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

ABSTRACT

The purpose of this paper is to propose models for finding the cumulative amount of paid capital up to a certain point in time for a discrete uniform series of cash flows, and also the unpaid capital amount of a similar series of cash flows. An exponential model for finding the present worth for a series of cash flows that is increasing exponentially is used to find the sum of paid capital. The second model that is used for finding the unpaid capital is developed based on the first model. The major advantages of the first model over the existing models is that it provides a direct tool for finding the cumulative amount of paid capital or unrecovered investment. The proposed models are easy to use since they are independent of finding the amount of the cash flow A. The cumulative paid interest up to a certain point in time can be found by subtracting the total repaid capital from the amount of total paid money up to that point.

INTRODUCTION

In many cases of a discrete uniform series of cash flows, a question often arises on the cumulative amount of capital or interest, in case of a loan, that has been repaid up to a certain point in time in the project's life corresponding to a given number of payments. From the investor's point of view, the question is how much profit has been received or how much capital has been recovered up to a certain point in time. In the literature of engineering economy, there are several methods that answer these questions. In specific, two of these will be presented in this paper. The purpose of this paper is to propose a simple model for calculating the cumulative amount of paid capital in a uniform cash flow series after a specified period of time has elapsed. Based on this model, another simple model for finding the amount of unpaid or unrecovered capital (remaining balance) at any point of project's time horizon is derived. Using the proposed model, the cumulative sum of interest that is received or paid up to a c ertain period in time can also be easily calculated.

First, the uniform and the exponential cash flow models will first be presented. This is followed by presenting the development of the first proposed model together with the underlying assumptions and the verification of these assumptions. The second proposed model is then developed based on the first one. This is followed by showing how the cumulative paid interest can be calculated. Finally, an illustrative example on the use of the first model is presented.

THE UNIFORM AND THE EXPONENTIAL CASH FLOW MODELS

UNIFORM CASH FLOW MODEL

The present value of a uniform series of cash flows can be calculated using the following well-known formula found in all engineering economy literature.

P = A [(1 + i).sup.n] - 1/[(1 + i).sup.n] * i (1)

Or as often written in functional form:

P = A (P/A,i,n)

where:

P: present value of a series of cash flows.

A: an end of period uniform cash flow.

i: interest rate per period.

n: project life, in periods.

The amount of A at any given time consists of two parts, interest [I.sub.t] and capital [C.sub.t], so that:

A = [I.sub.t] + [C.sub.t]. (2)

The portion of interest [I.sub.t] is found as follows:

[I.sub.t] = [P.sub.t-1] * i,

where [P.sub.t-1] is the discounted value at time (t-1) of the uniform cash flows A extending from time t to n, and it is given by:

[P.sub.t-1] = A [(1+i).sup.n-(t-1)]-1/[(1+i).sup.n-(t-1)] * i (3)

To find the amount of capital that remains invested or borrowed at any point of time during the project's life, the concept of "project balance pattern" (Park and Sharp-Bette, 1990) can be used. The project balance pattern is also known as: investment balance diagram (Degarmo et. al., 1993), and capital invested function (Oakford et. al. 1977)

The concept of project balance pattern is to find the net equivalent amount of money tied up in the project or committed to it at any point in time over the life of the project (Park and Sharp-Bette, 1990). At any point in time, t, in the life of the project, the amount of outstanding borrowed capital or unrecovered investment is found as the future value ([F.sub.t]) of all cash flows between time zero and t when compounded at interest rate i. It is obvious that finding the amount of remaining capital (invested or borrowed) at any time t, incorporates every cash flow from time zero up to that point in time. If the series of cash flows is uniform, the unpaid (or the remaining) balance [U.sub.t] at any time t is:

[U.sub.t] = P(F / P,i,t) - A(F / A,i,t). (4)

Alternatively, the unpaid balance or unrecovered investment at any point of project life can be found as the present value of the remaining cash flows between time t and n, discounted back to time t. For a uniform cash flow series the unpaid or unrecovered balance is:

[U.sub.t] = A(P / A,i,n-t). (5)

Using the above two methods, the cumulative amount of repaid or recovered capital [sigma][C.sub.t], which is the main subject of this paper, is found by subtracting unpaid or remaining capital the from the initial amount of the loan or investment:

[[[sigma].sup.t].sub.j=1] [C.sub.j] = P - [U.sub.t] (6)

In most literature, the cumulative interest [sigma]I, that has been paid up to a certain point in time (is found by subtracting the total paid capital, [[[sigma].sup.t].sub.j=1] [C.sub.j], from the total paid amount [simga]A). The value of [U.sub.t] is then substituted in the above formula for calculating the cumulative paid interest. That is:

[[[sigma].sup.t].sub.j=1] [I.sub.j] = A * t - (P - [U.sub.t]) (7)

From this formula, it is evident that finding the sum of paid interest at a certain point in time, requires calculating the uniform payment A, and the value of unpaid balance up to that point.

EXPONENTIAL CASH FLOW MODEL

A cash flow series, [S.sub.t], that is increasing or decreasing at a rate of [delta], is said to increase or decrease exponentially if the relationship between two consecutive cash flow values, [S.sub.t] and [S.sub.t+1], is such that:

[S.sub.t+1] = [S.sub.t] (1 + [delta]) (8)

The present value of this series of cash flows is found by summing all individual cash flows discounted back to time 0 at a discount rate i. The final form of this model as presented in most relevant literature (Degarmo et al) is:

P = [S.sub.1] 1 - [(1 + [delta]).sup.n]/[(1 + i).sup.n]/i - [delta] (9)

Which is the present worth of a geometric series in which the present worth factor is given by:

(P / A,[delta],i,n) = 1 - [(1 + [delta]).sup.n]/[(1 + i).sup.n]/1 - [delta]

As a special case, this equation can be used to find the algebraic sum of all cash flows from 0 to any time t for a series of cash flows that is increasing or decreasing exponentially. This is done by equating the interest rate of Eq. (9) to zero. The following equation is obtained:

P = [S.sub.1] [(1 + [delta]).sup.t] - 1/[delta] (10)

It can be noted that the right term would be the future worth of a uniform series of cash flows [S.sub.j] in which the interest rate equals to [delta].

MODEL DEVELOPMENT

CUMULATIVE PAID OR RECOVERED CAPITAL

Assumptions

The model that is developed for finding the cumulative amount of paid or recovered capital is based on utilizing the exponential model. The development of the model is also based on the fact that each uniform payment A consists of interest and capital portions as shown in FIGURE 2a. Each is assumed to change exponentially as shown in FIGURE 2b.

Assumption Verification

The fact that the capital portion in a uniform series of cash fi exponentially is demonstrated as follows:

In a uniform series of cash flows with interest rate i, the part o the first cash flow is:

[I.sub.1] = P*i

and the unpaid capital is

[U.sub.1] = P (1 + i) - ([C.sub.1] + [I.sub.1]) = P + p*I - [C.sub.1] - [I.sub.1]

and since

[I.sub.1] = P*i

then

[U.sub.1] = P - [C.sub.1]

The part of interest [I.sub.2] in the second payment is:

[I.sub.2] = [U.sub.1]*i.

Then

[I.sub.1] - [I.sub.2] = P*i-([U.sub.1]*i),

[I.sub.1] - [I.sub.2] = P*i - (P - [C.sub.1])*i

and after rearrangement, this equation becomes:

[I.sub.1] - [I.sub.2] = [C.sub.1]*i

and in general, we can write

[I.sub.t] - [I.sub.t+1] = [C.sub.t]*i

Now substituting the values of [I.sub.1]=A-[C.sub.1] and [I.sub.2]=A-[C.sub.2] in this equation, we obtain:

[C.sub.t+1] - [C.sub.t] = [C.sub.t]*i,

and therefore the general form for the capital portion is:

[C.sub.t+1] = [C.sub.t]*(1+i). (11)

From this equation it can be seen that the portion of capital in a uniform cash flow series grows exponentially at a rate i same as in Eq. (8).

The Proposed Model

Now since the capital portion of a uniform cash flow A follows an exponential model, Eq. (9) is used to find the algebraic sum of paid or unrecovered capital. In other words, the cumulative amount of capital [[[sigma].sup.t].sub.i=1] [C.sub.j] that has been paid up to a certain point in time (t) can be found by applying Eq. (10) in which the value of i is used instead of [delta], [C.sub.1] instead of [S.sub.1]. That is,

[[[sigma].sup.t].sub.j=1] [C.sub.j] = [C.sub.1] [(1+i).sup.t] - 1/i. (12)

Now since the capital portion of the first cash flow [C.sub.1] is:

[C.sub.1] = A - P*i (13)

therefore,

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

If the value of A as presented in Eq. (1) is substituted, then

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