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.

The desk's loss tail is much fatter than a normal distribution. For a normal, 2.326 over 1.645 gives a ratio of 1.41, so a 95% VaR of Rs 5 crore implies about Rs 7.07 crore at 99%. The desk shows Rs 11 crore, a ratio of 2.2, fatter even than a Student t with 3 degrees of freedom at 1.93. Rare losses are far larger than volatility alone would suggest.

Why does the ratio say anything about the shape of the tail?

If a city's 1-in-20 rainy day brings 5 cm of rain and its 1-in-100 day brings 11 cm, the storms are of a different kind from the drizzle, not just more of it. Scaling up the whole distribution would move both numbers together. Volatility stretches every quantile by the same factor, so the ratio between two quantiles is a pure measure of shape, and a ratio well above 1.41 means the tail is fatter than normal.

What would you do with that finding?

Two things. First, distrust any risk number on this desk that is built from volatility and a normal multiplier, such as a parametric VaR or a stress scaled from the 95% figure, because it will understate the 99% loss by about a third. Second, look at the P&L on the worst days: fat tails on a desk usually come from positions that pay small amounts often and lose large amounts rarely, such as sold options or carry trades.

The limitation is sample size. A 99% historical VaR from a year of data rests on two or three observations, so one extreme day could produce the 2.2 by itself. Check the ratio across several windows before concluding the desk's business is fat-tailed rather than unlucky.

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.

The fund ends at about 98.18, down 1.82% while the index is flat. Day one is plus 20%, taking 100 to 120. Day two is twice minus 9.09%, minus 18.18%, and 18.18% of 120 is 21.82, leaving 98.18. Each repeat of the up and down pair multiplies the fund by 0.9818, so after five pairs it sits at 91.23.

Why does doubling each day not double the two-day result?

Think of walking up an escalator that runs down. If you climb 10 steps and slip back 10, you are where you started. Now imagine every climb is measured as a share of your height above the ground, and so is every slip: slipping 18% from a higher point loses more steps than climbing 20% from a lower one gained. A daily-leveraged fund resets its exposure every day, so each day's percentage move is applied to a new base, and the losses land on the larger base.

How big is the drag, and what makes it worse?

Expand the product: for a leverage of L and an index that goes up r and then back to where it started, the fund loses about L times (L minus 1) times r squared on each pair, divided by 1 plus r. The drag grows with the square of the daily move and roughly the square of the leverage, so it is small in calm markets and fierce in choppy ones. With 2x and 10% moves that is 2 times 1 times 0.01 over 1.1, and each pair costs 1.82%; at 3x the same pair would leave 94.55. In a steady trending market the effect can run the other way and the fund beats twice the index.

Say where a risk manager meets this. A client who holds a daily-leveraged product for months is not holding twice the index; the product's prospectus usually says as much, and the gap is a suitability question as well as a maths one.

Uncorrelated, the portfolio's volatility is 4%; with a correlation of 0.3 it is 11.45%, above a floor of 10.95%. Uncorrelated risk shrinks with the square root of the count, 20% over 5. With correlation, the shared risk stays: variance is 20% squared times (1/25 plus 24/25 times 0.3). As the count grows the floor is 20% times the square root of 0.3.

Why does adding stocks stop helping?

Picture 25 shops in one town. Each has its own bad luck, a broken freezer or a rude cashier, and across 25 shops those mishaps average out. But if the town's main factory closes, every shop loses customers on the same day, and owning more shops in the same town does not help. Diversification removes each stock's own risk but cannot touch the risk the stocks share, and correlation is the measure of that shared part.

How do you reach the numbers in your head?

Uncorrelated first: variance divides by n, so volatility divides by the square root of n, 20% over 5 is 4%. For the floor, let n run to infinity and the 1/n term vanishes, leaving variance of sigma squared times rho. The square root of 0.3 is about 0.55, so the floor is about 11%, and 25 stocks already capture almost all the diversification available. The 25-stock figure of 11.45% is only half a point above the 10.95% floor.

Then say the limitation. Correlations are estimated in normal markets and tend to rise in a sell-off, exactly when diversification is wanted. A portfolio sized at 11% volatility on a correlation of 0.3 can behave like one at 15% or more if the correlation jumps to 0.6.

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.

Duration alone gives a -7.00% fall; adding convexity gives -6.70%. Duration is the straight-line estimate: minus 7 times 1%. Convexity adds half of 60 times 1% squared, which is 0.30%. So the bond falls about 6.7%, not 7%. The gap looks small at 100 basis points, but it grows with the square of the move.

What does each number measure?

Think of a car's speedometer and its acceleration. Speed tells you how far you will go in the next minute if nothing changes; acceleration tells you how the speed itself is changing. Duration is the speedometer of a bond: the percentage price change for a small yield move. Convexity is the acceleration: how duration itself shifts as yields move. A modified duration of 7 means about 7% of price per 1% of yield, and a convexityThe second-order sensitivity of a bond price to yield; it measures how much the price curve bends away from the duration line. of 60 says the price curve bends away from that straight line.

When does the convexity term stop being a rounding error?

Square the move and see. At 100 basis points convexity is worth 0.30%; at 300 basis points it is nine times that, 2.70%, against a duration effect of 21%. Because the convexity term grows with the square of the move, it is small for daily risk and large in a stress scenario. A treasury stress test that uses duration alone will overstate losses on a large rate rise for a plain bond and understate them for a callable bond or a mortgage book, where convexity is negative.

Close with the sign. Positive convexity is something a bond holder pays for through a slightly lower yield, and negative convexity, from options sold to borrowers, is what makes prepayable loans harder to hedge.

About Rs 3.4 crore: the Rs 3 crore VaR plus Rs 0.4 crore to get out. The position is marked at mid, but selling means taking the bid, half the spread below mid. Half of 40 basis points is 0.20%, and 0.20% of Rs 200 crore is Rs 0.4 crore. The exit cost adds 13% to the risk number, and more if spreads widen in a stress.

Why is ordinary VaR missing a cost?

A second-hand car dealer will quote you two prices for the same car: what he pays and what he sells for. Your car is worth the middle on paper, but if you need cash today you get the lower one. VaR is computed on mid prices, so it measures how far the value might move, not what it costs to actually leave the position. For a liquid government bond the difference is small; for a corporate bond the spread can be a meaningful share of the risk.

What makes the adjustment larger than it looks?

Two things, and both arrive in a stress. Spreads widen exactly when you need to sell, and a large position moves the price against you as you sell it. If the spread triples to 120 basis points, the exit cost is Rs 1.2 crore and the adjusted figure is Rs 4.2 crore, 40% above plain VaR. A risk team would also ask how many days it takes to exit Rs 200 crore without moving the market; if the answer is five days rather than one, the market VaR itself should be scaled to that horizon.

Say the limit of the simple version: it treats the spread as a fixed number. A fuller treatment uses the spread's own volatility, adding a multiple of its standard deviation, so the adjustment reflects how bad the spread gets in a bad week, not how it looks on an average one.

Two weighings. Put three balls on each pan and leave two aside. If one pan drops, the heavy ball is among its three: weigh one against another, and a balance points to the third. If the pans balance, weigh the two set aside against each other. A balance has three outcomes, so two weighings separate up to nine balls.

Why is halving the wrong instinct?

A guessing game where someone answers only yes or no halves the possibilities with each question. A balance answers in three ways: left heavier, right heavier or level. Each weighing has three outcomes, so the best split is into three groups, and the balanced outcome is information, not a wasted turn. Four against four uses only two of the three outcomes, because the pans can never balance, and so it wastes a third of what the scale can tell you.

How do you prove two is the minimum and find the limit?

One weighing has three outcomes, and eight balls need eight different answers, so one weighing cannot be enough. Two weighings have 3 times 3, nine outcomes. With w weighings you can separate at most 3 to the power w balls, so nine balls also need only two, and a tenth ball needs a third weighing. Saying the bound, not only the procedure, is what turns a trick into reasoning.

Why would a risk interviewer ask this?

It tests whether you count what a test can tell you before you run it. The same habit applies to a control check: a review that can only return pass or fail tells you less than one that also flags items for a closer look. Name the three outcomes first and the answer follows.