Private Wealth Management puzzles, solved step by step

About 7.19%. Quarterly compounding pays 7 / 4 = 1.75% each quarter, and each quarter's interest earns interest in the quarters that follow. Rs 100 grows to 101.75, 103.53, 105.34 and 107.19. The effective yield is 1.0175 to the fourth, less 1, which is 7.19%.

Why is the effective yield above the quoted 7%?

If a friend repays you in four instalments and you lend each instalment on at once, you end the year with more than if the whole amount came back at year end. The quoted rate is the per-period rate times the number of periods; the effective yield adds the interest earned on interest paid earlier in the year. The more often it compounds, the wider the gap.

When does the 7.19% not apply?

The effective yield assumes the interest stays in the deposit; on a payout deposit that sends interest to the client each quarter, he earns 7% unless he reinvests it himself. Tax is the second gap: interest on deposits is typically taxed as income each year at the client's slab rate, so the after-tax figure is lower still. Confirm the current treatment before comparing a deposit with a fund.

Use the effective rate whenever you compare products that compound at different frequencies, such as a quarterly deposit against an annual-coupon bond. Comparing quoted rates across frequencies is comparing different units.

He needs 66.7%, and the plan anchors on a price the market does not care about. Getting from Rs 300 to Rs 500 is a rise of 200 on 300. At 12% a year that is about 4.5 years. The purchase price is history; the question is whether he would buy the stock at Rs 300 today.

Why is 66.7% bigger than the 40% he lost?

A shirt marked down from Rs 500 to Rs 300 is 40% off. Marking it back up to Rs 500 is a 66.7% rise, because the markup is taken on the lower price. A percentage recovery is measured from the lower base, so the gain needed to recover always exceeds the loss suffered.

What is wrong with waiting to get back to Rs 500?

AnchoringLeaning on a reference number, here the purchase price, when judging a value that should not depend on it. on the purchase price makes a past number decide a future choice. The stock's next return is the same whether he bought at Rs 500 or Rs 200, so the only live question is whether Rs 300 of this stock is the best use of Rs 300 today. Holding a loser to avoid admitting a loss, while selling winners early, is a well documented pattern called the disposition effect.

Say it kindly. Nobody enjoys booking a loss, and the job is not to tell the client he was wrong. Ask him whether he would buy it today at 300; if the answer is no, he is holding it only because of a number from the past. A booked loss may also offset other gains for tax, which is a real reason to act rather than wait.

Roughly Rs 210 crore a year, on stated assumptions. Take 10 crore people, 2.5 crore households at four each, 40% with property or savings worth protecting, and two adults each: 2 crore people who need a will. If 15% ever write one over a 30-year window, that is 1 lakh wills a year, at a blended Rs 21,000 each.

How do you structure the estimate before any number?

Think of estimating how many cakes a bakery sells for birthdays: you would go from people, to birthdays a year, to the share celebrated with a bought cake, to the price. A market size is a chain of counts and rates, and saying the chain out loud before any number is most of the marks. Here the chain runs from people, to households with something to leave, to adults who need a will, to how many write one each year, to the fee.

Every number in the tree is an assumption to state, not a fact: the population of the ten cities, the household size and the 40% share should be replaced with census and survey figures when you have them. The fee split reflects two very different products: a simple will drafted online or by a local lawyer, and a complex will or estate plan prepared alongside a private wealth adviser.

Which step would you defend least, and what does that tell you?

The 15% lifetime rate of writing a formal will. The estimate is only as good as its least defensible step, so name that step and show what happens when it moves. At 30% the market doubles to Rs 420 crore; at 7.5% it halves. The interviewer will usually push on the weak step, so get there first.

Close with a sense check from the other side: 1 lakh wills a year across ten cities is about 40 wills a working day in each city, which feels plausible for the number of lawyers and advisers who offer the service.

About 18.3% of his wealth. In a flat market the puts, struck 10% below, never pay out, so each year he simply loses 2%. Ten years leave 0.98 to the power 10, which is 0.817 of the starting wealth. That is slightly under 20% because each premium is taken from a pot the earlier premiums have already shrunk.

Why is the cost close to 18% and not 20%?

Paying 2% of your salary every year for a subscription feels like a small fixed charge, but if it were 2% of your savings each year, the savings would shrink and each charge would be a little smaller in rupees. A yearly premium set as a share of the portfolio compounds downwards, so ten years of 2% cost 1 minus 0.98 to the 10th, about 18.3%, not a flat 20%. In rupees, a Rs 10 crore portfolio would stand at about Rs 8.17 crore.

So is the insurance a bad idea?

Not by itself. A put optionA contract that gives the holder the right to sell an asset at a set price, so it pays out when the price falls below that level. pays out in a crash, which is when a client most needs cash and is most tempted to sell at the bottom. Permanent insurance is a permanent drag, so the question is whether the protection is worth about 2 points of return every year, not whether it pays out in any one year. A flat decade is the worst case for the buyer, because he pays every premium and collects nothing.

Say the limitation: in a rising market the drag is the same 2% a year taken from a growing pot, and in a crash the puts pay. The honest comparison sets the yearly cost against the losses avoided in the bad years, and the answer depends on how often and how deep those falls are.

14% at a 12% portfolio return, and only 2% at 6%. Rs 2 crore at 12% earns Rs 24 lakh, less Rs 10 lakh of interest, leaving Rs 14 lakh on his Rs 1 crore. At 6% the portfolio earns Rs 12 lakh, less the same Rs 10 lakh, leaving Rs 2 lakh. His return is twice the portfolio return less 10, so leverage helps only above the 10% borrowing rate.

Where does the 14% come from?

A shopkeeper who borrows to double her stock doubles her sales margin, but pays the lender first. If the stock earns more than the loan costs, she keeps the difference on the borrowed half as well as her own margin. On borrowed money the client keeps only the spread between the portfolio return and the borrowing rate, so leverage adds 12 minus 10, two points, to his own 12%.

What happens below the crossing point?

Leverage turns against him. Below the borrowing rate every point the portfolio falls short costs him two, so a flat year loses 10% and a 10% fall loses 30%. The loan interest is due whatever the market does, which is why the downside is steeper than the upside is generous.

Then add what the arithmetic leaves out. A loan against a portfolio usually has a margin requirement: if prices fall far enough, the lender asks for more collateral or sells holdings at the worst moment. The expected 12% is also not certain; a borrowing plan that works only if the portfolio beats 10% needs the client to understand how often it will not.

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.

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.

Sister A, with about Rs 190 lakh against Rs 109 lakh, ahead by Rs 81 lakh. A pays in only Rs 10 lakh, but her pot is Rs 17.5 lakh at 35 and then compounds untouched for 25 years, about 10.8 times. B pays in Rs 26 lakh, yet her last payments have little time to grow. The early years carry the compounding.

How do you get the answer without a spreadsheet?

Work A in two stages. Ten payments of Rs 1 lakh at 10% build to about Rs 15.9 lakh by the last payment at 34, and to Rs 17.5 lakh a year later at 35. From 35 to 60 that pot compounds for 25 years. At 10% money doubles about every 7.2 years, so 25 years is about three and a half doublings, roughly 11 times, which takes Rs 17.5 lakh to about Rs 190 lakh.

Why does B never catch up, even with 26 payments?

Picture a snowball rolled from the top of a long slope against a bigger pile of snow thrown on halfway down. The first has more distance to gather snow. At 10% a rupee invested at 25 is worth about 28 rupees at 60, while a rupee invested at 50 is worth about 2.6, so a decade of early payments outweighs a quarter century of later ones. For B to match A she would need to pay about Rs 1.74 lakh a year from 35.

Then add the honest limits. A steady 10% is an illustration, not a forecast; a real sequence of returns will differ, and fees and taxes reduce both pots. The ratio between the sisters, though, comes from time in the market, and that holds at any positive rate: the lower the rate, the smaller A's lead.

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.

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.

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.

About 12.6 years, not 12. The rupees double in about 6.1 years at 12%, but purchasing power grows at the real rate. The real rate is 1.12 divided by 1.06, less 1, which is 5.66%, not 6%. At 5.66% money doubles in about 12.6 years; 72 / 6 gives 12, a few months short.

Why is the real rate 5.66% and not 6%?

If your money grows 12% while a thali's price rises 6%, you can afford 1.12 / 1.06 = 1.0566 times as many thalis, not 1.06 times. Real return is found by dividing out inflation, not subtracting it, because inflation reduces what each rupee of the gain buys as well as the principal. Subtraction overstates the real rate slightly, and the error grows with inflation.

Is the difference between 12 and 12.6 years worth making?

In the room, yes, briefly: 12 years from the shortcut, 12.6 done properly, and the reason is division against subtraction. The bigger point is the gap between 6 years and 12.6: a client who hears his money doubles every six years believes he will be twice as well off, when in spending power he will need more than twice as long. That gap is what goal planning is built around.

Note the limits: inflation is not steady, the client's own inflation, school fees or medical costs, may run above the headline rate, and tax on the nominal return lowers the real rate further. Each of those pushes the doubling time out, not in.