The document discusses the time value of money, which is the concept that money available at the present time is worth more than the same amount in the future due to its potential earning capacity. It provides examples of calculating present and future values using interest formulas, discount rates, and tables. Specifically, it shows how to determine the present value of a future payment, like a bond that will pay $100 in two years when the rate of return is 12%.

1. Valore del Denaro
(Time Value of money)

2. Time Value of Money
• Business investmentsBusiness investments
extend over long periodsextend over long periods
of time, so we mustof time, so we must
recognize the time valuerecognize the time value
of money.of money.
• Investments that promiseInvestments that promise
returns earlier in time arereturns earlier in time are
preferable to those thatpreferable to those that
promise returns later inpromise returns later in
time.time.

3. Time Value of Money
A dollar today is worthA dollar today is worth
more than a dollar amore than a dollar a
year from now since ayear from now since a
dollar received todaydollar received today
can be invested,can be invested,
yielding more than ayielding more than a
dollar a year from now.dollar a year from now.

4. If $100 is invested today at 8% interest,
how much will you have in two years?
At the end of one year:At the end of one year:
$100 + 0.08$100 + 0.08 ×× $100 = (1.08)$100 = (1.08) ×× $100 = $108$100 = $108
At the end of two years:
(1.08)×$108 = $116.64$116.64
or
(1.08)2
× $100 = $116.64
Interest and the Time Value of Money

5. Interest and the Time Value of Money
IfIf PP dollars are invested today at thedollars are invested today at the
annual interest rateannual interest rate rr, then in, then in nn yearsyears
you would haveyou would have FFnn dollars computed asdollars computed as
follows:follows:
FFnn = P(1 + r)= P(1 + r)nn

6. TheThe present valuepresent value of any sum to beof any sum to be
received in the future can be computedreceived in the future can be computed
by turning the interest formula aroundby turning the interest formula around
and solving for P:and solving for P:
(1 + r)(1 + r)nnP = FP = Fnn
11
Interest and the Time Value of Money

7. A bond will pay $100 in two years. What isA bond will pay $100 in two years. What is
the present value of the $100 if an investorthe present value of the $100 if an investor
can earn a return of 12% on investments?can earn a return of 12% on investments?
Interest and the Time Value of Money
(1 + .12)(1 + .12)22P = 100P = 100
11
P = $100 (0.797)P = $100 (0.797)
P = $79.70P = $79.70
P = $100 (0.797)P = $100 (0.797)
P = $79.70P = $79.70

8. What does this mean?What does this mean?
If $79.70 is put in the bank today,If $79.70 is put in the bank today,
it will be worth $100 in two years.it will be worth $100 in two years.
In that sense, $79.70 today isIn that sense, $79.70 today is
equivalent to $100 in two years.equivalent to $100 in two years.
What does this mean?What does this mean?
If $79.70 is put in the bank today,If $79.70 is put in the bank today,
it will be worth $100 in two years.it will be worth $100 in two years.
In that sense, $79.70 today isIn that sense, $79.70 today is
equivalent to $100 in two years.equivalent to $100 in two years.
Interest and the Time Value of Money
Present Value = $79.70Present Value = $79.70
A bond will pay $100 in two years. What isA bond will pay $100 in two years. What is
the present value of the $100 if an investorthe present value of the $100 if an investor
can earn a return of 12% on investments?can earn a return of 12% on investments?

9. Let’s verify that if we put $79.70 in the bankLet’s verify that if we put $79.70 in the bank
today at 12% interest that it would grow totoday at 12% interest that it would grow to
$100 at the end of two years.$100 at the end of two years.
Year 1 Year 2
Beginning balance 79.70$ 89.26$
Interest @ 12% 9.56$ 10.71$
Ending balance 89.26$ 99.97$
Interest and the Time Value of Money
We can also determine the presentWe can also determine the present
value usingvalue using present value tablespresent value tables..
We can also determine the presentWe can also determine the present
value usingvalue using present value tablespresent value tables..

10. Time Value of Money
Rate
Periods 10% 12% 14%
1 0.909 0.893 0.877
2 0.826 0.797 0.769
3 0.751 0.712 0.675
4 0.683 0.636 0.592
5 0.621 0.567 0.519
Excerpt fromExcerpt from Present Value of $1Present Value of $1 Table inTable in
the Appendix to Chapter 14the Appendix to Chapter 14

11. Rate
Periods 10% 12% 14%
1 0.909 0.893 0.877
2 0.826 0.797 0.769
3 0.751 0.712 0.675
4 0.683 0.636 0.592
5 0.621 0.567 0.519
Time Value of Money
$100$100 ×× 0.797 = $79.70 present value0.797 = $79.70 present value
Present value factor of $1 for 2 periods at 12%.Present value factor of $1 for 2 periods at 12%.Present value factor of $1 for 2 periods at 12%.Present value factor of $1 for 2 periods at 12%.

12. Quick Check 
How much would you have to put in the bank
today to have $100 at the end of five years if the
interest rate is 10%?
a. $62.10
b. $56.70
c. $90.90
d. $51.90
How much would you have to put in the bank
today to have $100 at the end of five years if the
interest rate is 10%?
a. $62.10
b. $56.70
c. $90.90
d. $51.90

13. How much would you have to put in the bank
today to have $100 at the end of five years if the
interest rate is 10%?
a. $62.10
b. $56.70
c. $90.90
d. $51.90
How much would you have to put in the bank
today to have $100 at the end of five years if the
interest rate is 10%?
a. $62.10
b. $56.70
c. $90.90
d. $51.90
Quick Check 
$100$100 ×× 0.621 = $62.100.621 = $62.10$100$100 ×× 0.621 = $62.100.621 = $62.10

14. Time Value of Money
11 22 33 44 55 66
$100$100 $100$100 $100$100 $100$100 $100$100 $100$100
An investment that involves a series
of identical cash flows at the end of
each year is called an annuityannuity.

15. Time Value of Money
Lacey Inc. purchased a tract of land on
which a $60,000 payment will be due
each year for the next five years. What is
the present value of this stream of cash
payments when the discount rate is 12%?

16. Time Value of Money
We could solve the problem like this . . .
Look in Appendix C of this Chapter for the
Present Value of an Annuity of $1 Table
Periods 10% 12% 14%
1 0.909 0.893 0.877
2 1.736 1.690 1.647
3 2.487 2.402 2.322
4 3.170 3.037 2.914
5 3.791 3.605 3.433

17. Time Value of Money
We could solve the problem like this . . .
Periods 10% 12% 14%
1 0.909 0.893 0.877
2 1.736 1.690 1.647
3 2.487 2.402 2.322
4 3.170 3.037 2.914
5 3.791 3.605 3.433
$60,000 × 3.605 = $216,300$60,000 × 3.605 = $216,300

18. Quick Check 
If the interest rate is 14%, how much would you
have to put in the bank today so as to be able to
withdraw $100 at the end of each of the next
five years?
a. $34.33
b. $500.00
c. $343.30
d. $360.50
If the interest rate is 14%, how much would you
have to put in the bank today so as to be able to
withdraw $100 at the end of each of the next
five years?
a. $34.33
b. $500.00
c. $343.30
d. $360.50

19. If the interest rate is 14%, how much would you
have to put in the bank today so as to be able to
withdraw $100 at the end of each of the next
five years?
a. $34.33
b. $500.00
c. $343.30
d. $360.50
If the interest rate is 14%, how much would you
have to put in the bank today so as to be able to
withdraw $100 at the end of each of the next
five years?
a. $34.33
b. $500.00
c. $343.30
d. $360.50
Quick Check 
$100$100 ×× 3.433 = $343.303.433 = $343.30$100$100 ×× 3.433 = $343.303.433 = $343.30

20. Quick Check 
If the interest rate is 14%, what is the present
value of $100 to be received at the end of the
3rd, 4th, and 5th years?
a. $866.90
b. $178.60
c. $ 86.90
d. $300.00
If the interest rate is 14%, what is the present
value of $100 to be received at the end of the
3rd, 4th, and 5th years?
a. $866.90
b. $178.60
c. $ 86.90
d. $300.00

21. If the interest rate is 14%, what is the present
value of $100 to be received at the end of the
3rd, 4th, and 5th years?
a. $866.90
b. $178.60
c. $ 86.90
d. $300.00
If the interest rate is 14%, what is the present
value of $100 to be received at the end of the
3rd, 4th, and 5th years?
a. $866.90
b. $178.60
c. $ 86.90
d. $300.00
Quick Check 
$100×(3.433-1.647)= $100×1.786 = $178.60
or
$100×(0.675+0.592+0.519)= $100×1.786 = $178.60
$100×(3.433-1.647)= $100×1.786 = $178.60
or
$100×(0.675+0.592+0.519)= $100×1.786 = $178.60