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 



Text 

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 tenyear 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 endoftheperiod
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. 

Return to Top of Page 