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. $2,000 Future Value (Graphic) 6% FV 0 1 2
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? 6%
$2,000 FV

6. Future Value (Formula)
FV1 = PV (1+r)n
= $2,000 (1.06)2
= $2,247.20
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: (2nd yx) 2 x 2000 =

7. Future Value (Example)
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. 8%
$5,000
FV5

8. Future Value Solution
Calculation based on general formula: FVn = PV (1+r)n
FV5 = $5,000 ( )5
= $7,346.64
Calculator keystrokes: nd 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. $4,000 Present Value (Graphic) 6% PV0 0 5 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? 6%
$4,000
PV0

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

12. Present Value Example 0 1 2 3 4 5 4% $2,500 PV0
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. 4%
$2,500
PV0

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

14. Frequency of Compounding
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

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 Computing Cash Flow per period in annuity
Examples
Computing Cash Flow per period in annuity

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