A portfolio manager runs Rs 500 crore with a soft stop at minus 5%, where risk is halved, and a hard stop at minus 10%. The book is down 7%. What return recovers the loss, and how does halving risk change the time it takes?

The book needs a 7.5% gain, which takes about 6 months at full risk but about 12 months at half risk. Rs 35 crore must be earned on Rs 465 crore. Halving risk halves the expected return, so the recovery takes roughly twice as long. In exchange, the chance of hitting the hard stop in the next three months falls from about 59% to about 28%. Restoring full risk once back above minus 5% cuts the wait to about 8 months.

Because the base has shrunk. A shop whose takings fall from Rs 100 to Rs 93 needs a 7.5% rise, not 7%, to get back, since 7% of 93 is only 6.51. Losing Rs 35 crore from Rs 500 crore is 7%, but earning it back on Rs 465 crore is 7.53%. The gap is small at 7% and grows fast: at the hard stop of 10%, recovery needs 11.1%, and a 50% loss needs 100%. That asymmetry is the whole reason drawdown limits exist.

Use the expected return as a planning rate, and say clearly that it is an average, not a path. At 15% a year, 7.53% takes ln(1.0753) over ln(1.15), about 6.2 months. At half risk the expected return is 7.5%, and the same recovery takes about 12.0 months, roughly twice as long. Many desks restore full risk once the book is back above the soft stop. Then the first Rs 10 crore comes at half risk in about 3.5 months and the last Rs 25 crore at full risk in about 4.4 months, 7.9 months in all.

Survival. The book has Rs 15 crore of room before the hard stop, 3.23% of today's capital. Over three months, full-risk volatility of 12% a year is 6%, and half-risk volatility is 3%. Ignoring the small expected gain, the chance of touching the hard stop within three months is about 59% at full risk and about 28% at half risk. A manager who is closed out cannot recover at all, so a slower recovery with a much lower chance of being shut down is usually the better trade.

Two assumptions carry it. The expected return is an average: the actual path will wander, and a recovery that takes six months on paper can take a year or never come. And the probabilities assume returns are roughly normal and independent from day to day, when a book in drawdown often has losing positions that keep losing together. The stops are there precisely because the planning numbers can be wrong, which is also the right answer if the interviewer asks why the manager cannot simply keep full risk and trust the strategy.