Portfolio Management puzzles, solved step by step

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

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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?
021Analyst A has an information coefficient of 0.10 on 25 independent bets a year. Analyst B has an information coefficient of 0.03 on 400 independent bets a year. Whose information ratio is higher?Performance measurementHardQuantitative asset managementSystematic investing
Try it first
Which analyst has the higher information ratio?
Show the worked solution
Analyst B, with 0.6 against 0.5. By the fundamental law of active management, the information ratio is roughly the information coefficient times the square root of the number of independent bets. A gets 0.10 times the root of 25, which is 5, for 0.5. B gets 0.03 times the root of 400, which is 20, for 0.6. Breadth beats accuracy here, but only if B's bets really are independent.
How can a less accurate analyst add more value?
Think of two shopkeepers. One makes a large margin on a few sales a day; the other makes a thin margin on hundreds. The second can earn more, and with less swing day to day, because many small edges average into a steadier total. Active return grows with the number of bets while its noise grows only with the square root, so the ratio of the two rises with the square root of breadth. This is Grinold's fundamental lawRichard Grinold's result that a strategy's information ratio is approximately its information coefficient times the square root of its breadth. of active management. B's skill is under a third of A's, but B's root breadth is four times A's.
Analyst A's skill of 0.10 on a root breadth of 5 gives an information ratio of 0.5, while analyst B's skill of 0.03 on a root breadth of 20 gives 0.6, unless B's bets are correlated enough to count as only 100, which cuts B to 0.3.
The relationship
$IR\approx IC\times\sqrt{BR} \qquad A: 0.10\sqrt{25}=0.5 \qquad B: 0.03\sqrt{400}=0.6$
IC the information coefficient: the correlation between forecasts and outcomes
BR breadth: the number of independent bets a year
IR the information ratio: active return per unit of tracking error
What it says in wordsSkill times the square root of independent bets gives the information ratio.
Where does the argument break?
At the word independent. Four hundred bets that all lean on the same factor, say cheap stocks, behave like far fewer independent bets, and breadth has to be counted in independent bets, not trades. If B's 400 positions carry the information of only 100 independent ones, B's ratio is 0.03 times 10, which is 0.3, and A is ahead again. The law also ignores costs: 400 bets a year means more turnover, and a thin edge of 0.03 is the first thing trading costs eat. A portfolio manager hiring between the two would ask B how correlated the signals are before believing the 0.6.
Where candidates lose it
The instinctive answer is A, because a coefficient of 0.10 sounds far better than 0.03. The interviewer is testing whether you know breadth enters the formula at all.
The second trap is taking B's 400 at face value. The follow-up is almost always what if the bets are correlated, and the answer is that breadth shrinks and B's advantage can vanish.
What the interviewer asks next
How many independent bets would A need to match B?
How would you estimate the effective number of independent bets in a portfolio?
Why do trading costs hit analyst B harder than analyst A?
029A fund returned 14% with 18% volatility in a year when cash paid 6%. Its benchmark returned 12%, and its tracking error against that benchmark was 4%. What are its Sharpe ratio and its information ratio, and what does each one tell you?Performance measurementWarm upPerformance analysisAsset management
Try it first
Which number goes in the denominator of the information ratio?
Show the worked solution
The Sharpe ratio is about 0.44 and the information ratio is 0.5. Sharpe divides the excess over cash, 14 minus 6 or 8 points, by total volatility of 18: 0.44. The information ratio divides the excess over the benchmark, 14 minus 12 or 2 points, by the tracking error of 4: 0.5. Sharpe judges the whole portfolio's risk; the information ratio judges only the manager's active bet.
Why are there two ratios for one fund?
Picture judging a cook. One question is whether the whole meal was worth its price. Another is whether the chef's changes to the standard recipe made it better. The Sharpe ratio asks whether the fund's total return beat cash by enough to justify all its risk, and the information ratio asks whether the manager's departures from the benchmark earned enough to justify that active risk. An investor choosing an asset mix cares about the first; one who has already chosen the market and is picking a manager cares about the second.
The Sharpe ratio divides 8 points of return over cash by 18 points of total volatility and gives 0.44; the information ratio divides 2 points of return over the benchmark by 4 points of tracking error and gives 0.50. Each ratio pairs a reward with the risk that produced it.
How do you read 0.44 and 0.5 once you have them?
The tracking errorThe standard deviation of the difference between a fund's return and its benchmark's return, a measure of how far the fund strays. is small against total volatility because most of the fund's ups and downs are the market's, which the benchmark shares. A 0.5 information ratio from a single year is respectable on paper but statistically weak: one year of 2 points against a 4-point tracking error is half of one standard deviation. It would take many years at that rate before anyone could separate skill from luck with confidence. The Sharpe ratio is also best compared with the benchmark's own Sharpe over the same period, not read alone.
The relationship
$\text{Sharpe} = \frac{R_p - R_f}{\sigma_p} = \frac{8}{18} = 0.44 \qquad \text{IR} = \frac{R_p - R_b}{TE} = \frac{2}{4} = 0.5$
R_p the fund's return, 14%
R_f the cash rate, 6%
\sigma_p the fund's total volatility, 18%
R_b the benchmark return, 12%
TE the tracking error, 4%
What it says in wordsEach ratio is a reward divided by the risk taken to earn it; the two differ in which reward and which risk.
Say one limitation in the room: both ratios assume returns are roughly normal. A fund that sells insurance-like options can post a smooth, high Sharpe for years and then lose a large amount in one month, which neither ratio sees in advance.
Where candidates lose it
The frequent mix-up is putting total volatility under the information ratio, which gives 2 over 18, about 0.11, and makes a reasonable manager look poor. The other is measuring the Sharpe numerator against the benchmark instead of cash.
Say the pairing aloud before calculating: excess over cash with total risk, excess over benchmark with active risk. Then the numbers take ten seconds.
What the interviewer asks next
The benchmark had 16% volatility. What was its Sharpe ratio, and did the fund beat it on that measure?
How many years of a 0.5 information ratio before the excess return is statistically significant?
Why can a fund have a higher Sharpe ratio than its benchmark but a negative information ratio?
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?
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?
065A fund returned 18% in a year when its benchmark index returned 14% and cash returned 6%. The fund's beta to the index is 1.3. What is its Jensen's alpha?Performance measurementCorePerformance analysisAsset management
Try it first
What is the fund's alpha for the year?
Show the worked solution
1.6%, not 4%. A fund with a beta of 1.3 should earn cash plus 1.3 times the market's return over cash: 6% plus 1.3 times 8%, which is 16.4%. It earned 18%, so the return its market exposure does not explain is 1.6 points. Most of the 4-point lead over the index came from simply taking more market risk in a rising year.
Why is beating the index by 4 points not the answer?
Imagine two drivers on a downhill road. One coasts, the other presses the accelerator. The second arrives first, but that says nothing about who drives better. A fund with a beta above 1 is expected to beat a rising market, so raw outperformance mixes skill with extra market exposure. Jensen's alpha strips out the part of the return that the fund's beta alone would have delivered, and asks what is left.
The relationship
$\alpha = R_p - [R_f + \beta(R_m - R_f)] = 18 - [6 + 1.3 \times 8] = 1.6\%$
R_p the fund's return, 18%
R_f the cash, or risk-free, rate, 6%
R_m the index return, 14%
beta the fund's sensitivity to the index, 1.3
What it says in wordsAlpha is the fund's return less what its market exposure alone would have earned.
The security market line gives a 1.3-beta fund a required return of 16.4% in a year when the index made 14% and cash 6%, so the fund's 18% is only 1.6 points of alpha, well short of its 4-point lead over the index.
What would you warn about the 1.6%?
Three things. Beta is estimated from past returns, and an error of 0.1 in beta moves alpha by 0.8 points here, half the answer. A single year's alpha is mostly noise, so it describes what happened rather than proving skill. And the answer flips in a falling market: in a year when the index loses, a 1.3-beta fund is expected to lose more, and a fund that merely matches the index would show positive alpha. Always read alpha next to the market's direction and the number of years behind it.
Where candidates lose it
The trap is answering 4%, the gap to the index, which ignores beta completely. The interviewer gave you the beta precisely so you would use it.
The second trap is multiplying the whole 14% by 1.3 to get 18.2% and calling alpha negative. Beta scales the market's return over cash, not the cash part. Write the formula first and put the numbers in.
What the interviewer asks next
In a year when the index falls 10% and cash earns 6%, what return gives this fund zero alpha?
How many years of 1.6% alpha, with 4% tracking error, would you need before trusting it?
Why might a manager prefer to be measured on information ratio rather than Jensen's alpha?
085An investor puts Rs 10 lakh into a fund that gains 20% in year one. Just before year two, the investor adds Rs 40 lakh, and the fund then falls 10%. What is the fund's time-weighted return, and how did the investor actually do?Performance measurementCorePerformance analysisWealth management
Try it first
Over the two years, how did the investor do in rupees?
Show the worked solution
The fund's time-weighted return is plus 8% over two years, but the investor lost Rs 3.2 lakh. Time-weighted return chains the yearly returns, 1.20 x 0.90, and ignores cash flows. The investor had Rs 10 lakh in for the gain and Rs 52 lakh in for the fall, so Rs 50 lakh became Rs 46.8 lakh, a money-weighted return of about minus 5.4% a year.
How can both numbers be right?
A cricket team's run rate says how well it batted; a spectator who arrived only for the collapse saw a bad match. Both are true. Time-weighted return measures what the fund did with each rupee over each period, so it judges the manager; money-weighted return weights each period by how much money was in, so it judges the investor's own timing. Here the timing was poor: most of the money arrived just before the fall.
Rs 10 lakh grew to Rs 12 lakh, the investor added Rs 40 lakh, and the Rs 52 lakh then fell 10% to Rs 46.8 lakh, so the fund shows plus 8% time-weighted while the investor lost Rs 3.2 lakh, about minus 5.4% a year money-weighted.
How do you get the money-weighted return?
It is the internal rate of return on the investor's own cash flows. Find the single yearly rate at which Rs 10 lakh put in two years ago and Rs 40 lakh put in one year ago grow to exactly Rs 46.8 lakh today. That is a quadratic, and its root is a growth factor of about 0.9462, or minus 5.4% a year.
The relationship
$10(1+m)^2 + 40(1+m) = 46.8 \;\Rightarrow\; m \approx -5.4\%$
m the money-weighted return, a yearly rate
10, 40 the investor's contributions, Rs lakh, two years and one year before the end
46.8 the ending value, Rs lakh
What it says in wordsThe money-weighted return is the one rate that grows the investor's contributions into the ending value.
This is why fund factsheets quote time-weighted returns: the manager does not control when investors add money. It is also why many investors earn less than the funds they own, because inflows tend to follow good years. Annualised, the fund's figure is 3.9% a year against the investor's minus 5.4%.
Where candidates lose it
The trap is reporting plus 8% as the investor's experience. The interviewer wants to hear that the fund's return and the investor's return are different quantities, and why.
The second slip is computing the investor's figure as a simple return on the total, Rs 3.2 lakh over Rs 50 lakh. That ignores that Rs 10 lakh was invested for two years and Rs 40 lakh for one. Name the internal rate of return as the correct tool.
What the interviewer asks next
Reverse it: the investor adds Rs 40 lakh after the fall instead. What changes?
Which return should appear on a fund factsheet, and which on a client statement?
Why do investors in aggregate often earn less than the funds they hold?
098A portfolio holds 60% equities returning 10% and 40% bonds returning 6%. Its benchmark holds 50% equities returning 12% and 50% bonds returning 5%. Split the 0.1 point shortfall into allocation, selection and interaction effects.Performance measurementHardPerformance analysisMulti-asset
Try it first
Was the decision to overweight equities a good one?
Show the worked solution
Allocation added 0.70 points, selection cost 0.50 and interaction cost 0.30, netting to the -0.1 shortfall. Allocation prices the weight bets at benchmark returns: +10% x 12 and -10% x 5. Selection prices the return gaps at benchmark weights: 50% x (10 - 12) and 50% x (6 - 5). Interaction is the weight bets times the return gaps.
Why split the result at all, if it only lagged by 0.1?
A team can lose a match by one run after a brilliant bowling spell and a dismal batting collapse; the scoreline hides both. Attribution separates the decision of how much to hold in each asset class from the decision of what to hold within it, so each can be judged on its own. Here the allocation call was right and the equity picking was poor, which is a very different story from a flat 0.1 point miss.
From the benchmark's 8.5%, the equity overweight adds 0.70 points, weak equity selection takes 0.50 and the interaction takes 0.30, arriving at the portfolio's 8.4%.
Segment Allocation Selection Interaction Total
Equities +1.20 -1.00 -0.20 -0.00
Bonds -0.50 +0.50 -0.10 -0.10
Total +0.70 -0.50 -0.30 -0.10
Equities carry the story: the overweight earned +1.20 while picking cost -1.00 and the interaction -0.20; bonds lost -0.50 on the underweight but picked well.
What is the interaction term, and why does it bite here?
Interaction is the weight bet times the return gap. The manager overweighted equities by 10 points and underperformed within them by 2 points, so the extra equity weight magnified the poor picking by 0.2; underweighting bonds, where the picking was good, gave up another 0.1. Some firms fold interaction into selection, since the weights are the manager's choice; say which convention you are using.
One more check an interviewer may ask for. The Brinson-Fachler version measures allocation against the total benchmark return, (weight bet) x (segment return minus 8.5). It gives the same total, 0.70, here, because the weight bets sum to zero, but it splits the credit between segments differently.
The relationship
$A = \sum (w_p - w_b) R_b, \quad S = \sum w_b (R_p - R_b), \quad I = \sum (w_p - w_b)(R_p - R_b)$
w_p, w_b portfolio and benchmark weights
R_p, R_b portfolio and benchmark returns within each segment
What it says in wordsAllocation prices the weight bets, selection prices the return gaps, interaction prices the two together.
Where candidates lose it
The common slip is judging the allocation by the portfolio's own equity return, 10%, instead of the benchmark's 12%. That mixes the picking into the allocation and understates a good decision.
The second slip is losing track of signs on the underweight. Ten fewer points of bonds at 5% is minus 0.5, and in selection the bonds' 1 point outperformance at 50% weight is plus 0.5. Write the table before speaking.
What the interviewer asks next
If the manager had held benchmark weights, what would the portfolio have returned?
Why do some firms fold interaction into selection?
How does attribution work for a portfolio that holds a segment the benchmark does not?

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

IC	the information coefficient: the correlation between forecasts and outcomes
BR	breadth: the number of independent bets a year
IR	the information ratio: active return per unit of tracking error

R_p	the fund's return, 14%
R_f	the cash rate, 6%
\sigma_p	the fund's total volatility, 18%
R_b	the benchmark return, 12%
TE	the tracking error, 4%

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

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

How can a less accurate analyst add more value?

Where does the argument break?

Where candidates lose it

What the interviewer asks next

Try it first

Why are there two ratios for one fund?

How do you read 0.44 and 0.5 once you have them?

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

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 beating the index by 4 points not the answer?

What would you warn about the 1.6%?

Where candidates lose it

What the interviewer asks next

Try it first

How can both numbers be right?

How do you get the money-weighted return?

Where candidates lose it

What the interviewer asks next

Try it first

Why split the result at all, if it only lagged by 0.1?

What is the interaction term, and why does it bite here?

Where candidates lose it

What the interviewer asks next