The paper deals with the situation when in a complete continuous-time financial market an agent starts with initial capital \(x\) less than the amount \(C(0)=E[C/S_0(T)]\) required for perfect hedging the liability (without risk) at terminal time \(t=T\). The authors present a solution to the problem of minimizing the expected discounted loss as a solution to the relevant stochastic control problem. Also the supremum of the minimal expected loss, i.e. \[ \rho(x;C)=\sup_{\nu\in D} \inf_{\pi(\cdot)\in A(x)} E_{\nu}\left({{C-X^{x,\pi}(T)}\over{S_0(T)}}\right)^+, \] is proposed as a measure of the risk associated with hedging a given liability \(C\) at time \(t=T\). Here \(A(x)\) is the class of admissible portfolio strategies, \(S_0\) is a price of the risk-free instrument in the market; \({\mathcal P}=\{P_{nu}\), \(\nu\in D\}\) is a suitable family of probability measures (``scenarios''), \([0,T]\) is the temporal horizon during which economic activity take place. In addition to this ``max-min'' approach a related measure of risk in the ``Bayesian'' framework is discussed. Examples are worked out under various ``capital requirement'' and possible interpretations are analysed. Certain open problems are pointed out.