Consider the following two-stage stochastic optimization problem \[ \min f_{1}\left( x\right) +\mathcal{R}\left( f_{2}\left( x,\tilde{z}\right) \right) ,\tag{1} \] where \(f_{1}\left( x\right) \) and \(f_{2}\left( x,\tilde{z}\right) \) denote the first and second stage cost functions, \(\mathcal{R}\left( \cdot \right) \) is a coherent risk measure, including the expectation as a special case, the vector \(x\in \mathbb{R}^{n}\), \(\mathbb{R}^{n}\) is a finite \(n\)-dimensional real Hilbert space equipped with inner product \(<\cdot ,\cdot >\) and \(\tilde{z}\) denotes a random quantity. The authors develop an approach to deal with the case where \(f_{1}\left( \cdot \right) \) and \(f_{2}\left( \cdot ,\cdot \right) \) are convex linear-quadratic. They show that under a standard set of regularity assumptions this two-stage quadratic stochastic optimization problem with measures of risk is equivalent to a conic optimization problem that can be solved in polynomial time.