Private Wealth Management puzzles, solved step by step

About 9.6% a year. A 150% return means Rs 1 became Rs 2.5. The yearly rate is 2.5 to the power one tenth, minus one, which is 9.60%. Dividing 150 by 10 to get 15% ignores compounding: 15% a year for ten years would turn Rs 1 into 4.05, not 2.5.

Why is 15% the wrong annual figure?

Think of a child's height chart. If a child grew 50 centimetres over ten years, you could say 5 centimetres a year, because height adds. Money does not add; it multiplies, and each year's gain earns its own gain the next year. An absolute return over many years must be converted to a compounded yearly rate before it can be compared with anything else. Dividing by the number of years overstates the yearly rate, and the overstatement grows with the length of the period.

How do you get 9.6% without a calculator?

Use doubling as the anchor. At about 9.6% money doubles in roughly 7.5 years, so in ten years it goes a little past double, which matches 2.5 times. You can also bracket it: 1.10 to the tenth is 2.59, a touch above 2.5, so the answer is a touch below 10%. Saying a bracket out loud, just under 10% because 10% gives 2.59, is as convincing to an interviewer as the exact 9.60%.

Why does a wealth interviewer care? Because a client comparing a 150% ten-year fund with a fixed deposit quoting a yearly rate is comparing two different units. The 9.6% is also before any comparison with a benchmarkThe index or reference portfolio a fund is measured against, so its return can be judged relative to what the market gave., which is the next question worth asking.

About 32% more wealth. Price growth alone turns Rs 1 into 1.08 to the 20th, Rs 4.66. Reinvesting a 1.5% dividend each year makes the total return 9.5% a year, and 1.095 to the 20th is Rs 6.14. The ratio is 1.32. A 1.5 point yearly difference compounds into a gap of about a third of the price-only result.

Why does a small yield make such a large gap?

Think of a fruit tree whose fruit you plant every season instead of eating. Each season's seeds become trees that also bear fruit. Reinvested dividends buy more units, and those units then earn both the price growth and their own dividends for the rest of the period. So the 1.5% is not added twenty times to the starting amount; it raises the compounding rate on the whole holding from 8% to 9.5%.

What if the client takes the dividends as cash?

Then the dividends stop compounding. Each year's dividend is 1.5% of the price-only holding, and across 20 years they add up to about 0.69 of the starting rupee. Price plus cash dividends ends at about 5.35 times, well short of the 6.14 times from reinvesting. This is why comparing a fund's return with a price indexAn index that tracks only the prices of its stocks and ignores the dividends they pay. flatters the fund: the fair benchmark is the total return index.

State the simplification. Real dividend yields move from year to year and are taxed in the client's hands, which lowers what is reinvested. The 32% is the gap under the question's clean assumptions, not a promise about any index.

Barely: the risk-adjusted excess, or alpha, is about 0.2%. The market premium was 12 minus 6, or 6 points. A beta of 1.3 earns 1.3 times that, 7.8 points, on top of the 6% risk-free rate, so the fund should have made 13.8% just by holding more market risk. It made 14%. Of the 2-point beat, 1.8 points were paid for by extra risk and 0.2 came from the manager.

Why is beating the index not the same as adding value?

Picture two drivers who both arrive early. One drove carefully and knew a shortcut; the other simply drove 30% faster on the highway. You would not call the second one a better navigator. A fund with a beta of 1.3 is the fast driver: in a rising market it is expected to beat the index simply because it carries more of the market's risk. The client paid for that extra speed with extra drawdown risk, and would have got it from any higher-beta fund.

So the fair yardstick is not the index but what the index would have paid at 1.3 times the risk. That is the capital asset pricing modelA model that says a portfolio should earn the risk-free rate plus its beta times the market premium, the market return less the risk-free rate. expectation, and whatever sits above it is called Jensen's alpha.

What would you tell the client, and what can one year not tell you?

Tell the client the fund did roughly what a higher-risk version of the index would have done, with a sliver on top. One year of 0.2 points of alpha is indistinguishable from noise; it takes several years and a stable beta before anyone can call it skill. Also say the mirror image: in a year the index falls 10%, the same 1.3 beta implies a fall of about 6 + 1.3 x (minus 16), or minus 14.8%, before any skill at all. The client should expect to feel that.

The limitation is the beta itself. It is estimated from past returns, it moves, and a different benchmark gives a different number. Say that you are treating 1.3 as given for the puzzle.

The household earned 0%. The husband's 20% on Rs 1 crore is a gain of Rs 20 lakh. The wife's minus 5% on Rs 4 crore is a loss of Rs 20 lakh. Together they started with Rs 5 crore and ended with Rs 5 crore. The simple average of 7.5% is wrong because it gives the small account the same weight as the large one.

Why does the simple average mislead?

Two children score 100 on a 10-mark test and 40 on a 100-mark test. Their average percentage is 70, but they got 50 marks out of 110, which is 45%. A household's return is the change in its total rupees divided by the rupees it started with, so each account counts in proportion to its size. Averaging percentages silently treats a Rs 1 crore account and a Rs 4 crore account as equals.

Why does this matter in a wealth review?

Consolidated reporting is one of the first things a family office client asks for, and it is where this error shows up. If the adviser's report leads with the husband's 20%, the family feels rich while its total wealth has not moved at all. The same trap appears when a relationship manager quotes the average return across a client's funds instead of the return on the client's money.

One limit: this weighting assumes no money moved in or out during the year. If the wife added Rs 1 crore in March, you would need a time-weighted or money-weighted calculation, which is a different question.

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.

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.

Annualised, 10% in six months is 21%, not 20%, and neither is a return he has earned. Repeating 10% on the larger Rs 1.1 lakh base gives Rs 1.21 lakh in a year. But annualising assumes the next six months repeat the last six. If they give back 10% instead, the year ends at Rs 99,000, a 1% loss. Short windows are mostly noise.

Why is it 21% and not 20%?

A cricketer who scores 50 in the first ten overs is not guaranteed 250 in fifty, and even the projection has to use the right arithmetic. Annualising a half-year return means compounding it, because the second half would grow the bigger balance. Rs 1 lakh becomes Rs 1.1 lakh, and another 10% on Rs 1.1 lakh is Rs 11,000, not Rs 10,000, which is where the extra point comes from.

Why is annualising a short window risky?

Annualising quietly assumes the next six months will look like the last six. Over short windows, most of a fund's return is noise, and scaling noise up to a year makes it look like a trend. Look at the bottom of the figure: the same first half ends the year at plus 21%, plus 10% or minus 1%, depending on a second half nobody has seen. This is why fund documents commonly show returns for periods under a year as absolute figures rather than annualised ones; confirm the current SEBI presentation rules before you quote them.

In the room, give both halves of the answer: the corrected 21%, then the caution. A client who hears 21% a year will plan around it. A client who hears 10% so far, which would be 21% if it repeated, which it may not, has the number and its limits.

He lost Rs 32 lakh, about -5.4% a year, while the manager reports plus 8%. Year one made Rs 20 lakh on Rs 1 crore. The client then had Rs 5.2 crore invested when the 10% fall came, costing Rs 52 lakh. He put in Rs 5 crore and holds Rs 4.68 crore. Both numbers are honest: 8% measures the manager's skill, the money-weighted loss measures the client's outcome.

Why can both numbers be right?

A bus that averages 60 km an hour tells you about the driver, not about a passenger who boarded only for the slow stretch through traffic. A time-weighted returnA return that chains each period growth rate together, so it ignores when money was added or withdrawn. Used to judge a manager. chains the period returns and ignores the size of the balance, so it judges the manager; a money-weighted returnThe internal rate of return on the actual rupees the client put in and took out, so it depends on the timing and size of his flows. weighs each period by the rupees actually at work, so it measures the client. The manager did not choose when the Rs 4 crore arrived; the client did.

How do you get the client's yearly rate?

Find the rate that makes his flows add up. Rs 1 crore invested for two years plus Rs 4 crore invested for one year must grow to Rs 4.68 crore. Solving gives a money-weighted return of about -5.4% a year, the number that describes what actually happened to his money. The quadratic is quick: with x as one plus the rate, x squared plus 4x equals 4.68, so x is about 0.9462.

The wealth lesson is behavioural. Clients tend to add money after strong years and pull it after weak ones, so their money-weighted results often trail the funds they hold. Showing a client both numbers, and why they differ, is one of the most useful conversations an adviser can have.

Fund A, Rs 2.59 lakh against Rs 2.19 lakh. Each two-year pair of Fund B multiplies money by 1.3 x 0.9, which is 1.17, so ten years is 1.17 to the fifth, or 2.19. Fund A compounds 1.1 ten times to 2.59. Fund B's true annual rate is the square root of 1.17, less 1, which is 8.17%, not 10%.

Why does the average of the returns mislead?

A shopkeeper raises a price by 30% and then cuts it by 10%. The tag does not end 20% higher: 100 becomes 130, and 10% off 130 is 117. Returns multiply, so the rupees you keep depend on the product of the growth factors, not on the average of the percentages. The minus 10% bites on a larger base than the plus 30% started from, and that asymmetry is where Fund B leaks money.

What should the client take away from this?

Two funds with the same average return can leave the client with very different amounts. The more a fund's returns swing, the further its compound rate falls below its average, so a fund sold on its average return is being sold on the wrong number. Here the swing costs 1.83 points a year, and over ten years that is 0.40 lakh on every lakh invested.

Say the limitation too: real funds do not alternate neatly, and a steady 10% fund does not exist. The puzzle isolates one effect, the cost of volatility to compound growth, so that the client sees it in rupees.

About 8% a year. The compound, or geometric, return is roughly the arithmetic mean less half the variance. Volatility of 20% is a variance of 0.2 squared, 0.04, and half of that is 0.02, or 2 points. So an average year of 10% compounds at about 8%. The higher the volatility, the wider the gap.

Why does volatility pull the compound return below the average?

Walk up a hill 30 steps and back down 10, then repeat: you are fine. But returns multiply: a rise of 30% followed by a fall of 10% leaves 1.3 x 0.9, which is 1.17 over two years, an average of 10% that compounds at only 8.17%. A loss is taken from a larger base than the gain that preceded it, so the more returns swing, the further compound growth falls below the average return.

How far can you trust the rule?

Check it on a case you can work exactly. The fund that alternates plus 30% and minus 10% has an average of 10% and a volatility of exactly 20%, and its true compound rate is the square root of 1.17 less one, 8.17%. The half-variance rule gives 8% against an exact 8.17%, close enough to say out loud, and it gets less accurate as volatility rises. At 40% volatility the rule takes off 8 points, and the exact answer depends heavily on the shape of the returns.

Then say why a client should care. Two portfolios with the same 10% average but volatility of 10% and 20% compound at about 9.5% and 8%. Over 20 years on Rs 1 crore that is roughly Rs 6.1 crore against Rs 4.7 crore. Lower volatility at the same average is worth real money, which is part of the case for diversification.