Private Wealth Management puzzles, solved step by step
- Puzzles
- 100
- Traced to a firm
- 3
- Topics
- 13
- Hard
- 30
015An equity index rises 8% a year on price for 20 years and also pays a 1.5% dividend yield, reinvested every year. How much more wealth does the total return version end with than the price index?Mutual fund distributionIndian wealth management
Try it first
Pick the extra ending wealth from reinvesting the dividends.
Show the worked solution
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%.
Over 20 years Rs 1 grows to 4.66 on price alone at 8% a year but to 6.14 with a 1.5% dividend reinvested, so the total return version ends about 32% richer, with most of the gap opening late. 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.
The relationship1.095 total return a year: 8% price growth plus a 1.5% dividend yield on the opening price 1.08 price return a year 20 years What it says in wordsThe extra wealth from reinvesting is the ratio of the two compounded growth factors.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.
Where candidates lose it
The trap is adding the yield in a straight line: 1.5% times 20 years is 30 points, so the total return index ends at about 4.96 times. That treats the dividends as cash put in a drawer, not reinvested.
The mirror error is thinking the dividend makes little difference because 1.5% sounds small. Show the 6.14 against 4.66 and the point makes itself.
What the interviewer asks next
- What would the gap be over 30 years instead of 20?
- A fund reports beating the price index by 1% a year. What does that say about its skill?
- How does tax on dividends change the answer for a client in a high bracket?
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?Mutual fund distributionIndian wealth management
Try it first
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 relationshipA 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?
078Fund A returns 10% every year. Fund B alternates: plus 30% one year, minus 10% the next, so its average annual return is also 10%. After 10 years, which fund has turned Rs 1 lakh into more money, and what is Fund B's true annual rate?Wealth management
Try it first
Before you work it: which fund ends with more?
Show the worked solution
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.
Fund A climbs steadily to Rs 2.59 lakh while Fund B zig-zags to Rs 2.19 lakh, even though both average 10% a year, because B compounds at only 8.17% a year. The relationship1.3 the growth factor in B's up year 0.9 the growth factor in B's down year g_B the geometric, or compound, annual return of Fund B What it says in wordsB's true rate is the rate that, applied every year, gives the same money: the square root of one pair's growth, less one.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.
Where candidates lose it
The trap is answering "they end level" because the averages match. The interviewer is checking whether you know that returns compound by multiplying, and most people who say level have never tried a two-year example.
The second loss is getting B's final value right but calling its rate 10% anyway. The rate the client earned is 8.17%, the square root of 1.17 less one.
What the interviewer asks next
- What arithmetic average would Fund B need to end level with Fund A?
- Fund C goes plus 50%, minus 30%. What is its compound rate?
- Which of the two numbers should a fund factsheet show, and why?