Mutual Fund Mastery puzzles, solved step by step

One third of each withdrawal, about Rs 16,667, is gain, and only that part is taxable. Rs 50,000 at NAV 60 redeems 833.33 units that cost 40 each, so Rs 33,333 is the investor's own capital returning. Deposit interest of Rs 50,000 is income in full. At any single assumed rate t, tax on the SWP is t times Rs 16,667 against t times Rs 50,000.

Is a withdrawal income?

Think of selling three of your own mangoes that you bought at Rs 40 each and can now sell at Rs 60. You receive Rs 180, but only Rs 60 of it is profit; Rs 120 is your money coming back. An SWP is a monthly sale of units, and each unit sold returns its purchase cost plus the gain on it, so only the gain portion is a taxable gain. At cost 40 and NAV 60, the gain is 20 out of 60, one third of every rupee withdrawn.

Why does it hold at any tax rate?

Apply one assumed rate t to both. The SWP's taxable amount is Rs 16,667 and the deposit's is Rs 50,000, so whatever t is, the SWP's tax is one third of the deposit's. At an assumed 30%, that is Rs 5,000 against Rs 15,000 a month. In practice the two are taxed under different heads and can face different rates and holding-period rules, so confirm the current rules; the structural point is that a withdrawal is mostly capital and interest is entirely income.

Now say the limitation honestly. The comparison is not like with like: the deposit's Rs 50,000 leaves the principal untouched, while the SWP is selling units, so the holding shrinks unless the fund grows faster than the withdrawals. And the gain share rises over time: as the NAV climbs, a larger part of every unit sold is gain, so the tax advantage narrows the longer the plan runs. A light tax bill early on is not a measure of whether the withdrawal rate is sustainable.

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.

200 miles. The car covers 100 miles at 50 miles an hour, so the journey lasts two hours. The bird flies the whole time at 100 miles an hour, so it covers 200 miles however many times it turns. The long way agrees: the bird's round trips are 133.3, then 44.4, then 14.8 miles, each a third of the last, summing to 133.3 times 1.5, 200.

What is the question really asking?

Think of a dog running back and forth between you and your front door while you walk home. You could trace every turn, or you could notice the dog runs the whole time you are walking. When something moves at a constant speed for a known length of time, distance is speed times time, and the path it takes in between does not matter. The bird's speed is given; the only thing you need is how long it flies, and that is the car's travel time.

Can you prove it the long way, too?

Yes, and doing so briefly earns credit. The car and bird close a 100 mile gap at 150 miles an hour, meeting after two thirds of an hour, 66.7 miles from B. The bird flies 66.7 miles back to B, arriving at 1 hour 20 minutes, when the car is 66.7 miles from A. Each new chase starts with a gap a third of the last, so each round trip is a third as long: 133.3, 44.4, 14.8 and so on. A geometric series with ratio one third sums to the first term times 1.5, which is 200.

Why would a fund interviewer ask this? It tests whether you look for the quantity that is fixed before diving into detail, the same habit that makes you ask what an investor actually earned before reconciling every trade. The limitation is that it is a pure puzzle; turning instantly at each meeting is an idealisation that makes the infinite number of turns harmless.

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.

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 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.