Portfolio Management puzzles, solved step by step

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

Showing 31–40 of 69 · filtered from 100Clear filters

039A 60/40 portfolio holds equities with 18% volatility and bonds with 6% volatility, and the two correlate at 0.1. What is the portfolio's volatility, and what share of its risk comes from equities?Portfolio risk mathsHardMulti-assetAsset allocation
Try it first
Roughly what share of the portfolio's risk comes from the 60% in equities?
Show the worked solution
Volatility is about 11.3%, and equities contribute about 93% of the risk. The variance is 0.36 x 324 plus 0.16 x 36 plus 2 x 0.6 x 0.4 x 0.1 x 18 x 6, which is 116.64 plus 5.76 plus 5.18, or 127.58. The square root is 11.3%. Equities' own term plus half the cross term is 119.2, which is 93% of the total.
Why does 60% of the money carry more than 90% of the risk?
Picture a household with two earners, one a salaried clerk and one a commission-only salesperson. The salesperson may bring in 60% of the money but almost all of the month-to-month swings. Risk contribution depends on weight times volatility, and variance squares it, so a sleeve three times as volatile with more capital swamps the other. Equities' variance term is 0.36 times 324, or 116.6; bonds' is 0.16 times 36, or 5.76. Twenty to one before the cross term.
Capital is split 60/40, but equities supply 93.5% of the portfolio's variance and bonds only 6.5%. The portfolio's volatility is 11.30%, so a 60/40 portfolio behaves almost entirely like an equity portfolio with the volume turned down.
How do you split the risk between the two sleeves?
Give each asset its own variance term plus half of the cross term. An asset's risk contribution is its weight times its covariance with the whole portfolio, and the contributions add to the total variance. For equities that is 0.6 x (0.6 x 324 + 0.4 x 10.8), which is 0.6 x 198.72, or 119.23. Divided by 127.58, that is 93.5%. Bonds take the remaining 6.5%.
The relationship
$\sigma_p^2 = w_e^2\sigma_e^2 + w_b^2\sigma_b^2 + 2 w_e w_b \rho\,\sigma_e\sigma_b \qquad RC_e = \frac{w_e\,(w_e\sigma_e^2 + w_b\,\rho\,\sigma_e\sigma_b)}{\sigma_p^2}$
w_e, w_b the capital weights, 60% and 40%
\sigma_e, \sigma_b the volatilities, 18% and 6%
\rho the correlation, 0.1
RC_e the equity share of portfolio variance
What it says in wordsPortfolio variance is each asset's own variance plus the cross term, and each asset's share is its weight times its covariance with the portfolio.
This is the arithmetic behind risk parity, which sizes each sleeve so that the risk contributions are equal rather than the capital. To give bonds half the risk here, the portfolio would hold roughly three times as much bond capital as equity, and often borrow to lift the return. The limitation: correlation is not stable. In some years stocks and bonds fall together, and the 7% can grow quickly.
Where candidates lose it
The first slip is saying the portfolio volatility is 0.6 x 18 plus 0.4 x 6, which is 13.2%. That ignores diversification and would be right only at a correlation of one. The second is reporting 60% as the equity risk share because that is the capital share.
Write the three variance terms, take the square root, then split. The whole point of the question is the gap between 60 and 93, so say it in a sentence.
What the interviewer asks next
What equity weight gives equal risk contributions from the two sleeves?
How does the equity risk share change if the correlation rises to 0.5?
Why might a pension fund still describe itself as 60/40 despite this?
040A bet pays even money and you win it 55% of the time. You can bet any fraction of your capital, as often as you like. What fraction maximises long-run growth, and what happens to growth if you stake double that?Probability and expected valueCoreHedge fundsQuantitative asset management
Try it first
What happens to long-run growth if you stake twice the growth-maximising fraction?
Show the worked solution
Stake 10% of capital each time; staking 20% drives long-run growth to about zero. For an even-money bet the growth-maximising, or Kelly, stake is p minus q, 0.55 minus 0.45, which is 10%. That earns about 0.50% a bet in log terms. At 20%, 0.55 x log 1.2 plus 0.45 x log 0.8 is about zero, so twice the stake gives no growth at all despite double the expected profit per bet.
Why does betting more of a winning edge ever make you poorer?
A shopkeeper who wins more often than she loses still goes under if one bad month can wipe out half the shop, because rebuilding from half takes a doubling. Wealth compounds, so what matters over many bets is the average log return, and a loss hurts the log more than an equal-sized gain helps it. With a small stake, the edge dominates. With a large stake, the losses' extra damage grows faster than the edge, and past a point it wins.
Long-run growth peaks at 0.50% a bet when 10% is staked and falls back to about zero at a 19.9% stake, so doubling the growth-maximising bet throws away the whole benefit of the edge. Half the Kelly stake keeps about three quarters of the peak growth.
How do you find the 10% and check the 20%?
Write growth per bet as g(f) = p log(1 + f) + q log(1 minus f), set its slope to zero, and the answer for even money is f = p minus q. The Kelly fraction for an even-money bet is simply the edge: 55% minus 45% is 10%. At 10%, g is 0.55 x 0.0953 minus 0.45 x 0.1054, about 0.50%. At 20% it is 0.55 x 0.1823 minus 0.45 x 0.2231, about -0.014%: essentially zero, and in fact a touch below it. Over 100 bets at the Kelly stake, capital grows by a factor of about 1.65 on the typical path; at 20% it goes nowhere.
The relationship
$g(f) = p\ln(1+f) + q\ln(1-f) \qquad f^{*} = p - q = 0.10$
f the fraction of capital staked on each bet
p, q the chances of winning and losing, 55% and 45%
g(f) the long-run growth rate per bet
f^{*} the growth-maximising stake
What it says in wordsLong-run growth is the probability-weighted log of each outcome, and it peaks when the stake equals the edge.
Practitioners usually bet a fraction of Kelly. Half Kelly, 5%, keeps about 75% of the peak growth with half the volatility, and it protects against the real problem: in markets you never know p exactly. If the true edge were 52.5% rather than 55%, the 10% stake would already sit near zero growth.
Where candidates lose it
Candidates reason from expected profit, which rises in a straight line with the stake, and conclude that more is always better with a positive edge. Others say bet everything, or bet 55%, confusing the win probability with the stake.
Frame it as compounding from the first sentence, give f equals p minus q, and then show the 20% case going to zero. That second number is usually why the question is asked.
What the interviewer asks next
What is the Kelly stake if the bet pays 2 to 1 and wins 40% of the time?
Why do most practitioners bet half Kelly or less?
How does uncertainty about the true win probability change the stake you choose?
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
Try it first
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?
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
Try it first
The value rises about 14%. How much does the owner's equity rise?
Show the worked solution
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?
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
Try it first
Before working it: is the depositor's purchasing power growing?
Show the worked solution
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?
048Estimate how much cash is withdrawn from ATMs in India on an average day.Market sizing and estimationCoreIndian asset managementAsset management
Try it first
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
Try it first
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.
What the interviewer asks next
What persistence correlation would keep a top-decile fund in the top quartile on average?
How does the answer change if you look at five-year rankings instead of one?
Why do flows into last year's best funds tend to make this problem worse?
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
Try it first
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
What is the lowest possible correlation between B and C if both links are 0.5?
How would you check whether a 10 x 10 correlation matrix is valid?
Give a real-world example of three assets where this bound would bind.
052A fund's NAV at six successive year-ends is 100, 130, 91, 120, 84 and 140. What is its maximum drawdown?Performance measurementWarm upPerformance analysisRisk management
Try it first
Pick the maximum drawdown before you work it.
Show the worked solution
Minus 35.4%, from the peak of 130 to the trough of 84. Track the running peak: 100, then 130, which holds until 140 in the final year. The deepest fall below that peak is 84, and 84 over 130 minus 1 is -35.4%. The earlier dip to 91 was only minus 30%, and the fall from the starting 100 looks like just 16%.
Measured from where?
Picture a climber on a ridge. How far she has fallen is measured from the highest point she reached, not from the car park and not from the step she took a moment ago. A drawdown is the fall from the running peak, the highest NAV seen so far, and the peak only resets when the NAV climbs above it. Here the peak jumps to 130 in year 1 and then sits there through the dip to 91, the recovery to 120 and the slide to 84. Every one of those years is inside the same drawdown.
The running peak holds at 130 from year 1 until the NAV reaches 140 in year 5, and the deepest point under that peak is 84 in year 4, so the maximum drawdown is -35.4%, much deeper than the 16% fall the start value suggests.
The relationship
$\text{MDD} = \min_t \left(\frac{V_t}{\max_{s\le t} V_s} - 1\right) = \frac{84}{130} - 1 = -35.4\%$
V_t the NAV at year-end t
max over s up to t the running peak, the highest NAV seen so far
What it says in wordsAt each date, compare the NAV with the best level reached so far; the worst of those comparisons is the maximum drawdown.
Why does the recovery to 120 not end the drawdown?
Because 120 is still below 130. Anyone who invested at the peak was still under water at 120, and the slide to 84 took them further down. A partial recovery does not reset the peak, so two separate-looking dips can be one long drawdown. Treating 120 as a new start gives a fall of 30% to 84, which is wrong on the question asked, though it is a correct answer to how bad the single year was.
Say what the number misses as well. Maximum drawdown is one path and one worst point, so it tells you nothing about how long the fund stayed under water, here four years from year 1 to year 5, and a longer history almost always shows a deeper one. Allocators read it next to the recovery time for that reason.
Where candidates lose it
The fast answer is 16%, the fall from the starting 100 to 84. It measures the investor who bought at launch, not the worst experience, and interviewers ask this with the path deliberately chosen so the start value misleads.
The second trap is resetting the peak at 120 and answering 30%. Say the rule out loud before you calculate: the peak only moves when the NAV beats it.
What the interviewer asks next
How long was the fund under water, and why do allocators care about that as much as the depth?
What gain was needed from 84 to recover the 130 peak?
Why does a longer track record almost always show a deeper maximum drawdown?
053You have nine bags of coins that look identical. Eight weigh the same and one is lighter. Using a balance scale with no weights, what is the fewest number of weighings that always finds the light bag?Logic brainteasersWarm upAsset management
Try it first
What is the fewest weighings that is guaranteed to work?
Show the worked solution
Two weighings. Put three bags on each side and leave three aside. Whichever side rises holds the light bag; if the pans balance, it is among the three on the table. Take that group of three and weigh one bag against another: the one that rises is light, and if they balance, it is the third. A balance gives three outcomes per weighing, and three times three covers nine bags.
Why is halving the pile the wrong instinct?
Halving is what you would do with a question that answers yes or no, like guessing a number between 1 and 100. A balance tells you more than that. A balance scale has three outcomes, left pan rises, right pan rises, or level, so each weighing should split the suspects into three equal groups, not two. The group left on the table is not wasted: a level balance is information too. With four and four plus one aside you would learn little from a level result except that the odd bag is the spare one, and an unlucky run needs three weighings.
Weighing three bags against three splits nine suspects into three groups of three, whichever way the pans move, and weighing one bag against one then names the light bag, so two weighings give nine end points, one for each bag.
How do you prove two is the minimum?
Count outcomes. One weighing produces at most three different results, and three results cannot point to nine different bags. With k weighings you can separate at most 3 to the power k bags, so nine bags need at least two weighings, and the method above shows two is enough. The same count answers the natural follow-up: three weighings handle up to 27 bags, four handle 81.
The relationship
$3^k \ge n \quad\Rightarrow\quad 3^2 = 9 \ge 9$
k the number of weighings
n the number of bags, one of which is light
3 outcomes of one weighing: left light, right light, balanced
What it says in wordsEach weighing multiplies the number of distinguishable outcomes by three, so you need enough weighings for the outcomes to cover every bag.
Why would an asset manager ask this? It is a test of whether you use all the information a measurement gives you. A risk report, a performance attribution or a data screen can be read in more ways than pass and fail, and the candidate who spots the third outcome here is the one who reads the level result as a finding.
Where candidates lose it
The trap is answering three or four because you split into halves. It is the natural instinct, and the interviewer is watching whether you notice that a balance can come out level, which is a result in its own right.
The second loss is giving two without the counting proof. Say the 3 to the power k rule in one sentence: it shows you know the answer is a minimum, not just a method that happened to work.
What the interviewer asks next
How many weighings do you need for 27 bags? For 100?
Now you do not know whether the odd bag is heavier or lighter, and there are twelve. How many weighings?
You may take any number of coins from each bag and weigh once on a scale that shows grams. How do you find the light bag?

w_e, w_b	the capital weights, 60% and 40%
\sigma_e, \sigma_b	the volatilities, 18% and 6%
\rho	the correlation, 0.1
RC_e	the equity share of portfolio variance

f	the fraction of capital staked on each bet
p, q	the chances of winning and losing, 55% and 45%
g(f)	the long-run growth rate per bet
f^{*}	the growth-maximising stake

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

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

Try it first

Why does 60% of the money carry more than 90% of the risk?

How do you split the risk between the two sleeves?

Where candidates lose it

What the interviewer asks next

Try it first

Why does betting more of a winning edge ever make you poorer?

How do you find the 10% and check the 20%?

Where candidates lose it

What the interviewer asks next

Try it first

Why does a survivors-only average flatter the category?

How do you compute the true figure and the bias?

Where candidates lose it

What the interviewer asks next

Try it first

Why does a one-point fall in the cap rate lift value by 14%, not by one point?

How does leverage turn 14% into 36%?

Where candidates lose it

What the interviewer asks next

Try it first

Why can a positive-looking rate lose purchasing power?

How do you get the exact figure, and what does the tax really cost?

Where candidates lose it

What the interviewer asks next

Try it first

Why build the count from the machines and from the people?

Which assumptions need the most care?

Where candidates lose it

What the interviewer asks next

Try it first

Why should a top-decile fund be expected to fall back?

How do you turn a rank correlation into an expected rank?

Where candidates lose it

What the interviewer asks next

Try it first

Why can the third correlation not be anything you like?

How do you get 0.62 without drawing angles?

Where candidates lose it

What the interviewer asks next

Try it first

Measured from where?

Why does the recovery to 120 not end the drawdown?

Where candidates lose it

What the interviewer asks next

Try it first

Why is halving the pile the wrong instinct?

How do you prove two is the minimum?

Where candidates lose it

What the interviewer asks next