Mutual Fund Mastery puzzles, solved step by step

Puzzles: 100
Traced to a firm: 29
Topics: 13
Hard: 30

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022A fund falls 35%. What gain does it need to get back to where it started, and at 12% a year, how long would that take?Risk, volatility and drawdownWarm upRisk and complianceIndian AMCs
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What gain recovers a 35% fall?
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A gain of about 53.8%, which at 12% a year takes about 3.8 years. After the fall, 100 is 65. Getting back to 100 needs 35 more on a base of 65: 1 over 0.65, less 1, is 53.85%. At 12% a year, the time is the log of 1 over 0.65 divided by the log of 1.12, 3.80 years.
Why is the recovery larger than the fall?
Think of a shop that cuts a price by 50% in a sale, then raises it by 50% after. The shirt that was Rs 1,000 went to Rs 500 and comes back only to Rs 750. A loss is a percentage of the old, larger base, and the recovery is a percentage of the new, smaller base, so the same rupee gap is always a larger percentage on the way back up. A 35 point fall from 100 leaves 65, and 35 is a much bigger slice of 65 than of 100.
A fall of 35 from 100 is 35%, but the same 35 regained from a base of 65 is 53.8%, which takes about 3.8 years at 12% a year.
How do you get the time without a calculator?
Use the rule of 72 to bracket it. At 12% money doubles in about 6 years, and a 54% gain is about 0.62 of a doubling in log terms, so roughly 0.62 times 6, a little under 4 years. The exact figure is the log of 1.538 over the log of 1.12, 0.431 over 0.113, which is 3.80 years. A quick check: 1.12 to the third is 1.40 and to the fourth is 1.57, so the answer sits between three and four years, closer to four.
The relationship
$g = \frac{1}{1 - d} - 1 = \frac{1}{0.65} - 1 = 53.8\%, \qquad t = \frac{\ln(1/0.65)}{\ln 1.12} = \frac{0.431}{0.113} \approx 3.8 \text{ years}$
d the drawdown, 35%
g the gain needed to recover
t years to recover at 12% a year
What it says in wordsThe recovery needed is the inverse of what is left, less one, and the time is how many years of 12% growth it takes to multiply by that.
Fall Gain needed to recover
10% 11.1%
20% 25.0%
35% 53.8%
50% 100.0%
The gap between the fall and the recovery widens as the fall deepens; a halving needs a doubling.
Say what it means and what it leaves out. Deep drawdowns cost time, not just money, which is why fund risk teams watch maximum drawdown alongside volatility. The 12% is an assumption for the arithmetic; returns after a fall can be faster or slower, and an investor who sells during the fall locks in the 35% and never earns the recovery at all.
Where candidates lose it
The common slip is answering 35%, as if gains and losses were symmetric. Candidates who say it in a risk interview have shown they do not see why drawdowns matter more than their headline size.
The second slip is dividing 53.8 by 12 to get 4.5 years, which ignores compounding. Use logs, or step through 1.12 to the third and fourth powers, and say the answer is a little under four years.
What the interviewer asks next
What fall needs a 100% gain to recover?
If the fund then earns 8% a year instead of 12%, how long does recovery take?
Why might a risk team set a limit on drawdown rather than on volatility?
034An index fund has 16% volatility and 0.3% tracking error. An active fund in the same category has 17% volatility and 6% tracking error. For a client who judges herself against the index, which fund is riskier, and why is volatility the wrong measure here?Risk, volatility and drawdownCoreRisk and complianceIndian AMCs
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If the active fund has no edge, roughly how often will it trail the index by 5 points or more in a year?
Show the worked solution
For this client the active fund is far riskier, though the two volatilities look almost identical. Her pain is falling behind the index, and that gap is measured by tracking error: 6% a year against 0.3%. With no edge, the active fund trails the index by 5 points or more in about 20% of years; the index fund almost never does. Volatility of 16% and 17% describes how much each fund swings, which is nearly the same.
Why does the same volatility not mean the same risk?
Two students both score between 60 and 90 across a year of tests. One sits every test next to her twin and scores within a mark of him every time; the other drifts ten marks above or below him. If what upsets the family is doing worse than the twin, only the second student is a worry. Risk depends on what the investor measures herself against: total volatility answers how much the fund swings, and tracking error answers how far it strays from the index.
The two funds have nearly the same volatility, 16% and 17%, but the active fund's tracking error is 6% against the index fund's 0.3%, so with no edge the active fund trails the index by 5 points or more in about 20% of years and the index fund almost never does.
How do you put a number on the regret?
Treat the yearly gap to the index as roughly normal, centred on zero if the manager has no edge, with a standard deviation equal to the tracking errorThe standard deviation of the difference between a fund return and its benchmark return.. A shortfall of 5 points is 5 over 6, or 0.83 standard deviations below the centre, which the normal table puts at about 20%. For the index fund, 5 points is about 17 standard deviations away: effectively never.
The relationship
$P(r_{fund} - r_{index} \le -5\%) = \Phi\left(\frac{-5}{TE}\right) = \Phi(-0.83) \approx 20\%$
TE tracking error, 6% for the active fund
\Phi the standard normal cumulative probability
What it says in wordsThe chance of a large shortfall against the index depends on the tracking error, not on the fund's own volatility.
When is volatility the right measure?
Volatility is the right measure when the client cares about losing money outright, and tracking error when she cares about falling behind a benchmark. A retiree drawing income cares about the first; an investor who reads the index level in the newspaper every morning cares about the second. Most people care about both, which is why a suitability conversation asks which one hurts more. The normal assumption is a simplification and real gaps have fatter tails, so treat 20% as an order of magnitude.
Where candidates lose it
The fast answer compares 16% with 17% and says the funds carry about the same risk, or that the active fund is only slightly riskier. The question has told you the client judges herself against the index, which makes relative risk the measure.
The second loss is saying the index fund is riskless. It has 16% volatility and will fall with the market. It is low-risk only in the relative sense, and saying so shows you hold both measures in your head at once.
What the interviewer asks next
Over three years, how often would the active fund trail the index by 5 points a year on average?
What alpha would the active fund need so that it trails by 5 points only one year in ten?
How would you explain tracking error to a client in one sentence?
048A fund's NAV at the end of each quarter runs 100, 130, 110, 140, 91, 120. What is its maximum drawdown, and why is it not measured from the starting NAV of 100?Risk, volatility and drawdownWarm upRisk and complianceIndian AMCs
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What is the maximum drawdown?
Show the worked solution
The maximum drawdown is 35%, from the peak of 140 to the trough of 91. Drawdown is always measured from the highest value reached before the fall, because that is the value an investor held and then lost. Measured from the start, 91 looks like a mild 9% dip, which hides how much was lost on the way. The fund later climbs to 120, still 14.3% below its peak.
Why measure from the peak and not from the start?
A house bought for Rs 1 crore, valued at Rs 1.4 crore at the top of a boom and then sold for Rs 91 lakh, did not lose 9%. Its owner watched Rs 49 lakh disappear. A drawdown measures the pain of falling from the best point reached, so it runs from the running peak, which keeps rising whenever the NAV sets a new high. Anyone who invested at 140, or held through it, lost 35%, and that is the risk the measure is built to show.
The NAV path touches 140 and then falls to 91, a 35% drawdown from the running peak, while the earlier dip from 130 to 110 is only 15.4%; measured from the starting 100, the low of 91 would look like just a 9% fall.
How do you compute it step by step?
Walk along the path and keep two numbers: the highest NAV so far and the current NAV. At each point the drawdown is current over peak, less one. The maximum drawdown is the worst of those readings, here 91 over 140 less one, minus 35%. The dip from 130 to 110 gives minus 15.4%, a smaller drawdown. The final 120 against a peak of 140 is a drawdown of minus 14.3% still open.
The relationship
$DD_t = \frac{NAV_t}{\max_{s \le t} NAV_s} - 1, \qquad MDD = \min_t DD_t = \frac{91}{140} - 1 = -35\%$
NAV_t the NAV at time t
max NAV_s the running peak up to time t
MDD the maximum drawdown, the worst reading
What it says in wordsEach drawdown compares today's NAV with the best NAV so far; the maximum drawdown is the deepest of them.
Add the recovery maths: from 91 the fund needs a 53.8% rise just to get back to 140. And the measure depends on how often you look. These are quarter-end NAVs; a daily series could show a deeper trough between the quarter ends, so a maximum drawdown should always be quoted with its data frequency and period.
Where candidates lose it
The trap is measuring from the starting NAV and answering 9%, because the question opens with 100. The interviewer is checking that you know the reference point moves up with every new high.
The second slip is dividing by the trough, 49 over 91, and answering 54%. That is the gain needed to recover, not the drawdown. Say both numbers and label them.
What the interviewer asks next
What gain does the fund need from 120 to set a new high?
Why might a daily NAV series show a larger maximum drawdown than quarter-end NAVs?
Two funds have the same volatility; one has a much larger maximum drawdown. What could explain it?
060A fund has a tracking error budget of 4% a year and runs 20 active positions, each carrying the same amount of active risk, with the positions uncorrelated. How much active risk can each position carry?Risk, volatility and drawdownHardRisk and complianceIndian AMCs
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How much active risk can each position carry?
Show the worked solution
About 0.89% each. Uncorrelated risks add in squares. Twenty positions of active risk s give a total variance of 20 s squared, which must equal 4 squared, 16. So s squared is 0.8 and s is 0.89%. The straight split of 4% over 20, 0.20%, would use under a quarter of the budget, because it assumes every position fails at the same time.
Why can each position carry more than a twentieth of the budget?
Think of twenty friends each guessing the weight of a cake. Each guess is off by about 100 grams, but in random directions, so the errors partly cancel and the total of their errors is nowhere near 2 kilograms. Independent errors grow with the square root of how many there are, not in proportion, because some push up while others push down. Tracking errorThe standard deviation of the gap between a fund return and its benchmark return, usually quoted per year. works the same way. Twenty uncorrelated bets of equal size s give a total of s x sqrt(20), about 4.47 s.
Twenty uncorrelated positions of 0.89% active risk each have squares that add to 16, whose square root is the 4% budget, while the straight-line split of 0.20% each would use only about 22% of the risk allowed.
How do you solve it, and what happens if the bets are not independent?
Set the total equal to the budget and solve. 4 = s x sqrt(20), so s = 4 / 4.47 = 0.89%: each position may carry almost four and a half times the naive answer. The catch is the word uncorrelated. If every pair of bets has a correlation of 0.2, the variance picks up 20 x 19 cross terms, each worth 0.2 s squared. The total variance becomes 96 s squared and each position can carry only 0.41%. Correlated bets behave more like one large bet.
The relationship
$\sigma_{TE}^2 = N s^2 \;\Rightarrow\; s = \frac{\sigma_{TE}}{\sqrt{N}} = \frac{4\%}{\sqrt{20}} \approx 0.89\%$
sigma TE the tracking error budget, 4% a year
N the number of independent active positions, 20
s the active risk of each position
What it says in wordsWith independent bets the budget is shared out in variance, so each bet's risk is the budget divided by the square root of the count.
This is why risk teams care about the true number of independent bets more than the number of line items. Twenty stocks that are all quietly a bet on falling interest rates are close to one position, and they would breach the budget at a fraction of the size the arithmetic above allows. The limit: tracking error is a one-number summary of normal-times behaviour, and correlations between bets tend to rise in a sell-off, exactly when the budget matters.
Where candidates lose it
Almost everyone says 0.20%. It assumes risks add like rupees, which happens only when every position moves together. The interviewer wants to hear the word variance before any number.
The second trap is answering 0.89% and stopping. Say that it rests on zero correlation, and show how a modest correlation of 0.2 shrinks the allowance to 0.41%.
What the interviewer asks next
The fund adds 20 more uncorrelated positions. What can each one carry now?
One position is twice the size of the others in risk terms. How does that change the allowance for the rest?
Why do correlations between active bets tend to rise in a market sell-off?
073A fund has a downside capture of 80% and an upside capture of 95%. The index falls 20% and then rises 25%, ending exactly where it started. Where does the fund end?Risk, volatility and drawdownCoreRisk and complianceIndian AMCs
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Where does the fund end, relative to its start?
Show the worked solution
The fund ends up about 4.0%, while the index is flat. It falls 80% of the index's 20%, so 16%, to 84. It then gains 95% of the index's 25%, so 23.75%, which takes 84 to 103.95. Falling less matters more than it looks, because a smaller hole needs a smaller climb: the fund only needed 19% to get back to 100 and got 23.75%.
Why does missing part of the rally still leave the fund ahead?
If you fall into a 2 metre ditch you need to climb 2 metres out; fall into a 1.6 metre ditch and the same climbing gets you above the rim. Losses and gains compound on different bases: after a 20% fall, the index needs 25% just to get back, but a fund that fell only 16% needs just 19%. Downside captureThe fund return in falling markets as a share of the index return in the same periods. 80% means the fund fell 8% when the index fell 10%. of 80% means the fund lost 16% when the index lost 20%. Upside captureThe fund return in rising markets as a share of the index return in the same periods. 95% means the fund rose 9.5% when the index rose 10%. of 95% means it gained 23.75% when the index gained 25%.
The index falls to 80 and recovers to 100, while the fund with 80% downside and 95% upside capture falls only to 84 and ends at 103.95, and a fund that captures 110% both ways ends at 99.45.
The relationship
$(1 - 0.80 \times 0.20)(1 + 0.95 \times 0.25) = 0.84 \times 1.2375 = 1.0395$
0.80 downside capture
0.20 the index's fall
0.95 upside capture
0.25 the index's rise
What it says in wordsScale each index move by the capture ratio for that direction, then multiply the growth factors.
What does the comparison fund show, and what are the limits?
Take a fund that captures 110% in both directions, a bolder version of the index. It falls 22% to 78, then rises 27.5% to 99.45, ending below the index. A fund that amplifies both moves loses ground on a round trip, because the bigger fall needs an even bigger recovery; a fund that softens the falls more than the rises gains ground. That is why many research teams read the two capture ratios together and look for downside capture well below upside capture.
The limits matter. Capture ratios are measured over past periods and change with the manager's positioning, so a defensive fund in one cycle can be caught out in the next. Over a long bull market with few falls, the 95% upside capture costs more than the 80% downside capture saves. And the answer here depends on the index ending flat; the ratio between the two captures decides the outcome only for that kind of round trip.
Where candidates lose it
The trap is reasoning with the capture ratios as if returns add: 80% of the fall and 95% of the rise feels like a net loss of the rally. Returns compound, so the smaller fall leaves a smaller hole to climb out of.
The second miss is subtracting the capture ratios and calling the answer 15%. Work the two moves in order, 84 then 103.95, and the number is about 4%.
What the interviewer asks next
The index rises 25% first and then falls 20%. Does the fund end in the same place?
What downside capture would leave the fund exactly flat with a 95% upside capture?
Why might a fund with a low downside capture still trail its index over ten years?
085A fund's monthly returns have a standard deviation of 5%. What is that as an annual volatility, and is a month of minus 12% a rare event?Risk, volatility and drawdownWarm upRisk and complianceIndian AMCs
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What is the fund's annual volatility?
Show the worked solution
About 17.3% a year, and a minus 12% month is a 2.4 standard deviation event: rare, not freakish. Annual volatility is the monthly figure times the square root of 12: 5% x 3.46 is 17.3%. A minus 12% month is 12 divided by 5, or 2.4 monthly standard deviations. A normal curve puts that at about 0.8% of months, roughly once a decade, and real markets produce such months more often.
Why does volatility scale with the square root of time?
Take ten steps where each one goes left or right on a coin toss. You rarely end ten steps from where you began; lefts and rights cancel, and a typical distance is only about three steps, the square root of ten. Monthly returns behave the same way: good and bad months partly cancel, so variance adds up over time and the standard deviation grows only with the square root of the number of months. That is why the conversion uses the square root of 12, about 3.46, and not 12.
The relationship
$\sigma_{year} = \sigma_{month}\sqrt{12} = 5\% \times 3.464 = 17.3\% \qquad z = \frac{-12\%}{5\%} = -2.4$
\sigma_{month} the standard deviation of monthly returns, 5%
\sqrt{12} the square root of the number of months in a year
z how many monthly standard deviations the fall is from an average month
What it says in wordsScale volatility up by the square root of time, and measure a single month against the monthly figure, not the annual one.
With a monthly standard deviation of 5%, a minus 12% month lies 2.4 deviations below the centre, and the normal curve puts only 0.82% of months beyond it, about one in 122, while the same fund's annual volatility is 17.3%.
So how rare is a minus 12% month?
Measure it in monthly standard deviations. A minus 12% month is 2.4 monthly standard deviations below an average month taken as zero, which a normal curve puts at about 0.8% of months, roughly one in 122, or once a decade. If the fund's average month is plus 1%, the fall is 2.6 deviations and a normal curve makes it a little rarer, 0.47%. Against the 17.3% annual figure the fall sounds small, and that is the confusion to avoid: judge a month on a monthly scale.
Now the limitation. Equity returns have fatter tails than the normal curve: large falls cluster in crises and turn up more often than the bell shape allows. Treat the normal figure as a floor on how often such a month arrives, not as a promise that it will be rare. For a client the useful sentence is plain: a fund with annual volatility near 17% will, from time to time, lose more than a tenth of its value in a single month.
Where candidates lose it
The first loss is scaling the wrong way. Multiplying by 12 gives 60%, which describes a far wilder fund than this one, and it comes from treating risk as if it added up like returns. Say variance adds, then take the square root.
The second is judging the minus 12% month against the annual 17.3% and calling it ordinary. Put the move and the volatility on the same time scale before comparing them, and the month turns out to be a 2.4 deviation event.
What the interviewer asks next
What is this fund's volatility over a single week?
The fund's worst month in ten years was minus 20%. What does that tell you about the normal assumption?
Why might reported annual volatility understate the risk of a bad month?
099An equity fund is 95% invested, with 5% in cash, and the stocks it holds have a portfolio beta of 1.2. What is its effective market exposure, and what should an investor expect if the market falls 10%?Risk, volatility and drawdownCoreRisk and complianceIndian AMCs
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What is the fund's effective market exposure?
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About 114% of the market, so a 10% fall would be expected to cost the fund about 11.4%. Effective exposure is the invested weight times the beta of what is held: 0.95 x 1.2 = 1.14. The 5% in cash looks cautious, but the stocks are more sensitive than the market, and that more than cancels the cash. A typical 10% fall in the index maps to about -11.4% for the fund, before stock-specific moves.
Why can a fund with cash carry more market risk than the index?
Two drivers on the same road: one drives only 95% of the distance but at 1.2 times the speed of traffic, the other drives all the way at traffic speed. The first covers more ground per hour despite stopping short. Market exposure is how much money is in stocks multiplied by how strongly those stocks move with the market, so a small cash buffer can be more than undone by holding high-beta stocks. Here 95% invested at a beta of 1.2 gives an exposure of 114%, more than a fully invested index fund.
A fund 95% invested in stocks with a beta of 1.2 has an effective exposure of 114%, more than the index, while a fully invested fund with a beta of 0.9 has only 90%, so the cash level alone says little about market risk.
The relationship
$\beta_{fund} = w_{equity} \times \beta_{stocks} + w_{cash} \times 0 = 0.95 \times 1.2 = 1.14$
w_{equity} the share of the fund in stocks, 95%
\beta_{stocks} the beta of the stock portfolio, 1.2
w_{cash} the cash share, 5%, with a beta of zero
What it says in wordsA fund's beta is the weighted average of the betas of what it holds, and cash has a beta of zero.
What should the investor expect in a 10% fall, and how firm is that?
About -11.4%, as a central estimate. The cash adds a sliver of interest, about 0.025% over a month at an assumed 6% a year, too small to change the answer. Beta is an average relationship, not a promise: in any one fall, the fund's own stocks can do better or worse than the beta implies, and betas measured from calm periods often rise in a sell-off. So -11.4% is the expected fund move, with a range around it that depends on how much of the fund's risk is stock-specific.
For a risk desk the lesson is about which number to monitor. A fund manager who says 'we are defensive, we hold 5% cash' may be running more market risk than a fully invested peer. Report exposure as invested weight times beta, not cash level, when judging how a fund will behave in a fall. The fully invested fund with a beta of 0.9 in the figure has 90% exposure and would be expected to fall about 9% in the same move.
Where candidates lose it
The common answer is 95%, read straight from the cash level, which treats every stock as moving one for one with the market. It is the answer a fund's marketing would like you to give.
The other slip is quoting 120% and forgetting the cash. Both numbers matter, and the answer is their product. Then add one sentence on beta being an estimate, so the expected fall is a centre, not a forecast.
What the interviewer asks next
What cash level would bring this fund's effective exposure back to 100%?
The fund uses index futures worth 10% of assets on top. What is its exposure now?
Why might the measured beta of the stocks rise during a market fall?

d	the drawdown, 35%
g	the gain needed to recover
t	years to recover at 12% a year

Fall	Gain needed to recover
10%	11.1%
20%	25.0%
35%	53.8%
50%	100.0%

TE	tracking error, 6% for the active fund
\Phi	the standard normal cumulative probability

NAV_t	the NAV at time t
max NAV_s	the running peak up to time t
MDD	the maximum drawdown, the worst reading

Try it first

Why is the recovery larger than the fall?

How do you get the time without a calculator?

Where candidates lose it

What the interviewer asks next

Try it first

Why does the same volatility not mean the same risk?

How do you put a number on the regret?

When is volatility the right measure?

Where candidates lose it

What the interviewer asks next

Try it first

Why measure from the peak and not from the start?

How do you compute it step by step?

Where candidates lose it

What the interviewer asks next

Try it first

Why can each position carry more than a twentieth of the budget?

How do you solve it, and what happens if the bets are not independent?

Where candidates lose it

What the interviewer asks next

Try it first

Why does missing part of the rally still leave the fund ahead?

What does the comparison fund show, and what are the limits?

Where candidates lose it

What the interviewer asks next

Try it first

Why does volatility scale with the square root of time?

So how rare is a minus 12% month?

Where candidates lose it

What the interviewer asks next

Try it first

Why can a fund with cash carry more market risk than the index?

What should the investor expect in a 10% fall, and how firm is that?

Where candidates lose it

What the interviewer asks next