Jared Weinstein

On the computation of local components of a newform

The problem. Let f be a cuspidal newform for Γ1(N) with weight k ≥ 2 and character e. There are well-established methods for computing such forms using modular symbols; see [Ste07]. Let πf be the corresponding automorphic representation of the adele group GL2(AQ).

Beyond Value at Risk: Forecasting Portfolio Loss at Multiple Horizons

We develop a portfolio risk model that uses high-frequency data to forecast the loss surface, which is the set of loss distributions at future time horizons. Our model uses a fully automated, semi-parametric fitting procedure that has its basis in extreme value statistics. We take account of distributional asymmetry, heavy tails, heteroscedasticity and serial correlation. Loss distributions are time aggregated by taking products of characteristic functions. We test loss-surface-implied forecasts of value at risk and expected shortfall out of sample on a diverse set of portfolios and we compare our forecasts to industry-standard risk forecasts that are based on asset and factor covariance matrices. The empirical results make a compelling case for the application and further development of our approach.

Erratum: “On the computation of local components of a newform”

Proposition 4.1 (1) of this article is not correct as stated. For instance, if e is the quadratic Dirichlet character of conductor 15, then there are two newforms in Sk(Γ1(15), e) with coefficients in Q, and these are twists of each other by the quadratic character of conductor 3, whereas the quantity u in the proposition is 0 here. I am grateful to Steve Donnelly for pointing out this example. The correct statement is the following. Let f be a newform of level Np, with p N , and character e = eNep, where eN has conductor prime to p and the conductor of ep is p . Let u = min(b r 2c, r − c).