Case 074Strategy evaluation and backtestsCore
A book has a mean daily P&L of Rs 2 lakh with a daily standard deviation of Rs 10 lakh. Give 95% intervals for tomorrow's P&L, for the average daily P&L over the next 250 days, and for the year's total.
1The situation
Ochrelane Capital's head of desk is reviewing a trading book whose daily P&L has averaged Rs 2 lakh with a standard deviation of Rs 10 lakh over a long history. Take those two numbers as known, take the days as independent, and take the shape as roughly normal for the moment.
He asks for three 95% intervals: for tomorrow's P&L, for the average daily P&L over the next 250 trading days, and for the total P&L over those 250 days. Then he asks which of the three he should trust least.
2Your task
Compute the three intervals, explain why their widths differ by factors of the square root of 250, and say which assumptions each one leans on most.
Quick check
The average daily P&L over 250 days has a standard error of 10 / sqrt(250). Does the year's total P&L have the same uncertainty?
Worked solution
Try it on paper, then open one step at a time.
30-second answerThe answer to give first
Tomorrow: Rs -17.6 to 21.6 lakh. The 250-day average: Rs 0.76 to 3.24 lakh a day. The year's total: Rs 190 to 810 lakh, about Rs 1.9 to 8.1 crore. One day's spread is the full 10 lakh; an average shrinks it by sqrt(250) to 0.63; a total grows it by sqrt(250) to 158. Trust tomorrow's interval least: it depends on the normal shape, which daily P&L usually breaks, while the other two lean on the central limit theorem and on the days being independent.
Step 1Why do the three intervals have such different widths?
Weigh one mango and the scale's error is the whole error; weigh 250 and divide, and the errors mostly cancel; weigh 250 together and the errors pile up, though not as fast as the count. Independent days add in variance, so n days have variance n times one day's: the total's standard deviation is 10 x sqrt(250) = 158.1 lakh, the average's is 10 / sqrt(250) = 0.632 lakh, and a single day's is 10. Each 95% interval is the centre plus or minus 1.96 of its own standard deviation. Tomorrow: 2 plus or minus 19.6, so -17.6 to 21.6 lakh. The average: 2 plus or minus 1.24, so 0.76 to 3.24. The total: 500 plus or minus 310, so 190 to 810 lakh.
| \sigma_{day} | standard deviation of one day's P&L |
| \sigma_{avg} | standard error of the mean of 250 independent days |
| \sigma_{total} | standard deviation of the sum of 250 independent days |
Step 2What does each interval say about the book?
Tomorrow's interval is the risk statement: a loss of up to about Rs 18 lakh is an ordinary day, and the book loses money on about 42% of days. The average's interval is the evidence statement: 0.76 to 3.24 excludes zero comfortably, which is the same fact as an annualised Sharpe ratio of 2 / 10 x sqrt(250) = 3.2. The total's interval is the budget statement: the year most likely lands between Rs 1.9 crore and Rs 8.1 crore, and the chance of a losing year on these numbers is about 0.1%. Three questions, three intervals, and the mistake is to answer one with another, usually by quoting the tight average interval when the question was about one day or one year.
| Quantity | Centre, Rs lakh | Standard deviation | 95% interval | Leans most on |
|---|---|---|---|---|
| Tomorrow's P&L | 2 | 10 | -17.6 to 21.6 | the normal shape of one day |
| Average over 250 days | 2 | 0.63 | 0.76 to 3.24 | independence of days; central limit theorem |
| Total over 250 days | 500 | 158 | 190 to 810 | independence of days; the mean staying 2 |
Step 3Which interval should the head of desk trust least?
Tomorrow's. It assumes one day's P&L is normal, and daily trading P&L is almost never normal: fat tails make a Rs 30 lakh loss, three standard deviations, far more common than the 0.1% a normal distribution allows, so the lower end of the one-day interval is the least reliable number of the six. The average and the total are sums of many days, and the central limit theorem makes sums closer to normal even when the days are not, so their shapes are safer. What those two lean on instead is independence and a stable mean: if P&L is autocorrelated, as it is for a book that holds positions for days, the effective number of independent days is below 250 and both intervals are too narrow; and a mean of Rs 2 lakh measured from the past is not a guarantee of the next 250 days. The limitation runs through all three: they are intervals for outcomes given known parameters, and the parameters are themselves estimates.
Where candidates lose it
The classic mistake is one standard error for everything: dividing by sqrt(250) for tomorrow's P&L or for the year's total, which makes a wildly risky day or year look safe. The average shrinks with n; the total grows with it; one day does neither.
The second miss is treating the tight interval on the average as proof the strategy works. It assumes independent days and a stable mean, and a book that holds positions for a week has far fewer than 250 independent observations in a year.
What the interviewer asks next
- Daily P&L has an autocorrelation of 0.3. Roughly how does that change the interval for the average?
- How would you build the one-day interval without assuming a normal distribution?
- The mean of 2 lakh was itself estimated from 500 past days. What is its standard error, and how does that change the year's interval?
Asked at Old Mission Capital, Prop Trading, Chicago, 2025 (Wall Street Oasis): asked to create confidence intervals on many different things like average return
Company names and figures are illustrative.