Compatibility between shape equation and boundary conditions of lipid membranes with free edges

Only some special open surfaces satisfying the shape equation of lipid membranes can be compatible with the boundary conditions. As a result of this compatibility, the first integral of the shape equation should vanish for axisymmetric lipid membranes, from which two theorems of non-existence are verified: (i) There is no axisymmetric open membrane being a part of torus satisfying the shape equation; (ii) There is no axisymmetric open membrane being a part of a biconcave discodal surface satisfying the shape equation. Additionally, the shape equation is reduced to a second-order differential equation while the boundary conditions are reduced to two equations due to this compatibility. Numerical solutions to the reduced shape equation and boundary conditions agree well with the experimental data [A. Saitoh \emph{et al.}, Proc. Natl. Acad. Sci. USA \textbf{95}, 1026 (1998)].Comment: 6 journal pages, 4 figure

Who Bears the Balloon Risk in Commercial MBS?

Much of the literature on the pricing of commercial mortgages underlying commercial mortgage-backed securities pools focuses on the effect of term default (default during the term of the loan), and ignores the possibility of balloon risk, the borrower\u27s inability to pay off the mortgage at maturity through refinancing or property sale. A contingent-claims mortgage pricing model that includes two default triggers—a cash flow trigger and an asset value trigger—may be used to assess the effect of balloon risk on the pricing of CMBS tranches. Simulations of cash flows for individual loans in a CMBS framework reveal how individual tranches are affected by balloon risk. Balloon risk is low at the whole-loan level, but under a number of scenarios total credit risk and balloon risk creep into investment-grade CMBS tranches and significantly impact their valuation