Ratio-Balanced Maximum Flows
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When a loan is approved for a person or company, the bank is subject to credit risk; the risk that the lender defaults. To mitigate this risk, a bank will require some form of security, which will be collected if the lender defaults. Accounts can be secured by several securities and a security can be used for several accounts. The goal is to fractionally assign the securities to the accounts so as to balance the risk. This situation can be modelled by a bipartite graph. We have a set S of securities and a set A of accounts. Each security has a value v_i and each account has an exposure e_j. If a security i can be used to secure an account j, we have an edge from i to j. Let f_ij be part of security i's value used to secure account j. We are searching for a maximum flow that send at most v_i units out of node i ∈ S and at most e_j units into node j ∈ A. Then s_j = e_j - ∑_i f_ij is the unsecured part of account j. We are searching for the maximum flow that minimizes ∑_j s_j^2/e_j.
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