Editing Cashflow matching

'''Cash flow matching''' is a process of [[Hedge (finance)|hedging]] in which a company or other entity matches its cash outflows (i.e., financial obligations) with its cash inflows over a given time horizon.<ref>{{Cite news|title=Cash flow matching|url=https://www.washingtonpost.com/wp-srv/business/longterm/glossary/a_m/cash_flow_matching.htm|url-status=live|archive-url=https://web.archive.org/web/20121102185537/https://www.washingtonpost.com/wp-srv/business/longterm/glossary/a_m/cash_flow_matching.htm|archive-date=November 2, 2012|newspaper=The Washington Post}}</ref> It is a subset of [[Immunization (finance)|immunization]] strategies in [[finance]].<ref>{{cite web|url= https://excel.tv/understanding-financial-immunization-your-key-to-managing-interest-rate-risk/ |title= Understanding Financial Immunization: Your Key to Managing Interest Rate Risk
| website= excel.tv }}</ref> Cash flow matching is of particular importance to [[Defined benefit pension plan|defined benefit pension plans]].<ref>{{Cite web|url=https://www.gsam.com/content/dam/gsam/pdfs/institutions/en/articles/pension-solutions/2020/Cash_Flow_Matching_Feb_2020_locked.pdf?sa=n&rd=n|title=Cash Flow Matching: The Next Phase of Pension Plan Management|last=|first=|date=February 2020|website=Goldman Sachs Asset Management|archive-url=|archive-date=|access-date=}}</ref>

==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.

==See also==
*[[Cash flow hedge|Cash flow hedging]]
*[[Debt sculpting]]
*[[Duration gap]]
*[[Dedicated portfolio theory]]
*[[Fannie Mae]]
*[[Immunization (finance)]]
*{{slink|Financial risk management#Investment management}}

==References==
{{reflist}}

{{Financial_risk}}
[[Category:Cash flow]]
[[Category:Corporate finance]]
[[Category:Derivatives (finance)]]
[[Category:Financial risk management]]
[[Category:Linear programming]]