Convexity of a Risk Measure/Jensen's Inequality

Hi all,

Just to share with those who believe in rigour. Convexity of a risk measure is a simple Mathematical property called Jensen's Inequality i.e the Expectation of averages is greater than the function of averages.

Or E[F(X)]>F[E(X)]

Also;-

f(t(x1)+(1-t)f(x2)) <= tf(x1)+(1-t)f(x2)

This explains;-

1) Subadditivity of risk measures
2) Aggregation of risks
3) The reason why solvency II works or even why ERM works
4) Also applies to portfolio risk management, say why 20% cash verses 80% Derivatives is better than Exchange Traded Funds.

SEE BELOW;-



PS;- There is nothing special about Mathematics as a cognitive domain.