Case 054Signal research and data tasksCore
Your model of daily stock returns has an out-of-sample R-squared of 0.4%. A senior researcher asks whether that is useless. Convert it to a correlation and to a rough annual Sharpe ratio for a strategy trading it across 300 stocks.
1The situation
At Nakshatrix Capital's superday you spend the morning on a dataset of two years of daily returns for 300 stocks and build a model that predicts each stock's next-day return. Fitted on the first half of the data and tested on the second, it explains 0.4% of the variance of next-day returns: an out-of-sample R-squared of 0.004.
In the afternoon you present to a senior researcher, who looks at the number and asks: is 0.4% not useless? What would it be worth as a strategy trading all 300 stocks every day?
2Your task
Convert the R-squared into a correlation, then into an approximate annual Sharpe ratio for a daily strategy across 300 stocks, and say which number you would defend and why.
Quick check
Is an out-of-sample R-squared of 0.4% on daily stock returns useless?
Worked solution
Try it on paper, then open one step at a time.
30-second answerThe answer to give first
Not useless: an R-squared of 0.004 is a correlation of 0.063, and across 300 stocks a day that is worth a Sharpe of about 2.7 after sensible haircuts. The fundamental law multiplies the correlation by the square root of the number of independent bets. Taken naively, 300 bets a day gives 17, which is a red flag, not a result. Allowing for stocks sharing factors and for the portfolio not holding the signal exactly brings it to about 2.7 before costs.
Step 1Why does an R-squared of 0.4% look so much worse than it is?
Because R-squared is a squared number, and squaring shrinks anything small. The square root of 0.004 is 0.063, the correlation between your forecast and the next day's return, which is the information coefficientThe correlation between a forecast and the outcome it forecasts, the standard measure of a return signal’s skill.. A correlation of 0.063 means you call the sign of the next day's return right about 52.0% of the time. That sounds like nothing until you remember a casino: on an even-money roulette bet with one green zero, the house wins 19 spins in 37, about 51.4%, and it is a very good business because it takes that edge thousands of times.
Step 2How do you get from a correlation to a Sharpe ratio?
Use Grinold's fundamental law of active managementGrinold’s result that the information ratio is approximately the IC times the square root of the number of independent bets taken per period.: the information ratio is the IC times the square root of the number of independent bets. If each of the 300 stocks were an independent bet, the daily ratio would be 0.063 x sqrt 300 = 1.10, and scaling by sqrt 252 days gives 17.4 a year. No strategy you will meet has a Sharpe of 17 after costs. The arithmetic is right; the assumption of 300 independent bets is not.
Step 3Which haircuts turn the paper number into one you can defend?
Two, each with a reason you can say. First, stocks move together through sectors and factors, so 300 names might be only about 30 truly independent bets a day; that divides the ratio by sqrt 10 and gives 5.5. Second, a real portfolio cannot hold the pure signal weights: position limits, neutrality constraints and trading slowly to save cost dilute it. The transfer coefficientThe correlation between the weights a signal asks for and the weights the portfolio actually holds; 1 means the signal is implemented perfectly. measures that dilution, and 0.5 is a common planning figure, giving about 2.7. Costs then take their share, which depends on how much of the daily signal is new each day.
| IC | information coefficient, 0.063 from the square root of R-squared |
| BR | breadth, independent bets per year: about 30 a day for 252 days |
| TC | transfer coefficient, how much of the signal the portfolio actually holds |
Close the presentation with the checks you would run next, because that is what the researcher is really probing. Is the out-of-sample year long enough to trust 0.063? With 75,000 stock-days its naive standard error is about 0.004, and larger in truth because stocks move together. Does the correlation hold within sectors, or is it one sector bet? How much of each day's signal is new, which sets turnover and cost? A small R-squared is normal for returns; a large implied Sharpe is the thing that needs defending.
Where candidates lose it
Many candidates apologise for the 0.4% and never convert it, which lets the interviewer conclude the model is worthless. Take the square root out loud: 0.063 is a respectable daily IC.
The opposite trap is announcing a Sharpe of 17 from the fundamental law with 300 independent bets. Stocks are correlated and portfolios are constrained; a candidate who does not haircut breadth looks as naive as one who dismisses the signal.
What the interviewer asks next
- How would you estimate the effective number of independent bets from the data?
- The model's R-squared in sample was 1.5%. What does the gap to 0.4% tell you?
- Would the same correlation be worth more or less on weekly returns, and why?
Asked at Squarepoint Capital, Quantitative Research, Paris, 2025 (Wall Street Oasis): construct a predictive model from it. After that, you present your result to a researcher for one hour.
Company names and figures are illustrative.