This document outlines key concepts related to time value of money including: future value and present value calculations using formulas that take into account interest rate, time period, and frequency of compounding. It provides examples of how to determine future or present value of single and multiple cash flows, as well as annuities and perpetuities. Key terms defined include time value, compounding, discounting, and effective annual rate.

1. Time Value of Money

2. Outline
 Meaning of Time Value
 Concept of Future Value and Compounding (FV)
 Concept of Present Value and Discounting (PV)
 Frequency of Compounding
 Present Value versus Future Value
 Determining the Interest rate (r)
 Determining the Time Period (n)
 Future Value and Present Value of Multiple Cash Flows
 Annuities and Perpetuities

3. Time Value of Money
 Basic Problem:
– How to determine value today of cash flows that are expected in the
future?
 Time value of money refers to the fact that a dollar in hand today
is worth more than a dollar promised at some time in the future
 Which would you rather have -- $1,000 today or $1,000 in 5
years?
 Obviously, $1,000 today.
 Money received sooner rather than later allows one to use the
funds for investment or consumption purposes. This concept is
referred to as the TIME VALUE OF MONEY!!
 TIME allows one the opportunity to postpone consumption and
earn INTEREST.

4. Future Value and Compounding
 Future value refers to the amount of money an investment will grow to over some
length of time at some given interest rate
 To determine the future value of a single cash flows, we need:
 present value of the cash flow (PV)
 interest rate (r), and
 time period (n)
 FVn = PV0 × (1 + r)n
 Future Value Interest Factor at ‘r’ rate of interest for ‘n’ time
periods
 Examples on computation of future value of a single cash flow

5. If you invested $2,000 today in an account that
pays 6% interest, with interest compounded
annually, how much will be in the account at the
end of two years if there are no withdrawals?
Future Value (Graphic)
0 1 2
$2,000
FV
6%

6. FV1 = PV (1+r)n
= $2,000 (1.06)2
= $2,247.20
Future Value (Formula)
FV = future value, a value at some future point in time
PV = present value, a value today which is usually designated as time 0
r = rate of interest per compounding period
n = number of compounding periods
Calculator Keystrokes: 1.06 (2nd yx) 2 x 2000 =

7.  John wants to know how large his $5,000
deposit will become at an annual compound
interest rate of 8% at the end of 5 years.
Future Value (Example)
0 1 2 3 4 5
$5,000
FV5
8%

8. 
Future Value Solution
 Calculation based on general formula:
FVn = PV (1+r)n
 FV5 = $5,000 (1+ 0.08)5
 = $7,346.64
 Calculator keystrokes: 1.08 2nd yx x 5000 =

9. Present Value and Discounting
 The current value of future cash flows discounted at the appropriate
discount rate over some length of time period
 Discounting is the process of translating a future value or a set of future
cash flows into a present value.
 To compute present value of a single cash flow, we need:
 Future value of the cash flow (FV)
 Interest rate (r) and
 Time Period (n)
 PV0 = FVn / (1 + r)n
 PVIF (r,n)
 Examples

10. Assume that you need to have exactly $4,000 saved 10
years from now. How much must you deposit today in
an account that pays 6% interest, compounded annually,
so that you reach your goal of $4,000?
0 5 10
$4,000
6%
PV0
Present Value (Graphic)

11. PV0 = FV / (1+r)10
= $4,000 / (1.06)10
= $2,233.58
Present Value
(Formula)
0 5 10
$4,000
6%
PV0

12. Joann needs to know how large of a deposit to make
today so that the money will grow to $2,500 in 5 years.
Assume today’s deposit will grow at a compound rate
of 4% annually.
Present Value Example
0 1 2 3 4 5
$2,500
PV0
4%

13.  Calculation based on general formula:
PV0 = FVn / (1+r)n
PV0 = $2,500/(1.04)5
= $2,054.81
 Calculator keystrokes: 1.04 2nd yx 5 =
2nd 1/x X 2500 =
Present Value Solution

14. General Formula:
FVn = PV0(1 + [r/m])mn
n: Number of Years
m: Compounding Periods per Year
r: Annual Interest Rate
FVn,m: FV at the end of Year n
PV0: PV of the Cash Flow today
Frequency of
Compounding

15. Frequency of Compounding Example
 Suppose you deposit $1,000 in an account that pays
12% interest, compounded quarterly. How much will
be in the account after eight years if there are no
withdrawals?
PV = $1,000
r = 12%/4 = 3% per quarter
n = 8 x 4 = 32 quarters

16. Solution based on formula:
FV= PV (1 + r)n
= 1,000(1.03)32
= 2,575.10
Calculator Keystrokes:
1.03 2nd yx 32 X 1000 =

17. Present Value versus Future Value
 Present value factors are reciprocals of future value
factors
 Interest rates and future value are positively related
 Interest rates and present value are negatively related
 Time period and future value are positively related
 Time period and present value are negatively related

18. Determining the Interest Rate (r)
 At what rate of interest should we invest our
money today to get a desired amount of money
after a certain number of years?
 Essentially, we are trying to determine the
interest rate given present value (PV), future
value (FV), and time period (n)
 Examples
 The rate which money can be doubled/tripled

19. Determining the Time Period (n)
 For how long should we invest money today to get a
desired amount of money in the future at a given rate
of interest
 Determining the time period (n) for which a current
amount (PV) needs to be invested to get a certain
future value (FV) given a rate of interest (r).
 Examples
 The time period needed to double/triple our current
investment

20. Future Value of Multiple Uneven
Cash Flows
 Compute the future value of each single cash
flow using future value formula and add them
up over all the cash flows
 Example

21. Present Value of Multiple Uneven
Cash Flows
 Compute the present value of each single cash
flow using present value formula and add them
over all the cash flows
 Examples

22. Annuities
 A series of level/even/equal sized cash flows
that occur at the end of each time period for a
fixed time period
 Examples of Annuities:
 Car Loans
 House Mortgages
 Insurance Policies
 Some Lotteries
 Retirement Money

23.  Present Value of an Annuity
– Examples
 Computing Cash Flow per period in annuity
– Examples

24. Perpetuities
 A series of level/even/equal sized cash flows
that occur at the end of each period for an
infinite time period
 Examples of Perpetuities:
 Consoles issued by British Government
 Preferred Stock
 Present Value of a Perpetuity

25. Effective Annual Rate
 Compounding other than annual