BUSINESS AND FAITH INTEGRATION
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Lecture_Note_Chapter_5-The_Time_Value_of_Money.docxBUSI 530
Chapter 5: The Time Value of Money
Chapter 5 Learning Objectives
1. Calculate the future value to which money invested at a given interest rate will grow.
2. Calculate the present value of a future payment.
3. Calculate present and future values of a series of cash payments.
4. Find the interest rate implied by present and future values.
5. Compare interest rates quoted over different time intervals—for example, monthly versus annual rates.
6. Understand the difference between real and nominal cash flows and between real and nominal interest rates.
Time Value of Money
Money has a time value. It can be expressed in multiple ways:
A dollar today held in savings will grow.
A dollar received in a year is not worth as much as a dollar received today.
Chapter 5 Outline
Interest and Future Value
Future Value
Interest: Simple vs. Compound
Present Value
Discount Rates and Present Values
Multiple Cash Flows
Level Cash Flows: Perpetuities and Annuities
Annuities Due
Effective Annual Interest Rates
Inflation and the Time Value of Money
Basic idea of this chapter: Money has a time value and it can be expressed by interest rates.
Future Values
Let r = annual interest rate
Let t = # of years
Simple Interest Compound Interest
Future Value - Amount to which an investment will grow after earning interest.
Compound Interest - Interest earned on interest.
Simple Interest - Interest earned only on the original investment.
Simple Interest: Example
Interest earned at a rate of 7% for five years on a principle balance of $100.
Example - Simple Interest
Today Future Years
1 2 3 4 5
Interest Earned 7 7 7 7 7
Value 100 107 114 121 128 135
Value at the end of Year 5: $135
Future Value – Amount to which an investment will grow after earning interest.
Simple Interest – Interest earned only on the original investment.
Compound Interest: Example
Interest earned at a rate of 7% for five years on the previous year’s balance.
Example - Simple Interest
Today Future Years
1 2 3 4 5
Interest Earned 7 7.49 8.01 8.58 9.15
Value 100 107 114.49 122.50 131.08 140.26
Value at the end of Year 5: $140.26
Future Value – Amount to which an investment will grow after earning interest.
Compound Interest – Interest earned on interest.
The Power of Compounding
Interest earned at a rate of 7% for the first forty years on the $100 invested using simple and compound interest.
Present Value
What is it?
Why is it useful?
Present Value – Value today of a future cash flow
Present Value
Discount Rate – Interest rate used to compute present values of future cash flows
Discount Factor – Present value of a $1 future payment
Present Value – Value today of a future cash flow
Recall: t = number of years
Present Value: Example
Always ahead of the game, Tommy, at 8 years old, believes he will need $100,000 to pay for college. If he can invest at a rate of 7% per year, how much money should he ask his rich Uncle GQ to give him?
FV = $100,000 t = 10 yrs r = 7%
Note: Ignore inflation/taxes
Time Value of Money (applications)
The PV formula has many applications. Given any variables in the equation, you can solve for the remaining variable.
Present Values: Changing Discount Rates
The present value of $100 to be received in 1 to 20 years at varying discount rates:
PV of Multiple Cash Flows
The present value of multiple cash flows can be calculated:
Recall: r = the discount rate
Note: The present value of an entire investment can be computed by summing the present values of each individual cash flow.
Multiple Cash Flows: Example
Your auto dealer gives you the choice to pay $15,500 cash now or make three payments: $8,000 now and $4,000 at the end of the following two years. If your cost of money (discount rate) is 8%, which do you prefer?
* The initial payment occurs immediately and therefore would not be discounted.
The total present value of the financed transaction is $15,133.06. Therefore you would save money should you choose to make payments instead of paying it all up front in cash.
Perpetuities
What are they?
Let C = Yearly Cash Payment
PV of Perpetuity:
Recall: r = the discount rate
Perpetuity – A stream of level cash payments that never ends
Perpetuities: Example
In order to create an endowment, which pays $185,000 per year forever, how much money must be set aside today if the rate of interest is 8%?
What if the first payment won’t be received until 3 years from today?
Annuities
What are they?
Why are they useful?
Annuity – Equally spaced level stream of cash flows lasting for a limited period of time
Present Value of an Annuity
Let:
C = yearly cash payment
r = interest rate
t = number of years cash payment is received
The terms within the brackets are collectively called the “annuity factor.”
Annuity – Equally spaced level stream of cash flows lasting for a limited period of time
Annuity Factor - The present value of $1 paid every year for each of t years.
Annuities: Example
You are purchasing a home and are scheduled to make 30 annual installments of $10,000 per year. Given an interest rate of 5%, what is the price you are paying for the house (i.e. what is the present value)?
Future Value of Annuities
You plan to save $4,000 every year for 20 years and then retire. Given a 10% rate of interest, how much will you have saved by the time you retire?
Annuity Due
What is it?
How does it differ from an ordinary annuity?
How does the future value differ from an ordinary annuity?
Recall: r = the discount rate
Annuity Due : Level streams of cash flow starting immediately
Annuities Due: Example
Suppose you invest $429.59 annually at the beginning of each year at 10% interest. After 50 years, how much would your investment be worth?
Annuity Due : Level stream of cash flows starting immediately.
Interest Rates: EAR & APR
What is EAR?
What is APR?
How do they differ?
Effective annual interest rate : - Interest rate that is annualized using compound interest.
Annual percentage rate: Interest rate that is annualized using simple interest.
EAR and APR Calculations
Effective annual interest rate : - Interest rate that is annualized using compound interest.
Annual percentage rate : Interest rate that is annualized using simple interest.
*where MR = monthly interest rate
Effective annual interest rate : - Interest rate that is annualized using compound interest.
Annual percentage rate : Interest rate that is annualized using simple interest.
EAR and APR: Example
Given a monthly rate of 1%, what is the Effective Annual Rate (EAR)? What is the Annual Percentage Rate (APR)?
Inflation
What is it?
What determines inflation rates?
What is deflation?
Inflation: The rate at which prices as a whole are increasing.
Inflation and Real Interest
Exact calculation:
Approximation:
Inflation : The rate at which prices as a whole are increasing.
Nominal interest rate : The rate at which money invested grows.
Real interest rate : The rate at which the purchasing power of an investment increases.
Inflation: Example
If the nominal interest rate on your interest-bearing savings account is 2.0% and the inflation rate is 3.0%, what is the real interest rate?
Appendix A: Inflation
Year
Future Value
Page 1 of 10
FV = Initial investment(1)
Simple
rt
´+´
r
1
(1)
t
r
DF
+
=
1
(1)
t
r
PVFV
+
=´
10
1
(1.07)
1
$100,000$50,835
(1)
t
PVFV
r
=´=´»
+
0
20
40
60
80
100
120
0
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
Number of Years
PV of $100
0%
5%
10%
15%
1
2
:
The cash flow in year 1
The cash flow in year 2
The cash flow in year t (with any numbe
r of cash flows in between)
t
Denote
C
C
C
=
=
=
12
12
(1)(1)(1)
....
t
t
C
CC
rrr
PV
+++
=+++
1
2
4,000
1
(1.08)
4,000
2
(1.08)
Initial Payment*
8,000.00
3,703.70
3,429.36
Total PV $15,133.06
PVofC
PVofC
+
+
==
==
=
C
r
PV
=
185,000
.08
$2,312,500
PV
==
2
2,312,500
(1.08)
$1,982,596
PV
+
==
11
(1)
t
r
rr
PVC
´+
éù
=´-
ëû
30
11
.05
.05(1.05)
$10,000
$153,724.51
PV
PV
+
éù
=-
ëû
=
20
20
11
.10
.10(1.10)
$4,000(1.10)
$229,100
FV
FV
+
éù
=-´+
ëû
=
(1)
AnnuityDueAnnuity
PVPVr
=´+
(1)
AnnuityDueAnnuity
FVFVr
=´+
)
1
(
r
FV
FV
Annuity
AD
+
´
=
000
,
550
$
)
10
.
1
(
)
000
,
500
($
)
1
(
=
´
=
+
´
=
AD
AD
Annuity
AD
FV
FV
r
FV
FV
1
)
1
(
12
-
+
=
MR
EAR
12
´
=
MR
APR
%
00
.
12
)
12
(
)
01
.
0
(
%
68
.
12
1
)
01
.
1
(
12
=
´
=
=
-
=
APR
EAR
1+nominal interest rate
1+inflation rate
1real interest rate=
+
rate
inflation
-
rate
interest
nominal
rate
interest
Real
»
1+.02
1+.03
1real interest rate=
1real interest rate= 0.9903
real interest rate = -.0097 or -.97%
Approximation = .02-.03 =.011%
+
+
-=-
FV = Initial investment(1)
t
Compound
r
´+
