Appendix E. Time value of money
1
Prof. Sorah Park
Ewha Womans University
Appendix Preview
Would you rather receive NT$1,000 today or a year from now?
You should prefer to receive the NT$1,000 today because you can invest the NT$1,000
and earn interest on it. As a result, you will have more than NT$1,000 a year from now.
What this example illustrates is the concept of the time value of money. Everyone
prefers to receive money today rather than in the future because of the interest factor.
Copyright ©2019 John Wiley & Son, Inc.
2
Nature of Interest
•
•
•
Payment for the use of money.
Difference between the amount borrowed or invested (principal) and the amount
repaid or collected.
Three elements determine the amount of interest:
1. Principal (p): The original amount borrowed or invested.
2. Interest Rate (i): Annual percentage of the principal.
3. Time (n): The number of periods that the principal is borrowed or
invested.
LO 1
Copyright ©2019 John Wiley & Son, Inc.
3
Nature of Interest
Simple Interest
•
Interest is computed on the principal (p) only.
Assume: You borrowed NT$5,000 for 2 years at a simple interest rate of 6% annually.
Calculate: Annual interest.
LO 1
Copyright ©2019 John Wiley & Son, Inc.
4
Nature of Interest
Compound Interest
•
•
LO 1
Computes interest on
•
the principal and
•
any interest earned that has not been paid or withdrawn.
Business situations use compound interest when interest is not paid periodically during
the time of borrowing.
Copyright ©2019 John Wiley & Son, Inc.
5
Compound Interest
Assume: You deposit €1,000 in Bank Two, where it will earn simple interest of 9%
per year, and you deposit another €1,000 in Citizens Bank, where it will earn
compound interest of 9% per year compounded annually. Also assume that in both
cases you will not withdraw any cash until three years from the date of deposit.
LO 1
Copyright ©2019 John Wiley & Son, Inc.
6
Future Value Concepts
Future value of a single amount
•
LO 1
Value at a future date of a given amount invested, assuming compound interest
FV
=
future value of a single amount
p
= principal (or present value; the value today)
i
=
interest rate for one period
n
=
number of periods
Copyright ©2019 John Wiley & Son, Inc.
7
Future value of a single amount
Assume: You deposit €1,000 for three years. The annual interest rate is 9%.
Calculate: The future value after three years.
FV
=
p
=
€1,000
× (1 + .09)3
=
€1,000
×
=
€1,295.03
×
(1 + i)n
1.29503
LO 1
Copyright ©2019 John Wiley & Son, Inc.
8
Future value of a single amount
Assume: Again, you deposit €1,000 for three years. The annual interest rate is 9%.
Calculate: The future value after three years using a table.
LO 1
What factor do we use?
Present Value
x
Factor
=
Future Value
€1,000
x
1.29503
=
€1,295.03
Copyright ©2019 John Wiley & Son, Inc.
9
Present Value Concepts
Present value of a single amount
The present value is the value now of a given amount to be paid or received in the future,
assuming compound interest.
Present value variables:
1. Future Value (FV): Dollar amount to be received
2. Interest Rate (i): Called the discount rate
3. Time (n): length of time until amount is received (number of periods).
LO 2
Copyright ©2019 John Wiley & Son, Inc.
10
Present Value Concepts
Present value of a single amount:
Value now of a given future amount invested, assuming compound interest.
LO 2
PV
=
present value
FV
=
the dollar amount to be received in the future
i
=
interest rate for one period
n
=
number of periods
Copyright ©2019 John Wiley & Son, Inc.
11
Present value of a single amount
Assume: You make a deposit and want a 10% rate of return. The future value of the
deposit in one year is €1,000.
Calculate: The present value.
PV
=
FV
=
€1,000
÷ (1 + .10)1
=
€1,000
÷
=
€909.09
÷
(1 + i)n
1.10
LO 2
Copyright ©2019 John Wiley & Son, Inc.
12
Present value of a single amount
Assume: Again, you make a deposit and want a 10% rate of return. The future
value of the deposit in one year is €1,000.
Calculate: The present value using a table.
LO 2
What factor do we use?
Future Value
x
Factor
=
Present Value
€1,000
x
0.90909
=
€909.09
Copyright ©2019 John Wiley & Son, Inc.
13
Present value of a single amount
Assume: You make a deposit and want a 10% rate of return. The future value of the
deposit in two years is €1,000.
Calculate: The present value.
PV
=
FV
=
€1,000
÷ (1 + .10)2
=
€1,000
÷
=
€826.45
÷
(1 + i)n
1.10
LO 2
Copyright ©2019 John Wiley & Son, Inc.
14
Present value of a single amount
Assume: Again, you make a deposit and want a 10% rate of return. The future
value of the deposit in two years is €1,000.
Calculate: The present value using a table.
LO 2
What factor do we use?
Future Value
x
Factor
=
Present Value
€1,000
x
0.82645
=
€826.45
Copyright ©2019 John Wiley & Son, Inc.
15
Present value of a single amount
Assume: You have a winning lottery ticket. You have the option of taking
NT$100,000 three years from now or taking the present value of NT$100,000
now. The discount rate is 8%.
Calculate: How much will you receive if you accept your winnings now?
LO 2
Future Value x
NT$100,000
Factor
= Present Value
NT$79,383
x 0.79383 =
Copyright ©2019 John Wiley & Son, Inc.
16
Present value of a single amount
Assume: You want to accumulate £5,000 for a down payment on a new car 4 years
from now.
Calculate: How much do you have to deposit today in your super savings account,
paying 9% interest?
LO 2
Future Value
x
Factor
=
Present Value
£5,000
x
0.70843
=
£3,542.15
Copyright ©2019 John Wiley & Son, Inc.
17
Present Value Concepts
Present value of an annuity
The present value is the value now of a series of amounts to be paid or received in the
future, assuming compound interest.
Present value variables:
1. Interest Rate (i): Called the discount rate
2. Number of payments (n): the number of payments (receipts)
3. Payment: the amount of the periodic payments (receipts)
LO 2
Copyright ©2019 John Wiley & Son, Inc.
18
Present value of an annuity
Assume: You will receive €1,000 cash annually for three years at a time when the
discount rate is 10%.
Calculate: The present value in this situation.
LO 2
Copyright ©2019 John Wiley & Son, Inc.
19
Present value of an annuity
Assume: You will receive €1,000 cash annually for three years at a time when the
discount rate is 10%.
Calculate: The present value in this situation using a table.
LO 2
Future Amount x
€1,000
x
Factor
=
2.48685
=
Present Value
€2,486.85
Copyright ©2019 John Wiley & Son, Inc.
20
Present value of an annuity
Assume: Kildare Construction has just signed a finance lease contract for equipment
that requires rental payments of €6,000 each, to be paid at the end of each of the next
5 years. The appropriate discount rate is 12%.
Calculate: What is the present value of the rental payments—that is, the amount used
to finance the leased equipment?
LO 2
Future
Amount
€6,000
x
Factor
=
x
3.60478
=
Present Value
€21,628.68
Copyright ©2019 John Wiley & Son, Inc.
21
Time periods and discounting
When the time frame is less than one year, it is necessary to convert the annual interest
rate to the applicable time frame.
Assume: An investor received €500 semiannually for three years instead of €1,000
annually. The annual discount rate is 10%.
Calculate: The present value in this situation using a table.
LO 2
Copyright ©2019 John Wiley & Son, Inc.
22
Time periods and discounting
Comparing compounding results:
Two €500 payments per year -> PV = €2,537.85
One €1,000 payment per year -> PV = €2,486.86
The higher number of payments results in a higher present value.
Future Amount
x
Factor
=
Present Value
€500
x
5.07569
=
€2,537.85
Copyright ©2019 John Wiley & Son, Inc.
23