Private Wealth Management puzzles, solved step by step

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038A client borrows Rs 50 lakh against Rs 1 crore of shares. The lender makes a margin call when the loan reaches 60% of the collateral's value. How far can the shares fall before the call arrives?Leverage and borrowingWarm upPrivate banking
Try it first
How big a fall triggers the margin call?
Show the worked solution
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.
The Rs 50 lakh loan stays fixed while the collateral shrinks around it, so loan-to-value climbs from 50% to 60% when the shares fall only 16.7%, to Rs 83.3 lakh. After a 40% fall the loan is 83.3% of the collateral.
The relationship
$C^{*} = \frac{L}{m} = \frac{50}{0.60} = 83.3 \qquad \text{fall} = 1 - \frac{83.3}{100} = 16.7\%$
L the loan, Rs 50 lakh, fixed
m the loan-to-value at which the lender calls, 60%
C* the collateral value that triggers the call
What it says in wordsThe call comes when the collateral has shrunk to the loan divided by the trigger ratio.
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.
Where candidates lose it
The fast wrong answer is 10%, from subtracting 50 from 60. It treats loan-to-value as if it moved one for one with the share price, when the ratio's denominator is the thing falling.
The other slip is answering 50%, the fall that wipes out the client's equity, which ignores that the lender acts long before that. Solve for the collateral at the trigger and state the fall; then say what the client must do when it arrives.
What the interviewer asks next
What fall triggers the call if the client borrowed Rs 40 lakh instead?
How much must he repay at the trigger to restore a 50% loan-to-value?
Why do lenders apply bigger haircuts to mid-cap shares than to large-cap ones?

L	the loan, Rs 50 lakh, fixed
m	the loan-to-value at which the lender calls, 60%
C*	the collateral value that triggers the call

Try it first

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

What happens when the call arrives?

Where candidates lose it

What the interviewer asks next