Risk Management puzzles, solved step by step

About 69%. This is the gambler's ruin. With r = 0.48 / 0.52, starting 10 steps from each barrier, the chance of hitting the top first is 1 / (1 + r to the power 10). r to the 10 is about 0.449, so the answer is 1 / 1.449, or 69.0%. A 2-point daily edge becomes a 19-point edge because the race takes many days.

Why does a small daily edge grow so much?

A casino wins only a couple of points more than half its bets at roulette, yet over thousands of spins it almost never ends a month behind. An edge that barely shows in one step accumulates over many, because the random part grows with the square root of the number of steps while the edge grows in proportion to it. Reaching plus or minus 10 takes around 95 days on average here, long enough for the edge of 0.04 a day to push the walk decisively one way.

How do you check the answer, and what does it say about stop-losses?

Two sanity checks. At 0.50, r is 1 and the formula gives 50%, as symmetry demands. Widen both barriers to 20 and the answer at 0.52 rises to 83.2%. The wider the barriers relative to the daily step, the more the edge dominates the noise. That is the risk manager's reading: a tight stop-loss relative to daily volatility turns a trader with a genuine edge into a near coin flip, while a wide one lets the edge show but exposes more capital to a trader without one.

State the limits. The model assumes a constant edge, equal step sizes and independent days. Real P&L has fat tails, the edge decays, and a losing streak may itself be evidence that the edge was never there. The formula tells you how an edge would play out, not whether you have one; with drawdownThe fall from a running peak in cumulative profit, measured in rupees or as a percentage. limits, that second question is usually the harder one.

Rs 15 crore. The loss takes CET1 from Rs 100 crore to Rs 80 crore, and writing off the defaulted loans takes risk-weighted assets from Rs 1,000 crore to Rs 950 crore, a ratio of 8.42%. To be back at 10% the bank needs 10% of Rs 950 crore, which is Rs 95 crore, so it must raise Rs 15 crore of new equity, assuming the new cash carries no risk weight.

Why is the answer not simply the Rs 20 crore lost?

A cricket team that needs a run rate of 6 an over has lost two overs to rain. It needs fewer runs than before, not the same total, because the target is a rate. A capital ratio is a rate too. The loss cuts the numerator by 20%, but the write-off also cuts the denominator by 5%, and the target is 10% of the new, smaller denominator. That makes the gap Rs 15 crore.

Where does the Rs 50 crore of RWA come from, and is it realistic?

Think of Rs 50 crore of defaulted loans at a 100% risk weight. The bank loses Rs 20 crore on them, a 40% loss given default, and recovers Rs 30 crore in cash, which carries no risk weight. Both sides of the ratio move on the same event, but a loss always hits capital at full size while the RWA falls only by the risk weight of what left. A write-off never rescues the ratio: 80 over 950 is still well below 10%.

The limitations are worth one line each. The new equity is assumed to sit in cash or government bonds at a zero risk weight; if the bank lends it out, RWA grows and it needs more. And a real bank must also rebuild any buffer above the minimum, whose size depends on current rules you would check rather than assume.

Betting 20% grows capital about 2.0% a bet; betting 50% shrinks it about 3.3% a bet. Growth per bet is the average log return: 0.6 ln(1 + f) + 0.4 ln(1 - f). At 20%, the Kelly fraction 2p - 1, that is +2.01%. At 50% it is -3.40%. After 100 bets the typical 20% bettor has about 7.5 times the start; the 50% bettor about 0.03 times.

How can a bet with a positive edge lose money?

A shopkeeper who stakes half his stock on every festival season, and wins more seasons than he loses, can still go under: one bad season halves him, and a good one only adds half back. Losses on a shrunken base are expensive to recover. The typical outcome of repeated betting is set by the average log return, not the average return, and at 50% the losses dominate the logs.

Where does the 20% come from?

Set the slope of the growth curve to zero. For even odds the Kelly fractionThe fraction of capital to bet each time that maximises the long-run compound growth rate of the capital. is the probability of winning minus the probability of losing: 0.6 minus 0.4, or 20%. Betting less than Kelly gives up some growth for a much smoother ride; betting more than about twice Kelly gives up growth altogether. Here the curve crosses zero near 39%, just under twice Kelly.

For a risk manager the lesson is position sizing. A desk with a genuine edge can still destroy capital by running too much leverage, and the average P&L will look fine right up until the drawdowns compound. Real edges are also uncertain, which is why practitioners bet a fraction of Kelly.

It doubles, from about 0.89% to 1.79% a day, as a share of one leg. For equal and opposite legs with the same volatility, pair variance is 2 x sigma squared x (1 - rho). The unhedged part, 1 - rho, goes from 0.1 to 0.4, four times as large, so volatility doubles. On Rs 100 crore a leg, daily volatility goes from about Rs 0.89 crore to Rs 1.79 crore.

Why does a modest fall in correlation hit a hedged book so hard?

Two boats tied close together on a river rise and fall almost as one; the rope between them barely strains. Loosen the tie and every wave that hits one boat and not the other shows up as strain on the rope. A long-short pair is the rope. It only carries the part of the two stocks that does not move together, and that part is 1 minus rho, which grows fast as correlation slips from a high level.

What does this mean for the desk's risk numbers?

VaR scales with volatility, so the pair's VaR doubles with no change in position. On Rs 100 crore a leg the 99% one-day VaR moves from about Rs 2.1 crore to Rs 4.2 crore, entirely from a correlation the desk does not control. Correlations between related stocks do fall in exactly the markets where one company's news breaks from its peer's, which is when the pair was supposed to be safe.

Say the limitation. The formula assumes equal volatilities and equal notionals; a real pair is often beta-weighted, which changes the coefficients but not the lesson. A risk limit that assumes the historical correlation of 0.9 is a limit on a number that can quietly double.

The barbell wins a large parallel move either way; it loses when the curve steepens. Both portfolios have duration 5, but the barbell's convexity is 34 against the bullet's 25. For a 200 basis point rise the barbell loses 9.35% against 9.52%, and for a fall it gains 10.72% against 10.52%. If the curve steepens 10 basis points per year around 5 years, the bullet is flat and the barbell loses 0.88%.

If the durations match, why do the portfolios behave differently?

Two families each spend an average of Rs 50,000 a month. One spends it evenly; the other spends Rs 20,000 some months and Rs 80,000 others. A small price rise hits them alike, but a big shock hits the second family's heavy months much harder. Averages match; the spread does not. Duration is the average maturity, so it only describes small, parallel moves; the barbell spreads its money further from 5 years, which gives it more convexity and exposes it to changes in the curve's shape.

Why does the steepener hurt only the barbell?

In this steepener the 5-year yield does not move, so the bullet does not move. The 2-year yield falls 30 basis points and the 8-year yield rises 30. The barbell's long leg carries four times the duration of its short leg, so the loss on the 8-year bond swamps the gain on the 2-year. Roughly: half the money times 8 years times 0.30% is a 1.2% loss, and half times 2 years times 0.30% is a 0.3% gain, a net loss of about 0.9%.

The limitation: the gains from convexity are small for small moves and are usually priced, since a barbell typically yields less than a bullet of the same duration. In a quiet market that yield give-up can outweigh the convexity benefit, so the choice is a view on volatility and curve shape, not a free lunch.

It must sell Rs 800 crore, four fifths of the position. The haircut is the share of the position the fund funds itself. At 2%, Rs 20 crore supports Rs 1,000 crore of bonds, 50 times leverage. At 10%, the same Rs 20 crore supports only Rs 200 crore, 10 times leverage. Keeping the full position would need Rs 100 crore of capital, Rs 80 crore more than it has.

Why does a small change in haircut force such a big sale?

Buying a flat with a 5% down payment lets Rs 10 lakh of savings control a Rs 2 crore flat. If the bank suddenly demands 20% down, the same Rs 10 lakh controls only a Rs 50 lakh flat. The haircutThe share of a security value a lender will not lend against, so the borrower must fund that share with its own capital. sets leverage directly, at one over the haircut, so moving it from 2% to 10% cuts the position the capital can support from 50 times to 10 times.

What happens next in a real market?

The sale itself moves prices. If many leveraged holders face the same haircut increase, they all sell the same bonds, prices fall, their capital shrinks, and the lower capital supports an even smaller position. That feedback loop between haircuts, prices and forced selling is the liquidity spiral seen in past funding crises. A fund at 50 times leverage loses its entire Rs 20 crore of capital on a 2% price fall.

The limitation: the answer assumes the fund cannot raise capital or find another lender at the old haircut. In practice it would try both, and a risk manager's job is to know in advance how many days that would take and how much the forced sale would cost at stressed prices.

10,100 trading days on average. Let E0 be the expected wait from no current run and E1 the wait after one exception. Then E0 = 1 + 0.99 E0 + 0.01 E1 and E1 = 1 + 0.99 E0. Solving gives E0 = (1 + p) / p squared = 1.01 / 0.0001 = 10,100, about 40 years of trading days if exceptions really are independent.

Why is the answer not simply one over p squared?

Waiting for two heads in a row with a fair coin takes six tosses on average, not four, because each time you get one head and then a tail, you are back to the start and the first head was wasted. A run has memory: after one exception you are one step from the finish, and a miss throws you back, so the waiting time needs one equation per state, not one probability. For VaR exceptions at 1% the extra is 100 days, the time spent waiting for each first exception.

Why would a risk manager care about a 40-year wait?

Because it turns an observation into a test. If the model is right and exceptions are independent, two in a row should take about forty years of trading to appear; seeing them twice in one year says the exceptions cluster, and clustering means the model misses changes in volatility. Backtesting frameworks test independence for exactly this reason, alongside the plain count of exceptions.

The limitation is the independence assumption, which is the thing being tested. Real exceptions tend to arrive in bursts when volatility jumps, so the observed wait is usually far shorter, and that gap is evidence against the model rather than bad luck.

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.

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.

Bank B, with about 6.05% of assets in equity against 3.5% for bank A. Equity to assets is the CET1 ratio times the share of assets that count as risk-weighted: 14% times 25% for A, 11% times 55% for B. A's higher ratio rests on low risk weights, so it is levered about 29 times against B's 16.5 times.

Why can a higher capital ratio mean less capital?

Two people each say they save 20% of their income. One counts only the salary left after rent; the other counts the whole salary. The first sounds equally thrifty but saves far fewer rupees. A CET1 ratio divides equity by risk-weighted assets, not total assets, so a bank that assigns low risk weights to its loans can show a high ratio on a thin equity base. The share of assets that ends up risk-weighted is called {term('RWA density', 'Risk-weighted assets divided by total assets; a low figure means the bank judges most of its assets to be low risk.')}, and it is the number that reconciles the two views.

Which number should a credit analyst trust?

Both, because they answer different questions. The CET1 ratio is right if the risk weights are right. Equity to assets makes no judgement about risk, so it is the check on whether the risk weights are doing too much of the work. Test A's number: if its true risk density were 40% rather than 25%, perhaps because its internal models are optimistic, its CET1 ratio would be 3.5% over 40%, only 8.75%. That is why regulators set a leverage ratio floor alongside risk-based ratios; the current minimum is set by the regulator, so confirm it before quoting it.

What would make bank A's low density legitimate?

A book of home loans with low loan-to-value ratios, or large holdings of government bonds, genuinely carries low risk weights, and a 25% density can be honest. The question is whether the weights are earned. An analyst asks what the assets are, whether the density has fallen while the book stayed the same, and how the bank's model-based weights compare with the standardised ones.