Risk Management puzzles, solved step by step

About 5.26% a year with annual compounding, or 5.13% continuously compounded. Parity says call minus put equals stock minus the present value of the strike. Here 12 minus 7 is 5, so the present value of 100 must be 95. A one-year discount factor of 0.95 means 100 over 95 minus 1, about 5.26%.

Why must call minus put equal the stock minus the discounted strike?

Agreeing today to buy a house next year at a fixed price is the same as buying it now with money borrowed until then: either way you own the house next year and pay the fixed price. A long call plus a short put at the same strike is that agreement. At expiry the pair pays the stock minus the strike in every state, so today it must cost the same as owning the stock and owing the strike in a year, which is S minus the present value of K.

Why would a risk manager care about the rate hidden in option prices?

Because it is a check that comes free. If the rate implied by parity sits far from the funding rate the desk actually pays, either the marks are stale, a dividend has been missed, or the options are American and parity no longer holds exactly. Model validation teams run exactly this test on option books to catch mispriced marks before they show up as a loss.

The limitation: parity holds exactly only for European options on a stock paying no dividend before expiry. A dividend would lower the stock's forward and push the implied rate the other way, so state the assumption before you give the number.

About 81.8%. Retail defaults make up 60% x 3% = 1.8% of the book and corporate defaults 40% x 1% = 0.4%, so 2.2% of all loans default. Given that a loan is among the 2.2%, the chance it is retail is 1.8 divided by 2.2, which is 9 in 11, or about 82%.

Why is the answer not simply 60%?

Suppose 60% of the cars on a road are small hatchbacks and 40% are trucks, but trucks are far more likely to have a broken tail light. If a policeman stops a car for a broken tail light, it is more likely to be a truck than the 40% share suggests. The stop tells you something. Learning that a loan defaulted is evidence, and it shifts the odds towards the segment that defaults more often.

What is the quickest way to do this in your head?

Use 1,000 loans. There are 600 retail loans, of which 18 default, and 400 corporate loans, of which 4 default. Out of 22 defaults, 18 are retail, so the answer is 18 over 22, and natural numbers make the base rate impossible to forget. This is Bayes' rule without the notation, and it is the fastest way to be right out loud.

For a risk team this is not an exam trick. The mix of a defaulted pool drives which collections team gets the work, which recovery rate applies and which model needs recalibrating. A 60% share of the book becomes an 82% share of the defaults.

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 4.0 crore, or 0.4 standard deviations. Exposure is the forward's value when it is positive and zero when it is negative. For a normal distribution with mean zero, the expected value of the positive part is sigma divided by the square root of 2 pi, about 0.399 x Rs 10 crore. Half the outcomes contribute nothing; the other half average about Rs 8 crore.

Why is the exposure positive when the expected value is zero?

Lending a friend your bicycle is a one-sided risk: if the friend loses it, you are out a bicycle; if the friend buys you a better one, you gain nothing from the loan itself. Counterparty exposure has the same shape. When the contract is in your favour, a default costs you its value; when it is against you, you still owe the full amount, so negative outcomes count as zero, not as a gain. Cutting off the left half leaves a positive average.

Where does 0.4 come from, and what is it called?

Integrating v times the normal density from zero to infinity gives sigma times the density at zero, which is 1 over the square root of 2 pi, about 0.399. At a single future date desks usually call this number the expected exposure; averaged over the life of the trade it becomes expected positive exposure, the input to credit valuation adjustment. The tail measure, potential future exposureA high percentile of the exposure distribution at a future date, used to size limits on how much a counterparty could owe. at 97.5%, is 1.96 sigma, about Rs 19.6 crore.

The limitation: this assumes no collateral and no netting with other trades. A daily margin agreement would cut the exposure to the move over the few days it takes to call and receive margin, which is far smaller than a year's worth of drift.

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.

About Rs 7,000 crore of value lost for a 1 basis point rise. A bond's price falls by roughly its modified duration times the change in yield. For the whole stock that is Rs 100 lakh crore x 7 x 0.0001, which is 0.07 lakh crore, or Rs 7,000 crore. The same arithmetic makes a 100 basis point rise cost about Rs 7 lakh crore.

Why does a hundredth of a per cent move so much money?

A one paisa rise in the price of petrol means nothing to one driver and a great deal to an oil company selling crores of litres. The move is small; the base is enormous. The DV01The change in value of a bond or portfolio for a one basis point change in yield, sometimes called PV01. of a holding is value times duration times one basis point, so a stock worth Rs 100 lakh crore with a duration of 7 carries Rs 7,000 crore of value per basis point.

How do you keep the units straight under pressure?

Convert once, at the end, and say the conversion aloud. One lakh crore is 1,00,000 crore, so 0.07 lakh crore is 7,000 crore; do the multiplication in the big unit and translate only the answer. A useful check: duration 7 means a 1% move changes value by 7%, so 1 basis point, a hundredth of that, changes it by 0.07%, and 0.07% of 100 lakh crore is 0.07 lakh crore.

State the limits. The stock and duration are assumptions for the puzzle, not current figures, which you would look up. The duration rule ignores convexity, harmless at one basis point but not at a hundred, where the true fall is a little smaller than Rs 7 lakh crore. And the loss lands on whoever holds the bonds, which is why a bank's treasury watches the DV01 of its government bond book daily.

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 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.

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.