Private Wealth Management puzzles, solved step by step

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  1. 024A client has pledged Rs 100 of shares against a Rs 50 loan, a 50% loan-to-value. The stock gaps down 30% overnight, and the lender sells shares to bring the loan back to 50% of the collateral. How much stock is sold?Leverage and borrowingHardPrivate banking

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    How much of the remaining Rs 70 of shares does the lender sell?

    Show the worked solution

    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.

    A gap down forces a sale at the worst price to restore 50% LTVShares 100Loan 50BeforeLTV 50%Client's equity 50Shares 70Loan 50After a 30% gapLTV 71.4%Client's equity 20Shares 40Loan 20After the forced saleLTV 50%Client's equity 20sell 30,repay 30price -30%
    A 30% gap takes shares from 100 to 70 against a 50 loan, a loan-to-value of 71.4%, and the lender must sell 30 of shares and repay 30 of loan to reach 50% again, leaving 40 of shares against 20 of loan.

    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.

    The relationship
    50−S=0.5 (70−S)  ⇒  S=50−0.5×701−0.5=3050 - S = 0.5\,(70 - S) \;\Rightarrow\; S = \frac{50 - 0.5 \times 70}{1 - 0.5} = 30
    Sshares sold, used to repay the loan
    50the loan before the sale
    70the shares after the 30% gap
    0.5the loan-to-value the lender restores
    What it says in wordsThe sale is the excess loan divided by one minus the target loan-to-value.

    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.

    Where candidates lose it

    The trap is Rs 15: the loan reduction needed if fresh cash were used. It forgets that selling collateral shrinks the collateral too, and the lender who sells only Rs 15 is still above 50%.

    The second loss is treating the sale as neutral because equity is Rs 20 before and after it. The damage shows only in the recovery, and the interviewer wants that point made.

    What the interviewer asks next

    • What gap down would take the loan-to-value to 100%?
    • If the lender's target is 40% rather than 50%, how much is sold?
    • How would you structure a loan against a concentrated stock to survive a 30% gap?