Private Wealth Management puzzles, solved step by step

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

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012A product gives twice the daily return of an index, resetting every day. The index rises 10% one day and falls 10% the next, ending down 1%. Where does the 2x product end?Leverage and borrowingCorePrivate banking
Try it first
Pick the 2x product's result over the two days.
Show the worked solution
Down 4%. The index goes 100, 110, 99, a 1% loss. The 2x product goes up 20% to 120, then down 20% to 96. It resets its leverage each day, so over several days it compounds twice the daily moves rather than delivering twice the period's return. In a choppy market that difference always works against the holder.
Why is it not simply twice the index's 1% loss?
Imagine a shop that runs a 20% sale one day and a 20% price rise the next. The price does not come back to where it started: 100 becomes 80 and then 96. Equal percentage moves up and down do not cancel, and leverage doubles the size of each move, so it more than doubles the leak. The index loses 1% from the up-and-down, the 2x product loses 4%.
The index goes from 100 to 110 to 99, down 1%, while the 2x daily product goes from 100 to 120 to 96, down 4%, twice the naive expectation of 2%.
Where does the extra loss come from?
From volatility dragThe shortfall of compounded growth below the average return, caused by returns bouncing up and down. It grows roughly with the square of the size of the moves.. A move of plus x then minus x leaves you with 1 minus x squared. For the index that is 1 minus 0.01; for the 2x product it is 1 minus 0.04, four times the leak, because the drag grows with the square of the move. Double the leverage and you quadruple the drag.
The relationship
$(1 + Lx)(1 - Lx) = 1 - L^2x^2 \qquad L=2,\; x=0.10:\; 1 - 0.04 = 0.96$
L the leverage multiple, here 2
x the size of each daily move, 10%
What it says in wordsAn up move and an equal down move leave the position short by the square of the leveraged move.
Say what it means for a client. A daily reset product can be close to its promise over a day and far from it over a year, even if the index ends flat. It is built for short holding periods; held for months in a choppy market it can lose money while the index goes nowhere.
Where candidates lose it
The trap is answering minus 2%, doubling the period's result. That reading treats a daily product as if it gave twice the return over any horizon, which is exactly the misunderstanding that hurts clients who hold these products for months.
The second loss is getting 96 and not explaining why. The one-line reason is that leverage doubles each move and the leak grows with the square of the move.
What the interviewer asks next
The index goes up 10% and then up 10% again. How does the 2x product do against twice the index?
What happens to a 3x daily product over the same two days?
Why is a daily reset product a poor fit for a client planning to hold for a year?
049A client takes a Rs 50 lakh home loan for 20 years at 9% a year. Roughly what is the monthly instalment, and over the life of the loan does he pay more in interest than he borrowed?Leverage and borrowingCoreIndian wealth management
Try it first
Over 20 years, how does the total interest compare with the Rs 50 lakh borrowed?
Show the worked solution
About Rs 44,986 a month, and yes: total interest is about Rs 58.0 lakh, more than the Rs 50 lakh borrowed. At 0.75% a month over 240 months, the standard EMI formula gives Rs 44,986. Multiplied by 240 that is Rs 108.0 lakh paid in all. In the first year about 83% of the instalments go to interest, and principal only overtakes interest in year 13.
How do you get to the EMI quickly?
Start from the interest alone: Rs 50 lakh at 0.75% a month is Rs 37,500. The EMI has to cover that and chip away at the loan, so it must be more than Rs 37,500. The EMI is the monthly interest scaled up by a factor that spreads repayment over 240 months: (1 + i) to the n, divided by (1 + i) to the n minus 1. Here 1.0075 to the power 240 is about 6.01, so the factor is 6.01 / 5.01, about 1.20, and Rs 37,500 x 1.20 is about Rs 44,986.
The relationship
$EMI = P\,i\,\frac{(1+i)^n}{(1+i)^n - 1} = 50{,}00{,}000 \times 0.0075 \times \frac{6.009}{5.009} \approx 44{,}986$
P the loan, Rs 50 lakh
i the monthly rate, 9% / 12 = 0.75%
n the number of monthly instalments, 240
What it says in wordsThe instalment is the first month's interest, grossed up just enough to clear the loan by the last payment.
Each year the client pays Rs 5.40 lakh in EMIs; in year 1 Rs 4.46 lakh of it is interest, and principal repaid first exceeds interest only in year 13. Over 20 years he pays Rs 58.0 lakh of interest on Rs 50 lakh borrowed.
Why is so much of the early EMI interest?
A long loan is like paying rent on money: the rent is charged on whatever you still owe, and in the early years you still owe almost all of it. Because interest is charged on the outstanding balance and the balance falls slowly at first, the first years of a 20-year loan are mostly interest, and the total interest ends up larger than the loan. That is also why a prepayment in the early years saves far more interest than the same prepayment late in the loan.
For a wealth conversation, the point is the comparison a client actually faces: prepaying the loan earns him the loan rate, after any tax benefit on interest, with no risk. The illustrative 9% is not a quoted rate; floating-rate loans also reset, which changes both the EMI and the tenure.
Where candidates lose it
The fast wrong answer multiplies 9% by 20 years by Rs 50 lakh and says Rs 90 lakh of interest, ignoring that the balance falls. The opposite error treats the EMI as principal divided by months plus a little, about Rs 25,000, which is far too low.
Anchor on the first month's interest of Rs 37,500, say the EMI must exceed it, then give Rs {inr(P49['emi'])} and the total. That sequence is what makes the number believable in the room.
What the interviewer asks next
What is the EMI if the tenure is 30 years instead of 20?
The client prepays Rs 5 lakh at the end of year 2. Roughly how much interest does he save?
Should a client with spare cash prepay this loan or invest? What does the answer depend on?
052A Rs 2 crore flat rents for Rs 48,000 a month, a gross rental yield of about 2.9%. A home loan costs 8.5%. If the client buys it entirely with a loan, what is the extra annual cash cost of owning compared with renting the same flat?Leverage and borrowingCoreIndian wealth management
Try it first
Roughly how much more does owning cost in cash each year?
Show the worked solution
About Rs 11.2 lakh a year more to own. Rent is Rs 48,000 x 12, or Rs 5.76 lakh, a 2.88% yield. Interest at 8.5% on Rs 2 crore is Rs 17 lakh. The owner pays Rs 11.24 lakh more in cash each year, so the flat must rise about 5.6% a year in price just to match renting, before maintenance and tax effects.
Which two numbers are actually being compared?
Strip it down to the cost of living in the flat for one year. The tenant pays rent. The owner who borrowed the whole price pays interest to the bank; the part of the EMI that repays principal is not a cost, it is money moved from the bank account into the flat. So the comparison is rent against interest, and at a 2.9% rental yield against an 8.5% loan rate, interest is roughly three times the rent. Per month that is about Rs 141,667 of interest against Rs 48,000 of rent.
On the same Rs 2 crore flat, renting costs Rs 5.76 lakh a year and owning with a full 8.5% loan costs Rs 17.00 lakh of interest, so owning takes Rs 11.24 lakh more cash every year and needs price growth of about 5.6% a year to break even.
Is the Rs 17 lakh exact, given the loan amortises?
Close enough for an interview, and you should say why. On a 20 year loan the EMI is about Rs 173,565 a month, and first-year interest works out to about Rs 16.8 lakh because a little principal is repaid each month. The simple 8.5% x Rs 2 crore overstates year one by only about Rs 15,270, so it is the right number to say out loud. The gap narrows slowly over later years as the loan shrinks, but the owner's own money then sits in the flat and could have earned something elsewhere.
What would make owning worth it?
The owner is betting on the price. Owning with debt pays off only if the flat appreciates by more than about 5.6% a year, the gap divided by the price. Maintenance, property tax and registration costs push that hurdle higher; the tax deduction on home loan interest pulls it lower, within limits that you must confirm against current rules. Say the break-even rate and let the client judge whether that growth is likely. That is numeracy, not a view on property.
Where candidates lose it
The common error is comparing the EMI with the rent. The EMI mixes interest, which is a cost, with principal repayment, which is saving, so the comparison makes owning look even worse than it is and confuses the client.
The opposite error is saying owning is free because the flat is an asset. The asset only pays for the interest if its price rises fast enough; name the break-even growth rate instead of assuming it.
What the interviewer asks next
The client pays cash instead of borrowing. What is the cost of owning now?
At what rental yield does owning with a full loan cost the same cash as renting?
How do the home loan tax deduction and maintenance costs change the break-even appreciation?
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?

L	the leverage multiple, here 2
x	the size of each daily move, 10%

P	the loan, Rs 50 lakh
i	the monthly rate, 9% / 12 = 0.75%
n	the number of monthly instalments, 240

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

Try it first

Why is it not simply twice the index's 1% loss?

Where does the extra loss come from?

Where candidates lose it

What the interviewer asks next

Try it first

How do you get to the EMI quickly?

Why is so much of the early EMI interest?

Where candidates lose it

What the interviewer asks next

Try it first

Which two numbers are actually being compared?

Is the Rs 17 lakh exact, given the loan amortises?

What would make owning worth it?

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