Mutual Fund Mastery puzzles, solved step by step

Puzzles: 100
Traced to a firm: 29
Topics: 13
Hard: 30

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001A bank fixed deposit pays 7% a year. Your interest is taxed at a 30% slab, assumed here for the arithmetic, and inflation runs at 6%. What is your real return after tax, and roughly how many years until the deposit has lost 10% of its purchasing power?Compounding and time valueCoreIndian AMCsDistribution and sales
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Before you calculate: what does the deposit earn in real, after-tax terms?
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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.
A 7% deposit taxed at an assumed 30% slab keeps 4.9%, and 6% inflation turns that into a real return of about -1.0% a year, so Rs 1 lakh held in the deposit buys about 10% less after roughly 10 years.
The relationship
$r_{\text{real}} = \frac{1 + i(1-t)}{1 + \pi} - 1 = \frac{1.049}{1.06} - 1 \approx -1.04\%$
i the nominal deposit rate, 7%
t the assumed tax slab, 30%
\pi inflation, 6%
What it says in wordsGrow the money at the after-tax rate, shrink its buying power at the inflation rate, and the ratio is the real return.
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.
Where candidates lose it
The fast wrong answer is plus 1%: seven minus six. It forgets that tax is charged on the nominal 7%, including the 6% that only replaces lost buying power. Candidates who say it have shown the interviewer they would mis-sell a deposit to a client in a high bracket.
The second trap is getting minus 1% and then answering the time question linearly, ten years at 1% is exactly 10%. It is close here, but say that it compounds and give the log form; the interviewer is checking you know why it is close.
What the interviewer asks next
What deposit rate would just keep a 30% taxpayer level with 6% inflation?
How does the answer change for an investor with no taxable income?
Why might a debt fund held for several years be compared with a deposit on an after-tax basis, and what would you check first?

027A Rs 10,000 monthly SIP for 20 years at 12% a year ends near Rs 1 crore. If the SIP instead rises 10% every year, the corpus roughly doubles to about Rs 2 crore. Why does a step-up matter so much more than it sounds?Compounding and time valueCoreIndian AMCsDistribution and sales

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By year 20, how large is the step-up SIP's monthly instalment compared with the Rs 10,000 it started at?

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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.

The step-up instalment climbs from Rs 10,000 to about Rs 61,159 a month over 20 years, so the money put in rises 2.9 times, from Rs 24 lakh to Rs 68.7 lakh. The corpus rises only 2.0 times, from Rs 99.9 lakh to Rs 198.9 lakh, because most of the extra money arrives late and compounds for fewer years.

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.

The relationship

FV = \sum_{y=0}^{19} m\,(1.1)^{y} \cdot s_{12} \cdot (1.01)^{12(19-y)}

m	the first year's monthly instalment, Rs 10,000
(1.1)^y	the step-up applied y times
s_12	the value at year end of twelve monthly payments of 1 at 1% a month, about 12.81
(1.01)^{12(19-y)}	growth from the end of year y+1 to the end of year 20

What it says in wordsEach year's twelve instalments are a bigger block than the last, and each block compounds only from the year it is paid.

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.

Where candidates lose it

Candidates often say the step-up doubles the corpus because of compounding, as if the extra money were somehow compounding better. It is the reverse: the extra money arrives late and compounds less. The doubling comes from the money put in nearly tripling.

The second trap is guessing that a 10% step-up adds about 10% to the corpus, or adds 10% of Rs 24 lakh. The step-up compounds on the instalment, and after nineteen raises the instalment is six times the start. Say 1.1 to the power 19 out loud.

What the interviewer asks next

What step-up rate would you need for the corpus to reach Rs 1.5 crore?
Would a 10% step-up in the first ten years only get you most of the benefit? Why or why not?
How would you compare a step-up SIP with simply starting at Rs 15,000 flat?

059Would you rather receive Rs 50 lakh at 50 or Rs 1 crore at 60? At 8% the Rs 50 lakh wins, at 7% the crore wins. Find the rate at which you are indifferent, and say why it is the rule of 72 again.Compounding and time valueCoreIndian AMCsDistribution and sales
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Before you work it: where is the indifference rate?
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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.
Rs 50 lakh received at 50 grows to more than Rs 1 crore by 60 at any rate above 7.18%, reaching Rs 107.9 lakh at 8% and only Rs 98.4 lakh at 7%, so the choice turns on the rate that doubles money in ten years.
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 relationship
$50(1+r)^{10} = 100 \;\Rightarrow\; r = 2^{1/10} - 1 \approx 7.18\% \approx \frac{72}{10}\%$
50 the early sum, Rs lakh, received at 50
100 the later sum, Rs lakh, received at 60
r the yearly rate earned on the early money
What it says in wordsWhen the later sum is twice the earlier one, the indifference rate is simply the rate that doubles money over the wait.
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.
Where candidates lose it
The common slip is 10%: the money must double, ten years, so 10% a year. That is simple interest thinking, and it misses that compounding does part of the doubling. The other slip is 5%, from splitting the 50 lakh gain evenly over ten years.
The quieter loss is stopping at the number. Name the assumptions: the rate is after tax, the later payment is certain, and the person has no need for the money before 60.
What the interviewer asks next
The offer becomes Rs 1.5 crore at 60. What is the indifference rate now?
How does inflation change the comparison if both sums are in today's rupees?
Why does the rule of 72 work less well at a 25% rate?
091What lump sum today is equivalent to a pension of Rs 50,000 a month for 25 years, if money can earn 7% a year? And why is that so far below the Rs 1.5 crore that the pension will actually pay out?Compounding and time valueCoreIndian AMCsDistribution and sales
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Roughly what lump sum today matches the pension?
Show the worked solution
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.
Each year of the pension pays Rs 6 lakh, but discounted at 7% the first year is worth Rs 5.78 lakh today and the twenty-fifth only Rs 1.08 lakh, so 25 years of payments totalling Rs 1.5 crore are worth about Rs 70.7 lakh today.
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.
The relationship
$PV = W \times \frac{1 - (1+i)^{-n}}{i} = 50{,}000 \times \frac{1 - (1.005833)^{-300}}{0.005833} \approx 70.7 \text{ lakh}$
W the monthly pension, Rs 50,000
i the monthly rate, 7% divided by 12
n the number of monthly payments, 300
What it says in wordsThe lump sum is the monthly payment times a factor that adds up every payment's discount for its own wait.
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.
Where candidates lose it
The common error is quoting Rs 1.5 crore, or something near it, because that is what the client will receive. It treats a rupee in year 25 as worth a rupee today, which is exactly what the question is testing.
The overcorrection is discounting the whole Rs 1.5 crore by 25 years and getting about Rs 28 lakh, as if all the money arrived on the last day. Most of it arrives much earlier. Discount payment by payment, or use the annuity factor, and say the rate is an assumption.
What the interviewer asks next
The pension rises 5% a year. Roughly how does the lump-sum value change?
The client is offered Rs 60 lakh now instead of the pension. What discount rate makes the two equal, roughly?
Why might the right discount rate for a government pension differ from that for a private one?

i	the nominal deposit rate, 7%
t	the assumed tax slab, 30%
\pi	inflation, 6%

Try it first

Why does the order of tax and inflation matter?

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

Where candidates lose it

What the interviewer asks next

Try it first

Why does 10% a year sound small and turn out large?

Why does the corpus double when the money put in nearly triples?

What is the honest way to say this to a client?

Where candidates lose it

What the interviewer asks next

Try it first

What is the question really asking?

Why is the crossover the rule of 72?

Where candidates lose it

What the interviewer asks next

Try it first

Why is a rupee paid later worth less today?

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

Where does most of the gap come from?

Where candidates lose it

What the interviewer asks next