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.

He lost about 11.6% a year while the fund made 9.5% a year. Rs 1 lakh bought 10,000 units at 10 and Rs 5 lakh bought 33,333 units at 15. At NAV 12 his 43,333 units are worth Rs 5.2 lakh against Rs 6 lakh put in. The rate that makes those cash flows balance, his internal rate of return, is -11.6% a year.

How can the fund and the investor disagree on the return?

Picture a shop that sells ten mangoes at a profit on Monday, then stocks up on a hundred on Tuesday just before the price drops. The shop's price record looks fine; the owner's till does not. The fund's return measures what one rupee did if it stayed the whole time; the investor's return weights each period by how much of his money was there. Here only Rs 1 lakh enjoyed the rise from 10 to 15, and Rs 6.5 lakh suffered the fall to 12.

How do you compute the investor's figure without a calculator?

Count units, then value them. Rs 1 lakh at 10 buys 10,000 units; Rs 5 lakh at 15 buys 33,333. That is 43,333 units, worth Rs 520,000 at 12. The money-weighted return is the single rate r that makes the two payments, grown at r, equal the final value. With x as 1 plus r, x squared plus 5x equals 5.2, and the positive root is 0.8838, so r is -11.6%.

The fund's own figure is simple: 12 over 10 is 1.2 in two years, and the square root of 1.2 is 1.0954, so 9.54% a year. That is the number a factsheet shows, because the manager does not choose when investors arrive. The investor's IRR is the number his account statement should show, and it is the one his experience matches.

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.

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.

About Rs 16,691 a month now, and about Rs 27,048 after a three-year wait, 62% more. At 11% a year, taken as 0.917% a month and paid at the start of each month, 144 payments of Rs 16,691 grow to Rs 50 lakh. With 108 payments the SIP must be Rs 27,048. The early years carry the most growth, so losing them costs far more than their share of time.

How do you get the starting SIP?

A monthly SIP is a stream of equal payments, each growing until the goal date. The first rupee grows for 144 months, the last for one. Add up what one rupee a month becomes, and the required SIP is the goal divided by that sum. At 11% a year, taken as 11 over 12, about 0.917% a month, one rupee paid at the start of each month for 144 months grows to about 300. Rs 50 lakh divided by 299.6 is Rs 16,691.

Why is the cost of waiting not proportional to the wait?

Think of planting trees for shade in twelve years. A tree planted in year one is huge by the deadline; one planted in year ten is a sapling. Skip the first three years of planting and you cannot make it up by planting a few extra saplings later. The payments a delay removes are the ones that would have compounded the longest, so the later SIP must replace both the missing payments and the growth they would have earned. Starting now, Rs 24.0 lakh paid in becomes Rs 50 lakh; starting late, Rs 29.2 lakh has to be paid in, because growth contributes Rs 20.8 lakh instead of Rs 26.0 lakh.

The two guesses are worth naming. Twenty-five per cent comes from three years being a quarter of twelve. Thirty-three per cent comes from spreading the same total contributions over nine years instead of twelve, Rs 22,255 a month. Both ignore growth. The true figure, Rs 27,048, sits well above both.

Say the limits. Eleven per cent is an assumption, not a forecast, and equity returns arrive unevenly; a plan built on one smooth rate should be checked against lower rates and reviewed as the date nears. Monthly compounding at 11 over 12 slightly overstates an 11% annual rate; using the exact monthly equivalent raises the SIP to about Rs 17,300, and the ratio of about 1.6 survives either way.

About -10.5% with duration alone and about -9.6% with convexity. Duration gives -7 times 0.015, which is -10.5%. Convexity adds one half times 80 times 0.015 squared, or +0.90%. The price-yield curve bends upward, so a straight-line duration estimate overstates losses when yields rise and understates gains when they fall, and the gap grows with the square of the move.

Why is duration alone not enough for a big move?

Think of estimating how far a ball thrown upward will fall back by looking only at its speed at the start. For a moment the straight line is fine; over a longer stretch the curve of its path takes over. Duration is the slope of the price-yield curve at today's yield, so it is exact for tiny moves and drifts further from the true price the larger the move. The curve bends upward for an ordinary bond, so it always sits above the tangent.

How do you compute both estimates quickly?

Write the move as a decimal first, 0.015. The duration term is minus duration times the move, -7 times 0.015, which is -10.5%; the convexity term is half the convexity times the move squared, 40 times 0.000225, which is +0.90%. Add them: -9.60%. For a 150 basis point fall the same two terms give +10.5% plus 0.90%, a gain of 11.40%, so convexity helps in both directions.

The squared term is why size matters. At 50 basis points the convexity term is only +0.10%, easy to ignore; at 300 basis points it is +3.6%, too large to leave out. The limitation is that this is still a two-term approximation of a curve; for very large moves, or for bonds with call features whose convexity can turn negative, a desk reprices the bond in full rather than trusting the formula.

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.