PV: The value of a future cash stream discounted to present day.
The formula for PV can be written as follows:
Where C1 is cash flow at date 1 and r is the rate of return. It is sometimes referred to as the discount rate. It is
always less than 1.0 for positive i, indicating that a future amount has a smaller present value.
Using a Spreadsheet for Time Value of Money Calculations
To Find
Enter This Formula
FUTURE VALUE
FV (rate,nper,pmt,pv)
PRESENT VALUE
PV (rate,nper,pmt,fv)
DISCOUNT RATE
RATE (nper,pmt,pv,fv)
NUMBER OF PERIODS
NPER (rate,pmt,pv,fv)
In these formulas, pv and fv are present and future value, nper is the number of periods, and rate is the discount, or
interest, rate.
Two things are a little tricky here. First, unlike a financial calculator, the spreadsheet requires that the rate be entered
as a decimal. Second, as with most financial calculators, you have to put a negative sign on either the present value
or the future value to solve for the rate or the number of periods. For the same reason, if you solve for a present
value, the answer will have a negative sign unless you input a negative future value. The same is true when you
compute a future value.
To illustrate how you might use these formulas, we will go back to an example in the chapter. If you invest $25,000
at 12 percent per year, how long until you have 20B950,000? You might set up a spreadsheet like this:
Example 1 Kyle Mayer has won the Kentucky State Lottery and will receive the following set of cash flows over the
next two years:
Year
Cash Flow
1
$20,000
2
50,000
Mr. Mayer can currently earn 6 percent in his money market account, so the appropriate discount rate is 6 percent.
The present value of the cash flows is:
Year
Cash Flow × Present Value Factor = Present Value
1
2
In other words, Mr. Mayer is equally inclined toward receiving $63,367.7 today and receiving $20,000 and $50,000
over the next two years.
Example 2 Finance.com has an opportunity to invest in a new high-speed computer that costs $50,000. The
computer will generate cash flows (from cost savings) of $25,000 one year from now, $20,000 two years from now,
and $15,000 three years from now. The computer will be worthless after three years, and no additional cash flows
will occur. Finance.com has determined that the appropriate discount rate is 7 percent for this investment.
Should Finance.com make this investment in a new high-speed computer?
The cash flows and present value factors of the proposed computer are as follows:
Year
Present Value Factor
0
1 = 1
1
2
3
The present value of the cash flows is:
Cash Flows × Present value factor = Present value
Year
0
−$50,000 × 1
=
−$ 50,000
1
$25,000 × .9346
=
$ 23,364.49
2
$20,000 × .8734
=
$ 17,468.77
3
$15,000 × .8163
=
$ 12,244.47
Total:
$ 3,077.73
Finance.com should invest in the new high-speed computer because the present value of its future cash flows is
greater than its cost. The NPV is $3,077.73.
Continuous compounding: Interest compounded continuously rather than at fixed intervals. We could
compound semiannually, quarterly, monthly, daily, hourly, each minute, or even more often. The
limiting case would be to compound every infinitesimal instant, which is commonly called continuous
compounding.
With continuous compounding, the value at the end of T years is expressed as:
Where C0 is the initial investment, r is the APR, and T is the number of years over which the investment runs. The
number e is a constant and is approximately equal to 2.718.
Figure 4.11 illustrates the relationship among annual, semiannual, and continuous compounding. Semiannual
compounding gives rise to both a smoother curve and a higher ending value than does annual compounding.
Continuous compounding has both the smoothest curve and the highest ending value of all.
Present Value with Continuous Compounding
The Michigan State Lottery is going to pay you $100,000 at the end of four years. If the annual continuously
compounded rate of interest is 8 percent, what is the present value of this payment?
Figure 4.11
Annual, Semiannual, and Continuous Compounding
Perpetuity It is a constant stream of cash flows without end. There aren’t many real life examples of a perpetuity.