Editing Cashflow matching (section)

==Solution with linear programming==
It is possible to solve the simple cash flow matching problem using [[linear programming]].<ref>{{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}}</ref> Suppose that we have a choice of <math>j=1,...,n</math> [[Bond (finance)|bonds]] with which to receive cash flows over <math>t=1,...,T</math> time periods in order to cover liabilities <math>L_{1},...,L_{T}</math> for each time period.  The <math>j</math>th bond in time period <math>t</math> is assumed to have known cash flows <math>F_{tj}</math> and initial price <math>p_{j}</math>. It possible to buy <math>x_{j}</math> bonds and to run a surplus <math>s_{t}</math> in a given time period, both of which must be non-negative, and leads to the set of constraints:<math display="block">\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}</math>Our goal is to minimize the initial cost of purchasing bonds to meet the liabilities in each time period, given by <math>p^{T}x</math>. Together, these requirements give rise to the associated linear programming problem:<math display="block">\min_{x,s} \; p^{T}x, \quad \text{s.t.} \; Fx + Rs = L, \; x,s\geq 0</math>where <math>F\in\mathbb{R}^{T\times n}</math> and <math>R\in\mathbb{R}^{T\times T}</math>, with entries:<math display="block">R_{t,t} = -1, \quad R_{t+1,t} = 1</math>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|mixed-integer linear programming]] to select a discrete number of instruments with which to match liabilities.