Testing a model against the data
This capstone turns the skill into a test. A model is a probability statement about returns, and the data can check it. If daily returns were normal, the loss exceeded on 1% of days would be the 99% value at risk -(μ+zσ) with z=Φ⁻¹(0.01)≈-2.326. Counting how many real days exceed that number, and comparing with 1% of all days, is a direct test of the normal model's tail.
Two numbers summarise the shape. Skewness is the third standardised moment and is zero for symmetric distributions. Kurtosis is the fourth standardised moment, 3 for a normal, so the excess kurtosis (kurtosis minus 3) is 0 for a normal and positive when the tails are heavier. A Student-t with ν degrees of freedom has excess kurtosis 6/(ν-4) for ν>4, which is why it can mimic the tails of returns. To simulate it with unit variance you divide the draws by √(ν/(ν-2)).
- Moments: mean, standard deviation, skewness and excess kurtosis describe location, spread and shape.
- Backtest a VaR: count exceedances and compare with (1-level)·n.
- Student-t with unit variance is a drop-in fatter-tailed alternative to the normal.
Lab · what's graded
- Sample moments25 pts
- A normal value at risk15 pts
- Counting exceedances15 pts
- Student-t draws20 pts
- A tail report on real returns25 pts