Mutual Fund Mastery puzzles, solved step by step

It is a 0.3 percentage point cut, which is 30 basis points, and a 20% relative cut in the fee; "a 0.3% cut" is the sloppy one. Basis points remove the ambiguity. The 20% is true but invites a client to imagine returns rising 20%; on Rs 10 lakh the saving is Rs 3,000 a year, and a net return of 8.5% becomes 8.8%.

Why are there two different answers that are both right?

A shop raises the price of milk from Rs 50 to Rs 60. It went up by Rs 10, and it went up by 20%. Nobody confuses those, because rupees and percent look different. With rates, the change and the base are both percentages, so "percent" can mean the gap between two rates or the change relative to the old rate, and you must say which. The gap between 1.5% and 1.2% is 0.3 percentage pointsThe plain difference between two percentages. From 1.5% to 1.2% is a fall of 0.3 percentage points.. Relative to the old fee, 0.3 is a fifth of 1.5, a 20% cut.

Which description will a client get wrong, and what should you say instead?

"A 0.3% cut" fails one way: a careful reader can take it as 0.3% of the fee, which would leave 1.5% x 0.997 = 1.4955%, a cut a hundred times too small. "A 20% cut" fails the other way: it is true of the fee, but a client hears 20% and imagines the investment doing 20% better. The fee is a slice of a slice; a 20% smaller slice lifts a net return of 8.5% to 8.8%, only 3.5% better in relative terms. The clean form for a desk is basis pointsHundredths of a percentage point. 30 basis points equal 0.30 percentage points.; the clean form for a client is rupees: Rs 15,000 a year on Rs 10 lakh becomes Rs 12,000.

The same confusion runs through everything a fund desk reports: a yield moving from 7.0% to 7.5% is 50 basis points, not 0.5%, and a fund outperforming by 2% might mean two points or a fiftieth more. The limit worth stating: neither unit tells the client the money, so pair any rate change with the rupee effect on their actual holding.

The equity fund's fee takes 15% of its expected return; the liquid fund's takes about 3.8%. A fee is charged on the money invested, but it comes out of the return, so it should be measured against the return. 1.5 / 10 is 15%; 0.25 / 6.5 is 3.85%. Measured against the 3.5% extra that equity is expected to earn over cash, the 1.5% fee takes 43%.

Why does 1.5% sound smaller than it is?

An agent who charges you 1.5% of your house price to sell it sounds cheap until you remember you only made Rs 10 lakh of profit on a Rs 1 crore house: the Rs 1.5 lakh fee takes 15% of the profit. Fund fees are quoted on the money invested, but they are paid out of the return, so the honest size of a fee is the fee divided by the return it takes from. For the equity fund, 1.5 / 10 = 15%: almost one rupee in every seven the fund is expected to earn goes to costs. For the liquid fund, 0.25 / 6.5 = 3.85%.

What is the sharper way to measure the equity fee?

An investor holds equity instead of cash to earn the extra return, the equity risk premiumThe return equities are expected to earn above cash or short government bills, as pay for their extra risk.. If cash pays 6.5% and equity is expected to make 10%, the extra is 3.5%. A 1.5% fee takes 1.5 / 3.5 = 43% of that extra, so close to half of the reason for owning equity goes in costs. That is why low-cost index funds draw so much money: the fee matters most where the expected extra return is thin.

Two limits keep this honest. Expected returns are assumptions, not promises, and in a year when the fund falls the fee takes more than all of the return. And a higher fee can still be worth paying if the manager adds more than the fee in return after costs, which is rare enough that the evidence for it should be asked for, not assumed.

Because the extra ten years come first, and the earliest rupees compound the longest. At 1% a month, the early starter reaches about Rs 6.50 crore and the late starter about Rs 1.90 crore, 3.4 times less, though the gap in money paid in is only Rs 12 lakh. That Rs 12 lakh, saved from 25 to 35, grows to about Rs 4.60 crore by 60, 71% of the early starter's corpus.

Why is the early starter so far ahead for so little extra?

Plant a mango sapling at 25 and another at 35, and at 60 the first tree is not slightly bigger; it has had ten more seasons of growth on a trunk that kept getting larger. Money saved early does not just add ten more years of deposits; it gives every rupee in those years twenty-five more years of compounding after them. The late starter pays in Rs 30 lakh over 25 years and ends with Rs 1.90 crore. The early starter pays Rs 42 lakh over 35 years and ends with Rs 6.50 crore.

How much does the first decade contribute on its own?

Split the early starter's plan in two. From 25 to 35 she saves Rs 12 lakh, which is about Rs 23.2 lakh by 35. From then she saves exactly what the late starter saves, so that part ends at Rs 1.90 crore. The Rs 23.2 lakh from the first decade compounds for 25 more years at 1% a month and becomes about Rs 4.60 crore, 71% of her final corpus. Her first ten years are worth more at 60 than the late starter's entire 25 years of saving.

State the assumptions, because the size of the gap depends on them. A steady 12% every year is a simplification; real equity returns swing, and the order of good and bad years matters for a monthly saver. Inflation shrinks both corpora in today's rupees, though not the ratio. And the gap narrows if the late starter saves more each month; to match the early starter at 60 she would have to invest about 3.4 times as much, around Rs 34,000 a month.

52%, the same as on any other day. If each day's move is independent, the five falls carry no information about tomorrow. A run of five falls was unlikely before it began, 0.48 to the fifth power or about 2.5%, but that probability belonged to days that are now finished. Believing a rise is due is the gambler's fallacy.

Why does the streak not make a rise more likely?

A captain who has lost five tosses in a row walks out for the sixth feeling owed a win. The coin has no record of the first five and no sense of fairness to restore; it is still 50:50. Independence means past outcomes do not enter the calculation for the next one, so tomorrow's chance of a rise is the 52% it always was. The feeling that things must even out is real, and it has a name: the gambler's fallacy. It is strongest exactly when a streak is long, which is when it does the most damage to decisions.

Where does the tiny probability come from, and why is it the wrong number?

It answers a different question. Standing on Monday morning, the chance of five falls in a row was 0.48 multiplied by itself five times, about 2.5%, and the chance of six in a row was about 1.2%, roughly one in 82. Once five falls have happened, the only uncertainty left is tomorrow, and for independent days the chance of a rise given the streak equals the chance of a rise on any day. Six falls in a row is rare only when viewed from the start; viewed from Friday evening, it needs just one more fall.

Is the independence assumption true for real markets?

Not exactly, and saying so earns credit. Day-to-day direction in a broad index is very hard to predict from the previous days, but the size of moves does cluster: a run of falls often comes with bigger swings, so tomorrow's move may be larger even if its direction is a near coin toss. For an adviser the danger is not the arithmetic but the conversation: a client who believes a bounce is due will add money for the wrong reason, or hold out for a rebound the odds do not promise. A streak on its own is not a reason to act; any case for investing has to rest on the client's plan and horizon, not on the last five days.

2,333.333 units, and nobody gains or loses on the day. The holding is worth 1,000 x Rs 42, or Rs 42,000. Divided by B's NAV of Rs 18, that is 2,333.333 units, and 2,333.333 x Rs 18 is Rs 42,000 again. The swap ratio, 42 over 18, is 2.333 units of B per unit of A. A merger changes the unit count and the scheme, not the value held.

Why does a lower NAV not make scheme B cheaper?

Change a Rs 500 note into Rs 100 notes and you get five of them. You now hold more notes, but you are no richer. A NAV is the value of one unit, so the number of units you hold only means something when multiplied by the NAV; a merger at NAV swaps notes of one size for notes of another. Scheme B's Rs 18 NAV only says its units started at a lower price or have grown less since launch. It says nothing about whether B is cheap, good or bad.

So can a merger leave an investor worse off?

Not on the day of the swap, when both schemes are valued at their closing NAVs. What a merger can change is everything after the swap: a different portfolio, a different expense ratio, a different level of risk, and possibly a tax event. If scheme B charges more, holds riskier bonds or follows another strategy, the investor's future returns change even though the swap itself was fair. That is why investors in a scheme being merged are generally offered a window to exit without an exit load; confirm the current rules on how that window works before relying on it.

The three decimals are not decoration. Registrars in India commonly allot units to three decimal places, so the answer is 2,333.333 and not a rounded 2,333. Rounding down to whole units would take about Rs 6 from this investor, and across lakhs of folios that adds up, so fractional units exist precisely to keep the swap exact.

About 9 years at 8%; the exact figure is 9.006 years, so the rule of 72 is out by about 2 days. At 6% the rule gives 12 years against an exact 11.90, and at 12% it gives 6 years against an exact 6.12. In the range where fund returns usually sit, the rule is off by weeks, not years.

Why does 72 divided by the rate give the doubling time?

If the price of a Rs 100 thali rises 8% a year, it costs about Rs 200 in nine years. You did not need a calculator, only the fact that 72 over 8 is 9. The exact doubling time is ln 2 divided by ln(1 + r), and for rates of a few per cent ln(1 + r) is close to r, so the answer is close to 69.3 divided by the rate in per cent. Using 72 instead of 69.3 does two jobs: it nudges the answer up to correct for rates around 8%, and it divides cleanly by 2, 3, 4, 6, 8, 9 and 12, which is what makes it a mental tool.

Where does the rule start to slip?

At both ends, but slowly. At 6% it says 12 years and the truth is 11.90, so it is long by about 38 days. At 12% it says 6 years against 6.12, short by about 42 days. The rule is almost exact near 8% and drifts by a little over a month either side, which is far inside the uncertainty of any return assumption you would feed it. Only at high rates does the drift become worth mentioning: at 24% the rule says 3 years against an exact 3.22.

Two uses in a fund interview. Turn a return into something a client can feel: at 8%, money doubles roughly every nine years, so it quadruples in about eighteen. And run a claim backwards to test it: a fund that says it tripled money in ten years has compounded at about 11.6% a year, because tripling is about 1.6 doublings, one every 6.3 years, and 72 over 6.3 is a little over 11.

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.

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.

About 2.0% for the gilt fund and about 0.025% for the liquid fund. A bond fund's NAV moves by roughly minus its modified duration times the change in yield. With yields down 0.25 points, 8 x 0.25% is 2.0% and 0.1 x 0.25% is 0.025%, about 0.03%. The same rate move hits the gilt fund 80 times harder, which is the whole difference between the two products.

Why does a fall in yields raise bond prices?

Suppose you own a bond paying 7% and new bonds now pay only 6.75%. Anyone wanting 7% must buy yours, so they pay more for it, and they keep paying more until its yield matches the market's. Bond prices and yields move in opposite directions, and modified duration tells you by how much: roughly the percentage price change for each percentage point move in yield. A fund holding bonds with an average modified duration of 8 rises about 8% for a full point fall in yields, and about 2% for a quarter point.

What does this approximation leave out?

Two things, one small and one large. The small one is convexity: the price-yield curve bends, so for a fall in yields the gain is a little more than duration says. With an assumed convexity of 80, it adds only about 0.025% for a quarter-point move. The large one is the assumption that bond yields fall by exactly the cut: the repo rate is an overnight rate, and longer yields move on expectations, so if the market had already priced in the cut, long yields may barely move on the day. Gilt funds often gain before a widely expected cut and do little when it arrives.

In rupees, on Rs 10 lakh the gilt fund gains about Rs 20,000 and the liquid fund about Rs 250. That symmetry is the point to make to a client: the gilt fund that gains Rs 20,000 on a quarter-point fall loses about the same if yields rise a quarter point instead. Duration is the dial for how much rate risk the investor is buying, in either direction.