is the tax it saves, 25% of Rs 10 crore, Rs 2.5 crore. That tax shield is the whole answer on cash, and the three statements are the bookkeeping that proves it.

What order do you walk it in?

Income statement first, because everything starts from net income. Then the cash flow statement: net income down 7.5, add back the 10 of depreciation because no cash left, and cash from operations is up 2.5. Finish on the balance sheet and prove it balances: assets fall by 10 of fixed assets less 2.5 of extra cash, 7.5, and equity falls by the 7.5 of lower retained earnings. Stating that both sides moved by the same 7.5 is the check the interviewer is waiting for.

Why does a credit analyst care?

Because a lender is repaid in cash, not in profit. A company whose earnings fall because of higher depreciation may be generating slightly more cash, so interest cover measured on net income and on cash flow can move in opposite directions. The limit: the tax saving is real only if the company is paying tax; a loss-making company gets no cash benefit this year, and a higher depreciation charge often reflects heavy past capital spending that the analyst should look at directly.

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.