Private Wealth Management puzzles, solved step by step

About Rs 100, and doubling volatility roughly doubles it to about Rs 200. With no interest or dividends, Black-Scholes prices an at-the-money call at very nearly 0.4 x stock price x volatility x the square root of time: 0.4 x 1,000 x 0.25 x 1 = Rs 100. The exact formula gives Rs 99.5. At 50% volatility the exact price is Rs 197.4. The option is priced as the expected value of its payoff, and that expectation grows with how far the stock can move.

What is Black-Scholes actually doing?

Imagine a rain insurance policy for an outdoor wedding that pays only if it pours. Its fair price depends on how uncertain the weather is: in a steady dry season it is nearly worthless, in a volatile monsoon it is dear. Black-Scholes prices an option as the average payoff over every path the stock could take, where the spread of those paths is set by volatility. The owner of a call keeps the upside of big moves and loses at most the premium on the downside, so a wider spread of paths is worth more.

At the money, with rates set to zero, the formula collapses to a clean shortcut. The standard normal distributionThe bell curve with mean zero and standard deviation one, used to describe the spread of possible stock returns in the model. has height 1 over the square root of 2 pi at its centre, which is 0.399, and that is where the 0.4 comes from.

Why would a wealth interviewer care about this?

Because structured products sold to private clients are bundles of options, and the client is often the seller of volatility without knowing it. A note that pays a high coupon while volatility is high is usually paying for an option the client has written, and the coupon rises with volatility for exactly the reason on this chart. The same shortcut shows time matters by its square root: a three-month option on the same stock costs about half the one-year option, Rs 49.8, not a quarter.

The limits: the shortcut is only good at the money and for short maturities with low rates. The model assumes constant volatility and smooth prices, which real markets break, and that is why traders quote different volatilities for different strikes.

His average cost is Rs 7.50 a unit, against an average NAV of Rs 8.33. Rs 100 buys 10 units at NAV 10, 20 units at NAV 5 and 10 units at NAV 10: 40 units for Rs 300. Because a fixed amount buys more units when the price is low, the cost per unit is the harmonic average of the prices, which is always at or below the simple average. At NAV 10 his 40 units are worth Rs 400, up 33.3%.

Why is his cost below the average price?

Spend a fixed Rs 100 on tomatoes every week. In the week they are cheap, the same Rs 100 fills twice the bag, so the cheap week makes up more of your total tomatoes. A fixed rupee amount automatically buys more units when the price is low, so the low prices carry more weight in the average cost than in the average price. That is rupee cost averaging, and it is arithmetic, not a trading skill.

Does this mean a SIP beats investing a lump sum?

No, and saying so is what earns the point. Rupee cost averaging guarantees a cost below the average price, not a better result than investing everything at once. Take a rising path of NAVs 5, 10 and 15. The SIP buys 36.67 units, worth Rs 550 at the end; Rs 300 invested at NAV 5 on day one buys 60 units, worth Rs 900. When prices mostly rise, money invested earlier does better; the SIP's real value is discipline and avoiding one badly timed lump sum.

The limit of the puzzle is its tidy V-shaped path, which flatters the SIP. On a path that only falls, the SIP still loses money, just less than a lump sum would have.

About Rs 17,500 on every Rs 1 lakh of return, before the benefit of deferral. Paid out as IDCW, the Rs 1 lakh is taxed at his illustrative 30% slab: Rs 30,000. Left in the growth option and later sold as a long-term gain, it is taxed at an illustrative 12.5%: Rs 12,500. The return is identical; only its tax label changes. Over ten years on Rs 10 lakh the gap compounds to about Rs 4.27 lakh.

Is the IDCW payout extra money?

No, and that misunderstanding is where most of the damage starts. Taking water out of your own tank does not give you more water. An IDCW payout is paid out of the fund's NAV, so the NAV falls by the amount paid; the client receives part of his own money, and in the payout option that money is taxed as income. The growth option leaves the same return inside the fund, where it is taxed only when units are sold and, if held long enough, at the capital gains rate.

Why does the gap grow over time?

Because the payout option is taxed every year, so the reinvested amount compounds at 7% after tax, while the growth option compounds at the full 10% and is taxed once at the end. Rs 10 lakh for ten years becomes about Rs 19.67 lakh in the payout option, even with every payout reinvested, against about Rs 23.95 lakh in the growth option after its tax. The rate gap and the deferral work in the same direction.

The payout option can still suit a client who needs regular cash and sits in a low slab. The rates here are illustrative: slab rates, capital gains rates, holding periods, exemption limits and deduction of tax at source all change, so confirm the current rules before advising on them.

About 40%. The chance a single stock survives is 95%. The chance all ten survive, if they are independent, is 0.95 to the power 10, which is 0.599. So the chance that at least one goes to zero is 1 minus 0.599, or 40.1%. Small risks that each look negligible add up quickly across a portfolio.

Why work with the chance that nothing goes wrong?

If each of ten wedding vendors has a 5% chance of letting you down, the day goes perfectly only if every one of them turns up. The chance that at least one thing fails is one minus the chance that everything works, and the chance that everything works is the product of each piece working. Adding the 5%s does not work, because it counts the unlucky years where two vendors fail twice.

Does this mean the portfolio is riskier with more stocks?

No, and the distinction is the adviser's real point. The chance of seeing at least one zero rises with the number of stocks, but the damage from each zero shrinks, because each stock is a smaller slice. Ten stocks at 10% each expect 0.5 zeros a year, costing about 5% of the portfolio on average, and the chance of two or more zeros is only 8.6%. With one stock the same 5% chance means a 5% chance of losing everything.

The limit is the word unrelated. Real stocks fail together in a crisis, which makes a year with several zeros more likely than independence suggests, and a year with none also more likely. With 30 stocks at the same odds, the chance of at least one zero rises to 79%, which is why a diversified client should expect to see a disaster in the statement sometimes.

About Rs 44,986 a month, and yes: total interest is about Rs 58.0 lakh, more than the Rs 50 lakh borrowed. At 0.75% a month over 240 months, the standard EMI formula gives Rs 44,986. Multiplied by 240 that is Rs 108.0 lakh paid in all. In the first year about 83% of the instalments go to interest, and principal only overtakes interest in year 13.

How do you get to the EMI quickly?

Start from the interest alone: Rs 50 lakh at 0.75% a month is Rs 37,500. The EMI has to cover that and chip away at the loan, so it must be more than Rs 37,500. The EMI is the monthly interest scaled up by a factor that spreads repayment over 240 months: (1 + i) to the n, divided by (1 + i) to the n minus 1. Here 1.0075 to the power 240 is about 6.01, so the factor is 6.01 / 5.01, about 1.20, and Rs 37,500 x 1.20 is about Rs 44,986.

Why is so much of the early EMI interest?

A long loan is like paying rent on money: the rent is charged on whatever you still owe, and in the early years you still owe almost all of it. Because interest is charged on the outstanding balance and the balance falls slowly at first, the first years of a 20-year loan are mostly interest, and the total interest ends up larger than the loan. That is also why a prepayment in the early years saves far more interest than the same prepayment late in the loan.

For a wealth conversation, the point is the comparison a client actually faces: prepaying the loan earns him the loan rate, after any tax benefit on interest, with no risk. The illustrative 9% is not a quoted rate; floating-rate loans also reset, which changes both the EMI and the tenure.

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 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.

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.

A loss weight of 1.5. He is indifferent when half the gain equals half the weighted loss: 0.5 x 1.5 = 0.5 x weight x 1.0, so the weight is 1.5. Refusing the flip tells you losses hurt him at least one and a half times as much as equal gains please him. For a client with crores invested, a Rs 1 lakh swing cannot threaten his wealth, so the refusal is loss aversion, not prudence.

How do you turn a refusal into a number?

Think of a child who is offered a toffee if a coin lands heads and must give back one if it lands tails, and still says no. A refusal of a positive bet reveals a weight on losses larger than the ratio of gain to loss, here 1.5 to 1. Write the felt value as half the gain minus half the weight times the loss and set it to zero. The halves cancel, and the weight is simply Rs 1.5 lakh over Rs 1 lakh. The idea has a name, loss aversionThe tendency to feel a loss more strongly than a gain of the same size. Described by Daniel Kahneman and Amos Tversky in prospect theory, 1979., from Kahneman and Tversky's prospect theory.

Why is this a trap for the client and not just a preference?

Because he judges each bet alone. Paul Samuelson described a colleague who refused one such bet but said he would take a hundred of them. Over 100 independent flips the expected gain is Rs 25 lakh, and the chance of ending behind is about 1.8%, because he loses only if fewer than 40 of the 100 flips land heads. Loss aversion applied one decision at a time rejects a set of choices that, taken together, almost never loses. A client who checks his portfolio daily and feels every red day is making the same mistake with his own money.

In the room, the good answer gives 1.5, then the reframing: show the client the portfolio of decisions, not the single flip. That is not persuading him to gamble; it is making sure he rejects bets for reasons he would still accept after seeing the whole picture.

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.