Every concept, endless questions
Each concept is a small financial model. Read how it works, then draw a real interview question from it: forwards, backwards or what-if, with fresh numbers and a worked solution every time.
Forward runs a concept the usual way. Inverse runs it backwards, which is what separates understanding from memorizing.
Statistics
5 conceptsAnnualizing volatility
If each day’s return is independent of the last, the variances of the days add up — so a year of 252 days has 252 times the variance of one day, and √252 ≈ 15.9 times the volatility. That square root is the whole trick, and it runs in every direction: from daily to annual, from annual to a ten-day risk horizon, from a daily standard deviation to a daily value at risk. It is also where the assumption bites. Returns that trend make long-horizon risk larger than √t says; returns that mean-revert make it smaller. The rule is a default, not a law.
Regression beta and R-squared
Regressing a stock on the market splits its risk in two. The slope, beta, says how many percent the stock moves for each percent the market moves, and it is correlation rescaled by the ratio of the two volatilities — so a volatile stock can have a high beta with a modest correlation, or a low beta with a high one. R-squared, the correlation squared, says what share of the stock’s variance the market explains; the rest is idiosyncratic, the part a hedge cannot touch. Alpha is whatever return is left after paying beta its due. Mixing up beta and correlation is the most common mistake, and the formula makes it easy to avoid.
Standard error and confidence intervals
A sample average is itself a random number: run the strategy over a different stretch of days and you would get a different mean. Its standard error — the volatility of daily returns divided by the square root of the number of days — says how much. A confidence interval is just the mean plus or minus about two standard errors. The square root is the painful part: to halve the uncertainty you need four times the data. That is why daily strategy returns, with a mean of a few basis points buried under a standard deviation of a hundred, take years to tell apart from zero.
Two-asset portfolio variance
The variance of a sum is the sum of the variances plus twice the covariance, and every portfolio calculation is that identity with weights attached. Covariance is correlation with units: the correlation scaled up by both volatilities. Because variance is a quadratic in the weight, it has a minimum, and the weight that reaches it depends only on the two variances and their covariance — not on expected returns. Flip the sign on one asset and the cross term changes sign too, which is why a long-short spread between two highly correlated assets can be far less volatile than either leg, and a spread between two uncorrelated ones is more volatile than both.
Is a Sharpe ratio significant?
A Sharpe ratio is a mean divided by a volatility, so testing whether the mean is really above zero is testing the Sharpe — and the t-statistic turns out to be just the annual Sharpe times the square root of the number of years. That one line explains why track records are so hard to judge. A genuinely good strategy with a Sharpe of one needs about four years before it clears the usual bar, and a backtest that tried fifty variations is almost guaranteed to find one that clears it by luck. Significance is a property of the evidence, not of the strategy.