Private Wealth Management puzzles, solved step by step

About age 70.4 with no discounting, and about age 78.7 at 7%. Rs 50 lakh divided by Rs 40,000 is 125 months, a little over ten years. At 7% a year each later payment is worth less today, and the present value of the pension only reaches Rs 50 lakh after about 225 months. So the pension wins only if the retiree, or whoever it continues to, lives beyond the late seventies.

Why is the undiscounted answer too generous to the pension?

A friend offers you Rs 1,000 today or Rs 100 a month for a year. Even though Rs 1,200 is more, you would think about what the Rs 1,000 could do meanwhile. A rupee of pension received at 75 is worth less than a rupee in hand at 60, because the lump sum could have been invested for those fifteen years. Discounting at the rate the lump sum could earn puts both choices on the same footing.

What decides the choice beyond the break-even age?

Three things the formula does not see. Longevity: the pension is insurance against living long, which is exactly the risk a lump sum cannot cover; a life table for the client's age and health says how likely age 79 is. The rate matters most: the higher the return the client could earn on the lump sum, the later the break-even, and at 10% a year the pension's present value never reaches Rs 50 lakh at all, however long he lives. Then the details: whether the pension continues to a spouse, whether it rises with inflation, and how each option is taxed.

The ceiling explains the last point. At 7% the whole infinite stream is worth Rs 40,000 / (0.07 / 12), about Rs 68.6 lakh; at 10% it is worth Rs 48 lakh, less than the lump sum. Check the payer's strength too: a pension is only as good as the promise behind it.

The barbell, in both directions, because it has more convexity. Equal duration makes the two portfolios move alike for small changes in yield. For large moves the price curve's bend matters, and a barbell's is sharper: convexity of about 40 against 26. On Rs 100, a 4-point fall in yields leaves the barbell 1.41 ahead; a 4-point rise leaves it 0.90 ahead.

Why does equal duration not mean equal behaviour?

Two cars can be doing 60 at this instant and still be in different places a minute from now if one is accelerating. Duration is the speed at the current yield; convexity is how that speed changes as yields move, and it decides who is ahead after a large move. A 9-year zero's price bends much more sharply than a 5-year zero's, far more than twice as much, while the 1-year zero barely bends at all, so the barbell's mix bends more than the bullet.

If the barbell always wins, what is the catch?

Two catches. First, markets price convexity, so a barbell with the same duration usually yields a little less, and if yields sit still the bullet quietly earns more. Second, the edge assumes a parallel move. If short yields fall 1 point while 9-year yields rise 1 point, with the 5-year unchanged, the bullet is untouched and the barbell loses about Rs 3.54 per Rs 100. A barbell is a bet on large, parallel moves; a bullet is a bet on stability or on the curve's shape changing.

For a client portfolio, the practical reading is that duration alone does not describe interest rate risk. Both portfolios here have a modified duration of about 4.67, and a manager who reports only that number has hidden the shape of the bet.

About 3.3 losing runs of five or more days a year, and the streak says nothing about tomorrow. A run of five falls has probability 0.48 to the power 5, about 2.55%, and it can start on almost any of the 250 days, provided the day before was up. That gives roughly 3.3 runs a year, and a 97% chance of at least one. With independent days, tomorrow's chance of rising is still 52%.

How often should a five-day losing run turn up?

Toss a slightly unfair coin 250 times and look for five tails in a row. Any one stretch of five is unlikely, 2.55%, but there are about 245 stretches to look at. A run of five down days needs an up day, or the start of the year, followed by five falls, so the expected count is about 0.0255 x (1 + 245 x 0.52), roughly 3.3 runs a year. Across a year, the chance of seeing at least one is about 97%.

Why does the client feel the market is due?

Because people expect short sequences to look like long-run averages, so a run of losses feels like a debt the market must repay. If the days are independent, the market keeps no ledger: the chance of a rise after five falls is the same 52% as after five rises. The adviser's job is to take the streak out of the decision and bring the conversation back to the client's plan and time horizon.

The limit is the independence assumption. Real markets show some short-term momentum and some mean reversion at different horizons, and volatility clusters, so streaks are a little more common than a coin predicts. None of that makes five falls a reliable signal to buy.

A few hundred people a year; the build here gives about 477, with a sensible range of 250 to 700. Split the events into four branches: secondary sales in late-stage rounds, acquisitions, IPOs once lock-ins end, and ESOP buybacks. For each, multiply events a year by the share that pays out and by the people above Rs 25 crore per event. Illustrative assumptions give 530, less about 10% for people counted twice.

How do you structure an estimate with no data in front of you?

Estimating how many weddings a city hosts, you would not guess one number; you would split by season, by venue type and by guests per wedding. A market-sizing answer is a tree: each branch is a separate route to the answer, and each carries its own named assumption that the interviewer can challenge. Here the routes are the four ways a startup stake turns into cash: selling in a later funding round, the company being bought, the company listing, and the company buying back employee options.

Which assumptions would you defend, and which would you flag?

Flag the multipliers first. The number of people per IPO or secondary round who clear Rs 25 crore moves the total more than any event count, so say it is the weakest link. Then say what you would check: listing counts and lock-in dates, round sizes with a secondary component, and buyback announcements. Also name the double counting: a founder who sells in a secondary this year may list next year, and the same person can appear in two branches, which is why the build takes off about 10%.

Close with why a wealth desk asks this. A liquidity event is the moment new wealth appears and needs managing, and the few hundred people it creates each year are exactly the clients a private wealth team competes for. The limit: the answer swings with the funding cycle, so a boom year and a lean year can differ several times over.

About Rs 1.1 crore. First-year revenue is 0.8% of Rs 20 crore, Rs 16 lakh. Attrition works like extra discounting and asset growth offsets it, so the shortcut divides by 12% + 10% - 8% = 14%: Rs 16 lakh / 0.14 is about Rs 1.14 crore. Summing year by year, where growth and survival multiply rather than add, gives Rs 1.08 crore. Both round to Rs 1.1 crore.

Why does attrition belong in the discount rate?

A shopkeeper values a regular customer by the purchases he expects, but also by how likely the customer is to keep coming. A 10% chance of losing the client each year shrinks every future year's expected revenue by a further 10%, exactly as if the discount rate were 10 points higher; asset growth pushes the other way. So a growing, leaky revenue stream is valued like a perpetuity at the discount rate plus attrition minus growth.

Why do the shortcut and the annual sum differ, and what moves the answer most?

The shortcut adds the rates, which is exact only for continuous compounding; with annual steps, growth and survival multiply, 1.08 x 0.90 = 0.972 rather than 0.98, so the yearly sum is about 5% lower. Neither is wrong; say which you used. The lever is attrition: cutting it from 10% to 5% lifts the shortcut value from Rs 1.14 crore to Rs 1.78 crore, because the denominator falls from 14% to 9%. That is the arithmetic behind a private bank's spending on service and retention.

The limits: this values revenue, not profit, so the relationship manager's cost and the platform's cost must come off before anyone calls it value. Growth above the discount rate less attrition would make the formula break down, which is a warning that the assumptions, not the client, have become unrealistic.

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.

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.

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.

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.

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.