Private Wealth Management puzzles, solved step by step

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

Showing 41–50 of 100

041A client invests Rs 100 in a fund each month for three months. The NAV is 10 in month one, 5 in month two and back to 10 in month three. What is his average cost per unit, and how does it compare with the average NAV?Returns arithmeticCoreMutual fund distributionIndian wealth management
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What is his average cost per unit?
Show the worked solution
His average cost is Rs 7.50 a unit, against an average NAV of Rs 8.33. Rs 100 buys 10 units at NAV 10, 20 units at NAV 5 and 10 units at NAV 10: 40 units for Rs 300. Because a fixed amount buys more units when the price is low, the cost per unit is the harmonic average of the prices, which is always at or below the simple average. At NAV 10 his 40 units are worth Rs 400, up 33.3%.
Why is his cost below the average price?
Spend a fixed Rs 100 on tomatoes every week. In the week they are cheap, the same Rs 100 fills twice the bag, so the cheap week makes up more of your total tomatoes. A fixed rupee amount automatically buys more units when the price is low, so the low prices carry more weight in the average cost than in the average price. That is rupee cost averaging, and it is arithmetic, not a trading skill.
A fixed Rs 100 buys 10, 20 and 10 units at NAVs of 10, 5 and 10, so the client holds 40 units at an average cost of Rs 7.50, below the Rs 8.33 average NAV. At the final NAV of 10 those units are worth Rs 400, a 33.3% gain although the NAV only returned to where it started.
The relationship
$\bar{c} = \frac{\sum A}{\sum A/P_i} = \frac{300}{10+20+10} = 7.50 \;\le\; \bar{P} = \frac{10+5+10}{3} = 8.33$
A the fixed amount invested each month, Rs 100
P_i the NAV in month i
\bar{c} the average cost per unit, a harmonic mean of the prices
What it says in wordsAverage cost is total money over total units, which always sits at or below the plain average of the prices.
Does this mean a SIP beats investing a lump sum?
No, and saying so is what earns the point. Rupee cost averaging guarantees a cost below the average price, not a better result than investing everything at once. Take a rising path of NAVs 5, 10 and 15. The SIP buys 36.67 units, worth Rs 550 at the end; Rs 300 invested at NAV 5 on day one buys 60 units, worth Rs 900. When prices mostly rise, money invested earlier does better; the SIP's real value is discipline and avoiding one badly timed lump sum.
The limit of the puzzle is its tidy V-shaped path, which flatters the SIP. On a path that only falls, the SIP still loses money, just less than a lump sum would have.
Where candidates lose it
The quick wrong answer is Rs 8.33, averaging the three NAVs as if he bought the same number of units each month. He bought the same rupees, not the same units.
The second trap is overselling the result. Candidates who stop at Rs 7.50 sound as though SIPs beat the market; add the rising-path check and say the advantage is in behaviour, not in the arithmetic.
What the interviewer asks next
What is his average cost if the NAVs are 10, 20 and 10?
Why is the average cost always at or below the average price?
When would you advise a client with a lump sum to stagger it, and what does it cost him?
042A portfolio manager takes 20% of each year's gains as a performance fee, with no high-water mark. The portfolio returns plus 30%, minus 30%, then plus 30%. How much fee is paid on Rs 100, where does the client end, and what would a high-water mark have changed?Fee and cost dragHardPrivate banking
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Without a high-water mark, how do the total fees compare with the client's own three-year gain?
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Fees total Rs 11.21 and the client ends at Rs 107.63; a high-water mark would have cut fees to Rs 6 and left him Rs 112.84. Year 1 takes 130 to 124 after a fee of 6. Year 2 falls to 86.8. Year 3 rises to 112.84, and without a mark the manager takes 20% of that 26.04 rise, 5.21. With a mark at 124, the year-3 rise only recovers old ground, so no fee is due.
Why is the client charged twice?
A painter paid per wall painted repaints the same wall after the rain washes it off and bills you again. You have one painted wall and two invoices. Without a high-water mark, the manager is paid on every rise, including rises that only recover ground the client already paid a fee on. The client paid for the climb from 100 to 124 in year 1, lost it in year 2, and paid again for climbing from 86.8 to 112.84, most of which is the same ground.
The fund's NAV rises to 130, pays a fee of 6, falls to 86.8, then rises to 112.84 and pays a second fee of 5.21, ending at 107.63. With a high-water mark at 124 the year-3 rise earns no fee and the client keeps 112.84.
Year Return NAV before fee Fee, no mark NAV after, no mark Fee, with mark
1 +30% 130.00 6.00 124.00 6.00
2 -30% 86.80 0.00 86.80 0.00
3 +30% 112.84 5.21 107.63 0.00
Total 11.21 107.63 6.00
Fees on Rs 100 at 20% of gains. With a high-water mark the year-3 NAV of 112.84 stays below the 124 mark, so no fee is due and the client ends at 112.84.
What does the high-water mark fix, and what does it not?
It fixes double charging: a fee is due only on value above the highest level on which a fee was already paid. Here the mark saves the client 5.21 on Rs 100 and turns a gain of 7.63 into 12.84. It does not stop the manager being paid for a lucky year, and a mark can be reset when money is withdrawn or a new series is opened, so the terms matter. Hurdle rates and fee crystallisation periods are the other clauses worth reading.
Notice also that three years of plus 30, minus 30, plus 30 would compound to only 118.3 even with no fees, because a 30% loss needs a 42.9% gain to recover. The fees then take a large share of a small result.
Where candidates lose it
The common error is computing the fee on the three-year gain, 20% of 18.3, and missing that it is charged each year on each rise. The other is forgetting that the year-1 fee changes the base for years 2 and 3.
Walk the three years in order, state the NAV after each fee, and then run the high-water case. The sentence the interviewer wants is that without a mark the client pays twice for the same ground.
What the interviewer asks next
What if the fee carried a 6% hurdle as well as a high-water mark?
How does a high-water mark change the manager's incentive after a bad year?
The client withdraws half his money after year 2. What happens to his high-water mark?
043A client in an illustrative 30% tax slab holds an equity fund and earns Rs 1 lakh of return in a year. He can take it through the IDCW payout option, taxed at his slab, or leave it in the growth option and pay an illustrative 12.5% capital gains rate when he sells. How much does the option choice cost him?Tax arithmeticCoreMutual fund distributionIndian wealth management
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On Rs 1 lakh of return, how much more tax does the payout option cost him?
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About Rs 17,500 on every Rs 1 lakh of return, before the benefit of deferral. Paid out as IDCW, the Rs 1 lakh is taxed at his illustrative 30% slab: Rs 30,000. Left in the growth option and later sold as a long-term gain, it is taxed at an illustrative 12.5%: Rs 12,500. The return is identical; only its tax label changes. Over ten years on Rs 10 lakh the gap compounds to about Rs 4.27 lakh.
Is the IDCW payout extra money?
No, and that misunderstanding is where most of the damage starts. Taking water out of your own tank does not give you more water. An IDCW payout is paid out of the fund's NAV, so the NAV falls by the amount paid; the client receives part of his own money, and in the payout option that money is taxed as income. The growth option leaves the same return inside the fund, where it is taxed only when units are sold and, if held long enough, at the capital gains rate.
The same Rs 1 lakh of return costs Rs 30,000 in tax in the payout option at an illustrative 30% slab, against Rs 12,500 in the growth option at an illustrative 12.5% gains rate, a difference of Rs 17,500. Over ten years on Rs 10 lakh at 10% the gap compounds to Rs 4.27 lakh.
The relationship
$\Delta = R\,(t_{\text{slab}} - t_{\text{cg}}) = 1{,}00{,}000 \times (0.30 - 0.125) = 17{,}500$
R the return in the year, Rs 1 lakh
t_slab the client's illustrative income tax slab, 30%
t_cg the illustrative long-term capital gains rate, 12.5%
What it says in wordsThe cost of the payout option is the return times the gap between the two tax rates.
Why does the gap grow over time?
Because the payout option is taxed every year, so the reinvested amount compounds at 7% after tax, while the growth option compounds at the full 10% and is taxed once at the end. Rs 10 lakh for ten years becomes about Rs 19.67 lakh in the payout option, even with every payout reinvested, against about Rs 23.95 lakh in the growth option after its tax. The rate gap and the deferral work in the same direction.
The payout option can still suit a client who needs regular cash and sits in a low slab. The rates here are illustrative: slab rates, capital gains rates, holding periods, exemption limits and deduction of tax at source all change, so confirm the current rules before advising on them.
Where candidates lose it
Candidates often treat the payout as income on top of the return, or say the option choice cannot matter because the fund is the same. Both miss that the option changes the tax treatment of an identical return.
Give the Rs 17,500 per Rs 1 lakh, then add deferral as the second effect, and flag that the rates are illustrative. That last line matters on a desk that answers to a compliance team.
What the interviewer asks next
For a client in a 5% slab, which option costs less tax?
How would you set up regular cash for a retiree without using the IDCW option?
Why might a fund's NAV fall sharply on a record date?
044A client holds 10 unrelated stocks. Each has a 5% chance of going to zero this year, independently of the others. What is the chance that at least one of them goes to zero?Probability and risk of lossCoreWealth management
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Pick the closest answer.
Show the worked solution
About 40%. The chance a single stock survives is 95%. The chance all ten survive, if they are independent, is 0.95 to the power 10, which is 0.599. So the chance that at least one goes to zero is 1 minus 0.599, or 40.1%. Small risks that each look negligible add up quickly across a portfolio.
Why work with the chance that nothing goes wrong?
If each of ten wedding vendors has a 5% chance of letting you down, the day goes perfectly only if every one of them turns up. The chance that at least one thing fails is one minus the chance that everything works, and the chance that everything works is the product of each piece working. Adding the 5%s does not work, because it counts the unlucky years where two vendors fail twice.
Each stock added multiplies the chance of no zero by 0.95, so after ten stocks it has fallen to 0.599. The red share above the last bar shows a 40.1% chance that at least one stock goes to zero this year.
The relationship
$P(\text{at least one}) = 1 - (1-p)^{n} = 1 - 0.95^{10} = 0.401$
p the chance any one stock goes to zero, 5%
n the number of independent stocks, 10
What it says in wordsThe chance of at least one failure is one minus the chance that every stock survives.
Does this mean the portfolio is riskier with more stocks?
No, and the distinction is the adviser's real point. The chance of seeing at least one zero rises with the number of stocks, but the damage from each zero shrinks, because each stock is a smaller slice. Ten stocks at 10% each expect 0.5 zeros a year, costing about 5% of the portfolio on average, and the chance of two or more zeros is only 8.6%. With one stock the same 5% chance means a 5% chance of losing everything.
The limit is the word unrelated. Real stocks fail together in a crisis, which makes a year with several zeros more likely than independence suggests, and a year with none also more likely. With 30 stocks at the same odds, the chance of at least one zero rises to 79%, which is why a diversified client should expect to see a disaster in the statement sometimes.
Where candidates lose it
The two fast wrong answers are 5%, ignoring that there are ten chances, and 50%, adding ten 5%s. The second one fails loudly if the interviewer asks about 25 stocks, where addition gives 125%.
Go straight to the complement, give 40%, and then say what it means for the client: expect to see a loser, but diversification caps how much any one loser costs.
What the interviewer asks next
How many such stocks before the chance of at least one zero passes 90%?
What is the chance of exactly one zero among the ten?
How does correlation between the stocks change the answer?
045A retiring employee can take Rs 50 lakh as a lump sum or Rs 40,000 a month for life, starting at age 60. At what age does the pension overtake the lump sum if you ignore the time value of money, and at what age if you discount at 7% a year?Retirement and withdrawalHardIndian wealth management
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Discounting at 7% instead of 0% moves the break-even age by roughly how much?
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About age 70.4 with no discounting, and about age 78.7 at 7%. Rs 50 lakh divided by Rs 40,000 is 125 months, a little over ten years. At 7% a year each later payment is worth less today, and the present value of the pension only reaches Rs 50 lakh after about 225 months. So the pension wins only if the retiree, or whoever it continues to, lives beyond the late seventies.
Why is the undiscounted answer too generous to the pension?
A friend offers you Rs 1,000 today or Rs 100 a month for a year. Even though Rs 1,200 is more, you would think about what the Rs 1,000 could do meanwhile. A rupee of pension received at 75 is worth less than a rupee in hand at 60, because the lump sum could have been invested for those fifteen years. Discounting at the rate the lump sum could earn puts both choices on the same footing.
Counted without discounting, the Rs 40,000 pension overtakes the Rs 50 lakh lump sum at about age 70.4. Discounted at 7% its value rises more slowly and flattens towards Rs 68.6 lakh, passing Rs 50 lakh only at about age 78.7.
The relationship
$50{,}00{,}000 = 40{,}000 \cdot \frac{1-(1+i)^{-n}}{i}, \quad i = \frac{0.07}{12} \;\Rightarrow\; n \approx 225 \text{ months}$
i the monthly discount rate, 7% a year divided by 12
n the number of monthly payments needed for the pension's present value to equal the lump sum
What it says in wordsThe break-even is the number of payments whose value today adds up to the lump sum.
What decides the choice beyond the break-even age?
Three things the formula does not see. Longevity: the pension is insurance against living long, which is exactly the risk a lump sum cannot cover; a life table for the client's age and health says how likely age 79 is. The rate matters most: the higher the return the client could earn on the lump sum, the later the break-even, and at 10% a year the pension's present value never reaches Rs 50 lakh at all, however long he lives. Then the details: whether the pension continues to a spouse, whether it rises with inflation, and how each option is taxed.
The ceiling explains the last point. At 7% the whole infinite stream is worth Rs 40,000 / (0.07 / 12), about Rs 68.6 lakh; at 10% it is worth Rs 48 lakh, less than the lump sum. Check the payer's strength too: a pension is only as good as the promise behind it.
Where candidates lose it
The first trap is stopping at 125 months and age 70, which ignores that the lump sum could earn a return. The interviewer asked for both answers precisely to see if you can explain why they differ by eight years.
The second trap is treating the break-even as the decision. The pension is longevity insurance; the right answer names the break-even, the rate it depends on, and the client facts that tip it.
What the interviewer asks next
At what discount rate does the pension never break even?
How does a 50% spouse continuation change your view?
If the pension rose 3% a year, would the break-even age move earlier or later?
046A bullet portfolio holds 5-year zero-coupon bonds. A barbell holds equal amounts of 1-year and 9-year zeros, so both have a duration of 5 years at a 7% yield. If yields jump sharply, up or down, which portfolio comes out ahead, and why?Fixed income numeracyHardPrivate banking
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Yields move 4 points in parallel, either way. Which portfolio does better?
Show the worked solution
The barbell, in both directions, because it has more convexity. Equal duration makes the two portfolios move alike for small changes in yield. For large moves the price curve's bend matters, and a barbell's is sharper: convexity of about 40 against 26. On Rs 100, a 4-point fall in yields leaves the barbell 1.41 ahead; a 4-point rise leaves it 0.90 ahead.
Why does equal duration not mean equal behaviour?
Two cars can be doing 60 at this instant and still be in different places a minute from now if one is accelerating. Duration is the speed at the current yield; convexity is how that speed changes as yields move, and it decides who is ahead after a large move. A 9-year zero's price bends much more sharply than a 5-year zero's, far more than twice as much, while the 1-year zero barely bends at all, so the barbell's mix bends more than the bullet.
The bullet and barbell price curves both pass through 100 at 7% and nearly overlap, but the barbell's lead is positive for every move: +0.31 and +0.25 for 2-point moves, rising to +1.41 and +0.90 for 4-point moves.
Yield move Bullet value Barbell value Barbell lead
-4 points 120.99 122.39 +1.41
-2 points 109.89 110.21 +0.31
none 100.00 100.00 +0.00
+2 points 91.16 91.41 +0.25
+4 points 83.23 84.13 +0.90
Rs 100 in each portfolio of zero-coupon bonds, flat 7% yield curve, annual compounding, parallel moves.
If the barbell always wins, what is the catch?
Two catches. First, markets price convexity, so a barbell with the same duration usually yields a little less, and if yields sit still the bullet quietly earns more. Second, the edge assumes a parallel move. If short yields fall 1 point while 9-year yields rise 1 point, with the 5-year unchanged, the bullet is untouched and the barbell loses about Rs 3.54 per Rs 100. A barbell is a bet on large, parallel moves; a bullet is a bet on stability or on the curve's shape changing.
For a client portfolio, the practical reading is that duration alone does not describe interest rate risk. Both portfolios here have a modified duration of about 4.67, and a manager who reports only that number has hidden the shape of the bet.
Where candidates lose it
The common answer is that equal duration means equal behaviour, which is true only for small moves. Others say the barbell wins when rates fall and loses when they rise, treating the long bond as the only thing that matters.
Say convexity, give the direction, both ways, and then name the cost: a lower yield if nothing moves and a loss if the curve twists. The interviewer wants to hear that there is no free lunch.
What the interviewer asks next
Why do long-dated bonds have so much more convexity than short ones?
What happens to the barbell if the yield curve steepens?
How would you explain convexity to a client who holds only fixed deposits?
047The index has fallen five days in a row and a client says it is now due to rise. Suppose each day is independent and a down day has a 48% chance. How many losing runs of five or more days should you expect in a 250-day year, and what does the streak say about tomorrow?Behavioural trapsHardWealth management
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After five down days in a row, what is the chance tomorrow is an up day, under these assumptions?
Show the worked solution
About 3.3 losing runs of five or more days a year, and the streak says nothing about tomorrow. A run of five falls has probability 0.48 to the power 5, about 2.55%, and it can start on almost any of the 250 days, provided the day before was up. That gives roughly 3.3 runs a year, and a 97% chance of at least one. With independent days, tomorrow's chance of rising is still 52%.
How often should a five-day losing run turn up?
Toss a slightly unfair coin 250 times and look for five tails in a row. Any one stretch of five is unlikely, 2.55%, but there are about 245 stretches to look at. A run of five down days needs an up day, or the start of the year, followed by five falls, so the expected count is about 0.0255 x (1 + 245 x 0.52), roughly 3.3 runs a year. Across a year, the chance of seeing at least one is about 97%.
The relationship
$E[\text{runs}] = p^{5}\left[1 + (n-5)(1-p)\right] = 0.48^{5}\,[1 + 245 \times 0.52] \approx 3.3$
p the chance of a down day, 48%
n trading days in the year, 250
(1-p) the up day that must come just before a run for it to be a new run
What it says in wordsCount every day a new five-day losing run could start, and multiply by the chance it does.
In one simulated year of 250 independent days with a 48% chance of a fall, three losing runs of five or more days appear, of 7, 5, 5 days. Independence alone predicts about 3.3 such runs a year, and none of them tells you anything about the next day.
Why does the client feel the market is due?
Because people expect short sequences to look like long-run averages, so a run of losses feels like a debt the market must repay. If the days are independent, the market keeps no ledger: the chance of a rise after five falls is the same 52% as after five rises. The adviser's job is to take the streak out of the decision and bring the conversation back to the client's plan and time horizon.
The limit is the independence assumption. Real markets show some short-term momentum and some mean reversion at different horizons, and volatility clusters, so streaks are a little more common than a coin predicts. None of that makes five falls a reliable signal to buy.
Where candidates lose it
One trap is agreeing with the client that the market is due, which is the gambler's fallacy with a market label. The other is calling a five-day run rare because 0.48 to the fifth is small, forgetting how many days it has to appear on.
Give the expected count, about three a year, and the 52% for tomorrow. Then say how you would steer the client back to his plan, because that is what the desk actually does with the maths.
What the interviewer asks next
How many runs of ten or more down days would you expect in a year?
What would you look for in the data before believing streaks carry information?
The client wants to add money after every three-day fall. How do you respond?

Year	Return	NAV before fee	Fee, no mark	NAV after, no mark	Fee, with mark
1	+30%	130.00	6.00	124.00	6.00
2	-30%	86.80	0.00	86.80	0.00
3	+30%	112.84	5.21	107.63	0.00
Total			11.21	107.63	6.00

Yield move	Bullet value	Barbell value	Barbell lead
-4 points	120.99	122.39	+1.41
-2 points	109.89	110.21	+0.31
none	100.00	100.00	+0.00
+2 points	91.16	91.41	+0.25
+4 points	83.23	84.13	+0.90

048Estimate how many Indian startup founders and early employees receive a liquidity event of more than Rs 25 crore each in a year, from funding rounds, exits and buybacks.Estimation and sizingHardIndian wealth management

Try it first

Which branch of the estimate is most likely to decide whether you land in the hundreds or the thousands?

Show the worked solution

A few hundred people a year; the build here gives about 477, with a sensible range of 250 to 700. Split the events into four branches: secondary sales in late-stage rounds, acquisitions, IPOs once lock-ins end, and ESOP buybacks. For each, multiply events a year by the share that pays out and by the people above Rs 25 crore per event. Illustrative assumptions give 530, less about 10% for people counted twice.

How do you structure an estimate with no data in front of you?

Estimating how many weddings a city hosts, you would not guess one number; you would split by season, by venue type and by guests per wedding. A market-sizing answer is a tree: each branch is a separate route to the answer, and each carries its own named assumption that the interviewer can challenge. Here the routes are the four ways a startup stake turns into cash: selling in a later funding round, the company being bought, the company listing, and the company buying back employee options.

Four branches, each an assumed number of events times the share that pays out times the people above Rs 25 crore per event, add to 530; removing about 10% counted twice gives about 477. Every input is an illustrative assumption to be replaced with current data.

Branch	Events a year (assumed)	Share that pays out	People above Rs 25 crore per event	People
Secondary sales in late-stage rounds	400	25%	2	200
Acquisitions	200	15%	3	90
IPOs, once lock-ins end	20	100%	10	200
ESOP buybacks	40	100%	1	40
Total, before overlap				530

All counts and shares are illustrative assumptions chosen to show the structure, not data; a real answer takes current figures from a funding tracker and exchange filings.

Which assumptions would you defend, and which would you flag?

Flag the multipliers first. The number of people per IPO or secondary round who clear Rs 25 crore moves the total more than any event count, so say it is the weakest link. Then say what you would check: listing counts and lock-in dates, round sizes with a secondary component, and buyback announcements. Also name the double counting: a founder who sells in a secondary this year may list next year, and the same person can appear in two branches, which is why the build takes off about 10%.

Close with why a wealth desk asks this. A liquidity event is the moment new wealth appears and needs managing, and the few hundred people it creates each year are exactly the clients a private wealth team competes for. The limit: the answer swings with the funding cycle, so a boom year and a lean year can differ several times over.

Where candidates lose it

The trap is pulling a single number from memory and defending it, or refusing to answer because the data is not public. The interviewer is scoring the tree, the named assumptions and the sanity check, not the figure.

The quieter loss is ignoring double counting and the funding cycle. One sentence on each shows you know an estimate is a range with a known weak link, not a fact.

What the interviewer asks next

How would your estimate change in a year when late-stage funding halves?
How many of these people would a single private bank realistically win?
What share of the paid-out money do you think ends up in managed portfolios, and how would you estimate it?

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
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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?
050A Rs 20 crore client relationship earns the bank 0.8% a year in revenue. The assets grow 8% a year, there is a 10% chance each year that the client leaves, and the bank discounts at 12%. Roughly what is the relationship worth to the bank today?Wealth business economicsHardPrivate banking
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Which denominator turns the Rs 16 lakh of first-year revenue into a lifetime value?
Show the worked solution
About Rs 1.1 crore. First-year revenue is 0.8% of Rs 20 crore, Rs 16 lakh. Attrition works like extra discounting and asset growth offsets it, so the shortcut divides by 12% + 10% - 8% = 14%: Rs 16 lakh / 0.14 is about Rs 1.14 crore. Summing year by year, where growth and survival multiply rather than add, gives Rs 1.08 crore. Both round to Rs 1.1 crore.
Why does attrition belong in the discount rate?
A shopkeeper values a regular customer by the purchases he expects, but also by how likely the customer is to keep coming. A 10% chance of losing the client each year shrinks every future year's expected revenue by a further 10%, exactly as if the discount rate were 10 points higher; asset growth pushes the other way. So a growing, leaky revenue stream is valued like a perpetuity at the discount rate plus attrition minus growth.
The relationship
$V \approx \frac{R_1}{r + a - g} = \frac{16}{0.12 + 0.10 - 0.08} = 114.3 \text{ lakh} \qquad V_{\text{exact}} = \frac{R_1/(1+r)}{1 - \frac{(1+g)(1-a)}{1+r}} = 108.1 \text{ lakh}$
R_1 first-year revenue, Rs 16 lakh
r the bank's discount rate, 12%
a the yearly attrition rate, 10%
g the yearly growth of the client's assets, 8%
What it says in wordsA relationship is worth its first-year revenue divided by the discount rate plus attrition less growth; the exact annual sum is a little lower.
Revenue of Rs 16 lakh growing 8% a year is thinned by 10% annual attrition and discounted at 12%, so each year's present value is smaller than the last. The discounted bars add to about Rs 1.08 crore against Rs 1.14 crore from the shortcut, and the first ten years carry 76% of the value.
Why do the shortcut and the annual sum differ, and what moves the answer most?
The shortcut adds the rates, which is exact only for continuous compounding; with annual steps, growth and survival multiply, 1.08 x 0.90 = 0.972 rather than 0.98, so the yearly sum is about 5% lower. Neither is wrong; say which you used. The lever is attrition: cutting it from 10% to 5% lifts the shortcut value from Rs 1.14 crore to Rs 1.78 crore, because the denominator falls from 14% to 9%. That is the arithmetic behind a private bank's spending on service and retention.
The limits: this values revenue, not profit, so the relationship manager's cost and the platform's cost must come off before anyone calls it value. Growth above the discount rate less attrition would make the formula break down, which is a warning that the assumptions, not the client, have become unrealistic.
Where candidates lose it
The first trap is dividing Rs 16 lakh by 12% and calling the relationship worth Rs 1.33 crore, forgetting that clients leave and assets grow. The second is subtracting attrition instead of adding it, which inflates the answer several times.
Say the denominator in words, discount plus attrition minus growth, before any number. Then note that the annual sum is slightly lower and that attrition is the lever the business can pull.
What the interviewer asks next
What is the relationship worth if attrition falls to 5%?
How much would the bank rationally spend to win this client?
Why should the calculation use contribution after the relationship manager's cost, not revenue?

A	the fixed amount invested each month, Rs 100
P_i	the NAV in month i
\bar{c}	the average cost per unit, a harmonic mean of the prices

R	the return in the year, Rs 1 lakh
t_slab	the client's illustrative income tax slab, 30%
t_cg	the illustrative long-term capital gains rate, 12.5%

p	the chance any one stock goes to zero, 5%
n	the number of independent stocks, 10

i	the monthly discount rate, 7% a year divided by 12
n	the number of monthly payments needed for the pension's present value to equal the lump sum

Try it first

Why is his cost below the average price?

Does this mean a SIP beats investing a lump sum?

Where candidates lose it

What the interviewer asks next

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Why is the client charged twice?

What does the high-water mark fix, and what does it not?

Where candidates lose it

What the interviewer asks next

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Is the IDCW payout extra money?

Why does the gap grow over time?

Where candidates lose it

What the interviewer asks next

Try it first

Why work with the chance that nothing goes wrong?

Does this mean the portfolio is riskier with more stocks?

Where candidates lose it

What the interviewer asks next

Try it first

Why is the undiscounted answer too generous to the pension?

What decides the choice beyond the break-even age?

Where candidates lose it

What the interviewer asks next

Try it first

Why does equal duration not mean equal behaviour?

If the barbell always wins, what is the catch?

Where candidates lose it

What the interviewer asks next

Try it first

How often should a five-day losing run turn up?

Why does the client feel the market is due?

Where candidates lose it

What the interviewer asks next

Try it first

How do you structure an estimate with no data in front of you?

Which assumptions would you defend, and which would you flag?

Where candidates lose it

What the interviewer asks next

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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

Why does attrition belong in the discount rate?

Why do the shortcut and the annual sum differ, and what moves the answer most?

Where candidates lose it

What the interviewer asks next