Private Wealth Management puzzles, solved step by step

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

Showing 21–30 of 30 · filtered from 100Clear filters

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?
071Two retirees each start with Rs 1 crore and withdraw Rs 10 lakh at the end of every year. Both earn the same five returns: one year of minus 20% and four years of plus 10%. One gets the bad year first, the other gets it last. Where does each end after five years?Retirement and withdrawalHardWealth management
Try it first
Does the order of the same five returns change where they end?
Show the worked solution
Bad year first ends at about Rs 56.1 lakh; bad year last ends at Rs 70 lakh. Left untouched, both pots would reach Rs 117.1 lakh whatever the order. With Rs 10 lakh withdrawn every year, an early 20% fall shrinks the pot the good years grow from, so each good year earns Rs 6 to 7 lakh instead of Rs 10 lakh. Same average return, Rs 13.9 lakh apart.
Why does order not matter without withdrawals?
Multiplication does not care about order: 0.8 x 1.1 x 1.1 x 1.1 x 1.1 is the same as 1.1 x 1.1 x 1.1 x 1.1 x 0.8, and both give 1.1713. A pot left alone ends at Rs 117.1 lakh either way. Withdrawals turn the problem from multiplication into multiplication and subtraction, and subtraction does care about order. Taking a fixed Rs 10 lakh from a pot that has just fallen removes a bigger share of it, and that share never gets to recover.
With Rs 10 lakh withdrawn every year, the retiree who suffers the 20% fall first slides to Rs 56.1 lakh by year 5, while the one who gets it last holds Rs 100 lakh for four years and ends at Rs 70 lakh, although both earned the same five returns.
Where exactly does the Rs 14 lakh go?
Follow the rupees. Both retirees lose Rs 20 lakh in their bad year, because in both cases the fall hits a Rs 100 lakh pot. The difference is what the four good years earn: bad-last grows Rs 100 lakh by 10% four times, Rs 40 lakh in all, while bad-first grows a pot that starts at Rs 70 lakh and keeps shrinking, earning only about Rs 26.1 lakh. Each good year earns less than the Rs 10 lakh he withdraws, so the pot keeps falling even in good years.
Year Return, bad first Balance, bad first Return, bad last Balance, bad last
1 -20% 70.0 +10% 100.0
2 +10% 67.0 +10% 100.0
3 +10% 63.7 +10% 100.0
4 +10% 60.1 +10% 100.0
5 +10% 56.1 -20% 70.0
Rs lakh, returns earned during the year and Rs 10 lakh withdrawn at each year end. The same five returns leave the bad-first retiree with Rs 56.1 lakh and the bad-last retiree with Rs 70.0 lakh.
This is sequence-of-returns risk, and it is why the years just before and just after retirement matter most. Planners respond with a cash or short-bond bucket that funds a few years of withdrawals, so the retiree is not forced to sell growth assets after a fall. Say the mechanism; the right size of such a bucket depends on the client.
Where candidates lose it
The trap is saying the order cannot matter because multiplication is commutative. That is true for a pot left alone and false once money comes out every year, which is the whole point of retirement.
The second miss is getting the right direction without a number. Give both ending balances, then the reason: the good years compound a smaller pot.
What the interviewer asks next
What happens to the gap if the withdrawals are Rs 5 lakh instead of Rs 10 lakh?
What if the retiree were adding Rs 10 lakh a year instead of withdrawing it?
How would a two-year cash bucket have changed the bad-first retiree's outcome?
075A one-year reverse convertible, bought for Rs 100 on a share trading at Rs 100, pays a 12% coupon. At maturity it returns Rs 100 unless the share ends below Rs 80, in which case the investor gets shares worth the final price instead. What does he receive in total if the share ends at Rs 81, and at Rs 79?Options and structured productsHardPrivate banking
Try it first
How far apart are the two outcomes, Rs 81 and Rs 79?
Show the worked solution
Rs 112 at Rs 81 and Rs 91 at Rs 79: a Rs 2 move in the share costs him Rs 21. Above the Rs 80 barrier he gets his Rs 100 back plus the Rs 12 coupon. Below it, he receives shares worth the final price plus the coupon, so at Rs 79 he takes the full fall from Rs 100. The coupon is mostly payment for selling a put on the share that only switches on below Rs 80.
Why is there a cliff at Rs 80?
Think of travel insurance that pays nothing unless your flight is delayed four hours, and then pays in full: a delay of three hours fifty-nine and four hours one are worth completely different amounts. The barrier switches the note from paying Rs 100 to paying the share's value, so crossing it moves the payout by the whole distance from Rs 100 to the share price, not by the Rs 2 the share moved. At Rs 81 he gets Rs 100 plus Rs 12; at Rs 79 he gets Rs 79 plus Rs 12.
The note pays a flat Rs 112 whenever the share ends at or above Rs 80, but just below the barrier it pays shares worth the price plus the coupon, so at Rs 79 the investor gets Rs 91 and the payout drops Rs 21 for a Rs 2 move.
What is the investor actually selling for the 12%?
Split the note into pieces. He lends Rs 100 for a year, which on its own earns an ordinary interest rate. He also sells the bank a put on the share struck at Rs 100 that only comes alive if the share ends below Rs 80, a knock-in putA put option that exists only if the underlying price crosses a barrier level; once knocked in, it pays like an ordinary put.. The part of the 12% above the interest rate is the premium for that put, and the put is why the note's downside looks like owning the share while its upside is capped at Rs 112. Above Rs 112 a shareholder does better; below Rs 80 the investor does no better than a shareholder, apart from the coupon.
What to say about suitability, without making a call on the product. The note swaps an uncertain equity return for a fixed Rs 12 in most outcomes and a large loss in the others, and the loss arrives as a jump. A client should see the Rs 79 row, not only the Rs 112 headline, and understand that the chance of breaching Rs 80 depends on how volatile the share is.
Where candidates lose it
The trap is treating the barrier as a buffer: a 20% protection, so a fall to Rs 79 costs only Rs 1 below the barrier. It costs Rs 21 against Rs 100, because below the barrier the protection disappears entirely.
The second miss is calling the coupon income. Most of the excess over an ordinary interest rate is option premium, paid for taking the share's downside below Rs 80.
What the interviewer asks next
At what final share price is the investor exactly back to his Rs 100?
Why do reverse convertibles on more volatile shares offer higher coupons?
How would a barrier that is watched every day, instead of only at maturity, change the risk?
090Two sisters invest at 10% a year. Sister A puts in Rs 1 lakh a year from age 25 to 34 and then stops. Sister B puts in Rs 1 lakh a year from 35 to 60. Payments are made at the start of each year. Who has more at 60, and by how much?Compounding and doublingHardIndian wealth management
Try it first
Who has more at 60?
Show the worked solution
Sister A, with about Rs 190 lakh against Rs 109 lakh, ahead by Rs 81 lakh. A pays in only Rs 10 lakh, but her pot is Rs 17.5 lakh at 35 and then compounds untouched for 25 years, about 10.8 times. B pays in Rs 26 lakh, yet her last payments have little time to grow. The early years carry the compounding.
How do you get the answer without a spreadsheet?
Work A in two stages. Ten payments of Rs 1 lakh at 10% build to about Rs 15.9 lakh by the last payment at 34, and to Rs 17.5 lakh a year later at 35. From 35 to 60 that pot compounds for 25 years. At 10% money doubles about every 7.2 years, so 25 years is about three and a half doublings, roughly 11 times, which takes Rs 17.5 lakh to about Rs 190 lakh.
The relationship
$A_{60} = 1.1^{26} \cdot \frac{1.1^{10} - 1}{0.1} \approx 189.9 \qquad B_{60} = \frac{1.1^{26} - 1}{0.1} \approx 109.2$
(1.1^10 - 1) / 0.1 ten yearly payments of Rs 1 lakh added up with their growth, about 15.9 lakh
1.1^26 growth from the last payment at 34 to age 60
B_60 26 payments from 35 to 60, the last earning nothing
What it says in wordsA's small early pot grows for 26 more years; B's larger total has much less time in the market.
Sister A stops paying at 34 with Rs 17.5 lakh at 35 and still ends at Rs 189.9 lakh at 60, well above Sister B's Rs 109.2 lakh from 26 payments, because A's money has decades longer to compound.
Why does B never catch up, even with 26 payments?
Picture a snowball rolled from the top of a long slope against a bigger pile of snow thrown on halfway down. The first has more distance to gather snow. At 10% a rupee invested at 25 is worth about 28 rupees at 60, while a rupee invested at 50 is worth about 2.6, so a decade of early payments outweighs a quarter century of later ones. For B to match A she would need to pay about Rs 1.74 lakh a year from 35.
Then add the honest limits. A steady 10% is an illustration, not a forecast; a real sequence of returns will differ, and fees and taxes reduce both pots. The ratio between the sisters, though, comes from time in the market, and that holds at any positive rate: the lower the rate, the smaller A's lead.
Where candidates lose it
Most candidates answer B because she paid in more than twice as much. They count rupees and forget the time each rupee has to grow, which is exactly the mistake a wealth client makes when he delays starting.
The second loss is freezing because the sum looks like it needs a spreadsheet. Split A into two stages and use doublings for the second: an estimate within a few lakh is what the interviewer wants.
What the interviewer asks next
At what rate of return would the two sisters end level?
How much would B need to invest each year to match A at 60?
What does this puzzle say about an employer's retirement contribution that starts at 25?
091A portfolio has an arithmetic average annual return of 10% and a volatility of 20%. Roughly what compound annual return should the client expect to earn over the long run?Returns arithmeticHardPrivate banking
Try it first
Your estimate of the long-run compound return.
Show the worked solution
About 8% a year. The compound, or geometric, return is roughly the arithmetic mean less half the variance. Volatility of 20% is a variance of 0.2 squared, 0.04, and half of that is 0.02, or 2 points. So an average year of 10% compounds at about 8%. The higher the volatility, the wider the gap.
Why does volatility pull the compound return below the average?
Walk up a hill 30 steps and back down 10, then repeat: you are fine. But returns multiply: a rise of 30% followed by a fall of 10% leaves 1.3 x 0.9, which is 1.17 over two years, an average of 10% that compounds at only 8.17%. A loss is taken from a larger base than the gain that preceded it, so the more returns swing, the further compound growth falls below the average return.
The relationship
$g \approx \mu - \frac{\sigma^2}{2} = 0.10 - \frac{0.20^2}{2} = 0.08$
g the geometric, or compound, annual return the client actually earns
\mu the arithmetic mean of yearly returns, 10%
\sigma the volatility, the standard deviation of yearly returns, 20%
What it says in wordsThe compound return is roughly the average return less half the square of the volatility.
The average year sits at 10% but the compound return the client earns sits at about 8%, two points lower, and the gap is half the variance, 0.5 x 0.20 x 0.20, which is too small to see on the full bell and plain on the magnified ruler.
How far can you trust the rule?
Check it on a case you can work exactly. The fund that alternates plus 30% and minus 10% has an average of 10% and a volatility of exactly 20%, and its true compound rate is the square root of 1.17 less one, 8.17%. The half-variance rule gives 8% against an exact 8.17%, close enough to say out loud, and it gets less accurate as volatility rises. At 40% volatility the rule takes off 8 points, and the exact answer depends heavily on the shape of the returns.
Then say why a client should care. Two portfolios with the same 10% average but volatility of 10% and 20% compound at about 9.5% and 8%. Over 20 years on Rs 1 crore that is roughly Rs 6.1 crore against Rs 4.7 crore. Lower volatility at the same average is worth real money, which is part of the case for diversification.
Where candidates lose it
The weak answer is 10%, treating the average year as the rate the client compounds at. It is the same mistake as reading a fund's average return as the growth in the client's statement.
The second loss is knowing the rule but not the reason. Give the plus 30, minus 10 example in one line: it shows the interviewer you understand why the gap exists, not only its formula.
What the interviewer asks next
What compound return does a 10% average with 30% volatility give, roughly?
Two funds have the same compound return but different volatility. Which has the higher average?
Why do some fund factsheets show the average return rather than the compound one?
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?
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
Try it first
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
Try it first
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?
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
Try it first
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?

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

Year	Return, bad first	Balance, bad first	Return, bad last	Balance, bad last
1	-20%	70.0	+10%	100.0
2	+10%	67.0	+10%	100.0
3	+10%	63.7	+10%	100.0
4	+10%	60.1	+10%	100.0
5	+10%	56.1	-20%	70.0

(1.1^10 - 1) / 0.1	ten yearly payments of Rs 1 lakh added up with their growth, about 15.9 lakh
1.1^26	growth from the last payment at 34 to age 60
B_60	26 payments from 35 to 60, the last earning nothing

g	the geometric, or compound, annual return the client actually earns
\mu	the arithmetic mean of yearly returns, 10%
\sigma	the volatility, the standard deviation of yearly returns, 20%

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 order not matter without withdrawals?

Where exactly does the Rs 14 lakh go?

Where candidates lose it

What the interviewer asks next

Try it first

Why is there a cliff at Rs 80?

What is the investor actually selling for the 12%?

Where candidates lose it

What the interviewer asks next

Try it first

How do you get the answer without a spreadsheet?

Why does B never catch up, even with 26 payments?

Where candidates lose it

What the interviewer asks next

Try it first

Why does volatility pull the compound return below the average?

How far can you trust the rule?

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

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

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