1. Time Value of Money Lecture No.2 Professor C. S. Park
Fundamentals of Engineering Economics
Copyright © 2005

2. Chapter 2 Time Value of Money
Interest: The Cost of Money
Economic Equivalence
Interest Formulas – Single Cash Flows
Equal-Payment Series
Dealing with Gradient Series
Composite Cash Flows.
Power-Ball Lottery

3. Decision Dilemma—Take a Lump Sum or Annual Installments
A suburban Chicago couple won the Power-ball.
They had to choose between a single lump sum $104 million, or $198 million paid out over 25 years (or $7.92 million per year).
The winning couple opted for the lump sum.
Did they make the right choice? What basis do we make such an economic comparison?

4. Option A
(Lump Sum)
Option B
(Installment Plan) 1 2 3 25
$104 M
$7.92 M

5. What Do We Need to Know?
To make such comparisons (the lottery decision problem), we must be able to compare the value of money at different point in time.
To do this, we need to develop a method for reducing a sequence of benefits and costs to a single point in time. Then, we will make our comparisons on that basis.

6. Time Value of Money
Money has a time value because it can earn more money over time (earning power).
Money has a time value because its purchasing power changes over time (inflation).
Time value of money is measured in terms of interest rate.
Interest is the cost of money—a cost to the borrower and an earning to the lender

7. Delaying Consumption

10. Key terms Principal (P) Interest rate (i) Interest period (n)
Number of interest periods (N)
A plan for receipt (An)
Future amount of money (F)

11. Which Repayment Plan? End of Year Receipts Payments Plan 1 Plan 2
$20,000.00
$200.00
Year 1
5,141.85
Year 2
Year 3
Year 4
Year 5
30,772.48
The amount of loan = $20,000, origination fee = $200, interest rate = 9% APR (annual percentage rate)

12. Cash Flow Diagram

13. End-of-Period Convention
Interest Period 1
End of interest
period
Beginning of
Interest period 1

14. Methods of Calculating Interest
Simple interest: the practice of charging an interest rate only to an initial sum (principal amount).
Compound interest: the practice of charging an interest rate to an initial sum and to any previously accumulated interest that has not been withdrawn.

15. Simple Interest P = Principal amount i = Interest rate
N = Number of interest periods
Example:
P = $1,000
i = 8%
N = 3 years
End of Year
Beginning Balance
Interest earned
Ending Balance
$1,000 1
$80
$1,080 2
$1,160 3
$1,240

16. Simple Interest Formula

17. Compound Interest
Compound interest: the practice of charging an interest rate to an initial sum and to any previously accumulated interest that has not been withdrawn.

18. Compound Interest P = Principal amount i = Interest rate
N = Number of interest periods
Example:
P = $1,000
i = 8%
N = 3 years
End of Year
Beginning Balance
Interest earned
Ending Balance
$1,000 1
$80
$1,080 2
$86.40
$1,166.40 3
$93.31
$1,259.71

19. Compounding Process $1,080 $1,166.40 $1,259.71 1 $1,000 2 3 $1,080
$1,259.71 1
$1,000 2 3
$1,080
$1,166.40

20. $1,259.71 1 2 3
$1,000

21. Compound Interest Formula

22. Some Fundamental Laws
The Fundamental Law of Engineering Economy

23. “The greatest mathematical discovery of all time,” Albert Einstein
Compound Interest
“The greatest mathematical discovery of all time,”
Albert Einstein

24. Practice Problem: Warren Buffett’s Berkshire Hathaway
Went public in 1965: $18 per share
Worth today (August 22, 2003): $76,200
Annual compound growth: 24.58%
Current market value: $ Billion
If he lives till 100 (current age: 73 years as of 2003), his company’s total market value will be ?

25. Market Value
Assume that the company’s stock will continue to appreciate at an annual rate of 24.58% for the next 27 years.

26. EXCEL Template
In 1626 the Indians sold Manhattan Island to Peter Minuit
Of the Dutch West Company for $24.
If they saved just $1 from the proceeds in a bank account
that paid 8% interest, how much would their descendents
have now?
As of Year 2003, the total US population would be close to
275 millions. If the total sum would be distributed equally
among the population, how much would each person receive?

27. Excel Solution
FV(8%,377,0,1)
= $3,988,006,142,690

28. Excel Worksheet A B C 1 P 2 i 8% 3 N 377 4 FV 5 FV(8%,377,0,1)
= $3,988,006,142,690

29. Practice Problem Problem Statement
If you deposit $100 now (n = 0) and $200 two years from now (n = 2) in a savings account that pays 10% interest, how much would you have at the end of year 10?

30. Solution F
$100
$200

31. ? Practice problem Problem Statement
Consider the following sequence of deposits and withdrawals over a period of 4 years. If you earn 10% interest, what would be the balance at the end of 4 years? ?
$1,210 1 4 2 3
$1,500
$1,000
$1,000

32. ?
$1,210 1 3 2 4
$1,000
$1,000
$1,500
$1,100
$1,000
$1,210
$2,981
$2,100
$2,310
+ $1,500
-$1,210
$1,100
$2,710

33. Solution n = 0 n = 1 n = 2 n = 3 n = 4 End of Period Beginning balance
Deposit made
Withdraw
Ending
n = 0
$1,000
n = 1
$1,000( ) =$1,100
$2,100
n = 2
$2,100( ) =$2,310
$1,210
$1,100
n = 3
$1,100( ) =$1,210
$1,500
$2,710
n = 4
$2,710( ) =$2,981
$2,981