Private Wealth Management puzzles, solved step by step

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

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003A portfolio earns 10% a year before fees for 30 years. The client pays an annual fee of 1%. Roughly what share of his final wealth does the fee cost him?Fee and cost dragCoreWealth management
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Pick the share of final wealth the 1% fee takes.
Show the worked solution
About a quarter: 24% of the final wealth. At 10% a year, Rs 1 grows to 17.45 over 30 years. Net of a 1% fee it grows at 9% to 13.27. The gap of 4.18 is 24% of the gross figure. A fee that looks like one tenth of the return costs a quarter of the wealth because every rupee paid misses its future compounding.
Why does a 1% fee cost so much more than 1%?
Picture a small leak in a water tank that is also being filled faster every year. The leak takes a little each day, but the water it takes is water that would have been there to hold more tomorrow. A fee removes not only the rupee paid but everything that rupee would have earned for the rest of the plan. Over 30 years the growth rate falls from 10% to 9%, and the difference between the two compounding paths widens every single year.
Rs 1 grows to 17.4 at 10% a year over 30 years but only to 13.3 at 9% after a 1% fee, so the fee takes 24% of the final wealth, and the gap between the curves widens every year.
How do you estimate the share in your head?
Use the ratio of the two growth factors. Each year the net portfolio is 1.09 over 1.10, about 0.991, of the gross. Over 30 years that ratio compounds to roughly 0.991 to the 30th, near 0.76. A shortcut that is close enough: the share lost is about the fee times the number of years, less a little, here 30% shading down to about 24%. The shortcut breaks for very long horizons, so give the exact figure when you can.
The relationship
$\text{share lost} = 1 - \left(\frac{1.09}{1.10}\right)^{30} \approx 1 - 0.760 = 24\%$
1.10 gross growth factor, 10% a year
1.09 growth factor after a 1% fee taken off the return
30 years invested
What it says in wordsThe share of final wealth lost is one minus the net path divided by the gross path.
State the convention. Here the fee is taken as one point off the return. If it is charged as 1% of the year-end value, the net factor is 1.10 times 0.99 and the share lost is 26%. Either way the answer is about a quarter, and the limitation is worth saying: the maths treats the 10% as certain, while the fee is paid in bad years too.
Where candidates lose it
The fast wrong answer is 1%, or 10% of the gain. It treats the fee as a one-off slice rather than a rate that compounds. Clients make this mistake all the time, and an adviser who repeats it sounds as though they have never shown a client a fee illustration.
The second loss is giving 24% without saying how the fee is charged. Say the convention in one sentence and note that the other convention lands close by.
What the interviewer asks next
What share does a 2% fee cost over the same 30 years?
Over 10 years instead of 30, what share does the 1% fee take?
How would you show this to a client who says 1% is nothing?
016A portfolio management service charges a 2% fixed fee plus 20% of returns above an 8% hurdle. In a year when the gross return is 15% on Rs 1 crore, what is the client's net return, and what share of the gross gain went in fees?Fee and cost dragHardIndian wealth management
Try it first
Pick the client's net return, with the performance fee charged on the return after the fixed fee.
Show the worked solution
The client nets 12%, and fees take 20% of the gross gain. The 2% fixed fee brings 15% down to 13%. That is 5 points above the 8% hurdle, and 20% of 5 points is a 1% performance fee. Net return is 12%, Rs 12 lakh on Rs 1 crore. Fees are Rs 3 lakh of the Rs 15 lakh gross gain. The exact order of fees is set by the agreement.
Why do the two fees stack rather than sit side by side?
Think of a restaurant bill with a fixed cover charge and then a service charge worked out on what is left. Each charge is modest on its own, but together they take a real share of the meal. The fixed fee is paid in every year, good or bad, and the performance fee then takes a slice of whatever is left above the hurdle, so in a good year the two stack. Here 2 points and 1 point together take a fifth of a 15% year.
On Rs 1 crore a 15% gross year loses 2 points to the fixed fee and 1 point to the performance fee on the 5 points above the 8% hurdle, leaving the client 12%, so fees take Rs 3 lakh of the Rs 15 lakh gain.
What changes if the agreement charges the performance fee on the gross return?
Then the hurdle is measured before the fixed fee. The performance fee becomes 20% of 7 points, 1.4%, the net return falls to 11.6%, and fees take 22.7% of the gain instead of 20%. The same headline terms give two answers, which is why the order of calculation, the high-water markA rule that performance fees are only paid on gains above the highest value the account has previously reached, so the client does not pay twice for recovering a loss. and the basis for the fixed fee all have to be read in the agreement.
On Rs 1 crore Fee after fixed fee Fee on gross
Gross gain, Rs lakh 15.0 15.0
Fixed fee, 2% (2.0) (2.0)
Performance fee, 20% above 8% (1.0) (1.4)
Net gain, Rs lakh 12.0 11.6
Share of gross gain paid in fees 20.0% 22.7%
Charging the performance fee after the fixed fee leaves the client Rs 12.0 lakh; charging it on the gross return leaves Rs 11.6 lakh, so the same headline terms differ by Rs 40,000 on Rs 1 crore.
One more layer is worth a sentence: fees usually attract GST in India, charged on top, which widens the gap further. Confirm the current rate and treatment before quoting a client a net figure.
Where candidates lose it
The trap is taking the performance fee as 20% of the whole 15% gross return, or forgetting the fixed fee comes out first. Both give a wrong net figure with total confidence.
The quieter loss is quoting 12% without the convention. Say that you have assumed the performance fee is charged after the fixed fee, and give the other answer, 11.6%, in the same breath.
What the interviewer asks next
What is the net return in a year when the gross return is 8%?
At what gross return do fees take exactly a quarter of the gain?
How does a high-water mark change the fee in the year after a loss?
029Product X takes a 3% commission upfront and nothing after. Product Y takes no upfront fee but a 1% trail every year on the value of the investment. Ignoring compounding, after how many years has Y cost the client more? And where does compounding move that break-even?Fee and cost dragCoreMutual fund distributionIndian wealth management
Try it first
Once you let the fees compound, what happens to the 3-year break-even?
Show the worked solution
Simple addition says 3 years; with compounding it is 3.03 years, almost unchanged. Three years of 1% equals the 3% upfront. Compounded, the trail leaves the client 0.99 to the power t of his no-fee wealth and the upfront leaves 0.97 of it forever. They meet where 0.99^t = 0.97, at 3.03 years. The market return cancels out, because both fees are a share of the same growing balance.
Why does the market return drop out?
A shopkeeper can take his cut as one slice of the cake at the start or as a thin slice every year. However much the cake rises, each fee is a fraction of whatever cake there is. An upfront fee leaves the client 97% of the wealth he would otherwise have had, for ever; a 1% trail leaves him 99% after one year, 98.01% after two, and 0.99^t after t years. The growth rate multiplies both sides equally, so it cancels when you compare them.
The relationship
$0.99^{\,t} = 0.97 \;\Rightarrow\; t = \frac{\ln 0.97}{\ln 0.99} = 3.03 \text{ years}$
0.99^t the share of no-fee wealth left after t years of a 1% trail
0.97 the share left after a one-time 3% upfront fee
What it says in wordsThe trail costs more once its compounded bite exceeds the one-time bite of the upfront fee.
The upfront fee costs a flat 3% of the client's no-fee wealth for ever, while the trail's cost rises from 1% after one year to 2.97% after three and crosses the flat line at 3.03 years. Counting rupees of trail paid instead, the dashed line, crosses early at about 2.5 years, which is the misleading version.
Where do candidates go wrong when they try to add compounding?
They count rupees. On Rs 100 in a market rising 10% a year, the trail rupees add up to Rs 3 after only about 2.5 years, because 1% of a growing balance is more rupees each year. Rupees paid at different dates are not comparable; the right yardstick is how much wealth the client ends with. On that yardstick the upfront fee also cost him the growth on the 3% that never got invested, and the two effects cancel almost exactly.
The practical reading for a distributor conversation: for a holding period under about three years the trail is cheaper; beyond it, the upfront structure is. The limit is that real products often combine both, and exit loads, switch costs and the quality of ongoing service are not in this sum.
Where candidates lose it
The first trap is stopping at 3 years when the interviewer explicitly asks about compounding. The second, more common, is claiming compounding makes the trail much more expensive much sooner, which comes from adding rupees paid in different years as if they were the same money.
Frame both fees as a share of the no-fee outcome. Then the answer is one line of logarithms, and saying that the market return cancels is the insight the question is fishing for.
What the interviewer asks next
What is the break-even if the upfront fee is 5% and the trail is 1%?
How does an exit load in year 1 change the comparison?
Why might a client rationally prefer the trail even for a long holding?
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
Try it first
Without a high-water mark, how do the total fees compare with the client's own three-year gain?
Show the worked solution
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?
055Rs 50 lakh sits in the regular plan of a mutual fund, which charges 0.8% a year more than the direct plan of the same fund. If the direct plan compounds at 11% a year after its own costs, what is the gap in ending wealth after 20 years?Fee and cost dragWarm upMutual fund distributionIndian wealth management
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Roughly how large is the gap after 20 years?
Show the worked solution
About Rs 54 lakh. At 11% the direct plan grows Rs 50 lakh to about Rs 403.1 lakh in 20 years. At 10.2% the regular plan reaches about Rs 348.8 lakh. The gap is Rs 54.3 lakh, 13.5% of the direct plan's ending value, from a cost difference that sounds like less than one per cent.
Why is the gap so much larger than 0.8% sounds?
A slow leak in a water tank loses a little each hour, but it keeps leaking from a tank that is also being filled, and every litre lost is a litre that never reaches the tap. An annual fee is charged on the whole growing balance every year, and each rupee taken stops compounding for the rest of the period. Charged only on the starting Rs 50 lakh, 0.8% for 20 years would be Rs 8 lakh. Charged on a balance that grows several times over, it is Rs 54 lakh.
Rs 50 lakh in the direct plan at 11% and the regular plan at 10.2% ends Rs 3.0 lakh apart after 5 years, Rs 9.9 lakh after 10 and Rs 54.3 lakh after 20, because the fee gap compounds along with the money.
How do you estimate it without a calculator?
Use the ratio of the two growth factors. Each year the regular plan keeps 1.102 / 1.11 of what the direct plan keeps, about 0.9928, so it loses roughly 0.72% of the direct plan's value a year. Over 20 years that compounds to about 1 minus 0.9928 to the 20th, roughly 13.5% of the ending value. The direct plan ends near Rs 400 lakh, so the gap is a little over Rs 50 lakh. Two lines of arithmetic, and the answer is within a lakh or two.
The relationship
$\text{gap} = 50\,(1.11^{20} - 1.102^{20}) \approx 54.3\ \text{lakh}$
50 the investment, Rs lakh
1.11 one year of growth in the direct plan
1.102 one year of growth in the regular plan, 0.8% lower
What it says in wordsThe gap is the same money compounded at two rates for 20 years, subtracted.
Be fair to the regular plan when you say this to a client. The extra cost pays the distributor, and some investors value the advice and service they get for it. The numeracy point is only that the price of that service, over 20 years, is measured in tens of lakhs, and the client should see it in rupees before deciding.
Where candidates lose it
The fast wrong answer is 0.8% x Rs 50 lakh x 20 years, Rs 8 lakh. It charges the fee only on the starting amount and ignores that each rupee of fee also loses its future growth.
The second miss is presenting the gap as proof that one plan is bad. The interviewer wants the number and a fair sentence on what the extra cost buys.
What the interviewer asks next
What share of the ending value is lost if the gap is 1.5% a year instead of 0.8%?
How does the answer change for a monthly SIP of Rs 50,000 instead of a lump sum?
What would a client get for the extra cost in a regular plan, and how would you judge whether it is worth it?
067Fund A charges an expense ratio of 1.0% and turns over 150% of its portfolio a year. Fund B charges 1.4% and turns over 20%. Each round trip of buying and selling costs about 0.4% in brokerage and market impact. Which fund is really cheaper?Fee and cost dragCoreMutual fund distributionIndian wealth management
Try it first
Which fund costs the investor less in total?
Show the worked solution
Fund B, by about 0.12 points a year. Fund A's trading costs 150% turnover x 0.4%, which is 0.60%, so its all-in cost is about 1.60%. Fund B's trading costs 20% x 0.4%, 0.08%, for an all-in 1.48%. The lower visible fee belongs to the more expensive fund. On Rs 1 crore at 12% gross for ten years, the gap is about Rs 2.9 lakh.
Where does the hidden cost come from?
A shopkeeper who keeps swapping his stock pays the wholesaler's margin and the transport every time, even if the shelf price never changes. Every time a fund sells one holding and buys another, it pays brokerage and moves prices against itself, and that cost comes out of the fund's value rather than out of the stated expense ratio. Turnover of 150% means the fund replaces its whole portfolio one and a half times a year, so it pays the round-trip cost one and a half times.
Fund A's 1.00% expense ratio plus 0.60% of trading cost from 150% turnover comes to 1.60% a year, while Fund B's 1.40% expense ratio plus 0.08% from 20% turnover comes to 1.48%, so the fund with the higher visible fee is cheaper to own.
The relationship
$\text{all-in cost} = \text{expense ratio} + \text{turnover}\times\text{round-trip cost}$
turnover the share of the portfolio replaced in a year, 150% for A and 20% for B
round-trip cost brokerage and market impact on one sale and one purchase, 0.4% here
What it says in wordsWhat the investor really pays is the visible fee plus the cost of all the trading the fund does.
How large is 0.12 points in rupees?
Small each year, visible over a decade. On Rs 1 crore with a 12% gross return, Fund A compounds at 10.40% to about Rs 269.0 lakh in ten years and Fund B at 10.52% to about Rs 271.9 lakh. A 0.12 point gap is about Rs 2.9 lakh on Rs 1 crore over ten years, and the investor never sees it on a statement. The same method matters more for small-company funds, where the impact cost of each trade is larger than 0.4%.
Say the limits. Turnover is only a cost if the trading does not add return; a high-turnover manager may earn back more than 0.6%. And the 0.4% round trip is an assumption, which varies with the size of the fund and the stocks it trades. How expense ratios and trading costs are disclosed changes with regulation, so confirm the current rules before comparing real schemes.
Where candidates lose it
The trap is picking Fund A because its expense ratio is lower. The question hands you turnover precisely to see whether you know the expense ratio leaves trading out.
The second error is multiplying turnover by a one-way cost and halving the hidden drag. Turnover counts the portfolio replaced once; each replacement is a sale and a purchase.
What the interviewer asks next
At what turnover does Fund A cost exactly as much as Fund B?
Why is market impact larger for a small-company fund than for a large-company fund?
Where would you look to find a fund's turnover, and what would make you distrust the number?
079An equity fund keeps 10% of its money in cash, earning 6%, while the stocks it holds return 14%. How much does the cash cost the fund's return each year?Fee and cost dragWarm upMutual fund distributionIndian wealth management
Try it first
How much return does the cash cost?
Show the worked solution
About 0.8 of a point a year. The fund earns 90% x 14% plus 10% x 6%, which is 12.6 plus 0.6, or 13.2%. Fully invested it would earn 14%. The drag is the cash weight times the gap in returns: 10% x (14 - 6) = 0.8 of a point, in a year when stocks beat cash.
Where does the 0.8 of a point come from?
Picture a shop with ten shelves, nine stocked with goods that earn a good margin and one kept empty as a buffer for a delivery that may come. The empty shelf is not free: it costs the margin it would have earned. Cash drag is the cash weight times the gap between what cash earns and what the invested money earns. Here that is 10% times 8 points, or 0.8 of a point.
Stocks at 90% of the fund earning 14% contribute 12.6 points and cash at 10% earning 6% contributes 0.6, a blended 13.2% that is 0.8 of a point below the 14% a fully invested fund would earn.
The relationship
$\text{drag} = w_{cash} \times (r_{stocks} - r_{cash}) = 0.10 \times (14\% - 6\%) = 0.8\%$
w_cash the share of the fund held in cash, 10%
r_stocks the return on the invested stocks, 14%
r_cash the return on cash, 6%
What it says in wordsThe cost of idle cash is its weight times the return it gives up.
Is the cash ever worth it?
Yes, and say so. In a year when stocks fall 10%, the same fund loses 8.4% rather than 10%, so the cash cushions. It also pays redemptions without forcing the manager to sell into a falling market. Cash drag is a cost in rising markets and a cushion in falling ones, so the question for a client is whether the manager holds cash on purpose or by accident. A fund that is always 10% in cash is charging an equity fee on money sitting in a deposit.
Scale it up to show the stakes: 0.8 of a point a year compounds. Rs 1 crore at 14% for ten years grows to about Rs 3.71 crore; at 13.2% it grows to about Rs 3.45 crore.
Where candidates lose it
Candidates often answer 0.6 of a point, confusing what the cash earns with what it costs, or 1.4 points, a tenth of the stock return, forgetting the cash still earns something.
The other loss is calling cash pure waste. Say the drag, then say the cushion in a falling year: the interviewer wants to hear that you see both sides of the same holding.
What the interviewer asks next
Stocks return minus 10% this year. What does the fund return?
At what stock return does the cash stop being a drag?
How would you check whether a fund's cash level is deliberate?
092A client's portfolio earns 10% a year before fees, the adviser charges 1% a year, and inflation runs at 5%. What share of the client's real return does the fee take?Fee and cost dragCoreWealth management
Try it first
What share of his real return goes on the fee?
Show the worked solution
About one fifth, 20% of his real return. Of the 10% nominal return, about 5 points only keep pace with 5% inflation. The client's real gain is about 5 points, and the 1% fee takes 1 of them. Measured against the nominal return the fee looks like a tenth; measured against what the client actually gains, it is twice that. Done exactly, the share is still 20%.
Why measure the fee against the real return?
Suppose your salary rises 10% in a year when prices rise 5%. Your real raise is about 5%. If a new deduction then takes 1% of your salary, it takes a fifth of your real raise, not a tenth of your nominal one. A fee is paid out of the part of the return the client keeps after inflation, so its true weight is the fee divided by the real return.
The same 1 point fee is a tenth of a 10 point nominal return but a fifth of the 5 point real return left after inflation, which is the part of the return the client can actually spend or save.
The relationship
$\frac{1.10}{1.05} - 1 = 4.76\% \qquad \frac{1.09}{1.05} - 1 = 3.81\% \qquad \frac{4.76 - 3.81}{4.76} = 20\%$
1.10 / 1.05 the real return before the fee, dividing out inflation
1.09 / 1.05 the real return after the 1% fee
20% the share of the real return the fee takes
What it says in wordsDividing out inflation instead of subtracting it changes both real returns but leaves the fee's share at one fifth.
What happens to the share when things get harder?
The fee is fixed while the real return moves, so the fee's share rises as the real return falls. If inflation rises to 7% with the same 10% return, the real gain is about 3 points and the fee takes a third. Add tax on the nominal return and the share climbs further, because tax is charged on the part that only kept pace with prices too.
Say the limit fairly. The adviser's fee buys something: planning, tax work and behavioural coaching that may add more than a point. The puzzle is not an argument against fees; it is a way to show a client what a fee costs in the only currency that matters, his real gain.
Where candidates lose it
The common answer is 10%, fee over nominal return. It is arithmetically right and answers the wrong question: the client cannot spend the part of his return that inflation took.
The second slip is quoting 1% because "the fee is 1%". Say both ratios, a tenth of nominal and a fifth of real, so the interviewer hears that you chose the second on purpose.
What the interviewer asks next
Inflation rises to 7%. What share does the fee take now?
Add a 20% tax on the nominal gain. What share of the after-tax real return does the fee take?
How would you explain this to a client without sounding as if the fee is not worth paying?

1.10	gross growth factor, 10% a year
1.09	growth factor after a 1% fee taken off the return
30	years invested

On Rs 1 crore	Fee after fixed fee	Fee on gross
Gross gain, Rs lakh	15.0	15.0
Fixed fee, 2%	(2.0)	(2.0)
Performance fee, 20% above 8%	(1.0)	(1.4)
Net gain, Rs lakh	12.0	11.6
Share of gross gain paid in fees	20.0%	22.7%

0.99^t	the share of no-fee wealth left after t years of a 1% trail
0.97	the share left after a one-time 3% upfront fee

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

Try it first

Why does a 1% fee cost so much more than 1%?

How do you estimate the share in your head?

Where candidates lose it

What the interviewer asks next

Try it first

Why do the two fees stack rather than sit side by side?

What changes if the agreement charges the performance fee on the gross return?

Where candidates lose it

What the interviewer asks next

Try it first

Why does the market return drop out?

Where do candidates go wrong when they try to add compounding?

Where candidates lose it

What the interviewer asks next

Try it first

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

Try it first

Why is the gap so much larger than 0.8% sounds?

How do you estimate it without a calculator?

Where candidates lose it

What the interviewer asks next

Try it first

Where does the hidden cost come from?

How large is 0.12 points in rupees?

Where candidates lose it

What the interviewer asks next

Try it first

Where does the 0.8 of a point come from?

Is the cash ever worth it?

Where candidates lose it

What the interviewer asks next

Try it first

Why measure the fee against the real return?

What happens to the share when things get harder?

Where candidates lose it

What the interviewer asks next