Risk Management puzzles, solved step by step

The CET1 ratio falls from 11.25% to 8.75%, a drop of 2.5 percentage points. The Rs 200 crore loss comes straight out of CET1, taking it from Rs 900 crore to Rs 700 crore, while risk-weighted assets stay close to Rs 8,000 crore. A loss equal to 2.5% of risk-weighted assets wipes out 22% of the capital.

Why does a loss that looks small against the balance sheet hurt so much?

Think of a family with a Rs 80 lakh home loan and Rs 9 lakh in savings. A Rs 2 lakh medical bill is a rounding error against the loan and a painful bite out of the savings, because the savings are what pay for it. A bank works the same way. The CET1Common equity tier 1: the shareholders money and retained profits a bank can use to absorb losses while it keeps operating. ratio is capital over risk-weighted assetsThe bank assets, each scaled by a weight that reflects how risky it is, so a safe government bond counts for little and an unsecured loan counts in full.. Losses are paid from the numerator rupee for rupee, while the denominator moves only a little, so a ratio of 11% can lose a quarter of its height in one hit.

Is there a shortcut for the size of the drop?

Yes. With the denominator fixed, the ratio falls by the loss divided by risk-weighted assets: 200 over 8,000 is 2.5 points. Every Rs 80 crore of loss costs this bank one full point of CET1 ratio. That is the number a risk manager keeps in their head, because it turns a loss estimate from a stress test straight into a capital headline.

Say the limitation in one line. The written-off loans do leave the balance sheet, so risk-weighted assets fall a little too, and a tax credit on the loss can soften the hit. Both effects are small next to the Rs 200 crore coming out of capital. Where the result lands against the bank's minimum depends on the current regulatory figure and any buffers, which you would confirm rather than quote from memory.

Rs 45 crore without netting and Rs 15 crore with it. Without netting, each trade stands alone, so the bank is exposed to every trade in its favour, 30 plus 15, and must still pay the 30 it owes. With an enforceable netting agreement all four collapse into one net claim: 30 minus 20 plus 15 minus 10, which is Rs 15 crore.

Why does netting cut the exposure by two thirds?

Two flatmates keep a running tab: one owes the other Rs 3,000 for rent, the other owes Rs 2,000 for groceries. If they settle as one tab, Rs 1,000 changes hands. If one of them walks out, the other would want the tab settled as one, not to pay the grocery bill in full while chasing the rent. A netting agreementA legal contract under which all trades between two parties are combined into a single net amount if one of them defaults. turns many trades into one claim, so money you owe the defaulter is set against money it owes you.

What happens to the negative trades without netting?

They still get paid, by the bank. The administrator of a failed counterparty will collect every trade where the bank owes money and join the queue of creditors for every trade where the bank is owed. That asymmetry is called cherry-picking, and it is why exposure without netting is the sum of the positive values, never the net. Here the bank pays Rs 30 crore out and recovers only whatever the estate pays on Rs 45 crore.

The limitation to say out loud: netting only helps where it is enforceable in the counterparty's jurisdiction, which is why banks obtain legal opinions before counting it. Where that is uncertain, the risk system should fall back to the gross number.

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.

Loans must shrink by Rs 9.5 crore, from Rs 95 crore to Rs 85.5 crore. Deposits fall 10% to Rs 90 crore, and 95% of Rs 90 crore is Rs 85.5 crore. The Rs 10 crore that leaves is met by Rs 9.5 crore of loans running off and Rs 0.5 crore of liquid assets, so the bank's lending falls by the same 10% as its deposits.

Why does a deposit outflow become a lending cut?

A household that lives on its salary and lends a cousin money every month has to stop lending if the salary is cut. The cousin has done nothing wrong; the money simply is not there. A bank funding its loans from deposits is in the same position. Holding the loan-to-deposit ratio fixed means every rupee of deposit flight passes straight into less lending, scaled by the ratio.

What makes this harder in practice than on paper?

Loans do not shrink on command. Term loans run off only as they repay, and calling them early harms the borrower and the bank's franchise. In the short run the bank has to meet the outflow from liquid assets or new funding, and a buffer of Rs 5 crore against a Rs 10 crore outflow is not enough. That gap is why liquidity rules ask banks to hold enough high quality liquid assets to survive a stressed outflow without selling loans.

Say the limitation: a real bank also has equity and wholesale funding on the liability side, and it could replace lost deposits with borrowing at a higher cost. The puzzle shuts that door deliberately, to show how directly a deposit run reaches lending when no other funding is available.

Probably not: the ratio improved only because the book grew. The bad loan ratio fell from 3.0% to 2.8%, but the bad loans themselves rose 16.7%, from Rs 120 crore to Rs 140 crore. The Rs 1,000 crore of new lending is too young to have defaulted. Set against last year's book, bad loans are 3.5% of the loans that could have gone bad.

How can a ratio fall while the problem grows?

A school with 40 failing students out of 1,000 has a 4% failure rate. Admit 500 new students in April, before any exams, and the rate drops to 2.7% without a single student improving. Any ratio can fall because its denominator grew, and a fast-growing loan book dilutes its bad loan ratio with loans that have not yet had time to fail. Here bad loans rose Rs 20 crore while the book rose Rs 1,000 crore.

What would you check before calling it either way?

Loans take time to go bad, a process lenders call seasoningThe time a loan needs before its true default rate shows, because few borrowers default in the first months after taking a loan.. The fair test compares bad loans with the book that was old enough to produce them, which is why risk teams track lagged ratios and default rates by the year a loan was written. A lagged ratio of 3.5% against 3.0% says the old book is getting worse, not better.

The limitation: the lagged ratio assumes the new loans added nothing to the Rs 140 crore. Some of the extra Rs 20 crore could come from new loans that failed fast, which would itself be a warning about how they were underwritten. Either way, the headline ratio is the weakest of the three readings.

The bonds can fall 8%, to Rs 92 crore, before the loan is uncovered. The haircut is exactly that cushion: collateral of 100 less a loan of 92. A haircut is set to cover the largest price fall likely while you seize and sell the bonds; with 1% daily volatility and ten days to sell, a 99% move is about 7.4%, so 8% covers it with little to spare.

What is a haircut, in plain terms?

A pawnbroker lends Rs 8,000 against a gold chain worth Rs 10,000. The Rs 2,000 gap is there because gold prices move and because the chain has to be sold if the loan is not repaid. A haircut is the price fall the lender can absorb before the collateral is worth less than the loan. Here the gap is Rs 8 crore on Rs 100 crore, so the bonds can lose 8% before the lender is exposed.

How would a risk team have chosen 8%?

Ask two questions: how volatile is the collateral, and how long would it take to get out? The second one is called the margin period of riskThe time between the last good margin call and the moment the lender has sold the collateral after a default.. A haircut is roughly the collateral's daily volatility, scaled to the days needed to sell it, at a high confidence level. With 1% daily volatility, ten days and a 99% level, the move is 2.33 times 1% times the square root of 10, about 7.4%. Round up for the bid-ask cost of a forced sale and you reach about 8%.

Then name what breaks it. The haircut assumes the bonds keep their normal volatility and can be sold in ten days; in a stress both assumptions fail together. If the bond issuer is linked to the borrower, the collateral falls just as the borrower defaults, and no haircut sized on normal days is enough.

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.

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.

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 Rs 37.9 crore, Rs 12 crore times the square root of 10. The rule works because the variance of a sum of independent daily P&Ls is the sum of their variances, so volatility grows with the square root of time. It needs daily P&L to be independent, identically distributed with zero mean, and the position to stay unchanged for ten days.

Why the square root, not ten times?

Walk ten steps where each step is a coin toss, left or right. You rarely end ten steps away; the lefts and rights partly cancel, and the typical distance is about three steps, the square root of ten. Independent daily gains and losses partly offset each other, so the spread of the ten-day total grows with the square root of ten, not with ten. Multiplying by ten assumes all ten days are bad days in the same direction, which is the one path the independence assumption rules out.

What breaks the rule, and in which direction?

Four things, and a market risk interviewer wants at least two. If losses cluster, with one bad day tending to follow another, the square root understates ten-day risk: an autocorrelation of 0.2 lifts the figure from Rs 37.9 crore to about Rs 45.5 crore. Volatility that rises after a shock, fat tails that do not shrink toward normal over a few days, and a position that cannot be cut or that the desk keeps adding to all push the same way. Mean reversion in P&L pushes the other way. The rule also assumes the portfolio is fixed for ten days, which a desk that trades daily does not satisfy.

Say where the rule is used: regulatory market risk capital has long been built on a ten-day horizon, and many banks produce it by scaling one-day VaR. The standards now also require horizons that differ by the liquidity of the risk, which is a direct admission that ten days of independence is not true for every position; confirm the current rules before quoting any detail.