Mutual Fund Mastery puzzles, solved step by step

About Rs 3.5 crore. Year one's trail is 0.75% of Rs 50 crore, Rs 0.375 crore. The book grows 10% from markets and loses 8% to attrition, so it nets 1.10 x 0.92, about 1.2% growth a year, and the trail grows with it. A growing perpetuity at 12% values that at 0.375 divided by (12% minus 1.2%), about Rs 3.47 crore, roughly seven times one year's income.

Why is the book worth so little next to the Rs 50 crore it holds?

A distributor does not own the Rs 50 crore; the clients do. What the distributor owns is a stream of income, like the owner of a rented flat owns the rent, not the tenant's savings. The book is worth the present value of its trail income, and that income is only 0.75% of the assets each year. So the right starting point is Rs 37.5 lakh a year, and the question is how that stream grows and how fast it leaks away.

How do market growth and attrition combine?

They multiply. A client who stays sees the book grow 10%; then 8% of the grown book leaves. The net growth is 1.10 times 0.92 less 1, about 1.2%, not 10% minus 8%. That growth feeds a growing perpetuityA stream of payments that continues forever and grows at a constant rate; its value is the first payment divided by the discount rate less the growth rate., which is the standard way to value a stream that neither ends nor stays flat.

Which lever matters more, markets or client retention?

About equally, which is the point the interviewer wants. Cutting attrition from 8% to 4% lifts the value to about Rs 5.86 crore; lifting market growth from 10% to 14% lifts it to about Rs 5.27 crore. A distributor cannot control the market but can control service and retention, and retention moves the value as much as the market does. The limits: the formula needs growth below 12%, a trail rate is set by the fund house and can change, and the 10% market growth is an assumption, not a forecast.

The risky bond, 8.16% against 7.5%. Ninety six times in a hundred it pays 11%; four times it returns 40 of 100, a 60% loss. Weighted, 0.96 x 11 plus 0.04 x (minus 60) is 8.16%. A debt fund may still prefer the AAA because its investors treat it like a deposit: a single default is a sudden loss, triggers redemptions, and in a fund of twenty such bonds, one default a year is more likely than not.

How do you set up the expected return?

Think of lending Rs 100 to twenty-five friends at a good rate, knowing that one of them, on average, will repay only Rs 40. You have to count that friend before you celebrate the rate. Expected return weights every outcome by its chance, and the default outcome is a large loss, not a zero. Recovery of 40% means you lose 60 of your 100, and for simplicity the coupon is also lost in default.

If the risky bond pays more on average, why hold the AAA?

Because an average is not what a debt fund investor experiences. A debt fund is bought as a place for money that must not fall, so the question is not only the average but the chance and size of a loss. On one bond, the return has a standard deviation of about 13.9%, against zero for the AAA. Spread across twenty such bonds, the chance that at least one defaults in a year is 1 minus 0.96 to the twentieth, about 56%. Each default cuts the NAV overnight and can set off redemptions that force the fund to sell its better bonds.

There is also a margin-of-safety check: the risky bond only matches the AAA if the default chance rises to (11 minus 7.5) over 71, about 4.9%. A small error in the 4% estimate wipes out the advantage. Default probabilities are estimates, not facts, and they tend to rise together in a downturn, which is exactly when investors redeem.

A P/E of 20x, rising to 25x if ROE falls to 12% while the P/B stays at 3. Earnings are ROE times book value, so P/E equals P/B divided by ROE: 3 over 0.15 is 20. At 12% ROE the same P/B gives 3 over 0.12, or 25. The shares look no dearer on book, but each rupee of earnings now costs 25% more.

How do P/B and ROE give you the P/E?

Suppose a shop is sold for three times the money the owner has put into it, and the shop earns 15% a year on that money. A buyer paying 300 for every 100 put in gets 15 of profit a year for her 300, which is 20 years of profit. ROE is the bridge between book value and earnings, so P/E is simply P/B divided by ROE. Price sits on top of both ratios, and it cancels.

What does an unchanged P/B after a fall in ROE tell you?

If returns fall and the price-to-book does not, the shares have become more expensive per rupee of profit, even though nothing on the book-value screen moved. For a lender, book value is the natural anchor, and many investors screen on P/B alone. This is the case where that screen misleads: the same 3.0x now buys 12% returns instead of 15%. To hold a 20x P/E at 12% ROE, the P/B would have to fall to 2.4x.

There are fair reasons the market might hold the P/B: it may expect ROE to recover, or the fall may be a one-off provision. Say that, then say the test: if ROE stays at 12%, a P/B of 3.0 is paying for returns that are no longer there. The ratios here are illustrations; the relationship holds for any company whose book value is meaningful.

For this client the active fund is far riskier, though the two volatilities look almost identical. Her pain is falling behind the index, and that gap is measured by tracking error: 6% a year against 0.3%. With no edge, the active fund trails the index by 5 points or more in about 20% of years; the index fund almost never does. Volatility of 16% and 17% describes how much each fund swings, which is nearly the same.

Why does the same volatility not mean the same risk?

Two students both score between 60 and 90 across a year of tests. One sits every test next to her twin and scores within a mark of him every time; the other drifts ten marks above or below him. If what upsets the family is doing worse than the twin, only the second student is a worry. Risk depends on what the investor measures herself against: total volatility answers how much the fund swings, and tracking error answers how far it strays from the index.

How do you put a number on the regret?

Treat the yearly gap to the index as roughly normal, centred on zero if the manager has no edge, with a standard deviation equal to the tracking errorThe standard deviation of the difference between a fund return and its benchmark return.. A shortfall of 5 points is 5 over 6, or 0.83 standard deviations below the centre, which the normal table puts at about 20%. For the index fund, 5 points is about 17 standard deviations away: effectively never.

When is volatility the right measure?

Volatility is the right measure when the client cares about losing money outright, and tracking error when she cares about falling behind a benchmark. A retiree drawing income cares about the first; an investor who reads the index level in the newspaper every morning cares about the second. Most people care about both, which is why a suitability conversation asks which one hurts more. The normal assumption is a simplification and real gaps have fatter tails, so treat 20% as an order of magnitude.

A duration of 4.05, rising to 7.25 after the switch. Weight each duration by its share of the Rs 100 crore: 0.40 x 1 + 0.35 x 4 + 0.25 x 9 = 4.05. Move the Rs 40 crore of 1-year duration into the 9-year bonds and it becomes 0.35 x 4 + 0.65 x 9 = 7.25. A 1% rise in yields now costs about Rs 7.25 crore instead of Rs 4.05 crore: the rate risk is about 79% higher.

Why is portfolio duration a weighted average?

A household's average commute is not the average of the three commutes, it depends on who travels most. If the person with the one-kilometre walk makes most of the trips, the household's average is short. A portfolio's duration is the money-weighted average of its bonds' durations, because each bond's price change counts in proportion to the rupees held in it. Here the biggest holding is the shortest, so the portfolio sits at 4.05, below the simple average of 4.67.

What changes in the portfolio when its duration rises?

The portfolio becomes more sensitive to interest rates in both directions: a 1% fall in yields gains about 7.25% instead of 4.05%, and a 1% rise loses as much. On Rs 100 crore, that is a swing of about Rs 7.25 crore for each 1% move instead of Rs 4.05 crore. On an upward sloping curve the portfolio's yield usually rises too, because longer bonds pay more, so the switch buys extra income with extra rate risk. Convexity also rises, which slightly cushions large moves, and the portfolio loses the cash-like buffer the short bonds gave it for meeting redemptions.

One caveat worth saying: duration is a linear estimate. For a 1% move it is close; for a 3% move, convexity makes the true loss smaller and the true gain larger than duration alone suggests. The durations here are given, and they drift as bonds age and yields move, so a portfolio's duration has to be re-measured, not set once.

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 investor who gets 9% first and 15% later ends richer, about Rs 3.62 crore against Rs 2.61 crore. A lump sum sees every year's return, and multiplication does not care about order: 1.15 to the 15 times 1.09 to the 15 is 29.64 either way. An SIP's money arrives over time, so the pot is small early and large late. Whichever return lands on the large late pot decides the result.

Why does order not matter for a lump sum?

A shop that marks a price up 15% and then 9% ends at the same tag as one that marks up 9% and then 15%; both multiply the price by 1.15 and by 1.09. A lump sum is one block of money exposed to every year's return, and because returns multiply, their order cannot change the product. Rs 10 lakh grows by 29.64 times in either order, to Rs 2.96 crore.

What breaks the symmetry in an SIP?

Think of a savings jar you add Rs 100 to every week. A bonus that doubles the jar is worth far more in December than in January, because in December the jar holds a year of savings. In an SIP, each instalment only experiences the returns after it is paid, so a return applied when the pot is large moves more rupees than the same return applied when the pot is small. After 15 years of Rs 10,000 a month, Rs 18 lakh has gone in and the pot holds Rs 61.6 lakh or Rs 36.9 lakh. The next 15 years apply to that pot plus Rs 18 lakh more, and it is the 15% that lands on it in one order and the 9% in the other.

What does this mean for a real SIP investor?

For anyone still contributing, the returns in the final years carry the most weight, because that is when the most money is exposed. A poor decade at the start of an SIP mostly costs a little growth on a small pot; the same decade at the end hits the whole accumulated corpus. The investor who saw 15% first was ahead by Rs 24.7 lakh at year 15 and still finished behind. The limits: real returns do not arrive in two neat blocks, and nobody can choose the order. The point is what the order does, which is why a plan near its goal often moves money towards steadier assets.

About Rs 95,000 before costs, and the trade itself pushes the premium down. The participant buys the underlying basket at Rs 245.60 a unit, swaps it for 50,000 new ETF units, and sells them at Rs 247.50, keeping Rs 1.90 a unit. That is a 0.77% premium on Rs 1.23 crore. Selling the new units adds supply, so the price falls towards NAV until the premium no longer covers costs.

Where does the profit come from?

Picture a sweet shop that sells a box of twelve laddoos for more than the twelve laddoos cost loose. Someone will buy loose laddoos, box them, and sell the boxes until the gap closes. An ETF unit is a box of shares, and when the box trades above the value of what is in it, the participant who can make new boxes earns the gap. Only authorised participantsLarge brokers or market makers allowed by the fund house to create and redeem ETF units in bulk, by exchanging baskets of the underlying shares. can make boxes, and only in creation-unit sizes, here 50,000 units.

Why does the premium shrink rather than persist?

Every round of the trade adds new units to the market, and that extra supply pushes the ETF price down towards the value of its basket. Buying the basket nudges the shares up a little too. The participant stops when the gap no longer pays for brokerage, taxes, impact and the risk of prices moving mid-trade. If those costs are 0.25% of the basket, about Rs 30,700, the net gain is about Rs 64,300, and the trade stops once the price is within about Rs 0.61 of NAV. The reverse trade works on a discount: buy units cheap, redeem them for the basket, sell the shares.

When does the mechanism fail to hold the price?

The arbitrage is only as good as the participants' ability to trade the basket. If the underlying shares are illiquid, or a market is shut while the ETF trades, or participants step back in a stressed market, premiums and discounts can stay wide for days. The iNAV is itself an estimate, refreshed every few seconds, so a small gap may be noise. The 0.25% cost is an assumption for the arithmetic; real costs depend on the basket.

The surface is 6n² unit squares, and n³ minus (n minus 2)³ cubes show at least one face. Six faces each carry n by n squares. For the cubes, count what is hidden: peeling one layer off every side leaves an (n minus 2) cube inside, so the visible ones are n³ minus (n minus 2)³, which expands to 6n² minus 12n plus 8. For a 3 by 3 by 3 cube that is 54 squares and 26 cubes.

Why are there two different counts here?

A house with a corner room has windows on two walls of that room, but it is still one room. Counting windows and counting rooms give different numbers. The surface counts squares, and a corner cube carries three squares and an edge cube two, so the square count is always larger than the number of cubes that show. Keeping the two counts apart is half the problem; candidates who blur them give 54 for both.

The surface is the easy part. Each of six faces is an n by n grid, so the area is 6n² unit squares: 54 for a standard cube. A quick check is to rebuild it from the cube types: 8 corners showing 3 squares, 12(n minus 2) edge cubes showing 2, and 6(n minus 2)² centre cubes showing 1. For n of 3 that is 24 plus 24 plus 6, which is 54 again.

Why count the hidden cubes instead of the visible ones?

The hidden cubes form one clean block, (n minus 2) on every side, so counting them and subtracting from n³ avoids every double count. Counting the visible cubes directly means adding corners, edges and faces while remembering that each edge already lost its corners. Both routes give the same answer, and the second makes a good check.

The expanded form is worth a sentence because it links the two answers. 6n² is the square count; the minus 12n plus 8 removes the extra squares that edge and corner cubes contribute. Say the edge case too: the formula needs n of at least 2. For n of 1 it gives 2, but a single cube is one cube.

Because withdrawals turn the order of returns into a permanent difference. Without withdrawals both would end at the same Rs 108.1 lakh, since returns multiply in any order. But the first retiree takes Rs 6 lakh out of a pot that has just fallen, to Rs 33.5 lakh after two years, so later good years work on a small base and she ends with about Rs 2.0 lakh. The second takes withdrawals from a pot that has grown, and ends with about Rs 39.0 lakh.

Why would order not matter without withdrawals?

Returns multiply, and multiplication does not care about order. Rs 60 lakh left untouched through these ten years ends at Rs 108.1 lakh whichever year comes first. The withdrawals are what break the symmetry: a fixed rupee amount taken out of the pot is a bigger share of a small pot than of a large one.

What exactly happens in the first two years?

Think of a shopkeeper who must pay rent of Rs 6,000 on the first of every month by selling stock. If prices have just crashed, he sells many more items to raise the same Rs 6,000, and those items are gone when prices recover. A retiree withdrawing a fixed sum after a fall sells more units at low prices, and those units never take part in the recovery. The first retiree's Rs 54 lakh, after the first withdrawal, meets minus 20% and becomes Rs 43.2 lakh; Rs 37.2 lakh then meets minus 10% and becomes Rs 33.5 lakh. The second retiree's Rs 54 lakh meets plus 20% first and is Rs 64.8 lakh after year one.

What does an adviser do with this?

The years just before and just after withdrawals begin carry the most weight; the risk is called sequence risk, and it is managed, not predicted. Common responses are holding two or three years of withdrawals in a steadier, short-term bucket so that a bad year does not force selling equity at a low, or trimming withdrawals after a fall. Note the mirror image of an SIP: for a saver adding money, bad years early are the kinder order; for a retiree taking money out, bad years early are the cruel one. The returns here are illustrative, and a 10% withdrawal rate is high by most planning conventions, which is what makes the gap so stark.