Private Wealth Management puzzles, solved step by step

About Rs 124 crore a month, on these assumptions. Fifty lakh people in households of four is 12.5 lakh households. If half of the top 20% run a SIP of about Rs 8,000, 15% of the middle 40% run Rs 3,000 and 2% of the rest run Rs 1,000, that is 2.1 lakh SIP households and about Rs 123.5 crore. Every input is an illustrative assumption, and the structure matters more than the figure.

Why segment before you multiply?

Estimating the sweets a wedding hall sells, you would not multiply every guest by the same number of laddoos: children, adults and the groom's uncles eat very differently. An average taken across the whole city hides the fact that a small, well-off slice contributes most of the SIP money. Split households into income bands, give each band its own share investing and its own ticket size, and the estimate becomes something you can defend line by line.

How do you check the answer is sane?

Test it two ways. Per SIP household the average works out to about Rs 5,881 a month, which should look plausible against what families in those bands earn. Per resident it is about Rs 247 a month. A sizing answer is judged on whether each assumption is stated and each check is run, not on hitting a number the interviewer has in mind. If you know an official total for SIP flows, compare your city's share with its share of the country's income, and say which assumption you would move if the two disagree.

Then name the sensitive line. Upper income households supply about 81% of the total, so a change in their share investing or ticket size moves the answer far more than anything in the lower bands. That is where you would spend a real week of research.

Parity fails by about Rs 45.42: the call is cheap relative to the put. Parity says call minus put equals the stock minus the strike's present value, 1,000 minus 1,000 / 1.07, which is Rs 65.42. The market prices it at Rs 20. Buy the call, sell the put, short the stock and lend Rs 934.58: that collects Rs 45.42 today and nets to zero at expiry at any stock price, assuming no dividends.

Why must a call, a put and a bond be tied together?

Two routes to the same destination must cost the same, or everyone takes the cheaper one. Owning a call and lending the strike's present value gives you, in a year, the stock if it ends above Rs 1,000 and Rs 1,000 in cash if it ends below. Owning the stock plus a put gives exactly the same thing. Two portfolios with identical payoffs in every state must cost the same today, and that equality is put-call parity. Rearranged, call minus put must equal stock minus the present value of the strike.

How do you prove the trade has no risk left?

Check both ends. If the stock ends at Rs 1,300, the call pays Rs 300, the put expires, you buy back the stock for Rs 1,300 and the loan returns Rs 1,000: 300 minus 1,300 plus 1,000 is zero. If it ends at Rs 700, the call expires, the put costs you Rs 300, the stock buyback costs Rs 700 and the loan returns Rs 1,000: zero again. Every payoff cancels at expiry, so the Rs 45.42 collected today is kept, worth about Rs 48.60 a year later.

Then say what would dissolve the gap in real life. An expected dividend with a present value of about Rs 45.42 would make these prices consistent, because the short seller must pay it. Costly stock borrowing, early exercise of American-style options and wide dealing spreads also eat into it. For a private banking client, parity matters because structured notes are built from exactly these pieces, and it is how you check whether a note is fairly priced.

About 14.25 years by the rule, 14.27 years exactly. The rule of 114 says divide 114 by the rate: 114 / 8 = 14.25. The exact answer is log 3 over log 1.08, which is 14.27, about a week longer. The rule works because tripling takes log 3 over log 2, about 1.585 times, as long as doubling, and 72 x 1.585 is about 114.

Why does a tripling rule exist at all?

If a train takes 9 hours to cover some distance, it takes about 14 hours to cover 1.585 times that distance at the same speed. Compound growth works the same way on a log scale. The time to reach any multiple is the log of that multiple divided by the log of one year's growth, so every multiple's rule is the doubling rule scaled by log of the multiple over log 2. For tripling that scale is 1.585, and 72 x 1.585 is 114.1, rounded to 114.

Where does the shortcut stop being good enough?

Both rules are tuned to rates near 8%. Between 6% and 10% the rule of 114 is within about two months of the exact answer, which is fine for a client conversation; at 4% it is off by about half a year. At low rates it says too long, at high rates too short, as the bottom of the figure shows. When the rate is far from 8%, say the rule, then say which way it errs.

The client version: at 8%, money roughly triples in 14 years and roughly doubles in 9, so a 35 year old's savings can triple before retirement at 49 and nearly multiply ninefold by 63. Two rules, one sentence, no calculator.

He lost Rs 32 lakh, about -5.4% a year, while the manager reports plus 8%. Year one made Rs 20 lakh on Rs 1 crore. The client then had Rs 5.2 crore invested when the 10% fall came, costing Rs 52 lakh. He put in Rs 5 crore and holds Rs 4.68 crore. Both numbers are honest: 8% measures the manager's skill, the money-weighted loss measures the client's outcome.

Why can both numbers be right?

A bus that averages 60 km an hour tells you about the driver, not about a passenger who boarded only for the slow stretch through traffic. A time-weighted returnA return that chains each period growth rate together, so it ignores when money was added or withdrawn. Used to judge a manager. chains the period returns and ignores the size of the balance, so it judges the manager; a money-weighted returnThe internal rate of return on the actual rupees the client put in and took out, so it depends on the timing and size of his flows. weighs each period by the rupees actually at work, so it measures the client. The manager did not choose when the Rs 4 crore arrived; the client did.

How do you get the client's yearly rate?

Find the rate that makes his flows add up. Rs 1 crore invested for two years plus Rs 4 crore invested for one year must grow to Rs 4.68 crore. Solving gives a money-weighted return of about -5.4% a year, the number that describes what actually happened to his money. The quadratic is quick: with x as one plus the rate, x squared plus 4x equals 4.68, so x is about 0.9462.

The wealth lesson is behavioural. Clients tend to add money after strong years and pull it after weak ones, so their money-weighted results often trail the funds they hold. Showing a client both numbers, and why they differ, is one of the most useful conversations an adviser can have.

Fund B, by about 0.12 points a year. Fund A's trading costs 150% turnover x 0.4%, which is 0.60%, so its all-in cost is about 1.60%. Fund B's trading costs 20% x 0.4%, 0.08%, for an all-in 1.48%. The lower visible fee belongs to the more expensive fund. On Rs 1 crore at 12% gross for ten years, the gap is about Rs 2.9 lakh.

Where does the hidden cost come from?

A shopkeeper who keeps swapping his stock pays the wholesaler's margin and the transport every time, even if the shelf price never changes. Every time a fund sells one holding and buys another, it pays brokerage and moves prices against itself, and that cost comes out of the fund's value rather than out of the stated expense ratio. Turnover of 150% means the fund replaces its whole portfolio one and a half times a year, so it pays the round-trip cost one and a half times.

How large is 0.12 points in rupees?

Small each year, visible over a decade. On Rs 1 crore with a 12% gross return, Fund A compounds at 10.40% to about Rs 269.0 lakh in ten years and Fund B at 10.52% to about Rs 271.9 lakh. A 0.12 point gap is about Rs 2.9 lakh on Rs 1 crore over ten years, and the investor never sees it on a statement. The same method matters more for small-company funds, where the impact cost of each trade is larger than 0.4%.

Say the limits. Turnover is only a cost if the trading does not add return; a high-turnover manager may earn back more than 0.6%. And the 0.4% round trip is an assumption, which varies with the size of the fund and the stocks it trades. How expense ratios and trading costs are disclosed changes with regulation, so confirm the current rules before comparing real schemes.

About -1.1% a year, on a real gain of zero. Tax takes 1.2 points of the 6%, leaving 4.8%, and inflation of 6% then takes more than that: 1.048 over 1.06 less one is -1.13%. Before tax the client only kept pace with prices, so the whole tax bill fell on inflation. His effective tax rate on his real gain is not 20%; it is unbounded, because the real gain was nothing.

Why does a 20% tax take more than 20% of the real gain?

Imagine your salary rises exactly as fast as prices, and the tax office treats the whole rise as new income. You are no better off, yet you pay more tax, so you end the year worse off. A tax on nominal gains also taxes the part of the gain that only compensates for inflation, so the tax falls more heavily on the real gain than the headline rate suggests. Here the real gain before tax is zero, so every rupee of tax is a rupee of buying power lost.

What happens at higher nominal returns?

The distortion shrinks but does not vanish. The lower the real return, the larger the share of it the tax takes, because the tax is sized on the nominal gain. The table runs the same 6% inflation and 20% tax across nominal returns: at 10% the real gain before tax is 3.77% and after tax 1.89%, an effective tax of about 50% on the real gain.

Tax systems sometimes correct for this by indexing the cost of an asset to inflation before computing the gain, which taxes only the real part. Whether and where indexation applies has changed over time and differs by asset, so confirm the current rules. The interview answer is the mechanism: without indexation, inflation raises the true tax rate on savers.

About Rs 49,157 if he sells in ten years, not Rs 80,000. Buying back the same fund lowers his cost base by Rs 4 lakh, so his eventual gain, and the tax on it, rise by the amount he saved. The harvest mostly defers Rs 80,000 of tax rather than removing it. At a 10% return, paying Rs 80,000 in ten years instead of today is worth Rs 80,000 less Rs 30,843, about Rs 49,157.

Why does the saving come back?

Think of borrowing from a friend without interest: the cash is useful now, but the debt is still there. When the client buys the same fund back at the lower price, the cost the tax office will later subtract from his sale price is Rs 4 lakh lower, so the taxable gain on the sale is Rs 4 lakh higher. At the same 20% rate that is Rs 80,000 of extra tax, exactly the amount saved today. The harvest has not cancelled the tax; it has moved it.

So what is the harvest worth, and what does it depend on?

It is worth the value of an interest-free loan of Rs 80,000, which depends on how long the client holds before selling. The longer the holding period, the more the harvest is worth, because the repayment is further away. At 10%, the harvest is worth about Rs 7,273 if he sells after one year, Rs 30,326 after five, Rs 49,157 after ten and Rs 68,109 after twenty. It is worth more if the later gain is taxed at a lower rate than the gains offset today, and less if trading costs eat into it.

Two conditions to state out loud. The saving only exists if there are gains to offset now, or losses can be carried forward. And some tax systems disallow a loss when the same security is bought back within a set window; rules differ by country and change, so confirm the current position before recommending the move to anyone.

A loss of about Rs 84 lakh, or -16.75%, and in some years more. If returns are roughly normal, one year in twenty falls more than 1.65 volatilities below the average. That is 8% minus 1.65 x 15%, or -16.75%, which on Rs 5 crore is about Rs 83.75 lakh. It is a threshold, not a worst case: the average year beyond it loses about 23%.

Why translate volatility into rupees?

A weather forecast that says the standard deviation of rainfall is 40 mm helps nobody pack; a forecast that says one monsoon in twenty floods the ground floor tells a family what to prepare for. A client cannot feel 15% volatility, but he can feel a Rs 84 lakh loss in one bad year, and he can decide whether he could live with it. The job is to turn the statistic into a rupee figure and a frequency.

What should you say about the limits of this number?

Three things. It is a cut-off, not a floor: when a bad year comes, the average loss beyond the cut-off is about 23%, near Rs 1.15 crore. Real market returns have fatter tails than the normal curve, so the true one-in-twenty loss is usually worse than the formula says. And one in twenty does not mean once every twenty years on a schedule; two such years can arrive back to back.

For a sterner test, one year in a hundred sits about 2.33 volatilities down: 8% minus 34.95% is -26.95%, about Rs 1.35 crore. Giving the client both lines, one in twenty and one in a hundred, is more honest than a single number.

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.