# Chapter 5

```Chapter 05
McGraw-Hill/Irwin
Time Value of Money 2:
Analyzing Annuity Cash Flows
1
Introduction
• Time Value of Money calculations
– Can deal with either single cash flows (Chapter 4)
– or multiple cash flows over time (Chapter 5)
5-2
Future Value of Multiple
Cash Flows
• Multiple cash flows
– Regular, evenly-spaced
•
•
•
•
Car loans and home mortgage loans
Saving for retirement
Companies paying interest on debt
Companies paying dividends
5-3
Future Value – Several Cash Flows
• Concept: Compounding
– Value in the future
– Different cash flows paid in at different times
...
5-4
Finding FV – Several Cash Flows
Example
• Assumptions
– Invest \$100 today (compounds for 3 years)
– Invest \$125 at end of year 2 (compounds for 2
years)
– Invest \$150 at end of year 3 (compounds for 1
year)
– Interest rates: 7%
...
5-5
FV Several Cash Flows
Time Line Example
Several
(Different)
Cash Flow
Values
...
5-6
Future Value – Level Cash Flows
• Concept: Compounding
• Also called “annuities”
– Value in the future
– Same cash flows paid in every period
...
5-7
Finding FV – Level Cash Flows/Annuity
Example
• Assumptions:
– Invest \$100 at the end of each year for 5 years
– Interest rates: 8%
...
5-8
Level Cash Flows Time Line Example
Level (same)
Cash Flows Each
Period
...
5-9
Future Value – Multiple Annuities
• Concept: Compounding – annuity equation to
compute future value – two levels of cash
flows
• To solve for multiple annuities, compute FV for
each separately and add them together
...
5-10
Finding FV – Multiple Annuities
Example
• Assumptions:
– Invest \$100 at end of years 1 - 3 at 8%
– Invest \$150 at end of years 4 - 5 at 8%
...
5-11
Future Value – Multiple Annuities
Step 1 (same as FV of Level Cash Flows Calculation)
Step 2
Step 3
Add two sums together – FV of
both is \$690.66
5-12
Present Value of Multiple
Cash Flows
• Multiple cash flows:
– Car loans and home mortgage loans
– Determining value of business opportunities
5-13
Present Value – Several Cash Flows
• Concept: Discounting
– Value of future sum today
– Different cash flows paid in at different times
...
5-14
Finding PV – Several Cash Flows
Example
• Assumptions:
– Deposit \$100 today
– Deposit \$125 next year
– Deposit \$150 at end of year 2
– Interest rates: 7%
...
5-15
PV Several Cash Flows
Time Line Example
...
5-16
PV Several Cash Flows
0
\$100
\$116.82
\$131.02
\$0.00
1
2
\$125
\$150
3
\$0
\$125/(1.07)
\$150/ (1.07)2
\$0 / (1.07)3
\$347.84
5-17
Present Value – Level Cash Flows
• Concept: Discounting
– Value of future sum today
– Level cash flows paid in at different times
• Most loans set up with even payments throughout
life of loan
...
5-18
Finding PV – Level Cash Flows
Example
• Assumptions:
– \$100 payments at end of each year for 5 years
– Interest rates: 8% per year
...
5-19
PV Level Cash Flows
Time Line Example
...
5-20
Present Value – Multiple Annuities
• Concept: Discounting
– Changing level cash flows
– Ex: Alex Rodriguez’s baseball contract
...
5-21
PV – Multiple Annuities Example
• Assumptions (Alex Rodriguez’s Contract):
– \$10 million signing bonus
– \$21 million per year from 2001 – 2004
– \$25 million per year in 2005 and 2006
– \$27 million per year in 2007 - 2010
– Interest rates: 8% per year ...
5-22
PV Multiple Annuities Example (cont.)
5-23
Perpetuity – Special Annuity
• Concept: Discounting
– Stream of level cash flows paid forever
– Preferred stocks are an example
– Value of investment is present value of all
future annuity payments
...
5-24
Ordinary Annuities vs. Annuities Due
• Ordinary Annuity
– Payment occurs at the end of each period
• Annuity Due
– Payment occurs at the beginning of each period
5-25
Annuity Due
Time Line Example
• Cash flows at beginning, not at end of period
• Five annuity-due cash flows basically same as
payment today plus 4-year ordinary annuity
• Payments occur one period sooner than
ordinary annuity -- earn extra period of interest
...
5-26
Future Value of Annuity Due
• Concept: Compounding
– Value of future sum today
– Cash flows at beginning of each period
...
5-27
Future Value of Annuity Due
– Assumptions:
• Assumes cash flows at the beginning of each period
• 5 annuity-due cash flows of \$100 each
– First cash flow compounds for 5 years
– Last cash flow compounds for 1 year
• Interest rates: 8%
...
5-28
Present Value of Annuity Due
• Concept: Discounting
– Today’s value of future sum
– Cash flows at beginning of each period
...
5-29
Present Value of Annuity Due
• Assumptions:
– Cash flows at beginning of period
– 5 annuity-due cash flows of \$100
• First cash flow paid today – not discounted
• Last cash flow discounted 4 years
• All cash flows discounted for one year less than
ordinary annuity
• Interest rates: 8%
...
5-30
Compounding Frequency
• Used in situations that do not use yearly time
periods
– Semiannual bond payments
– Quarterly stock dividends
– Consumer loans – monthly payments
5-31
Effect of Compounding Frequency
• Assumptions:
– \$100 deposit today
– 12% annual interest rate
– Bank compounds interest at six months instead of end
of year
– Interest is earned on interest
...
5-32
EARS and APRS
• Quoted, or nominal rate called annual percentage
rate (APR)
• Rate that incorporates compounding called
effective annual rate (EAR)
• Relationship between APR and EAR:
...
5-33
EARS vs. APR Example
– Assumptions:
•
•
•
•
Borrow \$100 today
12% annual interest rate
APR: Loan compounds annually -- you pay 12.00%
EARS: Loan compounds monthly -- you pay 12.68%
– Formula to convert APR to EAR:
12
0.12 

EAR  1

12 

1
...
5-34
Annuity Loans
• Compares payments
• Compares implied interest rate
5-35
Finding Payments on
Amortized Loan
• Concept:
– Rearrange PV of annuity formula to solve for
payment
...
5-36
Payments on Amortized Loan
Example
• Assumptions:
– Need \$10,000 to buy car
– Loan term: 4 years
– Interest rate: 9%
• Use interest rate of 0.75 % (=9%/12) and 48 periods (=4 X 12)
...
5-37
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