  We consider the problems of estimation and optimization of utility-based
shortfall risk (UBSR), which is a popular risk measure in finance. In the
context of UBSR estimation, we derive a non-asymptotic bound on the
mean-squared error of the classical sample average approximation (SAA) of UBSR.
Next, in the context of UBSR optimization, we derive an expression for the UBSR
gradient under a smooth parameterization. This expression is a ratio of
expectations, both of which involve the UBSR. We use SAA for the numerator as
well as denominator in the UBSR gradient expression to arrive at a biased
gradient estimator. We derive non-asymptotic bounds on the estimation error,
which show that our gradient estimator is asymptotically unbiased. We
incorporate the aforementioned gradient estimator into a stochastic gradient
(SG) algorithm for UBSR optimization. Finally, we derive non-asymptotic bounds
that quantify the rate of convergence of our SG algorithm for UBSR
optimization.
