Private Wealth Management puzzles, solved step by step

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

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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
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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?
095A portfolio has an expected return of 10% a year and a volatility of 20%, with independent, normally distributed years. What is the chance that its average annual return over ten years is negative?Probability and risk of lossHardWealth management
Try it first
Your estimate of the chance of a negative ten-year average.
Show the worked solution
About 6%. The average of ten independent years has the same 10% mean but a volatility of 20 divided by the square root of 10, which is 6.3. Zero is then 10 / 6.3, about 1.58 standard deviations below the mean, and the normal table puts 5.7% of outcomes below that, against 31% for a single year.
Why does averaging over ten years narrow the range?
One cricket innings can be anything from a duck to a century; a batter's average over ten innings is far more stable. Good and bad innings cancel partly. Averaging independent years cancels part of their randomness, so the volatility of the average falls with the square root of the number of years, while the expected return stays the same. Ten years divide 20 by about 3.16.
The relationship
$\sigma_{\bar{R}} = \frac{20\%}{\sqrt{10}} = 6.32\% \qquad z = \frac{0 - 10}{6.32} = -1.58 \qquad \Phi(-1.58) \approx 5.7\%$
\sigma_{\bar{R}} the volatility of the ten-year average return
\sqrt{10} the square root of the number of independent years
\Phi the share of a normal distribution below a given z
What it says in wordsShrink the volatility by root ten, then read off how far below the mean zero now sits.
Both bells centre on 10%, but the ten-year average has a volatility of 6.3 instead of 20, so the red tail below zero shrinks from 31% of outcomes for one year to 5.7% for the ten-year average.
Does this mean ten years makes the portfolio safe?
No, and say why. Time narrows the range of the average return, but it widens the range of rupee outcomes, because each year's result compounds on a bigger pot. The chance of a bad decade is smaller than the chance of a bad year, and a bad decade still costs far more money when it comes.
Two other limits belong in the answer. Years are not fully independent, and real returns have fatter tails than the normal curve, so the true chance is likely higher than 5.7%. And the average return is not the compound return: using the roughly 8% compound rate from volatility drag, the chance of ending the decade with less money than you started with is closer to 10% on the same rough method.
Where candidates lose it
Candidates either keep the one-year 31%, missing that averaging narrows the spread, or say zero because ten years sounds long. Both skip the square root of ten, which is the whole point.
The second loss is claiming that time removes risk. Give the 6%, then say the rupee range widens and the tails are fatter than normal: the interviewer wants the number and its limit.
What the interviewer asks next
What is the chance over 25 years?
How many years until the chance of a negative average falls below 1%?
Why does the chance of ending with less money differ from the chance of a negative average return?
096A client wants Rs 18 lakh a year in today's money through retirement and expects a 2% real return, drawing each year's money at the start of the year. How much larger a corpus does he need to fund 35 years of retirement instead of 30?Retirement and withdrawalHardIndian wealth management
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Roughly how much more corpus do five extra years need?
Show the worked solution
About Rs 48 lakh more, 11.6% on top of the 30-year corpus. At a 2% real return Rs 18 lakh a year for 30 years needs Rs 411 lakh today, and for 35 years Rs 459 lakh. The five extra years are Rs 90 lakh of spending, but they come last, so they cost only about Rs 48 lakh today.
How do you price 30 and 35 years of spending?
A retirement corpus is the price today of a stream of future spending. Paying for a child's school fees years ahead costs less today than the fees themselves, because the money set aside earns something while it waits. The corpus needed is the present value of every year's withdrawal at the real return, so later years cost less today than earlier ones.
The relationship
$C_n = 18 \cdot \frac{1 - 1.02^{-n}}{0.02} \cdot 1.02 \qquad C_{30} = 411.2 \qquad C_{35} = 459.0$
18 yearly spending in today's money, Rs lakh
0.02 the real return
1.02 the adjustment for drawing each year's money at the start of the year
C_n the corpus needed today for n years, Rs lakh
What it says in wordsThe corpus equals the value today of n start-of-year withdrawals of Rs 18 lakh at a 2% real return.
Thirty years of Rs 18 lakh need a Rs 411 lakh corpus and thirty-five years need Rs 459 lakh, so the five extra years add only Rs 47.8 lakh, well below the Rs 68.5 lakh that scaling by 35 over 30 would suggest.
Why is the extra much less than Rs 90 lakh, and why is that not the whole story?
The years 31 to 35 are 30 to 34 years away, and at 2% real each rupee then costs only about 55 paise today. Adding years at the end of retirement is cheaper than it looks, which is why the corpus rises about 12% for a 17% longer retirement. The lower the real return, the closer the extra gets to the full Rs 90 lakh.
But if the five extra years come from retiring five years earlier rather than living longer, the client is hit twice: he needs the bigger corpus, and he has five fewer years of saving and growth to build it. That double effect, not the Rs 48 lakh alone, is what makes early retirement expensive. The 2% real return is an assumption; at 0% the extra would be the full Rs 90 lakh.
Where candidates lose it
The two fast answers are Rs 90 lakh, five more years of spending, and about Rs 69 lakh, scaling the corpus by 35 over 30. Both ignore that the added years sit at the far end of retirement and are discounted the most.
The second loss is stopping at the number. Ask whether the extra years come from a longer life or an earlier exit: the interviewer wants to hear that retiring earlier also shortens the saving years.
What the interviewer asks next
What would the extra be at a 0% real return, and at 4%?
The client retires five years early instead of living longer. What else changes?
How would you plan for not knowing whether retirement lasts 25 years or 40?
097Interest rates rise by 1 point overnight. A client's bond fund has a modified duration of 6 and was yielding 7.5%. It loses about 6% at once. Roughly how long before the investor is back to even?Fixed income numeracyHardWealth management
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How long until the fund is as far ahead as it would have been with no rate rise?
Show the worked solution
Two answers: back to its starting value in about 9 months, but back level with the no-rise path only after about 6.7 years, close to the fund's duration. The fund drops to 94 and now earns 8.5%, so 94 grows past 100 in under a year. Against staying at 7.5%, it gains about 1 point a year, which takes roughly six to seven years to repay the 6 point loss.
Why does a rate rise hurt now and help later?
Imagine you have just locked into a rent of Rs 75,000 a month, and next day rents rise to Rs 85,000. Your lease is worth less to anyone buying it, but the next lease you sign will pay more. A rate rise marks down the bonds the fund holds today, and then lets it reinvest coupons and maturing money at the higher yield, so the loss comes at once and the payback comes year by year.
The relationship
$\Delta P \approx -D_{mod} \times \Delta y = -6 \times 1\% = -6\% \qquad 0.94 \times \left(\frac{1.085}{1.075}\right)^n = 1 \Rightarrow n \approx 6.7$
D_mod modified duration, the percentage price change for a 1 point move in yield
\Delta y the change in yield, here plus 1 point
n years until the fund catches the no-rise path
What it says in wordsDuration sizes the immediate loss; the extra yield then earns it back at about 1 point a year.
The fund drops from 100 to 94 at once and, growing at the new 8.5% yield, passes 100 again after about 9 months, but it only draws level with the path it would have followed without the rise after about 6.7 years, close to its duration of 6.45.
Which answer does the client need to hear?
Both, clearly labelled. "Back to what you put in" is under a year; "back to where you would have been" is about the duration. An investor whose holding period is longer than the fund's duration is left better off by a rate rise, because the higher yield more than repays the loss by the end of it. That is the logic of matching a bond fund's duration to the client's horizon.
State the approximations. The 6% ignores convexity, which makes the actual fall a little smaller. The payback assumes the fund keeps its duration steady and reinvests everything at 8.5%, and that rates do not move again. The rule that payback is about the duration uses the Macaulay duration, here 6 x 1.075, about 6.45 years; 6.7 is close to both.
Where candidates lose it
The common slips are "never" and "a year". The first forgets that the fund reinvests at the higher yield; the second counts the extra point of yield as if it repaid 6 points in one go.
The quieter loss is giving one number without saying what "even" means. Split it into back to cost and back to the no-rise path, then tie the second to duration.
What the interviewer asks next
The fund's duration is 2 instead of 6. How do the loss and the payback change?
Why does the fund fall slightly less than 6% in practice?
A client needs the money in 18 months. Which fund duration suits him?
098A client keeps Rs 20 lakh in a 7% fixed deposit as his emergency fund, and at the same time carries a Rs 15 lakh personal loan at 14%. What does keeping the two apart cost him each year?Behavioural trapsCoreIndian wealth management
Try it first
What does the habit cost him each year, before tax?
Show the worked solution
About Rs 1.05 lakh a year before tax, and more after it. On the Rs 15 lakh that overlaps, the deposit earns 7%, Rs 1.05 lakh, while the loan charges 14%, Rs 2.1 lakh. Repaying the loan from the deposit saves the 7 point gap. Deposit interest is taxed and personal-loan interest usually is not deductible, so at an illustrative 30% slab the cost is about Rs 1.37 lakh.
Why does it feel sensible to keep both?
Many families keep a jar labelled "emergencies" and never touch it, even while paying interest on a credit card. The label makes the money feel spoken for. Mental accountingTreating money differently depending on the label or pocket it sits in, even though a rupee is worth the same everywhere. makes a client treat the deposit and the loan as separate stories, but money is fungible: a rupee in the deposit and a rupee owed on the loan cancel. Every rupee kept in the deposit while the loan is open is effectively borrowed at 14% to earn 7%.
The same Rs 15 lakh earns Rs 1.05 lakh a year in the deposit pocket and costs Rs 2.10 lakh a year in the loan pocket, so keeping the pockets apart costs the client Rs 1.05 lakh a year before tax.
The relationship
$15 \times (14\% - 7\%) = 1.05 \qquad 15 \times 14\% - 15 \times 7\% \times (1 - 0.30) = 1.365$
15 the overlapping amount, Rs lakh
14% - 7% the gap between the loan rate and the deposit rate
0.30 an illustrative tax slab on deposit interest
What it says in wordsThe cost is the overlap times the rate gap, and larger once tax on the deposit interest is counted.
But does he not need the emergency fund?
He needs access to money in an emergency, which is a different thing from holding Rs 20 lakh in a deposit. Repaying the loan from the deposit and keeping Rs 5 lakh as a buffer leaves him with a smaller emergency pot but no 14% debt, and the saving of about Rs 1 lakh a year rebuilds the buffer. If a real emergency outruns Rs 5 lakh, he could borrow again, which is what he is already doing today.
Check the practical frictions before suggesting it: a prepayment charge on the loan, a penalty for breaking the deposit early, and whether the client's income is steady enough that a smaller buffer is safe. Those can shrink the saving, but they rarely close a 7 point gap. Say it with respect: the habit is common, and it comes from caution, not carelessness.
Where candidates lose it
The common answer is that nothing is lost because one is savings and the other is debt. That is the mental account talking, and the interviewer is testing whether you see through it.
The second loss is recommending he empty the deposit without a word about the buffer, prepayment charges or tax. Give the Rs 1.05 lakh, then the practical checks.
What the interviewer asks next
The loan is a home loan at 8.5% with a tax deduction on interest. Does the answer change?
How big should the emergency buffer be, and how would you decide?
Name another everyday habit that comes from mental accounting.
099Estimate how many Indian households hold more than Rs 10 crore in financial assets. Work from population, household size and a stated assumption about how wealth is distributed.Estimation and sizingHardIndian wealth management
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Which assumption moves this estimate the most?
Show the worked solution
Roughly 35,000 households, within a range of about 19,000 to 63,000. Take about 140 crore people at 4.5 a household: 31 crore households. Assume the top 1%, 31 lakh, hold over Rs 50 lakh in financial assets. Above that, assume a Pareto tail with alpha 1.5: twenty times the wealth means 20 to the 1.5, about 89 times fewer households.
How do you get from households to the top slice?
Start with the count you can defend and narrow it one assumption at a time. A wealth estimate needs three things said out loud: how many households there are, where the top slice starts, and how quickly the numbers thin out above it. The first can be checked against a census; the other two are judgements, so state them as assumptions and confirm the population figure before relying on it.
From about 31 crore households, an assumed top 1% of 31 lakh hold over Rs 50 lakh, and a Pareto tail with alpha 1.5 leaves about 35,000 above Rs 10 crore, with the tail assumption alone moving the answer between about 19,000 and 63,000.
Why use a Pareto tail, and what does alpha do?
Wealth at the top thins out in a regular way: each step up in wealth has a fixed fraction as many people as the step below. Think of a city's buildings: many of three floors, fewer of ten, a handful of forty. A Pareto tail says the number of households above a wealth level falls as that level to the power minus alpha, so alpha sets how fast the count thins, and the answer is very sensitive to it. With alpha 1.5, twenty times the wealth means about 89 times fewer households.
The relationship
$N(>W) = N_0 \left(\frac{W}{W_0}\right)^{-\alpha} = 31.1\text{ lakh} \times 20^{-1.5} \approx 35{,}000$
N_0 households above the starting threshold, the assumed top 1%, 31.1 lakh
W / W_0 Rs 10 crore over the Rs 50 lakh threshold, 20
\alpha the Pareto tail exponent, assumed 1.5
What it says in wordsHouseholds above Rs 10 crore are the top slice thinned by twenty times the wealth raised to the power alpha.
Close with the checks. Financial assets exclude property and unlisted business stakes, which is where much Indian wealth sits, so the count of households worth Rs 10 crore in total would be much larger. Published wealth reports can anchor the threshold and alpha; use them to test the estimate, not to replace the reasoning.
Where candidates lose it
The weak answer jumps to a number heard somewhere, with no route to it. The interviewer cannot test a number with no assumptions, so it earns nothing even if it happens to be close.
The second loss is treating every step as equally firm. Say which assumption is the tail, show how threefold a swing it produces, and the estimate becomes a piece of reasoning.
What the interviewer asks next
How would the answer change if the top 1% threshold were Rs 1 crore instead of Rs 50 lakh?
How would you estimate how many of these households already use a private bank?
What data would you use to check the alpha assumption?
100A relationship manager costs the bank Rs 60 lakh a year all in. Client assets yield 0.9% a year in revenue, and the bank wants revenue of three times the adviser's cost. How large a book must the adviser manage?Wealth business economicsCorePrivate banking
Try it first
Pick the book size.
Show the worked solution
Rs 200 crore. The bank wants revenue of three times Rs 60 lakh, which is Rs 1.8 crore a year. Each rupee of client assets brings in 0.9 paise, so the book must be Rs 1.8 crore divided by 0.009, which is Rs 200 crore. If the revenue yield slips to 0.75%, the same adviser needs Rs 240 crore.
How do the three numbers fit together?
A shop assistant earning Rs 30,000 a month has to help sell enough goods, at the shop's margin, to cover her pay several times over, because the rent, stock and the owner's profit come from the same margin. An adviser's required book is cost times the coverage the bank wants, divided by the revenue each rupee of assets earns. Coverage above one pays for the office, the platform, compliance and the bank's profit.
The relationship
$\text{book} = \frac{\text{cost} \times \text{coverage}}{\text{yield}} = \frac{60 \text{ lakh} \times 3}{0.009} = \text{Rs } 200 \text{ crore}$
cost the adviser's all-in cost to the bank, Rs 60 lakh a year
coverage revenue as a multiple of that cost, here 3
yield revenue as a share of client assets, 0.9% a year
What it says in wordsThe book needed is the revenue the bank wants from the adviser divided by what each rupee of assets earns.
An adviser costing Rs 60 lakh at three times coverage must bring in Rs 1.8 crore a year, which at a 0.9% revenue yield needs a Rs 200 crore book, and a fall in yield to 0.75% raises that to Rs 240 crore.
Which of the three numbers is the most fragile?
The yield. Revenue yield falls as clients move to cheaper products and larger clients negotiate fees, so the same adviser needs a bigger book every year just to stand still. A drop from 0.9% to 0.75%, which a shift towards advisory fees on large accounts could produce, lifts the required book from Rs 200 crore to Rs 240 crore, 20% more.
Add the time dimension too. A new adviser rarely starts with Rs 200 crore; the bank funds a ramp of two or three years while the book is built, and that ramp is part of the true cost of hiring. The coverage ratio and yield here are illustrative, and a bank's own figures vary widely by segment.
Where candidates lose it
The common slip is dividing cost by yield and stopping at about Rs 67 crore, which only pays the adviser's salary and leaves nothing for the rest of the bank. The coverage multiple is the step people drop.
The second slip is misplacing the decimal: 0.9% is 0.009, not 0.09. Say "0.9 paise per rupee" out loud and the order of magnitude stays right.
What the interviewer asks next
The bank lowers the coverage it wants to 2.5 times. What book is needed?
How long would it take a new adviser to build Rs 200 crore at Rs 50 crore of net new money a year?
Why do revenue yields tend to fall as client size rises?

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

\sigma_{\bar{R}}	the volatility of the ten-year average return
\sqrt{10}	the square root of the number of independent years
\Phi	the share of a normal distribution below a given z

18	yearly spending in today's money, Rs lakh
0.02	the real return
1.02	the adjustment for drawing each year's money at the start of the year
C_n	the corpus needed today for n years, Rs lakh

D_mod	modified duration, the percentage price change for a 1 point move in yield
\Delta y	the change in yield, here plus 1 point
n	years until the fund catches the no-rise path

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

Try it first

Why does averaging over ten years narrow the range?

Does this mean ten years makes the portfolio safe?

Where candidates lose it

What the interviewer asks next

Try it first

How do you price 30 and 35 years of spending?

Why is the extra much less than Rs 90 lakh, and why is that not the whole story?

Where candidates lose it

What the interviewer asks next

Try it first

Why does a rate rise hurt now and help later?

Which answer does the client need to hear?

Where candidates lose it

What the interviewer asks next

Try it first

Why does it feel sensible to keep both?

But does he not need the emergency fund?

Where candidates lose it

What the interviewer asks next

Try it first

How do you get from households to the top slice?

Why use a Pareto tail, and what does alpha do?

Where candidates lose it

What the interviewer asks next

Try it first

How do the three numbers fit together?

Which of the three numbers is the most fragile?

Where candidates lose it

What the interviewer asks next