Economic equivalence and developing of interest formulae

Economic Equivalence

Calculations for determining the economic effects of one or more cash flows are based on the concept of economic equivalence. Economic Equivalence refers to the fact that a cash flow whether the single payment or a series of payment can be converted to an equivalent cash flow at any point in time. It exists between cash flows that have the same economic effect. The three general principles of Economic Equivalence Calculation are as follows:

1. Equivalence calculations made to compare alternatives requires a common time basis
2. Equivalence depends on interest rate
3. Equivalence calculations may need the conversion of multiple payment cash flows to a single cash flow

Development of Interest formulae

As we begin to compare the sequence of cash flows instead of single payments, the required analysis becomes more complicated as we have to make more complex comparisons of cash flows. However, when patterns in cash flow transactions can be identified, we can take advantages of these pattern by developing concise expressions for computing either the present or future worth of the series. We will classify the cash flow transactions into five major categories and develop interest formulas for them.

1. Single cash flow
2. Equal (Uniform) series
3. Linear Gradient Series
4. Geometric Gradient Series
5. Irregular (Mixed) Series

1. Single Cash Flow:

It is the simplest case which involves the equivalence of a single present amount and its future worth amount. Thus the single cash flow formulas deal with only two amounts: a single present amount ‘P’ and its future worth ‘F’.

As derived earlier in compound interest topic, the future sum ‘F’ accumulated on investing present sum ‘P’ for ‘N’ interest periods at interest rate ’i’ is given by:

F = P(1 + i)^N …….(a)

The factor (1 + i)^N is called as the compound amount factor.

The resulting compound interest factor is expressed in a conventional representation that can be substituted in a formula to indicate precisely which table factor to use in solving an equation using the interest tables for a given interest rate and the number of interest periods. To specify how the interest tables are to be used for taking the direct value, we may also express that factor in functional notation as (F/P, i, N) which is read as “Find F, Given P, i and N ”. This is known as the single payment compound amount factor. It is expressed as

F = P (F/P, i, N) ……(b)

In contrast, finding the present worth of a future sum is simply the reverse of compounding process and is known as the discounting process. The present sum ‘P’ is found, given a future sum ‘F’ by simply solving as

P = F/(1 + i)^N = F(P/F, i, N) ………(c)

The factor 1/(1 + i)^N is known as the single payment present worth factor and is designated as (P/F, i, N). Tables have been constructed for P/F factors and for various values of ‘i’ and N. The interest rate ‘i’ and the P/F factor are also stated to as the discount rate and discounting factor respectively.

2. Uniform cash flows:

Probably the most familiar category includes the transactions arranged as a series of equal cash flows at regular intervals, known as an equal payment series or uniform series. For example, this category describes the cash flows of the installment loan contract which manages the repayment of a loan in equal periodic installments. Rental overheads, bond interest payments, and commercial installment plans are based on uniform payment series. The equal cash flow formulas deal with the equivalence relations to P, F and A (the constant amount of the cash flow in series).

In case cash flows regularities are present within the stream then the use of shortcuts such as calculating the present worth of a uniform series may be possible.

a. Compound amount factor: [Find F, Given A, i and N]

If you are interested in the future amount ‘F’ of a fund to which contribute ‘A’ amount each ‘N’ equal periods and on which earn interest at a rate of ‘i’ per period. At the end of ‘N’ periods, the total amount ‘F’ that can be withdrawn will be the sum of the compound amounts of the individual deposits.

The ‘A’ dollars we put in the fund at the end of the first period will be worth A(1+i)^{N – 1} at the end of ‘N’ periods. Similarly, the ‘A’ dollars we put into the fund at the end of the second period will be worth A(1+i)^{N – 2} and so forth. Finally, the last ‘A dollars that contribute at the end of Nth period will be worth exactly ‘A’ dollars at that time.

Mathematically,

F = A(1 + i)^{N – 1} + A(1 + i)^{N – 2} + ……… + A(1 + i) + A

or expressed alternatively as,

F = A + A(1 + i) + ……… + A(1 + i)^{N – 2} + A(1 + i)^{N – 1} ……(i)

Multiplying (i) by (1 + i) results

(1 + i)F = A(1 + i) + A(1 + i)² + ……… + A(1 + i)^{N – 1} + A(1 + i)^N ……(ii)

Subtracting (i) from (ii), gives

F = A[{(1 + i)^N – 1}/ i ] = A(F/A, i, N) ……(iii)

The bracketed term in Eqn.(iii) is called uniform series compound amount factor. This interest factor has been calculated for various combinations of N and i in the interest tables.

b. Sinking fund factor: [Find A, Given F, i and N]

Solving eqn (iii), we obtain

A = F[ i /{(1 + i)^N – 1} ] = F(A/F, i, N) ……(iv)

The term within the bracket is called the sinking fund factor. A sinking fund is an account that bears interest into which a fixed sum of money is deposited each interest period, it is commonly established for the function of replacing fixed assets or retiring corporate bonds.

c. Capital Recovery Factor (Annuity Factor): [Find A, Given P, i and N]

We know, F = P(1 + i)^N which can be used in (iv) to relate P to A as;

A = P[{i(1 + i)^N}/{(1 + i)^N – 1}] = P(A/P, i, N) ……(v)

This equation helps to determine the value of series of end of period payments ‘A’ when the present sum ‘P’ is known.

d. Present Worth Factor: [Find P, Given A, i and N]

Solving (v) for ‘P’, we get,

P = A[{(1 + i)^N – 1}/{i(1 + i)^N}] = A(P/A, i, N) ……(vi)

The bracketed term is called as the equal payment series present worth factor.

3. Linear Gradient Series:

While series of cash flows are involved many transactions, the amounts are not always uniform, however, they may vary in some regular way. One common array of variation occurs when each cash flow in a series increases or decreases by a fixed amount ‘G’. This type of cash flow is called a linear gradient series because its cash flow diagram results an ascending or descending straight line. Along with the use of P, F, and A, the formulas employed in such problems involve a fixed amount ‘G’ of the change in each cash flow.

a. Present worth factor: Linear Gradient [Find P, Given G, N and i]

It is used to find the principal to be deposited now to withdraw the gradient amounts specified. The present worth is related to ‘G’ as;

P = G[{(1 + i)^N – iN – 1}/{i²(1 + i)^N}] = G(P/G, i, N) ……(vii)

The resulting factor in the bracket is called the gradient series present worth factor.

b. Gradient to Equal Payment Series Conversion Factor: [Find A, Given G, N and i]

Using equation (vi) in (vii), we obtain the relation between A and G as:

A = G[{(1 + i)^N – iN – 1}/{i[(1 + i)^N – 1]}] = G(A/G, i, N) ……(viii)

Where the resulting factor in brackets is called as the gradient to the equal payment series conversion factor.

4. Geometric Gradient Series:

Another kind of gradient series is formed when the series in a cash flow is determined not by some constant amount but by some fixed rate, stated as the percentage. The curving gradient in the diagram of such a series suggests its name as geometric gradient series. In the formulas dealing with such series, the rate of change is symbolized by a lowercase ‘g’.

The magnitude of the nth payment, A_nis related to the first payment A₁by the formula

A_n = A₁(1 + g)^{n – 1}, n = 1,2,3…….N ……(ix)

The variable ‘g’ can be either negative or positive, liable on the type of cash flow. If g > 0, the series will increase and if g < 0, the series will decrease.

a. Present worth factor: [Find P, Given A₁, g, N and i]

The factor within the brackets is called the geometric gradient series present worth factor.

5. Irregular (Uneven) cash flows:

A common cash flow transactions involve a series of disbursements or receipts. Examples of series payments are installments payment of car loans and home mortgages which typically involve identical sums to be paid at regular intervals. But there is no clear pattern over the series so we call the transaction an uneven cash flows series.

Uneven cash flow is the combination of single cash flow. We can find the present worth of any uneven stream of payments by computing the present worth of each individual payment considering the time and interest rate and summing the results. After the present value is found, we can make other equivalence calculations such as future value, annual worth can be calculated by using the interest factors calculated as per assumption.

BIBLIOGRAPHY:

Chan S.Park, Contemporary Engineering Economics, Prentice Hall, Inc.
E. Paul De Garmo, William G.Sullivan and James A. Bonta delli, Engineering
Economy, MC Milan Publishing Company.
James L. Riggs, David D. Bedworth and Sabah U. Randhawa,Engineering
Economics, Tata MCGraw Hill Education Private Limited.