Risk Management puzzles, solved step by step

Puzzles: 100
Traced to a firm: 17
Topics: 13
Hard: 30

Showing 11–20 of 83 · filtered from 100Clear filters

014A stock goes from 100 to 150 and back to 100. Compare the average of the simple returns with the average of the log returns.Compounding and drawdownsCoreAsset manager riskQuant risk
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What is the average simple return over the two periods?
Show the worked solution
The simple returns average +8.3%, the log returns average 0%. The simple returns are +50% and -33.3%, and their average suggests a gain although the price is back at 100. The log returns are +40.5% and -40.5%, which add to zero and match what happened. Log returns add over time; simple returns do not.
Why does the simple average show a gain that never happened?
Imagine a shop that raises a price from Rs 100 to Rs 150 and then cuts it back to Rs 100. The rise is 50 on a base of 100, and the cut is 50 on a base of 150. Simple returns are each measured against a different starting price, so averaging them mixes percentages of different bases and overstates growth whenever prices bounce around. The arithmetic mean of +50% and -33.3% is +8.3%, but the investor has exactly the money they started with.
For a price that goes 100, 150, 100, simple returns of +50% and -33.3% average +8.3% although the price made nothing, while log returns of +40.5% and -40.5% add to zero and average zero.
The relationship
$r_t = \ln\frac{P_t}{P_{t-1}}, \qquad r_1 + r_2 = \ln\frac{150}{100} + \ln\frac{100}{150} = \ln\frac{100}{100} = 0$
r_t the log return in period t
P_t the price at the end of period t
What it says in wordsLog returns add up across periods to the log of the total change, so a round trip sums to zero.
Which one should a risk manager use?
It depends on what you are adding up. Log returns add across time, so they are the natural choice for compounding a return over many days and for most statistical models of a single asset. Simple returns add across assets, so a portfolio's return for one period is the weighted average of its holdings' simple returns, which log returns do not give you. The gap between the two averages is the volatility dragThe shortfall of compound growth below the arithmetic average return, which grows with the variance of returns.: roughly half the variance, and here the swings are wild enough to make it 8 points.
Close with the practical warning. A fund that reports the arithmetic average of its yearly returns will look better than the growth its investors actually got, and the more volatile the fund, the larger the flattering gap. The honest single number for growth over time is the geometric average, which here is zero.
Where candidates lose it
The trap is quoting +8.3% as the average return and calling it performance. It is a correct average of the wrong thing: percentages taken on different bases.
The second miss is saying log returns are simply better. They add over time but not across assets, so a risk manager needs both and should say when each applies.
What the interviewer asks next
What is the geometric average return here, and how does it relate to the log returns?
A fund returns +20% and -20% in alternate years. What is its compound growth rate?
Why do most VaR models use log returns for single assets but simple returns to aggregate a portfolio?
015A 60/40 portfolio holds equities with 15% volatility and bonds with 6% volatility, and the correlation between them is minus 0.2. What is the portfolio volatility?Correlation and diversificationCoreAsset manager risk
Try it first
Pick the portfolio volatility.
Show the worked solution
About 8.84%. The equity term is 0.6 times 15, squared, which is 81. The bond term is 0.4 times 6, squared, 5.76. The cross term is 2 times 0.6 times 0.4 times minus 0.2 times 15 times 6, which is minus 8.64. The variance is 78.12 and its square root is 8.84%, well below the 11.4% weighted average.
Why is the answer not the weighted average of 15% and 6%?
Think of two friends walking a dog on separate leads. If they always pull the same way, the dog is dragged as far as their combined pull. If one sometimes pulls left while the other pulls right, the pulls partly cancel. Volatilities only add in a straight line when the correlation is exactly one; any lower correlation lets the swings offset, and a negative one subtracts risk outright. The weighted average of 15% and 6% is 11.4%, the answer for a correlation of one.
The relationship
$\sigma_p^2 = (w_1\sigma_1)^2 + (w_2\sigma_2)^2 + 2\,w_1 w_2\,\rho\,\sigma_1\sigma_2 = 81 + 5.76 - 8.64 = 78.12$
w_1, w_2 the weights, 0.6 in equities and 0.4 in bonds
sigma_1, sigma_2 the volatilities, 15% and 6%
rho the correlation, minus 0.2
What it says in wordsAdd each asset's own variance contribution, then add the cross term, which is negative when the correlation is negative.
Equities contribute 81 and bonds 5.76 squared percentage points of variance, and the minus 0.2 correlation subtracts 8.64, leaving 78.12, a volatility of 8.84% against 9.31% at zero correlation and 11.4% at a correlation of one.
What do you add after the number?
Two things. First, equities carry almost all the risk: 81 of the 86.76 squared points before the cross term, so a 60/40 portfolio is mostly an equity risk portfolio with a bond cushion. A risk contributionThe share of a portfolio total variance that comes from one holding, including its share of the cross terms. breakdown makes that visible and is usually the next question. Second, the minus 0.2 is an estimate from history, and correlations between equities and bonds have changed sign across decades. If it turned positive at plus 0.3, the volatility would rise to about 10%.
Say the limit too. Volatility treats upside and downside swings alike and assumes the correlation holds in a crisis. In a sharp sell-off correlations can move together, so the diversification shown here is the benefit in normal conditions, not a promise for the worst month.
Where candidates lose it
The trap is answering 11.4%, the weighted average of the volatilities, which ignores diversification entirely. The second is getting the sign of the cross term wrong and adding 8.64 instead of subtracting it.
Say the formula before the numbers, and square the weighted volatilities first: 9 squared and 2.4 squared are easier out loud than 0.36 times 225.
What the interviewer asks next
What bond weight minimises the portfolio volatility?
What is the portfolio volatility if the correlation is plus 0.3?
What share of the portfolio's risk comes from equities once the cross term is split between the two?
016A swap has an expected positive exposure of about Rs 5 crore across its five-year life. The counterparty's credit spread is 200 basis points. Roughly what is the credit valuation adjustment?Counterparty exposure and collateralHardCounterparty riskQuant risk
Try it first
Which rough estimate is closest?
Show the worked solution
Roughly Rs 45 lakh. The credit spread is what the market charges each year for bearing the counterparty's default risk. Rs 5 crore of expected exposure at 2% is Rs 10 lakh a year. Over five years that is Rs 50 lakh, and discounting at an assumed 4% brings it to about Rs 44.5 lakh. That is the value you give up for trading with a counterparty that can default.
Why does a spread times an exposure give you CVA?
Think of a friend who owes you money and a moneylender who would charge that friend 2% more a year than a safe borrower. That 2% is the market's price for the chance your friend does not pay. The credit valuation adjustmentThe reduction in the value of a derivative to reflect the chance that the counterparty defaults while owing you money. is roughly the counterparty's credit spread charged on the amount it is expected to owe you, year by year, discounted to today. The spread already folds together the chance of default and the loss if it happens.
Rs 5 crore of expected exposure charged at a 2% credit spread costs Rs 10 lakh a year, and discounting each year at 4% gives Rs 9.62, 9.25, 8.89, 8.55 and 8.22 lakh, a rough CVA of about Rs 44.5 lakh.
The relationship
$\text{CVA} \approx s \sum_{t=1}^{5} \text{EPE}_t \, D_t = 0.02 \times 5 \times \sum_{t=1}^{5} 1.04^{-t} \approx 0.445 \text{ crore}$
s the counterparty's credit spread, 2% a year
EPE_t expected positive exposure in year t, Rs 5 crore
D_t the discount factor for year t, at an assumed 4%
What it says in wordsCharge the spread on each year's expected exposure, discount it to today, and add the years.
What does the rough version leave out?
Three things, each worth a sentence. First, the rough formula ignores that a counterparty that defaults in year two cannot default again in year three, so it slightly overstates the charge. Modelled explicitly, with a 60% loss given default and a constant default rate implied by the spread, the CVA comes to about Rs 41 lakh. Second, exposure on a swap is rarely flat; it usually rises and then falls as payments are made. Third, if the counterparty's credit worsens exactly when your exposure grows, the charge is larger, which is wrong-way risk.
The discount rate here is an assumption for the illustration, and the market spread would come from the counterparty's bonds or credit default swaps where they trade. The structure is what the interviewer wants: exposure profile, times default cost, discounted and summed.
Where candidates lose it
The trap is answering Rs 10 lakh, one year of the spread, or treating the spread as a one-off probability of losing the whole exposure. The spread is an annual rate, and the exposure lasts five years.
The second loss is not naming the assumptions. Say that you used a flat exposure, a 4% discount rate and the spread as a stand-in for default probability times loss given default.
What the interviewer asks next
How does a collateral agreement with a zero threshold change this CVA?
The exposure rises from Rs 2 crore in year one to Rs 8 crore in year five, with the same average. Is CVA higher or lower?
What is debit valuation adjustment, and why do some people dislike booking it as profit?
018A bond callable at 102 trades at 101. If yields fall 100 basis points, does its price rise as much as an otherwise identical non-callable bond, and what happens to its duration?Duration and ratesCoreTreasury and ALMBank market risk
Try it first
When yields fall 100 basis points, roughly how does the callable bond move?
Show the worked solution
No. The callable bond rises only about 1.0 point, to about 102.0, against about 7.2 points for the straight bond, and its duration collapses. The issuer will call the bond once refinancing is cheaper, so investors will not pay much above 102. In this stylised example its effective duration is about 3.4 against 6.8 for the straight bond, and it shrinks further as yields fall.
Why can the callable bond not keep rising?
Think of a home loan with no prepayment penalty. When rates fall, the borrower refinances, and the lender who was enjoying a high rate gets the money back. A call option lets the issuer do the same: when yields fall enough, it buys the bond back at 102, so no investor will pay much more than 102 for it. The price is effectively capped at the call price, while a straight bond with the same coupon and maturity keeps gaining as yields fall.
In this stylised 10 year 8% bond, a 100 basis point fall in yield lifts the straight bond from 101.7 to 108.9 but the callable bond only from 101.0 to 102.0, because its price flattens under the 102 call price.
What happens to duration and convexity?
Duration is how much the price moves for a yield change, and near the cap the callable bond hardly moves. As yields fall towards the level where the call is exercised, the callable bond's effective duration shrinks, and its price curve bends the wrong way: negative convexityWhen a bond gains less from a fall in yields than it loses from an equal rise, because its price curve bends downward.. Here a 100 basis point rise costs both bonds about the same, roughly 6 points, while a 100 basis point fall gives the callable bond only about 1. That lopsided payoff is the price of the call the investor has sold to the issuer, and it is paid for through a higher yield.
Be clear about the model. The curve here is a stylised cap chosen for illustration, not a full option pricing model; a desk would use an interest rate model to value the call and compute effective duration by bumping the whole yield curve. The shape is what matters for the interview: gains capped, losses intact, duration that shortens exactly when you would want it long.
Where candidates lose it
The trap is applying the straight bond's duration and predicting a 7 point gain. Standard duration assumes the cash flows are fixed, and a callable bond's cash flows change when the call is exercised.
The second miss is saying duration rises because the price is near par. Effective duration falls as the call becomes more likely, because the bond starts to behave like a short bond ending at the call date.
What the interviewer asks next
Why do mortgage-backed securities show the same pattern?
How would you hedge a portfolio of callable bonds against falling yields?
What would a putable bond's price curve look like against the same straight bond?
019Three traders, Asha, Bilal and Chen, are asked who breached a limit. Exactly one of them did, and exactly one of them tells the truth. Asha says Bilal did it, Bilal says he did not, and Chen says he did not. Who breached the limit?Logic, estimation and brainteasersCoreRating agencyRisk GCC
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Who breached the limit?
Show the worked solution
Chen breached the limit, and Bilal is the one telling the truth. Assume each trader in turn is the culprit and count the true statements. Asha as culprit makes two statements true, Bilal's and Chen's. Bilal as culprit also makes two true, Asha's and Chen's. Only Chen as culprit leaves exactly one true statement, Bilal's denial, which matches the rules.
Where do you start when every statement might be a lie?
Think of a lost house key and three rooms. Rather than arguing about who saw what, you check each room in turn and stop when one fits. With three possible culprits, the fastest route is to assume each one did it, work out which statements are then true, and keep the case that matches the rule of exactly one truth. That turns an argument about who is lying into three short checks.
Assuming Asha is the culprit gives two true statements and assuming Bilal gives two, both breaking the rule, while assuming Chen gives exactly one true statement, Bilal's, so Chen breached the limit.
Is there a shortcut that finds it faster?
Yes: look for two statements that cannot both be false. Asha says Bilal did it and Bilal says he did not, so exactly one of those two is true whoever the culprit is. That uses up the single true statement, which means Chen's statement must be false. Chen saying he did not do it is false, so Chen did it. Saying this pairing out loud shows the interviewer you look for structure before grinding through cases.
Then connect it to the job, in one line. A limit breach investigation often starts with conflicting accounts, and the same discipline applies: list the possibilities, check each against the hard evidence, and discard the ones that contradict it. The trade records play the part of the rule here; the accounts are only as good as what they agree with.
Where candidates lose it
The trap is trusting the first statement and following it: Asha accuses Bilal, so candidates test Bilal first and then get tangled. Starting from an accusation lets the story steer you.
The other loss is solving it silently. Say each assumption and its count of true statements; the reasoning is what the interviewer scores.
What the interviewer asks next
Change the rule to exactly one liar. Who breached now?
Add a fourth trader, Dev, who says Asha is lying. Does the answer change?
How would you set up a truth table for this in a spreadsheet?
020An auditor samples 60 of 3,000 trades and finds no booking errors. What error rate can you rule out at 95% confidence?Operational loss and fraudHardOperational riskBig Four risk advisory
Try it first
What can you conclude from zero errors in 60?
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You can rule out error rates above about 4.9%, and nothing below. If the true error rate were p, the chance of 60 clean trades would be one minus p, to the power 60. That falls to 5% at p of about 4.87%. The rule of three gives the same answer quickly: 3 divided by 60 is 5%. With a real error rate of 1%, a clean sample happens 55% of the time.
Why does a clean sample not prove a clean book?
Think of tasting one spoonful from a large pot to check the salt. A good spoonful tells you the pot is not badly over-salted, but a few salty patches could easily be missed. Zero errors in a sample puts an upper bound on the error rate; it never proves the rate is zero. At a true error rate of 1%, one trade in a hundred is wrong, yet 60 random trades would all be clean 55% of the time. The sample simply is not large enough to see errors that rare.
The chance that a sample of 60 trades shows no errors is 55% when the true error rate is 1% and falls to 5% only at an error rate of 4.87%, so a clean sample rules out rates above about 4.9% but nothing below.
The relationship
$(1-p)^{60} = 0.05 \;\Rightarrow\; p = 1 - 0.05^{1/60} \approx 4.87\% \qquad \text{rule of three: } p \approx \frac{3}{n} = 5\%$
p the true error rate in the population of trades
60 the sample size
0.05 the chance you accept of being wrong, for 95% confidence
What it says in wordsFind the error rate at which a clean sample would be a one-in-twenty event; anything higher is ruled out.
Where does the rule of three come from, and does the 3,000 matter?
The chance of zero errors is roughly e to the power of minus n times p, and e to the minus 3 is about 5%. So n times p equal to 3 marks the 95% bound, which gives the rule of three: divide 3 by the sample size. The population of 3,000 barely matters here because the sample is only 2% of it. Sampling without replacement tightens the bound slightly: counting exactly, the book could hold at most 144 errors, about 4.80%, instead of the 4.87% the simple formula gives.
Then give the practical point. If the firm's tolerance for booking errors is 1%, a sample of 60 cannot confirm it; you need about 300 clean trades, 3 divided by 1%, to rule out 1%. The limit is that the sample must be random; a sample of the easiest trades to check says little about the ones that go wrong.
Where candidates lose it
The trap is reporting that the error rate is zero, or that it is below one in 60. Neither follows. A clean sample of 60 is quite likely even when one trade in a hundred is wrong.
The second miss is freezing on the exact formula. Give the rule of three first, then refine it to 4.87% if asked.
What the interviewer asks next
How many trades must you sample, all clean, to rule out a 0.5% error rate?
The sample of 60 finds one error. What upper bound can you now give?
Why might a random sample still understate errors in complex trades?
021A desk is short 10,000 put options on a Rs 500 stock. Each put has a delta of minus 0.3 and a gamma of 0.004 per rupee. The stock falls 10%. Compare the delta-only loss estimate with the delta-gamma estimate.Options and GreeksHardBank market riskModel validation
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How much larger is the delta-gamma loss than the delta-only loss?
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Delta alone says a Rs 1.5 lakh loss; delta and gamma say Rs 2.0 lakh, a third more. The stock falls Rs 50. Each put gains 0.3 times 50, Rs 15, from delta, plus half of 0.004 times 50 squared, Rs 5, from gamma. Short 10,000 puts, the desk loses Rs 20 a put. A full revaluation in this example gives about Rs 2.06 lakh, so the gamma term captures most of the gap.
Why does the delta estimate fall short?
Think of driving downhill with the brakes a little weak. The first hundred metres feel manageable, but every metre adds speed, so the second hundred covers more ground than the first. A put gains value faster the further the stock falls, because its delta grows more negative as it moves towards being in the money; a short put position therefore loses at an accelerating rate. Delta is the slope at today's price, so a straight line from it misses the curve. GammaHow much an option delta changes for a one rupee move in the underlying price. measures that bend.
The relationship
$\Delta V \approx \delta\,\Delta S + \tfrac{1}{2}\,\Gamma\,(\Delta S)^2 = (-0.3)(-50) + \tfrac{1}{2}(0.004)(2{,}500) = 15 + 5 = 20$
delta the put delta, minus 0.3
Gamma the put gamma, 0.004 per rupee
Delta S the stock move, minus Rs 50
Delta V the change in each put's value, Rs
What it says in wordsEach put gains Rs 15 from the slope and Rs 5 from the bend; the desk is short, so it loses both.
For a 10% fall in the stock, the straight delta line predicts a Rs 1.5 lakh loss on 10,000 short puts, but the true P&L curve bends away to about Rs 2.06 lakh, and the delta-gamma estimate of Rs 2.0 lakh lands close to it.
Why does this matter for VaR?
Because a VaR built on delta alone treats every option book as a straight line. For a book that is short options, the straight line understates the loss in exactly the large moves that VaR is meant to capture, and the understatement grows with the square of the move. On a 5% fall the gamma term adds Rs 1.25 a put against Rs 7.50 from delta, one sixth; on a 10% fall it adds a third; on a 20% fall it would add two thirds. A risk team would use delta-gamma at minimum and full revaluation for large scenarios.
Say the limit of the gamma fix too. Delta-gamma is still an approximation: gamma itself changes as the stock moves, and volatility usually jumps when a stock falls 10%, which adds a vega loss for a short option book. In this example, built with an assumed 30% volatility and a strike of about Rs 463, full revaluation gives Rs 2.06 lakh, a little above the delta-gamma figure, before any volatility move.
Where candidates lose it
The trap is stopping at the delta answer, Rs 1.5 lakh, and treating gamma as a rounding detail. On a 10% move it adds a third to the loss, and for a short option book it always adds, never subtracts.
The second slip is the sign. Gamma is positive for the option holder; the desk is short, so the gamma term increases its loss whichever way the stock moves.
What the interviewer asks next
What would the P&L be if the stock rose 10% instead?
How many shares would you trade to delta hedge the book, and does that remove the gamma loss?
Implied volatility rises 5 points as the stock falls. What else would you need to estimate the full loss?
023You estimate a correlation of 0.5 between two assets from 36 monthly observations. Roughly what is its standard error, and is a later estimate of 0.3 from the next 36 months evidence that the relationship has changed?Statistics and estimationHardQuant riskAsset manager risk
Try it first
Is the drop from 0.5 to 0.3 statistically meaningful?
Show the worked solution
The standard error is about 0.125, and 0.3 is not evidence of a change. A rough rule is one minus the correlation squared, over the square root of n: 0.75 over 6. Using Fisher's transformation, the 95% interval for 0.5 runs from about 0.21 to 0.71, and 0.3 sits inside it. Testing the difference between two 36 month estimates gives z of about 0.97, well below 1.96.
Why is a correlation from 36 months so imprecise?
Think of judging whether two friends like the same films after watching three dozen films together. They might agree on 20 by chance even if their tastes barely overlap. A correlation is estimated from how often two series move together, and with 36 points random coincidences move the estimate a lot: the standard error of 0.5 is about 0.125, a quarter of the estimate itself. The rough rule is one minus the correlation squared over the square root of the sample size, 0.75 over 6.
The relationship
$\text{SE}(\hat\rho) \approx \frac{1 - \rho^2}{\sqrt{n}} = \frac{0.75}{6} = 0.125 \qquad z = \tanh^{-1}(\hat\rho), \;\; \text{SE}(z) = \frac{1}{\sqrt{n - 3}}$
rho the estimated correlation, 0.5
n the number of observations, 36
tanh^{-1} Fisher's transformation, which makes the sampling error close to normal
What it says in wordsThe error of a correlation shrinks only with the square root of the sample size, and Fisher's transformation gives a fair interval around it.
The 95% interval for a correlation of 0.5 from 36 months runs from 0.21 to 0.71, and the interval for 0.3 from another 36 months runs from -0.03 to 0.57, so the two overlap heavily and the drop is within sampling noise.
How do you test whether the relationship changed?
Transform both estimates with Fisher's z, which turns 0.5 into 0.549 and 0.3 into 0.310. The difference, 0.240, divided by its standard error of the square root of 2 over 33, about 0.246, gives a z of 0.97: well short of the 1.96 needed for 95% confidence. A gap that size would appear by chance about 33% of the time even if nothing had changed. The honest statement is that the data cannot tell 0.5 from 0.3.
Then say why a risk manager cares. Hedge ratios, portfolio volatility and diversification benefits are all built on correlations like this one, and treating 0.5 as exact makes a model look more certain than its data allow. The practical responses are longer or higher-frequency data, shrinking noisy estimates towards a sensible prior, and stress testing the portfolio at correlations across the whole interval rather than at the point estimate.
Where candidates lose it
The trap is reading the drop from 0.5 to 0.3 as a regime change. It is a 40% fall in the number but well inside the noise that three years of monthly data carry.
The second miss is quoting a standard error without saying it is rough. The simple formula is an approximation; Fisher's transformation is what you use for an interval or a test.
What the interviewer asks next
How many monthly observations would you need to bring the standard error down to 0.05?
Would weekly data over the same three years help, and what could go wrong?
How would you stress test a portfolio whose risk depends on this correlation?
024A desk's daily P&L has a standard deviation of Rs 2 crore. The head of the desk wants a loss number that is exceeded about one day a month. Which confidence level is that, and what is the VaR?VaR and expected shortfallCoreBank market riskTreasury and ALM
Try it first
Which confidence level matches one bad day a month?
Show the worked solution
About 95% confidence, and a VaR of about Rs 3.3 crore. A month has about 21 trading days, so one breach a month is a tail of 1 in 21, about 4.8%, a confidence level of about 95.2%. For a normal P&L that cut sits 1.67 standard deviations out, so the VaR is 1.67 times Rs 2 crore, about Rs 3.34 crore. At a round 95% it is Rs 3.29 crore.
How do you turn a frequency into a confidence level?
Think of a weather forecast that says a heavy rain day comes about once a month. That is the same statement as saying about 1 day in 30 is that wet. A VaR confidence level is just a frequency of bad days written as a percentage: one breach in 21 trading days is a 4.8% tail, which is a 95.2% confidence level. Counting trading days matters; using 30 calendar days would make the tail 3.3% and the level about 96.7%.
For daily P&L with a Rs 2 crore standard deviation, one loss day in 21 sits beyond Rs 3.34 crore, 1.67 standard deviations out, which is close to the Rs 3.29 crore 95% VaR.
The relationship
$\text{VaR} = z_{c}\,\sigma = 1.668 \times 2 \approx 3.34 \text{ crore} \qquad z_{95\%} = 1.645 \Rightarrow 3.29 \text{ crore}$
z_c the number of standard deviations that leaves the chosen tail below it
sigma the daily P&L standard deviation, Rs 2 crore
What it says in wordsMultiply the daily standard deviation by the number of standard deviations that matches the chosen frequency of bad days.
Why is this translation useful on a desk?
Because a desk head can act on a frequency and cannot act on a percentage. Saying we expect to lose more than Rs 3.3 crore about once a month, roughly 12 days a year, tells the head of the desk what to expect and when to worry. It also sets up the backtest: if losses beyond Rs 3.3 crore start happening three or four times a month, the model is understating risk, and that is visible within a quarter rather than a year.
Add the limits. The VaR says nothing about how bad the bad day is; a loss of Rs 3.4 crore and one of Rs 10 crore both count as one breach. The normal assumption also understates fat tails, so the real one-in-21 loss may sit further out than Rs 3.3 crore. And the Rs 2 crore standard deviation is itself an estimate that moves with market conditions.
Where candidates lose it
The trap is picking 99% because it is the regulatory number people remember. A 99% VaR is breached about once in 100 trading days, closer to once every five months than once a month.
The second miss is counting 30 calendar days. Losses happen on trading days, so the tail is 1 in 21, not 1 in 30.
What the interviewer asks next
What loss would the desk expect to exceed once a year?
The desk had five breaches last month. What do you conclude, and what would you check first?
How would the answer change if the P&L had fat tails with the same standard deviation?
026A Rs 100 crore loan earns Rs 3 crore of net interest income a year, costs Rs 0.8 crore to run, carries an expected loss of Rs 1 crore and needs Rs 8 crore of economic capital. What is its risk-adjusted return on capital, and does it clear a 14% hurdle?Capital and leverageCoreBank credit riskRisk GCC
Try it first
Where does the Rs 1 crore of expected loss go in the calculation?
Show the worked solution
RAROC is 15.0%, so the loan clears a 14% hurdle, but only just. Income of Rs 3 crore less Rs 0.8 crore of cost and Rs 1 crore of expected loss leaves Rs 1.2 crore of risk-adjusted profit. Divided by Rs 8 crore of economic capital that is 15.0%. After a 14% charge on the capital, Rs 1.12 crore, the loan adds only Rs 0.08 crore a year.
Why does expected loss sit in the numerator and unexpected loss in the denominator?
Think of a restaurant that knows about two plates in a hundred get dropped. That breakage is priced into the menu; it is a running cost. What the owner keeps cash aside for is the night a whole shelf comes down. Expected loss is the average, so it is a cost charged against income; unexpected loss is the bad year above the average, so it is covered by capital. RAROCRisk-adjusted return on capital: profit after costs and expected loss, divided by the capital held against unexpected loss. puts each in its proper place, which is why a bank can compare a thin-margin safe loan with a fat-margin risky one on the same scale.
Net interest income of Rs 3.0 crore, less Rs 0.8 crore of cost and Rs 1.0 crore of expected loss, leaves Rs 1.2 crore of risk-adjusted profit; over Rs 8 crore of economic capital that is a 15.0% RAROC, one point above the 14% hurdle.
The relationship
$\text{RAROC} = \frac{\text{income} - \text{cost} - \text{EL}}{\text{economic capital}} = \frac{3.0 - 0.8 - 1.0}{8} = 15.0\%$
EL expected loss, the average annual credit loss, Rs 1 crore
economic capital the buffer held against losses worse than average, Rs 8 crore
What it says in wordsTake the average loss off the income, then ask how hard the remaining profit works the capital held against a bad year.
How much room is there before the loan fails the hurdle?
Turn the hurdle into rupees. A 14% return on Rs 8 crore of capital is Rs 1.12 crore a year, and the loan earns Rs 1.2 crore, so it clears by Rs 0.08 crore. A margin that thin means one small change flips the answer. If expected loss rises from Rs 1.0 crore to Rs 1.08 crore, or economic capital rises from Rs 8 crore to Rs 8.57 crore, RAROC falls to exactly 14%. Saying those two breakeven numbers is what turns a ratio into a credit judgement.
State the limits too. This is a pre-tax, single-year figure; many banks compute RAROC after tax and add the return earned on the capital itself, which would lift the number. And the Rs 8 crore depends on the bank's own capital model, so the answer is only as good as that model's view of this borrower.
Where candidates lose it
The common slip is to subtract expected loss and then also add it to the capital, or to leave it out of the numerator on the grounds that capital covers losses. Either way the ratio comes out wrong by several points, and the interviewer hears that you do not know what capital is for.
The second slip is stopping at 15% and saying yes. A one-point margin deserves the breakeven sentence: how much worse can the loss or the capital get before the loan stops paying for itself.
What the interviewer asks next
The bank prices the loan 20 basis points lower to win the deal. Does it still clear the hurdle?
How would you estimate the Rs 8 crore of economic capital in the first place?
Why might a bank still make a loan with a RAROC below its hurdle?

r_t	the log return in period t
P_t	the price at the end of period t

w_1, w_2	the weights, 0.6 in equities and 0.4 in bonds
sigma_1, sigma_2	the volatilities, 15% and 6%
rho	the correlation, minus 0.2

s	the counterparty's credit spread, 2% a year
EPE_t	expected positive exposure in year t, Rs 5 crore
D_t	the discount factor for year t, at an assumed 4%

p	the true error rate in the population of trades
60	the sample size
0.05	the chance you accept of being wrong, for 95% confidence

Try it first

Why does the simple average show a gain that never happened?

Which one should a risk manager use?

Where candidates lose it

What the interviewer asks next

Try it first

Why is the answer not the weighted average of 15% and 6%?

What do you add after the number?

Where candidates lose it

What the interviewer asks next

Try it first

Why does a spread times an exposure give you CVA?

What does the rough version leave out?

Where candidates lose it

What the interviewer asks next

Try it first

Why can the callable bond not keep rising?

What happens to duration and convexity?

Where candidates lose it

What the interviewer asks next

Try it first

Where do you start when every statement might be a lie?

Is there a shortcut that finds it faster?

Where candidates lose it

What the interviewer asks next

Try it first

Why does a clean sample not prove a clean book?

Where does the rule of three come from, and does the 3,000 matter?

Where candidates lose it

What the interviewer asks next

Try it first

Why does the delta estimate fall short?

Why does this matter for VaR?

Where candidates lose it

What the interviewer asks next

Try it first

Why is a correlation from 36 months so imprecise?

How do you test whether the relationship changed?

Where candidates lose it

What the interviewer asks next

Try it first

How do you turn a frequency into a confidence level?

Why is this translation useful on a desk?

Where candidates lose it

What the interviewer asks next

Try it first

Why does expected loss sit in the numerator and unexpected loss in the denominator?

How much room is there before the loan fails the hurdle?

Where candidates lose it

What the interviewer asks next