Private Wealth Management puzzles, solved step by step

The 12% monthly deposit pays more: an effective 12.68% against 12.50%. One per cent a month, compounded twelve times, is 1.01 to the 12th, which is 1.1268. On Rs 1 lakh that is Rs 1,12,683 after a year against Rs 1,12,500, Rs 183 more. The gap is small, but the method is the point: convert every quoted rate to an effective annual rate before comparing.

Why is 12% monthly not 12%?

Imagine a savings box where interest is dropped in every month rather than at year end. From the second month on, the interest already in the box also earns. A quoted rate with a compounding frequency is a label, not the return; the return is the effective annual rate, which rises with every extra compounding. For 12% compounded monthly, that effective annual rateThe rate that, compounded once a year, gives the same result as the quoted rate with its compounding frequency. is 12.683%.

How do you do 1.01 to the 12th in your head?

Use the binomial shortcut. 1.01 to the 12th is roughly 1 plus 12 times 0.01 plus 66 times 0.0001, the number of pairs of months times the interest on interest. That is 1 plus 0.12 plus 0.0066, about 1.1266, within a hair of the exact 1.1268. The 0.66 point of extra return is the interest on interest, and it is what closes most of the gap to 12.5%.

Give the two numbers that frame it. Compounded continuously, 12% would give 12.75%, the ceiling for a 12% quote. And a monthly deposit would need to quote only 11.84% to match 12.5% annual. The limitation for a client: tax, premature withdrawal terms and the credit of the issuer usually matter more than Rs 183.

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.

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.

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.

About 23%. Take 1,000 funds: 100 skilled and 900 unskilled. A skilled fund wins three years running with chance 0.7 cubed, 34.3%, so 34.3 funds. An unskilled fund does it with chance 0.5 cubed, 12.5%, so 112.5 funds. Of the 146.8 funds with a streak, 34.3 are skilled: 23.4%. The streak moves the odds from 10% to 23%, but most streak funds are still unskilled.

Why is the answer not close to 70%?

Think of a medical test that is good but not perfect, used for a rare condition. Most positive results come from the many healthy people, simply because there are so many more of them. How common skill is to begin with matters as much as how well skill shows up in results. Skilled funds are one in ten, so even though they streak more often, unskilled funds produce most of the streaks by sheer numbers.

How do you set it up without a formula sheet?

Use natural frequencies. Pick a round population, 1,000 funds, and count how many land in each branch; the answer is one count over the total count. That is Bayes ruleA way of updating a starting probability with new evidence, by weighing how likely the evidence is under each possible explanation. without the notation, and it is far harder to get wrong out loud. The formula version gives the same 23.4%.

The client version is one sentence: a three-year record is weak evidence on its own. It is also worth naming the limitation of the model: real skill is not a clean 70% and fund returns are not independent year to year, but the direction of the answer survives both.

No: for a 20-year life the return is about 3.4% a year. He pays 100 and receives 7 a year for 20 years, 140 in total, of which 100 is his own capital coming back and only 40 is return. The rate that makes 20 payments of 7 worth 100 today is 3.44%. The payout rate is not a yield, because the capital is never returned.

Why is a 7% payout not a 7% return?

Imagine lending a friend Rs 1 lakh and being repaid Rs 7,000 a year until one of you dies, with nothing more after that. The Rs 7,000 is partly interest and partly your own Rs 1 lakh coming back in slices. An annuity payout rate includes the return of the buyer's own capital, because nothing is paid back at death; only the interest part is return. A bank deposit paying 7% hands back the Rs 1 lakh at the end as well; the annuity does not.

How does the return change with how long he lives?

Lifespan is the whole trade. If he dies after 10 years the return is about -6.0%, negative, because he gets back only 70; at 20 years it is 3.4%; at 30 years it reaches 5.7%. An annuity is insurance against living long, priced so that the insurer wins on clients who die early. That is its purpose, and the client should buy it for longevity protectionIncome that continues however long the client lives, so the risk of outliving savings passes to the insurer., not for yield.

Add what a private banker would. Annuity income is usually taxed as income in the year received, and payouts are fixed in rupees, so inflation erodes them. The 7% here is an illustration; real annuity rates depend on age, the option chosen and the current rate environment, and have to be taken from a current quote.

About 6.8%, not 9%. The client pays 104 and, if the call is used, receives a coupon of 9 after one year and 9 plus 100 after two. The Rs 4 of premium above par is lost at the call. The yield that discounts 9 and 109 back to 104 is 6.79%. For a bond above par that is likely to be called, the yield to call is the honest figure.

Why does the call matter so much?

Think of paying extra for a flat on a lease you expect to run for ever, then learning the landlord can end it after two years and refund only the base price. The rent was fine; the premium you paid is gone. A bond bought above par loses the premium if it is called at par, so its yield must be measured to the call date, not as if the coupon ran for ever. Here 4 of premium is lost over just two years.

How do you estimate it in your head?

Spread the premium loss over the years to the call. Coupons of 9 a year less 4 of premium over two years is about 7 a year, on an average price of about 102, which is roughly 6.9%, close to the exact 6.79%. The shorter the time to the call, the bigger the drag per year, because the same premium is lost over fewer coupons.

Name the second risk. A perpetual bond is callable, not must-call: if rates rise or the issuer weakens, it may not be called, and the client then holds a bond with no maturity at all. Desks quote the yield to worstThe lowest yield across all the dates on which the bond could be called or mature, the conservative figure for a callable bond. for exactly this reason, and a client needs both scenarios before buying.

About Rs 7.2 crore a year. Assume 6 relationship managers, each covering 25 client families: 150 families. At Rs 6 crore of assets each, the branch manages Rs 900 crore. A blended revenue yield of 0.8% across fees, commissions and lending gives Rs 7.2 crore. A top-down check on the client count lands at 160 families, which supports the build.

Where do you start the build?

Think of estimating a local restaurant's takings: tables, covers per table, spend per head. Nobody guesses the total first. Build revenue from counts and rates you can picture, so each assumption can be challenged and fixed without redoing the whole estimate. For a wealth branch the natural chain is relationship managers, families per manager, assets per family and the revenue the bank earns on those assets.

How do you check the estimate a second way?

Rebuild the weakest link from a different direction. The client count is the shakiest number, so check it top down: 8 lakh households, 1 in 1,000 with more than Rs 5 crore to invest, and a 20% share for this branch gives 160 families against 150. Every figure in that chain is an assumption too, stated as one. Then sense-check the output: Rs 1.2 crore of revenue per relationship manager has to cover that manager, the team behind him and the branch, which tells you whether the bank would keep the branch open.

Give the sensitivity before you are asked. The blended yieldTotal revenue from a book divided by the assets in it, mixing fees, commissions, spreads and lending income into one rate. is the most uncertain link: at 0.6% revenue is Rs 5.4 crore, at 1% it is Rs 9 crore. A book heavy in advisory mandates and lending earns more per rupee than one parked in low-fee products, so the product mix moves the answer as much as the client count.

Rs 30 of the remaining Rs 70. After the gap the shares are worth Rs 70 against a Rs 50 loan, a loan-to-value of 71.4%. Selling S of shares repays S of loan, so the lender needs 50 minus S to equal half of 70 minus S, which gives S = 30. The client is left with Rs 40 of shares, a Rs 20 loan and Rs 20 of equity, sold out near the low.

Why does the sale have to be so large?

Think of a bucket with a hole that you are emptying to lower the water to a mark painted halfway up its side, while the bucket itself shrinks as you pour. Every rupee of shares sold repays a rupee of loan but also removes a rupee of collateral, so the sale has to be twice the gap it closes. At a 50% target, each rupee sold lowers the required collateral by only 50 paise, which is why Rs 15 of excess loan needs Rs 30 of sales.

What does the forced sale cost the client if the price recovers?

It locks the loss in. If the stock climbs back to where it started, a rise of 42.9%, his Rs 40 of shares becomes Rs 57.1, and after the Rs 20 loan his equity is Rs 37.1. Had he not been sold out, the same recovery would have put him back at Rs 50 of equity, so the forced sale turned a temporary fall into a permanent loss of about Rs 12.9. A top-up of Rs 15 of cash would have restored the ratio without selling anything.

Say what a private banker does with this. Lending against shares is sized for gaps, not for daily moves: a lender may sell at the open before the client can respond. The cushion before a margin callA demand from the lender to add cash or collateral, or accept a sale, when the loan grows too large relative to the value of the pledged securities. and the cash the client can raise overnight matter more than the interest rate on the loan.

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 step-up ends bigger, Rs 86.8 lakh against Rs 75.7 lakh, but it wins late and by saving more. Its instalment first beats Rs 15,000 in year 6, at Rs 16,105, and its corpus overtakes only in year 11. Over 15 years it puts in Rs 38.1 lakh against Rs 27.0 lakh; the growth on each is almost identical, about Rs 48.7 lakh.

Why does the step-up lag for so long?

Two students save for a trip: one puts in Rs 150 a week from the start, the other begins at Rs 100 and adds 10% every term. The steady saver is ahead for most of the year because her money arrived first. The step-up pays in less than the flat SIP for the first five years, and money put in early is the money that compounds longest, so its corpus trails until year 11. Its year-5 instalment is still only Rs 14,641.

So is the step-up the better plan?

It is a different plan, not a smarter one. The step-up ends ahead because it asks the client to save Rs 11.1 lakh more; per rupee saved, the flat SIP compounds better because its money arrives earlier. Where the step-up earns its place is fit: it matches a salary that rises each year, so the client can afford it without strain in year one. The honest comparison for a client is between what each plan asks of his budget in each year, not between two final numbers.

Limits: 1% every month is smooth, real returns are not, and the step-up puts its biggest instalments in the last years, so a bad market late in the plan hurts it more. The 10% step-up also assumes the client's income actually rises that fast.

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.

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.