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 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.

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 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 half: 49.5%. At 1% a month the corpus is about Rs 50.5 lakh after 15 years and Rs 99.9 lakh after 20, so the final five years add Rs 49.5 lakh. Only Rs 6 lakh of that is new instalments; about Rs 41.2 lakh is growth on the pot already built. An SIP's corpus is back-loaded, which is why stopping early costs far more than the missed instalments.

Why is the guess of a quarter so far off?

A quarter assumes the corpus grows by the same amount each year, like a piggy bank filled with the same coins every month. Compounding is not a piggy bank. Each year's growth is a percentage of a pot that is already larger than the year before, so the rupee gains get bigger every year and the corpus is heavily back-loaded. After 15 years of Rs 10,000 a month the pot holds about Rs 50.5 lakh, and five more years at 1% a month multiply that by 1.01 to the 60th, about 1.82, before a single new instalment is counted.

Where exactly does the last five years' money come from?

Split the Rs 49.5 lakh three ways. The 60 new instalments put in Rs 6 lakh. Those instalments earn about Rs 2.2 lakh of growth by year 20. The year-15 pot of Rs 50.5 lakh earns about Rs 41.2 lakh. The engine of the final years is the money already invested, not the money still to come. For contrast, the first ten years build only about Rs 23.2 lakh, 23% of the final corpus, though half the instalments are paid in them.

What does this mean for an investor thinking of stopping at year 15?

It depends what stopping means. Redeeming at year 15 gives up about Rs 49.5 lakh, half the eventual corpus, to save Rs 6 lakh of instalments. Pausing the instalments but staying invested is far cheaper: the year-15 pot alone grows to about Rs 91.7 lakh, so the cost is about Rs 8.2 lakh. The 12% is an assumption for the arithmetic; real returns vary year to year, and a bad last five years would cut the back-loaded gain sharply, which is the sequence risk of an SIP.

You are indifferent at about 7.18% a year. Rs 50 lakh at 50 matches Rs 1 crore at 60 only if it doubles in ten years, so (1 + r) to the 10th = 2, which gives 7.18%. The rule of 72 says money doubles in 72 / rate years, so doubling in 10 years needs about 72 / 10 = 7.2%. At 8% the early money grows to Rs 107.9 lakh; at 7% only Rs 98.4 lakh.

What is the question really asking?

An uncle offers you his old car now or a new one in ten years. You cannot answer until you know what you would do with the car meanwhile. Money received earlier can be invested, so the fair comparison is what the Rs 50 lakh grows to by 60, set against the Rs 1 crore paid then. At 8%, Rs 50 lakh x 1.08 to the 10th is Rs 107.9 lakh, more than a crore, so the early money wins. At 7%, it is Rs 98.4 lakh, just short, so the crore wins. The answer flips somewhere between.

Why is the crossover the rule of 72?

The crore is exactly twice the 50 lakh, and the wait is ten years. So the choice reduces to one question: can you double your money in ten years? The rate that does that is 7.18%, and the rule of 72A shortcut for compounding: money doubles in roughly 72 divided by the yearly rate in years, so 7.2% doubles in about ten years. gives 72 / 10 = 7.2% in your head. The rule works because the doubling time is ln 2 / ln(1 + r), and ln 2 is 0.693; using 72 instead of 69.3 corrects for the rates people usually quote being near 8%.

The arithmetic is not the whole decision, and the interviewer will want you to say so. The rate that matters is the after-tax rate you can actually earn, not a headline rate. Rs 1 crore promised at 60 carries the risk that the promiser does not pay; Rs 50 lakh in hand does not. And a person who needs the money at 50, for a child's education say, values it more than the arithmetic does.

Because the extra ten years come first, and the earliest rupees compound the longest. At 1% a month, the early starter reaches about Rs 6.50 crore and the late starter about Rs 1.90 crore, 3.4 times less, though the gap in money paid in is only Rs 12 lakh. That Rs 12 lakh, saved from 25 to 35, grows to about Rs 4.60 crore by 60, 71% of the early starter's corpus.

Why is the early starter so far ahead for so little extra?

Plant a mango sapling at 25 and another at 35, and at 60 the first tree is not slightly bigger; it has had ten more seasons of growth on a trunk that kept getting larger. Money saved early does not just add ten more years of deposits; it gives every rupee in those years twenty-five more years of compounding after them. The late starter pays in Rs 30 lakh over 25 years and ends with Rs 1.90 crore. The early starter pays Rs 42 lakh over 35 years and ends with Rs 6.50 crore.

How much does the first decade contribute on its own?

Split the early starter's plan in two. From 25 to 35 she saves Rs 12 lakh, which is about Rs 23.2 lakh by 35. From then she saves exactly what the late starter saves, so that part ends at Rs 1.90 crore. The Rs 23.2 lakh from the first decade compounds for 25 more years at 1% a month and becomes about Rs 4.60 crore, 71% of her final corpus. Her first ten years are worth more at 60 than the late starter's entire 25 years of saving.

State the assumptions, because the size of the gap depends on them. A steady 12% every year is a simplification; real equity returns swing, and the order of good and bad years matters for a monthly saver. Inflation shrinks both corpora in today's rupees, though not the ratio. And the gap narrows if the late starter saves more each month; to match the early starter at 60 she would have to invest about 3.4 times as much, around Rs 34,000 a month.

About 9 years at 8%; the exact figure is 9.006 years, so the rule of 72 is out by about 2 days. At 6% the rule gives 12 years against an exact 11.90, and at 12% it gives 6 years against an exact 6.12. In the range where fund returns usually sit, the rule is off by weeks, not years.

Why does 72 divided by the rate give the doubling time?

If the price of a Rs 100 thali rises 8% a year, it costs about Rs 200 in nine years. You did not need a calculator, only the fact that 72 over 8 is 9. The exact doubling time is ln 2 divided by ln(1 + r), and for rates of a few per cent ln(1 + r) is close to r, so the answer is close to 69.3 divided by the rate in per cent. Using 72 instead of 69.3 does two jobs: it nudges the answer up to correct for rates around 8%, and it divides cleanly by 2, 3, 4, 6, 8, 9 and 12, which is what makes it a mental tool.

Where does the rule start to slip?

At both ends, but slowly. At 6% it says 12 years and the truth is 11.90, so it is long by about 38 days. At 12% it says 6 years against 6.12, short by about 42 days. The rule is almost exact near 8% and drifts by a little over a month either side, which is far inside the uncertainty of any return assumption you would feed it. Only at high rates does the drift become worth mentioning: at 24% the rule says 3 years against an exact 3.22.

Two uses in a fund interview. Turn a return into something a client can feel: at 8%, money doubles roughly every nine years, so it quadruples in about eighteen. And run a claim backwards to test it: a fund that says it tripled money in ten years has compounded at about 11.6% a year, because tripling is about 1.6 doublings, one every 6.3 years, and 72 over 6.3 is a little over 11.

About Rs 70.7 lakh, under half of the Rs 1.5 crore paid out. Discount each of the 300 monthly payments of Rs 50,000 at 7% a year, about 0.58% a month, and add them up. Payments in the first years are worth almost their face value; payments in year 25 are worth less than a fifth of it. Lump sum and pension are equivalent because Rs 70.7 lakh invested at 7% would pay exactly Rs 50,000 a month for 25 years.

Why is a rupee paid later worth less today?

A relative offers you Rs 1 lakh either today or in ten years. Taking it today is better even with no inflation at all, because you could put it in a deposit and have well over Rs 1 lakh in ten years. The value today of a future payment is the amount you would need to invest now to produce it, so the longer the wait, the smaller that amount. A pension is a long string of future payments, and each one has its own wait. The lump-sum value is the sum of all those discounted payments.

How do you add up 300 discounted payments without a spreadsheet?

Use the annuity formula, which is the sum of a series that shrinks by the same factor each month. With a monthly rate of 7% divided by 12 and 300 payments, each rupee of monthly pension is worth about Rs 141.5 today, so Rs 50,000 a month is worth about Rs 70.7 lakh. A quick check by years: Rs 6 lakh a year for 25 years at 7%, discounted once a year, gives about Rs 69.9 lakh, close because monthly payments arrive a little earlier on average.

Where does most of the gap come from?

From the back half of the pension. At 7%, money doubles in about ten years, so every payment after roughly year 10 is worth less than half its face value today, and the last ones are worth under a fifth. The first ten years pay Rs 60 lakh and are worth about Rs 43.1 lakh; the last fifteen pay Rs 90 lakh and are worth only about Rs 27.7 lakh. For an adviser comparing a pension offer with a lump sum, this is the number that matters, and the rate is the assumption that moves it most: at a lower rate the lump sum needed rises, at a higher one it falls.

Two limits to state. The 7% must be a rate the client could realistically earn with similar safety; using a risky return to discount a guaranteed pension flatters the lump sum. And Rs 50,000 a month is fixed in rupees, so its buying power in year 25 is far smaller than today; if the pension rises with inflation, its value is much higher.