Risk Management puzzles, solved step by step

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.

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.

About 0.8% get through if the checkers are truly independent. Each misses 20% of what reaches them, so 100 errors become 20, then 4, then 0.8. The real figure is likely higher because the checkers are not independent: later checkers relax because someone already looked. If checkers two and three catch only 40%, 7.2% get through, nine times as many.

Why do independent checks multiply?

Think of three sieves stacked on top of each other. If each lets one grain in five through, the first passes 20 of 100 grains, the second 4 of those 20, and the third 0.8 of the 4. When checks are independent, the chance of an error getting past all of them is the product of each one's miss rate. That is why a three-check process looks almost watertight on paper: 0.2 cubed is 0.008.

Why is the real leakage higher than 0.8%?

Because people who know others are checking check less carefully. The second checker assumes the first caught the obvious problems, and the third signs off because two colleagues already did. Checks that rely on each other are not independent, and once they are correlated the multiplication rule overstates how much they catch. There is also a common cause: an error that looks like a normal payment, a correct-looking amount to a familiar name, fools all three checkers for the same reason.

What would an operational risk team do about it?

Measure each check on its own, not the chain. Seed known errors into the flow and count what each checker catches; rotate who checks first; give each checker a different thing to look for rather than the same whole payment. The goal is to make the checks genuinely independent, because three checks that behave like one are more dangerous than one check everyone knows is the only one.

About 27.2%. One manager beats the index five years running with probability one half to the fifth, 1 in 32. The chance that none of the ten does it is 31/32 to the power 10, about 0.728. So at least one streak appears 27.2% of the time with no skill anywhere. With 100 managers it is 95.8%.

Why is a streak among many managers weak evidence?

Ask a hall of 300 people to toss a coin five times, and about nine will throw five heads. Nobody would call them skilled tossers. The chance that one named person has a streak is small, but the chance that someone in a crowd has one is large, and a fund manager with a five-year record is usually picked from a crowd. Which question you are answering decides everything: the probability for manager G, or the probability for whoever turned out to have the streak.

How do you get there fast without a calculator?

Use the expected count first: ten managers times 1 in 32 is about 0.31 streaks. When the expected count is small, the chance of at least one is a little below it, so 31% is an upper bound and the exact answer is about 27%. The gap comes from the chance of two or more streaks, which the simple sum counts twice. Saying the bound and then the exact figure shows the interviewer you can check your own work.

What does an asset manager's risk team do with this?

It sets the bar for evidence. A five-year record of beating the index is common among hundreds of funds even if none has skill, so the team looks at how the returns were earned: the size of the edge against its volatility, whether it comes from one bet or many, and whether the process explains the outcome. Survivorship matters too: the managers whose streaks broke often closed their funds, so the crowd you see is already filtered toward winners.

Not precisely enough to claim skill. The standard error of the mean is the daily volatility over the square root of the number of days: 1% over the square root of 250, about 0.063%. The estimate of 0.05% is only 0.79 standard errors from zero, and a two standard error band runs from -0.076% to 0.176%. You would need about 1,600 days, 6.4 years, to clear zero.

Why is a year of daily data not enough?

Weigh yourself on a bathroom scale that jumps by two kilos each time you step on it. If you lost 100 grams last month, a week of readings will not show it; the jumps drown the signal. The precision of an average improves only with the square root of the number of observations, so a small edge buried in large daily noise takes years to show. Here the daily noise is 1% and the edge is 0.05%, twenty times smaller.

How long would it take, and what does that mean for judging desks?

Set the t-statistic to 2 and solve for n: n equals (2 times 1% over 0.05%) squared, which is 1,600 days. An annual Sharpe ratio of about 0.79 needs more than six years of data before it is statistically distinguishable from zero. That is longer than most desks keep the same strategy, so a risk team cannot rely on the P&L record alone; it looks at whether the edge has a reason, whether it survives out of sample, and how much the result depends on a few days.

State the assumptions. The calculation treats daily returns as independent with constant volatility. Fat tails and volatility clustering make the true uncertainty larger, so six years is a floor, not a promise.

Rs 63 crore. The monthly losses double: 1, 2, 4, 8, 16 and 32 crore, and they add to 2 to the power 6, less 1, which is 63. Each month's loss is one more than everything before it combined, so the hole grows as fast as the bets. For a control function, the pattern is the signal: position size growing geometrically with no matching approved limit.

Why does doubling down grow the hole so fast?

A gambler at a roulette table who doubles his bet after every loss plans to recover everything with one win. After six losses in a row he has lost 63 times his first stake and must bet 64 times it to break even, if the table allows it. Doubling means each new bet equals all past losses plus one, so the cumulative loss is always just under the next bet, and a short losing run produces an enormous number. The strategy fails when the losing run is longer than the capital or the limit.

What should a control function see before month six?

Several things, and none needs the trader's honesty. A position growing geometrically, a P&L that is flat or smooth while the gross position explodes, and large trades that are cancelled, amended or booked to unusual accounts are the classic signs of concealed losses. Gross notional limits catch the growth even when net risk looks small. Reconciling traded volumes with confirmations from counterparties catches fictitious offsetting trades. A mandatory two-week leave rule catches positions that need the trader to keep rolling them.

The operational risk lesson is that the fraud is the smaller part of the loss. A Rs 1 crore trading loss, reported on day one, is a normal cost of business; the controls failed on the other Rs 62 crore. That is why control reviews measure time to detection, not only whether a loss was eventually found.