Mutual Fund Mastery puzzles, solved step by step

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

Showing 1–4 of 4 · filtered from 100Clear filters

015A retiree holds units of a fund bought at an NAV of 40, now at 60, and withdraws Rs 50,000 a month through a systematic withdrawal plan. What part of each withdrawal is gain? And why is the SWP taxed more lightly than Rs 50,000 a month of fixed deposit interest, at any assumed tax rate?Withdrawals and after-tax arithmeticCoreWealth and advisoryDistribution and sales
Try it first
Of each Rs 50,000 withdrawn, how much is gain?
Show the worked solution
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 At cost 40 and NAV 60, the gain is 20 out of 60, one third of every rupee withdrawn.
Each Rs 50,000 SWP withdrawal is two thirds the investor's own capital and one third gain, so at an assumed 30% rate it bears Rs 5,000 of tax, while Rs 50,000 of deposit interest is taxable in full and bears Rs 15,000.
The relationship
$\text{Gain share} = \frac{NAV - \text{cost}}{NAV} = \frac{60 - 40}{60} = \tfrac{1}{3},\qquad 50{,}000 \times \tfrac{1}{3} = \text{Rs } 16{,}667$
NAV the price at which units are redeemed, 60
cost the price at which those units were bought, 40
What it says in wordsThe share of a withdrawal that is gain equals the share of the unit price that sits above what the unit cost.
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.
Where candidates lose it
The common slip is treating the whole Rs 50,000 as taxable income, as if the withdrawal were a dividend or interest. Candidates who say it would misstate the tax on almost every retirement plan they review.
The opposite slip is claiming the SWP is simply better than the deposit. It is lighter on tax because it hands back capital, and that same fact means the principal is being drawn down. The interviewer wants both halves.
What the interviewer asks next
After two years the NAV is 80. What share of each withdrawal is gain then?
How would you check whether Rs 50,000 a month is a sustainable withdrawal from this holding?
Which units are treated as sold first when the investor bought at several different NAVs?

040Two retirees each start with Rs 60 lakh and withdraw Rs 6 lakh at the start of every year. Both see the same ten yearly returns, averaging 6.8%: one meets minus 20% and minus 10% in the first two years, the other in the last two. After ten years one has about Rs 2 lakh left and the other about Rs 39 lakh. Why?Withdrawals and after-tax arithmeticHardWealth and advisoryDistribution and sales

Try it first

What explains the gap of about Rs 37 lakh?

Show the worked solution

Because withdrawals turn the order of returns into a permanent difference. Without withdrawals both would end at the same Rs 108.1 lakh, since returns multiply in any order. But the first retiree takes Rs 6 lakh out of a pot that has just fallen, to Rs 33.5 lakh after two years, so later good years work on a small base and she ends with about Rs 2.0 lakh. The second takes withdrawals from a pot that has grown, and ends with about Rs 39.0 lakh.

Why would order not matter without withdrawals?

Returns multiply, and multiplication does not care about order. Rs 60 lakh left untouched through these ten years ends at Rs 108.1 lakh whichever year comes first. The withdrawals are what break the symmetry: a fixed rupee amount taken out of the pot is a bigger share of a small pot than of a large one.

What exactly happens in the first two years?

Think of a shopkeeper who must pay rent of Rs 6,000 on the first of every month by selling stock. If prices have just crashed, he sells many more items to raise the same Rs 6,000, and those items are gone when prices recover. A retiree withdrawing a fixed sum after a fall sells more units at low prices, and those units never take part in the recovery. The first retiree's Rs 54 lakh, after the first withdrawal, meets minus 20% and becomes Rs 43.2 lakh; Rs 37.2 lakh then meets minus 10% and becomes Rs 33.5 lakh. The second retiree's Rs 54 lakh meets plus 20% first and is Rs 64.8 lakh after year one.

Both retirees withdraw Rs 6 lakh at the start of each year and see the same returns averaging 6.8%. The one who meets minus 20% and minus 10% first is down to Rs 33.5 lakh after two years and ends with Rs 2.0 lakh, while the one who meets them last ends with Rs 39.0 lakh.

Year	Return, bad first	Corpus, Rs lakh	Return, bad last	Corpus, Rs lakh
1	-20%	43.2	+20%	64.8
2	-10%	33.5	+13%	66.4
3	+18%	32.4	+5%	63.5
4	+15%	30.4	+7%	61.5
5	+11%	27.1	+9%	60.5
6	+9%	23.0	+11%	60.5
7	+7%	18.2	+15%	62.6
8	+5%	12.8	+18%	66.8
9	+13%	7.6	-10%	54.8
10	+20%	2.0	-20%	39.0

Year by year: Rs 6 lakh is withdrawn at the start of each year, then that year's return applies to what is left. The returns are the same set in reverse order.

The relationship

V_{t} = (V_{t-1} - W)(1 + r_t), \qquad V_0 = 60,\; W = 6

V_t	the corpus at the end of year t, Rs lakh
W	the withdrawal at the start of each year, Rs 6 lakh
r_t	the return in year t

What it says in wordsEach year the withdrawal comes out first and the return applies only to what is left, so a fall before a withdrawal makes the withdrawal bite harder.

What does an adviser do with this?

The years just before and just after withdrawals begin carry the most weight; the risk is called sequence risk, and it is managed, not predicted. Common responses are holding two or three years of withdrawals in a steadier, short-term bucket so that a bad year does not force selling equity at a low, or trimming withdrawals after a fall. Note the mirror image of an SIP: for a saver adding money, bad years early are the kinder order; for a retiree taking money out, bad years early are the cruel one. The returns here are illustrative, and a 10% withdrawal rate is high by most planning conventions, which is what makes the gap so stark.

Where candidates lose it

The trap is answering that the average return decides the outcome, or insisting the order cannot matter because returns multiply. Both are true only for money that sits untouched. The question tells you withdrawals happen; build from there.

The second loss is explaining it as bad luck in general. The interviewer wants the mechanism in one sentence: fixed withdrawals after a fall sell more units at low prices, and those units miss the recovery.

What the interviewer asks next

How large a cash bucket would have let the first retiree avoid selling equity in years one and two?
If withdrawals were 4% of the current corpus instead of a fixed Rs 6 lakh, would the order still matter?
How would you explain sequence risk to a client in two sentences?

065Rs 10 lakh can go into a deposit at 7.2%, taxed every year at an assumed 30% slab, or a debt fund earning 7.0%, taxed at the same slab only on redemption after three years. Which leaves more after three years, and how much is the tax deferral worth?Withdrawals and after-tax arithmeticCoreWealth and advisoryDistribution and sales
Try it first
After three years and tax, which is ahead?
Show the worked solution
The deposit, by about Rs 1,418 on Rs 10 lakh. Taxed yearly, the deposit compounds at 7.2% x 0.7 = 5.04% and reaches Rs 11,58,949. The fund grows to Rs 12,25,043 and keeps 70% of its gain: Rs 11,57,530. Deferral alone is worth about Rs 3,399, roughly the cost of a 0.14% yield gap, so a 0.2% lower yield more than uses it up. Tax rates here are assumptions; confirm the current rules.
Where does the value of deferral come from?
Think of a shop that lets you pay your bill at the end of the year instead of every month. You keep the money a little longer and can earn something on it, but over a short time that is a small favour. Tax deferral is the same favour: the tax you would have paid each year stays invested and earns until redemption, so the gain comes from interest on money that would otherwise have gone to the tax office. Over three years at 7% that extra earning is modest. Over twenty years it is large, because it compounds.
Work both. The deposit pays 7.2%, of which 30% goes in tax each year, so it compounds at 5.04%: Rs 10 lakh becomes Rs 11,58,949. The fund compounds untaxed at 7.0% to Rs 12,25,043; on redemption 30% of the Rs 2,25,043 gain goes in tax, leaving Rs 11,57,530. The deposit is Rs 1,418 ahead.
On Rs 10 lakh over three years the deposit ends at Rs 11,58,949 and the debt fund at Rs 11,57,530, because deferral is worth only about Rs 3,399 while the fund's 0.2% lower yield costs a little more.
How do you separate the deferral from the yield gap?
Give the fund the same 7.2% and the only difference left is the timing of tax. It would end at Rs 11,62,348, Rs 3,399 above the deposit: that is the value of deferral. Then the fund's 0.2% lower yield takes away Rs 4,818, slightly more than the deferral gave, which is why the deposit edges ahead. The fund would break even at a yield of about 7.06%, so the deferral is worth roughly 0.14% a year over three years.
The relationship
$P(1 + 0.7 \times 0.072)^3 \;\text{vs}\; P + 0.7\left[P(1.07)^3 - P\right]$
P the amount invested, Rs 10 lakh
0.7 the share kept after an assumed 30% tax
0.072 the deposit rate, taxed every year
1.07 the fund's growth factor, taxed only on the final gain
What it says in wordsThe deposit compounds on an after-tax rate; the fund compounds on the full rate and pays tax once on the whole gain.
The limits matter more than the arithmetic here. The tax treatment of debt funds has changed in India in recent years, and slab rates, surcharge and cess all move the answer, so confirm the current rules before using any rate. A deposit has no price risk, while a debt fund's NAV moves with interest rates and credit events. And deferral grows with time: over fifteen years the same set-up would give the fund a clear lead.
Where candidates lose it
The trap is assuming deferral is worth a lot by default. Over three years it is worth only about 0.14% a year here, smaller than the 0.2% yield gap, and candidates who say the fund wins have not run the numbers.
The second miss is quoting a tax rate as fact. Say that 30% is an assumed slab and that the rules on debt funds should be checked, then show how the comparison changes over a longer holding period.
What the interviewer asks next
Over how many years does the debt fund pull ahead at these yields?
What if the investor is in a 10% slab instead of 30%?
Why might an investor prefer the deposit even if the fund were slightly ahead after tax?
089A retiree has Rs 40 lakh invested at an assumed 8% a year and withdraws Rs 30,000 a month. How long does the money last, and what monthly withdrawal would last forever?Withdrawals and after-tax arithmeticHardWealth and advisoryDistribution and sales
Try it first
Roughly how long does Rs 40 lakh last at Rs 30,000 a month?
Show the worked solution
About 27.6 years, and about Rs 26,700 a month would last forever. At 8% a year, about 0.67% a month, Rs 40 lakh earns Rs 26,667 a month. Taking Rs 30,000 eats about Rs 3,333 of capital in month one, and the gap widens as the corpus shrinks, so the money runs out after about 331 months. Withdraw only the interest and the corpus never falls.
Why does a small overdraw turn into a countdown?
Think of a well that refills by 100 buckets a day. Draw 100 and it lasts forever; draw 110 and the level drops a little, so tomorrow it refills slightly less, and the drop speeds up until the well is dry. A corpus earning interest works the same way: withdraw no more than it earns and it lasts forever; withdraw a little more and each month's shortfall shrinks the base that earns next month's interest. Here she takes Rs 30,000 against Rs 26,667 of interest, only about Rs 3,333 too much at the start, and it still empties the account.
Withdrawing the Rs 26,667 a month the corpus earns keeps Rs 40 lakh intact forever, while Rs 30,000 a month empties it in about 27.6 years and Rs 35,000 in about 18.0, so a few thousand rupees a month decide whether the money is permanent or a countdown.
How do you work out the number of months?
Use the annuity formula and solve for the number of payments. With a monthly rate of 8% divided by 12, the corpus lasts as long as it takes the withdrawals to use up both the capital and the interest it earns along the way. The answer is about 331 months, or 27.6 years, and it is very sensitive to the withdrawal: Rs 28,000 a month lasts about 38 years and Rs 35,000 only about 18.0. The path is not a straight line: after ten years the corpus is still about Rs 33.9 lakh, and after twenty about Rs 20.4 lakh, because the fall speeds up near the end.
The relationship
$n = \frac{-\ln\!\left(1 - \dfrac{P\,i}{W}\right)}{\ln(1+i)} = \frac{-\ln\!\left(1 - \dfrac{26{,}667}{30{,}000}\right)}{\ln(1.00667)} \approx 331 \text{ months}$
P the starting corpus, Rs 40 lakh
i the monthly rate, 8% divided by 12
W the monthly withdrawal, Rs 30,000
P i the first month's interest, Rs 26,667
What it says in wordsThe closer the withdrawal is to the interest earned, the closer the fraction gets to one and the longer the money lasts; at exactly the interest it lasts forever.
What does this leave out?
Three things, each of which shortens the runway. Rs 30,000 today buys less every year, so a real retiree needs a rising withdrawal, not a flat one. The 8% is an assumption, not a promise; a market-linked portfolio delivers it unevenly, and a bad few years early on, while withdrawals continue, does more damage than the same years late. And tax comes out of the return. Even the 'forever' figure only preserves Rs 40 lakh in rupees; after inflation, the capital it protects is shrinking. She is withdrawing 9% of the corpus a year while it earns 8%, and that one comparison tells you the plan has an end date.
Where candidates lose it
The quick answer divides Rs 40 lakh by Rs 3.6 lakh a year and says about 11 years. That treats the corpus as cash in a drawer and throws away the 8% it earns while it waits; the true runway is more than twice as long.
The opposite slip is hearing 8% earned against 9% withdrawn and calling it roughly sustainable. Any withdrawal above the interest has an end date, and the gap compounds against her. Give the 27.6 years, then the Rs 26,700 that would never run out.
What the interviewer asks next
She wants the money to last exactly 25 years. What monthly withdrawal allows that?
Inflation runs at 6%. How would you change the calculation?
Why does a bad market in her first three years of retirement matter more than one in her last three?

P	the amount invested, Rs 10 lakh
0.7	the share kept after an assumed 30% tax
0.072	the deposit rate, taxed every year
1.07	the fund's growth factor, taxed only on the final gain

P	the starting corpus, Rs 40 lakh
i	the monthly rate, 8% divided by 12
W	the monthly withdrawal, Rs 30,000
P i	the first month's interest, Rs 26,667

Try it first

Is a withdrawal income?

Why does it hold at any tax rate?

Where candidates lose it

What the interviewer asks next

Try it first

Why would order not matter without withdrawals?

What exactly happens in the first two years?

What does an adviser do with this?

Where candidates lose it

What the interviewer asks next

Try it first

Where does the value of deferral come from?

How do you separate the deferral from the yield gap?

Where candidates lose it

What the interviewer asks next

Try it first

Why does a small overdraw turn into a countdown?

How do you work out the number of months?

What does this leave out?

Where candidates lose it

What the interviewer asks next