Mutual Fund Mastery puzzles, solved step by step

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

Showing 1–6 of 6 · filtered from 100Clear filters

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
Try it first
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?
018Fund A returns 16% a year with 20% volatility. Fund B returns 12% with 10% volatility. Cash pays 6%. Which has the better Sharpe ratio, and which would you rather hold if you cannot borrow?Performance measurement and returnsCoreFund research and ratingsIndian AMCs
Try it first
Which fund has the higher Sharpe ratio?
Show the worked solution
B has the better Sharpe ratio, 0.6 against 0.5, but without borrowing a client who needs a 16% expected return can only get it from A. A earns 10 points over cash for 20 of volatility; B earns 6 for 10. With borrowing, B scaled to 20% volatility would offer 18%, beating A. Without it, B tops out at 12%.
What does the Sharpe ratio measure?
Think of two delivery riders. One earns Rs 1,000 a day riding 200 km; the other earns Rs 600 riding 100 km. The first earns more, the second earns more per kilometre. The Sharpe ratio is return above cash per unit of volatility, so it ranks how efficiently a fund turns risk into reward, not how much reward it delivers. A earns 10 points over cash for 20 points of volatility, 0.5; B earns 6 for 10, 0.6.
Fund B's line from cash rises more steeply than A's, a Sharpe ratio of 0.6 against 0.5, so with borrowing it would beat A at the same risk, but without borrowing it cannot go beyond its own 12% return.
Why can the lower-Sharpe fund still be the one to hold?
The Sharpe ratio assumes you can slide along the line from cash. Mix A with cash half and half and you get 11% at 10% volatility, worse than B's 12% at the same risk; that is B's Sharpe advantage at work. Going the other way, beyond B's own risk, requires borrowing, so an investor who cannot borrow and needs more than 12% expected return has to take the less efficient fund. With borrowing at the cash rate, B levered to 20% volatility would return 6% plus 0.6 times 20, which is 18%, beating A's 16%.
The relationship
$S = \frac{R - R_f}{\sigma}: \quad S_A = \frac{16 - 6}{20} = 0.5, \qquad S_B = \frac{12 - 6}{10} = 0.6$
R the fund's return
R_f the cash rate, 6%
\sigma the fund's volatility
What it says in wordsSubtract what cash pays, then divide by the risk taken to earn the rest.
Say the limits. Volatility treats upside and downside swings alike, and a fund with rare large losses can show a flattering Sharpe ratio until one arrives. Figures from a few years of history are noisy estimates, so a 0.5 against 0.6 difference may not be meaningful. And the answer to which to hold depends on the client's required return and tolerance for swings; the Sharpe ratio ranks the funds, it does not choose for the client.
Where candidates lose it
The common slip is picking A because it returns more, or because its excess return of 10 points beats B's 6. Both ignore the risk taken. The interviewer asked for a ratio and wants to see you divide.
The opposite slip is saying B, full stop, to the second question. The Sharpe ranking assumes leverage is available; without it, the higher-return fund may be the only way to reach a client's target, and saying that is what the follow-up was set up to test.
What the interviewer asks next
What mix of fund A and cash matches fund B's volatility, and what does it return?
Why might a fund with a high Sharpe ratio still lose a client a lot of money in one year?
What would the Sortino ratio change about this comparison?
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
Try it first
Your first guess: how many years?
Show the worked solution
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?
042A fund that holds mostly mid caps says it beat the Nifty 50 by 4 percentage points over the year. A mid cap weighted index that matches its style beat the Nifty 50 by 5 points over the same year. Did the manager add value?Performance measurement and returnsWarm upFund research and ratingsIndian AMCs
Try it first
What was the manager's own contribution, against the right benchmark?
Show the worked solution
No. Against a benchmark that matches its style, the manager lagged by 1 point. The fund's 4-point lead over the Nifty 50 came from holding mid caps in a year when mid caps beat large caps by 5 points. A plain mid cap index would have delivered that 5 points with no stock picking. The manager's own choices, after costs, cost the investor 1 point. Beating the wrong benchmark hid an underperformance.
Why is the Nifty 50 the wrong yardstick here?
A runner who trains at altitude and races at sea level will post faster times; the stopwatch is right but the comparison flatters him. A benchmark should hold what the fund holds, so that the gap measures the manager's choices and not the market segment the fund happens to sit in. A mid cap heavy fund compared with a large cap index is mostly measuring whether mid caps beat large caps that year, which no manager controls once the mandate is set.
Starting from the Nifty 50's 12%, the mid cap style added 5 points to reach 17%, and the manager's own choices then took away 1 point, ending at the fund's 16%; the claimed 4-point lead is entirely style.
How do you split the lead into style and skill?
Insert the style benchmark between the two numbers. Fund minus broad index splits into style benchmark minus broad index, which is the style effect, plus fund minus style benchmark, which is what the manager added. Here that is 5 plus (minus 1), giving the claimed 4. Illustrating with a Nifty 50 return of 12%, the style index made 17% and the fund 16%.
The relationship
$\underbrace{R_f - R_{N50}}_{+4} = \underbrace{R_{style} - R_{N50}}_{+5} + \underbrace{R_f - R_{style}}_{-1}$
R_f the fund's return
R_N50 the Nifty 50's return
R_style the return of a mid cap weighted index matching the fund's holdings
What it says in wordsThe fund's lead over the broad index is the style's lead plus the manager's own lead over the style.
Two fairness points. The fund's return is after its expenses and an index's is not, so part of the minus 1 is cost; a passive mid cap fund would also have trailed its index by its own cost. And one year is a small sample: the right test of skill is the gap to the style benchmark over a full cycle, measured consistently. Regulators in India require schemes to show a benchmark that reflects their category; confirm the current rules before relying on any particular index choice.
Where candidates lose it
The trap is accepting the 4 points as skill because the number is true. It is true and irrelevant: the question is what the manager added beyond the segment the fund sits in, and that needs the style benchmark.
The second loss is the arithmetic sign. Candidates sometimes add the 5 and the 4, or subtract the wrong way. Write fund minus style benchmark, 4 minus 5, and the minus 1 is clear.
What the interviewer asks next
In a year when mid caps lag large caps by 8 points and the fund trails the Nifty 50 by 6, what did the manager add?
How would you pick a fair benchmark for a fund that holds 60% large caps and 40% mid caps?
Why might a fund house prefer to show the Nifty 50 as its comparison?
080A fund holds 30% in IT stocks against a 20% benchmark weight. The IT sector returned 5% while the whole benchmark returned 12%. Separately, its bank stocks, a 25% weight, beat the bank index by 3 points. Split the fund's active return into an allocation effect and a selection effect.Performance measurement and returnsCoreFund research and ratingsIndian AMCs
Try it first
What did the IT overweight do to relative performance?
Show the worked solution
Allocation cost 0.70% and selection added 0.75%, a net active return of about plus 0.05%. The IT overweight is 10 points in a sector that trailed the benchmark by 7 points: 0.10 x (5% minus 12%) is minus 0.7%. The bank stocks beat their index by 3 points on a 25% weight: 0.25 x 3% is plus 0.75%. Good stock picking almost exactly paid for a poor sector bet.
What is the difference between allocation and selection?
Picture a selector who picks four spinners for a pitch that suits pace, but whose four spinners bowl better than any other spinners in the country would have. Two separate decisions: how many of each kind, and which ones. Allocation measures the first decision, sector weights against the benchmark's weights; selection measures the second, how the stocks chosen inside a sector did against that sector. Splitting them tells a fund research team whether a manager's skill lies in calling sectors or in picking stocks, which matters more than the total when deciding what to trust next.
The relationship
$A = (w_p - w_b)(R_s - R_b) \qquad S = w_p\,(r_s - R_s)$
w_p, w_b the fund's and the benchmark's weight in the sector
R_s the sector index return
R_b the whole benchmark's return
r_s the return on the stocks the fund actually held in that sector
What it says in wordsAllocation is the extra weight times how the sector did against the whole benchmark; selection is the weight held times how the chosen stocks did against their sector.
The 10 point IT overweight cost 0.70% because IT trailed the benchmark by 7 points, and bank stocks that beat their index by 3 points on a 25% weight added 0.75%, leaving an active return of only plus 0.05%.
Why is allocation measured against the whole benchmark rather than against zero?
Because the extra 10% in IT had to come from somewhere, and the alternative was the benchmark itself. An overweight in a sector that makes money still costs you if that sector made less than everything else you could have held. Say the assumptions behind the split out loud: the bank weight matches the benchmark's, so banks carry no allocation effect; the IT stocks held matched the IT index, so IT carries no selection effect; and the 10 points taken from other sectors came from sectors that earned the benchmark's 12%.
One detail an interviewer may probe. Using the fund's 25% weight in the selection term folds in what the BrinsonThe Brinson method, named after the authors who set it out in the 1980s, splits a fund active return into allocation, selection and an interaction term. framework calls the interaction effect; the textbook version uses the benchmark's weight and reports interaction separately. With the bank weights equal here, both give the same 0.75%.
Where candidates lose it
Candidates multiply the overweight by IT's own return, 10% x 5%, and call the IT bet a gain of 0.5%. That ignores what the money would have earned in the rest of the benchmark, and it turns a costly decision into a profitable-looking one.
The other slip is netting everything into plus 0.05% and calling the manager roughly neutral. The split is the whole point: plus 0.75 on stocks and minus 0.70 on sectors describes a good picker whose sector calls are giving the gains away.
What the interviewer asks next
What if the fund had been underweight IT by 10 points instead?
The bank index itself beat the benchmark. Where does that show up?
Over three years, which of the two effects would you trust more as evidence of skill, and why?
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
Try it first
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

R	the fund's return
R_f	the cash rate, 6%
\sigma	the fund's volatility

\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

R_f	the fund's return
R_N50	the Nifty 50's return
R_style	the return of a mid cap weighted index matching the fund's holdings

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

What does the Sharpe ratio measure?

Why can the lower-Sharpe fund still be the one to hold?

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

Why is the Nifty 50 the wrong yardstick here?

How do you split the lead into style and skill?

Where candidates lose it

What the interviewer asks next

Try it first

What is the difference between allocation and selection?

Why is allocation measured against the whole benchmark rather than against zero?

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