Risk Management puzzles, solved step by step

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.

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.

About 0.87%. Oversampling inflates every score by the same factor on the odds. The sample's odds of bad are 50 to 50; the population's are 2 to 98, one forty-ninth as high. The applicant's sample odds are 30 to 70, 0.429; divide by 49 to get 0.00875, which is a probability of 0.87%. The model ranks correctly but must be recalibrated.

Why does oversampling change the score?

Suppose a doctor learns to spot a rare illness from a teaching ward where half the patients have it. In a village clinic, where only one patient in fifty has it, the same symptoms mean far less. A model trained on a 50% bad sample has learned a base rate 25 times too high, so every score it produces is inflated, even though the ordering from safest to riskiest is still right. Oversampling is done on purpose, to give the model enough bad accounts to learn from, so the correction is a routine step, not a sign of a broken model.

How do you sanity-check the answer?

Take an applicant the model scores at exactly 50%, the sample average. After correction that applicant should sit at the population average, and the formula gives exactly 2%, which confirms the factor. Then note that 30% is below the sample average, so the corrected figure should be below 2%, and 0.87% is. The shortcut of scaling the probability by 2% over 50% gives 1.2%, which is close for low scores but breaks down badly for high ones: a sample score of 90% would scale to 3.6%, while the correct answer is about 15.5%.

Say what a validator would do next. The correction assumes the good and bad accounts were each sampled at random within their class. If the bad accounts were drawn from a different period or channel, the model's ranking may also be off, and the fix is to check calibration on a recent, unsampled holdout, comparing predicted and actual bad rates by score band.

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.