Private Wealth Management puzzles, solved step by step

The typical client shrinks towards zero, even though the average grows 10% a flip. A head and a tail together multiply wealth by 1.6 x 0.6 = 0.96, so the client in the middle loses about 2% a flip. After 40 flips the average stands at 45 times the start, but the median client holds 0.44 of it and only 44% of clients are ahead. A handful of lucky paths carry the average.

Why does a positive average not help the client in the middle?

Think of a shop that raises a price 60% and then cuts it 40%. A Rs 100 item goes to Rs 160 and then to Rs 96, not back to Rs 100, because the cut is taken on the bigger number. When the whole portfolio is staked every time, returns multiply, and the growth one client actually lives through is the geometric average, not the arithmetic one. Here that is the square root of 1.6 x 0.6, which is 0.9798: a loss of about 2.0% a flip, while the arithmetic average says plus 10%.

Where has the average gone, then?

It sits with a very small number of clients who threw long runs of heads. The average is real, but it is an average across people, and no single client can collect it by waiting. To finish ahead after 40 flips a client needs at least 21 heads, which happens 43.7% of the time. Stretch it to 100 flips and the average reaches about 13,781 times while the middle client holds 0.13, about an eighth of what he began with.

What changes if he stakes only part of the portfolio?

Staking less cuts the damage of a tail more than it cuts the gain of a head. The Kelly fractionThe share of wealth to stake on a repeated favourable bet that maximises the long-run growth rate. Named after John Kelly, who derived it in 1956. for this coin is 0.5/0.4 minus 0.5/0.6, about 42% of the portfolio. At that stake a head and a tail multiply wealth by 1.25 x 0.833, which is 1.0417, so the typical client grows about 2.1% a flip instead of shrinking. The same favourable bet turns from ruinous to useful purely through position size. That is the point a wealth interviewer wants: volatility costs compound wealth even when the expected return is positive.

About Rs 11.2 lakh a year more to own. Rent is Rs 48,000 x 12, or Rs 5.76 lakh, a 2.88% yield. Interest at 8.5% on Rs 2 crore is Rs 17 lakh. The owner pays Rs 11.24 lakh more in cash each year, so the flat must rise about 5.6% a year in price just to match renting, before maintenance and tax effects.

Which two numbers are actually being compared?

Strip it down to the cost of living in the flat for one year. The tenant pays rent. The owner who borrowed the whole price pays interest to the bank; the part of the EMI that repays principal is not a cost, it is money moved from the bank account into the flat. So the comparison is rent against interest, and at a 2.9% rental yield against an 8.5% loan rate, interest is roughly three times the rent. Per month that is about Rs 141,667 of interest against Rs 48,000 of rent.

Is the Rs 17 lakh exact, given the loan amortises?

Close enough for an interview, and you should say why. On a 20 year loan the EMI is about Rs 173,565 a month, and first-year interest works out to about Rs 16.8 lakh because a little principal is repaid each month. The simple 8.5% x Rs 2 crore overstates year one by only about Rs 15,270, so it is the right number to say out loud. The gap narrows slowly over later years as the loan shrinks, but the owner's own money then sits in the flat and could have earned something elsewhere.

What would make owning worth it?

The owner is betting on the price. Owning with debt pays off only if the flat appreciates by more than about 5.6% a year, the gap divided by the price. Maintenance, property tax and registration costs push that hurdle higher; the tax deduction on home loan interest pulls it lower, within limits that you must confirm against current rules. Say the break-even rate and let the client judge whether that growth is likely. That is numeracy, not a view on property.

About 24 years. Rs 10 lakh to Rs 80 lakh is eight times the money, which is three doublings. At 9% the rule of 72 gives 72 / 9 = 8 years per doubling, so three doublings take about 24 years. The exact figure, the log of 8 over the log of 1.09, is 24.13 years, so the shortcut is off by under two months.

Why count doublings instead of years?

A staircase is easier to climb in your head than a ramp. Compound growth is a ramp, but every doubling takes the same number of years at a fixed rate, so you can turn it into equal steps. First ask how many times the money must double, then multiply by the years one doubling takes. Eight times is 2 x 2 x 2, three doublings, and at 9% each takes about 8 years.

How good is the rule of 72 at 9%?

Very good. The exact doubling time at 9% is the log of 2 over the log of 1.09, which is 8.04 years, against 8.00 from the rule. The rule of 72 is most accurate for rates around 8%, which is why it serves so well for Indian savings and equity assumptions. Three doublings carry the small error three times, which is how 24 becomes 24.13.

Then turn it into a client sentence. Rs 10 lakh at 9% needs a working lifetime, not a decade, to reach Rs 80 lakh, and the last doubling, from Rs 40 lakh to Rs 80 lakh, adds more rupees than the first two combined. That is the argument for starting early, said with numbers rather than slogans.

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.

About Rs 54 lakh. At 11% the direct plan grows Rs 50 lakh to about Rs 403.1 lakh in 20 years. At 10.2% the regular plan reaches about Rs 348.8 lakh. The gap is Rs 54.3 lakh, 13.5% of the direct plan's ending value, from a cost difference that sounds like less than one per cent.

Why is the gap so much larger than 0.8% sounds?

A slow leak in a water tank loses a little each hour, but it keeps leaking from a tank that is also being filled, and every litre lost is a litre that never reaches the tap. An annual fee is charged on the whole growing balance every year, and each rupee taken stops compounding for the rest of the period. Charged only on the starting Rs 50 lakh, 0.8% for 20 years would be Rs 8 lakh. Charged on a balance that grows several times over, it is Rs 54 lakh.

How do you estimate it without a calculator?

Use the ratio of the two growth factors. Each year the regular plan keeps 1.102 / 1.11 of what the direct plan keeps, about 0.9928, so it loses roughly 0.72% of the direct plan's value a year. Over 20 years that compounds to about 1 minus 0.9928 to the 20th, roughly 13.5% of the ending value. The direct plan ends near Rs 400 lakh, so the gap is a little over Rs 50 lakh. Two lines of arithmetic, and the answer is within a lakh or two.

Be fair to the regular plan when you say this to a client. The extra cost pays the distributor, and some investors value the advice and service they get for it. The numeracy point is only that the price of that service, over 20 years, is measured in tens of lakhs, and the client should see it in rupees before deciding.

About minus 0.6% a year: the deposit loses buying power. Tax at 30% takes 2.1 of the 7 points, leaving 4.9% after tax. Inflation of 5.5% is larger than that, so the real, after-tax return is roughly 4.9 minus 5.5, or minus 0.6%; exactly, 1.049 over 1.055 less one is -0.57%. The deposit is safe in rupees and still shrinks in what the rupees buy.

In what order do tax and inflation come out?

Tax first, because the tax office charges on the rupee interest, not on the real gain. The client keeps 70% of the 7%, which is 4.9%, and only then does inflation get its turn. Think of a salary rise of 7% in a year when your rent, school fees and groceries go up 5.5%: after tax you are behind. On Rs 10 lakh, the deposit pays Rs 70,000, tax takes Rs 21,000, and the client holds Rs 10,49,000 when the same basket of goods now costs Rs 10,55,000.

What rate would the deposit need to break even?

Work backwards. After tax the deposit must at least match inflation, so the pre-tax rate must be 5.5% divided by 0.7, which is 7.86%. At a 30% slab, a deposit has to pay about 7.9% just to stand still in real terms. That is the sentence a client remembers, and it frames every later conversation about where safety money sits. Slabs, surcharge and cess change; the 30% here is illustrative and you should confirm the current rates before using them with a client.

Say the limit too. The deposit still does its job if the job is to keep money stable in rupees for a known bill next year. The point is only that over long periods, a return below inflation after tax slowly shrinks what the money can buy.

Deferral leaves about 5.58 against 4.66, roughly 20% more. Paying 20% of each year's gain cuts the growth rate from 10% to 8%, and 1.08 to the 20th is 4.66. Deferred, Re 1 compounds at the full 10% to 6.73; tax of 20% on the 5.73 gain leaves 5.58. The unpaid tax stayed invested for 20 years, and its growth mostly belongs to the client.

Why does the timing of the same tax rate matter?

Picture a partner who takes their share of the shop's profit every evening, against one who leaves it in the till and settles once, years later. In the second case the partner's share keeps working for the shop in the meantime. A tax paid every year leaves the portfolio and stops compounding; a tax deferred to the end stays invested, and the growth on it accrues mostly to the client. It behaves like an interest-free loan from the tax office, repaid at the end.

Does the government lose from deferral?

Not in rupees. Add up the annual bills and the government collects about 0.92 over 20 years; under deferral it collects 1.15 in one go. Deferral grows the whole pie, so both the client and the tax office end with more nominal rupees; the tax office simply waits longer for its share. Expressed as a yearly rate, deferral turns an 8% after-tax return into about 8.98% a year.

Where does a wealth client meet this?

Anywhere tax is charged on realisation rather than accrual. Interest on a deposit is usually taxed as it accrues, while gains on a fund held for years are usually taxed only when units are sold. The rates are illustrative here, and the rules differ by product and change over time, so confirm them before comparing real products. The mechanism does not change: the longer the tax waits, the more the difference compounds.

About 8%, which is unusual but not rare. A clean year has probability 0.7, and seven independent clean years is 0.7 to the 7th, 0.082. Among 100 funds carrying the same 30% yearly chance of a loss, about eight would show a clean seven-year record by luck. The record makes a safer fund more likely, but it does not prove the fund is safe.

How do you get to 8% in your head?

Square, then keep multiplying. 0.7 x 0.7 is 0.49, call it a half. Two more years take it to about a quarter, 0.24, two more to about an eighth, 0.12, and the seventh year times 0.7 gives about 0.08. Each extra clean year multiplies the chance by 0.7, so the probability of an unbroken run falls away quickly but never reaches zero. It is the same arithmetic as a batsman who survives each over with probability 0.7 lasting seven overs.

So is the record evidence of anything?

Yes, but weigh it properly. Compare with a fund that truly has only a 10% chance of a losing year: it keeps a clean seven-year record 0.9 to the 7th of the time, 47.8%. A clean record is about 5.8 times more likely from the safer fund than from the riskier one, which shifts the odds but does not settle them. With hundreds of funds on sale, some risky ones will carry spotless records, and those are exactly the ones placed in front of clients.

There is a second problem the arithmetic hides. Years are not independent across funds: in a strong market decade most funds have no losing years at the same time, so seven clean years may say more about the decade than about the fund. Ask how the fund behaved in the worst year the market had, not how many years it avoided a loss.

Twenty years of spending, Rs 60 lakh short of 25 times. Rs 1 lakh a month is Rs 12 lakh a year, and Rs 2.4 crore over Rs 12 lakh is 20. Twenty-five times spending would be Rs 3 crore. Put another way, the client would withdraw 5% of the corpus in year one, against the 4% that the 25 times marker implies.

Why express a corpus in years of spending?

A client hears Rs 2.4 crore and feels rich; a client who hears twenty years of your current life hears a question: and after that? Dividing the corpus by annual spending turns an abstract balance into a length of time, which is what a retirement decision is actually about. It also removes the units: the same arithmetic works for a Rs 50 lakh corpus or a Rs 50 crore one.

Where does 25 times come from, and how firm is it?

Twenty-five times spending is the same thing as withdrawing 4% a year, the 4% ruleA planning guideline from US research by William Bengen in the 1990s: withdraw 4% of the starting portfolio, raised each year for inflation, and the money historically lasted about 30 years. from US retirement research. It is a starting point, not a law, because it came from one country's market history and a 30 year retirement. Higher inflation, longer lives and early retirement all argue for a larger multiple, and planners in India often use one. Say the marker, then say its limits.

Does twenty years of spending mean the money runs out in twenty years?

Only if the corpus earns exactly inflation. If it earns 2% a year above inflation and the client withdraws Rs 12 lakh in today's money at the start of each year, the money lasts about 25 years. Years of spending is a zero-real-return yardstick, so it is conservative when real returns are positive and optimistic when they are negative. That is why it is the right first number to say, and the wrong last one.

About Rs 107, exactly Rs 107.02 per Rs 100. The bond pays Rs 8 a year when the market now accepts Rs 7, so it carries an extra Rs 1 a year for ten years. At 7%, ten years of Rs 1 is worth about Rs 7.02 today, which is the premium over par. The duration shortcut, a modified duration of 6.71 times a one-point fall, gives nearly the same, Rs 106.71.

Why does the price rise when yields fall?

Imagine you rent out a flat at Rs 8,000 a month on a ten-year lease, and new flats in the building now rent for Rs 7,000. Your lease has become more valuable, and a buyer would pay extra for it. A bond's coupon is fixed, so when the market yield falls below it, the bond is worth more than par by the value of the extra coupon it carries. Here the extra is Rs 1 a year for ten years, discounted at the new 7%: an annuity factor of 7.02, so about Rs 7.

How does duration give the same answer?

Modified durationThe approximate percentage change in a bond price for a one percentage point change in yield. for this bond at 8% is 6.71, so a one-point fall lifts the price by about 6.71%. Duration is the slope of the price-yield curve, so it is a good straight-line guide for small moves and a slight underestimate for large ones. The gap here is Rs 0.31, the curve's bend, called convexity. The same bend works in the holder's favour the other way: at 9% the price falls only to Rs 93.58, less than the duration line predicts.

For a client, turn it into rupees on his holding. Rs 50 lakh of this bond gains about Rs 3.5 lakh from a one-point fall in yields, and loses a little less than that from a one-point rise. That is interest rate risk said in a sentence he can act on.