Cash Flows: Estimation and Diagramming

As mentioned in earlier sections, cash flows are the amounts of money estimated for future projects  or observed for project events that have taken place. All cash flows occur during specific time periods, such as 1 month, every 6 months, or 1 year. Annual is the most common time period. For example, a payment of $10,000 once every year in December for 5 years is a series of 5 outgoing cash flows. And an estimated receipt of $500 every month for 2 years is a series of 24 incoming cash flows. Engineering economy bases its computations on the timing, size, and direction of cash flows.  

Cash  inflows  are the receipts, revenues, incomes, and savings generated by project and business activity. A  plus sign  indicates a cash inflow.

Cash  outflows  are costs, disbursements, expenses, and taxes caused by projects and business activity. A negative or minus sign  indicates a cash outflow. When a  project involves only costs, the minus sign may be omitted for some techniques, such as benefit/cost analysis.  

Of all the steps in  Figure 1–1  that outline the engineering economy study, estimating cash flows (step 3) is the most difficult, primarily because it is an attempt to predict the future. Some examples of cash flow estimates are shown here. As you scan these, consider how the cash inflow  or outflow may be estimated most accurately.

  Figure 1–1  Steps in an engineering economy study.

Cash  Inflow Estimates

  Income:    $150,000 per year from sales of solar-powered watches 
  Savings:   $24,500 tax savings from capital loss on equipment salvage 
  Receipt:    $750,000 received on large business loan plus accrued interest 
  Savings:     $150,000 per year saved by installing more efficient air conditioning 
  Revenue: $50,000 to $75,000 per month in sales for extended battery life iPhones    

Cash  Outflow Estimates

  Operating  costs: $230,000 per year annual operating costs for software services 
  First  cost: $800,000 next year to purchase replacement earthmoving equipment 
  Expense: $20,000 per year for loan interest payment to bank 
  Initial  cost: $1 to $1.2 million in capital expenditures for a water recycling unit  

All of these are   point estimates  , that is,   single-value estimates  for cash flow elements of an alternative, except for the last revenue and cost estimates listed above. They provide a   range estimate,  because the persons estimating the revenue and cost do not have enough knowledge or experience  with the systems to be more accurate. For the initial chapters, we will utilize point estimates. The use  of risk and sensitivity analysis for range estimates is covered in the later chapters of this book.

Once all cash inflows and outflows are estimated (or determined for a completed project), the  net cash flow  for each time period is calculated.

 where NCF is net cash flow,   R  is receipts, and   D  is disbursements. 

At the beginning of this section, the   timing, size, and direction of cash flows  were mentioned  as important. Because cash flows may take place at any time during an interest period, as a matter of convention, all cash flows are assumed to occur at the end of an interest period.

The end-of-period convention means that all cash inflows and all cash outflows are assumed to  take place at the end of the interest period  in which they actually occur. When several inflows  and outflows occur within the same period, the net  cash flow is assumed to occur at the   end   of  the period.  

 In assuming end-of-period cash flows, it is important to understand that future  (F)  and uniform annual (A) amounts are located at the end of the interest period, which is not necessarily December 31. If in Example 1.7 the lump-sum deposit took place on July 1, 2011, the withdraw- als will take place on July 1 of each succeeding year for 6 years. Remember, end of the period  means end of interest period, not end of calendar year.

The   cash flow diagram  is a very important tool in an economic analysis, especially when the cash flow series is complex. It is a graphical representation of cash flows drawn on the   y  axis with a time scale on the   x  axis. The diagram includes what is known, what is estimated, and what is  needed. That is, once the cash flow diagram is complete, another person should be able to work the problem by looking at the diagram.

Cash flow diagram time   t  = 0 is the present, and   t  = 1 is the end of time period 1. We assume that the periods are in years for now. The time scale of  Figure 1–4  is set up for 5 years. Since the end-of-year convention places cash flows at the ends of years, the “1” marks the end of year 1.

 Figure 1–4  A typical cash flow time scale for 5 years.

While it is not necessary to use an exact scale on the cash flow diagram, you will probably avoid errors if you make a neat diagram to approximate scale for both time and relative cash flow magnitudes.

The direction of the arrows on the diagram is important to differentiate income from outgo. A vertical arrow pointing up indicates a positive cash flow. Conversely, a down-pointing arrow indicates a negative cash flow.   We will use a bold, colored arrow to indicate what is unknown and to be determined.  For example, if a future value   F  is to be determined in year 5, a wide, colored arrow with   F  = ? is shown in year 5. The interest rate is also indicated on the diagram.  Figure 1–5  illustrates a cash inflow at the end of year 1, equal cash outflows at the end of years 2 and 3, an interest rate of 4% per year, and the unknown future value F after 5 years. The arrow for the unknown value is generally drawn in the opposite direction from the other cash flows; however, the engineering economy computations will determine the actual sign on the   F  value. 

  Figure 1–5  Example of positive and negative cash flows.

Before the diagramming of cash flows, a perspective or vantage point must be determined so  that or – signs can be assigned and the economic analysis performed correctly. Assume you  borrow $8500 from a bank today to purchase an $8000 used car for cash next week, and you plan  to spend the remaining $500 on a new paint job for the car two weeks from now. There are several perspectives possible when developing the cash flow diagram—those of the borrower (that’s you), the banker, the car dealer, or the paint shop owner. The cash flow signs and amounts for these perspectives are as follows.

One, and only one, of the perspectives is selected to develop the diagram.   For your perspective,  all three cash flows are involved and the diagram appears as shown in  Figure 1–6  with a time scale  of weeks. Applying the end-of-period convention, you have a receipt of $8500 now (time 0) and  cash outflows of $8000 at the end of week 1, followed by $500 at the end of week 2.  

  Figure 1–6  Cash flows from perspective of borrower for loan and purchases.

EXAMPLE 1.9

Each year Exxon-Mobil expends large amounts of funds for mechanical safety features  throughout its worldwide operations. Carla Ramos, a lead engineer for Mexico and Central  American operations, plans expenditures of $1 million now and each of the next 4 years just  for the improvement of field-based pressure-release valves. Construct the cash flow diagram to  find the equivalent value of these expenditures at the end of year 4, using a cost of capital esti- mate for safety-related funds of 12% per year.

Solution 

Figure 1–7  indicates the uniform and negative cash  flow series (expenditures) for five periods,  and the unknown   F  value (positive cash flow equivalent) at exactly the same time as the fifth  expenditure. Since the expenditures start immediately, the first $1 million is shown at time 0,  not time 1. Therefore, the last negative cash flow occurs at the end of the fourth year, when   F   also occurs. To make this diagram have a full 5 years on the time scale, the addition of the  year 1 completes the diagram. This addition demonstrates that year 0 is the end-of-period point for the year 1.

EXAMPLE 1.10

An electrical engineer wants to deposit an amount   P  now such that she can withdraw an equal  annual amount of   A1 = $2000 per year for the first 5 years, starting 1 year after the deposit, and  a different annual withdrawal of   A2 = $3000 per year for the following 3 years. How would the  cash flow diagram appear if   i = 8.5% per year?

Solution

The cash flows are shown in Figure 1–8. The negative cash outflow   P  occurs now. The with- drawals (positive cash inflow) for the   A1  series occur at the end of years 1 through 5, and   A2  occurs in years 6 through 8.

EXAMPLE 1.11

A rental company spent $2500 on a new air compressor 7 years ago. The annual rental income  from the compressor has been $750. The $100 spent on maintenance the first year has in- creased each year by $25. The company plans to sell the compressor at the end of next year for  $150. Construct the cash flow diagram from the company’s perspective and indicate where the  present worth now is located.
 
Solution

Let now be time   t = 0. The incomes and costs for years 7 through 1 (next year) are tabulated  below with net cash flow computed using Equation [1.5]. The net cash flows (one negative,  eight positive) are diagrammed in  Figure 1–9 . Present worth   P  is located at year 0.