Private Wealth Management puzzles, solved step by step

Bad year first ends at about Rs 56.1 lakh; bad year last ends at Rs 70 lakh. Left untouched, both pots would reach Rs 117.1 lakh whatever the order. With Rs 10 lakh withdrawn every year, an early 20% fall shrinks the pot the good years grow from, so each good year earns Rs 6 to 7 lakh instead of Rs 10 lakh. Same average return, Rs 13.9 lakh apart.

Why does order not matter without withdrawals?

Multiplication does not care about order: 0.8 x 1.1 x 1.1 x 1.1 x 1.1 is the same as 1.1 x 1.1 x 1.1 x 1.1 x 0.8, and both give 1.1713. A pot left alone ends at Rs 117.1 lakh either way. Withdrawals turn the problem from multiplication into multiplication and subtraction, and subtraction does care about order. Taking a fixed Rs 10 lakh from a pot that has just fallen removes a bigger share of it, and that share never gets to recover.

Where exactly does the Rs 14 lakh go?

Follow the rupees. Both retirees lose Rs 20 lakh in their bad year, because in both cases the fall hits a Rs 100 lakh pot. The difference is what the four good years earn: bad-last grows Rs 100 lakh by 10% four times, Rs 40 lakh in all, while bad-first grows a pot that starts at Rs 70 lakh and keeps shrinking, earning only about Rs 26.1 lakh. Each good year earns less than the Rs 10 lakh he withdraws, so the pot keeps falling even in good years.

This is sequence-of-returns risk, and it is why the years just before and just after retirement matter most. Planners respond with a cash or short-bond bucket that funds a few years of withdrawals, so the retiree is not forced to sell growth assets after a fall. Say the mechanism; the right size of such a bucket depends on the client.

A 7.0% average yield, Rs 7 lakh of interest a year, and Rs 20 lakh free every year. Equal Rs 20 lakh rungs make the average yield the simple average of the five rates. Against locking the whole Rs 1 crore at 7.5%, the ladder gives up about Rs 50,000 a year at the start, in exchange for a fifth of the money maturing every year and a spread of reinvestment dates.

Why build a ladder instead of one deposit?

A family that stores rice in five sacks, opening one a year, never has to break into next year's supply for this year's needs. A ladder spreads maturities so that a slice of the money comes free every year, which gives liquidity without breaking a deposit early and spreads the risk of reinvesting all the money at one bad rate. The price is that the shorter rungs earn less than the longest one.

How do you compute the interest quickly?

Because every rung is the same size, the average yield is the plain average of the rates: 6.5, 6.75, 7, 7.25 and 7.5 add to 35, and 35 over 5 is 7.0%. Rs 1 crore at 7.0% is Rs 7 lakh a year, against Rs 7.5 lakh if all of it sat in the 5-year deposit, so the ladder costs about Rs 50,000 a year at the start. Rung by rung it is Rs 1.30, 1.35, 1.40, 1.45 and 1.50 lakh.

Is the Rs 50,000 a permanent cost?

Not if he keeps the ladder going and rates stay put. Each year the matured Rs 20 lakh goes into a new 5-year deposit at the top rate, so after five years every rung was bought at the 5-year rate while one still matures each year. A rolling ladder gives up yield mainly in its first years; once rebuilt, it earns close to the long rate and keeps its yearly liquidity. If rates fall, the rungs that come free get reinvested lower; if they rise, the ladder catches the rise sooner than one long deposit would. Tax on deposit interest is the same across rungs in this example; confirm the current treatment for the client's slab.

25%. If returns were pure luck, next year's quartile would be independent of this year's, so a top-quartile fund would have a one-in-four chance of landing in each quartile, top included. A quarter of the top funds repeat by chance alone, 6.25% of all funds. Persistence only counts as evidence of skill when the repeat rate is clearly above 25%, and with few funds that bar is higher than it looks.

Why is the luck baseline 25% and not zero?

Roll a die twice. The chance the second roll is a six does not care whether the first one was, so one in six of the first-roll sixes will be followed by another six. Luck does not avoid repeats, it ignores history, so under pure chance each quartile of this year's funds scatters evenly across next year's four quartiles. A quarter of the top quartile stays top, which is 6.25% of all funds.

How far above 25% is enough to mean something?

It depends on how many funds you are counting. With 100 funds there are 25 in the top quartile, and luck alone predicts about 6.25 repeats with a spread of about 2.2. Ten or more repeats, a 40% rate, still happens by luck about 7.1% of the time, so a single year's persistence among a small set of funds is weak evidence. Several years of repeats, or a very large sample, is what separates skill from a good draw.

Two more checks before believing a persistence table. Funds that closed or merged after bad years quietly vanish from later counts, which makes survivors look more persistent than they were. And funds in the same category share market conditions, so a style that is in favour can keep a whole group of funds on top together. Neither of those is manager skill.

About Rs 6,840 crore a year, with about 21% from performance fees, on these assumptions. Fixed fees of 2% on Rs 1.6 lakh crore, 1% on Rs 1.6 lakh crore and 0.75% on Rs 0.8 lakh crore give Rs 5,400 crore. A 15% share of returns above a 10% hurdle, in a 16% year, adds Rs 1,440 crore. That is a 1.71% blended rate, and the performance part can vanish in a weak year.

How do you structure a fee pool estimate?

A toll road's yearly takings are traffic times toll, lane by lane, because trucks and cars pay different rates. A fee pool is the asset base times the fee rate, computed separately for each strategy because each charges differently, then added. Here the assumptions split Rs 4 lakh crore into 40% fixed-fee equity, 40% equity charging a lower fixed fee plus a performance feeA share of the return above an agreed hurdle rate, paid to the manager only when that hurdle is beaten, usually subject to a high-water mark., and 20% debt and multi-asset strategies. Every share and rate is an illustrative assumption.

Why does the performance share matter to the business?

Because it is the unstable part. Fixed fees fall only as far as assets fall; performance fees can drop to zero in a single year below the hurdle. In that year the pool shrinks from Rs 6,840 crore to Rs 5,400 crore, a blended rate of 1.35%, and a high-water markA rule that no performance fee is paid until the portfolio value climbs back above its previous peak. can keep it at zero for several years after a fall. Say that sentence and you have shown you understand the economics, not just the multiplication. Fee structures and caps are set by regulation; confirm the current rules before quoting real ones.

Rs 112 at Rs 81 and Rs 91 at Rs 79: a Rs 2 move in the share costs him Rs 21. Above the Rs 80 barrier he gets his Rs 100 back plus the Rs 12 coupon. Below it, he receives shares worth the final price plus the coupon, so at Rs 79 he takes the full fall from Rs 100. The coupon is mostly payment for selling a put on the share that only switches on below Rs 80.

Why is there a cliff at Rs 80?

Think of travel insurance that pays nothing unless your flight is delayed four hours, and then pays in full: a delay of three hours fifty-nine and four hours one are worth completely different amounts. The barrier switches the note from paying Rs 100 to paying the share's value, so crossing it moves the payout by the whole distance from Rs 100 to the share price, not by the Rs 2 the share moved. At Rs 81 he gets Rs 100 plus Rs 12; at Rs 79 he gets Rs 79 plus Rs 12.

What is the investor actually selling for the 12%?

Split the note into pieces. He lends Rs 100 for a year, which on its own earns an ordinary interest rate. He also sells the bank a put on the share struck at Rs 100 that only comes alive if the share ends below Rs 80, a knock-in putA put option that exists only if the underlying price crosses a barrier level; once knocked in, it pays like an ordinary put.. The part of the 12% above the interest rate is the premium for that put, and the put is why the note's downside looks like owning the share while its upside is capped at Rs 112. Above Rs 112 a shareholder does better; below Rs 80 the investor does no better than a shareholder, apart from the coupon.

What to say about suitability, without making a call on the product. The note swaps an uncertain equity return for a fixed Rs 12 in most outcomes and a large loss in the others, and the loss arrives as a jump. A client should see the Rs 79 row, not only the Rs 112 headline, and understand that the chance of breaching Rs 80 depends on how volatile the share is.

Credit is the bigger risk on expected loss, Rs 9 crore a year against Rs 6 crore, but conduct is the bigger risk in a bad year. Rs 2,000 crore x 1% x 45% is 9; Rs 60 crore x one in ten is 6. If a bad credit year triples defaults to 3%, credit loses 27, still well short of the 60 crore fine landing whole, and the fine also damages the franchise.

Why is the Rs 60 crore headline the wrong number to rank on?

Think of two risks to a household. A leaking tap wastes a little every month; a burglary costs a lot but comes rarely. You cannot compare them by the size of one burglary against one month of leaking, because one is certain and the other is not. Risks are ranked first by expected loss, the chance of the loss times its size, so a large but rare fine and a small but steady credit loss can be put on one scale.

The conduct fine goes on the same scale the same way: Rs 60 crore with a one in ten chance each year is Rs 6 crore a year. On the average year credit costs half as much again as conduct. For a private bank the loan book is usually Lombard lendingLoans to wealthy clients secured against their investment portfolios, which the bank can sell if the loan is not repaid. against portfolios, which is why the loss given default here is well below what an unsecured book would suffer.

So what do you say is the greatest risk?

Say both rankings, then pick with a reason. Expected loss tells you what to price and provision for; the bad year tells you what can hurt capital and the name, and for a bank that lives on client trust the bad year usually decides. A conduct fine rarely comes alone: clients leave, regulators restrict new business, and the cost runs well past the Rs 60 crore. That is why many private banking interviewers expect you to land on conduct and reputation as the greatest risk, and to show the arithmetic that says credit costs more on an ordinary year. The 3% bad-year default rate is an assumption; say it as one.

About 12% a year. Rs 5 lakh to Rs 20 lakh is four times, which is two doublings. Two doublings in 12 years means one every 6 years, and the rule of 72 gives 72 divided by 6, which is 12%. The exact rate is 4 to the power one twelfth, less 1, which is 12.25%.

Why turn the multiple into doublings first?

If someone tells you a town's population went from 5,000 to 20,000, you naturally say it doubled and doubled again. Doublings are easy to count and hard to get wrong. A growth multiple that is a power of two converts straight into a number of doublings, and the rule of 72 turns years per doubling into a rate. Four times is two doublings; eight times would be three.

How good is the rule of 72 here, and when does it slip?

The rule of 72 is most accurate for rates around 8%, and it drifts at the edges: at 12% it undershoots slightly, and at 20% or more it undershoots by more. For interview purposes, 12% with the words "a shade over" is the answer that shows you know it is an approximation. If the multiple is not a clean power of two, say 5 times in 12 years, estimate the doublings: 5 is a bit over two doublings, about 2.3, so a doubling every 5.2 years and roughly 14%.

Then turn it back to the client. A 12% compound rate over 12 years is a good result, but ask what it was in and what the fees and taxes were, because the client quoted a pre-tax figure from memory.

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 0.8 of a point a year. The fund earns 90% x 14% plus 10% x 6%, which is 12.6 plus 0.6, or 13.2%. Fully invested it would earn 14%. The drag is the cash weight times the gap in returns: 10% x (14 - 6) = 0.8 of a point, in a year when stocks beat cash.

Where does the 0.8 of a point come from?

Picture a shop with ten shelves, nine stocked with goods that earn a good margin and one kept empty as a buffer for a delivery that may come. The empty shelf is not free: it costs the margin it would have earned. Cash drag is the cash weight times the gap between what cash earns and what the invested money earns. Here that is 10% times 8 points, or 0.8 of a point.

Is the cash ever worth it?

Yes, and say so. In a year when stocks fall 10%, the same fund loses 8.4% rather than 10%, so the cash cushions. It also pays redemptions without forcing the manager to sell into a falling market. Cash drag is a cost in rising markets and a cushion in falling ones, so the question for a client is whether the manager holds cash on purpose or by accident. A fund that is always 10% in cash is charging an equity fee on money sitting in a deposit.

Scale it up to show the stakes: 0.8 of a point a year compounds. Rs 1 crore at 14% for ten years grows to about Rs 3.71 crore; at 13.2% it grows to about Rs 3.45 crore.

About Rs 1.04 crore. At 10% a year costs double roughly every 7.2 years (72 / 10), so 15 years is just over two doublings: Rs 25 lakh becomes 50 and then 100 lakh, plus a little. Exactly, 25 x 1.1 to the power 15 is Rs 104.4 lakh. The goal to fund is that figure, not Rs 25 lakh.

Why is the answer four times today's fee and not two and a half?

A school that raises fees 10% every year raises them on last year's fee, not on the fee from when your child started. The rises pile on each other. Cost inflation compounds exactly like an investment return, so a goal's cost has to be grown at its own inflation rate before anyone asks how much to save. Adding 10% of today's fee each year gives Rs 62.5 lakh, which is simple interest and leaves the plan Rs 42 lakh short.

What do you tell the client after the number?

Two things. First, the inflation rate is an assumption, not a fact: education costs may rise faster or slower than 10%, and a two point change moves the answer by tens of lakhs, since 25 x 1.08 to the 15th is about Rs 79 lakh and 25 x 1.12 to the 15th is about Rs 1.37 crore. The inflation assumption matters as much as the return assumption, so state it and show the range.

Second, the saving plan must beat this cost growth, not general inflation. If the portfolio earns 10% and fees rise 10%, the money only keeps pace; it does not get ahead. That is why goal planning uses the goal's own inflation rate.