Portfolio Management puzzles, solved step by step

Puzzles: 100
Traced to a firm: 31
Topics: 13
Hard: 30

Showing 41–50 of 100

041Two stocks both have an 11% cost of equity. The value stock's dividends grow 3% a year forever and the growth stock's grow 8%. Using a constant-growth model, what is each stock's equity duration, and roughly how much does each fall if the cost of equity rises by 50 basis points?Valuation riddlesHardBlackRockNew York · 2026BlackRockNew York · 2026
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Which stock is more sensitive to a rise in the discount rate, and by roughly how much?
Show the worked solution
Durations of 12.5 and 33.3 years; the value stock falls about 6% and the growth stock about 14%. With price equal to D over (r minus g), duration is 1 over (r minus g): 1 over 0.08 and 1 over 0.03. A 50 basis point rise takes the spreads to 8.5 and 3.5 points, so prices fall by 1 minus 8/8.5, which is 5.9%, and 1 minus 3/3.5, which is 14.3%.
Why does a stock have a duration at all?
Think of two people valuing a lottery ticket: one pays out a small sum every year starting now, the other pays little now and a lot decades from now. If the interest rate rises, the second ticket loses far more value, because its money is further away and gets discounted for longer. An equity is a stream of future cash flows, so like a bond it has a duration: the weighted distance to its cash, and the growth stock's cash sits much further out. In the constant-growth model that distance comes out as 1 over (r minus g).
At an 11% cost of equity, the value stock growing 3% has a duration of 12.5 years and loses 5.9% if the rate rises half a point, while the growth stock growing 8% has a duration of 33.3 years and loses 14.3%. The dashed lines show that the straight duration estimate overstates both falls.
How do you get the price changes exactly, and why is the duration estimate too big?
The price is proportional to 1 over (r minus g), so compare spreads before and after. For the value stock the spread moves from 8 to 8.5 points, a 5.9% price fall; for the growth stock it moves from 3 to 3.5, a 14.3% fall. Duration times the rate change gives 6.25% and 16.7%, a little larger, because price is a curved function of the rate: like a bond with convexity, the stock loses less than the straight-line estimate. The curvature matters more the longer the duration.
The relationship
$P = \frac{D_1}{r-g} \qquad D_{eq} = -\frac{1}{P}\frac{dP}{dr} = \frac{1}{r-g} \qquad \frac{P_{new}}{P_{old}} = \frac{r-g}{r + \Delta r - g}$
D_1 next year's dividend
r the cost of equity, 11%
g the perpetual growth rate, 3% or 8%
D_{eq} equity duration, the percentage price change per point of rate change
What it says in wordsIn a constant-growth model, sensitivity to the discount rate is one over the gap between the rate and the growth rate.
This is the arithmetic behind a familiar market pattern: when real yields rise sharply, long-duration growth stocks usually fall more than value stocks. Say the limitation too. The model assumes growth is fixed while the rate moves; in practice rates often rise because growth is strong, which can offset part of the fall. And no real company grows at 8% forever, so the growth stock's duration is a stylised upper figure.
Where candidates lose it
Candidates often say both stocks move the same because they share a cost of equity, or they compute the percentage change of r, about 4.5%, and apply it to both prices. Neither uses the spread between r and g, which is the whole mechanism.
State the formula, name the spread, and give the two durations first. Then offer the exact falls and say why they are smaller than the duration estimate.
What the interviewer asks next
What happens to the growth stock's duration if its growth rate rises to 10%?
Why might a rate rise driven by stronger growth hurt growth stocks less than this model suggests?
How would you hedge the rate sensitivity of a growth-heavy portfolio?
Asked at BlackRock, Restructuring, New York, 2026 (Wall Street Oasis): Which equities have duration ? multiple stocks vs value stocks
Asked at BlackRock, Risk and Quantitative Analysis, New York, 2026 (Wall Street Oasis): Which equities have duration? VaR, market views, stock valuation.
042A fund category had 100 funds five years ago. Twenty of them were closed or merged away after averaging minus 4% a year; the 80 survivors averaged 11% a year. What was the true average return of the category, and how big is the survivorship bias in the survivors' figure?Performance measurementCoreFund selectionMutual funds
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What was the category's true average return?
Show the worked solution
About 8% a year, so the survivor average overstates the category by 3 points. Weight each group by its share of the 100 funds: 0.8 x 11% plus 0.2 x minus 4% is 8.8% minus 0.8%, or 8%. The survivors' 11% leaves out exactly the funds that did worst, so it describes the winners, not what the average investor in the category earned.
Why does a survivors-only average flatter the category?
Look at a school's alumni wall of fame and you would think every student became a success. The wall shows who is on it, not who was in the class. Funds that do badly are closed or merged into better ones, so a list of funds that exist today is a list of funds that did well enough to survive, and its average is biased upward. The bias is not random: the missing funds are exactly the ones with the worst numbers.
Of 100 funds, the 80 survivors averaged 11% while the 20 closed funds averaged minus 4%, so the full-category average was 8%. A table built only from survivors shows 11% and overstates what investors earned by 3 points a year.
How do you compute the true figure and the bias?
Take a weighted average by number of funds. The true category average is 80% of 11 plus 20% of minus 4, which is 8%, and the bias is the gap: 3 points a year. A quick way to see it: the closed funds were 15 points behind the survivors, and they were a fifth of the category, so they pull the average down by a fifth of 15, which is 3.
The relationship
$\bar{R} = \frac{80 \times 11 + 20 \times (-4)}{100} = 8\% \qquad \text{bias} = 11 - 8 = 3 \text{ points}$
80, 20 the survivors and the closed funds
11, -4 each group's average annual return, per cent
What it says in wordsThe honest average counts every fund that existed at the start, including the ones that disappeared.
State the simplifications. This is an equal-weighted average across funds; an asset-weighted figure could differ, since closed funds are often small. The closed funds also ran for less than five years, so a precise study would compound each fund's returns over the time it existed. Neither changes the direction of the bias, and in real data sets the bias is larger over longer windows because more funds disappear.
Where candidates lose it
Most candidates either quote 11% or take the simple midpoint of 11 and minus 4, landing on 3.5% or 7.5%. The first ignores the dead funds; the second ignores that there are four survivors for each dead fund.
Weight by count, give 8% and 3 points, then say why it matters: a fund selector comparing a manager with a survivor-only peer average is holding the manager to a bar that no real investor earned.
What the interviewer asks next
How would you build a peer group for a manager review that avoids this bias?
If the closed funds were mostly small, how would an asset-weighted average differ?
Where else in investing does survivorship bias show up?
043You have n cars, each fuelled to drive exactly 1,000 miles, and fuel can be moved from one car to another along the way. Tanks cannot be overfilled. How far can one car get, and how does that distance grow as n becomes very large?Logic brainteasersHardMillennium ManagementLondon · 2024
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With four cars, how far can one car get?
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1,000 x (1 + 1/2 + 1/3 + ... + 1/n) miles, which grows without limit but only like the logarithm of n. Drive all n cars 1,000/n miles; together they have burned one full tank, so one car can refill the rest and be abandoned. Repeat with n minus 1 cars for 1,000/(n minus 1) miles, and so on. Four cars reach 2,083 miles, 100 cars about 5,187, and the distance tracks 1,000 x (ln n + 0.577).
When should a car drop out of the convoy?
Think of friends sharing water on a long walk, where every bottle is full at the start and nobody can carry more than one. The moment the group has drunk exactly one bottle's worth, one friend can pour the rest of theirs into everyone else's bottles and head home. A car should drop out the instant the convoy has burned exactly one tank in total, because that is the first moment its remaining fuel exactly fills the others. With n cars that happens after 1,000/n miles, since n cars burn fuel n times as fast as one.
Four cars travel 250, 333, 500 and 1,000 miles in successive legs as one car at a time tops up the rest and drops out, reaching 2,083 miles. The distance with n cars keeps growing but ever more slowly, from 2,929 miles with 10 cars to 5,187 with 100.
Why does the distance grow only like a logarithm?
The total is 1,000 times the harmonic sum 1 + 1/2 + ... + 1/n. Each extra car adds the shortest leg of the journey, 1,000/n miles, so the gains shrink as the convoy grows, and the harmonic sum rises like ln n plus about 0.577. It never stops growing, so any distance is reachable in principle, but slowly: 10 cars reach about 2,929 miles, and getting past 5,000 miles takes 83 cars. Doubling the fleet adds only about 1,000 x ln 2, roughly 693 miles.
The relationship
$D(n) = 1000\sum_{k=1}^{n}\frac{1}{k} \approx 1000\,(\ln n + 0.577)$
n the number of cars at the start
1000/k the leg driven while k cars remain
0.577 the Euler-Mascheroni constant
What it says in wordsEach leg is one tank shared among the cars still running, and the legs add up to a harmonic series.
Why a hedge fund asks it: the structure is the same as scaling a strategy. Each extra unit of capital or effort buys a smaller increment, and a candidate who sees the diminishing returns and names the rate of decay is showing the instinct the desk wants. Also be ready to argue optimality in one sentence: any plan that drops a car earlier wastes fuel it cannot hand over, and dropping later wastes the fuel spent carrying a car that is no longer needed.
Where candidates lose it
The quick wrong answers are n times 1,000, as if all the fuel could be pooled into one car, or a flat 1,000 because tanks cannot be overfilled. Both skip the key idea that the convoy itself consumes fuel while carrying the reserve.
Work n equals 2 out loud first: drive 500, pour the rest of car 2 into car 1, and drive 1,000 more, for 1,500. Then generalise. The interviewer wants the harmonic series and the words grows like log n.
What the interviewer asks next
Roughly how many cars do you need to travel 10,000 miles?
What changes if cars can come back to a depot and cache fuel along the road?
Where do you see diminishing returns of this shape in portfolio construction?
Asked at Millennium Management, Investments, London, 2024 (Wall Street Oasis): Suppose you have n cars, each fueled so that they can drive for 1000 miles.
044A property earns Rs 8 crore a year of net operating income and is valued at an 8% cap rate. The owner borrowed 60% of that value. If cap rates fall to 7% with the income unchanged, what happens to the property's value, and to the owner's equity?Private and real asset mathsCoreReal estate investmentReal assets
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The value rises about 14%. How much does the owner's equity rise?
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Value rises about 14.3%, from Rs 100 crore to Rs 114.3 crore, and the owner's equity rises about 35.7%. Rs 8 crore divided by 0.07 is Rs 114.3 crore. Debt stays at Rs 60 crore, so equity goes from Rs 40 crore to Rs 54.3 crore. At 60% debt the equity is 2.5 times as sensitive as the building, and the same leverage works in reverse if cap rates rise.
Why does a one-point fall in the cap rate lift value by 14%, not by one point?
A cap rate is a yield, and value is income divided by it, just as a bond's price rises when its yield falls. Going from 8% to 7% multiplies value by 8 over 7, because the same income is now capitalised at a lower yield: the building is worth 14.3 years of income instead of 12.5. A one-point move in a single-digit cap rate is a large move in value.
With income fixed at Rs 8 crore and debt fixed at Rs 60 crore, a fall in the cap rate from 8% to 7% lifts value 14.3% and equity 35.7%, while a rise to 9% cuts value 11.1% and equity 27.8%. The debt does not move, so the equity absorbs the whole change in value.
How does leverage turn 14% into 36%?
Buy a Rs 50 lakh flat with Rs 10 lakh of your own and a Rs 40 lakh loan. If the flat rises 10% to Rs 55 lakh, the loan is unchanged and your stake has risen from 10 to 15 lakh, or 50%. Debt is a fixed claim, so the whole change in asset value lands on the equity, and the equity's percentage move is the asset's move times value over equity. Here value over equity is 100 over 40, or 2.5, so 14.3% becomes 35.7%.
The relationship
$\Delta E\% = \Delta V\% \times \frac{V}{E} = 14.3\% \times \frac{100}{40} \approx 35.7\%$
V the starting property value, Rs 100 crore
E the starting equity, Rs 40 crore
\Delta V\% the percentage change in property value
What it says in wordsWith fixed debt, equity moves by the asset's percentage change scaled up by the ratio of value to equity.
Always give the reverse case, because the interviewer is testing whether you see both directions. If cap rates rise to 9%, value falls to Rs 88.9 crore, 11.1% down, and equity falls to Rs 28.9 crore, 27.8% down. The loan-to-value ratio climbs from 60% to about 68%, which may breach a lending covenant. This ignores interest, fees and any loan amortisation, which change the numbers but not the shape.
Where candidates lose it
The frequent error is saying the equity rises 14%, the same as the property, forgetting that the debt does not share in the gain. A second is treating the cap rate change as a one-point change in value.
Give the value change as 8 over 7, then the equity change through the ratio of value to equity, and finish with the downside at 9%. A candidate who volunteers the reverse case and the covenant risk sounds like someone who has owned a leveraged asset.
What the interviewer asks next
At what cap rate would the owner's equity be wiped out?
How would a 3% annual rise in NOI change the picture over five years?
Why do cap rates tend to move with long-term interest rates?
045Two stock markets close at different times of day. Their same-day daily returns correlate at 0.30, and market B's return today correlates at 0.25 with market A's return yesterday. All other lagged correlations are zero, neither market's returns are autocorrelated, and both have the same daily variance. Roughly what is the correlation of their monthly returns?Statistics and forecastingHardAQR Capital ManagementGreenwich · 2022
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Is the monthly correlation higher, lower or the same as the daily 0.30?
Show the worked solution
About 0.54, close to 0.55. Add up 21 daily returns in each market. Each monthly variance is 21 times the daily variance. The monthly covariance collects 21 same-day terms of 0.30 and 20 lagged terms of 0.25, where A's day falls inside the same month as B's next day. So the correlation is (21 x 0.30 + 20 x 0.25) over 21, which is 0.538, tending to 0.55 for long windows.
Why would daily data understate how much two markets move together?
Two friends in different cities hear the same news; one hears it at lunch, the other the next morning. Compare their moods hour by hour on the same day and they look unrelated; compare their moods over the whole week and they look very similar. When two markets close at different times, news that arrives between the two closes shows up in one market's return today and the other's tomorrow, so same-day daily correlation misses part of the true co-movement. Longer windows add the delayed part back in.
News that arrives after market B has closed moves market A the same day and market B only the next day, which creates the 0.25 lagged correlation. Summing 21 days captures both the same-day 0.30 and 20 of the lagged pairs, so the monthly correlation is about 0.54.
How do you add up the covariance terms?
Write each month's return as the sum of its daily returns. The covariance of two sums is the sum of every pairwise covariance. With a unit daily variance, same-day pairs contribute 21 x 0.30, and the pairs of B's day t with A's day t minus 1 contribute 20 x 0.25, since only 20 such pairs sit inside a 21-day month. The variances are 21 each, with no autocorrelation to add. So the monthly correlation is (6.3 + 5.0) over 21, which is 0.538.
The relationship
$\rho_{M} = \frac{N\rho_0 + (N-1)\rho_1}{N} = 0.30 + 0.25 \times \frac{20}{21} \approx 0.54$
N trading days in the month, 21
\rho_0 the same-day daily correlation, 0.30
\rho_1 B today against A yesterday, 0.25
\rho_{M} the monthly correlation
What it says in wordsOver a month, the same-day and the one-day-lagged co-movement both count, less one lagged pair lost at the month's edge.
This matters for portfolio construction. A risk model built on daily same-day correlations between markets in different time zones will think they diversify each other far more than they do over the horizon an investor actually holds. The standard fixes are to use weekly or monthly returns, or to add lagged terms to the daily covariance, as this calculation does. The limitation: with only 12 monthly observations a year, the monthly estimate is itself noisy.
Where candidates lose it
Candidates often say correlation is a property of the two assets and does not change with the horizon, or that monthly data is noisier and so correlations must be lower. Both miss the timing effect, which is the entire question.
Draw the two closing times, say where the news lands, then add up the covariance terms. The interviewer wants to hear that you know lagged cross-correlation, and that you counted 20 lagged pairs rather than 21.
What the interviewer asks next
How would you estimate the lagged correlation from daily data?
If market B's returns were also positively autocorrelated, would the monthly correlation go up or down?
Why do daily correlations between Asian and US markets look lower than weekly ones?
Asked at AQR Capital Management, Quantitative Research, Greenwich, 2022 (Wall Street Oasis): what would be the difference between the correlation of daily vs monthly returns of a given year
046A fixed deposit pays 8% a year. The depositor is taxed at 30% on the interest, and inflation runs at 6%. What is the real after-tax return?Compounding and fee dragCoreIndian wealth managementWealth management
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Before working it: is the depositor's purchasing power growing?
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About -0.4% a year, a small real loss. Tax at 30% takes 2.4 points of the 8% interest, leaving 5.6%. Inflation at 6% then shrinks what that buys: 1.056 over 1.06 minus 1 is about -0.38%. The quick version, 5.6 minus 6, gives minus 0.4. Before tax the real return was about 1.9%, so the tax took more than all of it. Confirm current tax rates for any real case.
Why can a positive-looking rate lose purchasing power?
Imagine your rent rises 6% a year and your savings account adds 8% a year, but a third of that 8% goes to tax. You are left with 5.6% more money to meet 6% higher prices. Tax is charged on the nominal interest, including the part that only compensates for inflation, so it can take more than the whole real gain. The deposit looks like it earns 2 points above inflation; after tax it earns slightly less than inflation.
An 8% deposit loses 2.4 points to tax at 30%, and 6% inflation then takes the remaining 5.6 points and a little more, leaving a real after-tax return of about -0.38%. Before tax, the real return was 1.89%, so tax removed more than all of it.
How do you get the exact figure, and what does the tax really cost?
Divide growth factors rather than subtracting rates: the money grows by 1.056 while prices grow by 1.06, so real wealth changes by 1.056 over 1.06, a factor of 0.9962. Measured against the real return, the 30% tax rate works like a 120% tax, because it is levied on the inflation part of the interest too. Before tax, the real return was 1.08 over 1.06 minus 1, or 1.89%; after tax it is -0.38%. Rs 10 lakh on deposit buys about Rs 9.96 lakh of today's goods a year later.
The relationship
$r_{real} = \frac{1 + i(1-t)}{1 + \pi} - 1 = \frac{1.056}{1.06} - 1 \approx -0.38\%$
i the deposit rate, 8%
t the tax rate on interest, 30%
\pi inflation, 6%
What it says in wordsTake tax off the nominal interest first, then divide by the growth in prices.
Keep the numbers as illustrations. Tax slabs, deposit rates and inflation all change, and the treatment of interest depends on the depositor's total income, so confirm the current figures before advising anyone. The shape does not change: the higher inflation and the tax rate are, the more a taxed nominal return overstates what the saver keeps.
Where candidates lose it
The common answer is 2%, eight minus six, which forgets tax entirely. The next is to apply the tax to the real return, 30% of 2 points, and say 1.4%. Both miss that the tax is charged on the nominal interest.
Take the tax off first, then deflate. Give the exact figure and the one-line lesson: in a high-inflation, high-tax setting, a deposit can lose purchasing power while showing a positive rate on the statement.
What the interviewer asks next
What deposit rate would give a zero real after-tax return here?
How does the answer change for a depositor in a 10% tax slab?
Why might an investor still hold the deposit despite a negative real return?
047A Rs 100 crore bond portfolio holds Rs 40 crore of a bond with duration 2, Rs 35 crore with duration 5 and Rs 25 crore with duration 12. What is the portfolio's duration, and how much must move from the 2-year bond into the 12-year bond to lift it to 7?Bond mathsHardPIMCOLos Angeles · 2026
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How much must move from the duration-2 bond to the duration-12 bond to lift portfolio duration from 5.55 to 7?
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Duration is 5.55, and Rs 14.5 crore must move from the 2-year bond to the 12-year bond. Portfolio duration is the value-weighted average: (40 x 2 + 35 x 5 + 25 x 12) over 100, which is 555 over 100, or 5.55. Each Rs 1 crore switched gains 10 years of duration on a hundredth of the portfolio, adding 0.1. Closing a gap of 1.45 needs Rs 14.5 crore.
Why is portfolio duration a simple weighted average?
Think of the average age of people in a room: each person counts in proportion to how many of them there are. Duration measures how much a bond's price moves for a one-point change in yield, and for a small parallel move the portfolio's rupee loss is just the sum of each bond's rupee loss, so its duration is the value-weighted average of the bonds' durations. The Rs 25 crore in the 12-year bond is only a quarter of the money but supplies more than half the duration: 3.00 of the 5.55.
Before the switch, the three bonds contribute 0.80, 1.75 and 3.00 years for a portfolio duration of 5.55. Moving Rs 14.5 crore from the 2-year to the 12-year bond changes the contributions to 0.51, 1.75 and 4.74, which total exactly 7.00.
What changes for the portfolio when duration goes from 5.55 to 7?
Solve for the switch with one line: the shift x changes duration by x times (12 minus 2) over 100, and that must equal 1.45. After the switch, a one-point parallel rise in yields costs about 7% of the portfolio, Rs 7 crore, instead of about 5.55%, Rs 5.55 crore. The portfolio gains more if yields fall and loses more if they rise. Its cash-flow profile also becomes a barbell, heavier at the long end, which gives it more convexity than a single bond with the same duration but more exposure to the long end of the curve if the curve steepens.
The relationship
$D_p = \sum_i w_i D_i = \frac{40(2) + 35(5) + 25(12)}{100} = 5.55 \qquad x = \frac{(7 - 5.55) \times 100}{12 - 2} = 14.5$
w_i each bond's share of portfolio value
D_i each bond's duration, in years
x the Rs crore switched from the 2-year to the 12-year bond
What it says in wordsDuration averages by value, so a switch moves it by the amount moved times the duration gap, over the portfolio's size.
Say the limits: a weighted average of durations describes small, parallel shifts in yields. If short and long yields move by different amounts, a portfolio at duration 7 built from a barbell behaves differently from one built from 7-year bonds, and the switch changes the portfolio's yield and credit mix as well. For large moves, convexity adds a second-order correction.
Where candidates lose it
The most common slip is taking a simple average of the three durations, 19 over 3, about 6.3, ignoring the amounts held. The other is solving for the switch but forgetting to divide by the portfolio size, which gives Rs 1.45 crore or some other scale error.
Say weighted by value first, give 5.55, then set up the switch as one equation. Close with what the higher duration means in rupees for a one-point move in yields; that sentence is the part the question actually asks about.
What the interviewer asks next
How would you reach duration 7 without selling any of the 2-year bond?
Why might the barbell at duration 7 behave differently from a bullet 7-year bond if the curve steepens?
What does a one-point parallel fall in yields do to the portfolio after the switch?
Asked at PIMCO, Generalist, Los Angeles, 2026 (Wall Street Oasis): Given a portfolio of these 3 bonds (I forgot exactly what they were) explain how the portfolio changes if duration increases.
048Estimate how much cash is withdrawn from ATMs in India on an average day.Market sizing and estimationCoreIndian asset managementAsset management
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Which approach gives the strongest estimate in the room?
Show the worked solution
Roughly Rs 6,000 crore a day, on stated assumptions. Assume about 2.5 lakh ATMs, each handling about 80 withdrawals a day: 2 crore withdrawals. Check from demand: 30 crore regular users withdrawing twice a month is 60 crore a month, also about 2 crore a day. At an average of Rs 3,000 a withdrawal, the total is about Rs 6,000 crore a day. Confirm against published payment statistics for a real figure.
Why build the count from the machines and from the people?
If you wanted to know how many cups of tea a railway station sells, you could count the stalls and ask how many cups each pours, or count the passengers and ask how many buy one. A supply-side estimate and a demand-side estimate rest on different assumptions, so when they land close together, each one checks the other. Here both roads meet at about 2 crore withdrawals a day, which is the number that matters; the rupee value is one multiplication away.
2.5 lakh ATMs at 80 withdrawals a day and 30 crore users at two withdrawals a month both give about 2 crore withdrawals a day. At Rs 3,000 each that is about Rs 6,000 crore a day, and the ticket size alone moves the answer between Rs 4,000 crore and Rs 8,000 crore.
Which assumptions need the most care?
Every number here is an assumption to be stated, not a fact to be quoted. The average ticket size is the loosest input: Rs 2,000 against Rs 4,000 moves the answer from Rs 4,000 crore to Rs 8,000 crore a day, a factor of two. Withdrawals per machine vary widely, too, between a busy city branch and a rural machine, so treat 80 as an average across very different sites. The user count is the easiest to reason about from the number of adults with bank cards who still rely on cash.
Say what would change the answer over time. Growth in digital payments lowers withdrawal counts; festivals, salary days and month ends raise them sharply, so an average day hides large swings. Converting to a year, about 365 times the daily figure, is a useful sanity check against any published annual total the interviewer might mention.
For an asset management desk, the point is less the number than the method: a stock-and-flow structure, two independent builds, and a clear statement of which input you trust least. That is the same discipline as sizing a company's addressable market before building a revenue forecast.
Where candidates lose it
Candidates often start from currency in circulation, which is a stock of notes rather than a daily flow, or from the whole population, which counts people who never use an ATM. Both produce a number without a way to check it.
Build it once from the machines, once from the users, and show they agree before multiplying by the ticket size. Then name the ticket size as the input you are least sure of.
What the interviewer asks next
How would the estimate change on the day after salaries are paid?
How would you estimate the share of ATM withdrawals that digital payments have replaced?
How would a bank use this number to plan cash replenishment?
049Fund rankings persist from one year to the next with a correlation of 0.2. A fund was in the top decile of its category last year. Where do you expect it to rank this year?Behavioural and decision trapsHardFund selectionPerformance analysis
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Where should you expect last year's top-decile fund to land this year?
Show the worked solution
Around the 64th percentile, still above average but most of the way back to the middle. On a normal scale, the top decile averages about 1.75 standard deviations above the median. With a correlation of 0.2, the expected position this year is a fifth of that, 0.35, which is about the 64th percentile. The quick linear version, 50 plus 0.2 x 45, gives the 59th.
Why should a top-decile fund be expected to fall back?
Think of a student who topped one exam. Part of that was knowing the subject and part was that the questions happened to suit them. Next time, the knowledge carries over and the luck does not. An extreme result is usually a mix of skill and a large dose of luck, and only the skill is expected to repeat, so the best forecast moves back towards the average by however much luck there was. A correlation of 0.2 says four fifths of last year's spread in rankings was the non-repeating kind.
With a year-to-year correlation of 0.2, the expected rank this year is a flat curve close to the 50th percentile. A top-decile fund, averaging about the 96th percentile last year, is expected around the 64th this year, far below the diagonal that perfect persistence would give.
How do you turn a rank correlation into an expected rank?
Work on a normal-score scale, where the regression is a straight line through the middle. The expected score this year is the correlation times last year's score, so a correlation of 0.2 keeps one fifth of the distance from average. The top decile's average score is the normal density at its cut-off, 1.28, divided by 0.1, which is about 1.75. A fifth of that is 0.35, and the normal table turns it into the 64th percentile. If you skip the normal scale and pull the 95th percentile a fifth of the way from 50, you get about the 59th, a fair first answer.
The relationship
$E[z_{t+1} \mid z_t] = \rho\, z_t \qquad \bar{z}_{top\,10\%} = \frac{\varphi(1.28)}{0.10} \approx 1.75 \qquad \Phi(0.2 \times 1.75) \approx 0.64$
\rho the year-to-year correlation of rankings, 0.2
z_t last year's normal score
\varphi, \Phi the standard normal density and cumulative probability
What it says in wordsNext year's expected score keeps only the correlation's share of this year's distance from the middle.
Then say what a fund selector does with it. Chasing last year's top decile buys mostly luck at a high price, often just as new money floods in. The answer is not to ignore performance but to weight longer records, look for a process that explains the result, and set expectations that even a genuinely good fund will usually rank well below where it did in its best year.
Where candidates lose it
The two extreme answers both lose the point. Expecting the fund to stay in the top decile ignores that a correlation of 0.2 is weak. Expecting it to fall below average overshoots: regression pulls results back towards the middle, not past it.
Say one fifth of the distance, give the 64th percentile on the normal scale or the 59th with the quick version, and explain that the difference comes from how extreme the top decile really is. Then draw the selection lesson.
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050Asset A has a correlation of 0.9 with asset B and 0.9 with asset C. What is the lowest possible correlation between B and C?Portfolio risk mathsHardQuantitative asset managementRisk management
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Pick the lowest correlation B and C could have.
Show the worked solution
About 0.62. Think of each asset's returns as a vector and correlation as the cosine of the angle between two of them. A correlation of 0.9 is an angle of about 25.8 degrees. B and C are each within 25.8 degrees of A, so they are at most 51.7 degrees apart, and cos 51.7 degrees is 0.62. The formula gives the same: 0.81 minus the square root of 0.19 x 0.19.
Why can the third correlation not be anything you like?
If your office is 10 km from your home and the gym is also 10 km from your home, the office and the gym cannot be 50 km apart. Distances have to fit on a map. Correlations work the same way: they are cosines of angles between return vectors, and angles have to fit together in space, so two strong correlations force a third. A correlation matrix that breaks this rule does not describe any real set of assets, and it can make a risk model report a negative variance.
B and C each sit 25.8 degrees from A, because the cosine of that angle is 0.9. At most they are 51.7 degrees apart, so their correlation cannot fall below cos 51.7 degrees, which is 0.62.
How do you get 0.62 without drawing angles?
Split B and C into a part driven by A and a part independent of A. Each has 0.9 of A in it, and the independent parts carry the remaining variance, 1 minus 0.81, or 0.19. The correlation of B and C is 0.81 from the shared A part plus up to 0.19 either way from their independent parts, depending on whether those parts move together or against each other. So the range is 0.81 minus 0.19 to 0.81 plus 0.19: 0.62 to 1.00. The lowest value comes when the independent parts are perfectly opposed.
The relationship
$\rho_{BC} \ge \rho_{AB}\rho_{AC} - \sqrt{(1-\rho_{AB}^2)(1-\rho_{AC}^2)} = 0.81 - 0.19 = 0.62$
\rho_{AB}, \rho_{AC} the given correlations, 0.9 each
\rho_{BC} the correlation being bounded
1-\rho^2 the share of each asset's variance not explained by A
What it says in wordsThe shared link through A gives 0.81, and the parts unrelated to A can take away at most 0.19.
Why this matters on a risk desk: when analysts override individual correlations in a model, for a stress test or a view, they can create a matrix that no real market could produce. The fix is to check the matrix is positive semi-definite, and the lesson generalises. The bound is only strong when the given correlations are high; with two links of 0.7, the minimum for the third is -0.02, which forces almost nothing.
Where candidates lose it
The common answer is minus 1, from the idea that correlations are unrelated to each other, or 0.81, from multiplying the two links as if correlation were transitive. The first ignores the geometry; the second gives the answer only for one special case.
Say the angle picture in one sentence, give 0.62, and show the formula as a check. Then add that the same logic is why a hand-edited correlation matrix must be tested before it goes into a risk model.
What the interviewer asks next
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D_1	next year's dividend
r	the cost of equity, 11%
g	the perpetual growth rate, 3% or 8%
D_{eq}	equity duration, the percentage price change per point of rate change

80, 20	the survivors and the closed funds
11, -4	each group's average annual return, per cent

n	the number of cars at the start
1000/k	the leg driven while k cars remain
0.577	the Euler-Mascheroni constant

V	the starting property value, Rs 100 crore
E	the starting equity, Rs 40 crore
\Delta V\%	the percentage change in property value

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