LP in Financial Planning, Explained with Examples

UPI Study › Courses › Quantitative Analysis › Free lessons › Lesson 11.2 Free with a free account

LP in Financial Planning

Quantitative Analysis · Chapter 11: Linear Programming ApplicationsSection 11.2 · lesson 49 of 5510 min · free with a free account

← Where the Budget GoesNext lesson: Balancing Risk and Return →

Earn 3 college credits for Quantitative AnalysisRecommended for credit by NCCRS · Math 250 · self-paced · one payment, lifetime access
Enroll in Quantitative Analysis · $250 →Download the syllabus (PDF)

In this lesson

Optimizing budgets, cash flow, and resource allocation.

  • Construct an LP for departmental or project budget allocation
  • Write flow-balance constraints for multi-period cash flow
  • Apply LP to working capital across short-term instruments
  • Solve a three-period investment LP for terminal wealth

Transcript

Everything said in the video, section by section (1,506 words). Read along, or search it.

LP Applications in Business · Section 11.2

LP in Financial Planning

Welcome to section eleven point two of Quantitative Analysis. This is where linear programming meets the finance function. Every financial plan is an allocation problem wearing a different suit. You have limited capital, competing uses, and time. So we will not re-derive the simplex method here. Instead we build three families of financial model. Budget allocation, multi-period cash flow, and working capital. Then we solve a full three-period investment problem, end to end. One warning before we start. The printed solution to that problem does not survive recomputation, and we will show why.

One Problem, Three Faces

Why LP Belongs in Financial Planning

Start with what these problems have in common. A treasurer weighs a marketing budget request against a capital equipment proposal. A chief financial officer schedules debt service alongside dividend distributions. A portfolio manager rolls short-term cash across treasury bills and commercial paper. Three different desks, one underlying mathematical problem. Which combination of choices maximises an objective? That objective might be return, terminal wealth, or net present value. And it is always subject to a binding set of resource and policy constraints. A spreadsheet heuristic hides its assumptions inside cell formulas. A linear programme makes every assumption explicit and every constraint testable. That is exactly what governance, lenders and auditors increasingly demand.

Floors And Caps

The Budget Allocation Model

The budget allocation model is the simplest of the three. A fixed pool of capital, distributed across departments, projects or categories. The decision variables are the allocations themselves. Marketing, research and development, capital expenditure, and so on. The objective is usually to maximise expected return. A dual framing hits a target return while minimising risk exposure. The total budget constraint says the allocations cannot exceed the pool. Then come the two-sided bounds, and these are what make the model realistic. Floors keep business-critical lines alive. Payroll, for instance. Caps stop the model concentrating everything in a single place. Drop the floors and the model will happily starve a critical department. Drop the caps and it puts the whole budget in the highest-return option.

Time Enters The Model

Multi-Period Cash Flow Optimisation

Most real financial decisions span more than one period. Cash received in period one can be reinvested in period two. Debt service in period three must be funded from earlier returns. A single-period model simply cannot express those links. So we index everything by time. The amount invested in asset j during period t becomes its own variable. The objective is either terminal wealth or the present value of future cash flows. Both are linear, so both are legal objectives for a linear programme. Then comes the heart of it, one flow-balance equality per period. Inflows must equal outflows exactly, in every single period. And decisions taken in period t appear on the right-hand side of period t plus one.

The Backbone

The Flow-Balance Constraint

Look at that balance equation on its own for a moment. On the left sit the inflows. Returns from investments maturing in this period, plus any new borrowing. On the right sit the outflows. New investments, debt service, and required distributions. The two sides are joined by an equals sign, not an inequality. That matters, because money does not appear or vanish between periods. Write one such equation for each period in the horizon. Together they form the backbone of the whole model. Get a single balance equation wrong and every schedule downstream is wrong.

The Treasury Desk

Working Capital Management

Working capital management is the treasury cousin of the same idea. The treasurer holds liquid funds and has to park them somewhere. Treasury bills, certificates of deposit, money-market funds, commercial paper. Each carries a return, and each carries a maturity. The objective is total interest income across the horizon. It is weighted by each instrument's return and by its holding period. The binding constraint is the cash floor. In every period, the immediately available portion must clear an operational minimum. Two hundred thousand dollars is a typical figure. Maturity constraints are what give that floor its teeth. Money locked in a one-year bond is not liquid, however healthy the balance sheet looks.

Worked Example 11.2

A Three-Period Investment Problem

Now the worked example, and we will carry it all the way to a number. A firm has two million dollars to invest. The horizon is three six-month periods, so eighteen months in total. Two instruments are available. Six-month treasury bills return two per cent per six-month period. They mature after one period. One-year bonds return three point five per cent per year. They mature after two periods, which is one full year. And there is one operating constraint. The firm must hold at least two hundred thousand dollars in available cash. That test bites at the end of every period, not only at the end of the horizon.

Steps 1 And 2

Decision Variables and Objective

Step one is to name the decisions. In each period we choose how much new money goes into each instrument. T one, T two and T three are the treasury bill purchases. B one and B two are the bond purchases. Notice there is no B three, and that is deliberate. A bond bought at the start of period three matures after the horizon closes. It could never pay off inside the model, so it is excluded. C sub t is idle cash carried at the end of period t. Idle cash earns nothing. Its only job is to satisfy the liquidity floor. Step two is the objective function, terminal wealth. It is one point zero two times T three, plus one point zero three five times B two, plus C three.

Steps 3 And 4

Flow Balance, Floors, Non-Negativity

Step three writes a balance equation for each of the three periods. Period one is easy, because the firm starts with two million dollars. T one plus B one plus C one equals two million. Period two is where the dynamics appear. The treasury bills bought in period one mature with interest. So the money available is one point zero two times T one, plus the cash carried forward. That total funds the new investments and the new cash balance. Period three collects two maturities at once. The period-two treasury bills, and the bonds bought back in period one. Step four adds the liquidity floor to each of the three cash variables. Only idle cash counts. A bond that has not matured is not liquid. And every variable is non-negative, because you cannot invest a negative amount.

Step 5

Which Instrument Actually Wins?

Step five. Before reaching for a solver, reason about the rates. A bond earns three point five per cent over a full year. Two consecutive treasury bill rolls earn one point zero two, times one point zero two. That is one point zero four zero four, or four point zero four per cent. Four point zero four beats three point five. So the treasury bill chain has the better yield, not the bond. The printed section states this comparison the other way round. Its own arithmetic, in the very same sentence, shows the opposite. Substitute the three balance equations into the objective and it collapses. Terminal wealth becomes two million, plus two per cent of every treasury bill dollar. Plus three point five per cent of every bond dollar. And the bond carries a second disadvantage, because it locks money away for two periods.

Step 6

The Optimal Solution

So what does the optimum actually look like? In every period, put the maximum into treasury bills and hold exactly the floor. Period one, one point eight million into bills and two hundred thousand in cash. That closes period one at two million and thirty-six thousand dollars. Period two rolls the proceeds forward. One million, eight hundred and thirty-six thousand goes straight back into bills. Period two closes at two million, seventy-two thousand, seven hundred and twenty dollars. Terminal wealth is two million, one hundred and ten thousand, one hundred and seventy-four dollars and forty cents. That is a gain of about five point five per cent over eighteen months. An all-bond plan would have returned two million, one hundred thousand, two hundred and sixty dollars. Now the correction, and it is a large one. The published solution table buys more treasury bills in period three than period three actually holds. And its stated terminal wealth is higher than this model can possibly reach.

The Price Of Liquidity

Sensitivity: What the Cash Floor Costs

Step six. What does the liquidity policy actually cost? The cash floor is a hard constraint, so it carries a price. With the bond dominated, terminal wealth has a clean closed form. It is one point zero two cubed, times two million minus the floor, plus the floor. Read the slope off that, and every extra dollar of floor costs about six point one cents. Relax the floor to one hundred thousand dollars and terminal wealth rises. It reaches two million, one hundred and sixteen thousand, two hundred and ninety-five dollars and twenty cents. Raise the floor to four hundred thousand dollars and wealth falls instead. It drops to two million, ninety-seven thousand, nine hundred and thirty-two dollars and eighty cents. That single policy change costs over twelve thousand dollars in eighteen months. This is the real business value of the model. Liquidity policy stops being a matter of opinion and becomes a priced choice. Formulate carefully, recompute everything, and let the arithmetic settle the argument.

About Quantitative Analysis at UPI Study

Quantitative Analysis (Math 250) is a self-paced online course of 55 video lessons in 13 chapters, from decision analysis and simulation to inventory and queuing models, project scheduling, linear programming and transportation models. It carries a NCCRS credit recommendation for 3 semester credits.

Every lesson transcript is free on pages like this one, and the videos are free with a free UPI Study account. Enrolling adds a practice quiz after each lesson, the graded work and final exam that earn the credit, lifetime access, and an official transcript on completion. UPI Study has helped over 130,000 students with college credits. ACE and NCCRS have over 2,100 cooperating colleges that will consider accepting credits; the final decision is with the universities.

Enroll now → Course details Download the syllabus All Math courses Every course with PRO →

Compare: Statistics vs Quantitative Analysis

Why take it here

Why students take Quantitative Analysis with UPI Study

Decide with numbersQuantitative Analysis teaches the models managers use to decide under uncertainty: decision trees, simulation, inventory and queuing models, project scheduling and linear programming. It is part of many business, management, operations and supply-chain programs.
Credit at a fraction of the priceThe same 3 credits can cost $750 or more at a US college. Here it is one payment, self-paced, with lifetime access.
Finish your degree soonerTransfer credits reduce the courses you still owe, so you can graduate earlier or lighten a heavy semester.
Strengthen an applicationA completed college-level course on an official transcript shows admissions and scholarship committees you can do the work.
Study on your own scheduleNo start dates, no deadlines. Watch the lessons, practice, and take the final when you are ready.
130,000+students helped with college creditsACE and NCCRS have over 2,100 cooperating colleges that will consider accepting credits; the final decision is with the universities.

Where the credit goes

Universities that work with NCCRS

Quantitative Analysis carries a NCCRS credit recommendation. These universities work with NCCRS; some are UPI Study's signed partners. ACE and NCCRS have over 2,100 cooperating colleges that will consider accepting credits; the final decision is with the universities.

Thomas Edison State UniversityThomas Edison State UniversitySigned partnerUniversity of the PeopleUniversity of the PeopleSigned partnerCharter Oak State CollegeCharter Oak State CollegeSigned partnerStrayer UniversityStrayer UniversitySigned partnerColumbia Southern UniversityColumbia Southern UniversitySigned partnerIndiana TechIndiana TechSigned partnerPenn StatePenn StateWorks with NCCRSSouthern New Hampshire UniversitySouthern New Hampshire UniversityWorks with NCCRSSUNYSUNYWorks with NCCRS +many more cooperating universitiesFind my college →

Find my college → Not listed? Ask Athena in the corner of this page, or book a 15-minute call.

FAQ

Questions students ask us

About LP in Financial Planning

How is linear programming used for budget allocation?
The decision variables are the amounts given to departments, projects or categories, and the objective usually maximizes expected return. A total budget constraint caps the sum of the allocations. Two-sided bounds make the model realistic: floors keep critical lines such as payroll funded, and caps stop the model putting everything into the single highest-return option. Without them, the solution can starve essential areas or over-concentrate.
What is a flow-balance constraint in a multi-period financial model?
A flow-balance constraint is an equation for each period stating that cash inflows equal cash outflows. Inflows include maturing investments with their returns and any new borrowing; outflows include new investments, debt service, required distributions and cash carried forward. It is an equality because money cannot appear or vanish between periods, and decisions made in one period show up as inflows in later periods when investments mature.
How do you compare rolling treasury bills with a one-year bond in an investment LP?
Compare compounded returns over the same horizon. Two consecutive six-month T-bills at 2% each grow a dollar to 1.02^2 = 1.0404, or 4.04% a year, beating a one-year bond at 3.5%, which also locks money up longer. Investing $1.8 million in rolling T-bills for three periods while holding a $200,000 cash floor ends with about $2,110,174, versus about $2,100,260 if the money goes into the bond first.
How much does a minimum cash balance cost an investment plan?
A cash floor is a binding constraint, so it has a price. In a three-period model where T-bills earn 2% per period, terminal wealth is 1.02^3 times ($2,000,000 minus the floor) plus the floor, so each extra dollar of required cash costs about 6.1 cents. Raising the floor from $200,000 to $400,000 lowers terminal wealth from about $2,110,174 to $2,097,933, a cost of over $12,000 in 18 months.

About the Quantitative Analysis course

How many college credits is Quantitative Analysis?
Quantitative Analysis (Math 250) carries a NCCRS credit recommendation for 3 semester credits. ACE and NCCRS have over 2,100 cooperating colleges that will consider accepting credits; the final decision is with the universities.
How much does Quantitative Analysis cost and what is included?
$250, paid once, with lifetime access: all 55 video lessons, a practice quiz after each, the graded work and final exam, and an official transcript when you finish. Students who want several courses choose UPI Study PRO instead.
Which universities accept UPI Study credits?
UPI Study has signed partner agreements with universities such as Thomas Edison State University, University of the People and Charter Oak State College, and works with universities such as Penn State, SNHU and SUNY. ACE and NCCRS have over 2,100 cooperating colleges that will consider accepting credits; the final decision is with the universities. Use "Find my college" on this page to check yours.
How long does Quantitative Analysis take?
It is self-paced with no deadlines. The 55 lessons total about 11 hours of video; most students add reading and practice time and finish in a few weeks to a few months, on their own schedule.
How are the exams done?
Quizzes and assignments are online in the course. The final exam is taken online with proctoring; a person on the UPI Study team checks assignments, usually within two days.
How do my credits reach my college?
When the course shows 100% complete, request the official transcript from your dashboard. It is sent to the college you name, and you can send it to more than one. Check your college first with "Find my college" on this page, or ask Athena.

Ready for the full course? Earn 3 credits in Quantitative Analysis.

55 video lessons, a quiz after each one, and the graded work that earns the credit. Recommended for credit by NCCRS.

Enroll in Quantitative Analysis →Download the syllabusEvery course with PRO →
Quantitative Analysis
3 credits · NCCRSEnroll →