Case 033Risk management and limit breachesCore

A family office can buy 10% out-of-the-money one-year puts on its Rs 1,000 crore equity book for 2% a year. In a 30% fall the puts pay back 20%. How often must such falls occur for the puts to pay for themselves, and how does this compare with holding 10% less equity?

The situation

Sanrakshan Family Office holds Rs 1,000 crore of Indian equities, tracking the broad index. After a sharp fall last year, the family wants protection. A dealer quotes one-year index puts struck 10% below today's level, covering the whole book, for 2% of the book a year, Rs 20 crore, bought again each year.

If the market falls 30% in a year, the puts pay the fall beyond 10%, which is 20% of the book, Rs 200 crore. The alternative on the table is simply to hold 100 crore less equity in cash. For the comparison, assume equities are expected to return 12% a year and cash 6.5%.

Your task

How often must 30% falls occur for the puts to break even, and is buying them better or worse than holding 10% less equity?

Quick check

How often must a 30% fall happen for the puts to pay for themselves?

Worked solution

Try it on paper, then open one step at a time.

30-second answerThe answer to give first

A 30% fall must come about once every 10 years for the puts to pay for themselves. They cost Rs 20 crore a year and pay Rs 200 crore in a crash. Holding 10% less equity costs only about Rs 5.5 crore a year of expected return but saves just Rs 30 crore in a crash. Matching the puts' crash protection by selling would need about Rs 600 crore less equity, costing about Rs 33 crore a year.

Step 1How do you decide whether insurance is worth its premium?

House insurance loses money on average: the insurer collects more than it pays out, or it would not stay in business. People buy it anyway because a fire would be ruinous. A put is insurance, and it pays for itself on average only if the disaster it covers is more likely than its price implies. Here the price is Rs 20 crore a year and the payout Rs 200 crore, so the implied frequency is one 30% fall every 10 years.

The relationship
p×200=20⇒p=10% a year, or one 30% fall in ten yearsp \times 200 = 20 \quad\Rightarrow\quad p = 10\% \text{ a year, or one 30\% fall in ten years}
pyearly chance of a 30% fall
200payout in a 30% fall, Rs crore: (30% - 10%) of Rs 1,000 crore
20premium, Rs crore a year
What it says in wordsThe puts break even when the expected payout per year equals the premium paid per year.

That is a simplification worth saying out loud. The puts also pay something in falls between 10% and 30%, and more in falls deeper than 30%, so the true break-even uses the whole range of outcomes, not one scenario. The question to take to the data is how often your market has fallen 30% or more within a year: if that has happened less than one year in ten, the puts cost more than they return on average.

Step 2How does the protected book compare with simply holding less equity?

Put the three books on one chart. The puts give a floor: below a 10% fall the book loses at most Rs 120 crore including the premium, but in every year above that line it earns Rs 20 crore less than the unprotected book. Holding Rs 100 crore less equity gives no floor; it cuts every loss and every gain by a tenth, so a 30% fall costs Rs 270 crore instead of Rs 300 crore.

Puts cap the crash and cost 2% every year the crash does not come-400-300-200-1000+100+200+300-40%-30%-20%-10%0+10%+20%+30%Rs croreNo hedge: -300Puts: -12010% less equity: -270premium: Rs 20 crore lost in every calm yearNo hedgeWith puts10% less equityMarket move over the year
In a 30% fall the unprotected book loses Rs 300 crore, the book with 10% less equity loses Rs 270 crore and the book with puts loses Rs 120 crore, but the puts cost Rs 20 crore in every year the market does not fall more than 10%.
Rs croreMarket -30%Market -10%FlatMarket +20%Expected yearly cost
No hedge-300-1000+2000
Puts, 10% out of the money-120-120-20+18020 premium
Rs 100 crore less equity-270-900+1805.5 return given up
Rs 600 crore less equity-120-400+8033 return given up
To match the puts' Rs 120 crore loss in a 30% fall by selling equity, Sanrakshan would need to hold only Rs 400 crore, giving up about Rs 33 crore of expected return a year against Rs 20 crore of premium.
Step 3Which would you recommend to the family office?

Start with what the family is actually afraid of. If the fear is a deep crash, puts buy that protection more cheaply than selling enough equity to get the same floor, Rs 20 crore a year against about Rs 33 crore of forgone return. If the fear is ordinary volatility, a 10% or 15% fall, the puts do nothing until the market is down 10%, and holding a little less equity is cheaper and simpler. A middle path is to hold somewhat less equity and buy cheaper puts further out of the money, or to fund puts by selling upside calls, which gives up gains above a level in exchange for the premium.

Where candidates lose it

The common loss is judging the puts by last year. After a crash, protection feels obviously worth it, and candidates forget that the premium is paid in every calm year too. The break-even frequency is the question, not the last outcome.

The second is comparing puts with 10% less equity as if they were the same protection. They are not: one gives a floor, the other a smaller slope. Compare them at equal protection, or say clearly which risk the family is buying against.

What the interviewer asks next

  • Implied volatility doubles after a sell-off and the premium rises to 4%. What happens to the break-even frequency?
  • How would a collar, buying the put and selling a 15% out-of-the-money call, change the cost and the payoff?
  • Why do puts bought every year tend to lose money on average in most markets, and who earns that premium?

Company names and figures are illustrative.