Risk Management puzzles, solved step by step

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

Showing 1–5 of 5 · filtered from 100Clear filters

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?
049A fund has an expected annual return of 12% and annual volatility of 20%, on Rs 100 crore. What is its one-year 95% VaR with and without the expected return, and when does the mean matter?VaR and expected shortfallCoreAsset manager risk
Try it first
Including the 12% expected return, what is the one-year 95% VaR?
Show the worked solution
Rs 32.9 crore ignoring the mean and Rs 20.9 crore including it. The 95% cut is 1.645 standard deviations below the mean: 1.645 x 20% is 32.9%, and a 12% expected return lifts the cut to minus 20.9%. At one year the mean cuts VaR by more than a third. At one day it barely matters: Rs 2.07 crore against Rs 2.02 crore.
Why does the mean matter at one year but not at one day?
On a short walk the path you take wanders more than your average direction moves you; on a long journey the direction wins. Expected return grows in proportion to time while volatility grows with its square root, so over short horizons the mean is noise and over long ones it is material. At one day the mean is 12% / 252, about 0.048%, against a daily volatility of 1.26%; the VaR figures differ by 2.3%. At one year the gap is 36%.
Over one year a 12% expected return shifts the whole distribution right, moving the 5% cut from minus 32.9% to minus 20.9%, whereas over one day the same mean moves VaR from Rs 2.07 crore only to Rs 2.02 crore.
The relationship
$\text{VaR}_{95\%} = (1.645\,\sigma - \mu) \times W = (32.9\% - 12\%) \times 100 = 20.9$
\sigma annual volatility, 20%
\mu expected annual return, 12%
W portfolio value, Rs 100 crore
What it says in wordsVaR is how far the 5% worst outcome sits below zero: the volatility term pulls it down, the mean pushes it back up.
Which figure should a risk report quote?
Say which you are quoting, because both are used. Relative VaRVaR measured from the expected outcome rather than from zero, so it captures only the uncertainty and ignores the expected gain. ignores the mean and measures pure uncertainty; absolute VaR includes it and measures the loss from today's value. For a one-year horizon, many risk teams quote the zero-mean figure deliberately: a 12% expected return is an assumption, and counting it as a cushion lets an optimistic forecast shrink the risk number. Daily trading VaR usually ignores the mean for the simpler reason that it makes no difference.
Limits to name: over a year, compounding and fat tails matter, so a normal model with a constant 20% volatility understates the chance of a large loss, and volatility itself changes over the year. The one-year figure is a rough guide to the size of a bad year, not a promise about it.
Where candidates lose it
The most common slip is adding the mean to the loss, giving 44.9, which moves the cut the wrong way. The second is ignoring the question's own hint and giving only 32.9.
The quieter miss is not answering the when part. The square-root rule for volatility against the linear growth of the mean is the one sentence the interviewer is waiting for.
What the interviewer asks next
Over what horizon does the mean cut VaR by half?
Why might a regulator prefer the zero-mean figure?
How would you compute one-year VaR if returns were lognormal?
063From 250 days of daily P&L, the five worst losses are Rs 18, 14, 11, 9.5 and 8 crore. What is the 99% one-day historical VaR, and what is the expected shortfall beyond it?VaR and expected shortfallCoreUBSRemote · 2020
Try it first
How many of the worst days sit beyond the 99% line in a 250-day sample?
Show the worked solution
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.
In a 250-day sample the 99% line sits 2.5 observations from the worst end, between losses of Rs 14 crore and Rs 11 crore, so historical VaR is Rs 12.5 crore by interpolation or Rs 11 crore by the third-worst convention. Expected shortfall, the average of the worst 2.5 observations, is Rs 15 crore.
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 relationship
$ES_{99\%} = \frac{18 + 14 + 0.5 \times 11}{2.5} = \frac{37.5}{2.5} = 15$
2.5 the number of observations in the worst 1% of 250 days
0.5 x 11 half of the third worst loss, the fraction of it inside the tail
What it says in wordsExpected shortfall is the average loss across the worst 1% of days, counting the third worst at half weight.
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.
Where candidates lose it
The first trap is picking the worst loss, Rs 18 crore, as the 99% VaR. That is closer to a 99.6% number. Count 1% of the sample, 2.5 days, then read off the list.
The second is giving a single number without the convention. Say 2.5 observations, say which rule you use, and show that expected shortfall is larger than VaR because it averages what lies beyond.
What the interviewer asks next
What would the 97.5% expected shortfall be from the same list?
The worst day drops out of the window tomorrow. What happens to VaR and ES?
Why might a regulator prefer expected shortfall to VaR for setting capital?
Asked at UBS, Risk Management, Remote, 2020 (Wall Street Oasis): Calculate VaR
074Under a normal distribution, 99% VaR should be about 1.41 times 95% VaR. A desk's historical VaR is Rs 5 crore at 95% and Rs 11 crore at 99%. What does the ratio tell you about the desk's P&L?VaR and expected shortfallCoreBank market riskModel validation
Try it first
The desk's ratio is 2.2 against a normal 1.41. What is the most likely reading?
Show the worked solution
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.
A normal distribution and a fat-tailed one with the same 95% VaR of Rs 5 crore diverge in the tail: the normal's 99% VaR is Rs 7.07 crore and the fat-tailed one's is Rs 11 crore. The desk's ratio of 2.2 exceeds the 1.41 of a normal and the 1.93 of a Student t with 3 degrees of freedom.
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 relationship
$\frac{\text{VaR}_{99}}{\text{VaR}_{95}}\Big|_{\text{normal}} = \frac{2.326}{1.645} = 1.41 \qquad \frac{11}{5} = 2.2$
2.326, 1.645 the one-sided 99% and 95% points of a standard normal
11, 5 the desk's historical 99% and 95% VaR, Rs crore
What it says in wordsFor a normal the ratio is fixed at 1.41 whatever the volatility, so a larger ratio signals a heavier tail.
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.
Where candidates lose it
The trap is reading the ratio as higher volatility. Volatility changes the size of both numbers but not their ratio; only the shape of the distribution changes the ratio.
The second slip is concluding the desk is fine because the 95% number looks normal. A fat tail hides at the 95% level and shows up only further out, which is exactly why regulators moved capital towards expected shortfall.
What the interviewer asks next
What would a ratio below 1.41 suggest?
How would you expect the desk's expected shortfall at 97.5% to compare with a normal's?
Which kinds of trading positions tend to produce fat left tails?
088A treasury holds Rs 500 crore of government bonds with a modified duration of 6. Daily changes in yield have a standard deviation of 6 basis points. What is the one-day 99% VaR?VaR and expected shortfallCoreUBSAnonymous employee in · 2020
Try it first
What is the position's DV01, the loss for a one basis point rise in yield?
Show the worked solution
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.
A Rs 500 crore position with duration 6 loses Rs 30 lakh per basis point; a 99% daily yield rise is 2.33 times 6 basis points, 14.0 basis points, so the one-day 99% VaR is Rs 4.19 crore, the loss on the worst 1% of days.
The relationship
$\text{VaR}_{99} = P \cdot D \cdot 0.0001 \cdot z_{99} \cdot \sigma_{bp} = 500 \times 6 \times 0.0001 \times 2.33 \times 6 = 4.19$
P position value, Rs 500 crore
D modified duration, 6
z 2.33, the one-sided 99% point
sigma_bp daily standard deviation of yield, 6 bp
What it says in wordsMultiply rupees lost per basis point by the yield rise that is exceeded only one day in a hundred.
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.
Where candidates lose it
The usual slip is a units error: forgetting that a basis point is 0.0001 and producing a VaR a hundred times too large or too small. Say DV01 out loud first, Rs 30 lakh a basis point, and the rest follows.
The other is using 2.58 because 99% sounds like a two-sided number. VaR is a one-sided loss measure, so the multiplier is 2.33.
What the interviewer asks next
What is the 99% expected shortfall under the same normal assumption?
The book holds 2-year and 10-year bonds with the same total DV01. What risk does one number hide?
How would convexity change the VaR for a 300 basis point stress?
Asked at UBS, Risk Management, Anonymous employee in, 2020 (Wall Street Oasis): Calculate VAR

z_c	the number of standard deviations that leaves the chosen tail below it
sigma	the daily P&L standard deviation, Rs 2 crore

\sigma	annual volatility, 20%
\mu	expected annual return, 12%
W	portfolio value, Rs 100 crore

2.5	the number of observations in the worst 1% of 250 days
0.5 x 11	half of the third worst loss, the fraction of it inside the tail

2.326, 1.645	the one-sided 99% and 95% points of a standard normal
11, 5	the desk's historical 99% and 95% VaR, Rs crore

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 the mean matter at one year but not at one day?

Which figure should a risk report quote?

Where candidates lose it

What the interviewer asks next

Try it first

Why is historical VaR a position in a list rather than a formula?

Which convention is right, and does it matter?

Where candidates lose it

What the interviewer asks next

Try it first

Why does the ratio say anything about the shape of the tail?

What would you do with that finding?

Where candidates lose it

What the interviewer asks next

Try it first

Why start from DV01 instead of a price volatility?

What does this number leave out?

Where candidates lose it

What the interviewer asks next