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.

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.

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.