Mutual Fund Mastery puzzles, solved step by step

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.

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%.

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.

Waiting saves Rs 5,000, and it puts the whole Rs 5 lakh at five more weeks of market risk, about Rs 27,900 either way at one standard deviation. The load is 1% of the redemption value. At an assumed 18% annual volatility, a five-week move has a standard deviation of 5.6%. With no drift assumed, there is about a 43% chance of falling more than the 1% saved. The load is a known cost to weigh, not an automatic reason to wait.

What exactly does waiting buy?

Think of a train ticket with a cancellation fee that disappears if you wait until tomorrow. If you were going to travel anyway, waiting costs nothing. If you need the money for something else tonight, waiting has a cost the fee schedule does not show. An exit load is a certain cost on one side; staying invested for the waiting period is an uncertain gain or loss on the other, and the two have to be sized against each other. The certain side here is 1% of Rs 5 lakh, Rs 5,000.

How big is the uncertain side?

Scale annual volatility to five weeks with the square root of time. Eighteen per cent times the square root of 5 over 52 is 5.58%, about Rs 27,900 on Rs 5 lakh: the possible swing is several times the load. If the market has no drift over those weeks, the chance of losing more than 1% is the chance of a move below -1%, which is 43%. Waiting is close to a coin toss with a small Rs 5,000 tilt in its favour.

The right answer depends on why the money is leaving. If the investor would stay invested anyway, waiting is close to free and the Rs 5,000 is worth having. If the money is needed for a fixed purpose, or he has decided he no longer wants this exposure, five weeks of risk on the full amount is a real cost. Check one more thing: holding periods can also change how a gain is taxed, so confirm whether the same date crosses a tax threshold under current rules. The 18% volatility is an assumption for the arithmetic.

About 23%. Picture 1,000 managers. The 100 skilled ones produce 100 times 0.7 cubed, 34.3 three-year streaks. The 900 unskilled produce 900 times 0.5 cubed, 112.5. Of 146.8 streak holders, 34.3 are skilled, 23.4%. The streak more than doubles the odds of skill, from 10%, yet most streak holders are still lucky.

Why does a three-year streak prove so little?

Think of a test for a rare condition that catches most real cases but also flags plenty of healthy people. If the condition is rare, most positive results are false alarms. When the thing you are looking for is rare, even good evidence leaves most positives coming from the larger group, so the starting proportion matters as much as the evidence. Here skill is the rare condition, at 10%, and a three-year streak is a test that unskilled managers pass one time in eight.

How do you set it up fast in the room?

Use counts, not formulas. Pick 1,000 managers, split them by skill, then split each group by whether it produced the streak, and the answer is one box over the sum of two boxes. Skilled: 100 times 0.343 is 34.3. Unskilled: 900 times 0.125 is 112.5. The answer is 34.3 over 146.8, 23.4%. The same result in odds form: prior odds of 1 to 9, times a likelihood ratiohow many times more likely the evidence is if the manager is skilled than if not of 0.343 over 0.125, about 2.74, gives 2.74 to 9.

Push it once to show judgement. A ten-year streak changes the picture: 100 times 0.7 to the tenth against 900 times 0.5 to the tenth gives about 76% skilled, because the unskilled pass a ten-year test only about once in a thousand. The limitation is that the 10% and 70% are assumptions, and real years are not independent coin tosses; managers with a style that suits a market phase can string wins together without skill.

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.

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.

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.

1,968.406 units. Stamp duty of 0.005% on Rs 50,000 is Rs 2.50, so Rs 49,997.50 is actually invested. Divided by the NAV of 25.40, that is 1,968.4055, rounded to 1,968.406. The figure 1,968.504 divides the full Rs 50,000 by the NAV and ignores the duty; the 0.098 extra units are worth about Rs 2.49, the duty give or take rounding.

Why does the charge come off before the units are counted?

Think of buying petrol with a Rs 500 note when the pump adds a Rs 5 card fee: you get Rs 495 worth of litres, not Rs 500 worth. A purchase is converted to units only after any charge on the transaction is deducted, so the units equal the net amount divided by the NAV, not the gross amount. Here the charge is a stamp duty of 0.005%, as stated in the question; the rate is set by law and can change, so confirm the current rate before using it.

How do you do the division cleanly?

Work in two steps and check the gap. Fifty thousand over 25.40 is 1968.5039; the duty removes Rs 2.50, which at 25.40 is about 0.0984 of a unit, so the answer is 1,968.4055, shown as 1,968.406. Checking it that way tells you the difference between the two answers is exactly the duty converted into units, which is a useful sense check in any operations role.

Two practical points complete the answer. Which NAV applies depends on when the application and the money reach the fund relative to the cut-off time, so 25.40 is the NAV for the day the purchase qualifies, not necessarily the day it was submitted. And units are shown to three decimals on the statement; the exact rounding convention is set out in the scheme's documents, which is where an operations team would check it.

About 9.0%, not 7.6%. You collect the 7.6% coupon. A year later the bond has four years left, and on an unchanged curve four-year bonds yield 7.2%, so its yield has fallen 0.40% with no market move at all. With a duration of about 3.4, that lifts the price about 1.35%, from 100 to 101.35. Coupon plus this roll-down gain is about 8.95%.

Why does the bond's yield change if the curve does not move?

Stand halfway down a slope and take one step towards the bottom. The slope has not moved, but you are lower than you were. A bond on an upward sloping yield curve is doing the same thing every day it is held. The curve stays put, but the bond's own maturity shortens by a year, so a year later it is priced off a lower point on the same curve. Nothing happened in the market; the bond simply got older.

That slide has a name, roll-downThe price gain a bond earns as it ages and its yield moves down an unchanged, upward sloping yield curve., and a fund manager counts it as part of expected return just as surely as the coupon. Today the bond is a 5-year bond at 7.6%. A year from now it is a 4-year bond, and 4-year bonds yield 7.2%. A 7.6% coupon discounted at 7.2% is worth more than par.

How do you size the roll-down gain in your head?

Use duration. A bond's price moves by roughly its modified duration times the change in its yield, and the duration that matters is the bond's duration at the end of the year, when it is a 4-year bond. A 4-year bond with a 7.6% coupon has a modified duration of about 3.36. Multiply by the 0.40% fall and you get about 1.34%. Pricing the bond exactly, four coupons of 7.6 and 100 at maturity discounted at 7.2%, gives 101.35, a gain of 1.35%. The shortcut is within a hundredth of a per cent.

When does the roll-down vanish or turn against you?

Roll-down is only as good as the slope. On a flat curve there is nothing to slide down and the return is the coupon. On an inverted curve, where shorter bonds yield more, the bond rolls up to a higher yield and loses price as it ages. And the curve rarely stays still: if the 4-year yield ends the year at 7.6% instead of 7.2%, the roll-down is gone and you earn roughly the coupon alone. Say this limit out loud; it shows you treat roll-down as an expected return on an assumption, not a promise.

Because the step-up compounds the instalment itself, not just the corpus. Rising 10% a year, the instalment reaches about Rs 61,159 a month by year 20, so the money put in grows from Rs 24 lakh to about Rs 69 lakh, 2.9 times as much. The corpus rises from about Rs 100 lakh to about Rs 199 lakh, 2.0 times. It grows less than the money in because the extra money arrives late.

Why does 10% a year sound small and turn out large?

Think of a salary that rises 10% a year. Nobody feels rich in year two, but after nineteen raises the salary is more than six times where it started. A step-up SIP works the same way: the 10% is applied to the instalment every year, so the instalment grows geometrically, and so does the money going in. The flat SIP puts in Rs 1.2 lakh every year for 20 years, Rs 24 lakh in all. The step-up SIP puts in Rs 1.2 lakh in year one and about Rs 7.3 lakh in year 20, Rs 68.7 lakh in all.

Why does the corpus double when the money put in nearly triples?

Look at when the extra money arrives. The step-up adds nothing in year one and a lot in the final years: 84% of all the extra contributions are paid in the last ten years. Money paid late has little time to compound, so the step-up's extra rupees earn less growth each than the flat SIP's early rupees did. In the flat SIP, growth is about 76% of the corpus; in the step-up SIP it is about 65%. The step-up wins on sheer volume of money, not on better compounding.

What is the honest way to say this to a client?

Say both halves. The step-up roughly doubles the end corpus at the same assumed return, which is a large effect for a small yearly decision. But it does so by asking for much more money, most of it in later years, and it assumes income rises enough to carry a Rs 61,000 monthly instalment. The 12% return is an assumption for the arithmetic, not an expectation, and the doubling holds at any steady return only in rough terms.

They gain exactly the same: Rs 20,000 each. Rs 1 lakh buys 10,000 units at Rs 10 or about 555.6 units at Rs 180. The same portfolio rising 20% lifts the NAVs to Rs 12 and Rs 216. Both holdings are worth Rs 1.2 lakh. A low NAV is not cheap and a high NAV is not expensive: what you earn is a percentage of the money you put in.

Why does a Rs 10 unit feel cheaper than a Rs 180 unit?

A pizza cut into twelve slices is not more pizza than the same pizza cut into four. Each slice is smaller, that is all. An NAV is the fund's portfolio value divided by the number of units in issue, so a lower NAV means the same money is cut into more, smaller slices. The instinct that Rs 10 is cheap comes from shares, where a low price is also misleading, and from shopping, where a lower price usually buys the same thing for less. Here it buys a smaller slice of the same thing.

What actually decides the gain?

Only the portfolio. A fund's return is the percentage change in its NAV, and the NAV moves by exactly the percentage the portfolio moves, less costs, whatever level it starts from. An NAV of 180 means the older fund has grown eighteen-fold since its own launch at 10; it tells you about the past and nothing about the next move. Two funds holding the same portfolio at the same cost will deliver the same return from here.

Is there any real difference between the two funds?

Yes, but none of it is about the NAV level. A new fund has no track record, may take weeks to deploy the money it raises, and may carry a different expense ratio. An older fund may be larger, which matters in small caps. Those are real reasons to prefer one or the other. A lower NAV is not one of them, and saying so plainly is what the interviewer is listening for.

About 2.8% a year of active risk, which leaves little room to earn back a 1.8% fee. An R-squared of 0.97 means 97% of the fund's variance moves with the index. With a beta of 1.0, the index part is 16% squared; total variance is that over 0.97, and the remaining 3% is the fund's own, about 2.8% a year as a volatility. To beat the index after a 1.8% fee, the manager needs an information ratio of about 0.64, which few sustain.

What does an R-squared of 0.97 actually say?

Picture a cover band that plays a famous album note for note, with one improvised solo per concert. Most of what you hear is the original. An R-squared of 0.97 says 97% of the fund's ups and downs are the index's ups and downs; only 3% of its variance is the manager doing something different. The fee, though, is charged on 100% of the money. That mismatch is what the interviewer wants you to find.

The step people miss is that R-squared is a share of variance, and variance is volatility squared. So you cannot take 3% of 16%. Work in squares, then come back with a square root. The fund's own volatility, measured against the index, is its tracking errorThe standard deviation of the gap between a fund return and its benchmark return; the size of the fund active bets, measured as a volatility., and when beta is 1.0 it equals this residual risk.

Why does a 2.8% active risk make a 1.8% fee hard to earn back?

Think of the active risk as the size of the bets. A manager who deviates by about 2.8% a year and wants to beat the index after costs needs gross excess returns above 1.8% a year. That is an information ratio, excess return over active risk, of about 0.64, and managers who hold even 0.5 for a decade are rare. Put simply, the fund charges an active fee for a portfolio that is mostly the index, and the small active part has to work very hard to pay for all of it. Some analysts call the pattern closet indexing.

State the limits. R-squared and beta are estimated from past returns against one chosen index, and a different index can give a different R-squared. A high R-squared is also not proof of low skill; it only says the room for skill to show is small. The fair conclusion is a question for the fund, not a verdict on it.

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.

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.