Mutual Fund Mastery puzzles, solved step by step

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

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004A fund's NAV goes from 10 to 15 in year one and falls to 12 by the end of year two. An investor put in Rs 1 lakh at the start and another Rs 5 lakh after year one. The fund returned about 9.5% a year. What did the investor earn?Performance measurement and returnsHardFund research and ratingsIndian AMCs
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The fund made money over the two years. Did this investor?
Show the worked solution
He lost about 11.6% a year while the fund made 9.5% a year. Rs 1 lakh bought 10,000 units at 10 and Rs 5 lakh bought 33,333 units at 15. At NAV 12 his 43,333 units are worth Rs 5.2 lakh against Rs 6 lakh put in. The rate that makes those cash flows balance, his internal rate of return, is -11.6% a year.
How can the fund and the investor disagree on the return?
Picture a shop that sells ten mangoes at a profit on Monday, then stocks up on a hundred on Tuesday just before the price drops. The shop's price record looks fine; the owner's till does not. The fund's return measures what one rupee did if it stayed the whole time; the investor's return weights each period by how much of his money was there. Here only Rs 1 lakh enjoyed the rise from 10 to 15, and Rs 6.5 lakh suffered the fall to 12.
The NAV rose from 10 to 15 and fell to 12, a fund return of 9.5% a year, but the investor put Rs 5 of his Rs 6 lakh in at 15, so his holding ends at Rs 5.2 lakh and his money-weighted return is -11.6% a year.
How do you compute the investor's figure without a calculator?
Count units, then value them. Rs 1 lakh at 10 buys 10,000 units; Rs 5 lakh at 15 buys 33,333. That is 43,333 units, worth Rs 520,000 at 12. The money-weighted return is the single rate r that makes the two payments, grown at r, equal the final value. With x as 1 plus r, x squared plus 5x equals 5.2, and the positive root is 0.8838, so r is -11.6%.
The relationship
$1\cdot x^2 + 5\cdot x = 5.2 \;\Rightarrow\; x = \frac{-5 + \sqrt{25 + 20.8}}{2} = 0.884,\quad r = x - 1 \approx -11.6\%$
1 and 5 the payments in Rs lakh at the start and after year one
x one plus the investor's annual return
5.2 the holding's value in Rs lakh at the end of year two
What it says in wordsGrow each payment at one unknown rate to the end date and solve for the rate that matches what the holding is worth.
The fund's own figure is simple: 12 over 10 is 1.2 in two years, and the square root of 1.2 is 1.0954, so 9.54% a year. That is the number a factsheet shows, because the manager does not choose when investors arrive. The investor's IRR is the number his account statement should show, and it is the one his experience matches.
Where candidates lose it
Most candidates answer 9.5% or a smaller positive number, because they assume the investor must share the fund's result. The interviewer is testing whether you know two returns exist and which one belongs to whom.
The second trap is dividing the total loss by the total invested, Rs 0.8 lakh on Rs 6 lakh. That ignores that Rs 1 lakh was in for two years and Rs 5 lakh for one; the IRR handles the timing.
What the interviewer asks next
Which of the two numbers should a fund manager be judged on, and why?
If the Rs 5 lakh had gone in at the start instead, what would the investor have earned?
Why do investors in a fund often earn less than the fund's reported return over long periods?
030A fund manager has delivered 2% a year of alpha with a tracking error of 5%. How many years of data do you need before a two-standard-error test says the alpha is unlikely to be luck?Performance measurement and returnsHardFund research and ratingsIndian AMCs
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Your first guess: how many years?
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About 25 years. The yearly signal is the 2% alpha and the yearly noise is the 5% tracking error, an information ratio of 0.4. Over n years the average alpha's standard error falls with the square root of n, so the t-statistic is 0.4 times root n. Setting that equal to 2 gives root n of 5, or 25 years. Skill of a realistic size takes longer to prove than most careers last.
Why does it take so long?
Think of judging a batsman from a few innings. A player who averages 45 against a league average of 40 is better, but any single innings swings by 30 runs, so a handful of innings cannot separate him from an average player on a good run. A fund manager's alpha is a small average edge buried in large year-to-year noise, and averaging only shrinks the noise with the square root of time. Four times the data halves the noise; it does not quarter it.
Here the edge is 2% a year and the noise, the tracking error, is 5% a year. One year of data is a signal of 2 against noise of 5, a ratio of 0.4, which is the information ratio. After n years the average alpha is still about 2%, but its standard error has fallen to 5% over root n.
The relationship
$t = \frac{\alpha}{\sigma_{TE}/\sqrt{n}} = IR\,\sqrt{n} \quad\Rightarrow\quad n = \left(\frac{2}{0.4}\right)^2 = 25$
\alpha the average alpha, 2% a year
\sigma_{TE} tracking error, 5% a year
IR information ratio, alpha over tracking error, 0.4
n years of data
What it says in wordsThe confidence in an alpha grows with the square root of the years, so the years needed are the square of 2 divided by the information ratio.
The t-statistic of a 2% alpha on 5% tracking error rises only with the square root of time: it is 0.89 after 5 years and 1.26 after 10, and it crosses the two-standard-error line only at 25 years.
What does this mean for how funds are judged?
Most track records are far too short to separate skill from luck at conventional confidence, so a strong five-year record is weak evidence on its own. At 5 years the t-statistic here is only 0.89. That is why fund researchers lean on the process, the consistency of the style and the source of the returns, not the headline number alone. The same arithmetic runs the other way: a manager with an information ratio of 1.0 would need only 4 years, and one at 0.5 would need 16.
Say the assumptions. The test treats yearly alphas as independent draws from a stable process with a constant edge. Real managers change style, teams leave, and markets change, so a 25-year record rarely describes one unchanged process. The number is a sense of scale, not a rule.
Where candidates lose it
Candidates often answer 2 or 3 years, reasoning that 2% beats zero every year on average. They are thinking about the edge and forgetting the noise around it. The ratio of the two, the information ratio, is what sets the clock.
The other loss is the square root. Candidates who see that noise shrinks with more data sometimes divide by n instead of root n and answer 6 or 7 years. Say root n out loud and the 25 follows.
What the interviewer asks next
With monthly data instead of yearly, does the answer change?
What information ratio would let you reach t = 2 in 10 years?
Why might a fund with a 10-year record and a t of 1.3 still be worth backing?
092A fund returned 60%, 2% and 3% in three successive years while the median peer returned 25%, 10% and 10%. The fund ranked in the bottom quartile in two of the three years, yet sits in the top quartile on its three-year return. How?Performance measurement and returnsHardFund research and ratingsIndian AMCs
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What are the fund's and the median peer's three-year returns, a year?
Show the worked solution
One outsized year carries the whole period: 18.9% a year against the peers' 14.8%. Compounded, 1.60 x 1.02 x 1.03 is 1.681 for the fund and 1.25 x 1.10 x 1.10 is 1.513 for the median peer. The 35-point lead in year 1 is larger than the 8 and 7 points given back later. A single period's rank says how much was earned, not how consistently, and here the latest two years run at 2.5% a year against 10.0%.
How can losing two years out of three still win the period?
A batter who scores 150 in one innings and 5 and 8 in the next two has a better series total than one who scores 40, 45 and 45, but nobody would call him the more reliable player. A multi-year return adds up the size of each year's result, not the number of years won, so one very large year can outweigh several small losses. Here the fund was 35 points ahead in year one, then 8 and 7 points behind. Compounding the three years keeps most of that first lead intact.
The fund trailed the median peer in two of three years, yet its 60% first year lifts its compounded three-year return to 18.9% a year against 14.8%, while the last two years alone show 2.5% against 10.0%.
The relationship
$\text{CAGR} = \left(1.60 \times 1.02 \times 1.03\right)^{1/3} - 1 = 1.681^{1/3} - 1 = 18.9\%$
1.60, 1.02, 1.03 the fund's growth factor in each year
1/3 the cube root, which turns three years of growth into a yearly rate
CAGR compound annual growth rate
What it says in wordsMultiply the yearly growth factors, then take the cube root to turn the three-year growth into an average yearly rate.
What would a fund researcher look at instead?
Consistency, measured several ways. Shift the start date by one year and the same fund drops from top to bottom: over the last two years it made 2.5% a year against the peers' 10.0%. That is the case for rolling returns, which compute the return over every possible window of a given length and show how often the fund beat its peers, not just whether it did over one window ending today. A researcher would also ask what produced the 60% year: one concentrated bet that paid off, a sector that ran, or something repeatable.
One more point about averaging. The simple average of the fund's three years is 21.7%, well above the 18.9% it actually compounded at; volatile return streams always show a bigger gap between the two. Quote compounded returns for anything an investor actually experienced, and treat a period return as one data point, not a verdict. The quartile labels here are illustrative: they assume a peer group in which 2% and 3% fell in the bottom quarter and 18.9% in the top.
Where candidates lose it
The tempting wrong answer is that the numbers must be inconsistent, since a fund that trails in most years should not be able to come out ahead over the period. Candidates who say this have counted wins instead of adding up sizes, and missed that one 60% year outweighs two years of trailing by 7 or 8 points.
The second loss is getting 18.9% right and stopping. The question invites the point that period returns hide consistency; say that rolling returns, and starting one year later, tell a very different story.
What the interviewer asks next
In what order would the three years have to come for the fund to look worst on a three-year view?
How would you measure consistency across ten years of monthly data?
A fund house advertises its fund's five-year rank. What would you ask before believing it says anything about skill?

1 and 5	the payments in Rs lakh at the start and after year one
x	one plus the investor's annual return
5.2	the holding's value in Rs lakh at the end of year two

\alpha	the average alpha, 2% a year
\sigma_{TE}	tracking error, 5% a year
IR	information ratio, alpha over tracking error, 0.4
n	years of data

1.60, 1.02, 1.03	the fund's growth factor in each year
1/3	the cube root, which turns three years of growth into a yearly rate
CAGR	compound annual growth rate

Try it first

How can the fund and the investor disagree on the return?

How do you compute the investor's figure without a calculator?

Where candidates lose it

What the interviewer asks next

Try it first

Why does it take so long?

What does this mean for how funds are judged?

Where candidates lose it

What the interviewer asks next

Try it first

How can losing two years out of three still win the period?

What would a fund researcher look at instead?

Where candidates lose it

What the interviewer asks next