Private Wealth Management puzzles, solved step by step

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

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013A client with Rs 5 crore can pay an adviser a flat 1% advisory fee on all his assets, or use a distributor who earns a 1.2% trail commission on the 70% of his money held in mutual funds. Which costs him less each year?Wealth business economicsWarm upIndian wealth management
Try it first
Which annual bill is smaller, in rupees?
Show the worked solution
The commission route costs less on these numbers: Rs 4.2 lakh a year against Rs 5.0 lakh. The advisory fee is 1% of the whole Rs 5 crore. The trail is 1.2% of only the Rs 3.5 crore held in funds. The two bills would match if 83.3% of his assets earned trail, and the cheaper bill says nothing yet about the quality or independence of the advice.
Why does the higher rate give the smaller bill?
A phone plan charging Rs 2 a minute on calls only can cost less than one charging Rs 1.50 a minute on calls and data together, if you rarely use data. A fee rate means nothing until it is multiplied by the base it is charged on, so fee models are compared in rupees on the client's actual asset mix. Here 1.2% applies to Rs 3.5 crore and 1% applies to Rs 5 crore.
On a Rs 5 crore portfolio the 1% advisory fee is a Rs 5.0 lakh bill, while a 1.2% trail on the Rs 3.5 crore in funds is Rs 4.2 lakh, and the two only match when 83.3% of assets pay trail.
What else should the comparison include?
Visibility and incentives. A trail is paid out of the fund's expense ratio, so the client never sees it as a bill, while an advisory fee arrives as an invoice he has to approve. The trail also pays more when more money sits in trail-paying funds, which is a pull away from direct equity, bonds or direct plansVersions of a mutual fund scheme bought without a distributor, with a lower expense ratio because no commission is paid out of them.. If the distributor moved another Rs 1 crore into funds, the trail bill would rise to Rs 5.4 lakh.
The relationship
$\text{Fee} = 0.01 \times 5 = \text{Rs } 5.0\text{ lakh} \qquad \text{Trail} = 0.012 \times 0.70 \times 5 = \text{Rs } 4.2\text{ lakh}$
5 the client's assets, Rs crore; 0.01 of a crore is Rs 1 lakh
0.70 the share of assets held in trail-paying funds
0.012 the trail rate a year
What it says in wordsMultiply each rate by the assets it is charged on, then compare rupees.
Add the regulatory point as a framework, not a fact from memory: Indian rules separate registered advisers who charge fees from distributors who earn commissions, and limit doing both for the same client. The current regulations need checking before any of this reaches a client.
Where candidates lose it
The trap is comparing 1% with 1.2% and declaring the fee cheaper. The rates apply to different bases, and the rupee bill is the only honest comparison.
The second loss is stopping at the cheaper bill. The interviewer wants to hear that the trail is invisible and that it rewards keeping money in funds, because that is how the two models shape advice differently.
What the interviewer asks next
At what share of assets in funds do the two models cost the same?
The distributor proposes moving the other Rs 1.5 crore into funds. What happens to his income and the client's bill?
How would you explain the invisible trail to a client in one sentence?
025A private bank charges 1% a year on the first Rs 10 crore of a client's assets, 0.6% on the next Rs 20 crore and 0.4% on everything above Rs 30 crore. What is the blended fee rate on a Rs 45 crore client?Wealth business economicsCorePrivate banking
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Pick the blended rate on Rs 45 crore.
Show the worked solution
About 0.62%. Each slice pays its own rate: 1% on the first Rs 10 crore is Rs 10 lakh, 0.6% on the next Rs 20 crore is Rs 12 lakh, and 0.4% on the last Rs 15 crore is Rs 6 lakh. The total is Rs 28 lakh, and Rs 28 lakh divided by Rs 45 crore is 0.622%. The blend falls as the client grows but never reaches 0.4%.
Why is the answer not 0.4%?
Income tax slabs work the same way. Someone whose income crosses into a higher slab pays the higher rate only on the part above the threshold, not on the whole income. A tiered fee charges each slice of assets at its own rate, so the top rate applies only to the top slice and the blended rate sits between the highest and lowest tiers. The first Rs 30 crore of this client's money is charged exactly as it would be for a Rs 30 crore client.
Each tier's area is its fee: Rs 10 lakh on the first Rs 10 crore, Rs 12 lakh on the next Rs 20 crore and Rs 6 lakh on the last Rs 15 crore, so Rs 28 lakh on Rs 45 crore blends to 0.62%.
What does the blend do as the client grows?
It drifts down toward the top tier's rate without reaching it. At Rs 30 crore the blend is 22 over 3,000, about 0.73%; at Rs 45 crore it is 0.62%; at Rs 100 crore it would be 50 over 10,000, 0.50%. For the bank this is the cost of winning large clients: revenue grows more slowly than assets, which is why the revenue marginRevenue divided by assets under management for a book or a whole business, the per-rupee earning rate of the wealth franchise. of a book tells you about its client mix.
The relationship
$\text{Blend} = \frac{0.01 \times 10 + 0.006 \times 20 + 0.004 \times 15}{45} = \frac{0.28}{45} = 0.622\%$
10, 20, 15 the rupee crore in each tier
0.01, 0.006, 0.004 each tier's fee rate
0.28 the total fee in Rs crore, Rs 28 lakh
What it says in wordsAdd the fee on each slice, then divide by the total assets.
Mention the practical point a banker would. Clients compare the headline top-tier rate across banks, while the bank's income depends on the blend, and moving assets across tiers or between family accounts can change which slices apply. Fee schedules are negotiated, so the tiers here are an illustration.
Where candidates lose it
The trap is 0.4%: applying the top-tier rate to the whole balance, as if the tiers were price bands rather than slabs. It understates the fee by more than a third.
The second trap is averaging the three rates to 0.67%. It ignores that the slices are different sizes, the same error as averaging averages.
What the interviewer asks next
What is the blended rate on a Rs 25 crore client?
At what asset level does the blended rate fall to 0.5%?
The client splits Rs 45 crore across two family accounts of Rs 22.5 crore each. What happens to his total fee?
039A relationship manager looks after 80 client families and loses 15% of them each year to moves, deaths and competitors. How many new families must he win every year just to keep his book the same size?Wealth business economicsWarm upWealth management
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Quick: how many new families a year keep the book at 80?
Show the worked solution
12 new families a year. Losing 15% of 80 families means 12 walk out each year, so 12 must come in to keep the book at 80. The general rule: a book settles where new families won equals the attrition rate times the book. Win only 8 a year and the book drifts toward 8 / 0.15, about 53 families; it does not collapse, but it shrinks by a third.
Why does attrition set the minimum pace?
A water tank with a leak at the bottom stays level only if the tap fills it as fast as it drains. A client book is the same tank: attrition drains a fixed share each year, so the new families needed are the leak rate times the size of the book. For 80 families at 15% that is 12, which is more than one new relationship a month before the adviser has grown at all.
At 15% attrition an 80-family book loses 12 families a year, so winning 12 keeps it flat. Winning 8 lets it slide to about 59 in ten years on the way to 53, while winning 16 lifts it toward 107.
The relationship
$N^{*} = \frac{\text{new per year}}{a} \qquad \frac{12}{0.15} = 80, \quad \frac{8}{0.15} = 53.3$
N* the book size where gains and losses balance
a the yearly attrition rate, 15%
What it says in wordsA book settles at the size where the families lost each year exactly equal the families won.
What does the number tell a wealth business?
Two things. First, the average family stays about 1 / 0.15, roughly 6.7 years, which caps what each relationship is worth. Cutting attrition from 15% to 10% lowers the new families needed from 12 to 8 a year, the same effect on book size as a 50% jump in new-client wins. That is why wealth firms spend so much on retention and service: it is usually cheaper to keep a family than to find one.
The limitation: families are not equal. Losing a large family and winning a small one keeps the count flat and shrinks the assets. A proper review runs the same arithmetic on assets and revenue, not just on heads.
Where candidates lose it
The slip is answering 15, reading the 15% as a count, or answering zero, as if a good adviser loses nobody. Neither survives a follow-up about how the book behaves over time.
Give 12, then show you see the steady state: the book settles where wins equal attrition times the book. That turns a percentage question into a business insight about retention.
What the interviewer asks next
How many years does the average family stay at 15% attrition?
If attrition drops to 10%, where does a book winning 12 families a year settle?
Why might a relationship manager's assets shrink even while his family count holds steady?
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
Try it first
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?
089A relationship manager's book grew from Rs 400 crore to Rs 500 crore over a year in which his clients' portfolios rose 12%. How much net new money did he bring in?Wealth business economicsWarm upWealth management
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How much of the Rs 100 crore growth is new money?
Show the worked solution
About Rs 52 crore. The market alone would have taken the book from Rs 400 crore to Rs 448 crore, 12% of 400 being Rs 48 crore. The rest of the growth to Rs 500 crore, Rs 52 crore, is net new money: fresh money in less withdrawals. If the new money arrived through the year and earned some of the rise, the figure is nearer Rs 49 crore.
Why separate the market from the money raised?
A shop's sales rise 25% in a year when prices across the market rose 12%. Most of that rise is inflation, not new customers. Assets under management grow from two sources, the market and net new money, and only the second measures what the adviser did. A banker who quotes 25% growth in a year when markets rose 12% is quoting a number he did not earn in full.
The book of Rs 400 crore gains Rs 48 crore from a 12% market and Rs 52 crore of net new money to close at Rs 500 crore, so only about half of the Rs 100 crore growth came from the adviser.
The relationship
$NNM = 500 - 400 \times 1.12 = 500 - 448 = 52$
NNM net new money: new client money in, less money withdrawn
400 x 1.12 where the opening book would be from market moves alone
What it says in wordsNet new money is the closing book less what the opening book would have grown to on its own.
When is Rs 52 crore not quite right?
The simple version assumes the new money arrived at the year end, so it earned nothing. If new money came in steadily, part of the market's rise was earned on it, so the true net new money is a little lower than the simple subtraction. With money arriving evenly and earning about half the year's 12%, it is 52 divided by 1.06, about Rs 49 crore.
Say also what the figure hides: net new money is inflows less outflows, so Rs 52 crore could be Rs 80 crore raised and Rs 28 crore lost to clients who left. A manager asking this question usually wants both halves.
Where candidates lose it
The common slip is calling all Rs 100 crore new money, or taking 12% off the closing figure instead of the opening one. The market acts on money the adviser already had, so the 12% applies to Rs 400 crore.
The second loss is missing the timing caveat. Give 52, then say it is an upper figure if the money arrived through the year.
What the interviewer asks next
The market fell 8% and the book still grew to Rs 420 crore. What was net new money?
Why do wealth firms report net new money separately from assets?
How would you split net new money into new clients and existing clients?
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?

5	the client's assets, Rs crore; 0.01 of a crore is Rs 1 lakh
0.70	the share of assets held in trail-paying funds
0.012	the trail rate a year

10, 20, 15	the rupee crore in each tier
0.01, 0.006, 0.004	each tier's fee rate
0.28	the total fee in Rs crore, Rs 28 lakh

N*	the book size where gains and losses balance
a	the yearly attrition rate, 15%

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%

Try it first

Why does the higher rate give the smaller bill?

What else should the comparison include?

Where candidates lose it

What the interviewer asks next

Try it first

Why is the answer not 0.4%?

What does the blend do as the client grows?

Where candidates lose it

What the interviewer asks next

Try it first

Why does attrition set the minimum pace?

What does the number tell a wealth business?

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

Try it first

Why separate the market from the money raised?

When is Rs 52 crore not quite right?

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