Equity Research puzzles, solved step by step
- Puzzles
- 100
- Traced to a firm
- 19
- Topics
- 11
- Hard
- 30
029A stock's daily closes run 100, 120, 90, 110, 130, 91, 95, 140, 126, 133. Find its largest drawdowns, in order of size.Balyasny Asset ManagementLondon · 2025
Try it first
Which drop is the largest drawdown?
Show the worked solution
30% (130 to 91), then 25% (120 to 90), then 10% (140 to 126). Walk the series keeping the highest close so far. Each time the price falls below that running peak, track the lowest point until a new peak is set. 120 falls to 90 before 130 sets a new high; 130 falls to 91 before 140; 140 falls to 126 and has not recovered. Divide each fall by its own peak.
Why measure from the running peak rather than the start?
Picture a savings account you check every evening. If it once showed Rs 1,30,000 and now shows Rs 91,000, the loss you feel is from the best balance you ever saw, not from what you first deposited. A drawdown is the fall from the highest value reached so far to the lowest value before that high is beaten, divided by the high. The start of the series only matters until the price first rises above it.
Measured from the running peak, the three drawdowns are 130 to 91 at 30%, 120 to 90 at 25% and 140 to 126 at 10%, so the largest drawdown does not contain the lowest price in the series. How do you find them in one pass?
Carry two numbers as you read left to right: the running peak and the lowest price since that peak. When the price sets a new high, close the open drawdown, record it, and reset both numbers to the new high. At the end, close whatever is still open, then sort by size and return the first n. That is the programming version of the question, and it runs in a single pass over the prices.
Peak Trough Fall Drawdown Recovered? 130 (day 5) 91 (day 6) 39 30% Yes 120 (day 2) 90 (day 3) 30 25% Yes 140 (day 8) 126 (day 9) 14 10% Not yet Each drawdown is divided by its own peak, which is why a smaller rupee fall from a higher peak can rank lower. State the edge cases before the interviewer does. An open drawdown at the end of the series counts, even though it has not recovered. Two drops inside one drawdown episode, such as 130 to 91 and a later dip to 93 before a new high, are one drawdown, not two.
Where candidates lose it
The fast wrong answer picks the lowest price, 90, and measures from the start or from 120. The largest drawdown here does not contain the lowest price at all: 91 comes after a higher peak, so the fall is bigger.
In the coding version, candidates lose the open drawdown at the end, because it is only recorded when a new high arrives. Close it after the loop.
What the interviewer asks next
- How long did the 30% drawdown take to recover, and why do some funds report that number too?
- Write the maximum drawdown in one line using a running maximum.
- Why do portfolio managers often stop out on drawdown rather than on a price level?
Asked at Balyasny Asset Management, Quantitative Trading, London, 2025 (Wall Street Oasis):
an OA with a programming and data science problem. Programming asked to return the n largest drawdowns
054A stock's daily volatility is 1.5%. What is its volatility over a month of 21 trading days, and what is a rough 95% one-month value at risk on a Rs 10 crore position?AQR Capital ManagementGreenwich · 2022
Try it first
What is the monthly volatility?
Show the worked solution
About 6.9% a month, and a 95% one-month value at risk of about Rs 1.13 crore. Variance adds across independent days, so volatility scales with the square root of time: 1.5% x the square root of 21, which is 4.58, gives 6.87%. The one-tailed 95% point of a normal distribution is 1.645 standard deviations, so a bad month costs 1.645 x 6.87% = 11.3% of Rs 10 crore, about Rs 1.13 crore.
Why does volatility grow with the square root of time?
Think of someone taking random steps left or right along a lane. After 100 steps they are not 100 steps from where they began; many steps cancelled, and the typical distance is about 10 steps, the square root of 100. Daily returns behave the same way when one day does not predict the next. Variances add across days, so the standard deviation, which is the square root of the variance, grows with the square root of the number of days.
The relationship\sigma_1 daily volatility, the standard deviation of one day's return \sigma_{21} volatility over 21 trading days \sqrt{21} the square root of the number of days, 4.58 What it says in wordsMonthly volatility is daily volatility times the square root of the number of trading days in the month.With a monthly volatility of 6.9%, the worst one month in twenty sits beyond 1.645 standard deviations, a fall of 11.3%, which on a Rs 10 crore position is a loss of more than Rs 1.13 crore. How do you turn a volatility into a rupee value at risk?
Value at riskA loss threshold that a position should exceed only with a stated small probability over a stated period, such as 5% over one month. answers one question: how much would a bad month cost, where bad means the worst month in twenty. Under a normal distribution 5% of outcomes fall below the mean minus 1.645 standard deviations, so the 95% one-month VaR is 1.645 x 6.87% = 11.3% of the position. On Rs 10 crore that is about Rs 1.13 crore. Assume a zero expected return for the month; over one month the drift is small against the volatility.
What does the number leave out?
Two things, and saying them separates an analyst from a calculator. The square root rule needs independent days; when bad days cluster, as they do in a sell-off, real monthly risk is higher than 6.9%. And VaR marks the edge of the tail, not what sits inside it: it says nothing about how bad the worst 5% of months can get. Real returns have fatter tails than the normal curve, so a one-in-twenty month can cost well over Rs 1.13 crore.
Where candidates lose it
The classic error is multiplying by 21 and quoting 31.5%, a monthly volatility that would make an ordinary stock look like a lottery ticket. The interviewer asks daily against monthly precisely to see whether you know that variance, not volatility, adds across time.
The second loss comes at VaR: using 1.96, the two-tailed figure, instead of 1.645. VaR is about losses only, one tail of the curve. Say one-tailed out loud and the right multiplier follows.
What the interviewer asks next
- What is the annual volatility, using 252 trading days? (about 23.8%)
- What would the 99% one-month VaR be, and why is it not simply double?
- Why might the square root of time rule understate risk in a crisis?
Asked at AQR Capital Management, Quantitative Research, Greenwich, 2022 (Wall Street Oasis):
Specific statistics questions on financial concepts. daily vs monthly return, VAR, more that i don't remember
