1. Chapter 2. Time Value of Money

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.

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 $116.5 million, or $198 million paid out over 25 years (or $7.92 million per year after taxes).
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
$116.5 M
$7.92 M
Total $198M

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 points 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 having money available for use — a cost to the borrower and an earning to the lender.

8. Delaying Consumption

10. Elements of Transactions Involving Interest
P (principal): initial amount of money invested or borrowed
i (interest rate): cost or price of money; expressed as a percentage per period of time
n (interest period): determines how frequently interest is calculated
N (number of interest periods): duration of transaction
An (a plan for receipts or disbursements): a particular cash flow pattern
F (future amount): results from the cumulative effects of the interest

11. Which Repayment Plan? End of Year Receipts ($) Payments ($) Plan 1
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. “The greatest mathematical discovery of all time” Albert Einstein
Compound Interest
The Fundamental Law of
Engineering Economy
“The greatest mathematical discovery of all time”
Albert Einstein

23. 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 ?

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

25. Example
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 (year 2003)?
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?

26. Solution
F=1(1+0.08)377= $3,988,006,142,690

27. 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

28. Practice Problem
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?

29. Solution F
$100
$200

30. Practice problem
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 3 1 2 4
$1,500
$1,000
$1,000

31. ?
$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

32. 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

33. Economic Equivalence

34. Economic Equivalence What do we mean by “economic equivalence?”
Why do we need to establish an economic equivalence?
How do we establish an economic equivalence?

35. Economic Equivalence
Economic equivalence exists between cash flows that have the same economic effect and could therefore be traded for one another.
Even though the amounts and timing of the cash flows may differ, the appropriate interest rate makes them equal.
Any cash flow – whether a single payment or a series of payments – can be converted to an equivalent cash flow at any point in time.

36. Economic Equivalence

37. Equivalence from Personal Financing Point of View
If you deposit P dollars today for N periods at i, you will have F dollars at the end of period N.

38. Alternate Way of Defining EquivalenceP
F dollars at the end of period N is equal to a single sum P dollars now, if your earning power is measured in terms of interest rate i. N = F N

39. = Practice Problem F At 8% interest, what is the equivalent worth
of $2,042 now 5 years from now?
$2,042
If you deposit $2,042 today in a savings
account that pays 8% interest annually.
how much would you have at the end of
5 years? 1 2 3 4 5 3 F = 5

40. Solution

41. would these two amounts be equivalent?
Example
At what interest rate
would these two amounts be equivalent?
i = ?
$3,000
$2,042 5

42. Solution F=P(1+i)N 3000=2042(1+i)5 1.469=(1+i)5 1.4691/5=1.07995=1+i