Mutual Fund Mastery puzzles, solved step by step

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

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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?
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?
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?
Show the worked solution
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?

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

0.80	downside capture
0.20	the index's fall
0.95	upside capture
0.25	the index's rise

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

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 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 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