Tent Cash Flow Designs And Analysis For Gradient Cash Flow Lectures

Students learn better if fun and motivational aids are incorporated into instructional materials. This paper presents an innovative method of introducing students to the design and analysis of arithmetic gradient series cash flows in engineering economy courses. Engineering economy students are typically intimidated by arithmetic gradient series cash flows. Hence, it is beneficial to develop creative ways to make the material interesting and less formidable. Several profiles of arithmetic gradient series cash flows are presented along with techniques of deriving closed form equations for their net present values. The designs are based on classroom lecture notes of the authors and have been used extensively over several years to motivate students and ease their fear of the analysis of arithmetic gradient series cash flows. The pedagogical benefits derived from the tent models help students to have a better understanding of other cash flow profiles. The models are given interesting names such as “The Executive Tent,” “Saw-Tooth Tent,” “Cathedral tent,” and so on. MATLAB software modules were developed for a computer implementation of the general present value equation of the Basic Tent cash flow. Introduction Learning experiences should not only be challenging, but also fun. In the teaching of engineering economy, instructors continue to strive to develop new and interesting ways to present new concepts. Analysis of arithmetic gradient series cash flows is one of the most dreaded problems by many engineering economy students. In the attempt to put students at ease, the authors have developed an interesting graphical representation of arithmetic gradient series cash flows. The approach presents arithmetic gradient series cash flows in familiar shapes of tents. The cash flows are fun to design and analyze. Cash flow diagrams are, in general, used in the teaching and practice of engineering economic analysis to evaluate present and future worth receipts and disbursements of investment alternatives. Periodic receipts and payments usually occur in five different series: uniform (or equal) amount series, single present or future receipt, arithmetic gradient series, geometric series, and irregular series. However, arithmetic gradient series has many applications in real life P ge 8.095.1 Proceedings of the 2003 American Society for Engineering Education Annual Conference & Exposition Copyright © 2003, American Society for Engineering Education investment and more emphasis is usually placed on its understanding when teaching engineering economy. However, the analysis of this type of cash flow profile usually intimidates students. The understanding and analysis of cash flow diagrams is an integral part of engineering economy education especially at the undergraduate level. Several innovative methods of enhancing the teaching of engineering economy are in the literature, including incorporating spreadsheets into the classroom lectures , the need for curriculum enhancement by integrating research advances into course materials , and practical factors that increase the efficacy of teaching engineering economy. In this paper, several designs and a close-form analysis of some real-life arithmetic gradient series cash flow profiles are presented to help students have a better understanding of this and other cash flow profiles and their analysis. The pedagogical benefits derived from the tent models help students to have a better understanding of other cash flow profiles. The models are given interesting names such as “The Executive Tent,” “Saw-Tooth Tent,” “Cathedral tent,” and so on. MATLAB software modules were developed for a computer implementation of the general present value equation of the Basic Tent cash flow. Background of the Analysis Several real life personal and corporate financial dealings are beyond the single sums and the uniform series cash flow models; they tend more towards the arithmetic gradient series and the geometric gradient series models. The arithmetic gradient series cash flow involves an increase or decrease of a constant amount in the cash flow of each analysis period. Thus, the receipt or disbursement at a particular time is greater or lower than the receipt or disbursement at the preceding period by a constant amount. This constant amount is denoted by G, for “Gradient” amount. The total present worth of the arithmetic gradient series is calculated by using the present amount factor to convert each individual amount from time t to time 0 at an interest rate of i% per period and taking the summation to obtain the present worth. The finite summation results in a closed form expression for the particular cash flow profile under consideration. Real-life Applications of Arithmetic Gradient Series Arithmetic gradient series cash flows feature prominently in many contract payments. Good examples can be found in the contracts of sports professionals. The pervasiveness and high publicity of such contracts make the analysis of arithmetic gradient series quite appealing and economically necessary. A case study, which the first author frequently cites in his lectures, is the 1984 highly publicized contract of Steve Young, a quarterback for the LA Express in the former USFL (United States Football League). The contract was widely reported as being worth $40 million dollars at that time. The cash flow profile of the contract revealed an intricate use of various segments of arithmetic gradient series cash flows. In fact, a sports reporter commented that “the most amazing part of this whole deal is how little, relatively speaking, the Express is paying to get both Young and an enormous amount of publicity.” He concluded that an athlete “needs not only an agent to watch what the club is doing, but also an attorney to watch what the agent is doing” with the contract cash flow profiles. We might add that an economic analyst is also needed to verify the cash flow claims of agents and attorneys. Finding Young’s contract curious, Badiru assigned it as a case study for computational analysis of the present value. It was P ge 8.095.2 Proceedings of the 2003 American Society for Engineering Education Annual Conference & Exposition Copyright © 2003, American Society for Engineering Education found that, contrary to the $40 million touted in the press, the present worth of the contract was only about $5 million at that time. The trick is that the club included some deferred payments stretching over 37 years (1990-2027) at a 1984 present cost of only $2.9 million. The deferred payments were reported as being worth $34 million, which was the raw sum of the amounts in the deferred cash flow profile. It turns out that clever manipulations of an arithmetic gradient series cash flow can create unfounded perceptions of the worth of a professional sports contract. This explains why some sports professionals suddenly find themselves bankrupt shortly after receiving what they assume to be multi-million dollar contracts. Similar examples were found in reviewing the contracts of other sports professionals. Other examples: Barry Sanders’ deal with the Detroit Lions in the mid 1990s was reportedly worth $34 million. The contract negotiation went on for several months because there was a sticking point regarding the “structuring” of the payments. Likewise, the Brian Bosworth contract with the Seattle Seahawks in 1987 made a clever use of the structuring of the arithmetic gradient series. So, the use of the arithmetic gradient series is more common in real life than what one might suspect. Consequently, engineering economy students need to be taught to overcome the fear of the arithmetic gradient series. In pedagogical terms, the teaching of the topic should be made more interesting and less threatening. The tent cash flow concept has been found to be one suitable approach as discussed in a later section on outcome assessment. Design and Analysis of Tent Cash Flow Profiles Real-life arithmetic gradient series cash flows usually start with some base amount at the end of the first period and then increase or decrease by a constant amount thereafter. The nonzero base amount is denoted as AT starting at period T. The analysis of the present worth for such cash flows requires breaking the cash flow into a uniform series cash flow of amount AT starting at period T and an arithmetic gradient series cash flow with a zero base amount. The uniform series present worth formula is used to calculate the present worth of the uniform series portion while the basic arithmetic gradient series formula is used to calculate the arithmetic gradient series part of the cash flow profile. The overall present worth is then calculated as: uniform series arithmetic gradient series P P P = ± Figure 1 presents a conventional arithmetic gradient series (GS) cash flow. Each cash flow amount at time t is defined as At = (t-1)G. The standard formula for this basic arithmetic gradient series profile is derived as: