Cash Matching with Cumulative Portfolio Cash Flows

Summary

The document outlines a linear programming formulation for selecting securities to fund a known schedule of future payments at minimum initial cost. It recommends expressing required payments as a cumulative total through each month, then doing the same for the cash outflows generated by a candidate portfolio. The feasibility constraints require cumulative portfolio cash flows to cover cumulative obligations at every point in time, so the portfolio remains sufficiently funded throughout the payment horizon.

The security purchase amounts are the decision variables. Both the initial portfolio cost and each period’s cumulative cash flow are linear functions of those amounts, making the setup a linear program. The source gives the conceptual objective and constraints but omits the image containing the detailed securities and payment schedule, so it does not specify coefficients, timing conventions, or additional restrictions such as nonnegative holdings. Those details must be supplied from the actual instruments and liabilities.

Key ideas

• Minimize the portfolio’s cost at the start of the funding horizon.
• Represent scheduled payments as cumulative obligations through each period.
• Constrain cumulative portfolio cash flows to meet or exceed cumulative payments at every period.
• Use security purchase amounts as decision variables, with costs and cash flows expressed linearly.

Tags

# Linear programming cash match portfolio - how to formulate?


# Linear programming cash match portfolio - how to formulate?












How would you formulate this linear program in standard form? (ie objective function and constraints).

any help would be appreciated. I don't understand how to formulate this without having an equation for each month, but I'm not sure what that would equate to.

https://imgur.com/a/VzFWv

Could anyone give me any hints to begin?

## Answer by Alex C (score 0)

https://quant.stackexchange.com/a/36460

You are asked to find "The least expensive portfolio that meets or exceeds the scheduled tuition payments". So you have known payments $P_t$ in each future month. You can construct the known "cumulative payment curve" $CP_t=\sum_{i=1}^t P_i$ giving all payments due up to and including month t.

Any portfolio will produce cash outflows $O_t$ in each month, with cumulative curve $CO_t=\sum_{i=1}^t O_t$. You must minimize the cost of the portfolio today, suject to the inequalities $CO_t \ge CP_t$ for all t, which basically says that each step along the way the portfolio has generated enough cash for you to have been able to keep up with the required tuition payments so far.

Both the "cost of the portfolio today" and the "cumulative outflows" are linear functions of the decision variables, the amounts $w_j$ to be used to purchase the available securities $S_j$ for $j=1,\cdots,N$

Shown in full with attribution under the source's licence. Licence: CC BY-SA 4.0 (Stack Exchange)

This summary was written by Stratmill's research agent from the original; it is not a copy of the source.