Risk Management puzzles, solved step by step

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.

About 21%, and volatility would need to fall to about 6.2%. A loss means a return below zero, which is 8 points, or 0.8 standard deviations, under the mean. About 21.2% of a normal distribution lies below that, roughly one year in five. For a 10% chance, zero must sit 1.28 standard deviations below the mean, so volatility must be 8 divided by 1.28, about 6.2%.

How do a return target and a volatility target fix the chance of loss?

Think of a commute that takes 40 minutes on average but varies from day to day. Whether you are ever late for a 50 minute deadline depends on how much it varies, not only on the average. The chance of a losing year depends on how many standard deviations the expected return sits above zero: here 8 divided by 10, which is 0.8. Look up 0.8 in the normal table and about 21.2% of years fall below zero. The investor who hears 8% and thinks losses are rare is wrong one year in five.

What would you actually set as targets, and what is wrong with this model?

Set the targets as a pair, and state the trade-off. If the investor cannot tolerate losing more than one year in ten, then either volatility must come down to about 6.2%, which usually lowers the expected return too, or the loss tolerance must be stated over a longer horizon. Over five years the mean grows five times but the volatility only by the square root of five, so the chance of a losing five-year stretch is much lower. Asking about the horizon is the question a good risk manager raises first.

Then name the model's limits. Real returns have fatter left tails than a normal curve, so the chance of a large loss is understated; returns are not independent from year to year; and the 8% expected return is an assumption, not a promise. A drawdown limit, such as no more than a 15% fall from peak, is often more useful to an institution than a probability of a losing year.

About 5.73%. Tracking error is the volatility of the fund's return minus the benchmark's. Its variance is 18 squared plus 16 squared minus 2 x 0.95 x 18 x 16, which is 324 plus 256 minus 547.2, or 32.8. The square root is 5.73%, nearly three times the 2-point gap in volatilities.

What exactly is tracking error measuring?

Two friends walk to the same office. How far apart they are at any moment depends less on how fast each walks than on whether they take the same streets. Tracking error is the volatility of the return difference, fund minus benchmark, so it depends on how much the two move apart, not on how much each moves. The fund and index can both swing wildly and still track closely if they swing together. That is why the formula needs the correlationA number from minus 1 to 1 describing how closely two returns move together; 1 means perfect lockstep., not just the two volatilities.

Why does the correlation matter more than the volatilities?

Look at the table in the figure. Keeping 18 and 16 fixed, moving correlation from 0.99 to 0.90 takes tracking error from 3.12% to 7.85%, more than doubling it. At high correlations each hundredth of correlation moves tracking error a lot, because the large shared term 2 x rho x 18 x 16 almost cancels the two variances. Now hold correlation at 0.95 and give both sides 16% volatility: tracking error is still 5.06%. The volatility gap contributes a little; the imperfect correlation contributes most.

Close with the limit. The formula uses a correlation estimated from history, and correlations drift, often falling in stressed markets. A fund reporting 5.7% tracking error in calm years can run well above it in a sell-off, so a risk team watches realised tracking error alongside the model figure.

Forecaster A, with a mean squared error of 5 against 6.25. Mean squared error splits into bias squared plus variance. A pays 1 squared for its bias and 2 squared for its spread, 5 in all. B pays nothing for bias but 2.5 squared, 6.25, for its spread. A's typical error, the square root, is 2.24 against 2.50.

How can a biased forecaster beat an unbiased one?

Two archers. One groups every arrow tightly but slightly left of centre; the other is centred on average but scatters arrows all over the target. Ask which one lands closer to the bullseye on a typical shot, and the tight grouping wins. Mean squared error charges for two things, how far off you are on average and how much you scatter, and a small, steady bias can cost far less than a large scatter. Being unbiased only removes the first charge.

When would your answer flip, and what would you do with A?

Solve for the tie: A matches B when bias squared plus 4 equals 6.25, so a bias of 1.50. Below that, A wins. More useful still, a bias that is stable can be measured and subtracted: correct A by one point and its MSE falls to 4, better than either original. That is the practical lesson for a risk team: a model that is consistently off in one direction is fixable, while a noisy model is not. The trade-off is also why risk teams use shrinkagePulling a noisy estimate towards a simpler, steadier target, accepting a little bias in return for much lower variance. on covariance matrices built from short histories.

The limit: MSE punishes large errors heavily because it squares them, and it treats over-forecasts and under-forecasts alike. A risk manager forecasting losses may care more about under-forecasting than over-forecasting, in which case a symmetric score is the wrong yardstick and the ranking could change.

Dividing by 4 is unbiased; both estimators are consistent; dividing by 5 is low by one fifth. The sample mean is estimated from the same five points, which uses up one degree of freedom, so the divide-by-n estimator averages (n - 1)/n of the true variance, 80% here. If the true variance is 4, it centres on 3.2, a bias of -0.8. As n grows that factor tends to 1, so both estimators converge on the truth.

Why does dividing by n come out too low?

Measure how spread out five friends' heights are by comparing each to the group's own average, and you will understate the spread, because that average was pulled towards those five people. Deviations measured from the sample mean are smaller on average than deviations from the true mean, so their sum of squares understates the spread by exactly one observation's worth. Dividing by n minus 1, the degrees of freedomThe number of independent pieces of information left after estimating something from the same data; estimating the mean uses up one., corrects it exactly.

What is the difference between unbiased and consistent?

Unbiased is about the average over many repeated samples of the same size; consistent is about what happens to one estimate as the sample grows. Divide-by-5 fails the first: repeat the five-day exercise many times and the estimates average 3.2, not 4. It passes the second: with 250 days the bias is only -0.016, and it shrinks to zero with more data. An estimator can be unbiased but inconsistent too, such as using only the first observation to estimate a mean: right on average, never improving.

Now the twist a model validator should add. For normal data the unbiased estimator has variance 8.00 here, while the divide-by-5 version has 5.12 plus a squared bias of 0.64, a mean squared error of 5.76. The biased estimator is closer to the truth on a typical sample. Which you prefer depends on the use: unbiasedness matters when estimates are averaged across many desks; a smaller typical error matters for a single desk's limit. With five data points, neither is reliable, and that is the more important thing to say.

The 8% grower falls 20%; the 2% grower falls about 5.9%. Under Gordon growth, price is next year's dividend over r minus g. For the fast grower that gap widens from 2% to 2.5%, so the price falls to 0.02 over 0.025, or 80% of what it was. For the slow grower the gap goes from 8% to 8.5%, and the price keeps 0.08 over 0.085 of its value.

Why does the fast grower react so much more?

Think of two ways to be paid Rs 10 lakh: most of it next year, or a trickle that grows for decades. If someone doubles the rate at which you discount the future, the trickle loses far more, because most of its money is far away. A fast-growing dividend is that trickle: its value sits in cash flows many years out. A stock whose value rests on distant cash flows behaves like a long bond, so the same rise in the discount rate cuts its price far more.

How do you turn this into a duration number?

Differentiate the price with respect to r and divide by price: the answer is 1 over (r minus g). That gives the fast grower an equity duration of 50 years and the slow grower 12.5 years. Duration times 0.5% predicts falls of 25% and 6.25%; the exact falls are a little smaller, 20% and 5.9%, because the price curve bends, the same convexity a bond has.

The limitation is that Gordon growth assumes growth never changes and runs for ever, which exaggerates duration for a fast grower that will slow. The direction survives any sensible model: growth stocks carry more rate risk than stocks priced on today's cash.

The new estimate is 11.6 with variance 0.8. The Kalman gain is the prior variance over the total, 4 over 5, or 0.8. The estimate moves 80% of the way from 10 towards 12, landing at 11.6. The variance becomes (1 - 0.8) x 4 = 0.8, smaller than either the prior's 4 or the measurement's 1.

How does the filter decide how far to move?

Two friends guess your commute time. One has ridden with you a hundred times, the other once. You would average their guesses, but lean heavily on the first. A Kalman filterA method that updates an estimate of something you cannot see directly each time a noisy measurement arrives, weighting old estimate and new data by how precise each is. does exactly that: it weights the prior and the measurement by their precision, one over variance. Precision 0.25 against 1 gives the measurement 80% of the weight.

Why is the new variance smaller than both inputs?

Because two independent pieces of evidence together know more than either alone. Precisions add: 1 over 4 plus 1 over 1 is 1.25, and one over 1.25 is a variance of 0.8. That is the part candidates skip. The filter does not just move the estimate; it becomes more confident with every measurement, until new data carry little weight and the estimate settles.

Say the limitation: this single step assumes both errors are normal and the hidden price did not move between the prior and the measurement. A full filter adds a prediction step that lets the price drift and widens the variance before each update. Risk teams use the idea to track hidden quantities such as a hedge ratio that changes over time.

99% VaR is about Rs 12.5 crore and expected shortfall about Rs 15 crore. One per cent of 250 days is 2.5 observations, so VaR sits between the second worst loss, 14, and the third, 11: Rs 12.5 crore by interpolation, or Rs 11 crore if the desk takes the third worst. Expected shortfall averages the worst 2.5 days: (18 + 14 + half of 11) / 2.5 = 15.

Why is historical VaR a position in a list rather than a formula?

If you rank a class of 250 students by exam score, the student at the 99th percentile is not computed, they are counted: roughly the second or third from the top. Historical VaR works the same way with losses. Rank the daily P&L from worst to best and count 1% of the sample in from the bad end; the loss you land on is the 99% VaR. With 250 days you land 2.5 places in, between two real days.

Which convention is right, and does it matter?

Neither is wrong; both are used, and the gap here is Rs 1.5 crore on a Rs 12.5 crore number. What is wrong is not saying which one you used, because two desks can report different VaR from identical data. Expected shortfall moves with the convention too: averaging only the two losses beyond Rs 11 crore gives Rs 16 crore rather than Rs 15 crore. Name the rule, then give the number.

The limitation is size. A 99% number from 250 days rests on two or three losses, so one bad day entering or leaving the window can move it by several crore. That is why regulators ask for backtesting and why expected shortfall, which uses the whole tail, is preferred where the tail is thin.

About Rs 30,000 crore, 1.5 times book. For a bank growing steadily, price to book equals ROE less growth, over cost of equity less growth: (15 - 6) / (12 - 6) = 1.5. Book of Rs 20,000 crore times 1.5 gives Rs 30,000 crore. A cross-check: earnings of Rs 3,000 crore at a P/E of 10 gives the same figure.

Why does ROE against cost of equity decide the answer?

Imagine a fixed deposit that pays exactly the return you demand. Nobody would pay more than its face value, and nobody would sell it for less. A deposit paying more than you demand is worth a premium; one paying less sells at a discount. A bank's book equity is that deposit. A bank that earns exactly its cost of equity is worth its book, and every point of ROE above that cost is priced as a premium to book. Here the bank earns 15% on capital that shareholders price at 12%, so you already know the answer is above Rs 20,000 crore before touching the arithmetic.

Where does the formula come from, so you can rebuild it under pressure?

Start from a dividend growth model. To grow book at 6% while earning 15%, the bank must keep 6 over 15, or 40%, of its profit, and it can pay out the other 60%. That means the dividend is book times ROE less growth, and dividing by cost of equity less growth gives the price. Earnings are Rs 3,000 crore, the payout is Rs 1,800 crore, and Rs 1,800 crore divided by 12% less 6% is Rs 30,000 crore.

What would a risk interviewer want you to add?

Say how fragile the number is. The denominator is only 6 points wide, so a one point rise in the cost of equity to 13% cuts price to book from 1.5 to 1.29, a fall of 14% in market value. The word sustainable is also doing work: a 15% ROE earned in a benign credit year, before loan losses normalise, is not the same as 15% through a cycle. A stress-testing team cares because a bank trading below book is telling you the market doubts its ROE or its asset values.

About Rs 4.19 crore. The position loses Rs 500 crore times 6 times 0.0001, Rs 30 lakh, for each basis point rise in yield. A 99% one-day rise is 2.33 times 6 basis points, about 14 basis points. Rs 30 lakh times 13.96 is Rs 4.19 crore. It assumes normal yield changes and a linear price response.

Why start from DV01 instead of a price volatility?

A taxi fare is a rate per kilometre times the distance. You would not guess the fare directly; you would multiply. For a bond, DV01 is the rupee rate per basis point and the yield move is the distance, so VaR is DV01 times the yield move at the chosen confidence. Yield volatility is what the market data gives you, and duration converts it to rupees. Guessing a price volatility for the bond skips the step the interviewer wants to see.

What does this number leave out?

Three things, and a treasury risk manager names them unprompted. The estimate assumes yield changes are normal, that the price responds in a straight line, and that every bond in the book moves with the same yield. Fat tails make a 14 basis point day more common than the normal says. Convexity makes the true loss slightly smaller than the linear figure. And a book spread along the curve has curve risk: if short yields rise and long yields do not, one DV01 figure misses it. Over ten days, the square-root rule would scale this to about Rs 13.2 crore, if daily moves are independent.

Also say which way hurts. A holder of bonds loses when yields rise, so the one-sided 99% point on the upside of yields is the one that matters, which is why the figure uses 2.33 and not the two-sided 2.58.