0. 5 Time Value of Money Introduction to Finance Chapter
Lawrence J. Gitman
Jeff Madura
Introduction to Finance
Time Value of Money

1. Learning Goals
Discuss the role of time value in finance and the use of computational aids to simplify its application.
Understand the concept of future value and its calculation for a single amount; understand the effects on future value and the true rate of interest of compounding more frequently than annually.
Understand the concept of present value, its calculation for a single amount, and the relationship of present to future cash flow.

2. Learning Goals
Find the future value and present value of an ordinary annuity, the future value of an annuity due, and the present value of a perpetuity.
Calculate the present value of a mixed stream of cash flows, describe the procedures involved in:
Determining deposits to accumulate to a future sum
Loan amortization
Finding interest or growth rates

3. The Role of Time Value in Finance
Most financial decisions involve costs and benefits that are spread out over time.
Time value of money allows comparison of cash flows from different periods.
Question
Would it be better for a company to invest $100,000 in a product that would return a total of $200,000 in one year, or one that would return $500,000 after two years?
Answer
It depends on the interest rate!

4. Basic Concepts Future Value Present Value
Compounding or growth over time
Present Value
Discounting to today’s value
Single cash flows and series of cash flows can be considered
Time lines are used to illustrate these relationships

5. Computational Aids Use the equations Use the financial tables
Use financial calculators
Use spreadsheets

6. Computational Aids
Figure 5.1

7. Computational Aids
Figure 5.2

8. Computational Aids
Figure 5.3

9. Computational Aids
Figure 5.4

10. Simple Interest
With simple interest, you don’t earn interest on interest.
Year 1: 5% of $100 = $5 + $100 = $105
Year 2: 5% of $100 = $5 + $105 = $110
Year 3: 5% of $100 = $5 + $110 = $115
Year 4: 5% of $100 = $5 + $115 = $120
Year 5: 5% of $100 = $5 + $120 = $125

11. Compound Interest
With compound interest, a depositor earns interest on interest!
Year 1: 5% of $ = $ $ = $105.00
Year 2: 5% of $ = $ $ = $110.25
Year 3: 5% of $ = $ $ = $115.76
Year 4: 5% of $ = $ $ = $121.55
Year 5: 5% of $ = $ $ = $127.63

12. Time Value Terms PV0 = present value or beginning amount
k = interest rate
FVn = future value at end of “n” periods
n = number of compounding periods
A = an annuity (series of equal payments or receipts)

13. Four Basic Models FVn = PV0(1+k)n = PV(FVIFk,n)
PV0 = FVn[1/(1+k)n] = FV(PVIFk,n)
FVAn = A (1+k)n - 1 = A(FVIFAk,n) k
PVA0 = A 1 - [1/(1+k)n] = A(PVIFAk,n)

14. Future Value Example Algebraically and Using FVIF Tables
You deposit $2,000 today at 6% interest. How much will you have in 5 years?
$2,000 x (1.06)5 = $2,000 x FVIF6%,5
$2,000 x = $2,676.40

15. Microsoft® Excel Function
Future Value Example
Using Microsoft® Excel
You deposit $2,000 today at 6% interest. How much will you have in 5 years?
Microsoft® Excel Function
= FV(interest, periods, pmt, PV)
= FV(.06, 5, , 2000)

16. A Graphic View of Future Value
Figure 5.5

17. Compounding More Frequently Than Annually
Compounding more frequently than once a year results in a higher effective interest rate because you are earning on interest on interest more frequently.
As a result, the effective interest rate is greater than the nominal (annual) interest rate.
Furthermore, the effective rate of interest will increase the more frequently interest is compounded.

18. Compounding More Frequently Than Annually
For example, what would be the difference in future value if I deposit $100 for 5 years and earn 12% annual interest compounded (a) annually, (b) semiannually, (c) quarterly, and (d) monthly?
Annually: 100 x ( )5 = $176.23
Semiannually: 100 x ( )10 = $179.09
Quarterly: 100 x ( )20 = $180.61
Monthly: 100 x ( )60 = $181.67

19. Compounding More Frequently Than Annually

20. Continuous Compounding
With continuous compounding the number of compounding periods per year approaches infinity.
Through the use of calculus, the equation thus becomes:
FVn (continuous compounding) = PV x (ekxn)
where “e” has a value of
Continuing with the previous example, find the future value of the $100 deposit after 5 years if interest is compounded continuously.
FVn = 100 x (2.7183).12x5 = $182.22

21. Nominal and Effective Rates
The nominal interest rate is the stated or contractual rate of interest charged by a lender or promised by a borrower.
The effective interest rate is the rate actually paid or earned.
In general, the effective rate is greater than the nominal rate whenever compounding occurs more than once per year.
EAR = (1 + k/m)m - 1

22. Nominal and Effective Rates
For example, what is the effective rate of interest on your credit card if the nominal rate is 18% per year, compounded monthly?
EAR = ( /12)12 - 1
EAR = 19.56%

23. Present Value
Present value is the current dollar value of a future amount of money.
It is based on the idea that a dollar today is worth more than a dollar tomorrow.
It is the amount today that must be invested at a given rate to reach a future amount.
It is also known as discounting, the reverse of compounding.
The discount rate is often also referred to as the opportunity cost, the discount rate, the required return, and the cost of capital.

24. Present Value Example Algebraically and Using PVIF Tables
How much must you deposit today in order to have $2,000 in 5 years if you can earn 6% interest on your deposit?
$2,000 x [1/(1.06)5] = $2,000 x PVIF6%,5
$2,000 x = $1,494.52

25. Microsoft® Excel Function
Present Value Example
Using Microsoft® Excel
How much must you deposit today in order to have $2,000 in 5 years if you can earn 6% interest on your deposit?
Microsoft® Excel Function
=PV(interest, periods, pmt, FV)
=PV(.06, 5, , 2000)

26. A Graphic View of Present Value
Figure 5.6

27. Annuities Annuities are equally-spaced cash flows of equal size.
Annuities can be either inflows or outflows.
An ordinary (deferred) annuity has cash flows that occur at the end of each period.
An annuity due has cash flows that occur at the beginning of each period.
An annuity due will always be greater than an otherwise equivalent ordinary annuity because interest will compound for an additional period.

28. Annuities
Table 5.1

29. Future Value of an Ordinary Annuity
Using the FVIFA Tables
An annuity is an equal annual series of cash flows.
Example
How much will your deposits grow to if you deposit $100 at the end of each year at 5% interest for three years?
FVA = 100(FVIFA,5%,3) = $315.25
Year 1 $100 deposited at end of year = $100.00
Year 2 $100 x .05 = $ $100 + $100 = $205.00
Year 3 $205 x .05 = $ $205 + $100 = $315.25

30. Future Value of an Ordinary Annuity
Using Microsoft® Excel
An annuity is an equal annual series of cash flows.
Example
How much will your deposits grow to if you deposit $100 at the end of each year at 5% interest for three years?
Microsoft® Excel Function
=FV(interest, periods, pmt, PV)
=FV(.06,5,100, )

31. Future Value of an Annuity Due
Using the FVIFA Tables
An annuity is an equal annual series of cash flows.
Example
How much will your deposits grow to if you deposit $100 at the beginning of each year at 5% interest for three years.
FVA = 100(FVIFA,5%,3)(1+k) = $330.96
FVA = 100(3.152)(1.05) = $330.96

32. Future Value of an Annuity Due
Using Microsoft® Excel
An annuity is an equal annual series of cash flows.
Example
How much will your deposits grow to if you deposit $100 at the beginning of each year at 5% interest for three years.
Microsoft® Excel Function
=FV(interest, periods, pmt, PV)
=FV(.06, 5,100, )
=315.25*(1.05)

33. Present Value of an Ordinary Annuity
Using PVIFA Tables
An annuity is an equal annual series of cash flows.
Example
How much could you borrow if you could afford annual payments of $2,000 (which includes both principal and interest) at the end of each year for three years at 10% interest?
PVA = 2,000(PVIFA,10%,3) = $4,973.70

34. Present Value of an Ordinary Annuity
Using Microsoft® Excel
An annuity is an equal annual series of cash flows.
Example
How much could you borrow if you could afford annual payments of $2,000 (which includes both principal and interest) at the end of each year for three years at 10% interest?
Microsoft® Excel Function
=PV(interest, periods, pmt, FV)
=PV(.10, 3, 2000, )

35. Present Value of a Mixed Stream
Using Microsoft® Excel
A mixed stream of cash flows reflects no particular pattern
Find the present value of the following mixed stream assuming a required return of 9%.
Microsoft® Excel Function
=NPV(interest, cells containing CFs)
=NPV(.09,B3:B7)

36. Present Value of a Perpetuity
A perpetuity is a special kind of annuity.
With a perpetuity, the periodic annuity or cash flow stream continues forever.
PV = Annuity/k
For example, how much would I have to deposit today in order to withdraw $1,000 each year forever if I can earn 8% on my deposit?
PV = $1,000/.08 = $12,500

37. Loan Amortization
Table 5.7

38. Determining Interest or Growth Rates
At times, it may be desirable to determine the compound interest rate or growth rate implied by a series of cash flows.
For example, you invested $1,000 in a mutual fund in 1994 which grew as shown in the table below?
It is important to note that although there are 7 years shown, there are only 6 time periods between the initial deposit and the final value.

39. Determining Interest or Growth Rates
At times, it may be desirable to determine the compound interest rate or growth rate implied by a series of cash flows.
For example, you invested $1,000 in a mutual fund in which grew as shown in the table below?
Thus, $1,000 is the present value, $5,525 is the future value, and 6 is the number of periods.

40. Determining Interest or Growth Rates
At times, it may be desirable to determine the compound interest rate or growth rate implied by a series of cash flows.
For example, you invested $1,000 in a mutual fund in which grew as shown in the table below?

41. Determining Interest or Growth Rates
At times, it may be desirable to determine the compound interest rate or growth rate implied by a series of cash flows.
For example, you invested $1,000 in a mutual fund in which grew as shown in the table below?
Microsoft® Excel Function
=Rate(periods, pmt, PV, FV)
=Rate(6, ,1000, 5525)

42. Using Microsoft® Excel
The Microsoft® Excel Spreadsheets used in the this presentation can be downloaded from the Introduction to Finance companion web site:

43. 5 End of Chapter Introduction to Finance Chapter Lawrence J. Gitman
Jeff Madura
Introduction to Finance
End of Chapter