Private Wealth Management puzzles, solved step by step

Down 4%. The index goes 100, 110, 99, a 1% loss. The 2x product goes up 20% to 120, then down 20% to 96. It resets its leverage each day, so over several days it compounds twice the daily moves rather than delivering twice the period's return. In a choppy market that difference always works against the holder.

Why is it not simply twice the index's 1% loss?

Imagine a shop that runs a 20% sale one day and a 20% price rise the next. The price does not come back to where it started: 100 becomes 80 and then 96. Equal percentage moves up and down do not cancel, and leverage doubles the size of each move, so it more than doubles the leak. The index loses 1% from the up-and-down, the 2x product loses 4%.

Where does the extra loss come from?

From volatility dragThe shortfall of compounded growth below the average return, caused by returns bouncing up and down. It grows roughly with the square of the size of the moves.. A move of plus x then minus x leaves you with 1 minus x squared. For the index that is 1 minus 0.01; for the 2x product it is 1 minus 0.04, four times the leak, because the drag grows with the square of the move. Double the leverage and you quadruple the drag.

Say what it means for a client. A daily reset product can be close to its promise over a day and far from it over a year, even if the index ends flat. It is built for short holding periods; held for months in a choppy market it can lose money while the index goes nowhere.

Rs 30 of the remaining Rs 70. After the gap the shares are worth Rs 70 against a Rs 50 loan, a loan-to-value of 71.4%. Selling S of shares repays S of loan, so the lender needs 50 minus S to equal half of 70 minus S, which gives S = 30. The client is left with Rs 40 of shares, a Rs 20 loan and Rs 20 of equity, sold out near the low.

Why does the sale have to be so large?

Think of a bucket with a hole that you are emptying to lower the water to a mark painted halfway up its side, while the bucket itself shrinks as you pour. Every rupee of shares sold repays a rupee of loan but also removes a rupee of collateral, so the sale has to be twice the gap it closes. At a 50% target, each rupee sold lowers the required collateral by only 50 paise, which is why Rs 15 of excess loan needs Rs 30 of sales.

What does the forced sale cost the client if the price recovers?

It locks the loss in. If the stock climbs back to where it started, a rise of 42.9%, his Rs 40 of shares becomes Rs 57.1, and after the Rs 20 loan his equity is Rs 37.1. Had he not been sold out, the same recovery would have put him back at Rs 50 of equity, so the forced sale turned a temporary fall into a permanent loss of about Rs 12.9. A top-up of Rs 15 of cash would have restored the ratio without selling anything.

Say what a private banker does with this. Lending against shares is sized for gaps, not for daily moves: a lender may sell at the open before the client can respond. The cushion before a margin callA demand from the lender to add cash or collateral, or accept a sale, when the loan grows too large relative to the value of the pledged securities. and the cash the client can raise overnight matter more than the interest rate on the loan.

About 16.7%. The loan stays at Rs 50 lakh, so the call comes when 50 is 60% of the collateral, which means collateral of 50 / 0.6 = Rs 83.3 lakh. Falling from Rs 1 crore to Rs 83.3 lakh is a drop of 16.7%. A 50% loan-to-value sounds like a large cushion, but the lender's trigger sits well inside it.

Why is the buffer smaller than the loan-to-value suggests?

A family pledges gold worth Rs 1 lakh for a Rs 50,000 loan and feels safe because the loan is half the gold. But the jeweller will call for more gold well before the gold is worth Rs 50,000. The margin call is set on the ratio of loan to collateral, and only the collateral moves, so the question is how far the denominator can shrink before the ratio hits the trigger. Going from 50% to 60% needs the collateral to lose one sixth of its value, not a tenth and not a half.

What happens when the call arrives?

The client must bring the ratio back, usually to where it started. At the trigger, restoring a 50% loan-to-value needs either Rs 8.3 lakh of repayment or Rs 16.7 lakh of extra shares pledged, within days. If he has neither, the lender sells shares at the low price, which locks in the loss. In a sharp market fall, many borrowers face the same call at once, which is why the forced selling tends to arrive at the worst prices.

One limit worth saying: interest that is added to the loan rather than paid pushes the loan up over time, so the real buffer is a little smaller than 16.7%. And a single volatile stock can move 16.7% in a week.

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.

14% at a 12% portfolio return, and only 2% at 6%. Rs 2 crore at 12% earns Rs 24 lakh, less Rs 10 lakh of interest, leaving Rs 14 lakh on his Rs 1 crore. At 6% the portfolio earns Rs 12 lakh, less the same Rs 10 lakh, leaving Rs 2 lakh. His return is twice the portfolio return less 10, so leverage helps only above the 10% borrowing rate.

Where does the 14% come from?

A shopkeeper who borrows to double her stock doubles her sales margin, but pays the lender first. If the stock earns more than the loan costs, she keeps the difference on the borrowed half as well as her own margin. On borrowed money the client keeps only the spread between the portfolio return and the borrowing rate, so leverage adds 12 minus 10, two points, to his own 12%.

What happens below the crossing point?

Leverage turns against him. Below the borrowing rate every point the portfolio falls short costs him two, so a flat year loses 10% and a 10% fall loses 30%. The loan interest is due whatever the market does, which is why the downside is steeper than the upside is generous.

Then add what the arithmetic leaves out. A loan against a portfolio usually has a margin requirement: if prices fall far enough, the lender asks for more collateral or sells holdings at the worst moment. The expected 12% is also not certain; a borrowing plan that works only if the portfolio beats 10% needs the client to understand how often it will not.