Portfolio Management puzzles, solved step by step

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

Showing 1–10 of 50 · filtered from 100Clear filters

005Which is the better bet: at least one six in four rolls of a single die, or at least one double six in twenty-four rolls of a pair of dice?Probability and expected valueCoreQuantitative asset managementHedge funds
Try it first
Which bet has the better odds?
Show the worked solution
The single six in four rolls, which wins 51.8% of the time against 49.1% for the double six in 24. Work each through its complement. No six in four rolls has chance (5/6) to the 4th, 48.2%. No double six in 24 rolls has chance (35/36) to the 24th, 50.9%. The naive count, trials times chance, gives two thirds for both and is wrong for both.
Why does multiplying the trials by six not keep the odds the same?
A double six is six times rarer than a six, so 24 rolls look like a fair swap for four. The intuition treats the chance of success as growing in a straight line with the number of tries. It does not. What compounds is the chance of failing every time, and a rare event's failure chance, raised to a high power, falls more slowly than the straight line suggests. Think of looking for a friend in a crowd: glancing four times at a small crowd and twenty-four times at a crowd six times the size are not the same search, because each extra glance adds less as the misses pile up.
The chance of no six in four rolls is 48.2%, so that bet wins 51.8%, while the chance of no double six in 24 rolls is 50.9%, so that bet wins only 49.1%, even though the naive count gives two thirds for both.
How many rolls would the double six bet need to be favourable?
Solve for the number of rolls where the chance of no double six drops below a half. (35/36) to the 24th is 0.5086, still above a half; (35/36) to the 25th is 0.4945, just below. So 25 rolls, not 24, is where the double six bet turns favourable, a gap of a single roll between a losing and a winning bet. That is the second way to show you understand the question: the naive rule is off by a small amount, and in a repeated game a small edge is everything.
The relationship
$1-\left(\tfrac{5}{6}\right)^{4}\approx 0.518 \qquad 1-\left(\tfrac{35}{36}\right)^{24}\approx 0.491$
5/6 the chance a single roll is not a six
35/36 the chance a roll of two dice is not a double six
What it says in wordsEach bet wins with one minus the chance of missing on every roll.
On a desk the lesson is the difference between a rough count and an exact one: a strategy that looks equivalent on a back-of-envelope scaling can sit on the wrong side of break-even once you do the compounding properly.
Where candidates lose it
The trap is the naive count. Four sixths and twenty-four thirty-sixths are both two thirds, and a candidate who answers "the same" has fallen for the exact error the puzzle was built to catch.
The other way to lose it is getting the complements right but rounding both to about 50% and calling it a tie. The whole answer lives in the gap between 51.8% and 49.1%, so say both to one decimal.
What the interviewer asks next
How many rolls of one die give better than even odds of at least one six?
What is the expected number of sixes in four rolls, and why is it not the same as the chance of at least one?
If you win Rs 100 on the double six bet and lose Rs 100 otherwise, what is your expected value over 24 rolls?
006What are the Macaulay duration and the modified duration of a three-year bond paying a 6% annual coupon and priced at par?Bond mathsCorePIMCOLos Angeles · 2024
Try it first
Before you calculate: where does the Macaulay duration sit?
Show the worked solution
Macaulay duration about 2.83 years and modified duration about 2.67. At par the yield equals the 6% coupon, so the cash flows discount to 5.66, 5.34 and 89.00, which add to 100. Weight each year by its share of the price: 1 x 0.0566 + 2 x 0.0534 + 3 x 0.8900 gives 2.83. Divide by 1.06 for modified duration: a one point rise in yield cuts the price by roughly 2.67%.
What is duration, if not the time to maturity?
Picture a seesaw with three children sitting at the one, two and three metre marks. If the child at three metres is much heavier, the pivot that balances the seesaw sits close to three, not at the middle. Macaulay duration is that pivot: the average time you wait for your money, with each payment weighted by its present value. Here the three weights are the discounted coupons of 5.66 and 5.34 and the discounted final payment of 89.00. The last payment is so heavy that the balance point, 2.83 years, sits only two months short of maturity.
The bond's discounted cash flows of 5.66, 5.34 and 89.00 sit at years 1, 2 and 3 and balance at 2.83 years, which is the Macaulay duration; dividing by 1.06 gives a modified duration of 2.67.
Year Cash flow Present value at 6% Share of price Year x share
1 6 5.66 0.0566 0.0566
2 6 5.34 0.0534 0.1068
3 106 89.00 0.8900 2.6700
Total 118 100.00 1.0000 2.8334
Weighting each payment date by its share of the price gives a Macaulay duration of 2.8334 years.
Why divide by 1.06 to get modified duration?
Macaulay duration is a time. Modified duration is a price sensitivity: the percentage change in price for a one point change in yield. With annual compounding the two differ by a factor of one plus the yield, so 2.833 divided by 1.06 is 2.673. Say what it means: if yields rise from 6% to 7%, the bond loses about 2.67% of its price, a little less in reality because the price-yield curve bends. That bend is convexity, and it is the natural next question.
The relationship
$D_{mac}=\sum_t t\,\frac{PV_t}{P}=\frac{1(5.66)+2(5.34)+3(89.00)}{100}\approx 2.83 \qquad D_{mod}=\frac{D_{mac}}{1+y}\approx 2.67$
PV_t the present value of the payment at year t
P the bond's price, 100 at par
y the yield, 6%
What it says in wordsMacaulay duration is the value-weighted average payment date; modified duration divides it by one plus the yield to turn it into a price sensitivity.
Where candidates lose it
The fast wrong answer is three years, which is true only of a zero coupon bond. The second is averaging the dates without weighting them, which gives two. The interviewer wants the words present value weighted before any number.
Candidates also mix up the two durations. Say which is a time and which is a sensitivity, and use modified duration for any question about how much the price moves.
What the interviewer asks next
What happens to the duration if the coupon rises to 10% and the bond still trades at par?
Estimate the price if yields jump to 7%, then say whether the true price is higher or lower.
What is the duration of a three-year zero coupon bond?
Asked at PIMCO, Product & Strategy, Los Angeles, 2024 (Wall Street Oasis): Lots of random bond math questions -- duration of this bond with x coupon sold at par
007A company trades at 20 times earnings. It uses cash that was earning 3% after tax to buy back 10% of its shares at the market price. Is the buyback accretive to earnings per share, and by how much?Valuation riddlesCoreFundamental asset managementAsset management
Try it first
Which comparison decides whether the buyback lifts earnings per share?
Show the worked solution
Yes, it is accretive, by about 4.4%. Take Rs 50 crore of earnings on 10 crore shares at Rs 100, so EPS is Rs 5.00. The buyback spends Rs 100 crore, which was earning Rs 3 crore, so earnings fall to Rs 47 crore while the share count falls to 9 crore. EPS becomes Rs 5.22. It is accretive because the 5% earnings yield beats the 3% return on cash.
What is the company actually swapping?
Suppose you hold a fixed deposit paying 3% after tax and use it to buy out a partner's share of a shop that earns 5% on its price. Your income goes up, because you replaced a 3% asset with a 5% one. A buyback is the same trade. The company gives up the after-tax return on its cash and gets back a slice of its own earnings, priced at the earnings yield, which is one over the P/E. At 20 times earnings that yield is 5%, so the swap raises earnings per share. At 33.3 times earnings the yield would be 3% and the buyback would leave EPS unchanged.
The buyback gives up cash earning 3% after tax and buys shares carrying 5% of earnings, so earnings fall from Rs 50 crore to Rs 47 crore while shares fall from 10 crore to 9 crore, and EPS rises from Rs 5.00 to Rs 5.22.
How do you get the exact number quickly?
Pick round figures and the answer falls out: Rs 1,000 crore of market value, Rs 50 crore of earnings and 10 crore shares. Earnings drop by 10% of the market value times 3%, which is Rs 3 crore, and shares drop by 10%, so EPS moves by 0.94 divided by 0.90. That is 1.0444, an accretion of 4.44%. The shortcut works for any size of company because only the ratios matter.
The relationship
$\frac{EPS_1}{EPS_0}=\frac{1-f\cdot PE\cdot y_c}{1-f}=\frac{1-0.1\times 20\times 0.03}{0.9}\approx 1.044$
f the fraction of shares bought back, 10%
PE the price to earnings multiple, 20
y_c the after-tax return on the cash spent, 3%
What it says in wordsEPS rises when the lost interest, as a share of earnings, is smaller than the share of the company bought back.
Does accretive mean the buyback was a good idea?
No, and a portfolio manager is expected to say so. Accretion is arithmetic about earnings per share; value is created only if the company paid less for its shares than they are worth. A company on a low P/E almost always gets an accretive buyback, even if the shares are overpriced, and one on a high P/E can create value with a dilutive buyback if the shares are cheap relative to future growth. Accretion also ignores risk: swapping cash for shares makes the remaining equity more levered.
Where candidates lose it
Candidates answer that a buyback always raises EPS because there are fewer shares. That forgets the lost income on the cash, and at a high enough P/E the buyback dilutes.
The second trap is stopping at accretive. On a buy-side desk the interviewer usually follows with whether it creates value, and the answer that separates candidates is that the two are different questions.
What the interviewer asks next
At what P/E does the same buyback become dilutive?
What if the buyback is funded with new debt at 8% before a 25% tax rate?
Why might a buyback that is accretive still destroy value for the remaining shareholders?
008A fund manager has a true information ratio of 0.5: genuine skill, with annual active returns averaging half their volatility. How many years of returns do you need before the track record is statistically significant at the 5% level?Performance measurementCoreFund selectionPerformance analysis
Try it first
Roughly how long must you wait?
Show the worked solution
About 15 years. The t-statistic of a track record is the information ratio times the square root of the number of years. Set 0.5 x root T equal to 1.96: the root of T is 3.92, so T is about 15.4 years. A manager twice as good, with an information ratio of 1.0, would still need nearly four years, and most careers and fund lives are shorter than this test demands.
Why does proof take so long even for a good manager?
Think of a cricketer whose true average is a little above the team's. In one season, luck swamps that small edge; only after many seasons does the average clearly separate. Skill adds up in proportion to time, but noise adds up in proportion to the square root of time, so the ratio between them grows only with the square root. An information ratio of 0.5 means one year's excess return is half a standard deviation of noise. It takes four years to reach a t-statistic of 1.0 and about 15.4 years to reach 1.96.
A manager with an information ratio of 0.5 sees the t-statistic of the track record reach the 1.96 significance line only after about 15.4 years, and even a manager with an information ratio of 1.0 needs about 3.8 years.
The relationship
$t=IR\sqrt{T}\ge 1.96 \;\Rightarrow\; T\ge\left(\frac{1.96}{0.5}\right)^2\approx 15.4$
IR the information ratio: mean active return over tracking error, per year
T years of track record
1.96 the two-sided 5% critical value
What it says in wordsThe years needed are the critical value divided by the information ratio, squared.
What should a fund selector do with that number?
Accept that statistics alone will not settle the question within a useful time. A fund selector who waits for significance hires managers after their best years are behind them; one who does not wait must lean on evidence other than the return series. That means the process, the people, turnover, whether the returns came from the bets the manager says they make, and how much of the record is explained by factors that could be bought cheaply. State the limitation: the calculation assumes independent years and a constant information ratio, and real skill decays as assets grow.
Where candidates lose it
The common answer is three to five years, because that is how long most reviews look back. The interviewer wants to see you compute rather than guess, and then notice how uncomfortable the answer is.
The other slip is squaring the wrong thing. Write t equals IR times root T, solve for root T first, then square.
What the interviewer asks next
Using monthly data, does the answer change?
How many years for a 90% confidence level instead?
If 1,000 managers have no skill, how many will look significant after 15 years?
009You buy a property for Rs 100 crore, collect Rs 7 crore of net rent at the end of each year for five years, and sell it at the end of year five for Rs 110 crore. What is the IRR?Private and real asset mathsCoreReal estate investmentReal assets
Try it first
Pick the closest IRR before you calculate.
Show the worked solution
About 8.7%. The IRR is the discount rate at which Rs 7 crore a year for five years plus Rs 110 crore at year five is worth exactly the Rs 100 crore paid. A quick estimate adds the 7% income yield to the annual growth in value, 100 to 110 over five years or 1.9% a year, which gives 8.9%. The exact answer is a little lower, 8.68%.
What is the IRR actually solving for?
Think of a savings account that pays you Rs 7 of interest each year on Rs 100 and then hands back Rs 110. You want the single interest rate that account must have been paying. The IRR is the one discount rate that makes the present value of everything you receive equal to what you paid. Try 9%: the rents are worth about 27.2 and the sale about 71.5, a total of 98.7, below 100, so 9% is too high. Try 8.5%: the total is about 100.7, so the answer sits between them, at 8.68%.
The property pays out 100 at year 0 and brings in 7 a year of rent plus 110 at the sale, and its IRR of 8.7% sits just under the rough sum of a 7.0% income yield and 1.9% a year of value growth.
The relationship
$100=\sum_{t=1}^{5}\frac{7}{(1+r)^t}+\frac{110}{(1+r)^5}\;\Rightarrow\; r\approx 8.7\%$
r the internal rate of return
7 net rent each year, Rs crore
110 the sale price at year five, Rs crore
What it says in wordsThe IRR is the rate that discounts the rents and the sale price back to the Rs 100 crore paid.
Why is the rough split a little too high?
The rule of thumb, IRR roughly equals income yield plus growth, is exact only when the rent grows at the same rate as the value, so the yield stays at 7%. Here the rent is flat while the value rises, so by year five the rent is only 6.4% of the property's worth, and the average yield across the hold is below 7%. That is why the answer is 8.68% rather than 8.92%. In the room, give the rough split first to show the structure, then the exact figure. Say the limitation too: an IRR assumes the rents can be reinvested at the IRR itself, and it says nothing about how much money was at work.
Where candidates lose it
The common error is adding 7% of rent to 2% a year of price gain, calling it 9%, and stopping. That treats the gain as simple interest and ignores that it arrives only at the end.
The other way to lose the point is to try to solve the equation exactly out loud. Bracket it: 9% gives less than 100, 8.5% gives more, so the answer is about 8.7%.
What the interviewer asks next
What is the IRR if the property sells for Rs 100 crore instead?
What if the rent rises 2% a year in step with the value?
How would 60% debt at 8% change the equity IRR?
010An ETF trades at 101.5 on the exchange while its indicative NAV is 100.0, and creating new units costs 0.3% of NAV. What does an authorised participant do, and what does it make per unit?Funds, ETFs and implementationCoreMutual fundsPortfolio implementation
Try it first
What does the authorised participant do?
Show the worked solution
It creates new units and sells them, keeping about 1.2 per unit, or 1.2% of NAV. The authorised participant buys the underlying basket at the NAV of 100, delivers it to the fund, receives new ETF units and sells them on the exchange at 101.5. Revenue of 101.5 less 100 for the basket and 0.3 of creation costs leaves 1.2. Its selling pushes the ETF price back toward NAV until the gap no longer covers the cost.
Why can anyone profit from the gap at all?
Picture a shop selling gift hampers for Rs 101.5 when the items inside cost Rs 100 at the market next door, and assembling a hamper costs Rs 0.3. Someone will buy the items, pack hampers and sell them until the hamper price falls. An ETF unit is a claim on a basket, and the creation and redemption window lets a large dealer convert one into the other, so the unit's price cannot drift far from the basket's value. The authorised participantA large dealer appointed by the fund to create and redeem ETF units in bulk directly with the fund. is that hamper maker.
The authorised participant buys the basket at 100, pays 0.3 in creation costs, receives new units and sells them at 101.5, keeping 1.2 per unit; the trade stops paying once the ETF price is back inside the band from 99.7 to 100.3.
Where does the arbitrage stop?
Every new unit sold adds supply on the exchange and every basket bought adds demand for the underlying shares, so the gap closes from both sides. The trade keeps paying until the premium falls to the creation cost of 0.3, so in a calm market the ETF price sits inside a band of roughly 99.7 to 100.3. Below 99.7 the trade reverses: buy cheap units, redeem them for the basket and sell the shares. The band is only as tight as the cost: an ETF holding illiquid bonds or foreign shares that trade in another time zone can show a wider gap for days, and that gap is a cost to whoever trades against it.
The relationship
$\text{gain}=P_{ETF}-NAV-c\cdot NAV=101.5-100-0.3=1.2$
P_{ETF} the exchange price of the ETF unit, 101.5
NAV the indicative net asset value per unit, 100
c the creation cost, 0.3% of NAV
What it says in wordsThe authorised participant keeps the premium over NAV less the cost of creating the unit.
The portfolio management point: this machinery is why an ETF can trade close to its holdings without the fund itself selling anything. Mutual fund investors transact at NAV once a day; ETF investors transact at a market price that stays near NAV only because this arbitrage is open.
Where candidates lose it
Candidates reach for the wrong direction, buying units and redeeming them, because redeem sounds like cashing in a profit. Name the cheap side and the dear side first, and the direction follows.
The quieter miss is forgetting the cost. A premium of 1.5 is not the profit; 1.2 is, and a premium of 0.2 is not an opportunity at all.
What the interviewer asks next
The ETF trades at 99.0. Walk through the trade.
Why do bond ETFs sometimes trade at large discounts in a stressed market?
Who bears the cost when an ETF persistently trades at a premium?
011A fund compounds at 15% a year for ten years. What share of the total gain over the decade arrives in the last three years?Compounding and fee dragCoreAsset managementWealth management
Try it first
Guess before you calculate.
Show the worked solution
About 45%. One rupee at 15% grows to 2.66 after seven years and 4.05 after ten. The total gain is 3.05, and the last three years add 4.05 minus 2.66, which is 1.39. That is 45.5% of the decade's gain in 30% of the time, because each year's 15% is earned on a larger base than the year before.
Why is the gain not spread evenly across the years?
Think of a snowball rolled down a long slope. In the first few metres it picks up a little snow; near the bottom each turn picks up much more, because the ball itself is bigger. Compounding earns the same rate on a growing base, so every year's rupee gain is 15% larger than the year before. Year 1 adds 0.15 on each rupee. Year 10 adds 0.53, three and a half times as much. The first three years together add only 0.52; the last three add 1.39.
At an assumed 15% a year, each rupee gains 0.15 in year 1 but 0.53 in year 10, so the last three years add 1.39 of the 3.05 total gain, 45.5% of it.
The relationship
$\frac{(1.15)^{10}-(1.15)^{7}}{(1.15)^{10}-1}=\frac{4.046-2.660}{3.046}\approx 0.455$
(1.15)^{10} the value of one rupee after ten years
(1.15)^{7} the value after seven years
(1.15)^{10}-1 the total gain over the decade
What it says in wordsThe last three years' share is the growth from year seven to year ten, divided by the growth over all ten.
What does this mean for an investor who leaves early?
Someone who exits after seven years has sat through 70% of the time but collected only 55% of the decade's gain. Because compounding back-loads the reward, leaving a long plan early costs far more than the fraction of time given up. The same arithmetic runs against the investor with fees: a charge taken every year compounds too, and its cost is also concentrated at the end. Say the limitation plainly: 15% is an assumed rate for the arithmetic, not a forecast, and real returns arrive unevenly, so the actual last three years could be the worst three.
Where candidates lose it
The trap is answering 30%, proportional to time, because the question sounds like a fraction of a decade. The interviewer is checking whether you picture compounding as a curve.
The other slip is dividing the last three years' gain by the final value, {P11_END:.2f}, rather than by the total gain, {P11_TOT:.2f}. Read the question again: it asks for a share of the gain, not of the ending pot.
What the interviewer asks next
At what rate would the last three years carry exactly half the gain?
A 1.5% annual fee is taken throughout. What share of the lost wealth falls in the last three years?
Why do long-horizon savers care more about the final years' return than the first years'?
012Estimate how many tonnes of gold Indian households buy in a year.Market sizing and estimationCoreRothschild & CoParis · 2026
Try it first
Which split makes this estimable in two minutes?
Show the worked solution
About 800 tonnes a year, on stated assumptions. Weddings: about 1 crore a year at 35 grams each is 350 tonnes. Festivals and gifting: 30 crore households, a quarter buying 4 grams, is 300 tonnes. Investment coins and bars: 5% of households buying 10 grams is 150 tonnes. The total is 800 tonnes. Check it against published industry demand figures before using it anywhere.
How do you start when you have no idea of the answer?
Ask when a family actually walks into a jeweller. It is not random: a wedding, Dhanteras or Akshaya Tritiya, a birth, or a decision to save in coins. Splitting a total into the occasions that drive it turns one number nobody knows into several that anyone can picture. State the base first: about 140 crore people at four to five per household is roughly 30 crore households. That figure and every weight below are assumptions for the estimate, said out loud so the interviewer can challenge them.
Splitting household gold buying into weddings at 350 tonnes, festivals and gifting at 300 tonnes and investment coins and bars at 150 tonnes gives an estimate of about 800 tonnes a year, with every input stated as an assumption.
Occasion Count Grams each Tonnes
Weddings 1 crore a year 35 350
Festivals and gifting 30 crore households x 25% 4 300
Coins and bars 30 crore households x 5% 10 150
Total 800
The three occasions add to about 800 tonnes, remembering that 1 crore grams is 10 tonnes.
How do you sanity check it, and where is it weakest?
Divide back: 800 tonnes over 30 crore households is about 2.7 grams per household per year, a small coin's worth on average, which feels plausible given that most households buy nothing in a typical year and a few buy a lot at a wedding. The weakest input is grams per wedding, because the spread is huge and the average is dragged up by a few large weddings. Moving it by 10 grams moves the total by 100 tonnes. Say that, and say you would check the result against published demand data from the industry before quoting it: an estimate is a structure plus assumptions, not a statistic.
On a multi-asset or wealth desk, the follow-up is usually why it matters: household gold is a large part of Indian savings, so its demand affects imports, the rupee and how much money is left for financial assets.
Where candidates lose it
Candidates start with a number they half remember and then try to justify it. The interviewer cannot tell whether you reasoned or recalled, and a wrong remembered number sinks the answer. Build it from drivers instead.
The other loss is unit confusion: grams, kilograms and tonnes across crore and lakh. Say the conversion once, 1 crore grams is 10 tonnes, and use it every time.
What the interviewer asks next
How would a sharp rise in the gold price change your estimate, and through which driver?
How much of this might be old gold exchanged rather than new buying?
Estimate the rupee value of the same purchases, stating the price you assume.
Asked at Rothschild & Co, Asset Management, Paris, 2026 (Wall Street Oasis): interview were with 2 seperate analysts, first part was more about market sizing and logic reasoning
014You backtest 20 independent trading strategies, none of which has any real edge, and test each one at the 5% significance level. What is the chance at least one looks significant, and what per-test threshold would hold that overall false alarm rate at 5%?Statistics and forecastingCoreQuantitative researchSystematic investing
Try it first
Chance that at least one of the 20 worthless strategies passes?
Show the worked solution
About 64%, and a per-test threshold of about 0.25%. Each worthless strategy passes by luck 5% of the time, so all 20 fail with chance 0.95 to the 20th, 35.8%, and at least one passes 64.2% of the time. To hold the overall rate at 5%, test each at 5% divided by 20, which is 0.25%; the exact version, 1 minus 0.95 to the power of one twentieth, is 0.256%.
Why does testing more ideas throw up a false winner?
Ask a room of 20 people to each flip a coin five times, and there is a fair chance someone gets five heads. Nobody in the room has a lucky hand; there were simply enough tries. A 5% test lets one worthless idea in twenty through by chance, so a researcher who tests twenty ideas should expect about one false winner, not be impressed by it. The expected number of false positives here is 20 x 0.05, exactly 1, and the chance of at least one is 64.2%.
Testing each worthless strategy at 5%, the chance that at least one looks significant reaches 64.2% at 20 strategies, while testing each at 0.25% holds it near 4.9%.
The relationship
$P(\ge 1 \text{ false}) = 1-(0.95)^{20}\approx 0.642 \qquad \alpha_{each}=\frac{0.05}{20}=0.25\%$
0.95 the chance a worthless strategy fails a 5% test
20 the number of independent strategies tested
\alpha_{each} the per-test threshold that caps the overall false alarm rate near 5%
What it says in wordsThe chance of at least one false winner is one minus the chance that every test correctly fails; dividing the level by the number of tests caps it.
What does a quant desk actually do about it?
Dividing the threshold by the number of tests is called the Bonferroni correctionA rule that divides the significance level by the number of tests run, so the chance of any false positive across all of them stays near the original level.. The real discipline is counting every test you ran, including the ones you dropped quietly, because the correction is only as honest as that count. A researcher who tried 200 variants and reports the best 20 has a far bigger multiple testing problem than the 20 suggest. Desks also hold out data the research never touched and demand a reason for the edge before the backtest. Say the limitation: the correction assumes independent tests, and for correlated strategies it is too strict, which costs real ideas.
Where candidates lose it
The fast wrong answers are 5%, which ignores that there are 20 tests, and 100%, which adds the chances. Say the complement and the answer arrives in one line.
The second trap is naming the fix without the cost. A tighter threshold throws away some genuine strategies too, and an interviewer on a systematic desk expects you to say that trade-off out loud.
What the interviewer asks next
If the 20 strategies are highly correlated, is the true chance of a false winner higher or lower than 64%?
Of 1,000 strategies, how many worthless ones pass at 5%?
Why is out-of-sample testing a better defence than a stricter threshold?
016A stock's prices over six days are 7, 1, 5, 3, 6 and 4. What is the most you can make with one buy followed by one later sell? And what is the most with any number of buy and sell trades, if you can hold at most one share and cannot short?Logic brainteasersCoreMan GroupLondon · 2019
Try it first
With unlimited trades, what is the maximum profit?
Show the worked solution
5 with one trade and 7 with any number. With one trade, buy at the lowest price that comes before a higher one: buy at 1 on day 2 and sell at 6 on day 5, for 5. With unlimited trades, collect every day-to-day rise and sit out every fall: 1 to 5 earns 4, 3 to 6 earns 3, a total of 7. The general answer is the sum of the positive daily changes.
How do you find the best single trade without checking every pair?
Walk through the prices once, like a shopper who notes the cheapest price seen so far and asks each day how much they would make selling today. The best single trade is the largest gap between today's price and the lowest price seen before today, found in one pass. At day 2 the low is 1. Day 3 offers 5 minus 1, which is 4; day 5 offers 6 minus 1, which is 5; no later day beats it. The trap is subtracting the lowest price from the highest overall: the high of 7 comes before the low of 1, so it cannot be sold after buying.
On prices of 7, 1, 5, 3, 6 and 4, one trade catches the widest later gap, buying at 1 and selling at 6 for 5, while unlimited trades catch each rise, 4 and then 3, for 7.
Why is the unlimited answer just the sum of the rises?
Any profitable trade from a low to a later high can be split into daily steps, and it gains only on the up days inside it while paying for every down day it sits through. With no limit on trades and no shorting, the best strategy holds the stock on every day it rises and nothing on every day it falls, so profit equals the sum of positive daily changes. Here that is 4 plus 3, which is 7. One pass through the prices gives the answer, which is the point of asking a coding-flavoured candidate.
The relationship
$\text{one trade}=\max_{j>i}(p_j-p_i)=6-1=5 \qquad \text{many}=\sum_t \max(p_{t+1}-p_t,0)=4+3=7$
p_t the price on day t
\max(p_{t+1}-p_t, 0) a day's rise, or zero on a falling day
What it says in wordsOne trade is the widest later gap; unlimited trades collect every daily rise.
Say the limitation as a portfolio manager would: this is perfect hindsight with no costs. Add a transaction cost per trade and the two answers move toward each other, because catching a small rise is no longer worth paying for.
Where candidates lose it
The common mistake is answering 6, the highest price minus the lowest, without checking that the high comes after the low. The interviewer is testing whether you respect the order of time.
For the second part, candidates sometimes add the falls as well, answering 9 or more, as if they could short. Reread the constraint: no shorting means falls are only avoided, never earned.
What the interviewer asks next
What if you are allowed at most two trades?
Each trade now costs 1. What is the best total?
Write the one-pass algorithm for the single trade and say its running time.
Asked at Man Group, Alternative Investments, London, 2019 (Wall Street Oasis): Given a series of prices, find the one buy/sell trade pair which gives the maximum profit

5/6	the chance a single roll is not a six
35/36	the chance a roll of two dice is not a double six

Year	Cash flow	Present value at 6%	Share of price	Year x share
1	6	5.66	0.0566	0.0566
2	6	5.34	0.0534	0.1068
3	106	89.00	0.8900	2.6700
Total	118	100.00	1.0000	2.8334

PV_t	the present value of the payment at year t
P	the bond's price, 100 at par
y	the yield, 6%

f	the fraction of shares bought back, 10%
PE	the price to earnings multiple, 20
y_c	the after-tax return on the cash spent, 3%

Try it first

Why does multiplying the trials by six not keep the odds the same?

How many rolls would the double six bet need to be favourable?

Where candidates lose it

What the interviewer asks next

Try it first

What is duration, if not the time to maturity?

Why divide by 1.06 to get modified duration?

Where candidates lose it

What the interviewer asks next

Try it first

What is the company actually swapping?

How do you get the exact number quickly?

Does accretive mean the buyback was a good idea?

Where candidates lose it

What the interviewer asks next

Try it first

Why does proof take so long even for a good manager?

What should a fund selector do with that number?

Where candidates lose it

What the interviewer asks next

Try it first

What is the IRR actually solving for?

Why is the rough split a little too high?

Where candidates lose it

What the interviewer asks next

Try it first

Why can anyone profit from the gap at all?

Where does the arbitrage stop?

Where candidates lose it

What the interviewer asks next

Try it first

Why is the gain not spread evenly across the years?

What does this mean for an investor who leaves early?

Where candidates lose it

What the interviewer asks next

Try it first

How do you start when you have no idea of the answer?

How do you sanity check it, and where is it weakest?

Where candidates lose it

What the interviewer asks next

Try it first

Why does testing more ideas throw up a false winner?

What does a quant desk actually do about it?

Where candidates lose it

What the interviewer asks next

Try it first

How do you find the best single trade without checking every pair?

Why is the unlimited answer just the sum of the rises?

Where candidates lose it

What the interviewer asks next