Net Worth

Projects involving an investment followed by returns may be described by specifying a cash flow for each compounding period. For purposes of analysis we desire to express the cash flow as a single value at some point of time. The usual time specified is the present, or time 0, and we call the single equivalent value at time zero the net present worth, NPW, of the cash flow.

Alternatively, we may choose to express the cash flow as an equivalent uniform series of payments over the life of the investment. When the compounding period is one year, the value of each payment is called the equivalent net annual worth, NAW, over the life of the investment.

This lesson will show that a multiperiod cash flow can be partitioned into simpler cash flows. The present worth or annual worth of each component is computed using equivalence factors. The equivalent values of the simpler cash flows are then added to obtain the NPW or NAW of the more complex one. There are two pages in this lesson. We consider some simple cases on this page and more complex cases on the next.

 When comparing or evaluating economic projects it is necessary to reduce the cash flow for the project to a single number. The Net Present Worth and Net Annual Worth are measures often used.

 Goals
 Write expressions for NPW and NAW using as few equivalence factors as possible. Evaluate NPW and NAW using factor tables. Evaluate NPW and NAW using the Cash Flow Calculator and Project Calculator. Compute the single value equivalent at any point in time.
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Section 3.2.1 especially relates to the materials for this lesson.

 3.2 Compound Interest Formulas
 Net Present Worth
 Consider the cash flow below. An investment of \$30 earns \$10 per year for ten years. At the end of ten years the amount invested, \$30, is returned. We are to compute the NPW of this cash flow at 10% annual interest. With interest compounded annually cash flows are restricted to integer times. It is common practice to place the annual returns at the end of each year. Then we can use our equivalence factors directly.

To analyze the cash flow we partition the cash flow into three components. They are shown below along with the computation of the present value of each. Our factors use the notation i for the interest rate.

 The initial investment is a single payment at time 0, so no adjustment is necessary. P(1) = -30 The annual income is a uniform series of payments, so we move the series to the present with the P/A factor. P(2) = 10(P/A, i, 10) The final payment is a single amount at time 10, so we move it to the present with the P/F factor. P(3) = 30(P/F, i, 10)

The present value of the entire cash flow is the sum of the present values of the components. We use the term net present worth (NPW) because the value is the result of a sum of terms.

NPW = P(1) + P(2) + P(3) = -30 + 10(P/A, i, 10) + 30(P/F, i, 10)

Usually there are alternate ways to partition a cash flow. An equivalent formulation combines the two payments at time 10.

NPW = -30 + 10(P/A, i, 9) + 40(P/F, i, 10)

Both formulas are correct and yield the same value. The best formula is the one that most closely represents the source of the individual cash flows. The statement of the problem separates the annual returns from the final payment, so the first formula is better for the example. In exercises and exams we seek formulas with the fewest equivalence factors. The present value of any complex cash flow can be expressed as the sum of the present values of the cash flows at each period. For the example:

NPW = -30 + 10(P/F, i, 1) + 10(P/F, i, 2) + ... + 10(P/F, i, 9) + 40(P/F, i, 10)

This expression yields the same value as the others but it would be judged poorly in this course because it has 11 terms rather than 3.

 For this course, expressions for the time value of money are judged by the number of terms in the expression.

For most applications, our purpose is to compute the NPW for a given interest rate. Choosing 10% as the interest rate:

NPW = -30 + 10(P/A, 0.1, 10) + 30(P/F,0.1, 10) = -30 + 10*6.145 + 30*0.3855 = -30 + 61.45 + 11.57 = 43.01

When solving homework and exam problems, it is recommended that the NPW be evaluated as in this example. First write the expression using symbolic equivalence factors. Then substitute the factor values for the symbols. Then evaluate and sum the terms. This strategy simplifies checking the results.

Although the data for practical problems often does not warrant great accuracy of computation, in this academic setting we will express equivalent values to at least two decimal places (to one cent accuracy). For banking applications, more accurate results might be necessary.

 Equivalent Values at Other Points in Time

In some situations, it is necessary to compute the equivalent value at times other than the present. Once the value at one time is determined, the F/P factor moves the value to a later time while the P/F factor moves the value to an earlier time. For the example, say we require the future value of the investment at the end of the ten-year period. The net future worth is:

NFW = NPW(F/P, 0.1, 10) = 43.01*2.593 = 111.56.

Of course, the NFW depends on the time selected for the value and the interest rate.

 Net Annual Worth

For some applications, it is desirable to express the complex cash flow as a uniform series. For simple cash flows, we compute the uniform equivalent of each component and add the results. Computations for the example are below.

 For the initial investment, we compute the annual equivalent by spreading the investment at time 0 over the following ten years. A(1) = -30(A/P, i, 10) The annual returns already have the form of a uniform series. A(2) = 10 The final payment is to be spread over the previous ten years. A(3) = 30(A/F, i, 10)

We sum the annual equivalents of the components to find the net annual worth (NAW).

NAW = A(1) + A(2) + A(3) = -30(A/P, i, 10) + 10 + 30(A/F, i, 10)

Evaluating this expression for a 10% interest rate gives

NAW = -30(A/P, 0.1, 10) + 10 + 30(A/F, 0.1, 10) = -30*0.1627 + 10 + 30*0.0627 = -4.88 + 10 + 1.88 = 7.00

One can always compute the NAW from the NPW or NFW. For complex cases this may be the best way. In general:

NAW = NPW(A/P, i, N) = NFW(A/F, i, N)

The value of N is the life of the project. For the example:

NAW = NPW(A/P, 0.1, 10) = 43.01*0.1627 = 7.00
NAW = NFW(A/F, 0.1, 10) = 111.56*0.0627 = 7.00

Notice that the equivalent value computed is at the end of each year, not the beginning. This is an important distinction.

 The NAW is an end-of-the-period equivalent value.
 Compounding Period

When the compounding period is other than one year, the interest rate is expressed as percent per compounding period, and the project life is expressed in number of compounding periods. Then we compute the equivalent uniform worth as the amount per compounding period. The name NAW is not appropriate for periods other than whole years since "A" stands for annual or annuity. If the compounding period is one month then one would compute the uniform equivalent worth with the formulas above. This could be abbreviated as EUW but NAW is often used in all cases. Either name is satisfactory if the period of the analysis is clear.

Whatever the compounding period, the payments of the uniform series are at the end of each compounding period.

Engineering Finance
by Paul A. Jensen
Copyright 2005 - All rights reserved