Chapter 4 Lecture (Part I)
Time Value of Money
Chapter 4 is the most important chapter covered in FIN 301. The skills learned in this
chapter will be applied in chapters 6 through 10, so time spent on chapter 4 is a great
investment in your FIN 301 grade.
Money has time value because of two main assumptions made about rational investors:
1) investors would prefer to consume now rather than later; 2) investors prefer more to
less. Because of assumption 2, we might be able to convince investors to consume less
now (and invest in our idea) if we can offer them the prospect (not the promise) of
consuming more later.
Before continuing further, it’s a good idea to review some key terms. A present value
refers to the value of amount of money right now (the present). If something is
happening to the money right now (it is being borrowed, or spent or lent, or invested)
that amount is a present value. An abbreviation for present value is PV. A future value
refers to the value of money at some later time. If something is happening to the money
in the future (a loan is being repaid, a dividend is being paid, it is being borrowed or
invested) it is called a future value. An abbreviation for future value is FV.
In the discussion that follows, I will be referring to specific slide numbers in the
PowerPoint presentation for chapter 4. An easy way to find a particular slide, for
example slide 5, is to right click on any slide, then left click on “edit slides” then left
click on “slide sorter view” (appears as 4 tiny squares) in the bottom left corner of the
screen.
The PowePoint for chapter 4 instructs you on how to set up your Texas Instruments BAII Plus calculator. This is shown on slide 5 of the PowerPoint. Take some time to review
this slide.
Future Values
Slides 7 through 10 provides a derivation for the future value formula which is used to
find the future value of a present amount. For example, if you put $100 in to a bank
today (a present value) and left it in the bank earning interest of 10% per year (an
unlikely interest rate at a bank today – 1.5% to 2% would be more realistic) the amount
you would have at the end of the year (a future value) would be $110.
The future formula derived in the PowerPoint presentation is:
FV = PV(1 + k)n where
FV = future value
PV = present value
K = interest rate
N = number of periods (in this case years)
This formula is programmed into your calculator. The calculator will be discussed
shortly
Formulas are more than just ways to find numerical answers. They also provide an
invaluable tool showing how the variables in the formula relate to one another. Ask
yourself what would happen to the future value if the interest rate, k, rose. What would
happen if the amount of time the funds were left in the bank, n, rose? The answer to both
questions is that the future value would rise. The higher k and n are the higher the FVIF
will be and the higher the future value will be. Page 145 of your text contains and
interesting graph (Figure 4.5). It shows the future value interest factor increases at an
increasing rate for any positive, nonzero, rate of interest, k, as n increases. In other
words, future values increase at increasing rates as money is invested longer and longer.
Why is this? The answer is that as you leave money invested, you are not just getting
interest on the original amount put in the bank, you are also getting interest on the interest
earned in preceding years. The longer your funds are in the bank, the more interest on
interest you are getting and the faster your money grows. This is called the
compounding effect.
The item contained within the parentheses (1 + k)n is called the future value interest
factor (FVIF). The FVIF is the engine that drives the future value formula.
To find the future value in one year of $100 invested today (a present value) earning 10%
per year, we just plug $100 in as the PV, 10% (or .10) in as the interest rate and 1 as n
and we get:
FV = $100(1.10)1 = $110
When all I want is a numerical answer, I write down a reminder telling me what buttons
to push on the calculator to get the answer I’m seeking.
FV = PV(FVIF, n, k%) This tells me that if I want to find a future value, I need to tell the
calculator what PV, n and k% are so it can plug these into the FV formula. To find the
FV of $100 invested at 10% for one year, I would need to:
Punch in 100 on the number pad
Hit the PV button on the third row down (the third row contains the financial buttons)
Punch in a 1 on the number pad
Hit the N button on the third row down
Punch in a 10 on the number pad
Hit the I/Y (interest rate per year) button on the third row down
Finally hit CPT (upper left corner) then FV (far right on the third row)
You should be getting $110. If you are not getting $110 as your answer, review the setup
procedures for your calculator on slide 5.
You may notice the answer is reported as a negative number on the calculator. The
reason for this is that the calculator would prefer we entered a -$100 as the present value
since we would be giving up $100 now (a negative amount) for the right to get $110 later
(a positive amount). I don’t worry about the negative sign, because I know the answer
really is supposed to be positive.
Take a moment to work through PowerPoint slides 12 – 17 and see if you can get the
same answers I got.
Present values
Present values represent what you (a rational investor) would be willing to pay now to
receive a future amount. Since we already have a formula for finding future values, we
can use simple algebra to find a formula for present values:
FV = PV(1 + k)n
Divide both sides by (1 + k)n
To get
PV = FV
(1 + k)n
This formula tells rational investors the most they would be willing to pay today for the
right to receive a future amount given the interest rate, k, and the time they would have to
wait to receive that future amount, n.
The denominator on the right-hand side of the equal sign is called the present value
interest factor (PVIF) and it is the engine that drives present values. This formula tells us
that as k and n rise, present values fall. The reverse is also true. These are very
important relationships! For more insight on this phenomenon, see PowerPoint slides
18 and 19.
This present value formula is also contained in your calculator. All we need to do to
make the calculator find a present value is to tell it what FV, k and n are. The reminder
is:
PV = FV(PVIF, n, k%)
For example, if you wanted to know the maximum you would be willing to pay now for
the right to receive $1,000 in 10 years, given your required return of 10% you could
either use the formula or the calculator.
PV = $1,000
(1.10)10 = $385.54
Or use the calculator
FV
n k%)
PV = $1,000( PVIF, 10, 10%)
Before proceeding with the calculator solution, it is important that you clear out (convert
to zero) your finance keys on the third row. Do this by hitting the 2nd (left second row
from top) then the clr tvm (far right third row; you will see the clr tvm above the FV key).
Proceed as follows:
Punch in 1,000 then hit FV
Punch in 10 then hit N
Punch in 10 then hit I/Y
Hit CPT then hit PV
You should be getting $385.54.
What does this answer mean? It means that the most you would be willing to pay now
for the right to receive $1,000 in 10 years given your required return is 10% is $385.54.
Would you be willing to pay less? Sure you would. If you paid less than $385.54 your
return would be higher that 10%. Would you be willing to pay more? No, because then
your return would be less than 10%.