Mutual Fund Mastery puzzles, solved step by step

No. Tarvela's share fell from 8.0% to about 7.4%. Its AUM grew 30% while the industry grew 40%. Share is 40,000 over 5,00,000 before and 52,000 over 7,00,000 after. To hold 8% Tarvela needed Rs 56,000 crore, so it is Rs 4,000 crore short of standing still. Growing slower than your market means losing share while growing.

Why can a business grow and still shrink?

A child who grows 3 cm in a year while every classmate grows 5 cm is taller than last year and lower in the line-up. Market share is a ratio, so it rises only when your growth beats the market's, and a large rupee gain means nothing on its own. Tarvela's Rs 12,000 crore gain sounds big until you set it next to the industry's Rs 2 lakh crore.

What is the fastest way to answer without dividing big numbers?

Compare growth factors. New share equals old share times Tarvela's growth factor over the industry's: 8% times 1.30 over 1.40. 1.3 over 1.4 is about 0.93, so the share falls by about 7% of itself, from 8.0% to 7.43%. You never need to divide 52,000 by 7,00,000 unless asked for the decimal.

Add the analyst's question. AUM growth has two parts, market movement and net flows. If Tarvela is heavier in a segment that rose less, it could lose share with no problem in its sales. Splitting growth into the part the market gave and the part investors gave is the next thing an interviewer will ask for.

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.

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.

Rs 39 a share. Enterprise value is the house: Rs 1,000 crore for the business itself. The debt is the mortgage, Rs 300 crore owed to the bank, which leaves the owner Rs 700 crore of the house. The cash is money in the drawer that the owner keeps on top, Rs 80 crore. Together that is Rs 780 crore for the shareholders, which over 20 crore shares is Rs 39 each.

What does the house stand for, and what does the mortgage stand for?

A family owns a house worth Rs 1 crore with a Rs 30 lakh home loan, and keeps Rs 8 lakh in a drawer. If they sold everything and settled up, they would walk away with the house's value, less the loan, plus the drawer: Rs 78 lakh. Enterprise value is the value of the operating business, the house; lenders are paid first out of it, and whatever cash the company already holds belongs to the shareholders on top. Scale the same family up a thousand times and you have this company.

Why is cash added back rather than subtracted?

Because enterprise value was built to leave it out. A buyer of the whole company would pay for the business and inherit the cash, so the price of the business alone nets the cash away. Going from enterprise value back to equity, you reverse that step: take off what the lenders are owed and add back the cash the shareholders already own. Subtracting cash as well, the common slip, gives 1,000 minus 300 minus 80, Rs 620 crore, or Rs 31 a share, which hands the drawer to the bank.

Add the refinements an interviewer will probe. Anything else with a claim ahead of the ordinary shareholders comes off too, such as preference shares or a minority partner's share of a subsidiary, the way a second loan on the house would. And the share count should include options and convertibles that are likely to become shares. The limitation of the analogy is that a house is easy to price on its own; a business's enterprise value is itself an estimate, and every rupee of error in it lands on the equity.

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 Rs 20 crore a year. A basis point is 0.01%, one rupee in every ten thousand. Rs 2 lakh crore is Rs 2,00,000 crore, and dividing by 10,000 gives Rs 20 crore. A handy anchor: one basis point on Rs 1 lakh crore is Rs 10 crore. To an investor holding Rs 1 lakh the same basis point is Rs 10 a year, which is why a fund house fights over fees its investors barely notice.

How do you get the zeros right under pressure?

A basis point is a hundredth of a per cent, so it is one rupee in every ten thousand, the way a paisa is one part in a hundred of a rupee. Convert the AUM into crore first, then move the decimal four places to the left; one basis point on Rs 1 lakh crore is Rs 10 crore, and every other size scales from that anchor. Rs 2 lakh crore is Rs 2,00,000 crore, so one basis point is Rs 20 crore.

Why does a fund house fight over a few basis points?

Scale it up. A 10 basis point cut on Rs 2 lakh crore is Rs 200 crore a year of revenue gone, every year, with no saving in cost. At fund-house scale a basis point is real money, and a fee change is a large change in revenue for a business whose costs barely move with it. If the house earns, say, an average 0.50% across its book, its revenue is Rs 1,000 crore, and a 10 basis point cut takes away 20% of it. Those figures are illustrations; the shape of the arithmetic is the point.

Why is the same basis point invisible to the investor?

Because it is spread thin. An investor with Rs 1 lakh pays Rs 10 a year per basis point, the price of a cup of tea. The benefit of a fee change is concentrated in one balance sheet and its cost is spread across lakhs of folios, so the fund house argues hard and each investor shrugs. That is also why fee caps exist: in India the regulator caps the total expense ratioThe yearly cost of running a scheme, charged to the fund as a percentage of its assets and taken out of the NAV. in slabs that fall as a scheme grows. Confirm the current slabs before quoting them; they have been revised before.

The gap says a few star funds are pulling the average up; the typical fund made about 11%. The median is the middle fund, so half the category earned 11% or less. A mean of 14% needs a small group far out on the right: here six funds between 24% and 42%, and only 9 of the 40 beat the mean. A client choosing a fund at random should hear 11%, together with the spread around it.

Why can the average sit above most of the funds?

Ten people sit in a tea stall, each earning about Rs 30,000 a month. A business owner earning Rs 30 lakh a month walks in. The average income in the room jumps above Rs 2.9 lakh; the median, the middle person, still earns about Rs 30,000. A mean is pulled by every extreme value, while a median only cares about the middle, so when a few values are very large the mean overstates what a typical member got. Fund returns behave the same way: a few funds that caught one theme early can drag the category average up.

How do you check that the stars explain the gap?

Take them out and recompute. Without the six stars the other 34 funds average 10.9%, almost exactly the median. The six funds alone add about 3.1 percentage points to the category mean, which is the whole gap. A second check is to count: if the mean described a typical fund, about half the funds would sit above it. Here only 9 of 40 do.

Which figure should a client hear, and what else?

The median, because it is the honest answer to what a typical fund in the category did. Then the spread, because 5% to 42% is the real range of outcomes, and the chance of picking a star in advance is small. Two further cautions belong in the same breath. Category figures usually include only funds that survived the five years, and funds that closed or merged were often the weak ones, so even the median flatters. And past five-year returns, mean or median, are not a forecast.

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.

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%.

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.

About 76%. The easy route is the opposite event. Five positive years in a row needs a 75% chance to come up five times: 0.75 to the fifth is about 23.7%. Every other outcome includes at least one losing year, so the chance is 1 minus 0.237, about 76.3%. A small yearly chance of loss becomes close to a sure thing over an ordinary holding period.

Why work with the opposite event?

Ask a cricket fan the chance that a bowler concedes at least one boundary in an over, and the quick way is to ask the chance he concedes none. At least one is messy to count directly, because it covers one loss, two losses and every combination; none is a single clean path, so compute none and subtract from one. Here none means five positive years, each with a 75% chance, independent of the others: 0.75 times itself five times is 23.7%.

Why does an adviser care about this number?

Because it sets the client's expectation before the first bad year arrives. Over five years a losing year is the likely case, not the unlucky one, so a client who has not been told this reads the first negative year as something gone wrong. On the same assumptions the expected number of losing years in five is 1.25, the chance of exactly one is about 40%, and over ten years the chance of at least one rises to about 94%. A losing year is a calendar-year label; it says nothing about whether the five-year return was positive.

State the two assumptions. The 25% is an illustration, not a measured figure for any market, and real yearly returns are not fully independent: bad years cluster in some periods. Clustering lowers the chance of at least one losing year slightly while making the losing stretches longer. The method survives both caveats; the exact figure does not.