PART 1: INTRODUCTION

As you no doubt suspect, there is a general formula for computing the present value of a stream of

future cash flows. Continuing to let CFt represent the cash flow at the end of year t, the present value of

an n-year mixed stream of cash flows (PV) is expressed as Equation 3.6:

Eq. 3.6

PVCF

r

CF

r

CF

r

CF

r

=

1

(1+)

1

(1+)

1

(1+)

1

(1+)

n n

t t

t

n

(^1122)
=1
∑
× ......





+×





++ ×






=×
Substitute the cash flows shown on the time line in Figure 3.10 and the 9% discount rate into
Equation 3.6 to obtain the present value, $19,047.58.^7
Let’s stop and consider the big picture for a moment. In the previous example, we are given a series
of cash flows that are spread out over time, and we want to place a value on the entire cash flow stream as
of a specific date. When we calculate the present value of the stream, we are determining the value of the
stream as of today. When we calculate the future value of the stream, we are determining the value of the
stream as of a specific date in the future. In both cases, we are placing a single value on the entire stream.
The following example illustrates how the present and future values of cash-flow streams are related.
Refer once more to the cash flow stream illustrated
on the time line in Figure 3.10. We’ve already seen
that the stream’s present value is $19,047.58. Now
let’s review what we covered in section 3-5a and
calculate the future value of this stream. Recall that
the $4,000 cash flow could earn 9% interest each
year for four years, the $8,000 cash flow could
earn interest each year for three years, and so on.
Applying Equation 3.3 to this stream, we obtain:
FV = $4,000 × (1 + 0.09)^4 + $8,000
× (1 + 0.09)^3 + $5,000(1 + 0.09)^2

• $4,000 × (1 + 0.09)^1 + $3,000
  FV = $5,646.33 + $10,360.23 + $5,940.50


• 
  

  $4,360.00 + $3,000.00
  = $29,307.06
  In other words, $29,307.06 is the amount of
  money you would have at the end of year 5 if you
  made annual deposits (shown in Figure 3.10) into
  an account earning 9% interest. In our example, the
  cash-flow stream represents a series of maintenance
  expenditures, not deposits. In that context, the
  calculation we just completed implies that making
  a lump sum payment of $29,307.06 in year 5 is
  equivalent to making the series of payments spread
  out over five years and depicted in Figure 3.10.
  Next, let’s calculate the present value of this lump
  sum as of today using Equation 3.2:

  
  


• 
  

=

+

PV =

FV

(1 r)

$29,307.06

(1 0.09)

n 5 $19,047.58

This equation says that making a lump sum

payment today of $19,047.58 is equivalent to paying

$29,307.06 five years from now. That shouldn’t

be a surprise, because $19,047.58 is precisely the

value that we obtained previously for the present

value of the mixed stream. Therefore, we have

three equivalent ways of expressing the costs of

maintaining the bed and breakfast hotel:

1 You can make the annual series of payments

shown in Figure 3.10.

2 You can make a lump sum payment of $19,047.58

today.

3 You can make a lump sum payment of $29,307.06

five years from today.

What the time value of money calculations are telling

us is that these three options are equivalent as long

as the interest rate is 9%.

example

7 A simple way to perform this calculation in Excel is to use the =function. To use that function, you simply enter the interest rate followed by

the series of annual cash flows. For example, entering =npv(0.09,–4,000,–8,000,–5,000,–4,000,–3,000) into Excel will generate the desired

result, $19,047.58.

LO3.4