Cashflow matching

Cash flow matching is a process of 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.^[1] It is a subset of immunization strategies in finance.^[2] Cash flow matching is of particular importance to defined benefit pension plans.^[3]

Solution with linear programming

It is possible to solve the simple cash flow matching problem using linear programming.^[4] Suppose that we have a choice of ${\displaystyle j=1,...,n}$ bonds with which to receive cash flows over ${\displaystyle t=1,...,T}$ time periods in order to cover liabilities ${\displaystyle L_{1},...,L_{T}}$ for each time period. The ${\displaystyle j}$th bond in time period ${\displaystyle t}$ is assumed to have known cash flows ${\displaystyle F_{tj}}$ and initial price ${\displaystyle p_{j}}$. It possible to buy ${\displaystyle x_{j}}$ bonds and to run a surplus ${\displaystyle s_{t}}$ in a given time period, both of which must be non-negative, and leads to the set of constraints:

$${\displaystyle {\begin{aligned}\sum _{j=1}^{n}F_{1j}x_{j}-s_{1}&=L_{1}\\\sum _{j=1}^{n}F_{tj}x_{j}+s_{t-1}-s_{t}&=L_{t},\quad t=2,...,T\end{aligned}}}$$

Our goal is to minimize the initial cost of purchasing bonds to meet the liabilities in each time period, given by ${\displaystyle p^{T}x}$. Together, these requirements give rise to the associated linear programming problem:

$${\displaystyle \min _{x,s}\;p^{T}x,\quad {\text{s.t.}}\;Fx+Rs=L,\;x,s\geq 0}$$

where ${\displaystyle F\in \mathbb {R} ^{T\times n}}$ and ${\displaystyle R\in \mathbb {R} ^{T\times T}}$, with entries:

$${\displaystyle R_{t,t}=-1,\quad R_{t+1,t}=1}$$

See also

• Cash flow hedging
• Debt sculpting
• Duration gap
• Dedicated portfolio theory
• Fannie Mae
• Immunization (finance)

References