ALTERNATIVE MODELS FOR FINDING CUMULATIVE PAID CAPITAL AND REMAINING BALANCE FOR A UNIFORM SERIES OF CASH FLOWS.

[[[sigma].sup.t].sub.j=1] [C.sub.j] = P(i/[(1 + i).sup.n] - 1)([(1 + i).sup.t] - 1/i)

and finally:

[[[sigma].sup.t].sub.j=1] [C.sub.j] = P [(1 + i).sup.t] - 1/[(1 + i).sup.n] - 1 (14)

It can be noted that

[(1 + i).sup.n] - 1/i is (F/ A,i,n)

and

[(1 + i).sup.t] - 1/i is (F/ A,i,t)

therefore, Eq. (14) be rewritten as follows:

[[[sigma].sup.t].sub.j=1] [C.sub.j] = P[(F/ A,i,t)/(F/ A,i,n)] (15)

Stated differently, the cumulative recovered investment or repaid capital at any time t of a project's life in a uniform series of a cash flow, is proportional to the initial payment. The proportionality factor is the ratio between the uniform series compound amount factor up to time t, and the same factor up to time n.

It can be noted that the developed model is independent of the value A. To the authors best knowledge no analysis similar to this model was found in literature [e.g. Fleischer et. al (1977), Wards and Sullivan (1981), Remer et. al. (1984), Grant et. al. (1990), Newman (1996) and Blank et. al. (1996)].

UNPAID OR UNRECOVERED CAPITAL (REMAINING BALANCE) [U.sub.t]

The amount of unpaid or unrecovered capital [U.sub.t] is

[U.sub.t] = P - [[[sigma].sup.t].sub.j=1] [C.sub.t]

Using Eqs. (14) and (15) above, the following two equations for [U.sub.t] are obtained:

a) Using Eq. (14):

[U.sub.t] = P - P[(1 + i).sup.t] - 1/[(1 + i).sup.n] - 1

Rearranging

[U.sub.t] = P [(1 + i).sup.n] - [(1+i).sup.t]/[(1 + i).sup.n] - 1 (16)

b) Using Eq. 15:

[U.sub.t] = P [1 - (F/ A,i,t)/(F/ A,i,n)] (17)

CUMULATIVE PAID INTEREST [sigma][I.sub.t]

The expression for finding the cumulative amount of interest of a uniform series of cash flows of a loan borrowed at rate i can be found in the literature (Park and Sharp-Bette, 1990, and Brooking and Burgess, 1976)

However, the cumulative interest paid up to time t can be found using the equations developed above as follows:

[[[sigma].sup.t].sub.j=1] [I.sub.j] = A*t - [[[sigma].sup.t].sub.j=1] [C.sub.j]

or

[[[sigma].sup.t].sub.j=1] [I.sub.j] = A*t - P [(F/ A,i,t)/F/ A,i,n)] (18)

APPLICATIONS

The above model (Eq. 14 or Eq. 15) can be used instead of the existing methods. It yields exactly the same results. The following example will demonstrate this.

EXAMPLE: Find the cumulative amount of paid capital after 20 payments for a loan of $1000 which is borrowed at monthly interest rate of 1% for 3 years to be repaid at equal end-of-the-month installments.

Solution:

a) Using the developed model as presented in Eq. (13):

[[[sigma].sup.20].sub.j=1] [C.sub.j] = P[(F/ A,1%,20)/(F/ A,1%,36)]

[[[sigma].sup.20].sub.j=1] [C.sub.j] = 1000[22.019004/43.076878] = $51156

b) Using the project balance pattern concept as presented in Eq. (4), then A has to be found first, followed by finding the unpaid capital [U.sub.t]. The cumulative paid capital [C.sub.t] is found by subtracting the unpaid capital from the initial capital. To illustrate this,

A = P (A/P, 1%, 36),

A = 1000(0.0332143) = $33.2143.

The value of unpaid capital

[U.sub.t] = 1000 (F/P, 1%, 20) - 33.214(F/A, 1%, 20),

= $488.844

and the cumulative paid capital [C.sub.t], is therefore $511.156, which is exactly the same as the result of the proposed model.

c) If Eq. (5) is to be used,

A = P (A/P, 1%, 36),

A = 1000(0.0332143) = $33.2143,

[U.sub.t] = A (P/A, i, n-t),

= 33.214(P/A, .01, 16) = $488.844, and [C.sub.t], = 1000-488.844 = 511.156

CONCLUSIONS

In this paper, a series of models that can be used in conjunction with series of uniform cash flows are developed. These models can be used to find, at any time of a project's life, the cumulative amount of paid or recovered capital, the unpaid or unrecovered capital (remaining balance) and the cumulative interest paid up to that point in time. These models offer an easy tool to use and a rather simple way for finding these variables. The major advantage of these models is that they are direct and do not require calculating the value A. The results which were obtained using these models are identical with the other available methods.

REFERENCES:

(1.) BLANK L.T. and A.J. TARQUIN, Engineering Economy, McGraw-Hill Book Company, 1996.

(2.) BROOKING, S.A. and A.R. BURGESS, "Present Worth of Interest Tax Credit," The Engineering Economist, Vol. 21, No 2, 1976.

(3.) DEGARMO, P.E., W.G. SULLIVAN and J.A. BONTADELLI, Engineering Economy, McGraw Hill Book Company, 1993.

(4.) FLEISCHER, G.A., and T.L.WARD, "Classification of Compound Interest Models in Economic Analysis," The Engineering Economist, Vol. 23 No. 1, 1977, pp. 13-29.

(5.) GRANT, E.L., W.G. IRESON, and R.S. LEAVENWORTH, Principles of Engineering Economy, Eighth Edition, John Wiley & Sons Inc., 1990.

(6.) NEWNAN, D.G., Engineering Economic Analysis, Engineering Press, 1996.

(7.) OAKFORD, R.V., S.A. BHIMJEE, and J.V. JUCKER, "The Internal Rate of Return, The Pseudo Internal Rate of Return, and the NPV and their use in Financial Decision Making," The Engineering Economist, Vol. 22, No. 3, 1977.

(8.) PARK, C.S., and SHARP-BETTE, Advanced Engineering Economy, John Wiley Publishers, 1990

(9.) REMER, D.S., J.G. TU, D.E. CARSON, and S.A.GANY, "The State of the Art of Present Worth Analysis of Cash Flow Distribution," Engineering Costs and Production Economics, Vol. 7, No. 4, 1984, pp. 257-278.

(10.) WARD, T.L., and W.G. SULLIVAN, "Equivalence of the Present Worth and Revenue Requirement Methods of Capital Investment Analysis," AIIE Transactions, Vol. 13, No. 1, 1981, pp. 29-40.

BIOGRAPHICAL SKETCHES

MOHAMMED A. SALEM HIYASSAT is an associate professor in the Civil Engineering Department at The University of Jordan, Amman, Jordan. He received his B.Sc. from Moscow Civil Engineering Institute, Moscow, and his M.S. and Ph.D. degrees from the University of Michigan, Ann Arbor. He teaches many courses in the area of construction project management and engineering economy. He has been chair of the Civil Engineering Department. His research interest is project management, resource leveling, bid evaluation, scheduling, workers motivation, construction industry in Jordan, and engineering economy.

KHALED M. RAWAJFEH is an associate professor in the Chemical Engineering Department at the University of Jordan. He received his B.Sc in Chemical Engineering from the South Bank Polytechnic (University of the South Bank), London, UK, his DEA from ENSIGC Toulouse, France, and his Diplome Docteur Ingenicur from ENSIGC Toulouse, France. He has worked as a project engineer at the Arab Potash Company, Head of Chemical Engineering Department, assistant dean for Student Affairs, vice dean at the Faculty of Engineering and Technology at the University of Jordan.

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