Private Wealth Management puzzles, solved step by step

Puzzles: 100
Traced to a firm: 3
Topics: 13
Hard: 30

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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?
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?

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

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

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

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

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