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© The McGraw-Hill Companies, Inc., 2003McGraw-Hill/Irwin
Time Value of Money
Business investments Business investments extend over long periods extend over long periods of time, so we must of time, so we must recognize the time value recognize the time value of money.of money.
Investments that promise Investments that promise returns earlier in time are returns earlier in time are preferable to those that preferable to those that promise returns later in promise returns later in time.time.
© The McGraw-Hill Companies, Inc., 2003McGraw-Hill/Irwin
Time Value of Money
A dollar today is worth A dollar today is worth more than a dollar a more than a dollar a
year from now since a year from now since a dollar received today dollar received today
can be invested, can be invested, yielding more than a yielding more than a
dollar a year from now.dollar a year from now.
© The McGraw-Hill Companies, Inc., 2003McGraw-Hill/Irwin
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 = $100 =
$108$108At 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
© The McGraw-Hill Companies, Inc., 2003McGraw-Hill/Irwin
Interest and the Time Value of Money
If If PP dollars are invested today at dollars are invested today at the annual interest rate the annual interest rate rr, then in , then in nn years you would have years you would have FFnn dollars dollars
computed as follows:computed as follows:
FFnn = P(1 + r) = P(1 + r)nn
© The McGraw-Hill Companies, Inc., 2003McGraw-Hill/Irwin
The The present valuepresent value of any sum to of any sum to be received in the future can be be received in the future can be computed by turning the interest computed by turning the interest formula around and solving for P:formula around and solving for P:
(1 + r)(1 + r)nnP = FP = Fnn11
Interest and the Time Value of Money
© The McGraw-Hill Companies, Inc., 2003McGraw-Hill/Irwin
Time Value of Money
RatePeriods 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
RatePeriods 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 from Excerpt from Present Value of $1Present Value of $1 Table in Table in the Appendix to Chapter 14the Appendix to Chapter 14
© The McGraw-Hill Companies, Inc., 2003McGraw-Hill/Irwin
A bond will pay $100 in two years. What is A bond will pay $100 in two years. What is the present value of the $100 if an investor the 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 = 10011
P = $100 (0.797)P = $100 (0.797)P = $79.70P = $79.70P = $100 (0.797)P = $100 (0.797)P = $79.70P = $79.70
© The McGraw-Hill Companies, Inc., 2003McGraw-Hill/Irwin
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 is In 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 is In 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 is A bond will pay $100 in two years. What is the present value of the $100 if an investor the present value of the $100 if an investor can earn a return of 12% on investments?can earn a return of 12% on investments?
© The McGraw-Hill Companies, Inc., 2003McGraw-Hill/Irwin
Let’s verify that if we put $79.70 in the bank Let’s verify that if we put $79.70 in the bank today at 12% interest that it would grow to today at 12% interest that it would grow to
$100 at the end of two years.$100 at the end of two years.
Year 1 Year 2Beginning balance 79.70$ 89.26$ Interest @ 12% 9.56$ 10.71$ Ending balance 89.26$ 99.97$
Year 1 Year 2Beginning 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 present We can also determine the present value using value using present value tablespresent value tables..
We can also determine the present We can also determine the present value using value using present value tablespresent value tables..
© The McGraw-Hill Companies, Inc., 2003McGraw-Hill/Irwin
Time Value of Money
RatePeriods 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
RatePeriods 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 from Excerpt from Present Value of $1Present Value of $1 Table in Table in the Appendix to Chapter 14the Appendix to Chapter 14
© The McGraw-Hill Companies, Inc., 2003McGraw-Hill/Irwin
RatePeriods 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
RatePeriods 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 value 0.797 = $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%.Atau dapat dihitung dengan mempergunakan kalkulator. Caranya: Tekan angka 1 dan tekan bagi 1,12, terus tekan sama dengan, yang akan menghasilkan 0.893 untuk tahun pertama. Lalu tekan sama dengan lagi untuk menghasikan 0.797, dan seterusnya.
© The McGraw-Hill Companies, Inc., 2003McGraw-Hill/Irwin
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.
© The McGraw-Hill Companies, Inc., 2003McGraw-Hill/Irwin
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
© The McGraw-Hill Companies, Inc., 2003McGraw-Hill/Irwin
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 = 0.621 = $62.10$62.10$100 $100 0.621 = 0.621 = $62.10$62.10
© The McGraw-Hill Companies, Inc., 2003McGraw-Hill/Irwin
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%?
© The McGraw-Hill Companies, Inc., 2003McGraw-Hill/Irwin
Time Value of Money
We could solve the problem like this . . .
Look in Appendix C of this Chapter for thePresent 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
© The McGraw-Hill Companies, Inc., 2003McGraw-Hill/Irwin
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
© The McGraw-Hill Companies, Inc., 2003McGraw-Hill/Irwin
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
© The McGraw-Hill Companies, Inc., 2003McGraw-Hill/Irwin
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 = $343.30 3.433 = $343.30$100 $100 3.433 = $343.30 3.433 = $343.30
© The McGraw-Hill Companies, Inc., 2003McGraw-Hill/Irwin
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
© The McGraw-Hill Companies, Inc., 2003McGraw-Hill/Irwin
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)= $1001.786 = $178.60or
$100(0.675+0.592+0.519)= $1001.786 = $178.60
$100(3.433-1.647)= $1001.786 = $178.60or
$100(0.675+0.592+0.519)= $1001.786 = $178.60
© The McGraw-Hill Companies, Inc., 2003McGraw-Hill/Irwin
Typical Cash Outflows
Repairs andRepairs andmaintenancemaintenance
IncrementalIncrementaloperatingoperating
costscosts
InitialInitialinvestmentinvestment
WorkingWorkingcapitalcapital
© The McGraw-Hill Companies, Inc., 2003McGraw-Hill/Irwin
Typical Cash Inflows
ReductionReductionof costsof costs
SalvageSalvagevaluevalue
IncrementalIncrementalrevenuesrevenues
Release ofRelease ofworkingworkingcapitalcapital