1. Chapter 2
Time Value of Money
(TVOM)

2. Today’s Objectives:
By the end of this lecture students will be able to
1- Apply ‘time value of money’ principles to personal and business situations.

3. Fundamental Concept
Money has a time value

4. Time Value of Money
Would you rather receive $1000 today or $1000 a year from today?
Would you rather receive $1000 today or $1500 a year from today?
Would you rather receive $1000 today or $5000 a year from today?
Would you rather receive $1000 today or $10,000 a year from today?
Regardless of the value of inflation, money has a time value due to its “earning power”

5. Principles of Engineering Economic Analysis
Money has a time value.
Make investments that are economically justified.
Choose the mutually exclusive investment alternative that maximizes economic worth.
Two investment alternatives are equivalent if they have the same economic worth.

6. Principles of Engineering Economic Analysis
Marginal revenue must exceed marginal cost.
Continue to invest as long as each additional increment of investment yields a return that is greater than the investor’s TVOM.
Consider only differences in cash flows among investment alternatives.

7. Principles of Engineering Economic Analysis
Compare investment alternatives over a common period of time.
Risks and returns tend to be positively correlated.
Past costs are irrelevant in engineering economic analyses, unless they impact future costs.

8. Cash Flow Diagrams
A cash flow diagrams is a powerful descriptive technique used in evaluating economic alternatives.
Horizontal line as time scale and vertical arrows to indicate cash flow.

9. Cash Flow Profiles for Two Investment Alternatives
Example 2.1
Cash Flow Profiles for Two Investment Alternatives
End of Year
(EOY)
CF(A)
CF(B)
CF(B-A)
-$100,000 $0 1
$10,000
$50,000
$40,000 2
$20,000 3
$30,000 4
-$20,000 5
-$40,000
Sum
Although the two investment alternatives have the same “bottom line,” there are obvious differences. Which would you prefer, A or B? Why?

10. Inv. A
Inv. B

11. Principle #7
Consider only differences in cash flows among investment alternatives

12. Inv. B – Inv. A

13. Example 2.2
$3,000
$3,000
$3,000
(+)
(-)
Alternative C
$6,000
$3,000
$3,000
$3,000
(+)
(-)
Alternative D
Which would you choose?
$6,000

14. Example 2.3 $3,000 $2,000 $2,000 $2,000 (+) (-) Alternative E 1 2 3 4
$4000
$3,000
$2,000
$2,000
$1,000
(+)
(-)
Alternative F
$2000
Alternative E-F 3
$4,000 4
Which would you choose?
$1000

15. Interest formulas Simple interest calculation:
Compound Interest Calculation:
Where
P = present value of single sum of money
Fn = accumulated value of P over n periods
i = interest rate per period
n = number of periods

16. Example 2.4: simple interest calculation
Robert borrows $4,000 from Susan and agrees to return it by the end of the fourth year. What should be the size of the payments if 8% simple interest is used?
F = ? ,P = $4000, i = 8%, n = 4
F4 = $4000(1+0.08×4) = $5280

17. Example : compound interest calculation
Robert borrows $4,000 from Susan and agrees to return it by the end of the fourth year. What should be the size of the payments if 8% compound interest is used?
F1 = 4000 (1+0.08) = $4320
F2 = 4320(1+0.08) = $4665.6
F3 = (1+.08) =$
F4 = (1+0.08) = $

18. RULES Discounting Cash Flow
Money has time value!
Cash flows cannot be added unless they occur at the same point(s) in time.
Multiply a cash flow by (1+i) to move it forward one time unit.
Divide a cash flow by (1+i) to move it backward one time unit.

19. Example (Lender’s Perspective) Value of $10,000 Investment Growing @ 10% per year
Start of Year
Value of Investment
Interest Earned
End of Year 1
$10,000.00
$1,000.00
$11,000.00 2
$1,100.00
$12,100.00 3
$1,210.00
$13,310.00 4
$1,331.00
$14,641.00 5
$1,464.10
$16,105.10

20. Example (Borrower’s Perspective) Value of $10,000 Investment Growing @ 10% per year

21. Compounding of Money Beginning of Period Amount Owed Interest Earned
End of Period 1 P Pi
P(1+i) 2
P(1+i)i
P(1+i)2 3
P(1+i)2i
P(1+i)3 4
P(1+i)3i
P(1+i)4 5
P(1+i)4i
P(1+i)5 .
n-1
P(1+i)n-2
P(1+i)n-2i
P(1+i)n-1 n
P(1+i)n-1i
P(1+i)n

22. Discounted Cash Flow Formulas
F = P (1 + i) n (2.1)
F = P (F|P i%, n) (2.2)
P = F (1 + i) -n (2.3)
P = F (P|F i%, n) (2.4)
Vertical line means “given”

23. Excel® DCF Worksheet Functions
F = P (1 + i) n (2.1)
F = P (F|P i%, n)
F =FV(i%,n,,-P)
P = F (1 + i) -n (2.3)
P = F (P|F i%, n)
P =PV(i%,n,,-F)

24. F = P(1+i)n F = P(F|P i%, n) F =FV(i%,n,,-P) P = F(1+i)-n
P = F(P|F i%, n)
P =PV(i%,n,,-F) F ….
n n P
P occurs n periods before F
(F occurs n periods after P)

25. Example 2.6
Dia St. John borrows $1,000 at 12% compounded annually. The loan is to be repaid after 5 years. How much must she repay in 5 years?

26. Example 2.6
Dia St. John borrows $1,000 at 12% compounded annually. The loan is to be repaid after 5 years. How much must she repay in 5 years?
F = P(F|P i%, n)
F = $1,000(F|P 12%,5)
F = $1,000(1.12)5
F = $1,000( )
F = $1,762.34

27. Example 2.6
Dia St. John borrows $1,000 at 12% compounded annually. The loan is to be repaid after 5 years. How much must she repay in 5 years?
F = P(F|P i, n)
F = $1,000(F|P 12%,5)
F = $1,000(1.12)5
F = $1,000( )
F = $1,762.34
F =FV(12%,5,,-1000)

28. Problem 4
How long does it take for money to double in value, if you earn (a) 2%, (b) 3%, (c) 4%, (d) 6%, (e) 8%, or (f) 12% annual compound interest?

29. Problem 4
How long does it take for money to double in value, if you earn (a) 2%, (b) 3%, (c) 4%, (d) 6%, (e) 8%, or (f) 12% annual compound interest?
I can think of six ways to solve this problem:
Solve using the Rule of 72
Use the interest tables; look for F|P factor equal to 2.0
Solve numerically; n = log(2)/log(1+i)
Solve using Excel® NPER function: =NPER(i%,,-1,2)
Solve using Excel® GOAL SEEK tool
Solve using Excel® SOLVER tool

30. (Quick, but not always accurate.)
Problem 4
How long does it take for money to double in value, if you earn (a) 2%, (b) 3%, (c) 4%, (d) 6%, (e) 8%, or (f) 12% annual compound interest?
RULE OF 72
Divide 72 by interest rate to determine how long it takes for money to double in value.
(Quick, but not always accurate.)

31. Problem 4
How long does it take for money to double in value, if you earn (a) 2%, (b) 3%, (c) 4%, (d) 6%, (e) 8%, or (f) 12% annual compound interest? Rule of 72 solution
(a) 72/2 = 36 yrs
(b) 72/3 = 24 yrs
(c) 72/4 = 18 yrs
(d) 72/6 = 12 yrs
(e) 72/8 = 9 yrs
(f) 72/12 = 6 yrs

32. Problem 4
How long does it take for money to double in value, if you earn (a) 2%, (b) 3%, (c) 4%, (d) 6%, (e) 8%, or (f) 12% annual compound interest?
Using interest tables & interpolating
(a) yrs
(b) yrs
(c) yrs
(d) yrs
(e) yrs
(f) yrs

33. Problem 4
How long does it take for money to double in value, if you earn (a) 2%, (b) 3%, (c) 4%, (d) 6%, (e) 8%, or (f) 12% annual compound interest? Mathematical solution
(a) log 2/log 1.02 = yrs
(b) log 2/log 1.03 = yrs
(c) log 2/log 1.04 = yrs
(d) log 2/log 1.06 = yrs
(e) log 2/log 1.08 = yrs
(f) log 2/log 1.12 = yrs

34. Problem 4
How long does it take for money to double in value, if you earn (a) 2%, (b) 3%, (c) 4%, (d) 6%, (e) 8%, or (f) 12% annual compound interest? Using the Excel® NPER function
(a) n =NPER(2%,,-1,2) = yrs
(b) n =NPER(3%,,-1,2) = yrs
(c) n =NPER(4%,,-1,2) = yrs
(d) n =NPER(6%,,-1,2) = yrs
(e) n =NPER(8%,,-1,2) = yrs
(f) n =NPER(12%,,-1,2) = yrs
Identical solution to that obtained mathematically

35. Problem 4
How long does it take for money to double in value, if you earn (a) 2%, (b) 3%, (c) 4%, (d) 6%, (e) 8%, or (f) 12% annual compound interest? Using the Excel® GOAL SEEK tool
(a) n = yrs
(b) n = yrs
(c) n = yrs
(d) n = yrs
(e) n =9.008 yrs
(f) n =6.116 yrs
Solution obtained differs from that obtained mathematically; red digits differ

37. Problem 4
How long does it take for money to double in value, if you earn (a) 2%, (b) 3%, (c) 4%, (d) 6%, (e) 8%, or (f) 12% annual compound interest? Using the Excel® SOLVER tool
(a) n = yrs
(b) n = yrs
(c) n = yrs
(d) n = yrs
(e) n =9.006 yrs
(f) n =6.116 yrs
Solution differs from mathematical solution, but at the 6th to 10th decimal place

38. Example 2.11
How much must you deposit, today, in order to accumulate $10,000 in 4 years, if you earn 5% compounded annually on your investment?

39. Example 2.11
How much must you deposit, today, in order to accumulate $10,000 in 4 years, if you earn 5% compounded annually on your investment?
P = F(P|F i, n)
P = $10,000(P|F 5%,4)
P = $10,000(1.05)-4
P = $10,000( )
P = $8,227.00
P =PV(5%,4,,-10000)
P = $

40. Example 2.11
How much must you deposit, today, in order to accumulate $10,000 in 4 years, if you earn 5% compounded annually on your investment?
P = F(P|F i, n)
P = $10,000(P|F 5%,4)
P = $10,000(1.05)-4
P = $10,000( )
P = $8,227.00
P =PV(5%,4,,-10000)
P = $8,227.02

41. Computing the Present Worth of Multiple Cash flows
(2.6)
(2.7)
A : denote the magnitude of cash flow at the end of time period.

42. Examples 2.8
Determine the present worth equivalent of the following series of cash flows. Use an interest rate of 6% per interest period.
P = $300(P|F 6%,1)- $300(P|F 6%,3)+$200(P|F 6%,4)+$400(P|F 6%,6)
+$200(P|F 6%,8) = $597.02
P =NPV(6%,300,0,-300,200,0,400,0,200)
P =$597.02
End of Period
Cash Flow $0 1
$300 2 3
-$300 4
$200 5 6
$400 7 8

43. Example
Determine the present worth equivalent of the CFD shown below, using an interest rate of 10% compounded annually.
P =NPV(10%,50000,40000,30000,40000,50000)
= $59,418.45

44. Computing the Future worth of Multiple cash Flows
(2.9)
(2.10)

45. Examples 2.9
Determine the future worth equivalent of the following series of cash flows. Use an interest rate of 6% per interest period.
F = $300(F|P 6%,7)-$300(F|P 6%,5)+$200(F|P 6%,4)+$400(F|P 6%,2)+$200
F = $951.59
F =FV(6%,8,,-NPV(6%,300,0,-300,200,0,400,0,200))
F =$951.56
End of Period
Cash Flow $0 1
$300 2 3
-$300 4
$200 5 6
$400 7 8
(The 3¢ difference in the answers is due to round-off error in the tables in Appendix A.)

46. Example
Determine the future worth equivalent of the CFD shown below, using an interest rate of 10% compounded annually.
F =10000*FV(10%,5,,-NPV(10%,5,4,3,4,5)+10)
= $95,694.00