Mutual Fund Mastery puzzles, solved step by step

About -1.0% a year, so the deposit loses about 10% of its purchasing power in roughly 10 years. Tax at 30% turns 7% into 4.9%. Inflation of 6% then shrinks what that buys: 1.049 divided by 1.06, less 1, is -1.04%. Compounding that loss, Rs 1 lakh buys what Rs 90,094 buys today after ten years.

Why does the order of tax and inflation matter?

Think of a salary rise that matches price rises but pushes you into a higher tax bracket. On paper you kept pace; in the shop you did not. Tax is charged on the whole nominal interest, including the part that only makes up for inflation, so the investor pays tax on money that is not a real gain. That is why you take tax off first, on the 7%, and only then compare what is left with inflation.

Tax of 30% on 7% is 2.1 points, leaving 4.9%. Inflation at 6% is bigger than 4.9%, so the deposit is already behind before any compounding. The quick answer is 4.9 minus 6, about minus 1.1%; the exact answer uses the ratio, because both rates compound.

How do you get from minus 1% a year to ten years?

Losing 1% a year compounds, but slowly. Ten years of losing about 1.04% a year leaves 0.9896 to the power 10, about 0.90, so roughly 10% of purchasing power is gone in about 10 years. The exact figure is the log of 0.9 over the log of 0.9896, which is 10.1 years. A rule of thumb works too: a 1% annual loss takes about 70 years to halve the money, so about a seventh of that for a tenth.

Say the limitation. The 30% slab and 6% inflation are assumptions for this arithmetic, and your own slab and the inflation you actually face may differ; confirm the current tax rules before using a slab in advice. The point survives any sensible inputs: a deposit is safe in rupees and can still lose ground in what those rupees buy.

About 14.2%, against a weighted average of 16.8%. Variance is 0.6 squared times 20% squared, plus 0.4 squared times 12% squared, plus twice 0.6 times 0.4 times 0.3 times 20% times 12%, which sums to 0.0202. Its square root is 14.2%. The 2.6 point gap exists only because the correlation is below 1.

Why is the mix not just the average of the two volatilities?

Two friends walking home on a windy night: if they stumble at exactly the same moments, holding hands does not steady them. If their stumbles come at different moments, each one's lean is partly caught by the other. Only the part of the two funds' swings that happens together adds up in full; the rest partly cancels, so the portfolio is calmer than the average of its parts. Correlation of 0.3 says most of their swings are not shared.

How do you work it quickly on paper?

Square the volatilities first, because variances are what add. Fund A's weighted variance is 0.36 times 0.04, which is 0.0144. Fund B's is 0.16 times 0.0144, which is 0.0023. The cross term is where correlation lives: 2 times 0.6 times 0.4 times 0.3 times 0.20 times 0.12 is 0.0035. The total is 0.0202, and the square root of 0.02 is about 0.141, so 14.2% is the answer.

Give the two edges to show you see the shape. At a correlation of 1 the formula collapses to the weighted average, 16.8%. At zero the cross term vanishes and the mix is 12.9%. At minus 1 it drops to 7.2%. The limitation is that correlations measured in calm years often rise in a selloff, so the benefit you computed can shrink exactly when it is wanted.

About 31 managers beat the index five years running by luck alone, and there is about a 62% chance that at least one beats it ten years running. Each year halves the survivors: 1,000, 500, 250, 125, 62.5, 31.25. For ten years, one manager's chance is 1 in 1,024, so the chance none of the thousand does it is (1023/1024) to the 1,000th, about 38%.

Why do perfect records appear even when nobody is skilled?

Ask a stadium of a thousand people to toss a coin and sit down on tails. After five rounds about 31 are still standing, and each of them has a perfect record. A streak that is rare for one person is expected somewhere in a large enough crowd, so the size of the starting group matters as much as the length of the streak. The five-year count is just 1,000 halved five times.

How do you get the ten-year figure without a calculator?

Go through the complement: work out the chance that nobody does it. Each manager fails with probability 1023 over 1024. For many small independent chances, (1 minus 1/n) to the power n is close to 1 over e, about 0.37, and here the power is 1,000 against 1,024, so the chance nobody does it is about 0.38. That leaves about 62% for at least one ten-year streak.

Say what it means for fund selection. A long record of beating the index is evidence, but weaker evidence than it looks when it is picked from a large universe after the fact. The limitation of the model is that real returns are not coin tosses and some managers do have skill; the puzzle only shows how much luck alone can produce.

About Rs 4.1 lakh a day, taken inside the NAV. 1.5% of Rs 1,000 crore is Rs 15 crore a year; divided by 365 it is about Rs 4.11 lakh a day. The fund books that as a liability before striking the NAV, so each unit is worth a little less, about 0.21 paise a day at an NAV of 50. The investor never sees a deduction because the cost is already in the price.

If nothing is deducted, how is the fee paid?

Think of a restaurant that folds the service charge into the menu price instead of adding it to the bill. You pay it with every dish; you just never see a line saying so. A fund's expenses are charged to the scheme itself every day, so they reduce net assets and the NAV, and the investor pays through a slightly lower price per unit rather than a visible deduction. The number of units you hold never changes because of the fee.

How small is it per unit, and why does that matter?

At an NAV of 50, one day's share is 50 times 1.5% over 365, about Rs 0.0021, or 0.21 paise. Nobody notices a fifth of a paise a day, which is exactly why the expense ratio has to be read off the factsheet rather than felt in the account. Across a year it is 1.5% of the money, and it compounds as a drag like any other cost.

Two practical points complete the answer. Because assets change every day, the accrual is recomputed on each day's net assets rather than fixed at Rs 4.1 lakh. And the published returns of a fund are already after this cost, so a fair comparison with an index must use the fund's NAV returns against the index's total return. Expense ratio limits are set by regulation and change; confirm the current SEBI framework before quoting one.

Selling B is cheaper on tax by up to about Rs 14,103, and the right question is which fund he would buy today. Rs 2 lakh of A carries a gain of Rs 46,154, so Rs 5,769 of tax at the assumed 12.5%. Rs 2 lakh of B carries a loss of Rs 66,667: no tax, and a loss that can offset other gains. The urge to sell the winner is the disposition effect.

Why does selling the winner feel right?

Picture someone clearing out a wardrobe who gives away the shirts he likes and keeps the ones that never fit, because giving those away would admit the purchase was a mistake. The disposition effect is the habit of selling what has gone up to enjoy the gain and holding what has gone down to avoid the regret, and it decides on the past price rather than on the future. The purchase price of each fund is a fact about the client, not about the funds.

How much does the instinct cost in tax?

Only the gain inside the units sold is taxed. A is worth 1.3 times its cost, so 0.3 over 1.3, about 23%, of any rupee sold is gain: Rs 46,154 on Rs 2 lakh, and Rs 5,769 of tax at the assumed 12.5%. B is worth 0.75 times its cost, so every Rs 2 lakh sold realises a loss of Rs 66,667, pays nothing now and can be set against other gains, worth up to Rs 8,333 at the same rate. The full swing between the two is up to Rs 14,103.

Is tax the whole answer?

No, and saying so is what earns the marks. The test that cuts through the feeling is this: if the client held only cash today, would he buy A, B, or neither in these amounts? If B fell because its strategy has stopped working, selling it fixes the portfolio and saves tax. If B fell with its whole category and A has simply had a good run, the portfolio question is about weights and concentration, not about which one he feels good selling.

State the limits. The 12.5% rate is an assumption for the arithmetic; real rules distinguish short and long holding periods, can carry exemption thresholds, and restrict which gains a loss may be set against. Confirm the current rules before using any rate with a client. And this is a framework for a conversation, not an instruction about any real fund.

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.

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.