Private Wealth Management puzzles, solved step by step

Puzzles: 100
Traced to a firm: 3
Topics: 13
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005A client needs Rs 10 lakh in hand from a mutual fund redemption. Forty per cent of every rupee he redeems is capital gain, taxed at an illustrative 12.5%. How much must he redeem?Tax arithmeticCoreMutual fund distributionIndian wealth management
Try it first
Which redemption gives exactly Rs 10 lakh after tax?
Show the worked solution
About Rs 10.53 lakh. Each rupee redeemed is 40 paise of gain, and 12.5% tax on that is 5 paise, so he keeps 95 paise per rupee. Dividing Rs 10 lakh by 0.95 gives Rs 10.5263 lakh. Tax is Rs 52,632 and he is left with exactly Rs 10 lakh. The rate is illustrative; check the current rules and any exemption before advising.
Why does adding the tax on top fall short?
Imagine a shop that adds a 5% delivery charge on the order total. If you want Rs 100 of goods delivered and add Rs 5 for delivery, the shop charges delivery on Rs 105, and you are short. When the tax falls on the amount you withdraw, the withdrawal must be grossed up by dividing, not by adding a percentage. Here every rupee redeemed carries 0.40 times 12.5%, which is 5 paise, of tax.
A redemption of Rs 10.53 lakh is 60% principal and 40% gain; tax at 12.5% of the gain is Rs 52,632, leaving exactly Rs 10 lakh, while redeeming Rs 10.50 lakh leaves Rs 2,500 short.
Why is the gain share of each rupee the same across the redemption?
Because the question fixes it: 40% of every rupee is gain. In practice the gain share depends on which units are sold first, which follows a first-in, first-out order for mutual fund units, and on the cost of acquisitionThe price paid for the units being sold, set against the sale value to find the capital gain. of each lot. The grossing up works for any tax rate and gain share: divide the need by one minus the gain share times the rate. Say that general form after the number.
The relationship
$G = \frac{N}{1 - g\,t} = \frac{10}{1 - 0.40 \times 0.125} = \frac{10}{0.95} \approx 10.53$
N the amount needed in hand, Rs 10 lakh
g the share of each redeemed rupee that is gain, 0.40
t the illustrative tax rate on the gain, 12.5%
G the gross redemption, Rs lakh
What it says in wordsDivide the cash needed by the share of each rupee that survives tax.
Say the boundary plainly. Real capital gains tax has holding-period rules, an annual exemption on some gains, and surcharge and cess on top, and the rates change with budgets. The desk wants the method; the client file needs the current rules confirmed before a single unit is sold.
Where candidates lose it
The trap is adding 5% and saying Rs 10.50 lakh. It feels right because 5 paise per rupee is the tax, but the extra Rs 50,000 is itself taxed, and the client ends up Rs 2,500 short on the day he needs the money.
The second loss is quoting a tax rate as fact. Call it illustrative, give the method, and say the current rate and exemption need to be checked.
What the interviewer asks next
If only 20% of each rupee were gain, how much would he redeem?
What if the first Rs 1 lakh of gain were exempt? How does the calculation change?
Why might you redeem from a different fund to raise the same Rs 10 lakh?
031A taxable bond yields 7.5% and a tax-free bond of similar quality and tenor yields 5.4%. For a client paying tax at an illustrative 30% slab, which one pays him more?Tax arithmeticWarm upIndian wealth management
Try it first
Pick one before you calculate.
Show the worked solution
The tax-free bond pays him more: 5.4% against 5.25%. The taxable bond's 7.5% loses 30% to tax, leaving 7.5 x 0.7 = 5.25%. Put the other way, the tax-free 5.4% is worth 5.4 / 0.7 = 7.71% to a taxable investor at this slab, which is more than 7.5%. On Rs 1 crore the difference is about Rs 15,000 a year.
What is the fair way to compare the two yields?
Two job offers: Rs 20 lakh with a company flat thrown in, or Rs 23 lakh with no flat. You do not compare 20 with 23; you compare what each leaves you after rent. Yields are compared on what the client keeps, which means after his own tax, not on the number printed on the term sheet. Either shrink the taxable yield by the tax rate, or gross the tax-free yield up to its taxable equivalent.
The relationship
$y_{\text{after}} = y\,(1-t) = 7.5 \times 0.7 = 5.25\% \qquad y_{\text{eq}} = \frac{y_{tf}}{1-t} = \frac{5.4}{0.7} = 7.71\%$
y the taxable bond's yield, 7.5%
t the client's marginal tax rate, an illustrative 30%
y_tf the tax-free bond's yield, 5.4%
What it says in wordsEither take the tax off the taxable yield or add it back onto the tax-free one; both routes pick the same winner.
At an illustrative 30% slab, the taxable bond's 7.5% leaves the client 5.25% after tax, while the tax-free bond keeps its full 5.4%. The taxable bond would need to yield at least 7.71% to match.
For which clients does the answer flip?
Solve for the tax rate at which the two are equal: 1 minus 5.4 / 7.5, which is 28%. Any client whose marginal rate is below 28% keeps more from the taxable bond; any client above it keeps more from the tax-free one. That is why the same product can be right for a senior partner and wrong for his retired mother, and why an adviser asks about the slab before quoting yields.
Limits worth saying: tax-free bonds are often thinly traded, so the exit price matters; surcharge and cess change the effective rate; and tax treatment depends on current law, so confirm the client's actual rate and the bond's status before you rely on the comparison.
Where candidates lose it
The trap is answering 7.5% because it is the bigger number, without asking about tax at all. For a wealth desk that is the one question you must never skip, because nearly every client comparison is an after-tax comparison.
The second slip is dividing when you should multiply, and grossing 7.5 up instead of 5.4. Name which yield you are converting, and why, before you touch the numbers.
What the interviewer asks next
What is the taxable-equivalent yield for a client at a 20% slab?
Why might a tax-free bond trade below its issue price in the secondary market?
How would the comparison change if the client expected his slab to fall after retirement?
043A client in an illustrative 30% tax slab holds an equity fund and earns Rs 1 lakh of return in a year. He can take it through the IDCW payout option, taxed at his slab, or leave it in the growth option and pay an illustrative 12.5% capital gains rate when he sells. How much does the option choice cost him?Tax arithmeticCoreMutual fund distributionIndian wealth management
Try it first
On Rs 1 lakh of return, how much more tax does the payout option cost him?
Show the worked solution
About Rs 17,500 on every Rs 1 lakh of return, before the benefit of deferral. Paid out as IDCW, the Rs 1 lakh is taxed at his illustrative 30% slab: Rs 30,000. Left in the growth option and later sold as a long-term gain, it is taxed at an illustrative 12.5%: Rs 12,500. The return is identical; only its tax label changes. Over ten years on Rs 10 lakh the gap compounds to about Rs 4.27 lakh.
Is the IDCW payout extra money?
No, and that misunderstanding is where most of the damage starts. Taking water out of your own tank does not give you more water. An IDCW payout is paid out of the fund's NAV, so the NAV falls by the amount paid; the client receives part of his own money, and in the payout option that money is taxed as income. The growth option leaves the same return inside the fund, where it is taxed only when units are sold and, if held long enough, at the capital gains rate.
The same Rs 1 lakh of return costs Rs 30,000 in tax in the payout option at an illustrative 30% slab, against Rs 12,500 in the growth option at an illustrative 12.5% gains rate, a difference of Rs 17,500. Over ten years on Rs 10 lakh at 10% the gap compounds to Rs 4.27 lakh.
The relationship
$\Delta = R\,(t_{\text{slab}} - t_{\text{cg}}) = 1{,}00{,}000 \times (0.30 - 0.125) = 17{,}500$
R the return in the year, Rs 1 lakh
t_slab the client's illustrative income tax slab, 30%
t_cg the illustrative long-term capital gains rate, 12.5%
What it says in wordsThe cost of the payout option is the return times the gap between the two tax rates.
Why does the gap grow over time?
Because the payout option is taxed every year, so the reinvested amount compounds at 7% after tax, while the growth option compounds at the full 10% and is taxed once at the end. Rs 10 lakh for ten years becomes about Rs 19.67 lakh in the payout option, even with every payout reinvested, against about Rs 23.95 lakh in the growth option after its tax. The rate gap and the deferral work in the same direction.
The payout option can still suit a client who needs regular cash and sits in a low slab. The rates here are illustrative: slab rates, capital gains rates, holding periods, exemption limits and deduction of tax at source all change, so confirm the current rules before advising on them.
Where candidates lose it
Candidates often treat the payout as income on top of the return, or say the option choice cannot matter because the fund is the same. Both miss that the option changes the tax treatment of an identical return.
Give the Rs 17,500 per Rs 1 lakh, then add deferral as the second effect, and flag that the rates are illustrative. That last line matters on a desk that answers to a compliance team.
What the interviewer asks next
For a client in a 5% slab, which option costs less tax?
How would you set up regular cash for a retiree without using the IDCW option?
Why might a fund's NAV fall sharply on a record date?
057Re 1 grows at 10% a year for 20 years. In case one, an illustrative 20% tax is paid on each year's gain as it is earned. In case two, 20% is paid once on the whole gain at the end. How much more does deferral leave the client?Tax arithmeticHardWealth management
Try it first
Same rate, same tax, same 20 years. Does the timing of the tax change the ending wealth?
Show the worked solution
Deferral leaves about 5.58 against 4.66, roughly 20% more. Paying 20% of each year's gain cuts the growth rate from 10% to 8%, and 1.08 to the 20th is 4.66. Deferred, Re 1 compounds at the full 10% to 6.73; tax of 20% on the 5.73 gain leaves 5.58. The unpaid tax stayed invested for 20 years, and its growth mostly belongs to the client.
Why does the timing of the same tax rate matter?
Picture a partner who takes their share of the shop's profit every evening, against one who leaves it in the till and settles once, years later. In the second case the partner's share keeps working for the shop in the meantime. A tax paid every year leaves the portfolio and stops compounding; a tax deferred to the end stays invested, and the growth on it accrues mostly to the client. It behaves like an interest-free loan from the tax office, repaid at the end.
Re 1 at 10% taxed at 20% every year grows at 8% to 4.66 after 20 years, while the same Re 1 taxed once at the end grows to 6.73 and keeps 5.58 after tax, about 20% more, and the gap opens mostly in the later years.
The relationship
$\underbrace{(1.08)^{20}}_{\text{annual}} = 4.66 \qquad \underbrace{1 + 0.8\,(1.10^{20} - 1)}_{\text{deferred}} = 5.58$
1.08 one year of growth after 20% tax on a 10% gain
1.10^20 twenty years of untaxed growth
0.8 the share of the gain the client keeps after the final tax
What it says in wordsAnnual tax lowers the compounding rate; deferred tax takes a slice of a larger final gain.
Does the government lose from deferral?
Not in rupees. Add up the annual bills and the government collects about 0.92 over 20 years; under deferral it collects 1.15 in one go. Deferral grows the whole pie, so both the client and the tax office end with more nominal rupees; the tax office simply waits longer for its share. Expressed as a yearly rate, deferral turns an 8% after-tax return into about 8.98% a year.
Where does a wealth client meet this?
Anywhere tax is charged on realisation rather than accrual. Interest on a deposit is usually taxed as it accrues, while gains on a fund held for years are usually taxed only when units are sold. The rates are illustrative here, and the rules differ by product and change over time, so confirm them before comparing real products. The mechanism does not change: the longer the tax waits, the more the difference compounds.
Where candidates lose it
The trap is saying the two cases are equal because the tax rate is the same. Candidates treat 20% of the gain as a fixed slice and miss that annual payment shrinks the base that compounds.
The second miss is getting the direction right but not the size. At 10% for 20 years the edge is about a fifth of ending wealth; say the number, because a client decides on the number.
What the interviewer asks next
What yearly after-tax return does the deferred case represent?
How does the edge from deferral change at 5 years, and at 40?
If the tax rate at the end is higher than the annual rate, when does deferral stop paying?
069A client sells a fund at a Rs 4 lakh loss to offset gains elsewhere, saving Rs 80,000 of tax at an illustrative 20% rate, then buys the same fund straight back. His cost base is now Rs 4 lakh lower. What is the harvest really worth?Tax arithmeticHardWealth management
Try it first
If he sells the fund for good in ten years, what is the harvest worth today, at a 10% return?
Show the worked solution
About Rs 49,157 if he sells in ten years, not Rs 80,000. Buying back the same fund lowers his cost base by Rs 4 lakh, so his eventual gain, and the tax on it, rise by the amount he saved. The harvest mostly defers Rs 80,000 of tax rather than removing it. At a 10% return, paying Rs 80,000 in ten years instead of today is worth Rs 80,000 less Rs 30,843, about Rs 49,157.
Why does the saving come back?
Think of borrowing from a friend without interest: the cash is useful now, but the debt is still there. When the client buys the same fund back at the lower price, the cost the tax office will later subtract from his sale price is Rs 4 lakh lower, so the taxable gain on the sale is Rs 4 lakh higher. At the same 20% rate that is Rs 80,000 of extra tax, exactly the amount saved today. The harvest has not cancelled the tax; it has moved it.
Harvesting the Rs 4 lakh loss saves Rs 80,000 today, but the lower cost base adds Rs 80,000 of tax when the fund is sold in year 10, which is worth only Rs 30,843 today at 10%, so the harvest is worth about Rs 49,157.
So what is the harvest worth, and what does it depend on?
It is worth the value of an interest-free loan of Rs 80,000, which depends on how long the client holds before selling. The longer the holding period, the more the harvest is worth, because the repayment is further away. At 10%, the harvest is worth about Rs 7,273 if he sells after one year, Rs 30,326 after five, Rs 49,157 after ten and Rs 68,109 after twenty. It is worth more if the later gain is taxed at a lower rate than the gains offset today, and less if trading costs eat into it.
The relationship
$V = 80{,}000 - \frac{80{,}000}{1.10^{10}} \approx 49{,}157$
80,000 tax saved today and extra tax paid on the later sale
1.10^10 ten years of growth at 10%
V the value today of paying the tax ten years later
What it says in wordsA harvest is worth the tax saved now minus the present value of the same tax paid later.
Two conditions to state out loud. The saving only exists if there are gains to offset now, or losses can be carried forward. And some tax systems disallow a loss when the same security is bought back within a set window; rules differ by country and change, so confirm the current position before recommending the move to anyone.
Where candidates lose it
The trap is valuing the harvest at the full Rs 80,000. Candidates forget that buying back the same fund resets the cost base, so the saved tax returns at the sale.
The overcorrection is saying the harvest is worthless. Deferral has value, and the answer should give it in rupees with the holding period stated.
What the interviewer asks next
If the later gain is taxed at 10% instead of 20%, what is the harvest worth?
The client buys a similar but different fund instead of the same one. What changes?
Why is a harvest worth less to a client who expects to sell within a year?
081A client holds three lots of the same mutual fund, 1,000 units each, bought at NAV 100, 150 and 180. The NAV is now 170 and he redeems one lot's worth of units. What is his taxable gain under first in, first out, and what would it be if he could choose which lot to sell?Tax arithmeticCoreMutual fund distributionIndian wealth management
Try it first
Under first in, first out, what gain does he book?
Show the worked solution
Rs 70,000 under first in, first out, against a loss of Rs 10,000 if he could pick the lot bought at 180. FIFO sells the NAV 100 units: 70 a unit on 1,000 units. Picking the newest lot books 170 minus 180, a loss of 10 a unit. The gap is Rs 80,000 of taxable gain, Rs 16,000 of tax at an illustrative 20%, though only a shift in timing.
Why does it matter which lot is sold?
Think of a shop selling rice it bought in three batches at three prices. Its profit on today's sale depends on which batch it says it sold. Units of one fund are identical in the market but not in the tax records: each lot carries its own cost, so the lot matched to a sale decides the gain. The oldest lot here has the lowest cost and the biggest gain.
Against today's NAV of 170 the lot bought at 100 carries a Rs 70,000 gain, the lot at 150 a Rs 20,000 gain and the lot at 180 a Rs 10,000 loss, so first in, first out books Rs 70,000 of gain where picking the newest lot would book a loss.
In India, redemptions of units of the same scheme are generally matched first in, first out, so the client cannot usually pick the lot; confirm the current rule before relying on it. The puzzle asks for both numbers so you can show the cost of the rule, Rs 80,000 of gain brought forward.
Does picking the lot actually save tax?
Mostly it moves tax in time. The three lots carry Rs 80,000 of gain in total at 170, whatever the order. Choosing the high-cost lot defers gains rather than erasing them, so its value is the time value of the deferred tax plus any loss he can set against other gains now. Two things can change that: the oldest lot may qualify for a lower long-term rate because it has been held longer, and a loss booked today may offset gains elsewhere in the same year.
The relationship
$\text{gain} = (170 - \text{cost}) \times 1{,}000: \; 70{,}000,\; 20{,}000,\; -10{,}000$
170 today's NAV, the sale price per unit
cost the NAV each lot was bought at: 100, 150 or 180
1,000 units per lot
What it says in wordsEach lot's gain is the sale price less that lot's cost, times its units.
Where candidates lose it
Candidates reach for the average cost, 143.3, and book a gain of about Rs 26,667. That is how many people track a holding, but it is not how the redemption is matched here.
The second slip is saying lot choice saves Rs 16,000 of tax outright. It mostly defers it; say the total gain is Rs 80,000 whichever order, and name the holding period as the thing that can change the rate.
What the interviewer asks next
The NAV falls to 140. Which lots are at a loss, and what does first in, first out book now?
How could the client use the loss on the newest lot this year?
Why might selling the oldest lot be cheaper in tax even with the largest gain?
094A client's Rs 1 crore fund holding carries a Rs 60 lakh unrealised gain. A new fund is expected to beat the old one by 1 point a year, 11% against 10%, but switching triggers Rs 12 lakh of tax at an illustrative 20%. How many years before the switch pays for itself?Tax arithmeticHardWealth management
Try it first
Treating the Rs 12 lakh as gone for good, roughly how long to break even?
Show the worked solution
About 14 years if the Rs 12 lakh is treated as lost for good, and about 7 if the client would sell and pay the tax some day anyway. The switch starts at Rs 88 lakh against Rs 1 crore and gains about 0.9% a year relative to staying, so it needs about 14.1 years to catch up. Counting the tax the stay path still owes cuts that to about 7.3 years.
How do you get the simple break-even?
Think of changing to a faster train that leaves 15 minutes later. You only arrive earlier if the journey is long enough for the extra speed to make up the late start. A switch that triggers tax starts behind and must earn back the tax through its edge in return, so the break-even is the late start divided by the speed gain. Here the late start is 100 / 88, a 13.6% gap, and the speed gain is 1.11 / 1.10, about 0.9% a year: roughly 14 years.
The relationship
$88 \times 1.11^n = 100 \times 1.10^n \;\Rightarrow\; n = \frac{\ln(100/88)}{\ln(1.11/1.10)} \approx 14.1$
88 what the client can reinvest after Rs 12 lakh of tax, Rs lakh
1.11, 1.10 one plus the expected returns of the new and old funds
n years until the switch catches the stay path
What it says in wordsThe switch catches up when its higher growth has made up the tax it paid at the start.
Treating the Rs 12 lakh as lost for good, the switch starts at 88% of the stay path and levels only after 14.1 years; if the client would sell at the end anyway, the stay path owes its own tax and the switch levels after 7.3 years.
Why does the answer halve if the client sells one day anyway?
Because the Rs 60 lakh gain is taxed whenever it is realised; staying defers the tax, it does not remove it. The fair comparison is after-tax wealth on the day the client will finally sell, and on that basis the switch loses only the growth on the deferred tax, so it breaks even in about 7 years. If the client means to hold until the gain passes to heirs, or will sell in a year of lower tax, the 14-year view is the closer one.
Then put the edge itself under pressure. A 1 point expected advantage sustained for 7 to 14 years is a strong claim for any fund. If the edge is uncertain, a long break-even is a reason to leave the holding alone, or to switch gradually using any tax-free gain allowance each year. The tax rate here is illustrative; confirm the current rate and holding-period rules before advising.
Where candidates lose it
The fast answers are one year, Rs 1 lakh a year on Rs 1 crore against the tax, or 12 years, Rs 12 lakh at Rs 1 lakh a year. The first forgets the tax, and the second forgets that the switch reinvests only Rs 88 lakh.
The deeper loss is stopping at 14 years. Say that staying only defers the tax, give the second answer, and ask how long the client plans to hold: the interviewer wants to hear you frame the comparison, not just solve it.
What the interviewer asks next
What yearly edge would the new fund need for the switch to pay within five years?
How would you stage the switch over several years to reduce the tax?
The client plans to leave the holding to his children. How does that change your view?

N	the amount needed in hand, Rs 10 lakh
g	the share of each redeemed rupee that is gain, 0.40
t	the illustrative tax rate on the gain, 12.5%
G	the gross redemption, Rs lakh

y	the taxable bond's yield, 7.5%
t	the client's marginal tax rate, an illustrative 30%
y_tf	the tax-free bond's yield, 5.4%

R	the return in the year, Rs 1 lakh
t_slab	the client's illustrative income tax slab, 30%
t_cg	the illustrative long-term capital gains rate, 12.5%

1.08	one year of growth after 20% tax on a 10% gain
1.10^20	twenty years of untaxed growth
0.8	the share of the gain the client keeps after the final tax

Try it first

Why does adding the tax on top fall short?

Why is the gain share of each rupee the same across the redemption?

Where candidates lose it

What the interviewer asks next

Try it first

What is the fair way to compare the two yields?

For which clients does the answer flip?

Where candidates lose it

What the interviewer asks next

Try it first

Is the IDCW payout extra money?

Why does the gap grow over time?

Where candidates lose it

What the interviewer asks next

Try it first

Why does the timing of the same tax rate matter?

Does the government lose from deferral?

Where does a wealth client meet this?

Where candidates lose it

What the interviewer asks next

Try it first

Why does the saving come back?

So what is the harvest worth, and what does it depend on?

Where candidates lose it

What the interviewer asks next

Try it first

Why does it matter which lot is sold?

Does picking the lot actually save tax?

Where candidates lose it

What the interviewer asks next

Try it first

How do you get the simple break-even?

Why does the answer halve if the client sells one day anyway?

Where candidates lose it

What the interviewer asks next