# Cholesky updates of A’A

Updating the Cholesky factorization of when one or more columns are added to or removed from matrix can be done very efficiently obviating the re-factorization from scratch. Continue reading →

# Zero risk

What does it mean for a random variable to exhibit zero risk? It of course depends on the risk measure we are using to quantify it. Continue reading →

# Average value-at-risk of simple quadratic form

The average value-at-risk of a quadratic form , where is given by a particularly complex closed-loop formula which I’ll describe below.

# Strict monotonicity of expected shortfall

The **expected shortfall**, also known as **average value-at-risk** or **conditional value-at-risk**, is a coherent risk measure defined as

for for some .