cashflow matching

It is possible to solve the simple cash flow matching problem using linear programming.{{Cite book|last1=Cornuéjols|first1=Gérard|url=https://www.cambridge.org/us/academic/subjects/mathematics/mathematical-finance/optimization-methods-finance-2nd-edition?format=HB|title=Optimization Methods in Finance|last2=Peña|first2=Javier|last3=Tütüncü|first3=Reha|publisher=Cambridge University Press|year=2018|isbn=9781107056749|edition=2nd|location=Cambridge, UK|pages=35–37}} Suppose that we have a choice of $j=1,...,n$ bonds with which to receive cash flows over $t=1,...,T$ time periods in order to cover liabilities $L\_\{1\},...,L\_\{T\}$ for each time period. The $j$th bond in time period $t$ is assumed to have known cash flows $F\_\{tj\}$ and initial price $p\_\{j\}$. It possible to buy $x\_\{j\}$ bonds and to run a surplus $s\_\{t\}$ in a given time period, both of which must be non-negative, and leads to the set of constraints:$$\backslash begin\{aligned\}$$

\sum_{j=1}^{n}F_{1j}x_{j} - s_{1} &= L_{1} \\

\sum_{j=1}^{n}F_{tj}x_{j} + s_{t-1} - s_{t} &= L_{t}, \quad t = 2,...,T

\end{aligned}Our goal is to minimize the initial cost of purchasing bonds to meet the liabilities in each time period, given by $p^\{T\}x$. Together, these requirements give rise to the associated linear programming problem:$$\backslash min\_\{x,s\}\; \backslash ;\; p^\{T\}x,\; \backslash quad\; \backslash text\{s.t.\}\; \backslash ;\; Fx\; +\; Rs\; =\; L,\; \backslash ;\; x,s\backslash geq\; 0$$where $F\backslash in\backslash mathbb\{R\}^\{T\backslash times\; n\}$ and $R\backslash in\backslash mathbb\{R\}^\{T\backslash times\; T\}$, with entries:$$R\_\{t,t\}\; =\; -1,\; \backslash quad\; R\_\{t+1,t\}\; =\; 1$$In the instance when fixed income instruments (not necessarily bonds) are used to provide the dedicated cash flows, it is unlikely to be the case that fractional components are available for purchase. Therefore, a more realistic approach to cash flow matching is to employ mixed-integer linear programming to select a discrete number of instruments with which to match liabilities.

• Cash flow hedging
• Debt sculpting
• Duration gap
• Dedicated portfolio theory
• Fannie Mae
• Immunization (finance)
• {{slink|Financial risk management#Investment management}}

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Category:Cash flow

Category:Corporate finance

Category:Derivatives (finance)

Category:Financial risk management

Category:Linear programming