Private Wealth Management puzzles, solved step by step

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

Showing 41–47 of 47 · filtered from 100Clear filters

086Estimate the annual market, in rupees, for will-drafting services across India's ten largest cities.Estimation and sizingCoreTrust and estate administration
Try it first
Which step in this estimate deserves the most scrutiny?
Show the worked solution
Roughly Rs 210 crore a year, on stated assumptions. Take 10 crore people, 2.5 crore households at four each, 40% with property or savings worth protecting, and two adults each: 2 crore people who need a will. If 15% ever write one over a 30-year window, that is 1 lakh wills a year, at a blended Rs 21,000 each.
How do you structure the estimate before any number?
Think of estimating how many cakes a bakery sells for birthdays: you would go from people, to birthdays a year, to the share celebrated with a bought cake, to the price. A market size is a chain of counts and rates, and saying the chain out loud before any number is most of the marks. Here the chain runs from people, to households with something to leave, to adults who need a will, to how many write one each year, to the fee.
Ten crore people become 2 crore adults who need a will, 1 lakh of whom write one each year at an average Rs 21,000, a market of about Rs 210 crore, and the 15% will-making rate is the step that moves it most.
Every number in the tree is an assumption to state, not a fact: the population of the ten cities, the household size and the 40% share should be replaced with census and survey figures when you have them. The fee split reflects two very different products: a simple will drafted online or by a local lawyer, and a complex will or estate plan prepared alongside a private wealth adviser.
Which step would you defend least, and what does that tell you?
The 15% lifetime rate of writing a formal will. The estimate is only as good as its least defensible step, so name that step and show what happens when it moves. At 30% the market doubles to Rs 420 crore; at 7.5% it halves. The interviewer will usually push on the weak step, so get there first.
The relationship
$\text{market} = \frac{2\text{ crore} \times 15\%}{30} \times \text{Rs } 21{,}000 \approx \text{Rs } 210 \text{ crore}$
2 crore adults in households with something to protect
15% / 30 the share who ever write a will, spread over a 30-year window
Rs 21,000 the blended fee, 80% simple and 20% complex wills
What it says in wordsWills written each year times the average fee gives the yearly market.
Close with a sense check from the other side: 1 lakh wills a year across ten cities is about 40 wills a working day in each city, which feels plausible for the number of lawyers and advisers who offer the service.
Where candidates lose it
The common loss is diving into numbers with no structure, so each figure arrives from nowhere and the interviewer cannot follow or challenge it. Say the chain first.
The second loss is defending every assumption equally. Name the will-making rate as the weak step, show the range, and the answer becomes a judgement rather than a guess.
What the interviewer asks next
How would you size the market for estate planning for families above Rs 25 crore?
What data would you use to replace the 15% assumption?
How does a rise in online will services change the fee mix and the market?
087A client buys a put option 10% below the market every year to protect his portfolio, at a cost of 2% of the portfolio a year. If the market goes nowhere for ten years, how much wealth has the insurance cost him?Options and structured productsCorePrivate banking
Try it first
After ten flat years, what share of his wealth has gone on premiums?
Show the worked solution
About 18.3% of his wealth. In a flat market the puts, struck 10% below, never pay out, so each year he simply loses 2%. Ten years leave 0.98 to the power 10, which is 0.817 of the starting wealth. That is slightly under 20% because each premium is taken from a pot the earlier premiums have already shrunk.
Why is the cost close to 18% and not 20%?
Paying 2% of your salary every year for a subscription feels like a small fixed charge, but if it were 2% of your savings each year, the savings would shrink and each charge would be a little smaller in rupees. A yearly premium set as a share of the portfolio compounds downwards, so ten years of 2% cost 1 minus 0.98 to the 10th, about 18.3%, not a flat 20%. In rupees, a Rs 10 crore portfolio would stand at about Rs 8.17 crore.
In a flat market the uninsured portfolio stays at 100 while the insured one pays 2% a year for puts that never pay and falls to 81.7, a gap that widens every year to 18.3% of wealth.
The relationship
$W_{10} = (1 - 0.02)^{10} = 0.817 \qquad \text{cost} = 1 - 0.817 = 18.3\%$
0.02 the yearly premium as a share of the portfolio
10 years the insurance is bought
W_10 wealth after ten years, as a share of the start
What it says in wordsEach year keeps 98% of the pot, so ten years keep 0.98 to the tenth power.
So is the insurance a bad idea?
Not by itself. A put optionA contract that gives the holder the right to sell an asset at a set price, so it pays out when the price falls below that level. pays out in a crash, which is when a client most needs cash and is most tempted to sell at the bottom. Permanent insurance is a permanent drag, so the question is whether the protection is worth about 2 points of return every year, not whether it pays out in any one year. A flat decade is the worst case for the buyer, because he pays every premium and collects nothing.
Say the limitation: in a rising market the drag is the same 2% a year taken from a growing pot, and in a crash the puts pay. The honest comparison sets the yearly cost against the losses avoided in the bad years, and the answer depends on how often and how deep those falls are.
Where candidates lose it
The fast answer is 20%, ten times 2%. It ignores that each year's premium is a share of a smaller pot, and the interviewer uses it to see whether you compound costs as naturally as returns.
The bigger loss is judging the insurance by the flat decade alone. Give the 18.3% and then say what the client gets for it, and when.
What the interviewer asks next
Over the same ten years the market falls 30% once. What would the puts have paid?
How could the client cut the premium, and what does he give up?
What is a collar, and why do clients use one?
088A portfolio is expected to return 12% a year. A client borrows at 10% to double his exposure, putting in Rs 1 crore of his own and Rs 1 crore of borrowed money. What does he earn on his own money? What if the portfolio returns 6% instead?Leverage and borrowingCorePrivate banking
Try it first
At a 12% portfolio return, what does he earn on his own Rs 1 crore?
Show the worked solution
14% at a 12% portfolio return, and only 2% at 6%. Rs 2 crore at 12% earns Rs 24 lakh, less Rs 10 lakh of interest, leaving Rs 14 lakh on his Rs 1 crore. At 6% the portfolio earns Rs 12 lakh, less the same Rs 10 lakh, leaving Rs 2 lakh. His return is twice the portfolio return less 10, so leverage helps only above the 10% borrowing rate.
Where does the 14% come from?
A shopkeeper who borrows to double her stock doubles her sales margin, but pays the lender first. If the stock earns more than the loan costs, she keeps the difference on the borrowed half as well as her own margin. On borrowed money the client keeps only the spread between the portfolio return and the borrowing rate, so leverage adds 12 minus 10, two points, to his own 12%.
The relationship
$r_{own} = 2r - (2 - 1) \times 10\% \qquad r = 12\%: 14\% \qquad r = 6\%: 2\%$
r the portfolio's return
2 exposure as a multiple of his own money
10% the rate on the borrowed crore
What it says in wordsHis return is twice the portfolio's, less the interest on the borrowed crore as a share of his own crore.
The levered line rises twice as steeply as the unlevered one and crosses it at the 10% borrowing rate, so a 12% portfolio return gives the client 14% while a 6% return gives him only 2%.
What happens below the crossing point?
Leverage turns against him. Below the borrowing rate every point the portfolio falls short costs him two, so a flat year loses 10% and a 10% fall loses 30%. The loan interest is due whatever the market does, which is why the downside is steeper than the upside is generous.
Then add what the arithmetic leaves out. A loan against a portfolio usually has a margin requirement: if prices fall far enough, the lender asks for more collateral or sells holdings at the worst moment. The expected 12% is also not certain; a borrowing plan that works only if the portfolio beats 10% needs the client to understand how often it will not.
Where candidates lose it
The fast wrong answer is 24%: double the exposure, double the return. It forgets the interest, and the interviewer uses it to check whether you net the cost of money before calling leverage a win.
The second loss is stopping at 14%. The second half of the question, 6%, is where the lesson is: leverage pays only above the borrowing rate, and below it the losses double.
What the interviewer asks next
What portfolio return leaves him with exactly zero on his own money?
The lender asks for more margin if the portfolio falls 25%. What does the client lose at that point?
How does interest being tax deductible, or not, change the break-even?
092A client's portfolio earns 10% a year before fees, the adviser charges 1% a year, and inflation runs at 5%. What share of the client's real return does the fee take?Fee and cost dragCoreWealth management
Try it first
What share of his real return goes on the fee?
Show the worked solution
About one fifth, 20% of his real return. Of the 10% nominal return, about 5 points only keep pace with 5% inflation. The client's real gain is about 5 points, and the 1% fee takes 1 of them. Measured against the nominal return the fee looks like a tenth; measured against what the client actually gains, it is twice that. Done exactly, the share is still 20%.
Why measure the fee against the real return?
Suppose your salary rises 10% in a year when prices rise 5%. Your real raise is about 5%. If a new deduction then takes 1% of your salary, it takes a fifth of your real raise, not a tenth of your nominal one. A fee is paid out of the part of the return the client keeps after inflation, so its true weight is the fee divided by the real return.
The same 1 point fee is a tenth of a 10 point nominal return but a fifth of the 5 point real return left after inflation, which is the part of the return the client can actually spend or save.
The relationship
$\frac{1.10}{1.05} - 1 = 4.76\% \qquad \frac{1.09}{1.05} - 1 = 3.81\% \qquad \frac{4.76 - 3.81}{4.76} = 20\%$
1.10 / 1.05 the real return before the fee, dividing out inflation
1.09 / 1.05 the real return after the 1% fee
20% the share of the real return the fee takes
What it says in wordsDividing out inflation instead of subtracting it changes both real returns but leaves the fee's share at one fifth.
What happens to the share when things get harder?
The fee is fixed while the real return moves, so the fee's share rises as the real return falls. If inflation rises to 7% with the same 10% return, the real gain is about 3 points and the fee takes a third. Add tax on the nominal return and the share climbs further, because tax is charged on the part that only kept pace with prices too.
Say the limit fairly. The adviser's fee buys something: planning, tax work and behavioural coaching that may add more than a point. The puzzle is not an argument against fees; it is a way to show a client what a fee costs in the only currency that matters, his real gain.
Where candidates lose it
The common answer is 10%, fee over nominal return. It is arithmetically right and answers the wrong question: the client cannot spend the part of his return that inflation took.
The second slip is quoting 1% because "the fee is 1%". Say both ratios, a tenth of nominal and a fifth of real, so the interviewer hears that you chose the second on purpose.
What the interviewer asks next
Inflation rises to 7%. What share does the fee take now?
Add a 20% tax on the nominal gain. What share of the after-tax real return does the fee take?
How would you explain this to a client without sounding as if the fee is not worth paying?
093A client's portfolio returns 12% a year and inflation is 6%. How long before his purchasing power doubles?Inflation and real returnCoreIndian wealth management
Try it first
How long until what his money buys has doubled?
Show the worked solution
About 12.6 years, not 12. The rupees double in about 6.1 years at 12%, but purchasing power grows at the real rate. The real rate is 1.12 divided by 1.06, less 1, which is 5.66%, not 6%. At 5.66% money doubles in about 12.6 years; 72 / 6 gives 12, a few months short.
Why is the real rate 5.66% and not 6%?
If your money grows 12% while a thali's price rises 6%, you can afford 1.12 / 1.06 = 1.0566 times as many thalis, not 1.06 times. Real return is found by dividing out inflation, not subtracting it, because inflation reduces what each rupee of the gain buys as well as the principal. Subtraction overstates the real rate slightly, and the error grows with inflation.
The relationship
$r_{real} = \frac{1.12}{1.06} - 1 = 5.66\% \qquad n = \frac{\ln 2}{\ln 1.0566} = 12.6 \text{ years}$
r_real the real return, growth in what the money buys
1.12 one plus the nominal return
1.06 one plus inflation
n years for purchasing power to double
What it says in wordsDivide out inflation to get the real rate, then find how many years at that rate make two.
At 12% the rupees in the account double in about 6.1 years, but at the 5.66% real rate what they buy takes about 12.6 years to double, a little longer than the 12 years that 72 / 6 suggests.
Is the difference between 12 and 12.6 years worth making?
In the room, yes, briefly: 12 years from the shortcut, 12.6 done properly, and the reason is division against subtraction. The bigger point is the gap between 6 years and 12.6: a client who hears his money doubles every six years believes he will be twice as well off, when in spending power he will need more than twice as long. That gap is what goal planning is built around.
Note the limits: inflation is not steady, the client's own inflation, school fees or medical costs, may run above the headline rate, and tax on the nominal return lowers the real rate further. Each of those pushes the doubling time out, not in.
Where candidates lose it
The fast wrong answer is six years, the rule of 72 on the nominal 12%. It answers when the rupees double, and the question asked about purchasing power.
The second slip is presenting exactly 12 years as the answer. It is a sound shortcut, but say that the real rate is 5.66% by division and that the true figure is about 12.6 years.
What the interviewer asks next
With inflation at 8% and the same 12% return, how long does purchasing power take to double?
How does a 20% tax on the nominal return change the answer?
Why might a retired client's personal inflation differ from the headline figure?
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.
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?

2 crore	adults in households with something to protect
15% / 30	the share who ever write a will, spread over a 30-year window
Rs 21,000	the blended fee, 80% simple and 20% complex wills

0.02	the yearly premium as a share of the portfolio
10	years the insurance is bought
W_10	wealth after ten years, as a share of the start

r	the portfolio's return
2	exposure as a multiple of his own money
10%	the rate on the borrowed crore

1.10 / 1.05	the real return before the fee, dividing out inflation
1.09 / 1.05	the real return after the 1% fee
20%	the share of the real return the fee takes

Try it first

How do you structure the estimate before any number?

Which step would you defend least, and what does that tell you?

Where candidates lose it

What the interviewer asks next

Try it first

Why is the cost close to 18% and not 20%?

So is the insurance a bad idea?

Where candidates lose it

What the interviewer asks next

Try it first

Where does the 14% come from?

What happens below the crossing point?

Where candidates lose it

What the interviewer asks next

Try it first

Why measure the fee against the real return?

What happens to the share when things get harder?

Where candidates lose it

What the interviewer asks next

Try it first

Why is the real rate 5.66% and not 6%?

Is the difference between 12 and 12.6 years worth making?

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 the three numbers fit together?

Which of the three numbers is the most fragile?

Where candidates lose it

What the interviewer asks next