Private Wealth Management puzzles, solved step by step
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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?Wealth management
Try it first
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 relationship1.10 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?
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?Mutual 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 relationship0.99^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?
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?Mutual 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 relationshipturnover 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?
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?Wealth 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 relationship1.10 / 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?