Ratio-Balanced Maximum Flows

Abstract 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 modeled 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 i j be the part of security i's value used to secure account j. We are searching for a maximum flow that sends 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 i j is the unsecured part of account j and r j = s j / e j is the risk ratio of account j. Balancing the risk means to determine a maximum flow with the following property: if f i j > 0 and there is an edge from i to l then r j ≥ r l . In particular, if f i j > 0 and f i l > 0 then r j = r l . We give a polynomial time algorithm for finding such a maximum flow and also give an alternative characterization of the risk balancing maximum flow. It is the maximum flow minimizing ∑ j r j 2 e j .