Risk Management puzzles, solved step by step

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

Showing 21–30 of 50 · filtered from 100Clear filters

039A retiree holds Rs 1 crore and withdraws Rs 10 lakh at the end of each year. Compare two years of minus 20% then plus 20% with plus 20% then minus 20%. Where does each order leave the retiree?Compounding and drawdownsCoreAsset manager risk
Try it first
Without withdrawals, both orders end at Rs 96 lakh. With the Rs 10 lakh withdrawals, which order ends better?
Show the worked solution
Up first ends at Rs 78 lakh; down first ends at Rs 74 lakh. Without withdrawals both orders end at Rs 96 lakh, because returns multiply. With withdrawals, the Rs 10 lakh taken out after year one sits out year two: it escapes a 20% fall in one order and misses a 20% gain in the other, a gap of 10 x 0.4, Rs 4 lakh.
Why does order matter only once money is withdrawn?
Multiplication does not care about order: 0.8 x 1.2 is the same as 1.2 x 0.8, 0.96 either way. A withdrawal breaks the chain, because rupees taken out before a return do not experience it. Think of a farmer who sells part of the harvest each year. A good year followed by a bad one lets him sell from a big crop first; a bad year first forces him to sell from a small one, and the portion sold never gets the chance to recover. This is sequence riskThe risk that the order of returns, not just their average, changes the outcome, which happens whenever money is being added or withdrawn..
With Rs 10 lakh withdrawn each year, a fall of 20% followed by a rise of 20% leaves Rs 74 lakh, while the same returns in the opposite order leave Rs 78 lakh; without withdrawals both orders would end at Rs 96 lakh.
Where exactly does the Rs 4 lakh gap come from?
Track the first withdrawal. In both orders the retiree takes Rs 10 lakh at the end of year one. That Rs 10 lakh then skips year two: in the up-first order it skips a 20% fall and saves Rs 2 lakh, in the down-first order it skips a 20% rise and loses Rs 2 lakh. The difference is 10 x (1.2 minus 0.8), Rs 4 lakh. The second withdrawal is taken at the very end and does not depend on the order at all.
The relationship
$W_2 = (W_0(1+r_1) - D)(1+r_2) - D \;\Rightarrow\; W_2^{\text{up first}} - W_2^{\text{down first}} = D\,(r_{\text{up}} - r_{\text{down}}) = 10 \times 0.4 = 4$
W_0 starting wealth, Rs 100 lakh
D the yearly withdrawal, Rs 10 lakh
r_1, r_2 the two years' returns
What it says in wordsThe starting wealth sees both returns in any order; only the withdrawn money depends on which return it misses.
Scale it up and the effect grows. Over twenty years of withdrawals, a bad run in the first few years forces sales at low prices from a pot that then has less left to recover, and two retirees with the same average return can end decades apart. The limit of this puzzle is its size: two years and one gap of Rs 4 lakh undersell how much the order of returns matters over a long retirement.
Where candidates lose it
The instinct is to say order cannot matter because multiplication is commutative. That is true only without cash flows, and the question gave you a withdrawal precisely to break it.
The second miss is getting 74 and 78 but being unable to say why. The single sentence about the first withdrawal skipping year two is what the interviewer wants to hear.
What the interviewer asks next
What if the retiree added Rs 10 lakh each year instead of withdrawing it?
How would you reduce sequence risk for a new retiree without changing the expected return?
Over thirty years, why does a bad first five years matter more than a bad last five?
042You want a curve steepener that is neutral to parallel moves: buy the 2-year bond and short the 10-year. The 10-year has a DV01 of 0.09 per 100 of face and the 2-year has 0.019. How much 2-year do you buy for each Rs 100 crore of 10-year you short?Duration and ratesCoreTreasury and ALMBank market risk
Try it first
Roughly how much 2-year face value balances Rs 100 crore of 10-year?
Show the worked solution
About Rs 474 crore of 2-year for each Rs 100 crore of 10-year. A parallel-neutral trade matches rupees per basis point, not face value. Rs 100 crore of 10-year moves Rs 9 lakh per basis point. The 2-year moves 0.019 per 100, so matching Rs 9 lakh needs 100 x 0.09 / 0.019, Rs 473.7 crore. A parallel shift then leaves the book flat; only a change in the gap between the two yields makes or loses money.
Why size by DV01 rather than by face value?
Balancing a see-saw is about weight times distance from the pivot, not the number of children on each end. DV01 is each leg's weight: the rupees it gains or loses when its yield moves one basis point. A 10-year bond is about 4.7 times as sensitive per rupee of face as a 2-year, so a parallel-neutral trade needs about 4.7 times as much 2-year face. Equal face value would leave the book mostly a bet on the 10-year, and a parallel move would swamp the curve view.
Rs 473.7 crore of 2-year bonds bought and Rs 100 crore of 10-year bonds sold each carry Rs 9 lakh of DV01, so the trade is level under a parallel move even though the 2-year face is almost five times larger.
The relationship
$N_2 = N_{10} \times \frac{DV01_{10}}{DV01_{2}} = 100 \times \frac{0.09}{0.019} = 473.7$
N face value of each leg, Rs crore
DV01 price change per 100 of face for a one basis point yield move
What it says in wordsSet the two legs' rupees per basis point equal, then solve for the face value of the hedge leg.
What does the trade make, and what can still go wrong?
If the 10-year yield rises 10 basis points while the 2-year stays put, the curve steepens and the short 10-year gains Rs 90 lakh. If both yields rise 10 basis points together, the 2-year loses Rs 90 lakh and the 10-year short gains Rs 90 lakh: flat, which is the design. Had you used equal face, that same parallel rise would have made about Rs 71 lakh, a large outright bet you did not mean to place.
Name the limits. DV01 is a local measure and drifts as yields move and time passes, so the ratio must be rebalanced. CarryThe income a position earns or pays while it is held unchanged: coupons received less the cost of funding and of shorting. differs across the two legs and can quietly dominate a slow trade. And short and long yields rarely move by the same amount; some desks weight the 2-year leg by its historical beta to the 10-year instead of one for one, which gives a different ratio.
Where candidates lose it
The fast wrong answer is equal face value, Rs 100 crore against Rs 100 crore. It leaves the trade roughly four-fifths an outright short of the 10-year, and the first parallel sell-off or rally decides the P&L, not the curve.
Candidates also invert the ratio and answer about Rs 21 crore, dividing 0.019 by 0.09. Check the direction: the less sensitive bond always needs the bigger face.
What the interviewer asks next
The trade should also be neutral to a 1-for-0.8 move between 2-year and 10-year yields. How does the ratio change?
How much does the position make if the curve steepens by 15 basis points?
Why might a treasury desk prefer futures to cash bonds for this trade?
044Estimate the cash a city's ATMs must dispense each weekday, given a population of 40 lakh, 75% of them adults, 30% of adults withdrawing once a week, an average withdrawal of Rs 3,000, and withdrawals spread evenly over five weekdays.Logic, estimation and brainteasersCoreTreasury and ALMRisk GCC
Try it first
Which order of magnitude is right for cash dispensed per weekday?
Show the worked solution
About Rs 54 crore a weekday. 40 lakh people x 75% adults is 30 lakh adults. 30% of them withdraw weekly, 9 lakh withdrawals. At Rs 3,000 each that is Rs 270 crore a week, and spread over five weekdays, Rs 54 crore a day. Moving the weekly share between 20% and 40% puts the answer between Rs 36 and 72 crore.
How do you structure the estimate so the interviewer can follow it?
Planning food for a wedding, you do not guess the total rice; you count guests, portions per guest and grams per portion. A sizing answer is a chain of multipliers, each one a stated assumption, so the interviewer can challenge any link without the whole answer collapsing. Say the chain before you multiply: people, adults, weekly users, ticket size, days. Then the arithmetic is almost an afterthought.
Multiplying 40 lakh people by 75% adults, 30% weekly users and Rs 3,000 a withdrawal gives Rs 270 crore a week, about Rs 54 crore a weekday, and the answer stays between Rs 36 and 72 crore while the weekly share sits between 20% and 40%.
How do you sanity check Rs 54 crore?
Test it from a second angle. Rs 270 crore a week across 30 lakh adults is Rs 900 per adult per week, a plausible cash habit in a city where digital payments carry much of the load. It also means about 1.8 lakh transactions a weekday; if you assumed a number of ATMs, you could check whether each would handle a sensible number of withdrawals a day. Then name the weakest link: the 30% weekly share. Doubling it doubles the answer; the population and adult share are far less uncertain.
The relationship
$40\text{ lakh} \times 0.75 \times 0.30 \times \text{Rs } 3{,}000 \div 5 = \text{Rs } 54 \text{ crore}$
0.75 the share of the population who are adults
0.30 the share of adults who withdraw once a week
5 weekdays the withdrawals are spread across
What it says in wordsMultiply the chain of assumptions, then divide by the days the total is spread over.
Finish with why a treasury desk cares. Cash is a liquidityThe ability to meet a payment obligation, here physical cash in machines, on time and without loss. obligation that cannot be deferred, and the average is not the peak. Salary days, festivals and long weekends bunch demand, so the cash plan is sized to a peak day plus a buffer, and an empty machine is an operational failure as much as a customer one. Rs 54 crore is the average the peaks are built on.
Where candidates lose it
The common loss is racing to a number without stating the chain, then being unable to defend it when the interviewer challenges one input. The interviewer cares more about the structure than whether you land on 54.
The other slip is forgetting the last step and quoting the weekly Rs 270 crore as the daily figure. Say the units at every link: people, withdrawals a week, rupees a week, rupees a day.
What the interviewer asks next
How would you size the peak day, around the first of the month?
If 10% of the city's ATMs are down at any time, how does the cash plan change?
How would you estimate the number of ATMs the city needs?
048You estimate a desk's daily P&L variance from five observations, once dividing the sum of squared deviations by 5 and once by 4. Which estimator is unbiased, which is consistent, and how large is the bias?Statistics and estimationCoreUBSZurich · 2021
Try it first
Which statement is true?
Show the worked solution
Dividing by 4 is unbiased; both estimators are consistent; dividing by 5 is low by one fifth. The sample mean is estimated from the same five points, which uses up one degree of freedom, so the divide-by-n estimator averages (n - 1)/n of the true variance, 80% here. If the true variance is 4, it centres on 3.2, a bias of -0.8. As n grows that factor tends to 1, so both estimators converge on the truth.
Why does dividing by n come out too low?
Measure how spread out five friends' heights are by comparing each to the group's own average, and you will understate the spread, because that average was pulled towards those five people. Deviations measured from the sample mean are smaller on average than deviations from the true mean, so their sum of squares understates the spread by exactly one observation's worth. Dividing by n minus 1, the degrees of freedomThe number of independent pieces of information left after estimating something from the same data; estimating the mean uses up one., corrects it exactly.
With five observations and a true variance of 4, the divide-by-4 estimator is centred on 4 while the divide-by-5 estimator is centred on 3.2, 80% of the truth, yet the biased version is narrower and has a lower mean squared error, 5.76 against 8.00.
What is the difference between unbiased and consistent?
Unbiased is about the average over many repeated samples of the same size; consistent is about what happens to one estimate as the sample grows. Divide-by-5 fails the first: repeat the five-day exercise many times and the estimates average 3.2, not 4. It passes the second: with 250 days the bias is only -0.016, and it shrinks to zero with more data. An estimator can be unbiased but inconsistent too, such as using only the first observation to estimate a mean: right on average, never improving.
The relationship
$E\!\left[\tfrac{1}{n}\textstyle\sum (x_i - \bar{x})^2\right] = \frac{n-1}{n}\,\sigma^2 = 0.8 \times 4 = 3.2$
n the number of observations, 5
\bar{x} the sample mean, estimated from the same five points
\sigma^2 the true variance, 4 in the illustration
What it says in wordsDividing by n recovers only (n minus 1) over n of the true variance on average.
Now the twist a model validator should add. For normal data the unbiased estimator has variance 8.00 here, while the divide-by-5 version has 5.12 plus a squared bias of 0.64, a mean squared error of 5.76. The biased estimator is closer to the truth on a typical sample. Which you prefer depends on the use: unbiasedness matters when estimates are averaged across many desks; a smaller typical error matters for a single desk's limit. With five data points, neither is reliable, and that is the more important thing to say.
Where candidates lose it
The usual slip is to treat unbiased and consistent as the same thing, and so to call the divide-by-5 estimator inconsistent. The interviewer asked both words together precisely to hear you separate them.
The second miss is answering from memory without the reason. One sentence on the sample mean using up a degree of freedom shows you know why n minus 1 exists, not just that it does.
What the interviewer asks next
Give an example of an estimator that is unbiased but not consistent.
Is the sample standard deviation, the square root of the unbiased variance, itself unbiased?
With 250 days of P&L, does the choice between n and n - 1 matter for VaR?
Asked at UBS, Risk Management, Zurich, 2021 (Wall Street Oasis): And several other questions on econometrics - what is an unbiased estimator vs consistent estimator?
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?
050An EWMA volatility model with a decay factor of 0.94 had yesterday's daily volatility estimate at 1.2%. Today's return is minus 3%. What is the updated volatility estimate?Statistics and estimationCoreBank market riskQuant risk
Try it first
Roughly where does the new daily volatility land?
Show the worked solution
About 1.38%. EWMA updates the variance, not the volatility: new variance is 0.94 x 1.2 squared plus 0.06 x 3 squared, which is 1.3536 plus 0.54, or 1.8936. The square root is 1.376%. The shock has only a 6% weight but enters squared, so it supplies 29% of the new variance.
Why blend variances rather than volatilities?
A household tracking how much its grocery bill swings would be misled if it averaged the size of the swings in rupees but ignored that one big swing matters far more than several small ones. Variance is the average squared move, so the model updates squared returns, and a move 2.5 times normal size counts 6.25 times as much. An EWMAExponentially weighted moving average: each day the estimate keeps a fixed share of yesterday and adds the rest from today, so older days fade geometrically. of variances is what makes one large shock move the estimate quickly. Blending the volatilities directly gives 1.31%, understating the jump.
Today's minus 3% return carries only a 6% weight in the EWMA update, but because it enters squared it supplies 28.5% of the new variance of 1.8936, lifting daily volatility from 1.20% to 1.38%.
The relationship
$\sigma_t^2 = \lambda\,\sigma_{t-1}^2 + (1-\lambda)\,r_t^2 = 0.94(1.44) + 0.06(9) = 1.8936, \quad \sigma_t = 1.376\%$
\lambda the decay factor, 0.94
\sigma_{t-1} yesterday's volatility estimate, 1.2%
r_t today's return, minus 3%
What it says in wordsKeep 94% of yesterday's variance, add 6% of today's squared return, then take the square root.
What happens next, and what are the model's limits?
If tomorrow is flat, the estimate decays to the square root of 0.94 x 1.8936, about 1.33%. Each day of calm keeps 94% of the variance, so a shock's influence halves in about 11 trading days: ln 0.5 over ln 0.94. That is the design choice behind 0.94: fast enough to react to a new regime within days, slow enough not to swing on every move.
The limits: EWMA has no pull towards a long-run average, so after a calm spell it can drift very low and understate risk just before volatility returns, which GARCH-type models address with a mean-reversion term. It also treats up and down moves alike, while falling markets often raise volatility more than rising ones. And the choice of 0.94 is a convention for daily data, not a law, so the estimate should be backtested against realised moves.
Where candidates lose it
The usual slip is to blend the two volatilities directly, 0.94 x 1.2 plus 0.06 x 3, and answer 1.31%. It looks like the formula but applies it to the wrong quantity and understates the effect of the shock.
The second is forgetting the square root at the end and quoting 1.89% as a volatility. Say the units at each step: variance in, variance out, then volatility.
What the interviewer asks next
What would the estimate be with lambda of 0.97 instead?
How many quiet days until the estimate is back below 1.25%?
Why might a risk manager prefer GARCH to EWMA for a ten-day VaR?
052An asset has an arithmetic average return of 10% a year and a volatility of 30%. Roughly what is its compound annual growth rate?Compounding and drawdownsCoreAsset manager risk
Try it first
Pick the closest before you calculate.
Show the worked solution
About 5.5% a year. Compound growth is roughly the arithmetic mean minus half the variance. The variance is 0.30 squared, 0.09, and half of it is 4.5 points, so 10% becomes about 5.5%. A quick check: +40% then -20% averages 10% with a 30% spread, and Rs 100 ends at Rs 112, which is 5.8% a year.
Why is the average return not the growth rate?
A shopkeeper who marks a shirt up 40% and later cuts it 20% has not made 20% on the pair of moves. He has made 12%, because the cut is taken from the higher price. Returns work the same way. A loss is always taken from whatever the gain left you, so ups and downs of the same average size leave you with less than steady growth at that average. The more the returns swing, the bigger the shortfall.
Rs 100 rising 40% and then falling 20% ends at Rs 112, a compound 5.8% a year, even though the two returns average 10%. The rule of thumb puts compound growth at the 10% average less half the variance of 0.09, which is about 5.5%.
Where does half the variance come from, and how good is it?
Take logs. The log of one plus a return is roughly the return minus half its square, so averaging the logs knocks off about half the variance. The drag grows with the square of volatility: at 15% volatility it is about 1.1 points, at 30% it is 4.5. The rule is an approximation that works best for small returns. The two-year example lands at 5.8%, and other return patterns with the same average and spread give slightly different answers.
The relationship
$g \approx \mu - \frac{\sigma^2}{2} = 0.10 - \frac{0.30^2}{2} = 0.055$
g the compound annual growth rate
\mu the arithmetic average annual return, 10%
\sigma the annual volatility, 30%
What it says in wordsCompound growth is the average return less half the variance.
For a risk manager this is why two funds with the same average return are not the same fund. The one with double the volatility has four times the drag, and its investors end up with less money even though the average looks identical.
Where candidates lose it
The trap is answering 10% because the question gave you 10%. The interviewer is checking whether you know that averages of returns overstate what an investor compounds, and that the gap is driven by volatility.
The second slip is subtracting the whole variance, or subtracting half the volatility, and landing at 1% or -5%. Square first, then halve: 0.09 over 2 is 4.5 points.
What the interviewer asks next
What volatility would make the compound growth zero with a 10% average?
If you lever this asset 2x, what happens to the average and to the compound growth?
Which number should a fund report to investors, and why?
053A hedge instrument has a correlation of 0.8 with your position. If you put on the best possible hedge, what share of the position's variance does it remove, and how much of the volatility is left?Correlation and diversificationCoreBank market riskQuant risk
Try it first
Your gut first: how much of the volatility does a 0.8 correlated hedge leave behind?
Show the worked solution
The best hedge removes 64% of the variance and leaves 60% of the volatility. With the minimum variance hedge ratio, the share of variance removed is the correlation squared, 0.8 squared or 0.64. That leaves 36% of the variance, and because volatility is its square root, 60% of the original volatility is still there.
Why does a correlation of 0.8 leave so much behind?
Two friends walk home along roughly the same road. Most of the time they are close, but each takes a detour now and then, and the gap between them on those days is what a hedge cannot touch. A hedge only cancels the part of your position that moves with the instrument. The part it cancels is rho squared of the variance, and whatever is left, called basis riskThe risk that the hedge and the position do not move together, so the hedge gains or loses a different amount from the position., is all yours. At 0.8 that residual is 36% of variance.
A hedge with a 0.8 correlation removes 64 of every 100 units of variance and leaves 36. Because volatility is the square root of variance, the same hedge leaves 60 of every 100 units of volatility, so most of the swing you feel day to day survives.
How do you get from variance left to volatility left?
Take the square root. Risk managers quote variance when they add risks and volatility when they talk about losses, and the square root between them is where candidates lose the answer. A hedge that removes 64% of variance sounds impressive; saying it leaves 60% of the volatility is the honest version, and it is the number that matters for a VaR limit, which scales with volatility.
The relationship
$\sigma_{\text{hedged}} = \sigma_P\sqrt{1-\rho^2} = \sigma_P\sqrt{1-0.64} = 0.6\,\sigma_P$
\sigma_P the volatility of the unhedged position
\rho the correlation between position and hedge, 0.8
What it says in wordsThe best hedge leaves the position's volatility times the square root of one minus the correlation squared.
The limitation: the 0.8 was measured on past data. In a stress the correlation can fall, and the residual grows just when you need the hedge most. That is why desks watch the stability of a hedge correlation as closely as its level.
Where candidates lose it
The fast wrong answer is 20% of the risk left, one minus the correlation. It treats correlation as the share of risk removed, which it is not. Squaring the correlation gives the share of variance explained.
The second trap is stopping at 36% and forgetting that volatility is the square root. Give both numbers, 64% of variance removed and 60% of volatility left, and say which one a VaR limit sees.
What the interviewer asks next
What correlation do you need to cut volatility in half?
What hedge ratio achieves this if the position has twice the hedge instrument's volatility?
Why might a proxy hedge's correlation drop in a crisis?
056Two stocks both have a 10% cost of equity. One grows its dividends at 8% a year, the other at 2%. Using the Gordon growth model, how much does each price fall if the discount rate rises by 50 basis points?Duration and ratesCoreBlackRockNew York · 2026
Try it first
Which stock falls more when the discount rate rises half a point?
Show the worked solution
The 8% grower falls 20%; the 2% grower falls about 5.9%. Under Gordon growth, price is next year's dividend over r minus g. For the fast grower that gap widens from 2% to 2.5%, so the price falls to 0.02 over 0.025, or 80% of what it was. For the slow grower the gap goes from 8% to 8.5%, and the price keeps 0.08 over 0.085 of its value.
Why does the fast grower react so much more?
Think of two ways to be paid Rs 10 lakh: most of it next year, or a trickle that grows for decades. If someone doubles the rate at which you discount the future, the trickle loses far more, because most of its money is far away. A fast-growing dividend is that trickle: its value sits in cash flows many years out. A stock whose value rests on distant cash flows behaves like a long bond, so the same rise in the discount rate cuts its price far more.
Both stocks are priced at 100 with a 10% cost of equity. A rise to 10.5% takes the 8% grower to 80, a fall of 20%, and the 2% grower to 94.1, a fall of 5.9%, because the fast grower's price rests on a gap of only 2 points between r and g.
How do you turn this into a duration number?
Differentiate the price with respect to r and divide by price: the answer is 1 over (r minus g). That gives the fast grower an equity duration of 50 years and the slow grower 12.5 years. Duration times 0.5% predicts falls of 25% and 6.25%; the exact falls are a little smaller, 20% and 5.9%, because the price curve bends, the same convexity a bond has.
The relationship
$P = \frac{D_1}{r-g} \qquad D_{\text{eq}} = -\frac{1}{P}\frac{dP}{dr} = \frac{1}{r-g}$
D_1 next year's dividend
r the cost of equity, 10%
g the constant dividend growth rate, 8% or 2%
What it says in wordsAn equity's sensitivity to the discount rate is one over the gap between the discount rate and growth.
The limitation is that Gordon growth assumes growth never changes and runs for ever, which exaggerates duration for a fast grower that will slow. The direction survives any sensible model: growth stocks carry more rate risk than stocks priced on today's cash.
Where candidates lose it
The trap is answering that both fall by about the same amount because the rate change is the same. The rate change is the same; the base it lands on is not. The fast grower's r minus g is a quarter of the slow grower's, so the same half point is four times as large relative to it.
The second slip is quoting the duration answer, 25%, as exact. Give 20% and say duration overstates it because the price curve bends.
What the interviewer asks next
What happens to each price if growth expectations for the fast grower fall to 7% at the same time?
Why might a portfolio of growth stocks behave like a long-duration bond fund?
What does equity duration mean for a pension fund that holds equities against long liabilities?
Asked at BlackRock, Risk and Quantitative Analysis, New York, 2026 (Wall Street Oasis): Which equities have duration? Technical and behavioural on VaR, market views and stock valuation.
059Failed trades on a settlement desk average 12 a day with a standard deviation of 3. An amber alert fires at 18. If nothing about the process has changed, how many false amber alerts should you expect in a 250-day year?Operational loss and fraudCoreOperational risk
Try it first
Your instinct first: how many false ambers a year?
Show the worked solution
About 6 false ambers a year. The threshold of 18 is two standard deviations above the mean of 12. If daily fails are roughly normal, about 2.3% of days land above two standard deviations on the high side, and 2.28% of 250 days is 5.7. Roughly one quiet day in every 44 will trip the alert with nothing wrong.
Why does a sensible threshold still fire when nothing is wrong?
A smoke alarm set sensitive enough to catch every real fire also goes off when someone burns toast. Set it deaf enough never to react to toast, and it may miss a real fire. Operational alerts face the same trade. Any threshold drawn on a noisy count carries a false alarm rate you can compute in advance, and the only way to cut it is to accept missing more real problems.
Daily fails centred on 12 with a standard deviation of 3 put the amber line at 18, two standard deviations up. The shaded tail holds 2.3% of days, which over a 250-day year means about 5.7 false ambers with the process unchanged.
How precise is the answer of about six?
It rests on two assumptions worth saying. First, fails are whole numbers: if a count of exactly 18 triggers the alert, a continuity correction moves the tail to about 3.3% and the count to about 8. Second, real operational counts often have fatter tails than a bell curve, with bad days clustering around month ends and system changes. Both push the true false alarm count up, so six is a floor, not a ceiling.
The relationship
$z = \frac{18-12}{3} = 2 \qquad P(Z>2) \approx 2.3\% \qquad 0.023 \times 250 \approx 5.7$
z how many standard deviations the threshold sits above the mean
P(Z>2) the share of a normal distribution beyond two standard deviations on one side
What it says in wordsConvert the threshold to standard deviations, read off the tail, and multiply by the number of days.
In the room, close with what you would do: tell the operations team to expect about one amber every two months from noise alone, so that a cluster of ambers in one week, not a single one, is what triggers an investigation.
Where candidates lose it
The trap is answering none, or close to none, because 18 looks far from 12. Six fails is only two standard deviations, and two standard deviations is not rare over 250 tries.
The second slip is using the two-sided 5% and getting 12 or 13 a year. Only the high side triggers an amber, so the tail is about 2.3%, not about 5%.
What the interviewer asks next
Where would you set the threshold to get about one false amber a year?
Three ambers arrive in one week. How surprised should you be if nothing has changed?
Why might failed trade counts not follow a normal distribution?

W_0	starting wealth, Rs 100 lakh
D	the yearly withdrawal, Rs 10 lakh
r_1, r_2	the two years' returns

N	face value of each leg, Rs crore
DV01	price change per 100 of face for a one basis point yield move

0.75	the share of the population who are adults
0.30	the share of adults who withdraw once a week
5	weekdays the withdrawals are spread across

n	the number of observations, 5
\bar{x}	the sample mean, estimated from the same five points
\sigma^2	the true variance, 4 in the illustration

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Why does order matter only once money is withdrawn?

Where exactly does the Rs 4 lakh gap come from?

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Why size by DV01 rather than by face value?

What does the trade make, and what can still go wrong?

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How do you structure the estimate so the interviewer can follow it?

How do you sanity check Rs 54 crore?

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Why does dividing by n come out too low?

What is the difference between unbiased and consistent?

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

Which figure should a risk report quote?

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Why blend variances rather than volatilities?

What happens next, and what are the model's limits?

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Why is the average return not the growth rate?

Where does half the variance come from, and how good is it?

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Why does a correlation of 0.8 leave so much behind?

How do you get from variance left to volatility left?

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Why does the fast grower react so much more?

How do you turn this into a duration number?

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Why does a sensible threshold still fire when nothing is wrong?

How precise is the answer of about six?

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